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

Optimal. Leaf size=66 \[ -\frac{2 \sqrt{b} (c x)^{3/2} \left (\frac{a}{b x^2}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{\sqrt{a} c^2 \left (a+b x^2\right )^{3/4}} \]

[Out]

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

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Rubi [A]  time = 0.165701, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ -\frac{2 \sqrt{b} (c x)^{3/2} \left (\frac{a}{b x^2}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{\sqrt{a} c^2 \left (a+b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 19.5147, size = 61, normalized size = 0.92 \[ - \frac{2 \sqrt{b} \left (c x\right )^{\frac{3}{2}} \left (\frac{a}{b x^{2}} + 1\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{2}\middle | 2\right )}{\sqrt{a} c^{2} \left (a + b x^{2}\right )^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0278824, size = 55, normalized size = 0.83 \[ \frac{2 x \left (\frac{a+b x^2}{a}\right )^{3/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};-\frac{b x^2}{a}\right )}{\sqrt{c x} \left (a+b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 9.17152, size = 31, normalized size = 0.47 \[ - \frac{{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4} \\ \frac{3}{2} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{2}}} \right )}}{b^{\frac{3}{4}} \sqrt{c} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-hyper((1/2, 3/4), (3/2,), a*exp_polar(I*pi)/(b*x**2))/(b**(3/4)*sqrt(c)*x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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