3.1592 \(\int \frac{1}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=276 \[ \frac{3 b e^2 (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^4}+\frac{e^2 (a+b x)}{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^3}+\frac{6 b^2 e^2 (a+b x) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}-\frac{6 b^2 e^2 (a+b x) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}-\frac{b^2}{2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}+\frac{3 b^2 e}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4} \]

[Out]

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

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Rubi [A]  time = 0.41236, antiderivative size = 276, 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 \[ \frac{3 b e^2 (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^4}+\frac{e^2 (a+b x)}{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^3}+\frac{6 b^2 e^2 (a+b x) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}-\frac{6 b^2 e^2 (a+b x) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}-\frac{b^2}{2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}+\frac{3 b^2 e}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 64.6363, size = 272, normalized size = 0.99 \[ - \frac{6 b^{2} e^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (a + b x \right )}}{\left (a + b x\right ) \left (a e - b d\right )^{5}} + \frac{6 b^{2} e^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (d + e x \right )}}{\left (a + b x\right ) \left (a e - b d\right )^{5}} + \frac{6 b e^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{\left (d + e x\right ) \left (a e - b d\right )^{5}} - \frac{3 e^{2} \left (2 a + 2 b x\right )}{2 \left (d + e x\right )^{2} \left (a e - b d\right )^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} + \frac{2 e}{\left (d + e x\right )^{2} \left (a e - b d\right )^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} + \frac{2 a + 2 b x}{4 \left (d + e x\right )^{2} \left (a e - b d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} \]

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)**(3/2),x)

[Out]

-6*b**2*e**2*sqrt(a**2 + 2*a*b*x + b**2*x**2)*log(a + b*x)/((a + b*x)*(a*e - b*d
)**5) + 6*b**2*e**2*sqrt(a**2 + 2*a*b*x + b**2*x**2)*log(d + e*x)/((a + b*x)*(a*
e - b*d)**5) + 6*b*e**3*sqrt(a**2 + 2*a*b*x + b**2*x**2)/((d + e*x)*(a*e - b*d)*
*5) - 3*e**2*(2*a + 2*b*x)/(2*(d + e*x)**2*(a*e - b*d)**3*sqrt(a**2 + 2*a*b*x +
b**2*x**2)) + 2*e/((d + e*x)**2*(a*e - b*d)**2*sqrt(a**2 + 2*a*b*x + b**2*x**2))
+ (2*a + 2*b*x)/(4*(d + e*x)**2*(a*e - b*d)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)
)

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Mathematica [A]  time = 0.176027, size = 163, normalized size = 0.59 \[ \frac{(a+b x) \left (-12 b^2 e^2 (a+b x)^2 \log (d+e x)+6 b^2 e (a+b x) (b d-a e)+b^2 \left (-(b d-a e)^2\right )+12 b^2 e^2 (a+b x)^2 \log (a+b x)+\frac{6 b e^2 (a+b x)^2 (b d-a e)}{d+e x}+\frac{e^2 (a+b x)^2 (b d-a e)^2}{(d+e x)^2}\right )}{2 \left ((a+b x)^2\right )^{3/2} (b d-a e)^5} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x)*(-(b^2*(b*d - a*e)^2) + 6*b^2*e*(b*d - a*e)*(a + b*x) + (e^2*(b*d - a
*e)^2*(a + b*x)^2)/(d + e*x)^2 + (6*b*e^2*(b*d - a*e)*(a + b*x)^2)/(d + e*x) + 1
2*b^2*e^2*(a + b*x)^2*Log[a + b*x] - 12*b^2*e^2*(a + b*x)^2*Log[d + e*x]))/(2*(b
*d - a*e)^5*((a + b*x)^2)^(3/2))

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Maple [B]  time = 0.03, size = 508, normalized size = 1.8 \[ -{\frac{ \left ( 12\,\ln \left ( bx+a \right ){x}^{2}{a}^{2}{b}^{2}{e}^{4}+12\,\ln \left ( bx+a \right ){x}^{2}{b}^{4}{d}^{2}{e}^{2}+24\,\ln \left ( bx+a \right ){x}^{3}{b}^{4}d{e}^{3}-24\,\ln \left ( ex+d \right ){x}^{3}a{b}^{3}{e}^{4}-48\,\ln \left ( ex+d \right ){x}^{2}a{b}^{3}d{e}^{3}-24\,\ln \left ( ex+d \right ) x{a}^{2}{b}^{2}d{e}^{3}-24\,\ln \left ( ex+d \right ) xa{b}^{3}{d}^{2}{e}^{2}+48\,\ln \left ( bx+a \right ){x}^{2}a{b}^{3}d{e}^{3}-24\,\ln \left ( ex+d \right ){x}^{3}{b}^{4}d{e}^{3}+12\,\ln \left ( bx+a \right ){x}^{4}{b}^{4}{e}^{4}-12\,\ln \left ( ex+d \right ){x}^{4}{b}^{4}{e}^{4}-12\,{x}^{3}a{b}^{3}{e}^{4}+24\,\ln \left ( bx+a \right ) xa{b}^{3}{d}^{2}{e}^{2}-8\,{a}^{3}bd{e}^{3}-12\,\ln \left ( ex+d \right ){x}^{2}{b}^{4}{d}^{2}{e}^{2}+24\,\ln \left ( bx+a \right ){x}^{3}a{b}^{3}{e}^{4}-24\,x{a}^{2}{b}^{2}d{e}^{3}+24\,xa{b}^{3}{d}^{2}{e}^{2}+{a}^{4}{e}^{4}+4\,x{b}^{4}{d}^{3}e+12\,{x}^{3}{b}^{4}d{e}^{3}-18\,{x}^{2}{a}^{2}{b}^{2}{e}^{4}+18\,{x}^{2}{b}^{4}{d}^{2}{e}^{2}-4\,x{a}^{3}b{e}^{4}+24\,\ln \left ( bx+a \right ) x{a}^{2}{b}^{2}d{e}^{3}-12\,\ln \left ( ex+d \right ){a}^{2}{b}^{2}{d}^{2}{e}^{2}+12\,\ln \left ( bx+a \right ){a}^{2}{b}^{2}{d}^{2}{e}^{2}-{b}^{4}{d}^{4}+8\,a{b}^{3}{d}^{3}e-12\,\ln \left ( ex+d \right ){x}^{2}{a}^{2}{b}^{2}{e}^{4} \right ) \left ( bx+a \right ) }{2\, \left ( ex+d \right ) ^{2} \left ( ae-bd \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \]

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)^(3/2),x)

[Out]

-1/2*(12*ln(b*x+a)*x^2*a^2*b^2*e^4+12*ln(b*x+a)*x^2*b^4*d^2*e^2+24*ln(b*x+a)*x^3
*b^4*d*e^3-24*ln(e*x+d)*x^3*a*b^3*e^4-48*ln(e*x+d)*x^2*a*b^3*d*e^3-24*ln(e*x+d)*
x*a^2*b^2*d*e^3-24*ln(e*x+d)*x*a*b^3*d^2*e^2+48*ln(b*x+a)*x^2*a*b^3*d*e^3-24*ln(
e*x+d)*x^3*b^4*d*e^3+12*ln(b*x+a)*x^4*b^4*e^4-12*ln(e*x+d)*x^4*b^4*e^4-12*x^3*a*
b^3*e^4+24*ln(b*x+a)*x*a*b^3*d^2*e^2-8*a^3*b*d*e^3-12*ln(e*x+d)*x^2*b^4*d^2*e^2+
24*ln(b*x+a)*x^3*a*b^3*e^4-24*x*a^2*b^2*d*e^3+24*x*a*b^3*d^2*e^2+a^4*e^4+4*x*b^4
*d^3*e+12*x^3*b^4*d*e^3-18*x^2*a^2*b^2*e^4+18*x^2*b^4*d^2*e^2-4*x*a^3*b*e^4+24*l
n(b*x+a)*x*a^2*b^2*d*e^3-12*ln(e*x+d)*a^2*b^2*d^2*e^2+12*ln(b*x+a)*a^2*b^2*d^2*e
^2-b^4*d^4+8*a*b^3*d^3*e-12*ln(e*x+d)*x^2*a^2*b^2*e^4)*(b*x+a)/(e*x+d)^2/(a*e-b*
d)^5/((b*x+a)^2)^(3/2)

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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)^(3/2)*(e*x + d)^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.227974, size = 1026, normalized size = 3.72 \[ -\frac{b^{4} d^{4} - 8 \, a b^{3} d^{3} e + 8 \, a^{3} b d e^{3} - a^{4} e^{4} - 12 \,{\left (b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} - 18 \,{\left (b^{4} d^{2} e^{2} - a^{2} b^{2} e^{4}\right )} x^{2} - 4 \,{\left (b^{4} d^{3} e + 6 \, a b^{3} d^{2} e^{2} - 6 \, a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x - 12 \,{\left (b^{4} e^{4} x^{4} + a^{2} b^{2} d^{2} e^{2} + 2 \,{\left (b^{4} d e^{3} + a b^{3} e^{4}\right )} x^{3} +{\left (b^{4} d^{2} e^{2} + 4 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 2 \,{\left (a b^{3} d^{2} e^{2} + a^{2} b^{2} d e^{3}\right )} x\right )} \log \left (b x + a\right ) + 12 \,{\left (b^{4} e^{4} x^{4} + a^{2} b^{2} d^{2} e^{2} + 2 \,{\left (b^{4} d e^{3} + a b^{3} e^{4}\right )} x^{3} +{\left (b^{4} d^{2} e^{2} + 4 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 2 \,{\left (a b^{3} d^{2} e^{2} + a^{2} b^{2} d e^{3}\right )} x\right )} \log \left (e x + d\right )}{2 \,{\left (a^{2} b^{5} d^{7} - 5 \, a^{3} b^{4} d^{6} e + 10 \, a^{4} b^{3} d^{5} e^{2} - 10 \, a^{5} b^{2} d^{4} e^{3} + 5 \, a^{6} b d^{3} e^{4} - a^{7} d^{2} e^{5} +{\left (b^{7} d^{5} e^{2} - 5 \, a b^{6} d^{4} e^{3} + 10 \, a^{2} b^{5} d^{3} e^{4} - 10 \, a^{3} b^{4} d^{2} e^{5} + 5 \, a^{4} b^{3} d e^{6} - a^{5} b^{2} e^{7}\right )} x^{4} + 2 \,{\left (b^{7} d^{6} e - 4 \, a b^{6} d^{5} e^{2} + 5 \, a^{2} b^{5} d^{4} e^{3} - 5 \, a^{4} b^{3} d^{2} e^{5} + 4 \, a^{5} b^{2} d e^{6} - a^{6} b e^{7}\right )} x^{3} +{\left (b^{7} d^{7} - a b^{6} d^{6} e - 9 \, a^{2} b^{5} d^{5} e^{2} + 25 \, a^{3} b^{4} d^{4} e^{3} - 25 \, a^{4} b^{3} d^{3} e^{4} + 9 \, a^{5} b^{2} d^{2} e^{5} + a^{6} b d e^{6} - a^{7} e^{7}\right )} x^{2} + 2 \,{\left (a b^{6} d^{7} - 4 \, a^{2} b^{5} d^{6} e + 5 \, a^{3} b^{4} d^{5} e^{2} - 5 \, a^{5} b^{2} d^{3} e^{4} + 4 \, a^{6} b d^{2} e^{5} - a^{7} d e^{6}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(b^4*d^4 - 8*a*b^3*d^3*e + 8*a^3*b*d*e^3 - a^4*e^4 - 12*(b^4*d*e^3 - a*b^3*
e^4)*x^3 - 18*(b^4*d^2*e^2 - a^2*b^2*e^4)*x^2 - 4*(b^4*d^3*e + 6*a*b^3*d^2*e^2 -
6*a^2*b^2*d*e^3 - a^3*b*e^4)*x - 12*(b^4*e^4*x^4 + a^2*b^2*d^2*e^2 + 2*(b^4*d*e
^3 + a*b^3*e^4)*x^3 + (b^4*d^2*e^2 + 4*a*b^3*d*e^3 + a^2*b^2*e^4)*x^2 + 2*(a*b^3
*d^2*e^2 + a^2*b^2*d*e^3)*x)*log(b*x + a) + 12*(b^4*e^4*x^4 + a^2*b^2*d^2*e^2 +
2*(b^4*d*e^3 + a*b^3*e^4)*x^3 + (b^4*d^2*e^2 + 4*a*b^3*d*e^3 + a^2*b^2*e^4)*x^2
+ 2*(a*b^3*d^2*e^2 + a^2*b^2*d*e^3)*x)*log(e*x + d))/(a^2*b^5*d^7 - 5*a^3*b^4*d^
6*e + 10*a^4*b^3*d^5*e^2 - 10*a^5*b^2*d^4*e^3 + 5*a^6*b*d^3*e^4 - a^7*d^2*e^5 +
(b^7*d^5*e^2 - 5*a*b^6*d^4*e^3 + 10*a^2*b^5*d^3*e^4 - 10*a^3*b^4*d^2*e^5 + 5*a^4
*b^3*d*e^6 - a^5*b^2*e^7)*x^4 + 2*(b^7*d^6*e - 4*a*b^6*d^5*e^2 + 5*a^2*b^5*d^4*e
^3 - 5*a^4*b^3*d^2*e^5 + 4*a^5*b^2*d*e^6 - a^6*b*e^7)*x^3 + (b^7*d^7 - a*b^6*d^6
*e - 9*a^2*b^5*d^5*e^2 + 25*a^3*b^4*d^4*e^3 - 25*a^4*b^3*d^3*e^4 + 9*a^5*b^2*d^2
*e^5 + a^6*b*d*e^6 - a^7*e^7)*x^2 + 2*(a*b^6*d^7 - 4*a^2*b^5*d^6*e + 5*a^3*b^4*d
^5*e^2 - 5*a^5*b^2*d^3*e^4 + 4*a^6*b*d^2*e^5 - a^7*d*e^6)*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)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.593425, 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)^(3/2)*(e*x + d)^3),x, algorithm="giac")

[Out]

sage0*x