3.989 \(\int \frac{1}{(c x)^{9/2} \left (a+b x^2\right )^{5/4}} \, dx\)

Optimal. Leaf size=83 \[ \frac{64 \left (a+b x^2\right )^{7/4}}{21 a^3 c (c x)^{7/2}}-\frac{16 \left (a+b x^2\right )^{3/4}}{3 a^2 c (c x)^{7/2}}+\frac{2}{a c (c x)^{7/2} \sqrt [4]{a+b x^2}} \]

[Out]

2/(a*c*(c*x)^(7/2)*(a + b*x^2)^(1/4)) - (16*(a + b*x^2)^(3/4))/(3*a^2*c*(c*x)^(7
/2)) + (64*(a + b*x^2)^(7/4))/(21*a^3*c*(c*x)^(7/2))

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Rubi [A]  time = 0.0886737, antiderivative size = 83, 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 \[ \frac{64 \left (a+b x^2\right )^{7/4}}{21 a^3 c (c x)^{7/2}}-\frac{16 \left (a+b x^2\right )^{3/4}}{3 a^2 c (c x)^{7/2}}+\frac{2}{a c (c x)^{7/2} \sqrt [4]{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((c*x)^(9/2)*(a + b*x^2)^(5/4)),x]

[Out]

2/(a*c*(c*x)^(7/2)*(a + b*x^2)^(1/4)) - (16*(a + b*x^2)^(3/4))/(3*a^2*c*(c*x)^(7
/2)) + (64*(a + b*x^2)^(7/4))/(21*a^3*c*(c*x)^(7/2))

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Rubi in Sympy [A]  time = 10.8446, size = 71, normalized size = 0.86 \[ \frac{2}{a c \left (c x\right )^{\frac{7}{2}} \sqrt [4]{a + b x^{2}}} - \frac{16 \left (a + b x^{2}\right )^{\frac{3}{4}}}{3 a^{2} c \left (c x\right )^{\frac{7}{2}}} + \frac{64 \left (a + b x^{2}\right )^{\frac{7}{4}}}{21 a^{3} c \left (c x\right )^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(c*x)**(9/2)/(b*x**2+a)**(5/4),x)

[Out]

2/(a*c*(c*x)**(7/2)*(a + b*x**2)**(1/4)) - 16*(a + b*x**2)**(3/4)/(3*a**2*c*(c*x
)**(7/2)) + 64*(a + b*x**2)**(7/4)/(21*a**3*c*(c*x)**(7/2))

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Mathematica [A]  time = 0.0490694, size = 52, normalized size = 0.63 \[ \frac{2 \sqrt{c x} \left (-3 a^2+8 a b x^2+32 b^2 x^4\right )}{21 a^3 c^5 x^4 \sqrt [4]{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((c*x)^(9/2)*(a + b*x^2)^(5/4)),x]

[Out]

(2*Sqrt[c*x]*(-3*a^2 + 8*a*b*x^2 + 32*b^2*x^4))/(21*a^3*c^5*x^4*(a + b*x^2)^(1/4
))

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Maple [A]  time = 0.008, size = 42, normalized size = 0.5 \[ -{\frac{2\,x \left ( -32\,{b}^{2}{x}^{4}-8\,ab{x}^{2}+3\,{a}^{2} \right ) }{21\,{a}^{3}}{\frac{1}{\sqrt [4]{b{x}^{2}+a}}} \left ( cx \right ) ^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c*x)^(9/2)/(b*x^2+a)^(5/4),x)

[Out]

-2/21*x*(-32*b^2*x^4-8*a*b*x^2+3*a^2)/(b*x^2+a)^(1/4)/a^3/(c*x)^(9/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{5}{4}} \left (c x\right )^{\frac{9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(5/4)*(c*x)^(9/2)),x, algorithm="maxima")

[Out]

integrate(1/((b*x^2 + a)^(5/4)*(c*x)^(9/2)), x)

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Fricas [A]  time = 0.211325, size = 62, normalized size = 0.75 \[ \frac{2 \,{\left (32 \, b^{2} x^{4} + 8 \, a b x^{2} - 3 \, a^{2}\right )}}{21 \,{\left (b x^{2} + a\right )}^{\frac{1}{4}} \sqrt{c x} a^{3} c^{4} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(5/4)*(c*x)^(9/2)),x, algorithm="fricas")

[Out]

2/21*(32*b^2*x^4 + 8*a*b*x^2 - 3*a^2)/((b*x^2 + a)^(1/4)*sqrt(c*x)*a^3*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)**(5/4),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{5}{4}} \left (c x\right )^{\frac{9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(5/4)*(c*x)^(9/2)),x, algorithm="giac")

[Out]

integrate(1/((b*x^2 + a)^(5/4)*(c*x)^(9/2)), x)