3.24.88 \(\int \frac {1}{(d+e x)^2 (a+b x+c x^2)^{3/2}} \, dx\) [2388]

Optimal. Leaf size=255 \[ -\frac {2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \sqrt {a+b x+c x^2}}-\frac {e \left (4 c^2 d^2+3 b^2 e^2-4 c e (b d+2 a e)\right ) \sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)}+\frac {3 e^2 (2 c d-b e) \tanh ^{-1}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{2 \left (c d^2-b d e+a e^2\right )^{5/2}} \]

[Out]

3/2*e^2*(-b*e+2*c*d)*arctanh(1/2*(b*d-2*a*e+(-b*e+2*c*d)*x)/(a*e^2-b*d*e+c*d^2)^(1/2)/(c*x^2+b*x+a)^(1/2))/(a*
e^2-b*d*e+c*d^2)^(5/2)-2*(b*c*d-b^2*e+2*a*c*e+c*(-b*e+2*c*d)*x)/(-4*a*c+b^2)/(a*e^2-b*d*e+c*d^2)/(e*x+d)/(c*x^
2+b*x+a)^(1/2)-e*(4*c^2*d^2+3*b^2*e^2-4*c*e*(2*a*e+b*d))*(c*x^2+b*x+a)^(1/2)/(-4*a*c+b^2)/(a*e^2-b*d*e+c*d^2)^
2/(e*x+d)

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Rubi [A]
time = 0.17, antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {754, 820, 738, 212} \begin {gather*} -\frac {e \sqrt {a+b x+c x^2} \left (-4 c e (2 a e+b d)+3 b^2 e^2+4 c^2 d^2\right )}{\left (b^2-4 a c\right ) (d+e x) \left (a e^2-b d e+c d^2\right )^2}-\frac {2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) (d+e x) \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}+\frac {3 e^2 (2 c d-b e) \tanh ^{-1}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{2 \left (a e^2-b d e+c d^2\right )^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(d + e*x)*Sqrt[a + b
*x + c*x^2]) - (e*(4*c^2*d^2 + 3*b^2*e^2 - 4*c*e*(b*d + 2*a*e))*Sqrt[a + b*x + c*x^2])/((b^2 - 4*a*c)*(c*d^2 -
b*d*e + a*e^2)^2*(d + e*x)) + (3*e^2*(2*c*d - b*e)*ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*
d*e + a*e^2]*Sqrt[a + b*x + c*x^2])])/(2*(c*d^2 - b*d*e + a*e^2)^(5/2))

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 738

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

Rule 754

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

Rule 820

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

Rubi steps

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

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1477\) vs. \(2(255)=510\).
time = 12.72, size = 1477, normalized size = 5.79 \begin {gather*} \frac {4 c^3 d e x^3 \left (-d^2+3 d e x+6 e^2 x^2\right ) \left (\sqrt {c} x-\sqrt {a+x (b+c x)}\right )-4 a^2 \sqrt {c} e \left (2 c d \left (d^2-2 d e x-4 e^2 x^2\right )+b e \left (-4 d^2-4 d e x+e^2 x^2\right )+\sqrt {c} e \left (4 d^2+4 d e x-e^2 x^2\right ) \sqrt {a+x (b+c x)}\right )+b^3 e \left (-\sqrt {c} x \left (8 d^3+8 d^2 e x+5 d e^2 x^2+9 e^3 x^3\right )+\left (2 d^3+2 d^2 e x+2 d e^2 x^2+3 e^3 x^3\right ) \sqrt {a+x (b+c x)}\right )+2 b c^2 x \left (2 \sqrt {c} x \left (d^4-3 d^3 e x+3 d^2 e^2 x^2+9 d e^3 x^3-3 e^4 x^4\right )+\left (-2 d^4+5 d^3 e x-3 d^2 e^2 x^2-12 d e^3 x^3+6 e^4 x^4\right ) \sqrt {a+x (b+c x)}\right )+b^2 c \left (\sqrt {c} x \left (4 d^4-16 d^3 e x-8 d^2 e^2 x^2+7 d e^3 x^3-21 e^4 x^4\right )+\left (-2 d^4+12 d^3 e x+10 d^2 e^2 x^2+2 d e^3 x^3+15 e^4 x^4\right ) \sqrt {a+x (b+c x)}\right )+a \left (4 c^2 d e x \left (\sqrt {c} x \left (-3 d^2+7 d e x+14 e^2 x^2\right )+\left (2 d^2-6 d e x-11 e^2 x^2\right ) \sqrt {a+x (b+c x)}\right )+b^2 e \left (\sqrt {c} \left (-8 d^3+8 d^2 e x+11 d e^2 x^2-13 e^3 x^3\right )+e \left (-4 d^2-4 d e x+e^2 x^2\right ) \sqrt {a+x (b+c x)}\right )+2 b c \left (2 \sqrt {c} \left (d^4-4 d^3 e x+8 d^2 e^2 x^2+15 d e^3 x^3-4 e^4 x^4\right )+e \left (6 d^3-8 d^2 e x-15 d e^2 x^2+5 e^3 x^3\right ) \sqrt {a+x (b+c x)}\right )\right )}{(d+e x) \left (-a^3 e^3 \left (2 d^2+2 d e x+e^2 x^2\right )+d^2 x \left (-c^3 d^2 e x^3-b^3 e \left (d^2+d e x+e^2 x^2\right )+b c^2 d x \left (d^2+2 e^2 x^2\right )+b^2 c \left (d^3+d e^2 x^2-e^3 x^3\right )\right )-a^2 e \left (b e \left (-3 d^3-d^2 e x+e^3 x^3\right )+c \left (2 d^4+2 d^3 e x+4 d^2 e^2 x^2+2 d e^3 x^3+e^4 x^4\right )\right )+a d \left (-c^2 d e x^2 \left (3 d^2+2 d e x+2 e^2 x^2\right )+b^2 e \left (-d^3+2 d^2 e x+2 d e^2 x^2+2 e^3 x^3\right )+b c \left (d^4-d^3 e x+3 d^2 e^2 x^2+d e^3 x^3+2 e^4 x^4\right )\right )\right ) \left (b^2+b \left (8 c x-4 \sqrt {c} \sqrt {a+x (b+c x)}\right )+4 c \left (a+2 c x^2-2 \sqrt {c} x \sqrt {a+x (b+c x)}\right )\right )}-\frac {6 e^2 \tan ^{-1}\left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+x (b+c x)}}{\sqrt {-c d^2+e (b d-a e)}}\right )}{\sqrt {-c d^2+e (b d-a e)} \left (c d^3+d e (-b d+a e)\right )}+\frac {3 e^2 (-b d+2 a e) \left (d^2 \sqrt {-c d^2+b d e-a e^2} \left (a e+b e x+c e x^2+\sqrt {c} d \sqrt {a+x (b+c x)}+\sqrt {c} e x \sqrt {a+x (b+c x)}\right )+e \sqrt {a+x (b+c x)} \left (c d^2 e x^2-b d \left (d^2+d e x+e^2 x^2\right )+a e \left (2 d^2+2 d e x+e^2 x^2\right )\right ) \tan ^{-1}\left (\frac {-\sqrt {c} (d+e x)+e \sqrt {a+x (b+c x)}}{\sqrt {-c d^2+e (b d-a e)}}\right )\right )}{d \sqrt {-c d^2+e (b d-a e)} \left (c d^2+e (-b d+a e)\right )^2 \sqrt {a+x (b+c x)} \left (b d \left (d^2+d e x+e^2 x^2\right )-e \left (c d^2 x^2+a \left (2 d^2+2 d e x+e^2 x^2\right )\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*c^3*d*e*x^3*(-d^2 + 3*d*e*x + 6*e^2*x^2)*(Sqrt[c]*x - Sqrt[a + x*(b + c*x)]) - 4*a^2*Sqrt[c]*e*(2*c*d*(d^2
- 2*d*e*x - 4*e^2*x^2) + b*e*(-4*d^2 - 4*d*e*x + e^2*x^2) + Sqrt[c]*e*(4*d^2 + 4*d*e*x - e^2*x^2)*Sqrt[a + x*(
b + c*x)]) + b^3*e*(-(Sqrt[c]*x*(8*d^3 + 8*d^2*e*x + 5*d*e^2*x^2 + 9*e^3*x^3)) + (2*d^3 + 2*d^2*e*x + 2*d*e^2*
x^2 + 3*e^3*x^3)*Sqrt[a + x*(b + c*x)]) + 2*b*c^2*x*(2*Sqrt[c]*x*(d^4 - 3*d^3*e*x + 3*d^2*e^2*x^2 + 9*d*e^3*x^
3 - 3*e^4*x^4) + (-2*d^4 + 5*d^3*e*x - 3*d^2*e^2*x^2 - 12*d*e^3*x^3 + 6*e^4*x^4)*Sqrt[a + x*(b + c*x)]) + b^2*
c*(Sqrt[c]*x*(4*d^4 - 16*d^3*e*x - 8*d^2*e^2*x^2 + 7*d*e^3*x^3 - 21*e^4*x^4) + (-2*d^4 + 12*d^3*e*x + 10*d^2*e
^2*x^2 + 2*d*e^3*x^3 + 15*e^4*x^4)*Sqrt[a + x*(b + c*x)]) + a*(4*c^2*d*e*x*(Sqrt[c]*x*(-3*d^2 + 7*d*e*x + 14*e
^2*x^2) + (2*d^2 - 6*d*e*x - 11*e^2*x^2)*Sqrt[a + x*(b + c*x)]) + b^2*e*(Sqrt[c]*(-8*d^3 + 8*d^2*e*x + 11*d*e^
2*x^2 - 13*e^3*x^3) + e*(-4*d^2 - 4*d*e*x + e^2*x^2)*Sqrt[a + x*(b + c*x)]) + 2*b*c*(2*Sqrt[c]*(d^4 - 4*d^3*e*
x + 8*d^2*e^2*x^2 + 15*d*e^3*x^3 - 4*e^4*x^4) + e*(6*d^3 - 8*d^2*e*x - 15*d*e^2*x^2 + 5*e^3*x^3)*Sqrt[a + x*(b
+ c*x)])))/((d + e*x)*(-(a^3*e^3*(2*d^2 + 2*d*e*x + e^2*x^2)) + d^2*x*(-(c^3*d^2*e*x^3) - b^3*e*(d^2 + d*e*x
+ e^2*x^2) + b*c^2*d*x*(d^2 + 2*e^2*x^2) + b^2*c*(d^3 + d*e^2*x^2 - e^3*x^3)) - a^2*e*(b*e*(-3*d^3 - d^2*e*x +
e^3*x^3) + c*(2*d^4 + 2*d^3*e*x + 4*d^2*e^2*x^2 + 2*d*e^3*x^3 + e^4*x^4)) + a*d*(-(c^2*d*e*x^2*(3*d^2 + 2*d*e
*x + 2*e^2*x^2)) + b^2*e*(-d^3 + 2*d^2*e*x + 2*d*e^2*x^2 + 2*e^3*x^3) + b*c*(d^4 - d^3*e*x + 3*d^2*e^2*x^2 + d
*e^3*x^3 + 2*e^4*x^4)))*(b^2 + b*(8*c*x - 4*Sqrt[c]*Sqrt[a + x*(b + c*x)]) + 4*c*(a + 2*c*x^2 - 2*Sqrt[c]*x*Sq
rt[a + x*(b + c*x)]))) - (6*e^2*ArcTan[(Sqrt[c]*(d + e*x) - e*Sqrt[a + x*(b + c*x)])/Sqrt[-(c*d^2) + e*(b*d -
a*e)]])/(Sqrt[-(c*d^2) + e*(b*d - a*e)]*(c*d^3 + d*e*(-(b*d) + a*e))) + (3*e^2*(-(b*d) + 2*a*e)*(d^2*Sqrt[-(c*
d^2) + b*d*e - a*e^2]*(a*e + b*e*x + c*e*x^2 + Sqrt[c]*d*Sqrt[a + x*(b + c*x)] + Sqrt[c]*e*x*Sqrt[a + x*(b + c
*x)]) + e*Sqrt[a + x*(b + c*x)]*(c*d^2*e*x^2 - b*d*(d^2 + d*e*x + e^2*x^2) + a*e*(2*d^2 + 2*d*e*x + e^2*x^2))*
ArcTan[(-(Sqrt[c]*(d + e*x)) + e*Sqrt[a + x*(b + c*x)])/Sqrt[-(c*d^2) + e*(b*d - a*e)]]))/(d*Sqrt[-(c*d^2) + e
*(b*d - a*e)]*(c*d^2 + e*(-(b*d) + a*e))^2*Sqrt[a + x*(b + c*x)]*(b*d*(d^2 + d*e*x + e^2*x^2) - e*(c*d^2*x^2 +
a*(2*d^2 + 2*d*e*x + e^2*x^2))))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(654\) vs. \(2(241)=482\).
time = 0.82, size = 655, normalized size = 2.57

method	result	size
default	\(\frac {-\frac {e^{2}}{\left (e^{2} a -b d e +c \,d^{2}\right ) \left (x +\frac {d}{e}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}-\frac {3 e \left (b e -2 c d \right ) \left (\frac {e^{2}}{\left (e^{2} a -b d e +c \,d^{2}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}-\frac {e \left (b e -2 c d \right ) \left (2 c \left (x +\frac {d}{e}\right )+\frac {b e -2 c d}{e}\right )}{\left (e^{2} a -b d e +c \,d^{2}\right ) \left (\frac {4 c \left (e^{2} a -b d e +c \,d^{2}\right )}{e^{2}}-\frac {\left (b e -2 c d \right )^{2}}{e^{2}}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}-\frac {e^{2} \ln \left (\frac {\frac {2 e^{2} a -2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (e^{2} a -b d e +c \,d^{2}\right ) \sqrt {\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}\right )}{2 \left (e^{2} a -b d e +c \,d^{2}\right )}-\frac {4 c \,e^{2} \left (2 c \left (x +\frac {d}{e}\right )+\frac {b e -2 c d}{e}\right )}{\left (e^{2} a -b d e +c \,d^{2}\right ) \left (\frac {4 c \left (e^{2} a -b d e +c \,d^{2}\right )}{e^{2}}-\frac {\left (b e -2 c d \right )^{2}}{e^{2}}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}}{e^{2}}\)	\(655\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(c*d^2-%e*b*d+%e^2*a>0)', see `
assume?` for

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1450 vs. \(2 (250) = 500\).
time = 4.10, size = 2943, normalized size = 11.54 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(3*sqrt(c*d^2 - b*d*e + a*e^2)*(((b^3*c - 4*a*b*c^2)*x^3 + (b^4 - 4*a*b^2*c)*x^2 + (a*b^3 - 4*a^2*b*c)*x)
*e^4 - (2*(b^2*c^2 - 4*a*c^3)*d*x^3 + (b^3*c - 4*a*b*c^2)*d*x^2 - (b^4 - 6*a*b^2*c + 8*a^2*c^2)*d*x - (a*b^3 -
4*a^2*b*c)*d)*e^3 - 2*((b^2*c^2 - 4*a*c^3)*d^2*x^2 + (b^3*c - 4*a*b*c^2)*d^2*x + (a*b^2*c - 4*a^2*c^2)*d^2)*e
^2)*log(-(8*c^2*d^2*x^2 + 8*b*c*d^2*x + (b^2 + 4*a*c)*d^2 - 4*sqrt(c*d^2 - b*d*e + a*e^2)*(2*c*d*x + b*d - (b*
x + 2*a)*e)*sqrt(c*x^2 + b*x + a) + (8*a*b*x + (b^2 + 4*a*c)*x^2 + 8*a^2)*e^2 - 2*(4*b*c*d*x^2 + 4*a*b*d + (3*
b^2 + 4*a*c)*d*x)*e)/(x^2*e^2 + 2*d*x*e + d^2)) - 4*(4*c^4*d^5*x + 2*b*c^3*d^5 + (a^2*b^2 - 4*a^3*c + (3*a*b^2
*c - 8*a^2*c^2)*x^2 + (3*a*b^3 - 10*a^2*b*c)*x)*e^5 - ((3*b^3*c - 4*a*b*c^2)*d*x^2 + (3*b^4 - 8*a*b^2*c - 4*a^
2*c^2)*d*x - (a*b^3 - 2*a^2*b*c)*d)*e^4 + ((7*b^2*c^2 - 4*a*c^3)*d^2*x^2 + (5*b^3*c - 16*a*b*c^2)*d^2*x - (2*b
^4 - 3*a*b^2*c - 4*a^2*c^2)*d^2)*e^3 - 2*(4*b*c^3*d^3*x^2 - 4*a*c^3*d^3*x - 3*(b^3*c - 2*a*b*c^2)*d^3)*e^2 + 2
*(2*c^4*d^4*x^2 - 3*b*c^3*d^4*x - (3*b^2*c^2 - 4*a*c^3)*d^4)*e)*sqrt(c*x^2 + b*x + a))/((b^2*c^4 - 4*a*c^5)*d^
7*x^2 + (b^3*c^3 - 4*a*b*c^4)*d^7*x + (a*b^2*c^3 - 4*a^2*c^4)*d^7 + ((a^3*b^2*c - 4*a^4*c^2)*x^3 + (a^3*b^3 -
4*a^4*b*c)*x^2 + (a^4*b^2 - 4*a^5*c)*x)*e^7 - (3*(a^2*b^3*c - 4*a^3*b*c^2)*d*x^3 + (3*a^2*b^4 - 13*a^3*b^2*c +
4*a^4*c^2)*d*x^2 + 2*(a^3*b^3 - 4*a^4*b*c)*d*x - (a^4*b^2 - 4*a^5*c)*d)*e^6 + 3*((a*b^4*c - 3*a^2*b^2*c^2 - 4
*a^3*c^3)*d^2*x^3 + (a*b^5 - 4*a^2*b^3*c)*d^2*x^2 + (a^3*b^2*c - 4*a^4*c^2)*d^2*x - (a^3*b^3 - 4*a^4*b*c)*d^2)
*e^5 - ((b^5*c + 2*a*b^3*c^2 - 24*a^2*b*c^3)*d^3*x^3 + (b^6 - a*b^4*c - 15*a^2*b^2*c^2 + 12*a^3*c^3)*d^3*x^2 -
(2*a*b^5 - 11*a^2*b^3*c + 12*a^3*b*c^2)*d^3*x - 3*(a^2*b^4 - 3*a^3*b^2*c - 4*a^4*c^2)*d^3)*e^4 + (3*(b^4*c^2
- 3*a*b^2*c^3 - 4*a^2*c^4)*d^4*x^3 + (2*b^5*c - 11*a*b^3*c^2 + 12*a^2*b*c^3)*d^4*x^2 - (b^6 - a*b^4*c - 15*a^2
*b^2*c^2 + 12*a^3*c^3)*d^4*x - (a*b^5 + 2*a^2*b^3*c - 24*a^3*b*c^2)*d^4)*e^3 - 3*((b^3*c^3 - 4*a*b*c^4)*d^5*x^
3 - (a*b^2*c^3 - 4*a^2*c^4)*d^5*x^2 - (b^5*c - 4*a*b^3*c^2)*d^5*x - (a*b^4*c - 3*a^2*b^2*c^2 - 4*a^3*c^3)*d^5)
*e^2 + ((b^2*c^4 - 4*a*c^5)*d^6*x^3 - 2*(b^3*c^3 - 4*a*b*c^4)*d^6*x^2 - (3*b^4*c^2 - 13*a*b^2*c^3 + 4*a^2*c^4)
*d^6*x - 3*(a*b^3*c^2 - 4*a^2*b*c^3)*d^6)*e), -1/2*(3*sqrt(-c*d^2 + b*d*e - a*e^2)*(((b^3*c - 4*a*b*c^2)*x^3 +
(b^4 - 4*a*b^2*c)*x^2 + (a*b^3 - 4*a^2*b*c)*x)*e^4 - (2*(b^2*c^2 - 4*a*c^3)*d*x^3 + (b^3*c - 4*a*b*c^2)*d*x^2
- (b^4 - 6*a*b^2*c + 8*a^2*c^2)*d*x - (a*b^3 - 4*a^2*b*c)*d)*e^3 - 2*((b^2*c^2 - 4*a*c^3)*d^2*x^2 + (b^3*c -
4*a*b*c^2)*d^2*x + (a*b^2*c - 4*a^2*c^2)*d^2)*e^2)*arctan(-1/2*sqrt(-c*d^2 + b*d*e - a*e^2)*(2*c*d*x + b*d - (
b*x + 2*a)*e)*sqrt(c*x^2 + b*x + a)/(c^2*d^2*x^2 + b*c*d^2*x + a*c*d^2 + (a*c*x^2 + a*b*x + a^2)*e^2 - (b*c*d*
x^2 + b^2*d*x + a*b*d)*e)) + 2*(4*c^4*d^5*x + 2*b*c^3*d^5 + (a^2*b^2 - 4*a^3*c + (3*a*b^2*c - 8*a^2*c^2)*x^2 +
(3*a*b^3 - 10*a^2*b*c)*x)*e^5 - ((3*b^3*c - 4*a*b*c^2)*d*x^2 + (3*b^4 - 8*a*b^2*c - 4*a^2*c^2)*d*x - (a*b^3 -
2*a^2*b*c)*d)*e^4 + ((7*b^2*c^2 - 4*a*c^3)*d^2*x^2 + (5*b^3*c - 16*a*b*c^2)*d^2*x - (2*b^4 - 3*a*b^2*c - 4*a^
2*c^2)*d^2)*e^3 - 2*(4*b*c^3*d^3*x^2 - 4*a*c^3*d^3*x - 3*(b^3*c - 2*a*b*c^2)*d^3)*e^2 + 2*(2*c^4*d^4*x^2 - 3*b
*c^3*d^4*x - (3*b^2*c^2 - 4*a*c^3)*d^4)*e)*sqrt(c*x^2 + b*x + a))/((b^2*c^4 - 4*a*c^5)*d^7*x^2 + (b^3*c^3 - 4*
a*b*c^4)*d^7*x + (a*b^2*c^3 - 4*a^2*c^4)*d^7 + ((a^3*b^2*c - 4*a^4*c^2)*x^3 + (a^3*b^3 - 4*a^4*b*c)*x^2 + (a^4
*b^2 - 4*a^5*c)*x)*e^7 - (3*(a^2*b^3*c - 4*a^3*b*c^2)*d*x^3 + (3*a^2*b^4 - 13*a^3*b^2*c + 4*a^4*c^2)*d*x^2 + 2
*(a^3*b^3 - 4*a^4*b*c)*d*x - (a^4*b^2 - 4*a^5*c)*d)*e^6 + 3*((a*b^4*c - 3*a^2*b^2*c^2 - 4*a^3*c^3)*d^2*x^3 + (
a*b^5 - 4*a^2*b^3*c)*d^2*x^2 + (a^3*b^2*c - 4*a^4*c^2)*d^2*x - (a^3*b^3 - 4*a^4*b*c)*d^2)*e^5 - ((b^5*c + 2*a*
b^3*c^2 - 24*a^2*b*c^3)*d^3*x^3 + (b^6 - a*b^4*c - 15*a^2*b^2*c^2 + 12*a^3*c^3)*d^3*x^2 - (2*a*b^5 - 11*a^2*b^
3*c + 12*a^3*b*c^2)*d^3*x - 3*(a^2*b^4 - 3*a^3*b^2*c - 4*a^4*c^2)*d^3)*e^4 + (3*(b^4*c^2 - 3*a*b^2*c^3 - 4*a^2
*c^4)*d^4*x^3 + (2*b^5*c - 11*a*b^3*c^2 + 12*a^2*b*c^3)*d^4*x^2 - (b^6 - a*b^4*c - 15*a^2*b^2*c^2 + 12*a^3*c^3
)*d^4*x - (a*b^5 + 2*a^2*b^3*c - 24*a^3*b*c^2)*d^4)*e^3 - 3*((b^3*c^3 - 4*a*b*c^4)*d^5*x^3 - (a*b^2*c^3 - 4*a^
2*c^4)*d^5*x^2 - (b^5*c - 4*a*b^3*c^2)*d^5*x - (a*b^4*c - 3*a^2*b^2*c^2 - 4*a^3*c^3)*d^5)*e^2 + ((b^2*c^4 - 4*
a*c^5)*d^6*x^3 - 2*(b^3*c^3 - 4*a*b*c^4)*d^6*x^2 - (3*b^4*c^2 - 13*a*b^2*c^3 + 4*a^2*c^4)*d^6*x - 3*(a*b^3*c^2
- 4*a^2*b*c^3)*d^6)*e)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (d + e x\right )^{2} \left (a + b x + c x^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\left (d+e\,x\right )}^2\,{\left (c\,x^2+b\,x+a\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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