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

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

[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])

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Rubi [A]  time = 1.26033, 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 \[ -\frac{e^3 (a+b x) (a B e-5 A b e+4 b B d)}{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^6}-\frac{e^3 (a+b x) (B d-A e)}{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^5}-\frac{5 b e^3 (a+b x) \log (a+b x) (a B e-3 A b e+2 b B d)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^7}+\frac{5 b e^3 (a+b x) \log (d+e x) (a B e-3 A b e+2 b B d)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^7}-\frac{2 b e^2 (2 a B e-5 A b e+3 b B d)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}+\frac{3 b e (a B e-2 A b e+b B d)}{2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}-\frac{b (2 a B e-3 A b e+b B d)}{3 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}-\frac{b (A b-a B)}{4 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3} \]

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])

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Rubi in Sympy [A]  time = 117.565, size = 488, normalized size = 1.06 \[ - \frac{5 b e^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (3 A b e - B a e - 2 B b d\right ) \log{\left (a + b x \right )}}{\left (a + b x\right ) \left (a e - b d\right )^{7}} + \frac{5 b e^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (3 A b e - B a e - 2 B b d\right ) \log{\left (d + e x \right )}}{\left (a + b x\right ) \left (a e - b d\right )^{7}} + \frac{5 e^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (3 A b e - B a e - 2 B b d\right )}{\left (d + e x\right ) \left (a e - b d\right )^{7}} - \frac{5 e^{2} \left (3 A b e - B a e - 2 B b d\right )}{2 \left (d + e x\right ) \left (a e - b d\right )^{5} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} - \frac{5 e \left (2 a + 2 b x\right ) \left (3 A b e - B a e - 2 B b d\right )}{12 \left (d + e x\right ) \left (a e - b d\right )^{4} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} - \frac{5 \left (3 A b e - B a e - 2 B b d\right )}{12 \left (d + e x\right ) \left (a e - b d\right )^{3} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} - \frac{\left (2 a + 2 b x\right ) \left (3 A b e - B a e - 2 B b d\right )}{8 e \left (d + e x\right ) \left (a e - b d\right )^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}} - \frac{\left (2 a + 2 b x\right ) \left (A e - B d\right )}{4 e \left (d + e x\right )^{2} \left (a e - b d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*b*e**3*sqrt(a**2 + 2*a*b*x + b**2*x**2)*(3*A*b*e - B*a*e - 2*B*b*d)*log(a + b
*x)/((a + b*x)*(a*e - b*d)**7) + 5*b*e**3*sqrt(a**2 + 2*a*b*x + b**2*x**2)*(3*A*
b*e - B*a*e - 2*B*b*d)*log(d + e*x)/((a + b*x)*(a*e - b*d)**7) + 5*e**4*sqrt(a**
2 + 2*a*b*x + b**2*x**2)*(3*A*b*e - B*a*e - 2*B*b*d)/((d + e*x)*(a*e - b*d)**7)
- 5*e**2*(3*A*b*e - B*a*e - 2*B*b*d)/(2*(d + e*x)*(a*e - b*d)**5*sqrt(a**2 + 2*a
*b*x + b**2*x**2)) - 5*e*(2*a + 2*b*x)*(3*A*b*e - B*a*e - 2*B*b*d)/(12*(d + e*x)
*(a*e - b*d)**4*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)) - 5*(3*A*b*e - B*a*e - 2*B*
b*d)/(12*(d + e*x)*(a*e - b*d)**3*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)) - (2*a +
2*b*x)*(3*A*b*e - B*a*e - 2*B*b*d)/(8*e*(d + e*x)*(a*e - b*d)**2*(a**2 + 2*a*b*x
+ b**2*x**2)**(5/2)) - (2*a + 2*b*x)*(A*e - B*d)/(4*e*(d + e*x)**2*(a*e - b*d)*
(a**2 + 2*a*b*x + b**2*x**2)**(5/2))

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Mathematica [A]  time = 0.607156, size = 302, normalized size = 0.66 \[ \frac{\frac{6 e^3 (a+b x)^3 (b d-a e)^2 (A e-B d)}{(d+e x)^2}+\frac{12 e^3 (a+b x)^3 (b d-a e) (-a B e+5 A b e-4 b B d)}{d+e x}-60 b e^3 (a+b x)^3 \log (a+b x) (a B e-3 A b e+2 b B d)+60 b e^3 (a+b x)^3 \log (d+e x) (a B e-3 A b e+2 b B d)+24 b e^2 (a+b x)^2 (b d-a e) (-2 a B e+5 A b e-3 b B d)-\frac{3 b (A b-a B) (b d-a e)^4}{a+b x}-18 b e (a+b x) (b d-a e)^2 (-a B e+2 A b e-b B d)-4 b (b d-a e)^3 (2 a B e-3 A b e+b B d)}{12 \left ((a+b x)^2\right )^{3/2} (b d-a e)^7} \]

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 = 0.048, size = 2420, normalized size = 5.3 \[ \text{result too large to display} \]

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*(-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^5-240*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+240*A*x^2*a*b^5*d^3*e^3+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^3-155*B*x^
2*a*b^5*d^4*e^2-570*A*x*a^4*b^2*d*e^5+300*A*x*a^3*b^3*d^2*e^4+360*A*x*a^2*b^4*d^
3*e^3-60*A*x*a*b^5*d^4*e^2+214*B*x*a^5*b*d*e^5+280*B*x*a^4*b^2*d^2*e^4-320*B*x*a
^3*b^3*d^3*e^3-60*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+60*B*l
n(e*x+d)*x^6*a*b^5*e^6+120*B*ln(e*x+d)*x^6*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-B*a*b^5*d^6+6*B*a^6*d*e^5-4*B*x*b^6*d^6+12*B*x*
a^6*e^6+1020*B*ln(e*x+d)*x^4*a*b^5*d^2*e^4+1440*A*ln(b*x+a)*x^2*a^3*b^3*d*e^5+60
0*B*ln(e*x+d)*x^5*a*b^5*d*e^5+1440*A*ln(b*x+a)*x^4*a*b^5*d*e^5-1440*A*ln(e*x+d)*
x^4*a*b^5*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+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+4
80*B*ln(e*x+d)*x^3*a*b^5*d^3*e^3-600*B*ln(b*x+a)*x^5*a*b^5*d*e^5+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+240*B*ln(e*x+d)*x^5*a^2
*b^4*e^6+240*B*ln(e*x+d)*x^5*b^6*d^2*e^4+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^4-1080*A*ln(e*x+d)*x^4*a^2*b^4*e^6-180*A*ln(e*x+d)*x^4*b
^6*d^2*e^4-360*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+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+720*A*ln(b*x+a)*x^3*a
^3*b^3*e^6-720*A*ln(e*x+d)*x^3*a^3*b^3*e^6-240*B*ln(b*x+a)*x^3*a^4*b^2*e^6+240*B
*ln(e*x+d)*x^3*a^4*b^2*e^6+180*A*ln(b*x+a)*x^2*a^4*b^2*e^6-180*A*ln(e*x+d)*x^2*a
^4*b^2*e^6-60*B*ln(b*x+a)*x^2*a^5*b*e^6+60*B*ln(e*x+d)*x^2*a^5*b*e^6+180*A*ln(b*
x+a)*a^4*b^2*d^2*e^4-180*A*ln(e*x+d)*a^4*b^2*d^2*e^4-60*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+60*B*ln(e*x+d)*a^5*b*d^2*e^4+120*B*ln(e*x+d)*a
^4*b^2*d^3*e^3-220*B*x*a^2*b^4*d^4*e^2+38*B*x*a*b^5*d^5*e+360*A*x^4*a*b^5*d*e^5+
300*B*x^4*a^2*b^4*d*e^5-330*B*x^4*a*b^5*d^2*e^4+60*B*x^5*a*b^5*d*e^5-180*A*x^3*a
^2*b^4*d*e^5+900*A*x^3*a*b^5*d^2*e^4+580*B*x^3*a^3*b^3*d*e^5-180*B*x^3*a^2*b^4*d
^2*e^4-620*B*x^3*a*b^5*d^3*e^3-840*A*x^2*a^3*b^3*d*e^5+990*A*x^2*a^2*b^4*d^2*e^4
-105*A*a^4*b^2*d^2*e^4+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+125*B*a^5*b*d^2*e^4-80*B*a^4*b^2*d^3*e^3-60*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^5-360*A*ln(e*x+d)*x*a^4*b^2*d*e^5-720*A*ln(e*x+d)*x*a^3*b^3*d
^2*e^4-120*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^4-3*A*b^6*d
^6+6*A*a^6*e^6-780*A*x^3*a^3*b^3*e^6+60*A*x^3*b^6*d^3*e^3+260*B*x^3*a^4*b^2*e^6-
40*B*x^3*b^6*d^4*e^2-375*A*x^2*a^4*b^2*e^6-15*A*x^2*b^6*d^4*e^2+125*B*x^2*a^5*b*
e^6+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+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^4-2160*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^4-1200*B*ln(b*x+a)*x^3*a^3*b^3*d*e^5-1080*A
*ln(e*x+d)*x^2*a^2*b^4*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+600*B*ln(e*x+d)*x^2*a^
4*b^2*d*e^5+1320*B*ln(e*x+d)*x^2*a^3*b^3*d^2*e^4+720*B*ln(e*x+d)*x^2*a^2*b^4*d^3
*e^3+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^4-1200*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+1200*B*ln(e*x+d)*x^
4*a^2*b^4*d*e^5+270*A*x^4*b^6*d^2*e^4+210*B*x^4*a^3*b^3*e^6-180*B*x^4*b^6*d^3*e^
3-180*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^4-36*A*x*a^5*b*e^6+180*A*ln(b*x+a)*x^6*b^6*e^6-180*A*ln(e*x+d)*x^6*b^6*e^6-48
0*B*ln(b*x+a)*x*a^3*b^3*d^3*e^3+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)*(b*x+a)/(e*x+d)^2/(a*e-b*d)^7
/((b*x+a)^2)^(5/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.354295, size = 3329, normalized size = 7.24 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(e*x + d)^3),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 - 3
0*(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 - 84*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 - 1
19*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 - 2
1*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. \[ \text{Timed out} \]

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/XCAS [A]  time = 0.670521, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x