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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]