3.1005 \(\int \frac{(c x)^{3/4}}{\left (a+b x^2\right )^{7/4}} \, dx\)

Optimal. Leaf size=61 \[ \frac{4 (c x)^{7/4} \left (\frac{b x^2}{a}+1\right )^{3/4} \, _2F_1\left (\frac{7}{8},\frac{7}{4};\frac{15}{8};-\frac{b x^2}{a}\right )}{7 a c \left (a+b x^2\right )^{3/4}} \]

[Out]

(4*(c*x)^(7/4)*(1 + (b*x^2)/a)^(3/4)*Hypergeometric2F1[7/8, 7/4, 15/8, -((b*x^2)
/a)])/(7*a*c*(a + b*x^2)^(3/4))

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Rubi [A]  time = 0.0666797, 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 \[ \frac{4 (c x)^{7/4} \left (\frac{b x^2}{a}+1\right )^{3/4} \, _2F_1\left (\frac{7}{8},\frac{7}{4};\frac{15}{8};-\frac{b x^2}{a}\right )}{7 a c \left (a+b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

(4*(c*x)^(7/4)*(1 + (b*x^2)/a)^(3/4)*Hypergeometric2F1[7/8, 7/4, 15/8, -((b*x^2)
/a)])/(7*a*c*(a + b*x^2)^(3/4))

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Rubi in Sympy [A]  time = 8.12389, size = 51, normalized size = 0.84 \[ \frac{4 \left (c x\right )^{\frac{7}{4}} \sqrt [4]{a + b x^{2}}{{}_{2}F_{1}\left (\begin{matrix} \frac{7}{4}, \frac{7}{8} \\ \frac{15}{8} \end{matrix}\middle |{- \frac{b x^{2}}{a}} \right )}}{7 a^{2} c \sqrt [4]{1 + \frac{b x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*(c*x)**(7/4)*(a + b*x**2)**(1/4)*hyper((7/4, 7/8), (15/8,), -b*x**2/a)/(7*a**2
*c*(1 + b*x**2/a)**(1/4))

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Mathematica [A]  time = 0.0512133, size = 62, normalized size = 1.02 \[ -\frac{2 x (c x)^{3/4} \left (\left (\frac{b x^2}{a}+1\right )^{3/4} \, _2F_1\left (\frac{3}{4},\frac{7}{8};\frac{15}{8};-\frac{b x^2}{a}\right )-7\right )}{21 a \left (a+b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*x*(c*x)^(3/4)*(-7 + (1 + (b*x^2)/a)^(3/4)*Hypergeometric2F1[3/4, 7/8, 15/8,
-((b*x^2)/a)]))/(21*a*(a + b*x^2)^(3/4))

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Maple [F]  time = 0.053, 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((c*x)^(3/4)/(b*x^2+a)^(7/4),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((c*x)^(3/4)/(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)**(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. \[ \int \frac{\left (c x\right )^{\frac{3}{4}}}{{\left (b x^{2} + a\right )}^{\frac{7}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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