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

Optimal. Leaf size=93 \[ \frac{4 \sqrt{b} \sqrt{c x} \sqrt [4]{\frac{a}{b x^2}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{a^{3/2} c^2 \sqrt [4]{a+b x^2}}-\frac{2}{a c \sqrt{c x} \sqrt [4]{a+b x^2}} \]

[Out]

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

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Rubi [A]  time = 0.118814, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{4 \sqrt{b} \sqrt{c x} \sqrt [4]{\frac{a}{b x^2}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{a^{3/2} c^2 \sqrt [4]{a+b x^2}}-\frac{2}{a c \sqrt{c x} \sqrt [4]{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{2}{a c \sqrt{c x} \sqrt [4]{a + b x^{2}}} + \frac{2 \sqrt{c x} \sqrt [4]{\frac{a}{b x^{2}} + 1} \int ^{\frac{1}{x}} \frac{1}{\left (\frac{a x^{2}}{b} + 1\right )^{\frac{5}{4}}}\, dx}{a c^{2} \sqrt [4]{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/(a*c*sqrt(c*x)*(a + b*x**2)**(1/4)) + 2*sqrt(c*x)*(a/(b*x**2) + 1)**(1/4)*Int
egral((a*x**2/b + 1)**(-5/4), (x, 1/x))/(a*c**2*(a + b*x**2)**(1/4))

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Mathematica [C]  time = 0.0656208, size = 76, normalized size = 0.82 \[ \frac{x \left (8 b x^2 \sqrt [4]{\frac{b x^2}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^2}{a}\right )-6 \left (a+2 b x^2\right )\right )}{3 a^2 (c x)^{3/2} \sqrt [4]{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(-6*(a + 2*b*x^2) + 8*b*x^2*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[1/4, 3/4,
7/4, -((b*x^2)/a)]))/(3*a^2*(c*x)^(3/2)*(a + b*x^2)^(1/4))

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Maple [F]  time = 0.082, size = 0, normalized size = 0. \[ \int{1 \left ( cx \right ) ^{-{\frac{3}{2}}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 132.303, size = 48, normalized size = 0.52 \[ \frac{\Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{5}{4} \\ \frac{3}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac{5}{4}} c^{\frac{3}{2}} \sqrt{x} \Gamma \left (\frac{3}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

gamma(-1/4)*hyper((-1/4, 5/4), (3/4,), b*x**2*exp_polar(I*pi)/a)/(2*a**(5/4)*c**
(3/2)*sqrt(x)*gamma(3/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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