3.24.5 \(\int \frac {\sqrt {d+e x}}{(a+b x+c x^2)^3} \, dx\) [2305]

Optimal. Leaf size=634 \[ -\frac {(b+2 c x) \sqrt {d+e x}}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {\sqrt {d+e x} \left (13 b^2 c d e-4 a c^2 d e-b^3 e^2-4 b c \left (3 c d^2+2 a e^2\right )-c \left (24 c^2 d^2+b^2 e^2-4 c e (6 b d-5 a e)\right ) x\right )}{4 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )}-\frac {\sqrt {c} \left (96 c^3 d^3+b^2 \left (b+\sqrt {b^2-4 a c}\right ) e^3-8 c^2 d e \left (18 b d-3 \sqrt {b^2-4 a c} d-13 a e\right )+2 c e^2 \left (23 b^2 d+10 a \sqrt {b^2-4 a c} e-2 b \left (6 \sqrt {b^2-4 a c} d+13 a e\right )\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{4 \sqrt {2} \left (b^2-4 a c\right )^{5/2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )}+\frac {\sqrt {c} \left (96 c^3 d^3+b^2 \left (b-\sqrt {b^2-4 a c}\right ) e^3-8 c^2 d e \left (18 b d+3 \sqrt {b^2-4 a c} d-13 a e\right )+2 c e^2 \left (23 b^2 d+12 b \sqrt {b^2-4 a c} d-26 a b e-10 a \sqrt {b^2-4 a c} e\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{4 \sqrt {2} \left (b^2-4 a c\right )^{5/2} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )} \]

[Out]

-1/2*(2*c*x+b)*(e*x+d)^(1/2)/(-4*a*c+b^2)/(c*x^2+b*x+a)^2-1/4*(13*b^2*c*d*e-4*a*c^2*d*e-b^3*e^2-4*b*c*(2*a*e^2
+3*c*d^2)-c*(24*c^2*d^2+b^2*e^2-4*c*e*(-5*a*e+6*b*d))*x)*(e*x+d)^(1/2)/(-4*a*c+b^2)^2/(a*e^2-b*d*e+c*d^2)/(c*x
^2+b*x+a)-1/8*arctanh(2^(1/2)*c^(1/2)*(e*x+d)^(1/2)/(2*c*d-e*(b-(-4*a*c+b^2)^(1/2)))^(1/2))*c^(1/2)*(96*c^3*d^
3+b^2*e^3*(b+(-4*a*c+b^2)^(1/2))-8*c^2*d*e*(18*b*d-13*a*e-3*d*(-4*a*c+b^2)^(1/2))+2*c*e^2*(23*b^2*d+10*a*e*(-4
*a*c+b^2)^(1/2)-2*b*(13*a*e+6*d*(-4*a*c+b^2)^(1/2))))/(-4*a*c+b^2)^(5/2)/(a*e^2-b*d*e+c*d^2)*2^(1/2)/(2*c*d-e*
(b-(-4*a*c+b^2)^(1/2)))^(1/2)+1/8*arctanh(2^(1/2)*c^(1/2)*(e*x+d)^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2)))^(1/2)
)*c^(1/2)*(96*c^3*d^3+b^2*e^3*(b-(-4*a*c+b^2)^(1/2))-8*c^2*d*e*(18*b*d-13*a*e+3*d*(-4*a*c+b^2)^(1/2))+2*c*e^2*
(23*b^2*d-26*a*b*e+12*b*d*(-4*a*c+b^2)^(1/2)-10*a*e*(-4*a*c+b^2)^(1/2)))/(-4*a*c+b^2)^(5/2)/(a*e^2-b*d*e+c*d^2
)*2^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2)))^(1/2)

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Rubi [A]
time = 3.62, antiderivative size = 634, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {750, 836, 840, 1180, 214} \begin {gather*} -\frac {\sqrt {c} \left (-8 c^2 d e \left (-3 d \sqrt {b^2-4 a c}-13 a e+18 b d\right )+2 c e^2 \left (-2 b \left (6 d \sqrt {b^2-4 a c}+13 a e\right )+10 a e \sqrt {b^2-4 a c}+23 b^2 d\right )+b^2 e^3 \left (\sqrt {b^2-4 a c}+b\right )+96 c^3 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{4 \sqrt {2} \left (b^2-4 a c\right )^{5/2} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )} \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {c} \left (-8 c^2 d e \left (3 d \sqrt {b^2-4 a c}-13 a e+18 b d\right )+2 c e^2 \left (12 b d \sqrt {b^2-4 a c}-10 a e \sqrt {b^2-4 a c}-26 a b e+23 b^2 d\right )+b^2 e^3 \left (b-\sqrt {b^2-4 a c}\right )+96 c^3 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{4 \sqrt {2} \left (b^2-4 a c\right )^{5/2} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )} \left (a e^2-b d e+c d^2\right )}-\frac {(b+2 c x) \sqrt {d+e x}}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {\sqrt {d+e x} \left (-c x \left (-4 c e (6 b d-5 a e)+b^2 e^2+24 c^2 d^2\right )-4 b c \left (2 a e^2+3 c d^2\right )-4 a c^2 d e+b^3 \left (-e^2\right )+13 b^2 c d e\right )}{4 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[d + e*x]/(a + b*x + c*x^2)^3,x]

[Out]

-1/2*((b + 2*c*x)*Sqrt[d + e*x])/((b^2 - 4*a*c)*(a + b*x + c*x^2)^2) - (Sqrt[d + e*x]*(13*b^2*c*d*e - 4*a*c^2*
d*e - b^3*e^2 - 4*b*c*(3*c*d^2 + 2*a*e^2) - c*(24*c^2*d^2 + b^2*e^2 - 4*c*e*(6*b*d - 5*a*e))*x))/(4*(b^2 - 4*a
*c)^2*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2)) - (Sqrt[c]*(96*c^3*d^3 + b^2*(b + Sqrt[b^2 - 4*a*c])*e^3 - 8*
c^2*d*e*(18*b*d - 3*Sqrt[b^2 - 4*a*c]*d - 13*a*e) + 2*c*e^2*(23*b^2*d + 10*a*Sqrt[b^2 - 4*a*c]*e - 2*b*(6*Sqrt
[b^2 - 4*a*c]*d + 13*a*e)))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]])/
(4*Sqrt[2]*(b^2 - 4*a*c)^(5/2)*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]*(c*d^2 - b*d*e + a*e^2)) + (Sqrt[c]*(96
*c^3*d^3 + b^2*(b - Sqrt[b^2 - 4*a*c])*e^3 - 8*c^2*d*e*(18*b*d + 3*Sqrt[b^2 - 4*a*c]*d - 13*a*e) + 2*c*e^2*(23
*b^2*d + 12*b*Sqrt[b^2 - 4*a*c]*d - 26*a*b*e - 10*a*Sqrt[b^2 - 4*a*c]*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*
x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(4*Sqrt[2]*(b^2 - 4*a*c)^(5/2)*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*
c])*e]*(c*d^2 - b*d*e + a*e^2))

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 750

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^m*(b + 2*c
*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] - Dist[1/((p + 1)*(b^2 - 4*a*c)), Int[(d + e*x)^(m
- 1)*(b*e*m + 2*c*d*(2*p + 3) + 2*c*e*(m + 2*p + 3)*x)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d
, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[p, -1] && GtQ[m
, 0] && (LtQ[m, 1] || (ILtQ[m + 2*p + 3, 0] && NeQ[m, 2])) && IntQuadraticQ[a, b, c, d, e, m, p, x]

Rule 836

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)
*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/((p + 1)*(b^2 - 4*a*
c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2
*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d -
b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b,
c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] ||
IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 840

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2,
Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /
; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 1180

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rubi steps

\begin {align*} \int \frac {\sqrt {d+e x}}{\left (a+b x+c x^2\right )^3} \, dx &=-\frac {(b+2 c x) \sqrt {d+e x}}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {\int \frac {-6 c d+\frac {b e}{2}-5 c e x}{\sqrt {d+e x} \left (a+b x+c x^2\right )^2} \, dx}{2 \left (b^2-4 a c\right )}\\ &=-\frac {(b+2 c x) \sqrt {d+e x}}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {\sqrt {d+e x} \left (13 b^2 c d e-4 a c^2 d e-b^3 e^2-4 b c \left (3 c d^2+2 a e^2\right )-c \left (24 c^2 d^2+b^2 e^2-4 c e (6 b d-5 a e)\right ) x\right )}{4 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )}-\frac {\int \frac {\frac {1}{4} \left (-48 c^3 d^3-b^3 e^3-b c e^2 (11 b d-16 a e)+4 c^2 d e (15 b d-13 a e)\right )-\frac {1}{4} c e \left (24 c^2 d^2+b^2 e^2-4 c e (6 b d-5 a e)\right ) x}{\sqrt {d+e x} \left (a+b x+c x^2\right )} \, dx}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {(b+2 c x) \sqrt {d+e x}}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {\sqrt {d+e x} \left (13 b^2 c d e-4 a c^2 d e-b^3 e^2-4 b c \left (3 c d^2+2 a e^2\right )-c \left (24 c^2 d^2+b^2 e^2-4 c e (6 b d-5 a e)\right ) x\right )}{4 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )}-\frac {\text {Subst}\left (\int \frac {\frac {1}{4} e \left (-48 c^3 d^3-b^3 e^3-b c e^2 (11 b d-16 a e)+4 c^2 d e (15 b d-13 a e)\right )+\frac {1}{4} c d e \left (24 c^2 d^2+b^2 e^2-4 c e (6 b d-5 a e)\right )-\frac {1}{4} c e \left (24 c^2 d^2+b^2 e^2-4 c e (6 b d-5 a e)\right ) x^2}{c d^2-b d e+a e^2+(-2 c d+b e) x^2+c x^4} \, dx,x,\sqrt {d+e x}\right )}{\left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {(b+2 c x) \sqrt {d+e x}}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {\sqrt {d+e x} \left (13 b^2 c d e-4 a c^2 d e-b^3 e^2-4 b c \left (3 c d^2+2 a e^2\right )-c \left (24 c^2 d^2+b^2 e^2-4 c e (6 b d-5 a e)\right ) x\right )}{4 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )}-\frac {\left (c \left (96 c^3 d^3+b^2 \left (b-\sqrt {b^2-4 a c}\right ) e^3-8 c^2 d e \left (18 b d+3 \sqrt {b^2-4 a c} d-13 a e\right )+2 c e^2 \left (23 b^2 d+12 b \sqrt {b^2-4 a c} d-26 a b e-10 a \sqrt {b^2-4 a c} e\right )\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{2} \sqrt {b^2-4 a c} e+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{8 \left (b^2-4 a c\right )^{5/2} \left (c d^2-b d e+a e^2\right )}+\frac {\left (c \left (96 c^3 d^3+b^2 \left (b+\sqrt {b^2-4 a c}\right ) e^3-8 c^2 d e \left (18 b d-3 \sqrt {b^2-4 a c} d-13 a e\right )+2 c e^2 \left (23 b^2 d+10 a \sqrt {b^2-4 a c} e-2 b \left (6 \sqrt {b^2-4 a c} d+13 a e\right )\right )\right )\right ) \text {Subst}\left (\int \frac {1}{-\frac {1}{2} \sqrt {b^2-4 a c} e+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{8 \left (b^2-4 a c\right )^{5/2} \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {(b+2 c x) \sqrt {d+e x}}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {\sqrt {d+e x} \left (13 b^2 c d e-4 a c^2 d e-b^3 e^2-4 b c \left (3 c d^2+2 a e^2\right )-c \left (24 c^2 d^2+b^2 e^2-4 c e (6 b d-5 a e)\right ) x\right )}{4 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )}-\frac {\sqrt {c} \left (96 c^3 d^3+b^2 \left (b+\sqrt {b^2-4 a c}\right ) e^3-8 c^2 d e \left (18 b d-3 \sqrt {b^2-4 a c} d-13 a e\right )+2 c e^2 \left (23 b^2 d+10 a \sqrt {b^2-4 a c} e-2 b \left (6 \sqrt {b^2-4 a c} d+13 a e\right )\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{4 \sqrt {2} \left (b^2-4 a c\right )^{5/2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )}+\frac {\sqrt {c} \left (96 c^3 d^3+b^2 \left (b-\sqrt {b^2-4 a c}\right ) e^3-8 c^2 d e \left (18 b d+3 \sqrt {b^2-4 a c} d-13 a e\right )+2 c e^2 \left (23 b^2 d+12 b \sqrt {b^2-4 a c} d-26 a b e-10 a \sqrt {b^2-4 a c} e\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{4 \sqrt {2} \left (b^2-4 a c\right )^{5/2} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )}\\ \end {align*}

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Mathematica [A]
time = 12.46, size = 580, normalized size = 0.91 \begin {gather*} -\frac {(b+2 c x) \sqrt {d+e x}}{2 \left (b^2-4 a c\right ) (a+x (b+c x))^2}-\frac {\sqrt {d+e x} \left (b^3 e^2+b^2 c e (-13 d+e x)+4 b c \left (2 a e^2+3 c d (d-2 e x)\right )+4 c^2 \left (6 c d^2 x+a e (d+5 e x)\right )\right )}{4 \left (b^2-4 a c\right )^2 \left (-c d^2+e (b d-a e)\right ) (a+x (b+c x))}-\frac {\sqrt {c} \left (\frac {\left (96 c^3 d^3+b^2 \left (b+\sqrt {b^2-4 a c}\right ) e^3+8 c^2 d e \left (-18 b d+3 \sqrt {b^2-4 a c} d+13 a e\right )+2 c e^2 \left (23 b^2 d+10 a \sqrt {b^2-4 a c} e-2 b \left (6 \sqrt {b^2-4 a c} d+13 a e\right )\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-b e+\sqrt {b^2-4 a c} e}}\right )}{\sqrt {2 c d+\left (-b+\sqrt {b^2-4 a c}\right ) e}}-\frac {\left (96 c^3 d^3+b^2 \left (b-\sqrt {b^2-4 a c}\right ) e^3-8 c^2 d e \left (18 b d+3 \sqrt {b^2-4 a c} d-13 a e\right )+2 c e^2 \left (23 b^2 d+12 b \sqrt {b^2-4 a c} d-26 a b e-10 a \sqrt {b^2-4 a c} e\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{4 \sqrt {2} \left (b^2-4 a c\right )^{5/2} \left (c d^2+e (-b d+a e)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[d + e*x]/(a + b*x + c*x^2)^3,x]

[Out]

-1/2*((b + 2*c*x)*Sqrt[d + e*x])/((b^2 - 4*a*c)*(a + x*(b + c*x))^2) - (Sqrt[d + e*x]*(b^3*e^2 + b^2*c*e*(-13*
d + e*x) + 4*b*c*(2*a*e^2 + 3*c*d*(d - 2*e*x)) + 4*c^2*(6*c*d^2*x + a*e*(d + 5*e*x))))/(4*(b^2 - 4*a*c)^2*(-(c
*d^2) + e*(b*d - a*e))*(a + x*(b + c*x))) - (Sqrt[c]*(((96*c^3*d^3 + b^2*(b + Sqrt[b^2 - 4*a*c])*e^3 + 8*c^2*d
*e*(-18*b*d + 3*Sqrt[b^2 - 4*a*c]*d + 13*a*e) + 2*c*e^2*(23*b^2*d + 10*a*Sqrt[b^2 - 4*a*c]*e - 2*b*(6*Sqrt[b^2
- 4*a*c]*d + 13*a*e)))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])/Sqrt
[2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e] - ((96*c^3*d^3 + b^2*(b - Sqrt[b^2 - 4*a*c])*e^3 - 8*c^2*d*e*(18*b*d + 3*
Sqrt[b^2 - 4*a*c]*d - 13*a*e) + 2*c*e^2*(23*b^2*d + 12*b*Sqrt[b^2 - 4*a*c]*d - 26*a*b*e - 10*a*Sqrt[b^2 - 4*a*
c]*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/Sqrt[2*c*d - (b + Sqr
t[b^2 - 4*a*c])*e]))/(4*Sqrt[2]*(b^2 - 4*a*c)^(5/2)*(c*d^2 + e*(-(b*d) + a*e)))

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Maple [A]
time = 0.96, size = 800, normalized size = 1.26

method	result	size
derivativedivides	\(128 e^{5} c^{3} \left (\frac {\frac {\frac {\left (-6 b e +12 c d +5 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right ) \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\, \left (e x +d \right )^{\frac {3}{2}}}{16 c \left (-b e +2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right )}-\frac {\left (-6 b e +12 c d +7 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right ) \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\, \sqrt {e x +d}}{32 c^{2}}}{\left (-e x -\frac {b e}{2 c}+\frac {\sqrt {e^{2} \left (-4 a c +b^{2}\right )}}{2 c}\right )^{2}}+\frac {\left (20 a c \,e^{2}-17 b^{2} e^{2}+48 b c d e -48 c^{2} d^{2}+18 b e \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}-36 d \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\, c \right ) \sqrt {2}\, \arctanh \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \left (-4 b e +8 c d +4 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right ) \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}}{16 \left (4 a c -b^{2}\right )^{2} e^{4} \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c}-\frac {\frac {-\frac {\left (-6 b e +12 c d -5 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right ) \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\, \left (e x +d \right )^{\frac {3}{2}}}{16 c \left (-b e +2 c d -\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right )}+\frac {\left (-6 b e +12 c d -7 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right ) \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\, \sqrt {e x +d}}{32 c^{2}}}{\left (-e x -\frac {b e}{2 c}-\frac {\sqrt {e^{2} \left (-4 a c +b^{2}\right )}}{2 c}\right )^{2}}-\frac {\left (-20 a c \,e^{2}+17 b^{2} e^{2}-48 b c d e +48 c^{2} d^{2}+18 b e \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}-36 d \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\, c \right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \left (4 b e -8 c d +4 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right ) \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}}{16 \left (4 a c -b^{2}\right )^{2} e^{4} \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c}\right )\)	\(800\)
default	\(128 e^{5} c^{3} \left (\frac {\frac {\frac {\left (-6 b e +12 c d +5 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right ) \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\, \left (e x +d \right )^{\frac {3}{2}}}{16 c \left (-b e +2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right )}-\frac {\left (-6 b e +12 c d +7 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right ) \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\, \sqrt {e x +d}}{32 c^{2}}}{\left (-e x -\frac {b e}{2 c}+\frac {\sqrt {e^{2} \left (-4 a c +b^{2}\right )}}{2 c}\right )^{2}}+\frac {\left (20 a c \,e^{2}-17 b^{2} e^{2}+48 b c d e -48 c^{2} d^{2}+18 b e \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}-36 d \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\, c \right ) \sqrt {2}\, \arctanh \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \left (-4 b e +8 c d +4 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right ) \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}}{16 \left (4 a c -b^{2}\right )^{2} e^{4} \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c}-\frac {\frac {-\frac {\left (-6 b e +12 c d -5 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right ) \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\, \left (e x +d \right )^{\frac {3}{2}}}{16 c \left (-b e +2 c d -\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right )}+\frac {\left (-6 b e +12 c d -7 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right ) \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\, \sqrt {e x +d}}{32 c^{2}}}{\left (-e x -\frac {b e}{2 c}-\frac {\sqrt {e^{2} \left (-4 a c +b^{2}\right )}}{2 c}\right )^{2}}-\frac {\left (-20 a c \,e^{2}+17 b^{2} e^{2}-48 b c d e +48 c^{2} d^{2}+18 b e \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}-36 d \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\, c \right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \left (4 b e -8 c d +4 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right ) \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}}{16 \left (4 a c -b^{2}\right )^{2} e^{4} \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c}\right )\)	\(800\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(1/2)/(c*x^2+b*x+a)^3,x,method=_RETURNVERBOSE)

[Out]

128*e^5*c^3*(1/16/(4*a*c-b^2)^2/e^4/(-e^2*(4*a*c-b^2))^(1/2)/c*((1/16*(-6*b*e+12*c*d+5*(-4*a*c*e^2+b^2*e^2)^(1
/2))*(-4*a*c*e^2+b^2*e^2)^(1/2)/c/(-b*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))*(e*x+d)^(3/2)-1/32*(-6*b*e+12*c*d+7*
(-4*a*c*e^2+b^2*e^2)^(1/2))*(-4*a*c*e^2+b^2*e^2)^(1/2)/c^2*(e*x+d)^(1/2))/(-e*x-1/2*b*e/c+1/2/c*(e^2*(-4*a*c+b
^2))^(1/2))^2+1/4*(20*a*c*e^2-17*b^2*e^2+48*b*c*d*e-48*c^2*d^2+18*b*e*(-4*a*c*e^2+b^2*e^2)^(1/2)-36*d*(-4*a*c*
e^2+b^2*e^2)^(1/2)*c)/(-4*b*e+8*c*d+4*(-4*a*c*e^2+b^2*e^2)^(1/2))*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2
))*c)^(1/2)*arctanh(c*(e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)))-1/16/(4*a*c-b^2)
^2/e^4/(-e^2*(4*a*c-b^2))^(1/2)/c*((-1/16*(-6*b*e+12*c*d-5*(-4*a*c*e^2+b^2*e^2)^(1/2))*(-4*a*c*e^2+b^2*e^2)^(1
/2)/c/(-b*e+2*c*d-(-4*a*c*e^2+b^2*e^2)^(1/2))*(e*x+d)^(3/2)+1/32*(-6*b*e+12*c*d-7*(-4*a*c*e^2+b^2*e^2)^(1/2))*
(-4*a*c*e^2+b^2*e^2)^(1/2)/c^2*(e*x+d)^(1/2))/(-e*x-1/2*b*e/c-1/2/c*(e^2*(-4*a*c+b^2))^(1/2))^2-1/4*(-20*a*c*e
^2+17*b^2*e^2-48*b*c*d*e+48*c^2*d^2+18*b*e*(-4*a*c*e^2+b^2*e^2)^(1/2)-36*d*(-4*a*c*e^2+b^2*e^2)^(1/2)*c)/(4*b*
e-8*c*d+4*(-4*a*c*e^2+b^2*e^2)^(1/2))*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan(c*(e*x+d)^
(1/2)*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(1/2)/(c*x^2+b*x+a)^3,x, algorithm="maxima")

[Out]

integrate(sqrt(x*e + d)/(c*x^2 + b*x + a)^3, x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 30590 vs. \(2 (579) = 1158\).
time = 49.60, size = 30590, normalized size = 48.25 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(1/2)/(c*x^2+b*x+a)^3,x, algorithm="fricas")

[Out]

-1/8*(sqrt(1/2)*((b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d^2*x^4 + 2*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d^2*x
^3 + (b^6*c - 6*a*b^4*c^2 + 32*a^3*c^4)*d^2*x^2 + 2*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c^3)*d^2*x + (a^2*b^4*
c - 8*a^3*b^2*c^2 + 16*a^4*c^3)*d^2 + (a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2 + (a*b^4*c^2 - 8*a^2*b^2*c^3 + 16*a^
3*c^4)*x^4 + 2*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c^3)*x^3 + (a*b^6 - 6*a^2*b^4*c + 32*a^4*c^3)*x^2 + 2*(a^2*
b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*x)*e^2 - ((b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d*x^4 + 2*(b^6*c - 8*a*b^4*
c^2 + 16*a^2*b^2*c^3)*d*x^3 + (b^7 - 6*a*b^5*c + 32*a^3*b*c^3)*d*x^2 + 2*(a*b^6 - 8*a^2*b^4*c + 16*a^3*b^2*c^2
)*d*x + (a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*d)*e)*sqrt((4608*c^7*d^7 - 16128*b*c^6*d^6*e + 672*(31*b^2*c^5
+ 20*a*c^6)*d^5*e^2 - 1680*(7*b^3*c^4 + 20*a*b*c^5)*d^4*e^3 + 70*(35*b^4*c^3 + 392*a*b^2*c^4 + 176*a^2*c^5)*d^
3*e^4 + 21*(b^5*c^2 - 360*a*b^3*c^3 - 880*a^2*b*c^4)*d^2*e^5 - 21*(b^6*c - 10*a*b^4*c^2 - 320*a^2*b^2*c^3 - 16
0*a^3*c^4)*d*e^6 - (b^7 - 35*a*b^5*c + 280*a^2*b^3*c^2 + 1680*a^3*b*c^3)*e^7 + ((b^10*c^3 - 20*a*b^8*c^4 + 160
*a^2*b^6*c^5 - 640*a^3*b^4*c^6 + 1280*a^4*b^2*c^7 - 1024*a^5*c^8)*d^6 - 3*(b^11*c^2 - 20*a*b^9*c^3 + 160*a^2*b
^7*c^4 - 640*a^3*b^5*c^5 + 1280*a^4*b^3*c^6 - 1024*a^5*b*c^7)*d^5*e + 3*(b^12*c - 19*a*b^10*c^2 + 140*a^2*b^8*
c^3 - 480*a^3*b^6*c^4 + 640*a^4*b^4*c^5 + 256*a^5*b^2*c^6 - 1024*a^6*c^7)*d^4*e^2 - (b^13 - 14*a*b^11*c + 40*a
^2*b^9*c^2 + 320*a^3*b^7*c^3 - 2560*a^4*b^5*c^4 + 6656*a^5*b^3*c^5 - 6144*a^6*b*c^6)*d^3*e^3 + 3*(a*b^12 - 19*
a^2*b^10*c + 140*a^3*b^8*c^2 - 480*a^4*b^6*c^3 + 640*a^5*b^4*c^4 + 256*a^6*b^2*c^5 - 1024*a^7*c^6)*d^2*e^4 - 3
*(a^2*b^11 - 20*a^3*b^9*c + 160*a^4*b^7*c^2 - 640*a^5*b^5*c^3 + 1280*a^6*b^3*c^4 - 1024*a^7*b*c^5)*d*e^5 + (a^
3*b^10 - 20*a^4*b^8*c + 160*a^5*b^6*c^2 - 640*a^6*b^4*c^3 + 1280*a^7*b^2*c^4 - 1024*a^8*c^5)*e^6)*sqrt((441*c^
4*d^4*e^10 - 882*b*c^3*d^3*e^11 + 21*(19*b^2*c^2 + 50*a*c^3)*d^2*e^12 + 42*(b^3*c - 25*a*b*c^2)*d*e^13 + (b^4
- 50*a*b^2*c + 625*a^2*c^2)*e^14)/((b^10*c^6 - 20*a*b^8*c^7 + 160*a^2*b^6*c^8 - 640*a^3*b^4*c^9 + 1280*a^4*b^2
*c^10 - 1024*a^5*c^11)*d^12 - 6*(b^11*c^5 - 20*a*b^9*c^6 + 160*a^2*b^7*c^7 - 640*a^3*b^5*c^8 + 1280*a^4*b^3*c^
9 - 1024*a^5*b*c^10)*d^11*e + 3*(5*b^12*c^4 - 98*a*b^10*c^5 + 760*a^2*b^8*c^6 - 2880*a^3*b^6*c^7 + 5120*a^4*b^
4*c^8 - 2560*a^5*b^2*c^9 - 2048*a^6*c^10)*d^10*e^2 - 10*(2*b^13*c^3 - 37*a*b^11*c^4 + 260*a^2*b^9*c^5 - 800*a^
3*b^7*c^6 + 640*a^4*b^5*c^7 + 1792*a^5*b^3*c^8 - 3072*a^6*b*c^9)*d^9*e^3 + 15*(b^14*c^2 - 16*a*b^12*c^3 + 81*a
^2*b^10*c^4 - 20*a^3*b^8*c^5 - 1120*a^4*b^6*c^6 + 3456*a^5*b^4*c^7 - 2816*a^6*b^2*c^8 - 1024*a^7*c^9)*d^8*e^4
- 6*(b^15*c - 10*a*b^13*c^2 - 30*a^2*b^11*c^3 + 760*a^3*b^9*c^4 - 3520*a^4*b^7*c^5 + 5376*a^5*b^5*c^6 + 2560*a
^6*b^3*c^7 - 10240*a^7*b*c^8)*d^7*e^5 + (b^16 + 10*a*b^14*c - 350*a^2*b^12*c^2 + 2380*a^3*b^10*c^3 - 3920*a^4*
b^8*c^4 - 17024*a^5*b^6*c^5 + 71680*a^6*b^4*c^6 - 66560*a^7*b^2*c^7 - 20480*a^8*c^8)*d^6*e^6 - 6*(a*b^15 - 10*
a^2*b^13*c - 30*a^3*b^11*c^2 + 760*a^4*b^9*c^3 - 3520*a^5*b^7*c^4 + 5376*a^6*b^5*c^5 + 2560*a^7*b^3*c^6 - 1024
0*a^8*b*c^7)*d^5*e^7 + 15*(a^2*b^14 - 16*a^3*b^12*c + 81*a^4*b^10*c^2 - 20*a^5*b^8*c^3 - 1120*a^6*b^6*c^4 + 34
56*a^7*b^4*c^5 - 2816*a^8*b^2*c^6 - 1024*a^9*c^7)*d^4*e^8 - 10*(2*a^3*b^13 - 37*a^4*b^11*c + 260*a^5*b^9*c^2 -
800*a^6*b^7*c^3 + 640*a^7*b^5*c^4 + 1792*a^8*b^3*c^5 - 3072*a^9*b*c^6)*d^3*e^9 + 3*(5*a^4*b^12 - 98*a^5*b^10*
c + 760*a^6*b^8*c^2 - 2880*a^7*b^6*c^3 + 5120*a^8*b^4*c^4 - 2560*a^9*b^2*c^5 - 2048*a^10*c^6)*d^2*e^10 - 6*(a^
5*b^11 - 20*a^6*b^9*c + 160*a^7*b^7*c^2 - 640*a^8*b^5*c^3 + 1280*a^9*b^3*c^4 - 1024*a^10*b*c^5)*d*e^11 + (a^6*
b^10 - 20*a^7*b^8*c + 160*a^8*b^6*c^2 - 640*a^9*b^4*c^3 + 1280*a^10*b^2*c^4 - 1024*a^11*c^5)*e^12)))/((b^10*c^
3 - 20*a*b^8*c^4 + 160*a^2*b^6*c^5 - 640*a^3*b^4*c^6 + 1280*a^4*b^2*c^7 - 1024*a^5*c^8)*d^6 - 3*(b^11*c^2 - 20
*a*b^9*c^3 + 160*a^2*b^7*c^4 - 640*a^3*b^5*c^5 + 1280*a^4*b^3*c^6 - 1024*a^5*b*c^7)*d^5*e + 3*(b^12*c - 19*a*b
^10*c^2 + 140*a^2*b^8*c^3 - 480*a^3*b^6*c^4 + 640*a^4*b^4*c^5 + 256*a^5*b^2*c^6 - 1024*a^6*c^7)*d^4*e^2 - (b^1
3 - 14*a*b^11*c + 40*a^2*b^9*c^2 + 320*a^3*b^7*c^3 - 2560*a^4*b^5*c^4 + 6656*a^5*b^3*c^5 - 6144*a^6*b*c^6)*d^3
*e^3 + 3*(a*b^12 - 19*a^2*b^10*c + 140*a^3*b^8*c^2 - 480*a^4*b^6*c^3 + 640*a^5*b^4*c^4 + 256*a^6*b^2*c^5 - 102
4*a^7*c^6)*d^2*e^4 - 3*(a^2*b^11 - 20*a^3*b^9*c + 160*a^4*b^7*c^2 - 640*a^5*b^5*c^3 + 1280*a^6*b^3*c^4 - 1024*
a^7*b*c^5)*d*e^5 + (a^3*b^10 - 20*a^4*b^8*c + 160*a^5*b^6*c^2 - 640*a^6*b^4*c^3 + 1280*a^7*b^2*c^4 - 1024*a^8*
c^5)*e^6))*log(1/2*sqrt(1/2)*(504*(b^6*c^5 - 12*a*b^4*c^6 + 48*a^2*b^2*c^7 - 64*a^3*c^8)*d^5*e^6 - 1260*(b^7*c
^4 - 12*a*b^5*c^5 + 48*a^2*b^3*c^6 - 64*a^3*b*c^7)*d^4*e^7 + 6*(157*b^8*c^3 - 1672*a*b^6*c^4 + 4992*a^2*b^4*c^
5 + 128*a^3*b^2*c^6 - 13568*a^4*c^7)*d^3*e^8 - 9*(17*b^9*c^2 + 8*a*b^7*c^3 - 1728*a^2*b^5*c^4 + 9088*a^3*b^3*c
^5 - 13568*a^4*b*c^6)*d^2*e^9 - (31*b^10*c - 926*a*b^8*c^2 + 7336*a^2*b^6*c^3 - 18976*a^3*b^4*c^4 - 2944*a^4*b
^2*c^5 + 51200*a^5*c^6)*d*e^10 - (b^11 - 53*a*b...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(1/2)/(c*x**2+b*x+a)**3,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 5117 vs. \(2 (579) = 1158\).
time = 4.04, size = 5117, normalized size = 8.07 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(1/2)/(c*x^2+b*x+a)^3,x, algorithm="giac")

[Out]

1/32*((b^4*c*d^2*e - 8*a*b^2*c^2*d^2*e + 16*a^2*c^3*d^2*e - b^5*d*e^2 + 8*a*b^3*c*d*e^2 - 16*a^2*b*c^2*d*e^2 +
a*b^4*e^3 - 8*a^2*b^2*c*e^3 + 16*a^3*c^2*e^3)^2*(24*c^2*d^2*e - 24*b*c*d*e^2 + (b^2 + 20*a*c)*e^3)*sqrt(-4*c^
2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e) + 2*(24*(b^2*c^4 - 4*a*c^5)*sqrt(b^2 - 4*a*c)*d^5*e - 60*(b^3*c^3 - 4*a
*b*c^4)*sqrt(b^2 - 4*a*c)*d^4*e^2 + 2*(23*b^4*c^2 - 64*a*b^2*c^3 - 112*a^2*c^4)*sqrt(b^2 - 4*a*c)*d^3*e^3 - 3*
(3*b^5*c + 16*a*b^3*c^2 - 112*a^2*b*c^3)*sqrt(b^2 - 4*a*c)*d^2*e^4 - (b^6 - 30*a*b^4*c + 72*a^2*b^2*c^2 + 128*
a^3*c^3)*sqrt(b^2 - 4*a*c)*d*e^5 + (a*b^5 - 20*a^2*b^3*c + 64*a^3*b*c^2)*sqrt(b^2 - 4*a*c)*e^6)*sqrt(-4*c^2*d
+ 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*abs(b^4*c*d^2*e - 8*a*b^2*c^2*d^2*e + 16*a^2*c^3*d^2*e - b^5*d*e^2 + 8*a*b^
3*c*d*e^2 - 16*a^2*b*c^2*d*e^2 + a*b^4*e^3 - 8*a^2*b^2*c*e^3 + 16*a^3*c^2*e^3) - (192*(b^6*c^6 - 12*a*b^4*c^7
+ 48*a^2*b^2*c^8 - 64*a^3*c^9)*d^8*e - 768*(b^7*c^5 - 12*a*b^5*c^6 + 48*a^2*b^3*c^7 - 64*a^3*b*c^8)*d^7*e^2 +
4*(299*b^8*c^4 - 3440*a*b^6*c^5 + 12576*a^2*b^4*c^6 - 12032*a^3*b^2*c^7 - 9472*a^4*c^8)*d^6*e^3 - 12*(75*b^9*c
^3 - 752*a*b^7*c^4 + 1824*a^2*b^5*c^5 + 2304*a^3*b^3*c^6 - 9472*a^4*b*c^7)*d^5*e^4 + (323*b^10*c^2 - 1960*a*b^
8*c^3 - 6880*a^2*b^6*c^4 + 64000*a^3*b^4*c^5 - 93440*a^4*b^2*c^6 - 38912*a^5*c^7)*d^4*e^5 - 2*(21*b^11*c + 184
*a*b^9*c^2 - 3616*a^2*b^7*c^3 + 12288*a^3*b^5*c^4 + 1280*a^4*b^3*c^5 - 38912*a^5*b*c^6)*d^3*e^6 - (b^12 - 150*
a*b^10*c + 948*a^2*b^8*c^2 + 2176*a^3*b^6*c^3 - 24960*a^4*b^4*c^4 + 38400*a^5*b^2*c^5 + 13312*a^6*c^6)*d^2*e^7
+ 2*(a*b^11 - 86*a^2*b^9*c + 832*a^3*b^7*c^2 - 2368*a^4*b^5*c^3 - 256*a^5*b^3*c^4 + 6656*a^6*b*c^5)*d*e^8 - (
a^2*b^10 - 64*a^3*b^8*c + 672*a^4*b^6*c^2 - 2560*a^5*b^4*c^3 + 3328*a^6*b^2*c^4)*e^9)*sqrt(-4*c^2*d + 2*(b*c -
sqrt(b^2 - 4*a*c)*c)*e))*arctan(2*sqrt(1/2)*sqrt(x*e + d)/sqrt(-(2*b^4*c^2*d^3 - 16*a*b^2*c^3*d^3 + 32*a^2*c^
4*d^3 - 3*b^5*c*d^2*e + 24*a*b^3*c^2*d^2*e - 48*a^2*b*c^3*d^2*e + b^6*d*e^2 - 6*a*b^4*c*d*e^2 + 32*a^3*c^3*d*e
^2 - a*b^5*e^3 + 8*a^2*b^3*c*e^3 - 16*a^3*b*c^2*e^3 + sqrt((2*b^4*c^2*d^3 - 16*a*b^2*c^3*d^3 + 32*a^2*c^4*d^3
- 3*b^5*c*d^2*e + 24*a*b^3*c^2*d^2*e - 48*a^2*b*c^3*d^2*e + b^6*d*e^2 - 6*a*b^4*c*d*e^2 + 32*a^3*c^3*d*e^2 - a
*b^5*e^3 + 8*a^2*b^3*c*e^3 - 16*a^3*b*c^2*e^3)^2 - 4*(b^4*c^2*d^4 - 8*a*b^2*c^3*d^4 + 16*a^2*c^4*d^4 - 2*b^5*c
*d^3*e + 16*a*b^3*c^2*d^3*e - 32*a^2*b*c^3*d^3*e + b^6*d^2*e^2 - 6*a*b^4*c*d^2*e^2 + 32*a^3*c^3*d^2*e^2 - 2*a*
b^5*d*e^3 + 16*a^2*b^3*c*d*e^3 - 32*a^3*b*c^2*d*e^3 + a^2*b^4*e^4 - 8*a^3*b^2*c*e^4 + 16*a^4*c^2*e^4)*(b^4*c^2
*d^2 - 8*a*b^2*c^3*d^2 + 16*a^2*c^4*d^2 - b^5*c*d*e + 8*a*b^3*c^2*d*e - 16*a^2*b*c^3*d*e + a*b^4*c*e^2 - 8*a^2
*b^2*c^2*e^2 + 16*a^3*c^3*e^2)))/(b^4*c^2*d^2 - 8*a*b^2*c^3*d^2 + 16*a^2*c^4*d^2 - b^5*c*d*e + 8*a*b^3*c^2*d*e
- 16*a^2*b*c^3*d*e + a*b^4*c*e^2 - 8*a^2*b^2*c^2*e^2 + 16*a^3*c^3*e^2)))/(((b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b
^2*c^5 - 64*a^3*c^6)*sqrt(b^2 - 4*a*c)*d^6 - 3*(b^7*c^2 - 12*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*sqrt(b
^2 - 4*a*c)*d^5*e + 3*(b^8*c - 11*a*b^6*c^2 + 36*a^2*b^4*c^3 - 16*a^3*b^2*c^4 - 64*a^4*c^5)*sqrt(b^2 - 4*a*c)*
d^4*e^2 - (b^9 - 6*a*b^7*c - 24*a^2*b^5*c^2 + 224*a^3*b^3*c^3 - 384*a^4*b*c^4)*sqrt(b^2 - 4*a*c)*d^3*e^3 + 3*(
a*b^8 - 11*a^2*b^6*c + 36*a^3*b^4*c^2 - 16*a^4*b^2*c^3 - 64*a^5*c^4)*sqrt(b^2 - 4*a*c)*d^2*e^4 - 3*(a^2*b^7 -
12*a^3*b^5*c + 48*a^4*b^3*c^2 - 64*a^5*b*c^3)*sqrt(b^2 - 4*a*c)*d*e^5 + (a^3*b^6 - 12*a^4*b^4*c + 48*a^5*b^2*c
^2 - 64*a^6*c^3)*sqrt(b^2 - 4*a*c)*e^6)*abs(b^4*c*d^2*e - 8*a*b^2*c^2*d^2*e + 16*a^2*c^3*d^2*e - b^5*d*e^2 + 8
*a*b^3*c*d*e^2 - 16*a^2*b*c^2*d*e^2 + a*b^4*e^3 - 8*a^2*b^2*c*e^3 + 16*a^3*c^2*e^3)*abs(c)) - 1/32*((b^4*c*d^2
*e - 8*a*b^2*c^2*d^2*e + 16*a^2*c^3*d^2*e - b^5*d*e^2 + 8*a*b^3*c*d*e^2 - 16*a^2*b*c^2*d*e^2 + a*b^4*e^3 - 8*a
^2*b^2*c*e^3 + 16*a^3*c^2*e^3)^2*(24*c^2*d^2*e - 24*b*c*d*e^2 + (b^2 + 20*a*c)*e^3)*sqrt(-4*c^2*d + 2*(b*c + s
qrt(b^2 - 4*a*c)*c)*e) - 2*(24*(b^2*c^4 - 4*a*c^5)*sqrt(b^2 - 4*a*c)*d^5*e - 60*(b^3*c^3 - 4*a*b*c^4)*sqrt(b^2
- 4*a*c)*d^4*e^2 + 2*(23*b^4*c^2 - 64*a*b^2*c^3 - 112*a^2*c^4)*sqrt(b^2 - 4*a*c)*d^3*e^3 - 3*(3*b^5*c + 16*a*
b^3*c^2 - 112*a^2*b*c^3)*sqrt(b^2 - 4*a*c)*d^2*e^4 - (b^6 - 30*a*b^4*c + 72*a^2*b^2*c^2 + 128*a^3*c^3)*sqrt(b^
2 - 4*a*c)*d*e^5 + (a*b^5 - 20*a^2*b^3*c + 64*a^3*b*c^2)*sqrt(b^2 - 4*a*c)*e^6)*sqrt(-4*c^2*d + 2*(b*c + sqrt(
b^2 - 4*a*c)*c)*e)*abs(b^4*c*d^2*e - 8*a*b^2*c^2*d^2*e + 16*a^2*c^3*d^2*e - b^5*d*e^2 + 8*a*b^3*c*d*e^2 - 16*a
^2*b*c^2*d*e^2 + a*b^4*e^3 - 8*a^2*b^2*c*e^3 + 16*a^3*c^2*e^3) - (192*(b^6*c^6 - 12*a*b^4*c^7 + 48*a^2*b^2*c^8
- 64*a^3*c^9)*d^8*e - 768*(b^7*c^5 - 12*a*b^5*c^6 + 48*a^2*b^3*c^7 - 64*a^3*b*c^8)*d^7*e^2 + 4*(299*b^8*c^4 -
3440*a*b^6*c^5 + 12576*a^2*b^4*c^6 - 12032*a^3*b^2*c^7 - 9472*a^4*c^8)*d^6*e^3 - 12*(75*b^9*c^3 - 752*a*b^7*c
^4 + 1824*a^2*b^5*c^5 + 2304*a^3*b^3*c^6 - 9472*a^4*b*c^7)*d^5*e^4 + (323*b^10*c^2 - 1960*a*b^8*c^3 - 6880*a^2
*b^6*c^4 + 64000*a^3*b^4*c^5 - 93440*a^4*b^2*c^6 - 38912*a^5*c^7)*d^4*e^5 - 2*(21*b^11*c + 184*a*b^9*c^2 - 361
6*a^2*b^7*c^3 + 12288*a^3*b^5*c^4 + 1280*a^4*b^...

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Mupad [B]
time = 69.78, size = 2500, normalized size = 3.94 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^(1/2)/(a + b*x + c*x^2)^3,x)

[Out]

log(- (2^(1/2)*((2^(1/2)*((c^2*e^3*(b*e - 2*c*d)*(b^2*e^2 - 12*c^2*d^2 - 16*a*c*e^2 + 12*b*c*d*e))/((4*a*c - b
^2)*(a*e^2 + c*d^2 - b*d*e)) - (2^(1/2)*c^2*e^2*(4*a*c - b^2)*(b*e - 2*c*d)*(d + e*x)^(1/2)*(-(b^17*e^7 + 4718
592*a^5*c^12*d^7 - 4608*b^10*c^7*d^7 + b^2*e^7*(-(4*a*c - b^2)^15)^(1/2) + 92160*a*b^8*c^8*d^7 - 1720320*a^8*b
*c^8*e^7 + 3440640*a^8*c^9*d*e^6 + 16128*b^11*c^6*d^6*e - 737280*a^2*b^6*c^9*d^7 + 2949120*a^3*b^4*c^10*d^7 -
5898240*a^4*b^2*c^11*d^7 + 1140*a^2*b^13*c^2*e^7 - 10160*a^3*b^11*c^3*e^7 + 34880*a^4*b^9*c^4*e^7 + 43776*a^5*
b^7*c^5*e^7 - 680960*a^6*b^5*c^6*e^7 + 1863680*a^7*b^3*c^7*e^7 + 13762560*a^6*c^11*d^5*e^2 + 12615680*a^7*c^10
*d^3*e^4 - 20832*b^12*c^5*d^5*e^2 + 11760*b^13*c^4*d^4*e^3 - 2450*b^14*c^3*d^3*e^4 - 21*b^15*c^2*d^2*e^5 - 21*
c^2*d^2*e^5*(-(4*a*c - b^2)^15)^(1/2) - 55*a*b^15*c*e^7 - 25*a*c*e^7*(-(4*a*c - b^2)^15)^(1/2) + 21*b^16*c*d*e
^6 + 21*b*c*d*e^6*(-(4*a*c - b^2)^15)^(1/2) - 3064320*a^2*b^8*c^7*d^5*e^2 + 1209600*a^2*b^9*c^6*d^4*e^3 + 1444
80*a^2*b^10*c^5*d^3*e^4 - 136080*a^2*b^11*c^4*d^2*e^5 + 11182080*a^3*b^6*c^8*d^5*e^2 - 2150400*a^3*b^7*c^7*d^4
*e^3 - 2576000*a^3*b^8*c^6*d^3*e^4 + 853440*a^3*b^9*c^5*d^2*e^5 - 18063360*a^4*b^4*c^9*d^5*e^2 - 6451200*a^4*b
^5*c^8*d^4*e^3 + 12454400*a^4*b^6*c^7*d^3*e^4 - 1908480*a^4*b^7*c^6*d^2*e^5 + 4128768*a^5*b^2*c^10*d^5*e^2 + 3
0965760*a^5*b^3*c^9*d^4*e^3 - 24729600*a^5*b^4*c^8*d^3*e^4 - 2128896*a^5*b^5*c^7*d^2*e^5 + 12328960*a^6*b^2*c^
9*d^3*e^4 + 15912960*a^6*b^3*c^8*d^2*e^5 - 322560*a*b^9*c^7*d^6*e - 630*a*b^14*c^2*d*e^6 - 16515072*a^5*b*c^11
*d^6*e + 403200*a*b^10*c^6*d^5*e^2 - 201600*a*b^11*c^5*d^4*e^3 + 21560*a*b^12*c^4*d^3*e^4 + 7980*a*b^13*c^3*d^
2*e^5 + 2580480*a^2*b^7*c^8*d^6*e + 840*a^2*b^12*c^3*d*e^6 - 10321920*a^3*b^5*c^9*d^6*e + 84000*a^3*b^10*c^4*d
*e^6 + 20643840*a^4*b^3*c^10*d^6*e - 846720*a^4*b^8*c^5*d*e^6 + 3472896*a^5*b^6*c^6*d*e^6 - 34406400*a^6*b*c^1
0*d^4*e^3 - 6236160*a^6*b^4*c^7*d*e^6 - 18923520*a^7*b*c^9*d^2*e^5 + 2580480*a^7*b^2*c^8*d*e^6)/((4*a*c - b^2)
^10*(a*e^2 + c*d^2 - b*d*e)^3))^(1/2))/2)*(-(b^17*e^7 + 4718592*a^5*c^12*d^7 - 4608*b^10*c^7*d^7 + b^2*e^7*(-(
4*a*c - b^2)^15)^(1/2) + 92160*a*b^8*c^8*d^7 - 1720320*a^8*b*c^8*e^7 + 3440640*a^8*c^9*d*e^6 + 16128*b^11*c^6*
d^6*e - 737280*a^2*b^6*c^9*d^7 + 2949120*a^3*b^4*c^10*d^7 - 5898240*a^4*b^2*c^11*d^7 + 1140*a^2*b^13*c^2*e^7 -
10160*a^3*b^11*c^3*e^7 + 34880*a^4*b^9*c^4*e^7 + 43776*a^5*b^7*c^5*e^7 - 680960*a^6*b^5*c^6*e^7 + 1863680*a^7
*b^3*c^7*e^7 + 13762560*a^6*c^11*d^5*e^2 + 12615680*a^7*c^10*d^3*e^4 - 20832*b^12*c^5*d^5*e^2 + 11760*b^13*c^4
*d^4*e^3 - 2450*b^14*c^3*d^3*e^4 - 21*b^15*c^2*d^2*e^5 - 21*c^2*d^2*e^5*(-(4*a*c - b^2)^15)^(1/2) - 55*a*b^15*
c*e^7 - 25*a*c*e^7*(-(4*a*c - b^2)^15)^(1/2) + 21*b^16*c*d*e^6 + 21*b*c*d*e^6*(-(4*a*c - b^2)^15)^(1/2) - 3064
320*a^2*b^8*c^7*d^5*e^2 + 1209600*a^2*b^9*c^6*d^4*e^3 + 144480*a^2*b^10*c^5*d^3*e^4 - 136080*a^2*b^11*c^4*d^2*
e^5 + 11182080*a^3*b^6*c^8*d^5*e^2 - 2150400*a^3*b^7*c^7*d^4*e^3 - 2576000*a^3*b^8*c^6*d^3*e^4 + 853440*a^3*b^
9*c^5*d^2*e^5 - 18063360*a^4*b^4*c^9*d^5*e^2 - 6451200*a^4*b^5*c^8*d^4*e^3 + 12454400*a^4*b^6*c^7*d^3*e^4 - 19
08480*a^4*b^7*c^6*d^2*e^5 + 4128768*a^5*b^2*c^10*d^5*e^2 + 30965760*a^5*b^3*c^9*d^4*e^3 - 24729600*a^5*b^4*c^8
*d^3*e^4 - 2128896*a^5*b^5*c^7*d^2*e^5 + 12328960*a^6*b^2*c^9*d^3*e^4 + 15912960*a^6*b^3*c^8*d^2*e^5 - 322560*
a*b^9*c^7*d^6*e - 630*a*b^14*c^2*d*e^6 - 16515072*a^5*b*c^11*d^6*e + 403200*a*b^10*c^6*d^5*e^2 - 201600*a*b^11
*c^5*d^4*e^3 + 21560*a*b^12*c^4*d^3*e^4 + 7980*a*b^13*c^3*d^2*e^5 + 2580480*a^2*b^7*c^8*d^6*e + 840*a^2*b^12*c
^3*d*e^6 - 10321920*a^3*b^5*c^9*d^6*e + 84000*a^3*b^10*c^4*d*e^6 + 20643840*a^4*b^3*c^10*d^6*e - 846720*a^4*b^
8*c^5*d*e^6 + 3472896*a^5*b^6*c^6*d*e^6 - 34406400*a^6*b*c^10*d^4*e^3 - 6236160*a^6*b^4*c^7*d*e^6 - 18923520*a
^7*b*c^9*d^2*e^5 + 2580480*a^7*b^2*c^8*d*e^6)/((4*a*c - b^2)^10*(a*e^2 + c*d^2 - b*d*e)^3))^(1/2))/16 + (c^3*e
^2*(d + e*x)^(1/2)*(b^6*e^6 + 4608*c^6*d^6 - 800*a^3*c^3*e^6 + 8832*a*c^5*d^4*e^2 + 1472*a^2*b^2*c^2*e^6 + 348
8*a^2*c^4*d^2*e^4 + 15072*b^2*c^4*d^4*e^2 - 7104*b^3*c^3*d^3*e^3 + 1226*b^4*c^2*d^2*e^4 - 34*a*b^4*c*e^6 - 138
24*b*c^5*d^5*e + 22*b^5*c*d*e^5 - 17664*a*b*c^4*d^3*e^3 - 2672*a*b^3*c^2*d*e^5 - 3488*a^2*b*c^3*d*e^5 + 11504*
a*b^2*c^3*d^2*e^4))/(8*(4*a*c - b^2)^4*(a*e^2 + c*d^2 - b*d*e)^2))*(-(b^17*e^7 + 4718592*a^5*c^12*d^7 - 4608*b
^10*c^7*d^7 + b^2*e^7*(-(4*a*c - b^2)^15)^(1/2) + 92160*a*b^8*c^8*d^7 - 1720320*a^8*b*c^8*e^7 + 3440640*a^8*c^
9*d*e^6 + 16128*b^11*c^6*d^6*e - 737280*a^2*b^6*c^9*d^7 + 2949120*a^3*b^4*c^10*d^7 - 5898240*a^4*b^2*c^11*d^7
+ 1140*a^2*b^13*c^2*e^7 - 10160*a^3*b^11*c^3*e^7 + 34880*a^4*b^9*c^4*e^7 + 43776*a^5*b^7*c^5*e^7 - 680960*a^6*
b^5*c^6*e^7 + 1863680*a^7*b^3*c^7*e^7 + 13762560*a^6*c^11*d^5*e^2 + 12615680*a^7*c^10*d^3*e^4 - 20832*b^12*c^5
*d^5*e^2 + 11760*b^13*c^4*d^4*e^3 - 2450*b^14*c^3*d^3*e^4 - 21*b^15*c^2*d^2*e^5 - 21*c^2*d^2*e^5*(-(4*a*c - b^
2)^15)^(1/2) - 55*a*b^15*c*e^7 - 25*a*c*e^7*(-(4*a*c - b^2)^15)^(1/2) + 21*b^16*c*d*e^6 + 21*b*c*d*e^6*(-(4*a*
c - b^2)^15)^(1/2) - 3064320*a^2*b^8*c^7*d^5*e^...

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