3.1008 \(\int \frac{1}{(c x)^{3/4} \left (a+b x^2\right )^{7/4}} \, dx\)

Optimal. Leaf size=59 \[ \frac{4 \sqrt [4]{c x} \left (\frac{b x^2}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{8},\frac{7}{4};\frac{9}{8};-\frac{b x^2}{a}\right )}{a c \left (a+b x^2\right )^{3/4}} \]

[Out]

(4*(c*x)^(1/4)*(1 + (b*x^2)/a)^(3/4)*Hypergeometric2F1[1/8, 7/4, 9/8, -((b*x^2)/
a)])/(a*c*(a + b*x^2)^(3/4))

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Rubi [A]  time = 0.0668793, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{4 \sqrt [4]{c x} \left (\frac{b x^2}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{8},\frac{7}{4};\frac{9}{8};-\frac{b x^2}{a}\right )}{a c \left (a+b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Int[1/((c*x)^(3/4)*(a + b*x^2)^(7/4)),x]

[Out]

(4*(c*x)^(1/4)*(1 + (b*x^2)/a)^(3/4)*Hypergeometric2F1[1/8, 7/4, 9/8, -((b*x^2)/
a)])/(a*c*(a + b*x^2)^(3/4))

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Rubi in Sympy [A]  time = 8.32128, size = 49, normalized size = 0.83 \[ \frac{4 \sqrt [4]{c x} \sqrt [4]{a + b x^{2}}{{}_{2}F_{1}\left (\begin{matrix} \frac{7}{4}, \frac{1}{8} \\ \frac{9}{8} \end{matrix}\middle |{- \frac{b x^{2}}{a}} \right )}}{a^{2} c \sqrt [4]{1 + \frac{b x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(c*x)**(3/4)/(b*x**2+a)**(7/4),x)

[Out]

4*(c*x)**(1/4)*(a + b*x**2)**(1/4)*hyper((7/4, 1/8), (9/8,), -b*x**2/a)/(a**2*c*
(1 + b*x**2/a)**(1/4))

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Mathematica [A]  time = 0.045259, size = 63, normalized size = 1.07 \[ \frac{2 \left (5 x \left (\frac{b x^2}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{8},\frac{3}{4};\frac{9}{8};-\frac{b x^2}{a}\right )+x\right )}{3 a (c x)^{3/4} \left (a+b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((c*x)^(3/4)*(a + b*x^2)^(7/4)),x]

[Out]

(2*(x + 5*x*(1 + (b*x^2)/a)^(3/4)*Hypergeometric2F1[1/8, 3/4, 9/8, -((b*x^2)/a)]
))/(3*a*(c*x)^(3/4)*(a + b*x^2)^(3/4))

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Maple [F]  time = 0.057, size = 0, normalized size = 0. \[ \int{1 \left ( cx \right ) ^{-{\frac{3}{4}}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c*x)^(3/4)/(b*x^2+a)^(7/4),x)

[Out]

int(1/(c*x)^(3/4)/(b*x^2+a)^(7/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{7}{4}} \left (c x\right )^{\frac{3}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(7/4)*(c*x)^(3/4)),x, algorithm="maxima")

[Out]

integrate(1/((b*x^2 + a)^(7/4)*(c*x)^(3/4)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b x^{2} + a\right )}^{\frac{7}{4}} \left (c x\right )^{\frac{3}{4}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(7/4)*(c*x)^(3/4)),x, algorithm="fricas")

[Out]

integral(1/((b*x^2 + a)^(7/4)*(c*x)^(3/4)), 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)**(3/4)/(b*x**2+a)**(7/4),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{7}{4}} \left (c x\right )^{\frac{3}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(7/4)*(c*x)^(3/4)),x, algorithm="giac")

[Out]

integrate(1/((b*x^2 + a)^(7/4)*(c*x)^(3/4)), x)