Optimal. Leaf size=127 \[ \frac{2 c \sqrt{c x} \sqrt{a+b x^2}}{3 b}-\frac{a^{3/4} c^{3/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{3 b^{5/4} \sqrt{a+b x^2}} \]
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Rubi [A] time = 0.201396, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{2 c \sqrt{c x} \sqrt{a+b x^2}}{3 b}-\frac{a^{3/4} c^{3/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{3 b^{5/4} \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
[In] Int[(c*x)^(3/2)/Sqrt[a + b*x^2],x]
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Rubi in Sympy [A] time = 17.58, size = 114, normalized size = 0.9 \[ - \frac{a^{\frac{3}{4}} c^{\frac{3}{2}} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}} \right )}\middle | \frac{1}{2}\right )}{3 b^{\frac{5}{4}} \sqrt{a + b x^{2}}} + \frac{2 c \sqrt{c x} \sqrt{a + b x^{2}}}{3 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x)**(3/2)/(b*x**2+a)**(1/2),x)
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Mathematica [C] time = 0.210603, size = 106, normalized size = 0.83 \[ \frac{2 c \sqrt{c x} \left (-\frac{i a \sqrt{x} \sqrt{\frac{a}{b x^2}+1} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}{\sqrt{x}}\right )\right |-1\right )}{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}+a+b x^2\right )}{3 b \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(c*x)^(3/2)/Sqrt[a + b*x^2],x]
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Maple [A] time = 0.015, size = 125, normalized size = 1. \[ -{\frac{c}{3\,{b}^{2}x}\sqrt{cx} \left ( \sqrt{-ab}\sqrt{{1 \left ( bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{{1 \left ( -bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{-{bx{\frac{1}{\sqrt{-ab}}}}}{\it EllipticF} \left ( \sqrt{{1 \left ( bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}},{\frac{\sqrt{2}}{2}} \right ) \sqrt{2}a-2\,{b}^{2}{x}^{3}-2\,abx \right ){\frac{1}{\sqrt{b{x}^{2}+a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x)^(3/2)/(b*x^2+a)^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x\right )^{\frac{3}{2}}}{\sqrt{b x^{2} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x)^(3/2)/sqrt(b*x^2 + a),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{c x} c x}{\sqrt{b x^{2} + a}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x)^(3/2)/sqrt(b*x^2 + a),x, algorithm="fricas")
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Sympy [A] time = 10.994, size = 44, normalized size = 0.35 \[ \frac{c^{\frac{3}{2}} x^{\frac{5}{2}} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt{a} \Gamma \left (\frac{9}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x)**(3/2)/(b*x**2+a)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x\right )^{\frac{3}{2}}}{\sqrt{b x^{2} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x)^(3/2)/sqrt(b*x^2 + a),x, algorithm="giac")
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