3.761 \(\int \frac{\left (a+b x^2\right )^{4/3}}{(c x)^{29/3}} \, dx\)

Optimal. Leaf size=85 \[ -\frac{27 \left (a+b x^2\right )^{13/3}}{910 a^3 c (c x)^{26/3}}+\frac{9 \left (a+b x^2\right )^{10/3}}{70 a^2 c (c x)^{26/3}}-\frac{3 \left (a+b x^2\right )^{7/3}}{14 a c (c x)^{26/3}} \]

[Out]

(-3*(a + b*x^2)^(7/3))/(14*a*c*(c*x)^(26/3)) + (9*(a + b*x^2)^(10/3))/(70*a^2*c*
(c*x)^(26/3)) - (27*(a + b*x^2)^(13/3))/(910*a^3*c*(c*x)^(26/3))

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Rubi [A]  time = 0.0888606, antiderivative size = 85, 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 \[ -\frac{27 \left (a+b x^2\right )^{13/3}}{910 a^3 c (c x)^{26/3}}+\frac{9 \left (a+b x^2\right )^{10/3}}{70 a^2 c (c x)^{26/3}}-\frac{3 \left (a+b x^2\right )^{7/3}}{14 a c (c x)^{26/3}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^(4/3)/(c*x)^(29/3),x]

[Out]

(-3*(a + b*x^2)^(7/3))/(14*a*c*(c*x)^(26/3)) + (9*(a + b*x^2)^(10/3))/(70*a^2*c*
(c*x)^(26/3)) - (27*(a + b*x^2)^(13/3))/(910*a^3*c*(c*x)^(26/3))

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Rubi in Sympy [A]  time = 10.7469, size = 73, normalized size = 0.86 \[ - \frac{3 \left (a + b x^{2}\right )^{\frac{7}{3}}}{14 a c \left (c x\right )^{\frac{26}{3}}} + \frac{9 \left (a + b x^{2}\right )^{\frac{10}{3}}}{70 a^{2} c \left (c x\right )^{\frac{26}{3}}} - \frac{27 \left (a + b x^{2}\right )^{\frac{13}{3}}}{910 a^{3} c \left (c x\right )^{\frac{26}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**(4/3)/(c*x)**(29/3),x)

[Out]

-3*(a + b*x**2)**(7/3)/(14*a*c*(c*x)**(26/3)) + 9*(a + b*x**2)**(10/3)/(70*a**2*
c*(c*x)**(26/3)) - 27*(a + b*x**2)**(13/3)/(910*a**3*c*(c*x)**(26/3))

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Mathematica [A]  time = 0.0635553, size = 52, normalized size = 0.61 \[ -\frac{3 \left (a+b x^2\right )^{7/3} \left (35 a^2-21 a b x^2+9 b^2 x^4\right )}{910 a^3 c^9 x^8 (c x)^{2/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^(4/3)/(c*x)^(29/3),x]

[Out]

(-3*(a + b*x^2)^(7/3)*(35*a^2 - 21*a*b*x^2 + 9*b^2*x^4))/(910*a^3*c^9*x^8*(c*x)^
(2/3))

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Maple [A]  time = 0.007, size = 42, normalized size = 0.5 \[ -{\frac{3\,x \left ( 9\,{b}^{2}{x}^{4}-21\,ab{x}^{2}+35\,{a}^{2} \right ) }{910\,{a}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{3}}} \left ( cx \right ) ^{-{\frac{29}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(4/3)/(c*x)^(29/3),x)

[Out]

-3/910*x*(b*x^2+a)^(7/3)*(9*b^2*x^4-21*a*b*x^2+35*a^2)/a^3/(c*x)^(29/3)

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Maxima [A]  time = 1.39171, size = 74, normalized size = 0.87 \[ -\frac{3 \,{\left (\frac{65 \,{\left (b x^{2} + a\right )}^{\frac{7}{3}} b^{2}}{x^{\frac{14}{3}}} - \frac{91 \,{\left (b x^{2} + a\right )}^{\frac{10}{3}} b}{x^{\frac{20}{3}}} + \frac{35 \,{\left (b x^{2} + a\right )}^{\frac{13}{3}}}{x^{\frac{26}{3}}}\right )}}{910 \, a^{3} c^{\frac{29}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(4/3)/(c*x)^(29/3),x, algorithm="maxima")

[Out]

-3/910*(65*(b*x^2 + a)^(7/3)*b^2/x^(14/3) - 91*(b*x^2 + a)^(10/3)*b/x^(20/3) + 3
5*(b*x^2 + a)^(13/3)/x^(26/3))/(a^3*c^(29/3))

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Fricas [A]  time = 0.225448, size = 92, normalized size = 1.08 \[ -\frac{3 \,{\left (9 \, b^{4} x^{8} - 3 \, a b^{3} x^{6} + 2 \, a^{2} b^{2} x^{4} + 49 \, a^{3} b x^{2} + 35 \, a^{4}\right )}{\left (b x^{2} + a\right )}^{\frac{1}{3}} \left (c x\right )^{\frac{1}{3}}}{910 \, a^{3} c^{10} x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(4/3)/(c*x)^(29/3),x, algorithm="fricas")

[Out]

-3/910*(9*b^4*x^8 - 3*a*b^3*x^6 + 2*a^2*b^2*x^4 + 49*a^3*b*x^2 + 35*a^4)*(b*x^2
+ a)^(1/3)*(c*x)^(1/3)/(a^3*c^10*x^9)

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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((b*x**2+a)**(4/3)/(c*x)**(29/3),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{\frac{4}{3}}}{\left (c x\right )^{\frac{29}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(4/3)/(c*x)^(29/3),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^(4/3)/(c*x)^(29/3), x)