Optimal. Leaf size=296 \[ -\frac{\sqrt [4]{b} \log \left (\frac{\sqrt{b} \sqrt{c} x}{\sqrt{a-b x^2}}-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a-b x^2}}+\sqrt{c}\right )}{2 \sqrt{2} c^{3/2}}+\frac{\sqrt [4]{b} \log \left (\frac{\sqrt{b} \sqrt{c} x}{\sqrt{a-b x^2}}+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a-b x^2}}+\sqrt{c}\right )}{2 \sqrt{2} c^{3/2}}+\frac{\sqrt [4]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt{c} \sqrt [4]{a-b x^2}}\right )}{\sqrt{2} c^{3/2}}-\frac{\sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt{c} \sqrt [4]{a-b x^2}}+1\right )}{\sqrt{2} c^{3/2}}-\frac{2 \sqrt [4]{a-b x^2}}{c \sqrt{c x}} \]
[Out]
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Rubi [A] time = 0.614862, antiderivative size = 296, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.45 \[ -\frac{\sqrt [4]{b} \log \left (\frac{\sqrt{b} \sqrt{c} x}{\sqrt{a-b x^2}}-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a-b x^2}}+\sqrt{c}\right )}{2 \sqrt{2} c^{3/2}}+\frac{\sqrt [4]{b} \log \left (\frac{\sqrt{b} \sqrt{c} x}{\sqrt{a-b x^2}}+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a-b x^2}}+\sqrt{c}\right )}{2 \sqrt{2} c^{3/2}}+\frac{\sqrt [4]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt{c} \sqrt [4]{a-b x^2}}\right )}{\sqrt{2} c^{3/2}}-\frac{\sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt{c} \sqrt [4]{a-b x^2}}+1\right )}{\sqrt{2} c^{3/2}}-\frac{2 \sqrt [4]{a-b x^2}}{c \sqrt{c x}} \]
Antiderivative was successfully verified.
[In] Int[(a - b*x^2)^(1/4)/(c*x)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 66.3904, size = 267, normalized size = 0.9 \[ - \frac{\sqrt{2} \sqrt [4]{b} \log{\left (- \frac{\sqrt{2} \sqrt [4]{b} \sqrt{c} \sqrt{c x}}{\sqrt [4]{a - b x^{2}}} + \frac{\sqrt{b} c x}{\sqrt{a - b x^{2}}} + c \right )}}{4 c^{\frac{3}{2}}} + \frac{\sqrt{2} \sqrt [4]{b} \log{\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c} \sqrt{c x}}{\sqrt [4]{a - b x^{2}}} + \frac{\sqrt{b} c x}{\sqrt{a - b x^{2}}} + c \right )}}{4 c^{\frac{3}{2}}} - \frac{\sqrt{2} \sqrt [4]{b} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt{c} \sqrt [4]{a - b x^{2}}} - 1 \right )}}{2 c^{\frac{3}{2}}} - \frac{\sqrt{2} \sqrt [4]{b} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt{c} \sqrt [4]{a - b x^{2}}} + 1 \right )}}{2 c^{\frac{3}{2}}} - \frac{2 \sqrt [4]{a - b x^{2}}}{c \sqrt{c x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-b*x**2+a)**(1/4)/(c*x)**(3/2),x)
[Out]
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Mathematica [C] time = 0.0646625, size = 72, normalized size = 0.24 \[ -\frac{2 x \left (b x^2 \left (1-\frac{b x^2}{a}\right )^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};\frac{b x^2}{a}\right )+3 a-3 b x^2\right )}{3 (c x)^{3/2} \left (a-b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
[In] Integrate[(a - b*x^2)^(1/4)/(c*x)^(3/2),x]
[Out]
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Maple [F] time = 0.05, size = 0, normalized size = 0. \[ \int{1\sqrt [4]{-b{x}^{2}+a} \left ( cx \right ) ^{-{\frac{3}{2}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-b*x^2+a)^(1/4)/(c*x)^(3/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-b*x^2 + a)^(1/4)/(c*x)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-b*x^2 + a)^(1/4)/(c*x)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 14.9041, size = 51, normalized size = 0.17 \[ \frac{\sqrt [4]{a} \Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, - \frac{1}{4} \\ \frac{3}{4} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )}}{2 c^{\frac{3}{2}} \sqrt{x} \Gamma \left (\frac{3}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-b*x**2+a)**(1/4)/(c*x)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.253768, size = 429, normalized size = 1.45 \[ \frac{2 \, \sqrt{2} b^{\frac{1}{4}} \sqrt{{\left | c \right |}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} b^{\frac{1}{4}} \sqrt{{\left | c \right |}} + \frac{2 \,{\left (-b c^{2} x^{2} + a c^{2}\right )}^{\frac{1}{4}} \sqrt{{\left | c \right |}}}{\sqrt{c x}}\right )}}{2 \, b^{\frac{1}{4}} \sqrt{{\left | c \right |}}}\right ) + 2 \, \sqrt{2} b^{\frac{1}{4}} \sqrt{{\left | c \right |}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} b^{\frac{1}{4}} \sqrt{{\left | c \right |}} - \frac{2 \,{\left (-b c^{2} x^{2} + a c^{2}\right )}^{\frac{1}{4}} \sqrt{{\left | c \right |}}}{\sqrt{c x}}\right )}}{2 \, b^{\frac{1}{4}} \sqrt{{\left | c \right |}}}\right ) + \sqrt{2} b^{\frac{1}{4}} \sqrt{{\left | c \right |}}{\rm ln}\left (\frac{\sqrt{2}{\left (-b c^{2} x^{2} + a c^{2}\right )}^{\frac{1}{4}} b^{\frac{1}{4}}{\left | c \right |}}{\sqrt{c x}} + \sqrt{b}{\left | c \right |} + \frac{\sqrt{-b c^{2} x^{2} + a c^{2}}{\left | c \right |}}{c x}\right ) - \sqrt{2} b^{\frac{1}{4}} \sqrt{{\left | c \right |}}{\rm ln}\left (-\frac{\sqrt{2}{\left (-b c^{2} x^{2} + a c^{2}\right )}^{\frac{1}{4}} b^{\frac{1}{4}}{\left | c \right |}}{\sqrt{c x}} + \sqrt{b}{\left | c \right |} + \frac{\sqrt{-b c^{2} x^{2} + a c^{2}}{\left | c \right |}}{c x}\right ) - \frac{8 \,{\left (-b c^{2} x^{2} + a c^{2}\right )}^{\frac{1}{4}} \sqrt{{\left | c \right |}}}{\sqrt{c x}}}{4 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-b*x^2 + a)^(1/4)/(c*x)^(3/2),x, algorithm="giac")
[Out]