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

Optimal. Leaf size=106 \[ \frac{256 b^3 (a+2 b x)}{21 a^6 \sqrt{a x+b x^2}}-\frac{32 b^2 (a+2 b x)}{21 a^4 \left (a x+b x^2\right )^{3/2}}+\frac{4 b}{7 a^2 x \left (a x+b x^2\right )^{3/2}}-\frac{2}{7 a x^2 \left (a x+b x^2\right )^{3/2}} \]

[Out]

-2/(7*a*x^2*(a*x + b*x^2)^(3/2)) + (4*b)/(7*a^2*x*(a*x + b*x^2)^(3/2)) - (32*b^2
*(a + 2*b*x))/(21*a^4*(a*x + b*x^2)^(3/2)) + (256*b^3*(a + 2*b*x))/(21*a^6*Sqrt[
a*x + b*x^2])

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Rubi [A]  time = 0.114943, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{256 b^3 (a+2 b x)}{21 a^6 \sqrt{a x+b x^2}}-\frac{32 b^2 (a+2 b x)}{21 a^4 \left (a x+b x^2\right )^{3/2}}+\frac{4 b}{7 a^2 x \left (a x+b x^2\right )^{3/2}}-\frac{2}{7 a x^2 \left (a x+b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

-2/(7*a*x^2*(a*x + b*x^2)^(3/2)) + (4*b)/(7*a^2*x*(a*x + b*x^2)^(3/2)) - (32*b^2
*(a + 2*b*x))/(21*a^4*(a*x + b*x^2)^(3/2)) + (256*b^3*(a + 2*b*x))/(21*a^6*Sqrt[
a*x + b*x^2])

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Rubi in Sympy [A]  time = 11.8109, size = 100, normalized size = 0.94 \[ - \frac{2}{7 a x^{2} \left (a x + b x^{2}\right )^{\frac{3}{2}}} + \frac{4 b}{7 a^{2} x \left (a x + b x^{2}\right )^{\frac{3}{2}}} - \frac{32 b^{2} \left (a + 2 b x\right )}{21 a^{4} \left (a x + b x^{2}\right )^{\frac{3}{2}}} + \frac{128 b^{3} \left (2 a + 4 b x\right )}{21 a^{6} \sqrt{a x + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/(7*a*x**2*(a*x + b*x**2)**(3/2)) + 4*b/(7*a**2*x*(a*x + b*x**2)**(3/2)) - 32*
b**2*(a + 2*b*x)/(21*a**4*(a*x + b*x**2)**(3/2)) + 128*b**3*(2*a + 4*b*x)/(21*a*
*6*sqrt(a*x + b*x**2))

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Mathematica [A]  time = 0.0600019, size = 73, normalized size = 0.69 \[ \frac{2 \left (-3 a^5+6 a^4 b x-16 a^3 b^2 x^2+96 a^2 b^3 x^3+384 a b^4 x^4+256 b^5 x^5\right )}{21 a^6 x^2 (x (a+b x))^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(-3*a^5 + 6*a^4*b*x - 16*a^3*b^2*x^2 + 96*a^2*b^3*x^3 + 384*a*b^4*x^4 + 256*b
^5*x^5))/(21*a^6*x^2*(x*(a + b*x))^(3/2))

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Maple [A]  time = 0.008, size = 77, normalized size = 0.7 \[ -{\frac{ \left ( 2\,bx+2\,a \right ) \left ( -256\,{b}^{5}{x}^{5}-384\,{b}^{4}{x}^{4}a-96\,{b}^{3}{x}^{3}{a}^{2}+16\,{b}^{2}{x}^{2}{a}^{3}-6\,bx{a}^{4}+3\,{a}^{5} \right ) }{21\,x{a}^{6}} \left ( b{x}^{2}+ax \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(b*x^2+a*x)^(5/2),x)

[Out]

-2/21*(b*x+a)*(-256*b^5*x^5-384*a*b^4*x^4-96*a^2*b^3*x^3+16*a^3*b^2*x^2-6*a^4*b*
x+3*a^5)/x/a^6/(b*x^2+a*x)^(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*x^2 + a*x)^(5/2)*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.221299, size = 112, normalized size = 1.06 \[ \frac{2 \,{\left (256 \, b^{5} x^{5} + 384 \, a b^{4} x^{4} + 96 \, a^{2} b^{3} x^{3} - 16 \, a^{3} b^{2} x^{2} + 6 \, a^{4} b x - 3 \, a^{5}\right )}}{21 \,{\left (a^{6} b x^{4} + a^{7} x^{3}\right )} \sqrt{b x^{2} + a x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/21*(256*b^5*x^5 + 384*a*b^4*x^4 + 96*a^2*b^3*x^3 - 16*a^3*b^2*x^2 + 6*a^4*b*x
- 3*a^5)/((a^6*b*x^4 + a^7*x^3)*sqrt(b*x^2 + a*x))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{2} \left (x \left (a + b x\right )\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**2/(b*x**2+a*x)**(5/2),x)

[Out]

Integral(1/(x**2*(x*(a + b*x))**(5/2)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a x\right )}^{\frac{5}{2}} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/((b*x^2 + a*x)^(5/2)*x^2), x)