3.1602 \(\int \frac{1}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=365 \[ \frac{5 b e^4 (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^6}+\frac{e^4 (a+b x)}{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^5}+\frac{15 b^2 e^4 (a+b x) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^7}-\frac{15 b^2 e^4 (a+b x) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^7}+\frac{10 b^2 e^3}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}-\frac{3 b^2 e^2}{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}+\frac{b^2 e}{(a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}-\frac{b^2}{4 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3} \]

[Out]

(10*b^2*e^3)/((b*d - a*e)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - b^2/(4*(b*d - a*e)^
3*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (b^2*e)/((b*d - a*e)^4*(a + b*x)^
2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (3*b^2*e^2)/((b*d - a*e)^5*(a + b*x)*Sqrt[a^2
+ 2*a*b*x + b^2*x^2]) + (e^4*(a + b*x))/(2*(b*d - a*e)^5*(d + e*x)^2*Sqrt[a^2 +
2*a*b*x + b^2*x^2]) + (5*b*e^4*(a + b*x))/((b*d - a*e)^6*(d + e*x)*Sqrt[a^2 + 2
*a*b*x + b^2*x^2]) + (15*b^2*e^4*(a + b*x)*Log[a + b*x])/((b*d - a*e)^7*Sqrt[a^2
+ 2*a*b*x + b^2*x^2]) - (15*b^2*e^4*(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 = 0.67147, antiderivative size = 365, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ \frac{5 b e^4 (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^6}+\frac{e^4 (a+b x)}{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^5}+\frac{15 b^2 e^4 (a+b x) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^7}-\frac{15 b^2 e^4 (a+b x) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^7}+\frac{10 b^2 e^3}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}-\frac{3 b^2 e^2}{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}+\frac{b^2 e}{(a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}-\frac{b^2}{4 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]

[Out]

(10*b^2*e^3)/((b*d - a*e)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - b^2/(4*(b*d - a*e)^
3*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (b^2*e)/((b*d - a*e)^4*(a + b*x)^
2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (3*b^2*e^2)/((b*d - a*e)^5*(a + b*x)*Sqrt[a^2
+ 2*a*b*x + b^2*x^2]) + (e^4*(a + b*x))/(2*(b*d - a*e)^5*(d + e*x)^2*Sqrt[a^2 +
2*a*b*x + b^2*x^2]) + (5*b*e^4*(a + b*x))/((b*d - a*e)^6*(d + e*x)*Sqrt[a^2 + 2
*a*b*x + b^2*x^2]) + (15*b^2*e^4*(a + b*x)*Log[a + b*x])/((b*d - a*e)^7*Sqrt[a^2
+ 2*a*b*x + b^2*x^2]) - (15*b^2*e^4*(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 = 91.654, size = 364, normalized size = 1. \[ - \frac{15 b^{2} e^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (a + b x \right )}}{\left (a + b x\right ) \left (a e - b d\right )^{7}} + \frac{15 b^{2} e^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (d + e x \right )}}{\left (a + b x\right ) \left (a e - b d\right )^{7}} + \frac{15 b e^{5} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{\left (d + e x\right ) \left (a e - b d\right )^{7}} - \frac{15 e^{4} \left (2 a + 2 b x\right )}{4 \left (d + e x\right )^{2} \left (a e - b d\right )^{5} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} + \frac{5 e^{3}}{\left (d + e x\right )^{2} \left (a e - b d\right )^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} + \frac{5 e^{2} \left (2 a + 2 b x\right )}{8 \left (d + e x\right )^{2} \left (a e - b d\right )^{3} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{e}{2 \left (d + e x\right )^{2} \left (a e - b d\right )^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{2 a + 2 b x}{8 \left (d + e x\right )^{2} \left (a e - b d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)**3/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

-15*b**2*e**4*sqrt(a**2 + 2*a*b*x + b**2*x**2)*log(a + b*x)/((a + b*x)*(a*e - b*
d)**7) + 15*b**2*e**4*sqrt(a**2 + 2*a*b*x + b**2*x**2)*log(d + e*x)/((a + b*x)*(
a*e - b*d)**7) + 15*b*e**5*sqrt(a**2 + 2*a*b*x + b**2*x**2)/((d + e*x)*(a*e - b*
d)**7) - 15*e**4*(2*a + 2*b*x)/(4*(d + e*x)**2*(a*e - b*d)**5*sqrt(a**2 + 2*a*b*
x + b**2*x**2)) + 5*e**3/((d + e*x)**2*(a*e - b*d)**4*sqrt(a**2 + 2*a*b*x + b**2
*x**2)) + 5*e**2*(2*a + 2*b*x)/(8*(d + e*x)**2*(a*e - b*d)**3*(a**2 + 2*a*b*x +
b**2*x**2)**(3/2)) + e/(2*(d + e*x)**2*(a*e - b*d)**2*(a**2 + 2*a*b*x + b**2*x**
2)**(3/2)) + (2*a + 2*b*x)/(8*(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 = 0.279659, size = 209, normalized size = 0.57 \[ \frac{-60 b^2 e^4 (a+b x)^3 \log (d+e x)+40 b^2 e^3 (a+b x)^2 (b d-a e)-12 b^2 e^2 (a+b x) (b d-a e)^2-\frac{b^2 (b d-a e)^4}{a+b x}+4 b^2 e (b d-a e)^3+60 b^2 e^4 (a+b x)^3 \log (a+b x)+\frac{20 b e^4 (a+b x)^3 (b d-a e)}{d+e x}+\frac{2 e^4 (a+b x)^3 (b d-a e)^2}{(d+e x)^2}}{4 \left ((a+b x)^2\right )^{3/2} (b d-a e)^7} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]

[Out]

(4*b^2*e*(b*d - a*e)^3 - (b^2*(b*d - a*e)^4)/(a + b*x) - 12*b^2*e^2*(b*d - a*e)^
2*(a + b*x) + 40*b^2*e^3*(b*d - a*e)*(a + b*x)^2 + (2*e^4*(b*d - a*e)^2*(a + b*x
)^3)/(d + e*x)^2 + (20*b*e^4*(b*d - a*e)*(a + b*x)^3)/(d + e*x) + 60*b^2*e^4*(a
+ b*x)^3*Log[a + b*x] - 60*b^2*e^4*(a + b*x)^3*Log[d + e*x])/(4*(b*d - a*e)^7*((
a + b*x)^2)^(3/2))

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Maple [B]  time = 0.038, size = 983, normalized size = 2.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^3/(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

-1/4*(-60*ln(e*x+d)*x^2*a^4*b^2*e^6-120*ln(e*x+d)*x^5*b^6*d*e^5-360*ln(e*x+d)*x^
4*a^2*b^4*e^6+720*ln(b*x+a)*x^3*a^2*b^4*d*e^5+240*ln(b*x+a)*x^3*a*b^5*d^2*e^4+48
0*ln(b*x+a)*x^2*a^3*b^3*d*e^5+360*ln(b*x+a)*x^2*a^2*b^4*d^2*e^4+120*ln(b*x+a)*x*
a^4*b^2*d*e^5+240*ln(b*x+a)*x*a^3*b^3*d^2*e^4+2*a^6*e^6-b^6*d^6-240*ln(e*x+d)*x^
3*a^3*b^3*e^6-60*x^5*a*b^5*e^6+60*x^5*b^6*d*e^5+20*x^3*b^6*d^3*e^3-260*x^3*a^3*b
^3*e^6-210*x^4*a^2*b^4*e^6-125*x^2*a^4*b^2*e^6-5*x^2*b^6*d^4*e^2-12*x*a^5*b*e^6+
2*x*b^6*d^5*e-190*x*a^4*b^2*d*e^5+240*ln(b*x+a)*x^3*a^3*b^3*e^6+60*ln(b*x+a)*x^2
*a^4*b^2*e^6+60*ln(b*x+a)*a^4*b^2*d^2*e^4+330*x^2*a^2*b^4*d^2*e^4+80*x^2*a*b^5*d
^3*e^3+8*d^5*a*b^5*e+80*a^3*b^3*d^3*e^3-30*d^4*e^2*a^2*b^4+90*x^4*b^6*d^2*e^4+60
*ln(b*x+a)*x^6*b^6*e^6-60*ln(e*x+d)*x^6*b^6*e^6-60*ln(e*x+d)*x^4*b^6*d^2*e^4+480
*ln(b*x+a)*x^4*a*b^5*d*e^5+100*x*a^3*b^3*d^2*e^4+120*x*a^2*b^4*d^3*e^3-20*x*a*b^
5*d^4*e^2+300*x^3*a*b^5*d^2*e^4+360*ln(b*x+a)*x^4*a^2*b^4*e^6+60*ln(b*x+a)*x^4*b
^6*d^2*e^4-280*x^2*a^3*b^3*d*e^5+120*x^4*a*b^5*d*e^5-60*x^3*a^2*b^4*d*e^5-240*ln
(e*x+d)*x^3*a*b^5*d^2*e^4-480*ln(e*x+d)*x^2*a^3*b^3*d*e^5-360*ln(e*x+d)*x^2*a^2*
b^4*d^2*e^4-120*ln(e*x+d)*x*a^4*b^2*d*e^5-35*b^2*a^4*d^2*e^4+240*ln(b*x+a)*x^5*a
*b^5*e^6-240*ln(e*x+d)*x*a^3*b^3*d^2*e^4-480*ln(e*x+d)*x^4*a*b^5*d*e^5-720*ln(e*
x+d)*x^3*a^2*b^4*d*e^5+120*ln(b*x+a)*x^5*b^6*d*e^5-240*ln(e*x+d)*x^5*a*b^5*e^6-2
4*a^5*b*d*e^5-60*ln(e*x+d)*a^4*b^2*d^2*e^4)*(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(1/((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 = 0.257185, size = 2113, normalized size = 5.79 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(e*x + d)^3),x, algorithm="fricas")

[Out]

-1/4*(b^6*d^6 - 8*a*b^5*d^5*e + 30*a^2*b^4*d^4*e^2 - 80*a^3*b^3*d^3*e^3 + 35*a^4
*b^2*d^2*e^4 + 24*a^5*b*d*e^5 - 2*a^6*e^6 - 60*(b^6*d*e^5 - a*b^5*e^6)*x^5 - 30*
(3*b^6*d^2*e^4 + 4*a*b^5*d*e^5 - 7*a^2*b^4*e^6)*x^4 - 20*(b^6*d^3*e^3 + 15*a*b^5
*d^2*e^4 - 3*a^2*b^4*d*e^5 - 13*a^3*b^3*e^6)*x^3 + 5*(b^6*d^4*e^2 - 16*a*b^5*d^3
*e^3 - 66*a^2*b^4*d^2*e^4 + 56*a^3*b^3*d*e^5 + 25*a^4*b^2*e^6)*x^2 - 2*(b^6*d^5*
e - 10*a*b^5*d^4*e^2 + 60*a^2*b^4*d^3*e^3 + 50*a^3*b^3*d^2*e^4 - 95*a^4*b^2*d*e^
5 - 6*a^5*b*e^6)*x - 60*(b^6*e^6*x^6 + a^4*b^2*d^2*e^4 + 2*(b^6*d*e^5 + 2*a*b^5*
e^6)*x^5 + (b^6*d^2*e^4 + 8*a*b^5*d*e^5 + 6*a^2*b^4*e^6)*x^4 + 4*(a*b^5*d^2*e^4
+ 3*a^2*b^4*d*e^5 + a^3*b^3*e^6)*x^3 + (6*a^2*b^4*d^2*e^4 + 8*a^3*b^3*d*e^5 + a^
4*b^2*e^6)*x^2 + 2*(2*a^3*b^3*d^2*e^4 + a^4*b^2*d*e^5)*x)*log(b*x + a) + 60*(b^6
*e^6*x^6 + a^4*b^2*d^2*e^4 + 2*(b^6*d*e^5 + 2*a*b^5*e^6)*x^5 + (b^6*d^2*e^4 + 8*
a*b^5*d*e^5 + 6*a^2*b^4*e^6)*x^4 + 4*(a*b^5*d^2*e^4 + 3*a^2*b^4*d*e^5 + a^3*b^3*
e^6)*x^3 + (6*a^2*b^4*d^2*e^4 + 8*a^3*b^3*d*e^5 + a^4*b^2*e^6)*x^2 + 2*(2*a^3*b^
3*d^2*e^4 + 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^11*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^10*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(1/(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 = 0.647323, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(e*x + d)^3),x, algorithm="giac")

[Out]

sage0*x