3.998 \(\int \frac{(c x)^{5/4}}{\sqrt [4]{a+b x^2}} \, dx\)

Optimal. Leaf size=58 \[ \frac{4 (c x)^{9/4} \sqrt [4]{\frac{b x^2}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{9}{8};\frac{17}{8};-\frac{b x^2}{a}\right )}{9 c \sqrt [4]{a+b x^2}} \]

[Out]

(4*(c*x)^(9/4)*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[1/4, 9/8, 17/8, -((b*x^2)
/a)])/(9*c*(a + b*x^2)^(1/4))

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Rubi [A]  time = 0.0673859, antiderivative size = 58, 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 \[ \frac{4 (c x)^{9/4} \sqrt [4]{\frac{b x^2}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{9}{8};\frac{17}{8};-\frac{b x^2}{a}\right )}{9 c \sqrt [4]{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(4*(c*x)^(9/4)*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[1/4, 9/8, 17/8, -((b*x^2)
/a)])/(9*c*(a + b*x^2)^(1/4))

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Rubi in Sympy [A]  time = 8.07022, size = 49, normalized size = 0.84 \[ \frac{4 \left (c x\right )^{\frac{9}{4}} \left (a + b x^{2}\right )^{\frac{3}{4}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{9}{8} \\ \frac{17}{8} \end{matrix}\middle |{- \frac{b x^{2}}{a}} \right )}}{9 a c \left (1 + \frac{b x^{2}}{a}\right )^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*(c*x)**(9/4)*(a + b*x**2)**(3/4)*hyper((1/4, 9/8), (17/8,), -b*x**2/a)/(9*a*c*
(1 + b*x**2/a)**(3/4))

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Mathematica [A]  time = 0.0578219, size = 69, normalized size = 1.19 \[ \frac{4 c \sqrt [4]{c x} \left (-a \sqrt [4]{\frac{b x^2}{a}+1} \, _2F_1\left (\frac{1}{8},\frac{1}{4};\frac{9}{8};-\frac{b x^2}{a}\right )+a+b x^2\right )}{7 b \sqrt [4]{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.058, size = 0, normalized size = 0. \[ \int{1 \left ( cx \right ) ^{{\frac{5}{4}}}{\frac{1}{\sqrt [4]{b{x}^{2}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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