3.80 \(\int \frac{x^{19/2}}{\left (a x+b x^3\right )^{9/2}} \, dx\)

Optimal. Leaf size=76 \[ -\frac{8 x^{3/2}}{105 b^3 \left (a x+b x^3\right )^{3/2}}-\frac{4 x^{9/2}}{35 b^2 \left (a x+b x^3\right )^{5/2}}-\frac{x^{15/2}}{7 b \left (a x+b x^3\right )^{7/2}} \]

[Out]

-x^(15/2)/(7*b*(a*x + b*x^3)^(7/2)) - (4*x^(9/2))/(35*b^2*(a*x + b*x^3)^(5/2)) -
(8*x^(3/2))/(105*b^3*(a*x + b*x^3)^(3/2))

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Rubi [A]  time = 0.182234, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ -\frac{8 x^{3/2}}{105 b^3 \left (a x+b x^3\right )^{3/2}}-\frac{4 x^{9/2}}{35 b^2 \left (a x+b x^3\right )^{5/2}}-\frac{x^{15/2}}{7 b \left (a x+b x^3\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]  Int[x^(19/2)/(a*x + b*x^3)^(9/2),x]

[Out]

-x^(15/2)/(7*b*(a*x + b*x^3)^(7/2)) - (4*x^(9/2))/(35*b^2*(a*x + b*x^3)^(5/2)) -
(8*x^(3/2))/(105*b^3*(a*x + b*x^3)^(3/2))

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Rubi in Sympy [A]  time = 18.5827, size = 68, normalized size = 0.89 \[ - \frac{x^{\frac{15}{2}}}{7 b \left (a x + b x^{3}\right )^{\frac{7}{2}}} - \frac{4 x^{\frac{9}{2}}}{35 b^{2} \left (a x + b x^{3}\right )^{\frac{5}{2}}} - \frac{8 x^{\frac{3}{2}}}{105 b^{3} \left (a x + b x^{3}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(19/2)/(b*x**3+a*x)**(9/2),x)

[Out]

-x**(15/2)/(7*b*(a*x + b*x**3)**(7/2)) - 4*x**(9/2)/(35*b**2*(a*x + b*x**3)**(5/
2)) - 8*x**(3/2)/(105*b**3*(a*x + b*x**3)**(3/2))

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Mathematica [A]  time = 0.042192, size = 55, normalized size = 0.72 \[ -\frac{\sqrt{x} \left (8 a^2+28 a b x^2+35 b^2 x^4\right )}{105 b^3 \left (a+b x^2\right )^3 \sqrt{x \left (a+b x^2\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(19/2)/(a*x + b*x^3)^(9/2),x]

[Out]

-(Sqrt[x]*(8*a^2 + 28*a*b*x^2 + 35*b^2*x^4))/(105*b^3*(a + b*x^2)^3*Sqrt[x*(a +
b*x^2)])

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Maple [A]  time = 0.008, size = 48, normalized size = 0.6 \[ -{\frac{ \left ( b{x}^{2}+a \right ) \left ( 35\,{x}^{4}{b}^{2}+28\,a{x}^{2}b+8\,{a}^{2} \right ) }{105\,{b}^{3}}{x}^{{\frac{9}{2}}} \left ( b{x}^{3}+ax \right ) ^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(19/2)/(b*x^3+a*x)^(9/2),x)

[Out]

-1/105*(b*x^2+a)*(35*b^2*x^4+28*a*b*x^2+8*a^2)*x^(9/2)/b^3/(b*x^3+a*x)^(9/2)

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Maxima [A]  time = 1.49293, size = 55, normalized size = 0.72 \[ -\frac{35 \,{\left (b x^{2} + a\right )}^{2} - 42 \,{\left (b x^{2} + a\right )} a + 15 \, a^{2}}{105 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(19/2)/(b*x^3 + a*x)^(9/2),x, algorithm="maxima")

[Out]

-1/105*(35*(b*x^2 + a)^2 - 42*(b*x^2 + a)*a + 15*a^2)/((b*x^2 + a)^(7/2)*b^3)

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Fricas [A]  time = 0.210718, size = 101, normalized size = 1.33 \[ -\frac{35 \, b^{2} x^{5} + 28 \, a b x^{3} + 8 \, a^{2} x}{105 \,{\left (b^{6} x^{6} + 3 \, a b^{5} x^{4} + 3 \, a^{2} b^{4} x^{2} + a^{3} b^{3}\right )} \sqrt{b x^{3} + a x} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(19/2)/(b*x^3 + a*x)^(9/2),x, algorithm="fricas")

[Out]

-1/105*(35*b^2*x^5 + 28*a*b*x^3 + 8*a^2*x)/((b^6*x^6 + 3*a*b^5*x^4 + 3*a^2*b^4*x
^2 + a^3*b^3)*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**(19/2)/(b*x**3+a*x)**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.229833, size = 68, normalized size = 0.89 \[ \frac{8}{105 \, a^{\frac{3}{2}} b^{3}} - \frac{35 \,{\left (b x^{2} + a\right )}^{2} - 42 \,{\left (b x^{2} + a\right )} a + 15 \, a^{2}}{105 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(19/2)/(b*x^3 + a*x)^(9/2),x, algorithm="giac")

[Out]

8/105/(a^(3/2)*b^3) - 1/105*(35*(b*x^2 + a)^2 - 42*(b*x^2 + a)*a + 15*a^2)/((b*x
^2 + a)^(7/2)*b^3)