3.1583 \(\int \frac{1}{(d+e x)^3 \sqrt{a^2+2 a b x+b^2 x^2}} \, dx\)

Optimal. Leaf size=182 \[ \frac{b (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^2}+\frac{a+b x}{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)}+\frac{b^2 (a+b x) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac{b^2 (a+b x) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3} \]

[Out]

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

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Rubi [A]  time = 0.216585, antiderivative size = 182, 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 \[ \frac{b (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^2}+\frac{a+b x}{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)}+\frac{b^2 (a+b x) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac{b^2 (a+b x) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 43.4718, size = 168, normalized size = 0.92 \[ - \frac{b^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (a + b x \right )}}{\left (a + b x\right ) \left (a e - b d\right )^{3}} + \frac{b^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (d + e x \right )}}{\left (a + b x\right ) \left (a e - b d\right )^{3}} + \frac{b e \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{\left (d + e x\right ) \left (a e - b d\right )^{3}} - \frac{2 a + 2 b x}{4 \left (d + e x\right )^{2} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.122721, size = 97, normalized size = 0.53 \[ \frac{(a+b x) \left (2 b^2 (d+e x)^2 \log (a+b x)+(b d-a e) (-a e+3 b d+2 b e x)-2 b^2 (d+e x)^2 \log (d+e x)\right )}{2 \sqrt{(a+b x)^2} (d+e x)^2 (b d-a e)^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.021, size = 162, normalized size = 0.9 \[ -{\frac{ \left ( bx+a \right ) \left ( 2\,\ln \left ( bx+a \right ){x}^{2}{b}^{2}{e}^{2}-2\,\ln \left ( ex+d \right ){x}^{2}{b}^{2}{e}^{2}+4\,\ln \left ( bx+a \right ) x{b}^{2}de-4\,\ln \left ( ex+d \right ) x{b}^{2}de+2\,\ln \left ( bx+a \right ){b}^{2}{d}^{2}-2\,\ln \left ( ex+d \right ){b}^{2}{d}^{2}-2\,xab{e}^{2}+2\,x{b}^{2}de+{a}^{2}{e}^{2}-4\,abde+3\,{b}^{2}{d}^{2} \right ) }{2\, \left ( ae-bd \right ) ^{3} \left ( ex+d \right ) ^{2}}{\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(b*x+a)*(2*ln(b*x+a)*x^2*b^2*e^2-2*ln(e*x+d)*x^2*b^2*e^2+4*ln(b*x+a)*x*b^2*
d*e-4*ln(e*x+d)*x*b^2*d*e+2*ln(b*x+a)*b^2*d^2-2*ln(e*x+d)*b^2*d^2-2*x*a*b*e^2+2*
x*b^2*d*e+a^2*e^2-4*a*b*d*e+3*b^2*d^2)/((b*x+a)^2)^(1/2)/(a*e-b*d)^3/(e*x+d)^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/(sqrt((b*x + a)^2)*(e*x + d)^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.21623, size = 327, normalized size = 1.8 \[ \frac{3 \, b^{2} d^{2} - 4 \, a b d e + a^{2} e^{2} + 2 \,{\left (b^{2} d e - a b e^{2}\right )} x + 2 \,{\left (b^{2} e^{2} x^{2} + 2 \, b^{2} d e x + b^{2} d^{2}\right )} \log \left (b x + a\right ) - 2 \,{\left (b^{2} e^{2} x^{2} + 2 \, b^{2} d e x + b^{2} d^{2}\right )} \log \left (e x + d\right )}{2 \,{\left (b^{3} d^{5} - 3 \, a b^{2} d^{4} e + 3 \, a^{2} b d^{3} e^{2} - a^{3} d^{2} e^{3} +{\left (b^{3} d^{3} e^{2} - 3 \, a b^{2} d^{2} e^{3} + 3 \, a^{2} b d e^{4} - a^{3} e^{5}\right )} x^{2} + 2 \,{\left (b^{3} d^{4} e - 3 \, a b^{2} d^{3} e^{2} + 3 \, a^{2} b d^{2} e^{3} - a^{3} d e^{4}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 4.79151, size = 381, normalized size = 2.09 \[ \frac{b^{2} \log{\left (x + \frac{- \frac{a^{4} b^{2} e^{4}}{\left (a e - b d\right )^{3}} + \frac{4 a^{3} b^{3} d e^{3}}{\left (a e - b d\right )^{3}} - \frac{6 a^{2} b^{4} d^{2} e^{2}}{\left (a e - b d\right )^{3}} + \frac{4 a b^{5} d^{3} e}{\left (a e - b d\right )^{3}} + a b^{2} e - \frac{b^{6} d^{4}}{\left (a e - b d\right )^{3}} + b^{3} d}{2 b^{3} e} \right )}}{\left (a e - b d\right )^{3}} - \frac{b^{2} \log{\left (x + \frac{\frac{a^{4} b^{2} e^{4}}{\left (a e - b d\right )^{3}} - \frac{4 a^{3} b^{3} d e^{3}}{\left (a e - b d\right )^{3}} + \frac{6 a^{2} b^{4} d^{2} e^{2}}{\left (a e - b d\right )^{3}} - \frac{4 a b^{5} d^{3} e}{\left (a e - b d\right )^{3}} + a b^{2} e + \frac{b^{6} d^{4}}{\left (a e - b d\right )^{3}} + b^{3} d}{2 b^{3} e} \right )}}{\left (a e - b d\right )^{3}} + \frac{- a e + 3 b d + 2 b e x}{2 a^{2} d^{2} e^{2} - 4 a b d^{3} e + 2 b^{2} d^{4} + x^{2} \left (2 a^{2} e^{4} - 4 a b d e^{3} + 2 b^{2} d^{2} e^{2}\right ) + x \left (4 a^{2} d e^{3} - 8 a b d^{2} e^{2} + 4 b^{2} d^{3} e\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b**2*log(x + (-a**4*b**2*e**4/(a*e - b*d)**3 + 4*a**3*b**3*d*e**3/(a*e - b*d)**3
- 6*a**2*b**4*d**2*e**2/(a*e - b*d)**3 + 4*a*b**5*d**3*e/(a*e - b*d)**3 + a*b**
2*e - b**6*d**4/(a*e - b*d)**3 + b**3*d)/(2*b**3*e))/(a*e - b*d)**3 - b**2*log(x
+ (a**4*b**2*e**4/(a*e - b*d)**3 - 4*a**3*b**3*d*e**3/(a*e - b*d)**3 + 6*a**2*b
**4*d**2*e**2/(a*e - b*d)**3 - 4*a*b**5*d**3*e/(a*e - b*d)**3 + a*b**2*e + b**6*
d**4/(a*e - b*d)**3 + b**3*d)/(2*b**3*e))/(a*e - b*d)**3 + (-a*e + 3*b*d + 2*b*e
*x)/(2*a**2*d**2*e**2 - 4*a*b*d**3*e + 2*b**2*d**4 + x**2*(2*a**2*e**4 - 4*a*b*d
*e**3 + 2*b**2*d**2*e**2) + x*(4*a**2*d*e**3 - 8*a*b*d**2*e**2 + 4*b**2*d**3*e))

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GIAC/XCAS [A]  time = 0.215969, size = 235, normalized size = 1.29 \[ \frac{1}{2} \,{\left (\frac{2 \, b^{3}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{4} d^{3} - 3 \, a b^{3} d^{2} e + 3 \, a^{2} b^{2} d e^{2} - a^{3} b e^{3}} - \frac{2 \, b^{2} e{\rm ln}\left ({\left | x e + d \right |}\right )}{b^{3} d^{3} e - 3 \, a b^{2} d^{2} e^{2} + 3 \, a^{2} b d e^{3} - a^{3} e^{4}} + \frac{3 \, b^{2} d^{2} - 4 \, a b d e + a^{2} e^{2} + 2 \,{\left (b^{2} d e - a b e^{2}\right )} x}{{\left (b d - a e\right )}^{3}{\left (x e + d\right )}^{2}}\right )}{\rm sign}\left (b x + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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