3.78 \(\int \frac{x^{23/2}}{\left (a x+b x^3\right )^{9/2}} \, dx\)

Optimal. Leaf size=101 \[ -\frac{16 \sqrt{x}}{35 b^4 \sqrt{a x+b x^3}}-\frac{8 x^{7/2}}{35 b^3 \left (a x+b x^3\right )^{3/2}}-\frac{6 x^{13/2}}{35 b^2 \left (a x+b x^3\right )^{5/2}}-\frac{x^{19/2}}{7 b \left (a x+b x^3\right )^{7/2}} \]

[Out]

-x^(19/2)/(7*b*(a*x + b*x^3)^(7/2)) - (6*x^(13/2))/(35*b^2*(a*x + b*x^3)^(5/2))
- (8*x^(7/2))/(35*b^3*(a*x + b*x^3)^(3/2)) - (16*Sqrt[x])/(35*b^4*Sqrt[a*x + b*x
^3])

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Rubi [A]  time = 0.249992, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ -\frac{16 \sqrt{x}}{35 b^4 \sqrt{a x+b x^3}}-\frac{8 x^{7/2}}{35 b^3 \left (a x+b x^3\right )^{3/2}}-\frac{6 x^{13/2}}{35 b^2 \left (a x+b x^3\right )^{5/2}}-\frac{x^{19/2}}{7 b \left (a x+b x^3\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]  Int[x^(23/2)/(a*x + b*x^3)^(9/2),x]

[Out]

-x^(19/2)/(7*b*(a*x + b*x^3)^(7/2)) - (6*x^(13/2))/(35*b^2*(a*x + b*x^3)^(5/2))
- (8*x^(7/2))/(35*b^3*(a*x + b*x^3)^(3/2)) - (16*Sqrt[x])/(35*b^4*Sqrt[a*x + b*x
^3])

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Rubi in Sympy [A]  time = 25.1243, size = 92, normalized size = 0.91 \[ - \frac{x^{\frac{19}{2}}}{7 b \left (a x + b x^{3}\right )^{\frac{7}{2}}} - \frac{6 x^{\frac{13}{2}}}{35 b^{2} \left (a x + b x^{3}\right )^{\frac{5}{2}}} - \frac{8 x^{\frac{7}{2}}}{35 b^{3} \left (a x + b x^{3}\right )^{\frac{3}{2}}} - \frac{16 \sqrt{x}}{35 b^{4} \sqrt{a x + b x^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(23/2)/(b*x**3+a*x)**(9/2),x)

[Out]

-x**(19/2)/(7*b*(a*x + b*x**3)**(7/2)) - 6*x**(13/2)/(35*b**2*(a*x + b*x**3)**(5
/2)) - 8*x**(7/2)/(35*b**3*(a*x + b*x**3)**(3/2)) - 16*sqrt(x)/(35*b**4*sqrt(a*x
+ b*x**3))

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Mathematica [A]  time = 0.0455262, size = 66, normalized size = 0.65 \[ -\frac{\sqrt{x} \left (16 a^3+56 a^2 b x^2+70 a b^2 x^4+35 b^3 x^6\right )}{35 b^4 \left (a+b x^2\right )^3 \sqrt{x \left (a+b x^2\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(23/2)/(a*x + b*x^3)^(9/2),x]

[Out]

-(Sqrt[x]*(16*a^3 + 56*a^2*b*x^2 + 70*a*b^2*x^4 + 35*b^3*x^6))/(35*b^4*(a + b*x^
2)^3*Sqrt[x*(a + b*x^2)])

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Maple [A]  time = 0.009, size = 59, normalized size = 0.6 \[ -{\frac{ \left ( b{x}^{2}+a \right ) \left ( 35\,{x}^{6}{b}^{3}+70\,a{x}^{4}{b}^{2}+56\,{a}^{2}{x}^{2}b+16\,{a}^{3} \right ) }{35\,{b}^{4}}{x}^{{\frac{9}{2}}} \left ( b{x}^{3}+ax \right ) ^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(23/2)/(b*x^3+a*x)^(9/2),x)

[Out]

-1/35*(b*x^2+a)*(35*b^3*x^6+70*a*b^2*x^4+56*a^2*b*x^2+16*a^3)*x^(9/2)/b^4/(b*x^3
+a*x)^(9/2)

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Maxima [A]  time = 1.52599, size = 74, normalized size = 0.73 \[ -\frac{35 \,{\left (b x^{2} + a\right )}^{3} - 35 \,{\left (b x^{2} + a\right )}^{2} a + 21 \,{\left (b x^{2} + a\right )} a^{2} - 5 \, a^{3}}{35 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(23/2)/(b*x^3 + a*x)^(9/2),x, algorithm="maxima")

[Out]

-1/35*(35*(b*x^2 + a)^3 - 35*(b*x^2 + a)^2*a + 21*(b*x^2 + a)*a^2 - 5*a^3)/((b*x
^2 + a)^(7/2)*b^4)

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Fricas [A]  time = 0.213751, size = 116, normalized size = 1.15 \[ -\frac{35 \, b^{3} x^{7} + 70 \, a b^{2} x^{5} + 56 \, a^{2} b x^{3} + 16 \, a^{3} x}{35 \,{\left (b^{7} x^{6} + 3 \, a b^{6} x^{4} + 3 \, a^{2} b^{5} x^{2} + a^{3} b^{4}\right )} \sqrt{b x^{3} + a x} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(23/2)/(b*x^3 + a*x)^(9/2),x, algorithm="fricas")

[Out]

-1/35*(35*b^3*x^7 + 70*a*b^2*x^5 + 56*a^2*b*x^3 + 16*a^3*x)/((b^7*x^6 + 3*a*b^6*
x^4 + 3*a^2*b^5*x^2 + a^3*b^4)*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**(23/2)/(b*x**3+a*x)**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.230306, size = 86, normalized size = 0.85 \[ \frac{16}{35 \, \sqrt{a} b^{4}} - \frac{35 \,{\left (b x^{2} + a\right )}^{3} - 35 \,{\left (b x^{2} + a\right )}^{2} a + 21 \,{\left (b x^{2} + a\right )} a^{2} - 5 \, a^{3}}{35 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(23/2)/(b*x^3 + a*x)^(9/2),x, algorithm="giac")

[Out]

16/35/(sqrt(a)*b^4) - 1/35*(35*(b*x^2 + a)^3 - 35*(b*x^2 + a)^2*a + 21*(b*x^2 +
a)*a^2 - 5*a^3)/((b*x^2 + a)^(7/2)*b^4)