3.82 \(\int \frac{x^{15/2}}{\left (a x+b x^3\right )^{9/2}} \, dx\)

Optimal. Leaf size=51 \[ -\frac{2 x^{5/2}}{35 b^2 \left (a x+b x^3\right )^{5/2}}-\frac{x^{11/2}}{7 b \left (a x+b x^3\right )^{7/2}} \]

[Out]

-x^(11/2)/(7*b*(a*x + b*x^3)^(7/2)) - (2*x^(5/2))/(35*b^2*(a*x + b*x^3)^(5/2))

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Rubi [A]  time = 0.121487, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ -\frac{2 x^{5/2}}{35 b^2 \left (a x+b x^3\right )^{5/2}}-\frac{x^{11/2}}{7 b \left (a x+b x^3\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]  Int[x^(15/2)/(a*x + b*x^3)^(9/2),x]

[Out]

-x^(11/2)/(7*b*(a*x + b*x^3)^(7/2)) - (2*x^(5/2))/(35*b^2*(a*x + b*x^3)^(5/2))

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Rubi in Sympy [A]  time = 12.4103, size = 44, normalized size = 0.86 \[ - \frac{x^{\frac{11}{2}}}{7 b \left (a x + b x^{3}\right )^{\frac{7}{2}}} - \frac{2 x^{\frac{5}{2}}}{35 b^{2} \left (a x + b x^{3}\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(15/2)/(b*x**3+a*x)**(9/2),x)

[Out]

-x**(11/2)/(7*b*(a*x + b*x**3)**(7/2)) - 2*x**(5/2)/(35*b**2*(a*x + b*x**3)**(5/
2))

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Mathematica [A]  time = 0.0338084, size = 44, normalized size = 0.86 \[ -\frac{\sqrt{x} \left (2 a+7 b x^2\right )}{35 b^2 \left (a+b x^2\right )^3 \sqrt{x \left (a+b x^2\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(15/2)/(a*x + b*x^3)^(9/2),x]

[Out]

-(Sqrt[x]*(2*a + 7*b*x^2))/(35*b^2*(a + b*x^2)^3*Sqrt[x*(a + b*x^2)])

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Maple [A]  time = 0.007, size = 37, normalized size = 0.7 \[ -{\frac{ \left ( b{x}^{2}+a \right ) \left ( 7\,b{x}^{2}+2\,a \right ) }{35\,{b}^{2}}{x}^{{\frac{9}{2}}} \left ( b{x}^{3}+ax \right ) ^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(15/2)/(b*x^3+a*x)^(9/2),x)

[Out]

-1/35*(b*x^2+a)*(7*b*x^2+2*a)*x^(9/2)/b^2/(b*x^3+a*x)^(9/2)

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Maxima [A]  time = 1.50049, size = 32, normalized size = 0.63 \[ -\frac{7 \, b x^{2} + 2 \, a}{35 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(15/2)/(b*x^3 + a*x)^(9/2),x, algorithm="maxima")

[Out]

-1/35*(7*b*x^2 + 2*a)/((b*x^2 + a)^(7/2)*b^2)

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Fricas [A]  time = 0.21041, size = 86, normalized size = 1.69 \[ -\frac{7 \, b x^{3} + 2 \, a x}{35 \,{\left (b^{5} x^{6} + 3 \, a b^{4} x^{4} + 3 \, a^{2} b^{3} x^{2} + a^{3} b^{2}\right )} \sqrt{b x^{3} + a x} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(15/2)/(b*x^3 + a*x)^(9/2),x, algorithm="fricas")

[Out]

-1/35*(7*b*x^3 + 2*a*x)/((b^5*x^6 + 3*a*b^4*x^4 + 3*a^2*b^3*x^2 + a^3*b^2)*sqrt(
b*x^3 + a*x)*sqrt(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(x**(15/2)/(b*x**3+a*x)**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.229927, size = 45, normalized size = 0.88 \[ -\frac{7 \, b x^{2} + 2 \, a}{35 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b^{2}} + \frac{2}{35 \, a^{\frac{5}{2}} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(15/2)/(b*x^3 + a*x)^(9/2),x, algorithm="giac")

[Out]

-1/35*(7*b*x^2 + 2*a)/((b*x^2 + a)^(7/2)*b^2) + 2/35/(a^(5/2)*b^2)