Optimal. Leaf size=91 \[ -\frac{c \sqrt{c x} \sqrt [4]{a-b x^2}}{b}-\frac{\sqrt{a} (c x)^{3/2} \left (1-\frac{a}{b x^2}\right )^{3/4} F\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{\sqrt{b} \left (a-b x^2\right )^{3/4}} \]
[Out]
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Rubi [A] time = 0.208862, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ -\frac{c \sqrt{c x} \sqrt [4]{a-b x^2}}{b}-\frac{\sqrt{a} (c x)^{3/2} \left (1-\frac{a}{b x^2}\right )^{3/4} F\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{\sqrt{b} \left (a-b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
[In] Int[(c*x)^(3/2)/(a - b*x^2)^(3/4),x]
[Out]
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Rubi in Sympy [A] time = 32.4337, size = 76, normalized size = 0.84 \[ - \frac{\sqrt{a} \left (c x\right )^{\frac{3}{2}} \left (- \frac{a}{b x^{2}} + 1\right )^{\frac{3}{4}} F\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{2}\middle | 2\right )}{\sqrt{b} \left (a - b x^{2}\right )^{\frac{3}{4}}} - \frac{c \sqrt{c x} \sqrt [4]{a - b x^{2}}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x)**(3/2)/(-b*x**2+a)**(3/4),x)
[Out]
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Mathematica [C] time = 0.0601226, size = 68, normalized size = 0.75 \[ \frac{c \sqrt{c x} \left (a \left (1-\frac{b x^2}{a}\right )^{3/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};\frac{b x^2}{a}\right )-a+b x^2\right )}{b \left (a-b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
[In] Integrate[(c*x)^(3/2)/(a - b*x^2)^(3/4),x]
[Out]
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Maple [F] time = 0.045, size = 0, normalized size = 0. \[ \int{1 \left ( cx \right ) ^{{\frac{3}{2}}} \left ( -b{x}^{2}+a \right ) ^{-{\frac{3}{4}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x)^(3/2)/(-b*x^2+a)^(3/4),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x\right )^{\frac{3}{2}}}{{\left (-b x^{2} + a\right )}^{\frac{3}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x)^(3/2)/(-b*x^2 + a)^(3/4),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{c x} c x}{{\left (-b x^{2} + a\right )}^{\frac{3}{4}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x)^(3/2)/(-b*x^2 + a)^(3/4),x, algorithm="fricas")
[Out]
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Sympy [A] time = 13.4456, size = 46, normalized size = 0.51 \[ \frac{c^{\frac{3}{2}} x^{\frac{5}{2}} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )}}{2 a^{\frac{3}{4}} \Gamma \left (\frac{9}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x)**(3/2)/(-b*x**2+a)**(3/4),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x\right )^{\frac{3}{2}}}{{\left (-b x^{2} + a\right )}^{\frac{3}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x)^(3/2)/(-b*x^2 + a)^(3/4),x, algorithm="giac")
[Out]