3.23.96 \(\int \frac {(d+e x)^{5/2}}{(a+b x+c x^2)^2} \, dx\) [2296]

Optimal. Leaf size=504 \[ \frac {e (2 c d-b e) \sqrt {d+e x}}{c \left (b^2-4 a c\right )}-\frac {(d+e x)^{3/2} (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\left (8 c^3 d^3+b^2 \left (b-\sqrt {b^2-4 a c}\right ) e^3-2 c^2 d e \left (6 b d-\sqrt {b^2-4 a c} d-8 a e\right )+2 c e^2 \left (b^2 d+3 a \sqrt {b^2-4 a c} e-b \left (\sqrt {b^2-4 a c} d+4 a e\right )\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} c^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {\left (8 c^3 d^3+b^2 \left (b+\sqrt {b^2-4 a c}\right ) e^3-2 c^2 d e \left (6 b d+\sqrt {b^2-4 a c} d-8 a e\right )+2 c e^2 \left (b^2 d+b \sqrt {b^2-4 a c} d-4 a b e-3 a \sqrt {b^2-4 a c} e\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} c^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \]

[Out]

-(e*x+d)^(3/2)*(b*d-2*a*e+(-b*e+2*c*d)*x)/(-4*a*c+b^2)/(c*x^2+b*x+a)+e*(-b*e+2*c*d)*(e*x+d)^(1/2)/c/(-4*a*c+b^
2)+1/2*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))*(8*c^3*d^3+b^2*e^3*(b-(-4
*a*c+b^2)^(1/2))-2*c^2*d*e*(6*b*d-8*a*e-d*(-4*a*c+b^2)^(1/2))+2*c*e^2*(b^2*d+3*a*e*(-4*a*c+b^2)^(1/2)-b*(4*a*e
+d*(-4*a*c+b^2)^(1/2))))/c^(3/2)/(-4*a*c+b^2)^(3/2)*2^(1/2)/(2*c*d-e*(b-(-4*a*c+b^2)^(1/2)))^(1/2)-1/2*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))*(8*c^3*d^3+b^2*e^3*(b+(-4*a*c+b^2)^(1/2
))-2*c^2*d*e*(6*b*d-8*a*e+d*(-4*a*c+b^2)^(1/2))+2*c*e^2*(b^2*d-4*a*b*e+b*d*(-4*a*c+b^2)^(1/2)-3*a*e*(-4*a*c+b^
2)^(1/2)))/c^(3/2)/(-4*a*c+b^2)^(3/2)*2^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2)))^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 2.81, antiderivative size = 504, 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 = {752, 838, 840, 1180, 214} \begin {gather*} \frac {\left (-2 c^2 d e \left (-d \sqrt {b^2-4 a c}-8 a e+6 b d\right )+2 c e^2 \left (-b \left (d \sqrt {b^2-4 a c}+4 a e\right )+3 a e \sqrt {b^2-4 a c}+b^2 d\right )+b^2 e^3 \left (b-\sqrt {b^2-4 a c}\right )+8 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 )}{\sqrt {2} c^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}-\frac {\left (-2 c^2 d e \left (d \sqrt {b^2-4 a c}-8 a e+6 b d\right )+2 c e^2 \left (b d \sqrt {b^2-4 a c}-3 a e \sqrt {b^2-4 a c}-4 a b e+b^2 d\right )+b^2 e^3 \left (\sqrt {b^2-4 a c}+b\right )+8 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 )}{\sqrt {2} c^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {(d+e x)^{3/2} (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {e \sqrt {d+e x} (2 c d-b e)}{c \left (b^2-4 a c\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(5/2)/(a + b*x + c*x^2)^2,x]

[Out]

(e*(2*c*d - b*e)*Sqrt[d + e*x])/(c*(b^2 - 4*a*c)) - ((d + e*x)^(3/2)*(b*d - 2*a*e + (2*c*d - b*e)*x))/((b^2 -
4*a*c)*(a + b*x + c*x^2)) + ((8*c^3*d^3 + b^2*(b - Sqrt[b^2 - 4*a*c])*e^3 - 2*c^2*d*e*(6*b*d - Sqrt[b^2 - 4*a*
c]*d - 8*a*e) + 2*c*e^2*(b^2*d + 3*a*Sqrt[b^2 - 4*a*c]*e - b*(Sqrt[b^2 - 4*a*c]*d + 4*a*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^(3/2)*(b^2 - 4*a*c)^(3/2)*Sqrt[2*c
*d - (b - Sqrt[b^2 - 4*a*c])*e]) - ((8*c^3*d^3 + b^2*(b + Sqrt[b^2 - 4*a*c])*e^3 - 2*c^2*d*e*(6*b*d + Sqrt[b^2
- 4*a*c]*d - 8*a*e) + 2*c*e^2*(b^2*d + b*Sqrt[b^2 - 4*a*c]*d - 4*a*b*e - 3*a*Sqrt[b^2 - 4*a*c]*e))*ArcTanh[(S
qrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[2]*c^(3/2)*(b^2 - 4*a*c)^(3/2)*S
qrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e])

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 752

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m - 1)*(d
*b - 2*a*e + (2*c*d - b*e)*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 - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2*c*d^2*(2*p + 3) + e*(b*e - 2*d*
c)*(m + 2*p + 2)*x, 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, 1] && IntQuadraticQ[a, b, c, d,
e, m, p, x]

Rule 838

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[g*
((d + e*x)^m/(c*m)), x] + Dist[1/c, Int[(d + e*x)^(m - 1)*(Simp[c*d*f - a*e*g + (g*c*d - b*e*g + c*e*f)*x, x]/
(a + b*x + c*x^2)), 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] && FractionQ[m] && GtQ[m, 0]

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 {(d+e x)^{5/2}}{\left (a+b x+c x^2\right )^2} \, dx &=-\frac {(d+e x)^{3/2} (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\int \frac {\sqrt {d+e x} \left (\frac {1}{2} \left (4 c d^2-5 b d e+6 a e^2\right )-\frac {1}{2} e (2 c d-b e) x\right )}{a+b x+c x^2} \, dx}{-b^2+4 a c}\\ &=\frac {e (2 c d-b e) \sqrt {d+e x}}{c \left (b^2-4 a c\right )}-\frac {(d+e x)^{3/2} (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {\int \frac {\frac {1}{2} \left (4 c^2 d^3-a b e^3-c d e (5 b d-8 a e)\right )+\frac {1}{2} e \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) x}{\sqrt {d+e x} \left (a+b x+c x^2\right )} \, dx}{c \left (b^2-4 a c\right )}\\ &=\frac {e (2 c d-b e) \sqrt {d+e x}}{c \left (b^2-4 a c\right )}-\frac {(d+e x)^{3/2} (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {2 \text {Subst}\left (\int \frac {\frac {1}{2} e \left (4 c^2 d^3-a b e^3-c d e (5 b d-8 a e)\right )-\frac {1}{2} d e \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right )+\frac {1}{2} e \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 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 )}{c \left (b^2-4 a c\right )}\\ &=\frac {e (2 c d-b e) \sqrt {d+e x}}{c \left (b^2-4 a c\right )}-\frac {(d+e x)^{3/2} (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\left (8 c^3 d^3+b^2 \left (b+\sqrt {b^2-4 a c}\right ) e^3-2 c^2 d e \left (6 b d+\sqrt {b^2-4 a c} d-8 a e\right )+2 c e^2 \left (b^2 d+b \sqrt {b^2-4 a c} d-4 a b e-3 a \sqrt {b^2-4 a c} e\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 )}{2 c \left (b^2-4 a c\right )^{3/2}}-\frac {\left (8 c^3 d^3+b^2 \left (b-\sqrt {b^2-4 a c}\right ) e^3-2 c^2 d e \left (6 b d-\sqrt {b^2-4 a c} d-8 a e\right )+2 c e^2 \left (b^2 d+3 a \sqrt {b^2-4 a c} e-b \left (\sqrt {b^2-4 a c} d+4 a 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 )}{2 c \left (b^2-4 a c\right )^{3/2}}\\ &=\frac {e (2 c d-b e) \sqrt {d+e x}}{c \left (b^2-4 a c\right )}-\frac {(d+e x)^{3/2} (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\left (8 c^3 d^3+b^2 \left (b-\sqrt {b^2-4 a c}\right ) e^3-2 c^2 d e \left (6 b d-\sqrt {b^2-4 a c} d-8 a e\right )+2 c e^2 \left (b^2 d+3 a \sqrt {b^2-4 a c} e-b \left (\sqrt {b^2-4 a c} d+4 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 )}{\sqrt {2} c^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {\left (8 c^3 d^3+b^2 \left (b+\sqrt {b^2-4 a c}\right ) e^3-2 c^2 d e \left (6 b d+\sqrt {b^2-4 a c} d-8 a e\right )+2 c e^2 \left (b^2 d+b \sqrt {b^2-4 a c} d-4 a b e-3 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^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C] Result contains complex when optimal does not.
time = 7.19, size = 796, normalized size = 1.58 \begin {gather*} \frac {-\frac {2 \sqrt {c} \sqrt {d+e x} \left (a b e^2+2 c^2 d^2 x+b^2 e^2 x+b c d (d-2 e x)-2 a c e (2 d+e x)\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}+\frac {2 \sqrt {2} e^2 \left (6 c d+\left (-3 b+\sqrt {b^2-4 a c}\right ) e\right ) \tan ^{-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 {b^2-4 a c} \sqrt {-2 c d+\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {\left (8 i c^3 d^3-b^2 \left (5 i b+\sqrt {-b^2+4 a c}\right ) e^3-2 c^2 d e \left (6 i b d+\sqrt {-b^2+4 a c} d+16 i a e\right )+2 c e^2 \left (7 i b^2 d+b \sqrt {-b^2+4 a c} d+8 i a b e+a \sqrt {-b^2+4 a c} e\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e-i \sqrt {-b^2+4 a c} e}}\right )}{\left (-b^2+4 a c\right )^{3/2} \sqrt {-c d+\frac {1}{2} \left (b-i \sqrt {-b^2+4 a c}\right ) e}}-\frac {\left (-8 i c^3 d^3-b^2 \left (-5 i b+\sqrt {-b^2+4 a c}\right ) e^3+2 c^2 d e \left (6 i b d-\sqrt {-b^2+4 a c} d+16 i a e\right )+2 c e^2 \left (-7 i b^2 d+b \sqrt {-b^2+4 a c} d-8 i a b e+a \sqrt {-b^2+4 a c} e\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e+i \sqrt {-b^2+4 a c} e}}\right )}{\left (-b^2+4 a c\right )^{3/2} \sqrt {-c d+\frac {1}{2} \left (b+i \sqrt {-b^2+4 a c}\right ) e}}+\frac {2 \sqrt {2} e^2 \left (-6 c d+\left (3 b+\sqrt {b^2-4 a c}\right ) e\right ) \tan ^{-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 {b^2-4 a c} \sqrt {-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}}}{2 c^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(5/2)/(a + b*x + c*x^2)^2,x]

[Out]

((-2*Sqrt[c]*Sqrt[d + e*x]*(a*b*e^2 + 2*c^2*d^2*x + b^2*e^2*x + b*c*d*(d - 2*e*x) - 2*a*c*e*(2*d + e*x)))/((b^
2 - 4*a*c)*(a + x*(b + c*x))) + (2*Sqrt[2]*e^2*(6*c*d + (-3*b + Sqrt[b^2 - 4*a*c])*e)*ArcTan[(Sqrt[2]*Sqrt[c]*
Sqrt[d + e*x])/Sqrt[-2*c*d + b*e - Sqrt[b^2 - 4*a*c]*e]])/(Sqrt[b^2 - 4*a*c]*Sqrt[-2*c*d + (b - Sqrt[b^2 - 4*a
*c])*e]) - (((8*I)*c^3*d^3 - b^2*((5*I)*b + Sqrt[-b^2 + 4*a*c])*e^3 - 2*c^2*d*e*((6*I)*b*d + Sqrt[-b^2 + 4*a*c
]*d + (16*I)*a*e) + 2*c*e^2*((7*I)*b^2*d + b*Sqrt[-b^2 + 4*a*c]*d + (8*I)*a*b*e + a*Sqrt[-b^2 + 4*a*c]*e))*Arc
Tan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + b*e - I*Sqrt[-b^2 + 4*a*c]*e]])/((-b^2 + 4*a*c)^(3/2)*Sqrt[-
(c*d) + ((b - I*Sqrt[-b^2 + 4*a*c])*e)/2]) - (((-8*I)*c^3*d^3 - b^2*((-5*I)*b + Sqrt[-b^2 + 4*a*c])*e^3 + 2*c^
2*d*e*((6*I)*b*d - Sqrt[-b^2 + 4*a*c]*d + (16*I)*a*e) + 2*c*e^2*((-7*I)*b^2*d + b*Sqrt[-b^2 + 4*a*c]*d - (8*I)
*a*b*e + a*Sqrt[-b^2 + 4*a*c]*e))*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + b*e + I*Sqrt[-b^2 + 4*a
*c]*e]])/((-b^2 + 4*a*c)^(3/2)*Sqrt[-(c*d) + ((b + I*Sqrt[-b^2 + 4*a*c])*e)/2]) + (2*Sqrt[2]*e^2*(-6*c*d + (3*
b + Sqrt[b^2 - 4*a*c])*e)*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e]])/(S
qrt[b^2 - 4*a*c]*Sqrt[-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e]))/(2*c^(3/2))

________________________________________________________________________________________

Maple [A]
time = 0.77, size = 692, normalized size = 1.37

method	result	size
derivativedivides	\(2 e^{3} \left (\frac {-\frac {\left (2 a c \,e^{2}-b^{2} e^{2}+2 b c d e -2 c^{2} d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{2 c \,e^{2} \left (4 a c -b^{2}\right )}+\frac {\left (a b \,e^{3}-2 a d \,e^{2} c -b^{2} d \,e^{2}+3 d^{2} e b c -2 c^{2} d^{3}\right ) \sqrt {e x +d}}{2 c \,e^{2} \left (4 a c -b^{2}\right )}}{\left (e x +d \right )^{2} c +b e \left (e x +d \right )-2 c d \left (e x +d \right )+e^{2} a -b d e +c \,d^{2}}+\frac {-\frac {\left (-8 a b c \,e^{3}+16 d \,e^{2} c^{2} a +b^{3} e^{3}+2 b^{2} d \,e^{2} c -12 b \,c^{2} d^{2} e +8 c^{3} d^{3}+6 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c \,e^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} e^{2}-2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b c d e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{2} d^{2}\right ) \sqrt {2}\, \arctanh \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}+\frac {\left (8 a b c \,e^{3}-16 d \,e^{2} c^{2} a -b^{3} e^{3}-2 b^{2} d \,e^{2} c +12 b \,c^{2} d^{2} e -8 c^{3} d^{3}+6 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c \,e^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} e^{2}-2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b c d e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{2} d^{2}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}}{e^{2} \left (4 a c -b^{2}\right )}\right )\)	\(692\)
default	\(2 e^{3} \left (\frac {-\frac {\left (2 a c \,e^{2}-b^{2} e^{2}+2 b c d e -2 c^{2} d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{2 c \,e^{2} \left (4 a c -b^{2}\right )}+\frac {\left (a b \,e^{3}-2 a d \,e^{2} c -b^{2} d \,e^{2}+3 d^{2} e b c -2 c^{2} d^{3}\right ) \sqrt {e x +d}}{2 c \,e^{2} \left (4 a c -b^{2}\right )}}{\left (e x +d \right )^{2} c +b e \left (e x +d \right )-2 c d \left (e x +d \right )+e^{2} a -b d e +c \,d^{2}}+\frac {-\frac {\left (-8 a b c \,e^{3}+16 d \,e^{2} c^{2} a +b^{3} e^{3}+2 b^{2} d \,e^{2} c -12 b \,c^{2} d^{2} e +8 c^{3} d^{3}+6 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c \,e^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} e^{2}-2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b c d e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{2} d^{2}\right ) \sqrt {2}\, \arctanh \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}+\frac {\left (8 a b c \,e^{3}-16 d \,e^{2} c^{2} a -b^{3} e^{3}-2 b^{2} d \,e^{2} c +12 b \,c^{2} d^{2} e -8 c^{3} d^{3}+6 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c \,e^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} e^{2}-2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b c d e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{2} d^{2}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}}{e^{2} \left (4 a c -b^{2}\right )}\right )\)	\(692\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(5/2)/(c*x^2+b*x+a)^2,x,method=_RETURNVERBOSE)

[Out]

2*e^3*((-1/2*(2*a*c*e^2-b^2*e^2+2*b*c*d*e-2*c^2*d^2)/c/e^2/(4*a*c-b^2)*(e*x+d)^(3/2)+1/2/c*(a*b*e^3-2*a*c*d*e^
2-b^2*d*e^2+3*b*c*d^2*e-2*c^2*d^3)/e^2/(4*a*c-b^2)*(e*x+d)^(1/2))/((e*x+d)^2*c+b*e*(e*x+d)-2*c*d*(e*x+d)+e^2*a
-b*d*e+c*d^2)+2/e^2/(4*a*c-b^2)*(-1/8*(-8*a*b*c*e^3+16*d*e^2*c^2*a+b^3*e^3+2*b^2*d*e^2*c-12*b*c^2*d^2*e+8*c^3*
d^3+6*(-e^2*(4*a*c-b^2))^(1/2)*a*c*e^2-(-e^2*(4*a*c-b^2))^(1/2)*b^2*e^2-2*(-e^2*(4*a*c-b^2))^(1/2)*b*c*d*e+2*(
-e^2*(4*a*c-b^2))^(1/2)*c^2*d^2)/c/(-e^2*(4*a*c-b^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))+1/8*(8*a*b*c*e^3-16*d*e
^2*c^2*a-b^3*e^3-2*b^2*d*e^2*c+12*b*c^2*d^2*e-8*c^3*d^3+6*(-e^2*(4*a*c-b^2))^(1/2)*a*c*e^2-(-e^2*(4*a*c-b^2))^
(1/2)*b^2*e^2-2*(-e^2*(4*a*c-b^2))^(1/2)*b*c*d*e+2*(-e^2*(4*a*c-b^2))^(1/2)*c^2*d^2)/c/(-e^2*(4*a*c-b^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))))

________________________________________________________________________________________

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)^(5/2)/(c*x^2+b*x+a)^2,x, algorithm="maxima")

[Out]

integrate((x*e + d)^(5/2)/(c*x^2 + b*x + a)^2, x)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 5133 vs. \(2 (461) = 922\).
time = 2.89, size = 5133, normalized size = 10.18 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(5/2)/(c*x^2+b*x+a)^2,x, algorithm="fricas")

[Out]

1/2*(sqrt(1/2)*(a*b^2*c - 4*a^2*c^2 + (b^2*c^2 - 4*a*c^3)*x^2 + (b^3*c - 4*a*b*c^2)*x)*sqrt((32*c^5*d^5 - 80*b
*c^4*d^4*e + 10*(5*b^2*c^3 + 12*a*c^4)*d^3*e^2 + 5*(b^3*c^2 - 36*a*b*c^3)*d^2*e^3 - 5*(b^4*c - 6*a*b^2*c^2 - 2
4*a^2*c^3)*d*e^4 - (b^5 - 15*a*b^3*c + 60*a^2*b*c^2)*e^5 + (b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c
^6)*sqrt((25*c^4*d^4*e^6 - 50*b*c^3*d^3*e^7 + 15*(b^2*c^2 + 6*a*c^3)*d^2*e^8 + 10*(b^3*c - 9*a*b*c^2)*d*e^9 +
(b^4 - 18*a*b^2*c + 81*a^2*c^2)*e^10)/(b^6*c^6 - 12*a*b^4*c^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)))/(b^6*c^3 - 12*a
*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6))*log(sqrt(1/2)*(10*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d^3*e^4 - 15*(
b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d^2*e^5 + 3*(b^6*c - 2*a*b^4*c^2 - 32*a^2*b^2*c^3 + 96*a^3*c^4)*d*e^6 +
(b^7 - 17*a*b^5*c + 88*a^2*b^3*c^2 - 144*a^3*b*c^3)*e^7 - (8*(b^6*c^5 - 12*a*b^4*c^6 + 48*a^2*b^2*c^7 - 64*a^3
*c^8)*d^2 - 8*(b^7*c^4 - 12*a*b^5*c^5 + 48*a^2*b^3*c^6 - 64*a^3*b*c^7)*d*e - (b^8*c^3 - 24*a*b^6*c^4 + 192*a^2
*b^4*c^5 - 640*a^3*b^2*c^6 + 768*a^4*c^7)*e^2)*sqrt((25*c^4*d^4*e^6 - 50*b*c^3*d^3*e^7 + 15*(b^2*c^2 + 6*a*c^3
)*d^2*e^8 + 10*(b^3*c - 9*a*b*c^2)*d*e^9 + (b^4 - 18*a*b^2*c + 81*a^2*c^2)*e^10)/(b^6*c^6 - 12*a*b^4*c^7 + 48*
a^2*b^2*c^8 - 64*a^3*c^9)))*sqrt((32*c^5*d^5 - 80*b*c^4*d^4*e + 10*(5*b^2*c^3 + 12*a*c^4)*d^3*e^2 + 5*(b^3*c^2
- 36*a*b*c^3)*d^2*e^3 - 5*(b^4*c - 6*a*b^2*c^2 - 24*a^2*c^3)*d*e^4 - (b^5 - 15*a*b^3*c + 60*a^2*b*c^2)*e^5 +
(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*sqrt((25*c^4*d^4*e^6 - 50*b*c^3*d^3*e^7 + 15*(b^2*c^2 +
6*a*c^3)*d^2*e^8 + 10*(b^3*c - 9*a*b*c^2)*d*e^9 + (b^4 - 18*a*b^2*c + 81*a^2*c^2)*e^10)/(b^6*c^6 - 12*a*b^4*c
^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)))/(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)) + 2*(80*c^5*d^6*e^
3 - 240*b*c^4*d^5*e^4 + (199*b^2*c^3 + 404*a*c^4)*d^4*e^5 + 2*(b^3*c^2 - 404*a*b*c^3)*d^3*e^6 - 6*(6*b^4*c - 4
7*a*b^2*c^2 - 108*a^2*c^3)*d^2*e^7 - (5*b^5 - 122*a*b^3*c + 648*a^2*b*c^2)*d*e^8 + (5*a*b^4 - 81*a^2*b^2*c + 3
24*a^3*c^2)*e^9)*sqrt(x*e + d)) - sqrt(1/2)*(a*b^2*c - 4*a^2*c^2 + (b^2*c^2 - 4*a*c^3)*x^2 + (b^3*c - 4*a*b*c^
2)*x)*sqrt((32*c^5*d^5 - 80*b*c^4*d^4*e + 10*(5*b^2*c^3 + 12*a*c^4)*d^3*e^2 + 5*(b^3*c^2 - 36*a*b*c^3)*d^2*e^3
- 5*(b^4*c - 6*a*b^2*c^2 - 24*a^2*c^3)*d*e^4 - (b^5 - 15*a*b^3*c + 60*a^2*b*c^2)*e^5 + (b^6*c^3 - 12*a*b^4*c^
4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*sqrt((25*c^4*d^4*e^6 - 50*b*c^3*d^3*e^7 + 15*(b^2*c^2 + 6*a*c^3)*d^2*e^8 + 10
*(b^3*c - 9*a*b*c^2)*d*e^9 + (b^4 - 18*a*b^2*c + 81*a^2*c^2)*e^10)/(b^6*c^6 - 12*a*b^4*c^7 + 48*a^2*b^2*c^8 -
64*a^3*c^9)))/(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6))*log(-sqrt(1/2)*(10*(b^4*c^3 - 8*a*b^2*c^
4 + 16*a^2*c^5)*d^3*e^4 - 15*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d^2*e^5 + 3*(b^6*c - 2*a*b^4*c^2 - 32*a^2*
b^2*c^3 + 96*a^3*c^4)*d*e^6 + (b^7 - 17*a*b^5*c + 88*a^2*b^3*c^2 - 144*a^3*b*c^3)*e^7 - (8*(b^6*c^5 - 12*a*b^4
*c^6 + 48*a^2*b^2*c^7 - 64*a^3*c^8)*d^2 - 8*(b^7*c^4 - 12*a*b^5*c^5 + 48*a^2*b^3*c^6 - 64*a^3*b*c^7)*d*e - (b^
8*c^3 - 24*a*b^6*c^4 + 192*a^2*b^4*c^5 - 640*a^3*b^2*c^6 + 768*a^4*c^7)*e^2)*sqrt((25*c^4*d^4*e^6 - 50*b*c^3*d
^3*e^7 + 15*(b^2*c^2 + 6*a*c^3)*d^2*e^8 + 10*(b^3*c - 9*a*b*c^2)*d*e^9 + (b^4 - 18*a*b^2*c + 81*a^2*c^2)*e^10)
/(b^6*c^6 - 12*a*b^4*c^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)))*sqrt((32*c^5*d^5 - 80*b*c^4*d^4*e + 10*(5*b^2*c^3 +
12*a*c^4)*d^3*e^2 + 5*(b^3*c^2 - 36*a*b*c^3)*d^2*e^3 - 5*(b^4*c - 6*a*b^2*c^2 - 24*a^2*c^3)*d*e^4 - (b^5 - 15*
a*b^3*c + 60*a^2*b*c^2)*e^5 + (b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*sqrt((25*c^4*d^4*e^6 - 50
*b*c^3*d^3*e^7 + 15*(b^2*c^2 + 6*a*c^3)*d^2*e^8 + 10*(b^3*c - 9*a*b*c^2)*d*e^9 + (b^4 - 18*a*b^2*c + 81*a^2*c^
2)*e^10)/(b^6*c^6 - 12*a*b^4*c^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)))/(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 6
4*a^3*c^6)) + 2*(80*c^5*d^6*e^3 - 240*b*c^4*d^5*e^4 + (199*b^2*c^3 + 404*a*c^4)*d^4*e^5 + 2*(b^3*c^2 - 404*a*b
*c^3)*d^3*e^6 - 6*(6*b^4*c - 47*a*b^2*c^2 - 108*a^2*c^3)*d^2*e^7 - (5*b^5 - 122*a*b^3*c + 648*a^2*b*c^2)*d*e^8
+ (5*a*b^4 - 81*a^2*b^2*c + 324*a^3*c^2)*e^9)*sqrt(x*e + d)) + sqrt(1/2)*(a*b^2*c - 4*a^2*c^2 + (b^2*c^2 - 4*
a*c^3)*x^2 + (b^3*c - 4*a*b*c^2)*x)*sqrt((32*c^5*d^5 - 80*b*c^4*d^4*e + 10*(5*b^2*c^3 + 12*a*c^4)*d^3*e^2 + 5*
(b^3*c^2 - 36*a*b*c^3)*d^2*e^3 - 5*(b^4*c - 6*a*b^2*c^2 - 24*a^2*c^3)*d*e^4 - (b^5 - 15*a*b^3*c + 60*a^2*b*c^2
)*e^5 - (b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*sqrt((25*c^4*d^4*e^6 - 50*b*c^3*d^3*e^7 + 15*(b
^2*c^2 + 6*a*c^3)*d^2*e^8 + 10*(b^3*c - 9*a*b*c^2)*d*e^9 + (b^4 - 18*a*b^2*c + 81*a^2*c^2)*e^10)/(b^6*c^6 - 12
*a*b^4*c^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)))/(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6))*log(sqrt(1
/2)*(10*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d^3*e^4 - 15*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d^2*e^5 + 3*(
b^6*c - 2*a*b^4*c^2 - 32*a^2*b^2*c^3 + 96*a^3*c^4)*d*e^6 + (b^7 - 17*a*b^5*c + 88*a^2*b^3*c^2 - 144*a^3*b*c^3)
*e^7 + (8*(b^6*c^5 - 12*a*b^4*c^6 + 48*a^2*b^2*c^7 - 64*a^3*c^8)*d^2 - 8*(b^7*c^4 - 12*a*b^5*c^5 + 48*a^2*b^3*
c^6 - 64*a^3*b*c^7)*d*e - (b^8*c^3 - 24*a*b^6*c...

________________________________________________________________________________________

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)**(5/2)/(c*x**2+b*x+a)**2,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1416 vs. \(2 (461) = 922\).
time = 2.00, size = 1416, normalized size = 2.81 \begin {gather*} -\frac {2 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{2} d^{2} e - 2 \, \sqrt {x e + d} c^{2} d^{3} e - 2 \, {\left (x e + d\right )}^{\frac {3}{2}} b c d e^{2} + 3 \, \sqrt {x e + d} b c d^{2} e^{2} + {\left (x e + d\right )}^{\frac {3}{2}} b^{2} e^{3} - 2 \, {\left (x e + d\right )}^{\frac {3}{2}} a c e^{3} - \sqrt {x e + d} b^{2} d e^{3} - 2 \, \sqrt {x e + d} a c d e^{3} + \sqrt {x e + d} a b e^{4}}{{\left ({\left (x e + d\right )}^{2} c - 2 \, {\left (x e + d\right )} c d + c d^{2} + {\left (x e + d\right )} b e - b d e + a e^{2}\right )} {\left (b^{2} c - 4 \, a c^{2}\right )}} - \frac {{\left (16 \, {\left (b^{2} c^{6} - 4 \, a c^{7}\right )} d^{4} e - 32 \, {\left (b^{3} c^{5} - 4 \, a b c^{6}\right )} d^{3} e^{2} + 16 \, {\left (b^{4} c^{4} - 2 \, a b^{2} c^{5} - 8 \, a^{2} c^{6}\right )} d^{2} e^{3} - {\left (2 \, c^{2} d^{2} e - 2 \, b c d e^{2} - {\left (b^{2} - 6 \, a c\right )} e^{3}\right )} {\left (b^{2} c e - 4 \, a c^{2} e\right )}^{2} - 32 \, {\left (a b^{3} c^{4} - 4 \, a^{2} b c^{5}\right )} d e^{4} - 2 \, {\left (2 \, \sqrt {b^{2} - 4 \, a c} c^{4} d^{3} e - 3 \, \sqrt {b^{2} - 4 \, a c} b c^{3} d^{2} e^{2} - \sqrt {b^{2} - 4 \, a c} a b c^{2} e^{4} + {\left (b^{2} c^{2} + 2 \, a c^{3}\right )} \sqrt {b^{2} - 4 \, a c} d e^{3}\right )} {\left | b^{2} c e - 4 \, a c^{2} e \right |} - {\left (b^{6} c^{2} - 12 \, a b^{4} c^{3} + 32 \, a^{2} b^{2} c^{4}\right )} e^{5}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x e + d}}{\sqrt {-\frac {2 \, b^{2} c^{2} d - 8 \, a c^{3} d - b^{3} c e + 4 \, a b c^{2} e + \sqrt {{\left (2 \, b^{2} c^{2} d - 8 \, a c^{3} d - b^{3} c e + 4 \, a b c^{2} e\right )}^{2} - 4 \, {\left (b^{2} c^{2} d^{2} - 4 \, a c^{3} d^{2} - b^{3} c d e + 4 \, a b c^{2} d e + a b^{2} c e^{2} - 4 \, a^{2} c^{2} e^{2}\right )} {\left (b^{2} c^{2} - 4 \, a c^{3}\right )}}}{b^{2} c^{2} - 4 \, a c^{3}}}}\right )}{\sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left (2 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} \sqrt {b^{2} - 4 \, a c} d - {\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3} + {\left (b^{3} c - 4 \, a b c^{2}\right )} \sqrt {b^{2} - 4 \, a c}\right )} e\right )} {\left | b^{2} c e - 4 \, a c^{2} e \right |} {\left | c \right |}} + \frac {{\left (16 \, {\left (b^{2} c^{6} - 4 \, a c^{7}\right )} d^{4} e - 32 \, {\left (b^{3} c^{5} - 4 \, a b c^{6}\right )} d^{3} e^{2} + 16 \, {\left (b^{4} c^{4} - 2 \, a b^{2} c^{5} - 8 \, a^{2} c^{6}\right )} d^{2} e^{3} - {\left (2 \, c^{2} d^{2} e - 2 \, b c d e^{2} - {\left (b^{2} - 6 \, a c\right )} e^{3}\right )} {\left (b^{2} c e - 4 \, a c^{2} e\right )}^{2} - 32 \, {\left (a b^{3} c^{4} - 4 \, a^{2} b c^{5}\right )} d e^{4} + 2 \, {\left (2 \, \sqrt {b^{2} - 4 \, a c} c^{4} d^{3} e - 3 \, \sqrt {b^{2} - 4 \, a c} b c^{3} d^{2} e^{2} - \sqrt {b^{2} - 4 \, a c} a b c^{2} e^{4} + {\left (b^{2} c^{2} + 2 \, a c^{3}\right )} \sqrt {b^{2} - 4 \, a c} d e^{3}\right )} {\left | b^{2} c e - 4 \, a c^{2} e \right |} - {\left (b^{6} c^{2} - 12 \, a b^{4} c^{3} + 32 \, a^{2} b^{2} c^{4}\right )} e^{5}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x e + d}}{\sqrt {-\frac {2 \, b^{2} c^{2} d - 8 \, a c^{3} d - b^{3} c e + 4 \, a b c^{2} e - \sqrt {{\left (2 \, b^{2} c^{2} d - 8 \, a c^{3} d - b^{3} c e + 4 \, a b c^{2} e\right )}^{2} - 4 \, {\left (b^{2} c^{2} d^{2} - 4 \, a c^{3} d^{2} - b^{3} c d e + 4 \, a b c^{2} d e + a b^{2} c e^{2} - 4 \, a^{2} c^{2} e^{2}\right )} {\left (b^{2} c^{2} - 4 \, a c^{3}\right )}}}{b^{2} c^{2} - 4 \, a c^{3}}}}\right )}{\sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left (2 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} \sqrt {b^{2} - 4 \, a c} d + {\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3} - {\left (b^{3} c - 4 \, a b c^{2}\right )} \sqrt {b^{2} - 4 \, a c}\right )} e\right )} {\left | b^{2} c e - 4 \, a c^{2} e \right |} {\left | c \right |}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(5/2)/(c*x^2+b*x+a)^2,x, algorithm="giac")

[Out]

-(2*(x*e + d)^(3/2)*c^2*d^2*e - 2*sqrt(x*e + d)*c^2*d^3*e - 2*(x*e + d)^(3/2)*b*c*d*e^2 + 3*sqrt(x*e + d)*b*c*
d^2*e^2 + (x*e + d)^(3/2)*b^2*e^3 - 2*(x*e + d)^(3/2)*a*c*e^3 - sqrt(x*e + d)*b^2*d*e^3 - 2*sqrt(x*e + d)*a*c*
d*e^3 + sqrt(x*e + d)*a*b*e^4)/(((x*e + d)^2*c - 2*(x*e + d)*c*d + c*d^2 + (x*e + d)*b*e - b*d*e + a*e^2)*(b^2
*c - 4*a*c^2)) - (16*(b^2*c^6 - 4*a*c^7)*d^4*e - 32*(b^3*c^5 - 4*a*b*c^6)*d^3*e^2 + 16*(b^4*c^4 - 2*a*b^2*c^5
- 8*a^2*c^6)*d^2*e^3 - (2*c^2*d^2*e - 2*b*c*d*e^2 - (b^2 - 6*a*c)*e^3)*(b^2*c*e - 4*a*c^2*e)^2 - 32*(a*b^3*c^4
- 4*a^2*b*c^5)*d*e^4 - 2*(2*sqrt(b^2 - 4*a*c)*c^4*d^3*e - 3*sqrt(b^2 - 4*a*c)*b*c^3*d^2*e^2 - sqrt(b^2 - 4*a*
c)*a*b*c^2*e^4 + (b^2*c^2 + 2*a*c^3)*sqrt(b^2 - 4*a*c)*d*e^3)*abs(b^2*c*e - 4*a*c^2*e) - (b^6*c^2 - 12*a*b^4*c
^3 + 32*a^2*b^2*c^4)*e^5)*arctan(2*sqrt(1/2)*sqrt(x*e + d)/sqrt(-(2*b^2*c^2*d - 8*a*c^3*d - b^3*c*e + 4*a*b*c^
2*e + sqrt((2*b^2*c^2*d - 8*a*c^3*d - b^3*c*e + 4*a*b*c^2*e)^2 - 4*(b^2*c^2*d^2 - 4*a*c^3*d^2 - b^3*c*d*e + 4*
a*b*c^2*d*e + a*b^2*c*e^2 - 4*a^2*c^2*e^2)*(b^2*c^2 - 4*a*c^3)))/(b^2*c^2 - 4*a*c^3)))/(sqrt(-4*c^2*d + 2*(b*c
- sqrt(b^2 - 4*a*c)*c)*e)*(2*(b^2*c^2 - 4*a*c^3)*sqrt(b^2 - 4*a*c)*d - (b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3 + (b
^3*c - 4*a*b*c^2)*sqrt(b^2 - 4*a*c))*e)*abs(b^2*c*e - 4*a*c^2*e)*abs(c)) + (16*(b^2*c^6 - 4*a*c^7)*d^4*e - 32*
(b^3*c^5 - 4*a*b*c^6)*d^3*e^2 + 16*(b^4*c^4 - 2*a*b^2*c^5 - 8*a^2*c^6)*d^2*e^3 - (2*c^2*d^2*e - 2*b*c*d*e^2 -
(b^2 - 6*a*c)*e^3)*(b^2*c*e - 4*a*c^2*e)^2 - 32*(a*b^3*c^4 - 4*a^2*b*c^5)*d*e^4 + 2*(2*sqrt(b^2 - 4*a*c)*c^4*d
^3*e - 3*sqrt(b^2 - 4*a*c)*b*c^3*d^2*e^2 - sqrt(b^2 - 4*a*c)*a*b*c^2*e^4 + (b^2*c^2 + 2*a*c^3)*sqrt(b^2 - 4*a*
c)*d*e^3)*abs(b^2*c*e - 4*a*c^2*e) - (b^6*c^2 - 12*a*b^4*c^3 + 32*a^2*b^2*c^4)*e^5)*arctan(2*sqrt(1/2)*sqrt(x*
e + d)/sqrt(-(2*b^2*c^2*d - 8*a*c^3*d - b^3*c*e + 4*a*b*c^2*e - sqrt((2*b^2*c^2*d - 8*a*c^3*d - b^3*c*e + 4*a*
b*c^2*e)^2 - 4*(b^2*c^2*d^2 - 4*a*c^3*d^2 - b^3*c*d*e + 4*a*b*c^2*d*e + a*b^2*c*e^2 - 4*a^2*c^2*e^2)*(b^2*c^2
- 4*a*c^3)))/(b^2*c^2 - 4*a*c^3)))/(sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*(2*(b^2*c^2 - 4*a*c^3)*sq
rt(b^2 - 4*a*c)*d + (b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3 - (b^3*c - 4*a*b*c^2)*sqrt(b^2 - 4*a*c))*e)*abs(b^2*c*e
- 4*a*c^2*e)*abs(c))

________________________________________________________________________________________

Mupad [B]
time = 4.68, size = 2500, normalized size = 4.96 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^(5/2)/(a + b*x + c*x^2)^2,x)

[Out]

atan(((((256*a^4*b*c^5*e^6 - 4*a*b^7*c^2*e^6 - 512*a^4*c^6*d*e^5 + 4*b^8*c^2*d*e^5 + 48*a^2*b^5*c^3*e^6 - 192*
a^3*b^3*c^4*e^6 - 512*a^3*c^7*d^3*e^3 + 8*b^6*c^4*d^3*e^3 - 12*b^7*c^3*d^2*e^4 + 384*a^2*b^2*c^6*d^3*e^3 - 576
*a^2*b^3*c^5*d^2*e^4 - 40*a*b^6*c^3*d*e^5 - 96*a*b^4*c^5*d^3*e^3 + 144*a*b^5*c^4*d^2*e^4 + 96*a^2*b^4*c^4*d*e^
5 + 768*a^3*b*c^6*d^2*e^4 + 128*a^3*b^2*c^5*d*e^5)/(b^6*c - 64*a^3*c^4 - 12*a*b^4*c^2 + 48*a^2*b^2*c^3) - (2*(
d + e*x)^(1/2)*((32*b^6*c^5*d^5 - 2048*a^3*c^8*d^5 - b^11*e^5 + b^2*e^5*(-(4*a*c - b^2)^9)^(1/2) - 384*a*b^4*c
^6*d^5 + 3840*a^5*b*c^5*e^5 - 7680*a^5*c^6*d*e^4 - 80*b^7*c^4*d^4*e + 1536*a^2*b^2*c^7*d^5 - 288*a^2*b^7*c^2*e
^5 + 1504*a^3*b^5*c^3*e^5 - 3840*a^4*b^3*c^4*e^5 - 7680*a^4*c^7*d^3*e^2 + 50*b^8*c^3*d^3*e^2 + 5*b^9*c^2*d^2*e
^3 - 5*c^2*d^2*e^3*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*e^5 - 9*a*c*e^5*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d
*e^4 + 5*b*c*d*e^4*(-(4*a*c - b^2)^9)^(1/2) + 960*a^2*b^4*c^5*d^3*e^2 + 2400*a^2*b^5*c^4*d^2*e^3 + 2560*a^3*b^
2*c^6*d^3*e^2 - 8960*a^3*b^3*c^5*d^2*e^3 + 960*a*b^5*c^5*d^4*e + 90*a*b^8*c^2*d*e^4 + 5120*a^3*b*c^7*d^4*e - 4
80*a*b^6*c^4*d^3*e^2 - 240*a*b^7*c^3*d^2*e^3 - 3840*a^2*b^3*c^6*d^4*e - 480*a^2*b^6*c^3*d*e^4 + 320*a^3*b^4*c^
4*d*e^4 + 11520*a^4*b*c^6*d^2*e^3 + 3840*a^4*b^2*c^5*d*e^4)/(8*(4096*a^6*c^9 + b^12*c^3 - 24*a*b^10*c^4 + 240*
a^2*b^8*c^5 - 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))^(1/2)*(4*b^7*c^3*e^3 - 48*a*b^5*c^4*e^
3 - 256*a^3*b*c^6*e^3 + 512*a^3*c^7*d*e^2 - 8*b^6*c^4*d*e^2 + 192*a^2*b^3*c^5*e^3 + 96*a*b^4*c^5*d*e^2 - 384*a
^2*b^2*c^6*d*e^2))/(b^4*c + 16*a^2*c^3 - 8*a*b^2*c^2))*((32*b^6*c^5*d^5 - 2048*a^3*c^8*d^5 - b^11*e^5 + b^2*e^
5*(-(4*a*c - b^2)^9)^(1/2) - 384*a*b^4*c^6*d^5 + 3840*a^5*b*c^5*e^5 - 7680*a^5*c^6*d*e^4 - 80*b^7*c^4*d^4*e +
1536*a^2*b^2*c^7*d^5 - 288*a^2*b^7*c^2*e^5 + 1504*a^3*b^5*c^3*e^5 - 3840*a^4*b^3*c^4*e^5 - 7680*a^4*c^7*d^3*e^
2 + 50*b^8*c^3*d^3*e^2 + 5*b^9*c^2*d^2*e^3 - 5*c^2*d^2*e^3*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*e^5 - 9*a*c*e
^5*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e^4 + 5*b*c*d*e^4*(-(4*a*c - b^2)^9)^(1/2) + 960*a^2*b^4*c^5*d^3*e^2
+ 2400*a^2*b^5*c^4*d^2*e^3 + 2560*a^3*b^2*c^6*d^3*e^2 - 8960*a^3*b^3*c^5*d^2*e^3 + 960*a*b^5*c^5*d^4*e + 90*a*
b^8*c^2*d*e^4 + 5120*a^3*b*c^7*d^4*e - 480*a*b^6*c^4*d^3*e^2 - 240*a*b^7*c^3*d^2*e^3 - 3840*a^2*b^3*c^6*d^4*e
- 480*a^2*b^6*c^3*d*e^4 + 320*a^3*b^4*c^4*d*e^4 + 11520*a^4*b*c^6*d^2*e^3 + 3840*a^4*b^2*c^5*d*e^4)/(8*(4096*a
^6*c^9 + b^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5 - 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8))
)^(1/2) - (2*(d + e*x)^(1/2)*(b^6*e^8 - 72*a^3*c^3*e^8 + 32*c^6*d^6*e^2 + 120*a*c^5*d^4*e^4 - 96*b*c^5*d^5*e^3
+ 74*a^2*b^2*c^2*e^8 + 80*a^2*c^4*d^2*e^6 + 90*b^2*c^4*d^4*e^4 - 20*b^3*c^3*d^3*e^5 - 10*b^4*c^2*d^2*e^6 - 16
*a*b^4*c*e^8 + 4*b^5*c*d*e^7 - 240*a*b*c^4*d^3*e^5 - 20*a*b^3*c^2*d*e^7 - 80*a^2*b*c^3*d*e^7 + 140*a*b^2*c^3*d
^2*e^6))/(b^4*c + 16*a^2*c^3 - 8*a*b^2*c^2))*((32*b^6*c^5*d^5 - 2048*a^3*c^8*d^5 - b^11*e^5 + b^2*e^5*(-(4*a*c
- b^2)^9)^(1/2) - 384*a*b^4*c^6*d^5 + 3840*a^5*b*c^5*e^5 - 7680*a^5*c^6*d*e^4 - 80*b^7*c^4*d^4*e + 1536*a^2*b
^2*c^7*d^5 - 288*a^2*b^7*c^2*e^5 + 1504*a^3*b^5*c^3*e^5 - 3840*a^4*b^3*c^4*e^5 - 7680*a^4*c^7*d^3*e^2 + 50*b^8
*c^3*d^3*e^2 + 5*b^9*c^2*d^2*e^3 - 5*c^2*d^2*e^3*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*e^5 - 9*a*c*e^5*(-(4*a*
c - b^2)^9)^(1/2) - 5*b^10*c*d*e^4 + 5*b*c*d*e^4*(-(4*a*c - b^2)^9)^(1/2) + 960*a^2*b^4*c^5*d^3*e^2 + 2400*a^2
*b^5*c^4*d^2*e^3 + 2560*a^3*b^2*c^6*d^3*e^2 - 8960*a^3*b^3*c^5*d^2*e^3 + 960*a*b^5*c^5*d^4*e + 90*a*b^8*c^2*d*
e^4 + 5120*a^3*b*c^7*d^4*e - 480*a*b^6*c^4*d^3*e^2 - 240*a*b^7*c^3*d^2*e^3 - 3840*a^2*b^3*c^6*d^4*e - 480*a^2*
b^6*c^3*d*e^4 + 320*a^3*b^4*c^4*d*e^4 + 11520*a^4*b*c^6*d^2*e^3 + 3840*a^4*b^2*c^5*d*e^4)/(8*(4096*a^6*c^9 + b
^12*c^3 - 24*a*b^10*c^4 + 240*a^2*b^8*c^5 - 1280*a^3*b^6*c^6 + 3840*a^4*b^4*c^7 - 6144*a^5*b^2*c^8)))^(1/2)*1i
- (((256*a^4*b*c^5*e^6 - 4*a*b^7*c^2*e^6 - 512*a^4*c^6*d*e^5 + 4*b^8*c^2*d*e^5 + 48*a^2*b^5*c^3*e^6 - 192*a^3
*b^3*c^4*e^6 - 512*a^3*c^7*d^3*e^3 + 8*b^6*c^4*d^3*e^3 - 12*b^7*c^3*d^2*e^4 + 384*a^2*b^2*c^6*d^3*e^3 - 576*a^
2*b^3*c^5*d^2*e^4 - 40*a*b^6*c^3*d*e^5 - 96*a*b^4*c^5*d^3*e^3 + 144*a*b^5*c^4*d^2*e^4 + 96*a^2*b^4*c^4*d*e^5 +
768*a^3*b*c^6*d^2*e^4 + 128*a^3*b^2*c^5*d*e^5)/(b^6*c - 64*a^3*c^4 - 12*a*b^4*c^2 + 48*a^2*b^2*c^3) + (2*(d +
e*x)^(1/2)*((32*b^6*c^5*d^5 - 2048*a^3*c^8*d^5 - b^11*e^5 + b^2*e^5*(-(4*a*c - b^2)^9)^(1/2) - 384*a*b^4*c^6*
d^5 + 3840*a^5*b*c^5*e^5 - 7680*a^5*c^6*d*e^4 - 80*b^7*c^4*d^4*e + 1536*a^2*b^2*c^7*d^5 - 288*a^2*b^7*c^2*e^5
+ 1504*a^3*b^5*c^3*e^5 - 3840*a^4*b^3*c^4*e^5 - 7680*a^4*c^7*d^3*e^2 + 50*b^8*c^3*d^3*e^2 + 5*b^9*c^2*d^2*e^3
- 5*c^2*d^2*e^3*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*e^5 - 9*a*c*e^5*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e^
4 + 5*b*c*d*e^4*(-(4*a*c - b^2)^9)^(1/2) + 960*a^2*b^4*c^5*d^3*e^2 + 2400*a^2*b^5*c^4*d^2*e^3 + 2560*a^3*b^2*c
^6*d^3*e^2 - 8960*a^3*b^3*c^5*d^2*e^3 + 960*a*b^5*c^5*d^4*e + 90*a*b^8*c^2*d*e^4 + 5120*a^3*b*c^7*d^4*e - 480*
a*b^6*c^4*d^3*e^2 - 240*a*b^7*c^3*d^2*e^3 - 384...

________________________________________________________________________________________