3.1783 \(\int \frac{A+B x}{(d+e x)^3 (a^2+2 a b x+b^2 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=460 \[ -\frac{e^3 (a+b x) (a B e-5 A b e+4 b B d)}{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^6}-\frac{e^3 (a+b x) (B d-A e)}{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^5}-\frac{2 b e^2 (2 a B e-5 A b e+3 b B d)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}-\frac{5 b e^3 (a+b x) \log (a+b x) (a B e-3 A b e+2 b B d)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^7}+\frac{5 b e^3 (a+b x) \log (d+e x) (a B e-3 A b e+2 b B d)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^7}+\frac{3 b e (a B e-2 A b e+b B d)}{2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}-\frac{b (2 a B e-3 A b e+b B d)}{3 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}-\frac{b (A b-a B)}{4 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3} \]

[Out]

(-2*b*e^2*(3*b*B*d - 5*A*b*e + 2*a*B*e))/((b*d - a*e)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (b*(A*b - a*B))/(4*(b
*d - a*e)^3*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (b*(b*B*d - 3*A*b*e + 2*a*B*e))/(3*(b*d - a*e)^4*(a +
b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (3*b*e*(b*B*d - 2*A*b*e + a*B*e))/(2*(b*d - a*e)^5*(a + b*x)*Sqrt[a^2
+ 2*a*b*x + b^2*x^2]) - (e^3*(B*d - A*e)*(a + b*x))/(2*(b*d - a*e)^5*(d + e*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2
]) - (e^3*(4*b*B*d - 5*A*b*e + a*B*e)*(a + b*x))/((b*d - a*e)^6*(d + e*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (5*
b*e^3*(2*b*B*d - 3*A*b*e + a*B*e)*(a + b*x)*Log[a + b*x])/((b*d - a*e)^7*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (5*b
*e^3*(2*b*B*d - 3*A*b*e + a*B*e)*(a + b*x)*Log[d + e*x])/((b*d - a*e)^7*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.55715, antiderivative size = 460, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061, Rules used = {770, 77} \[ -\frac{e^3 (a+b x) (a B e-5 A b e+4 b B d)}{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^6}-\frac{e^3 (a+b x) (B d-A e)}{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^5}-\frac{2 b e^2 (2 a B e-5 A b e+3 b B d)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}-\frac{5 b e^3 (a+b x) \log (a+b x) (a B e-3 A b e+2 b B d)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^7}+\frac{5 b e^3 (a+b x) \log (d+e x) (a B e-3 A b e+2 b B d)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^7}+\frac{3 b e (a B e-2 A b e+b B d)}{2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}-\frac{b (2 a B e-3 A b e+b B d)}{3 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}-\frac{b (A b-a B)}{4 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/((d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]

[Out]

(-2*b*e^2*(3*b*B*d - 5*A*b*e + 2*a*B*e))/((b*d - a*e)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (b*(A*b - a*B))/(4*(b
*d - a*e)^3*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (b*(b*B*d - 3*A*b*e + 2*a*B*e))/(3*(b*d - a*e)^4*(a +
b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (3*b*e*(b*B*d - 2*A*b*e + a*B*e))/(2*(b*d - a*e)^5*(a + b*x)*Sqrt[a^2
+ 2*a*b*x + b^2*x^2]) - (e^3*(B*d - A*e)*(a + b*x))/(2*(b*d - a*e)^5*(d + e*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2
]) - (e^3*(4*b*B*d - 5*A*b*e + a*B*e)*(a + b*x))/((b*d - a*e)^6*(d + e*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (5*
b*e^3*(2*b*B*d - 3*A*b*e + a*B*e)*(a + b*x)*Log[a + b*x])/((b*d - a*e)^7*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (5*b
*e^3*(2*b*B*d - 3*A*b*e + a*B*e)*(a + b*x)*Log[d + e*x])/((b*d - a*e)^7*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

Rule 770

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

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int \frac{A+B x}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx &=\frac{\left (b^4 \left (a b+b^2 x\right )\right ) \int \frac{A+B x}{\left (a b+b^2 x\right )^5 (d+e x)^3} \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{\left (b^4 \left (a b+b^2 x\right )\right ) \int \left (\frac{A b-a B}{b^3 (b d-a e)^3 (a+b x)^5}+\frac{b B d-3 A b e+2 a B e}{b^3 (b d-a e)^4 (a+b x)^4}+\frac{3 e (-b B d+2 A b e-a B e)}{b^3 (b d-a e)^5 (a+b x)^3}-\frac{2 e^2 (-3 b B d+5 A b e-2 a B e)}{b^3 (b d-a e)^6 (a+b x)^2}+\frac{5 e^3 (-2 b B d+3 A b e-a B e)}{b^3 (b d-a e)^7 (a+b x)}-\frac{e^4 (-B d+A e)}{b^5 (b d-a e)^5 (d+e x)^3}-\frac{e^4 (-4 b B d+5 A b e-a B e)}{b^5 (b d-a e)^6 (d+e x)^2}-\frac{5 e^4 (-2 b B d+3 A b e-a B e)}{b^4 (b d-a e)^7 (d+e x)}\right ) \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{2 b e^2 (3 b B d-5 A b e+2 a B e)}{(b d-a e)^6 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{b (A b-a B)}{4 (b d-a e)^3 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{b (b B d-3 A b e+2 a B e)}{3 (b d-a e)^4 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{3 b e (b B d-2 A b e+a B e)}{2 (b d-a e)^5 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{e^3 (B d-A e) (a+b x)}{2 (b d-a e)^5 (d+e x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{e^3 (4 b B d-5 A b e+a B e) (a+b x)}{(b d-a e)^6 (d+e x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 b e^3 (2 b B d-3 A b e+a B e) (a+b x) \log (a+b x)}{(b d-a e)^7 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 b e^3 (2 b B d-3 A b e+a B e) (a+b x) \log (d+e x)}{(b d-a e)^7 \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.391004, size = 302, normalized size = 0.66 \[ \frac{\frac{6 e^3 (a+b x)^3 (b d-a e)^2 (A e-B d)}{(d+e x)^2}+24 b e^2 (a+b x)^2 (b d-a e) (-2 a B e+5 A b e-3 b B d)+\frac{12 e^3 (a+b x)^3 (b d-a e) (-a B e+5 A b e-4 b B d)}{d+e x}-60 b e^3 (a+b x)^3 \log (a+b x) (a B e-3 A b e+2 b B d)+60 b e^3 (a+b x)^3 \log (d+e x) (a B e-3 A b e+2 b B d)-\frac{3 b (A b-a B) (b d-a e)^4}{a+b x}-18 b e (a+b x) (b d-a e)^2 (-a B e+2 A b e-b B d)-4 b (b d-a e)^3 (2 a B e-3 A b e+b B d)}{12 \left ((a+b x)^2\right )^{3/2} (b d-a e)^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/((d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]

[Out]

(-4*b*(b*d - a*e)^3*(b*B*d - 3*A*b*e + 2*a*B*e) - (3*b*(A*b - a*B)*(b*d - a*e)^4)/(a + b*x) - 18*b*e*(b*d - a*
e)^2*(-(b*B*d) + 2*A*b*e - a*B*e)*(a + b*x) + 24*b*e^2*(b*d - a*e)*(-3*b*B*d + 5*A*b*e - 2*a*B*e)*(a + b*x)^2
+ (6*e^3*(b*d - a*e)^2*(-(B*d) + A*e)*(a + b*x)^3)/(d + e*x)^2 + (12*e^3*(b*d - a*e)*(-4*b*B*d + 5*A*b*e - a*B
*e)*(a + b*x)^3)/(d + e*x) - 60*b*e^3*(2*b*B*d - 3*A*b*e + a*B*e)*(a + b*x)^3*Log[a + b*x] + 60*b*e^3*(2*b*B*d
- 3*A*b*e + a*B*e)*(a + b*x)^3*Log[d + e*x])/(12*(b*d - a*e)^7*((a + b*x)^2)^(3/2))

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Maple [B]  time = 0.031, size = 2419, normalized size = 5.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/(e*x+d)^3/(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

1/12*(-600*B*ln(e*x+d)*x^2*a^4*b^2*d*e^5+120*B*ln(b*x+a)*x*a^5*b*d*e^5+480*B*ln(b*x+a)*x*a^4*b^2*d^2*e^4-1080*
A*ln(b*x+a)*x^2*a^2*b^4*d^2*e^4+1440*A*ln(e*x+d)*x^2*a^3*b^3*d*e^5+1080*A*ln(e*x+d)*x^2*a^2*b^4*d^2*e^4-1440*A
*ln(b*x+a)*x^2*a^3*b^3*d*e^5+1200*B*ln(b*x+a)*x^3*a^3*b^3*d*e^5+1680*B*ln(b*x+a)*x^3*a^2*b^4*d^2*e^4+480*B*ln(
b*x+a)*x^3*a*b^5*d^3*e^3-300*A*x*a^3*b^3*d^2*e^4-214*B*x*a^5*b*d*e^5-280*B*x*a^4*b^2*d^2*e^4+840*A*x^2*a^3*b^3
*d*e^5-990*A*x^2*a^2*b^4*d^2*e^4-530*B*x^2*a^4*b^2*d*e^5-230*B*x^2*a^3*b^3*d^2*e^4+740*B*x^2*a^2*b^4*d^3*e^3+5
70*A*x*a^4*b^2*d*e^5+620*B*x^3*a*b^5*d^3*e^3-580*B*x^3*a^3*b^3*d*e^5+180*B*x^3*a^2*b^4*d^2*e^4+180*A*x^3*a^2*b
^4*d*e^5-900*A*x^3*a*b^5*d^2*e^4+330*B*x^4*a*b^5*d^2*e^4-38*B*x*a*b^5*d^5*e+320*B*x*a^3*b^3*d^3*e^3-60*B*x^5*a
*b^5*d*e^5-360*A*x^4*a*b^5*d*e^5-300*B*x^4*a^2*b^4*d*e^5+155*B*x^2*a*b^5*d^4*e^2-360*A*x*a^2*b^4*d^3*e^3+60*A*
x*a*b^5*d^4*e^2+220*B*x*a^2*b^4*d^4*e^2-1080*A*ln(b*x+a)*x^4*a^2*b^4*e^6-180*A*ln(b*x+a)*x^4*b^6*d^2*e^4-240*A
*x^2*a*b^5*d^3*e^3+180*A*ln(e*x+d)*x^4*b^6*d^2*e^4-240*B*ln(e*x+d)*x^5*b^6*d^2*e^4+240*B*ln(b*x+a)*x^5*a^2*b^4
*e^6+240*B*ln(b*x+a)*x^5*b^6*d^2*e^4+1080*A*ln(e*x+d)*x^4*a^2*b^4*e^6-240*B*ln(e*x+d)*x^5*a^2*b^4*e^6+720*A*ln
(e*x+d)*x^5*a*b^5*e^6+360*A*ln(e*x+d)*x^5*b^6*d*e^5-720*A*ln(b*x+a)*x^5*a*b^5*e^6-360*A*ln(b*x+a)*x^5*b^6*d*e^
5-120*B*ln(e*x+d)*x^6*b^6*d*e^5+60*B*ln(b*x+a)*x^6*a*b^5*e^6+120*B*ln(b*x+a)*x^6*b^6*d*e^5-180*A*ln(b*x+a)*x^2
*a^4*b^2*e^6-60*B*ln(e*x+d)*x^2*a^5*b*e^6-60*B*ln(e*x+d)*x^6*a*b^5*e^6-240*B*ln(e*x+d)*x^3*a^4*b^2*e^6+240*B*l
n(b*x+a)*x^3*a^4*b^2*e^6+180*A*ln(e*x+d)*x^2*a^4*b^2*e^6+360*B*ln(b*x+a)*x^4*a^3*b^3*e^6+120*B*ln(b*x+a)*x^4*b
^6*d^3*e^3+720*A*ln(e*x+d)*x^3*a^3*b^3*e^6-720*A*ln(b*x+a)*x^3*a^3*b^3*e^6-360*B*ln(e*x+d)*x^4*a^3*b^3*e^6-120
*B*ln(e*x+d)*x^4*b^6*d^3*e^3-120*B*ln(e*x+d)*a^4*b^2*d^3*e^3+60*B*ln(b*x+a)*a^5*b*d^2*e^4+120*B*ln(b*x+a)*a^4*
b^2*d^3*e^3+180*A*ln(e*x+d)*a^4*b^2*d^2*e^4-180*A*ln(b*x+a)*a^4*b^2*d^2*e^4-60*B*ln(e*x+d)*a^5*b*d^2*e^4+60*B*
ln(b*x+a)*x^2*a^5*b*e^6-12*B*x*a^6*e^6+4*B*x*b^6*d^6-6*B*a^6*d*e^5+B*a*b^5*d^6+180*A*x^5*a*b^5*e^6-180*A*x^5*b
^6*d*e^5-60*B*x^5*a^2*b^4*e^6+120*B*x^5*b^6*d^2*e^4-210*B*x^4*a^3*b^3*e^6-260*B*x^3*a^4*b^2*e^6-125*B*x^2*a^5*
b*e^6+180*A*ln(e*x+d)*x^6*b^6*e^6-180*A*ln(b*x+a)*x^6*b^6*e^6-60*A*x^3*b^6*d^3*e^3+40*B*x^3*b^6*d^4*e^2+15*A*x
^2*b^6*d^4*e^2-10*B*x^2*b^6*d^5*e-6*A*x*b^6*d^5*e+630*A*x^4*a^2*b^4*e^6-270*A*x^4*b^6*d^2*e^4+180*B*x^4*b^6*d^
3*e^3+780*A*x^3*a^3*b^3*e^6+375*A*x^2*a^4*b^2*e^6+36*A*x*a^5*b*e^6-240*A*a^3*b^3*d^3*e^3+90*A*a^2*b^4*d^4*e^2-
24*A*a*b^5*d^5*e+60*B*a^3*b^3*d^4*e^2-10*B*a^2*b^4*d^5*e+72*A*a^5*b*d*e^5+105*A*a^4*b^2*d^2*e^4-125*B*a^5*b*d^
2*e^4+80*B*a^4*b^2*d^3*e^3+720*A*ln(e*x+d)*x*a^3*b^3*d^2*e^4-360*A*ln(b*x+a)*x*a^4*b^2*d*e^5-720*A*ln(b*x+a)*x
*a^3*b^3*d^2*e^4-120*B*ln(e*x+d)*x*a^5*b*d*e^5-480*B*ln(e*x+d)*x*a^4*b^2*d^2*e^4-480*B*ln(e*x+d)*x*a^3*b^3*d^3
*e^3-1320*B*ln(e*x+d)*x^2*a^3*b^3*d^2*e^4+600*B*ln(b*x+a)*x^2*a^4*b^2*d*e^5+1320*B*ln(b*x+a)*x^2*a^3*b^3*d^2*e
^4+720*B*ln(b*x+a)*x^2*a^2*b^4*d^3*e^3+360*A*ln(e*x+d)*x*a^4*b^2*d*e^5-480*B*ln(e*x+d)*x^3*a*b^5*d^3*e^3-720*B
*ln(e*x+d)*x^2*a^2*b^4*d^3*e^3+1440*A*ln(e*x+d)*x^4*a*b^5*d*e^5-1440*A*ln(b*x+a)*x^4*a*b^5*d*e^5-1200*B*ln(e*x
+d)*x^4*a^2*b^4*d*e^5-1020*B*ln(e*x+d)*x^4*a*b^5*d^2*e^4+1200*B*ln(b*x+a)*x^4*a^2*b^4*d*e^5+1020*B*ln(b*x+a)*x
^4*a*b^5*d^2*e^4+2160*A*ln(e*x+d)*x^3*a^2*b^4*d*e^5+720*A*ln(e*x+d)*x^3*a*b^5*d^2*e^4-2160*A*ln(b*x+a)*x^3*a^2
*b^4*d*e^5-720*A*ln(b*x+a)*x^3*a*b^5*d^2*e^4-1200*B*ln(e*x+d)*x^3*a^3*b^3*d*e^5-1680*B*ln(e*x+d)*x^3*a^2*b^4*d
^2*e^4-6*A*a^6*e^6+3*A*b^6*d^6+480*B*ln(b*x+a)*x*a^3*b^3*d^3*e^3-600*B*ln(e*x+d)*x^5*a*b^5*d*e^5+600*B*ln(b*x+
a)*x^5*a*b^5*d*e^5)*(b*x+a)/(e*x+d)^2/(a*e-b*d)^7/((b*x+a)^2)^(5/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)^3/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.92231, size = 5027, normalized size = 10.93 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)^3/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")

[Out]

1/12*(6*A*a^6*e^6 - (B*a*b^5 + 3*A*b^6)*d^6 + 2*(5*B*a^2*b^4 + 12*A*a*b^5)*d^5*e - 30*(2*B*a^3*b^3 + 3*A*a^2*b
^4)*d^4*e^2 - 80*(B*a^4*b^2 - 3*A*a^3*b^3)*d^3*e^3 + 5*(25*B*a^5*b - 21*A*a^4*b^2)*d^2*e^4 + 6*(B*a^6 - 12*A*a
^5*b)*d*e^5 - 60*(2*B*b^6*d^2*e^4 - (B*a*b^5 + 3*A*b^6)*d*e^5 - (B*a^2*b^4 - 3*A*a*b^5)*e^6)*x^5 - 30*(6*B*b^6
*d^3*e^3 + (11*B*a*b^5 - 9*A*b^6)*d^2*e^4 - 2*(5*B*a^2*b^4 + 6*A*a*b^5)*d*e^5 - 7*(B*a^3*b^3 - 3*A*a^2*b^4)*e^
6)*x^4 - 20*(2*B*b^6*d^4*e^2 + (31*B*a*b^5 - 3*A*b^6)*d^3*e^3 + 9*(B*a^2*b^4 - 5*A*a*b^5)*d^2*e^4 - (29*B*a^3*
b^3 - 9*A*a^2*b^4)*d*e^5 - 13*(B*a^4*b^2 - 3*A*a^3*b^3)*e^6)*x^3 + 5*(2*B*b^6*d^5*e - (31*B*a*b^5 + 3*A*b^6)*d
^4*e^2 - 4*(37*B*a^2*b^4 - 12*A*a*b^5)*d^3*e^3 + 2*(23*B*a^3*b^3 + 99*A*a^2*b^4)*d^2*e^4 + 2*(53*B*a^4*b^2 - 8
4*A*a^3*b^3)*d*e^5 + 25*(B*a^5*b - 3*A*a^4*b^2)*e^6)*x^2 - 2*(2*B*b^6*d^6 - (19*B*a*b^5 + 3*A*b^6)*d^5*e + 10*
(11*B*a^2*b^4 + 3*A*a*b^5)*d^4*e^2 + 20*(8*B*a^3*b^3 - 9*A*a^2*b^4)*d^3*e^3 - 10*(14*B*a^4*b^2 + 15*A*a^3*b^3)
*d^2*e^4 - (107*B*a^5*b - 285*A*a^4*b^2)*d*e^5 - 6*(B*a^6 - 3*A*a^5*b)*e^6)*x - 60*(2*B*a^4*b^2*d^3*e^3 + (B*a
^5*b - 3*A*a^4*b^2)*d^2*e^4 + (2*B*b^6*d*e^5 + (B*a*b^5 - 3*A*b^6)*e^6)*x^6 + 2*(2*B*b^6*d^2*e^4 + (5*B*a*b^5
- 3*A*b^6)*d*e^5 + 2*(B*a^2*b^4 - 3*A*a*b^5)*e^6)*x^5 + (2*B*b^6*d^3*e^3 + (17*B*a*b^5 - 3*A*b^6)*d^2*e^4 + 4*
(5*B*a^2*b^4 - 6*A*a*b^5)*d*e^5 + 6*(B*a^3*b^3 - 3*A*a^2*b^4)*e^6)*x^4 + 4*(2*B*a*b^5*d^3*e^3 + (7*B*a^2*b^4 -
3*A*a*b^5)*d^2*e^4 + (5*B*a^3*b^3 - 9*A*a^2*b^4)*d*e^5 + (B*a^4*b^2 - 3*A*a^3*b^3)*e^6)*x^3 + (12*B*a^2*b^4*d
^3*e^3 + 2*(11*B*a^3*b^3 - 9*A*a^2*b^4)*d^2*e^4 + 2*(5*B*a^4*b^2 - 12*A*a^3*b^3)*d*e^5 + (B*a^5*b - 3*A*a^4*b^
2)*e^6)*x^2 + 2*(4*B*a^3*b^3*d^3*e^3 + 2*(2*B*a^4*b^2 - 3*A*a^3*b^3)*d^2*e^4 + (B*a^5*b - 3*A*a^4*b^2)*d*e^5)*
x)*log(b*x + a) + 60*(2*B*a^4*b^2*d^3*e^3 + (B*a^5*b - 3*A*a^4*b^2)*d^2*e^4 + (2*B*b^6*d*e^5 + (B*a*b^5 - 3*A*
b^6)*e^6)*x^6 + 2*(2*B*b^6*d^2*e^4 + (5*B*a*b^5 - 3*A*b^6)*d*e^5 + 2*(B*a^2*b^4 - 3*A*a*b^5)*e^6)*x^5 + (2*B*b
^6*d^3*e^3 + (17*B*a*b^5 - 3*A*b^6)*d^2*e^4 + 4*(5*B*a^2*b^4 - 6*A*a*b^5)*d*e^5 + 6*(B*a^3*b^3 - 3*A*a^2*b^4)*
e^6)*x^4 + 4*(2*B*a*b^5*d^3*e^3 + (7*B*a^2*b^4 - 3*A*a*b^5)*d^2*e^4 + (5*B*a^3*b^3 - 9*A*a^2*b^4)*d*e^5 + (B*a
^4*b^2 - 3*A*a^3*b^3)*e^6)*x^3 + (12*B*a^2*b^4*d^3*e^3 + 2*(11*B*a^3*b^3 - 9*A*a^2*b^4)*d^2*e^4 + 2*(5*B*a^4*b
^2 - 12*A*a^3*b^3)*d*e^5 + (B*a^5*b - 3*A*a^4*b^2)*e^6)*x^2 + 2*(4*B*a^3*b^3*d^3*e^3 + 2*(2*B*a^4*b^2 - 3*A*a^
3*b^3)*d^2*e^4 + (B*a^5*b - 3*A*a^4*b^2)*d*e^5)*x)*log(e*x + d))/(a^4*b^7*d^9 - 7*a^5*b^6*d^8*e + 21*a^6*b^5*d
^7*e^2 - 35*a^7*b^4*d^6*e^3 + 35*a^8*b^3*d^5*e^4 - 21*a^9*b^2*d^4*e^5 + 7*a^10*b*d^3*e^6 - a^11*d^2*e^7 + (b^1
1*d^7*e^2 - 7*a*b^10*d^6*e^3 + 21*a^2*b^9*d^5*e^4 - 35*a^3*b^8*d^4*e^5 + 35*a^4*b^7*d^3*e^6 - 21*a^5*b^6*d^2*e
^7 + 7*a^6*b^5*d*e^8 - a^7*b^4*e^9)*x^6 + 2*(b^11*d^8*e - 5*a*b^10*d^7*e^2 + 7*a^2*b^9*d^6*e^3 + 7*a^3*b^8*d^5
*e^4 - 35*a^4*b^7*d^4*e^5 + 49*a^5*b^6*d^3*e^6 - 35*a^6*b^5*d^2*e^7 + 13*a^7*b^4*d*e^8 - 2*a^8*b^3*e^9)*x^5 +
(b^11*d^9 + a*b^10*d^8*e - 29*a^2*b^9*d^7*e^2 + 91*a^3*b^8*d^6*e^3 - 119*a^4*b^7*d^5*e^4 + 49*a^5*b^6*d^4*e^5
+ 49*a^6*b^5*d^3*e^6 - 71*a^7*b^4*d^2*e^7 + 34*a^8*b^3*d*e^8 - 6*a^9*b^2*e^9)*x^4 + 4*(a*b^10*d^9 - 4*a^2*b^9*
d^8*e + a^3*b^8*d^7*e^2 + 21*a^4*b^7*d^6*e^3 - 49*a^5*b^6*d^5*e^4 + 49*a^6*b^5*d^4*e^5 - 21*a^7*b^4*d^3*e^6 -
a^8*b^3*d^2*e^7 + 4*a^9*b^2*d*e^8 - a^10*b*e^9)*x^3 + (6*a^2*b^9*d^9 - 34*a^3*b^8*d^8*e + 71*a^4*b^7*d^7*e^2 -
49*a^5*b^6*d^6*e^3 - 49*a^6*b^5*d^5*e^4 + 119*a^7*b^4*d^4*e^5 - 91*a^8*b^3*d^3*e^6 + 29*a^9*b^2*d^2*e^7 - a^1
0*b*d*e^8 - a^11*e^9)*x^2 + 2*(2*a^3*b^8*d^9 - 13*a^4*b^7*d^8*e + 35*a^5*b^6*d^7*e^2 - 49*a^6*b^5*d^6*e^3 + 35
*a^7*b^4*d^5*e^4 - 7*a^8*b^3*d^4*e^5 - 7*a^9*b^2*d^3*e^6 + 5*a^10*b*d^2*e^7 - a^11*d*e^8)*x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)**3/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)^3/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")

[Out]

sage0*x