3.1006 \(\int \frac{\sqrt [4]{c x}}{\left (a+b x^2\right )^{7/4}} \, dx\)

Optimal. Leaf size=61 \[ \frac{4 (c x)^{5/4} \left (\frac{b x^2}{a}+1\right )^{3/4} \, _2F_1\left (\frac{5}{8},\frac{7}{4};\frac{13}{8};-\frac{b x^2}{a}\right )}{5 a c \left (a+b x^2\right )^{3/4}} \]

[Out]

(4*(c*x)^(5/4)*(1 + (b*x^2)/a)^(3/4)*Hypergeometric2F1[5/8, 7/4, 13/8, -((b*x^2)
/a)])/(5*a*c*(a + b*x^2)^(3/4))

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Rubi [A]  time = 0.0655268, antiderivative size = 61, 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 \[ \frac{4 (c x)^{5/4} \left (\frac{b x^2}{a}+1\right )^{3/4} \, _2F_1\left (\frac{5}{8},\frac{7}{4};\frac{13}{8};-\frac{b x^2}{a}\right )}{5 a c \left (a+b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

(4*(c*x)^(5/4)*(1 + (b*x^2)/a)^(3/4)*Hypergeometric2F1[5/8, 7/4, 13/8, -((b*x^2)
/a)])/(5*a*c*(a + b*x^2)^(3/4))

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Rubi in Sympy [A]  time = 8.19319, size = 51, normalized size = 0.84 \[ \frac{4 \left (c x\right )^{\frac{5}{4}} \sqrt [4]{a + b x^{2}}{{}_{2}F_{1}\left (\begin{matrix} \frac{7}{4}, \frac{5}{8} \\ \frac{13}{8} \end{matrix}\middle |{- \frac{b x^{2}}{a}} \right )}}{5 a^{2} c \sqrt [4]{1 + \frac{b x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*(c*x)**(5/4)*(a + b*x**2)**(1/4)*hyper((7/4, 5/8), (13/8,), -b*x**2/a)/(5*a**2
*c*(1 + b*x**2/a)**(1/4))

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Mathematica [A]  time = 0.0488793, size = 62, normalized size = 1.02 \[ \frac{2 x \sqrt [4]{c x} \left (\left (\frac{b x^2}{a}+1\right )^{3/4} \, _2F_1\left (\frac{5}{8},\frac{3}{4};\frac{13}{8};-\frac{b x^2}{a}\right )+5\right )}{15 a \left (a+b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

(2*x*(c*x)^(1/4)*(5 + (1 + (b*x^2)/a)^(3/4)*Hypergeometric2F1[5/8, 3/4, 13/8, -(
(b*x^2)/a)]))/(15*a*(a + b*x^2)^(3/4))

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Maple [F]  time = 0.05, size = 0, normalized size = 0. \[ \int{1\sqrt [4]{cx} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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