3.191 \(\int \frac{x^3}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=133 \[ -\frac{3 a^2}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 a (a+b x) \log (a+b x)}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{x (a+b x)}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{a^3}{2 b^4 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.161543, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{3 a^2}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 a (a+b x) \log (a+b x)}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{x (a+b x)}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{a^3}{2 b^4 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 15.8666, size = 129, normalized size = 0.97 \[ - \frac{3 a \left (a + b x\right ) \log{\left (a + b x \right )}}{b^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} - \frac{x^{3} \left (2 a + 2 b x\right )}{4 b \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} - \frac{3 x^{2}}{2 b^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} + \frac{3 \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0417018, size = 71, normalized size = 0.53 \[ \frac{-5 a^3-4 a^2 b x+4 a b^2 x^2-6 a (a+b x)^2 \log (a+b x)+2 b^3 x^3}{2 b^4 (a+b x) \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.008, size = 89, normalized size = 0.7 \[ -{\frac{ \left ( 6\,\ln \left ( bx+a \right ){x}^{2}a{b}^{2}-2\,{b}^{3}{x}^{3}+12\,\ln \left ( bx+a \right ) x{a}^{2}b-4\,a{b}^{2}{x}^{2}+6\,{a}^{3}\ln \left ( bx+a \right ) +4\,{a}^{2}bx+5\,{a}^{3} \right ) \left ( bx+a \right ) }{2\,{b}^{4}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(6*ln(b*x+a)*x^2*a*b^2-2*b^3*x^3+12*ln(b*x+a)*x*a^2*b-4*a*b^2*x^2+6*a^3*ln(
b*x+a)+4*a^2*b*x+5*a^3)*(b*x+a)/b^4/((b*x+a)^2)^(3/2)

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Maxima [A]  time = 0.720522, size = 180, normalized size = 1.35 \[ \frac{x^{2}}{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}} - \frac{3 \, a \log \left (x + \frac{a}{b}\right )}{{\left (b^{2}\right )}^{\frac{3}{2}} b} - \frac{9 \, a^{3} b}{2 \,{\left (b^{2}\right )}^{\frac{7}{2}}{\left (x + \frac{a}{b}\right )}^{2}} - \frac{6 \, a^{2} x}{{\left (b^{2}\right )}^{\frac{5}{2}}{\left (x + \frac{a}{b}\right )}^{2}} + \frac{2 \, a^{2}}{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} b^{4}} - \frac{a^{3}}{{\left (b^{2}\right )}^{\frac{3}{2}} b^{3}{\left (x + \frac{a}{b}\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x^2/(sqrt(b^2*x^2 + 2*a*b*x + a^2)*b^2) - 3*a*log(x + a/b)/((b^2)^(3/2)*b) - 9/2
*a^3*b/((b^2)^(7/2)*(x + a/b)^2) - 6*a^2*x/((b^2)^(5/2)*(x + a/b)^2) + 2*a^2/(sq
rt(b^2*x^2 + 2*a*b*x + a^2)*b^4) - a^3/((b^2)^(3/2)*b^3*(x + a/b)^2)

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Fricas [A]  time = 0.22291, size = 112, normalized size = 0.84 \[ \frac{2 \, b^{3} x^{3} + 4 \, a b^{2} x^{2} - 4 \, a^{2} b x - 5 \, a^{3} - 6 \,{\left (a b^{2} x^{2} + 2 \, a^{2} b x + a^{3}\right )} \log \left (b x + a\right )}{2 \,{\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(2*b^3*x^3 + 4*a*b^2*x^2 - 4*a^2*b*x - 5*a^3 - 6*(a*b^2*x^2 + 2*a^2*b*x + a^
3)*log(b*x + a))/(b^6*x^2 + 2*a*b^5*x + a^2*b^4)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{\left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**3/((a + b*x)**2)**(3/2), x)

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GIAC/XCAS [A]  time = 0.563295, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x