3.81 \(\int \frac{x^{17/2}}{\left (a x+b x^3\right )^{9/2}} \, dx\)

Optimal. Leaf size=51 \[ \frac{2 x^{15/2}}{35 a^2 \left (a x+b x^3\right )^{5/2}}+\frac{x^{17/2}}{7 a \left (a x+b x^3\right )^{7/2}} \]

[Out]

x^(17/2)/(7*a*(a*x + b*x^3)^(7/2)) + (2*x^(15/2))/(35*a^2*(a*x + b*x^3)^(5/2))

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Rubi [A]  time = 0.120685, 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 \[ \frac{2 x^{15/2}}{35 a^2 \left (a x+b x^3\right )^{5/2}}+\frac{x^{17/2}}{7 a \left (a x+b x^3\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]  Int[x^(17/2)/(a*x + b*x^3)^(9/2),x]

[Out]

x^(17/2)/(7*a*(a*x + b*x^3)^(7/2)) + (2*x^(15/2))/(35*a^2*(a*x + b*x^3)^(5/2))

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Rubi in Sympy [A]  time = 12.2469, size = 42, normalized size = 0.82 \[ \frac{x^{\frac{17}{2}}}{7 a \left (a x + b x^{3}\right )^{\frac{7}{2}}} + \frac{2 x^{\frac{15}{2}}}{35 a^{2} \left (a x + b x^{3}\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(17/2)/(b*x**3+a*x)**(9/2),x)

[Out]

x**(17/2)/(7*a*(a*x + b*x**3)**(7/2)) + 2*x**(15/2)/(35*a**2*(a*x + b*x**3)**(5/
2))

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Mathematica [A]  time = 0.0456865, size = 44, normalized size = 0.86 \[ \frac{x^{9/2} \sqrt{x \left (a+b x^2\right )} \left (7 a+2 b x^2\right )}{35 a^2 \left (a+b x^2\right )^4} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(17/2)/(a*x + b*x^3)^(9/2),x]

[Out]

(x^(9/2)*Sqrt[x*(a + b*x^2)]*(7*a + 2*b*x^2))/(35*a^2*(a + b*x^2)^4)

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Maple [A]  time = 0.008, size = 37, normalized size = 0.7 \[{\frac{ \left ( b{x}^{2}+a \right ) \left ( 2\,b{x}^{2}+7\,a \right ) }{35\,{a}^{2}}{x}^{{\frac{19}{2}}} \left ( b{x}^{3}+ax \right ) ^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(17/2)/(b*x^3+a*x)^(9/2),x)

[Out]

1/35*(b*x^2+a)*x^(19/2)*(2*b*x^2+7*a)/a^2/(b*x^3+a*x)^(9/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{\frac{17}{2}}}{{\left (b x^{3} + a x\right )}^{\frac{9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(17/2)/(b*x^3 + a*x)^(9/2),x, algorithm="maxima")

[Out]

integrate(x^(17/2)/(b*x^3 + a*x)^(9/2), x)

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Fricas [A]  time = 0.210888, size = 103, normalized size = 2.02 \[ \frac{{\left (2 \, b x^{6} + 7 \, a x^{4}\right )} \sqrt{b x^{3} + a x} \sqrt{x}}{35 \,{\left (a^{2} b^{4} x^{8} + 4 \, a^{3} b^{3} x^{6} + 6 \, a^{4} b^{2} x^{4} + 4 \, a^{5} b x^{2} + a^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(17/2)/(b*x^3 + a*x)^(9/2),x, algorithm="fricas")

[Out]

1/35*(2*b*x^6 + 7*a*x^4)*sqrt(b*x^3 + a*x)*sqrt(x)/(a^2*b^4*x^8 + 4*a^3*b^3*x^6
+ 6*a^4*b^2*x^4 + 4*a^5*b*x^2 + a^6)

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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**(17/2)/(b*x**3+a*x)**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.234793, size = 39, normalized size = 0.76 \[ \frac{x^{5}{\left (\frac{2 \, b x^{2}}{a^{2}} + \frac{7}{a}\right )}}{35 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(17/2)/(b*x^3 + a*x)^(9/2),x, algorithm="giac")

[Out]

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