3.958 \(\int \frac{1}{(c x)^{9/2} \sqrt [4]{a-b x^2}} \, dx\)

Optimal. Leaf size=59 \[ \frac{8 \left (a-b x^2\right )^{7/4}}{21 a^2 c (c x)^{7/2}}-\frac{2 \left (a-b x^2\right )^{3/4}}{3 a c (c x)^{7/2}} \]

[Out]

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

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Rubi [A]  time = 0.0598506, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{8 \left (a-b x^2\right )^{7/4}}{21 a^2 c (c x)^{7/2}}-\frac{2 \left (a-b x^2\right )^{3/4}}{3 a c (c x)^{7/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((c*x)^(9/2)*(a - b*x^2)^(1/4)),x]

[Out]

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

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Rubi in Sympy [A]  time = 7.37526, size = 48, normalized size = 0.81 \[ - \frac{2 \left (a - b x^{2}\right )^{\frac{3}{4}}}{3 a c \left (c x\right )^{\frac{7}{2}}} + \frac{8 \left (a - b x^{2}\right )^{\frac{7}{4}}}{21 a^{2} c \left (c x\right )^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(c*x)**(9/2)/(-b*x**2+a)**(1/4),x)

[Out]

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

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Mathematica [A]  time = 0.0376486, size = 42, normalized size = 0.71 \[ -\frac{2 \sqrt{c x} \left (a-b x^2\right )^{3/4} \left (3 a+4 b x^2\right )}{21 a^2 c^5 x^4} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((c*x)^(9/2)*(a - b*x^2)^(1/4)),x]

[Out]

(-2*Sqrt[c*x]*(a - b*x^2)^(3/4)*(3*a + 4*b*x^2))/(21*a^2*c^5*x^4)

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Maple [A]  time = 0.007, size = 32, normalized size = 0.5 \[ -{\frac{2\,x \left ( 4\,b{x}^{2}+3\,a \right ) }{21\,{a}^{2}} \left ( -b{x}^{2}+a \right ) ^{{\frac{3}{4}}} \left ( cx \right ) ^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c*x)^(9/2)/(-b*x^2+a)^(1/4),x)

[Out]

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

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Maxima [A]  time = 1.39583, size = 54, normalized size = 0.92 \[ -\frac{2 \,{\left (\frac{7 \,{\left (-b x^{2} + a\right )}^{\frac{3}{4}} b}{x^{\frac{3}{2}}} + \frac{3 \,{\left (-b x^{2} + a\right )}^{\frac{7}{4}}}{x^{\frac{7}{2}}}\right )}}{21 \, a^{2} c^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x^2 + a)^(1/4)*(c*x)^(9/2)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.21291, size = 63, normalized size = 1.07 \[ \frac{2 \,{\left (4 \, b^{2} x^{4} - a b x^{2} - 3 \, a^{2}\right )}}{21 \,{\left (-b x^{2} + a\right )}^{\frac{1}{4}} \sqrt{c x} a^{2} c^{4} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x^2 + a)^(1/4)*(c*x)^(9/2)),x, algorithm="fricas")

[Out]

2/21*(4*b^2*x^4 - a*b*x^2 - 3*a^2)/((-b*x^2 + a)^(1/4)*sqrt(c*x)*a^2*c^4*x^3)

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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(1/(c*x)**(9/2)/(-b*x**2+a)**(1/4),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-b x^{2} + a\right )}^{\frac{1}{4}} \left (c x\right )^{\frac{9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x^2 + a)^(1/4)*(c*x)^(9/2)),x, algorithm="giac")

[Out]

integrate(1/((-b*x^2 + a)^(1/4)*(c*x)^(9/2)), x)