3.967 \(\int \frac{1}{(c x)^{5/2} \left (a+b x^2\right )^{3/4}} \, dx\)

Optimal. Leaf size=97 \[ \frac{4 b^{3/2} (c x)^{3/2} \left (\frac{a}{b x^2}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{3 a^{3/2} c^4 \left (a+b x^2\right )^{3/4}}-\frac{2 \sqrt [4]{a+b x^2}}{3 a c (c x)^{3/2}} \]

[Out]

(-2*(a + b*x^2)^(1/4))/(3*a*c*(c*x)^(3/2)) + (4*b^(3/2)*(1 + a/(b*x^2))^(3/4)*(c
*x)^(3/2)*EllipticF[ArcCot[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(3*a^(3/2)*c^4*(a + b*x^2
)^(3/4))

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Rubi [A]  time = 0.202989, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316 \[ \frac{4 b^{3/2} (c x)^{3/2} \left (\frac{a}{b x^2}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{3 a^{3/2} c^4 \left (a+b x^2\right )^{3/4}}-\frac{2 \sqrt [4]{a+b x^2}}{3 a c (c x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((c*x)^(5/2)*(a + b*x^2)^(3/4)),x]

[Out]

(-2*(a + b*x^2)^(1/4))/(3*a*c*(c*x)^(3/2)) + (4*b^(3/2)*(1 + a/(b*x^2))^(3/4)*(c
*x)^(3/2)*EllipticF[ArcCot[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(3*a^(3/2)*c^4*(a + b*x^2
)^(3/4))

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Rubi in Sympy [A]  time = 23.6327, size = 85, normalized size = 0.88 \[ - \frac{2 \sqrt [4]{a + b x^{2}}}{3 a c \left (c x\right )^{\frac{3}{2}}} + \frac{4 b^{\frac{3}{2}} \left (c x\right )^{\frac{3}{2}} \left (\frac{a}{b x^{2}} + 1\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{2}\middle | 2\right )}{3 a^{\frac{3}{2}} c^{4} \left (a + b x^{2}\right )^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(c*x)**(5/2)/(b*x**2+a)**(3/4),x)

[Out]

-2*(a + b*x**2)**(1/4)/(3*a*c*(c*x)**(3/2)) + 4*b**(3/2)*(c*x)**(3/2)*(a/(b*x**2
) + 1)**(3/4)*elliptic_f(atan(sqrt(a)/(sqrt(b)*x))/2, 2)/(3*a**(3/2)*c**4*(a + b
*x**2)**(3/4))

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Mathematica [C]  time = 0.0554217, size = 72, normalized size = 0.74 \[ -\frac{2 x \left (2 b x^2 \left (\frac{b x^2}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};-\frac{b x^2}{a}\right )+a+b x^2\right )}{3 a (c x)^{5/2} \left (a+b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((c*x)^(5/2)*(a + b*x^2)^(3/4)),x]

[Out]

(-2*x*(a + b*x^2 + 2*b*x^2*(1 + (b*x^2)/a)^(3/4)*Hypergeometric2F1[1/4, 3/4, 5/4
, -((b*x^2)/a)]))/(3*a*(c*x)^(5/2)*(a + b*x^2)^(3/4))

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Maple [F]  time = 0.046, size = 0, normalized size = 0. \[ \int{1 \left ( cx \right ) ^{-{\frac{5}{2}}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c*x)^(5/2)/(b*x^2+a)^(3/4),x)

[Out]

int(1/(c*x)^(5/2)/(b*x^2+a)^(3/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{3}{4}} \left (c x\right )^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(3/4)*(c*x)^(5/2)),x, algorithm="maxima")

[Out]

integrate(1/((b*x^2 + a)^(3/4)*(c*x)^(5/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b x^{2} + a\right )}^{\frac{3}{4}} \sqrt{c x} c^{2} x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(3/4)*(c*x)^(5/2)),x, algorithm="fricas")

[Out]

integral(1/((b*x^2 + a)^(3/4)*sqrt(c*x)*c^2*x^2), 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(1/(c*x)**(5/2)/(b*x**2+a)**(3/4),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{3}{4}} \left (c x\right )^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(3/4)*(c*x)^(5/2)),x, algorithm="giac")

[Out]

integrate(1/((b*x^2 + a)^(3/4)*(c*x)^(5/2)), x)