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