Optimal. Leaf size=269 \[ \frac{2 a^{5/4} \sqrt{c} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{5 b^{3/4} \sqrt{a+b x^2}}-\frac{4 a^{5/4} \sqrt{c} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{5 b^{3/4} \sqrt{a+b x^2}}+\frac{2 (c x)^{3/2} \sqrt{a+b x^2}}{5 c}+\frac{4 a \sqrt{c x} \sqrt{a+b x^2}}{5 \sqrt{b} \left (\sqrt{a}+\sqrt{b} x\right )} \]
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Rubi [A] time = 0.505198, antiderivative size = 269, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ \frac{2 a^{5/4} \sqrt{c} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{5 b^{3/4} \sqrt{a+b x^2}}-\frac{4 a^{5/4} \sqrt{c} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{5 b^{3/4} \sqrt{a+b x^2}}+\frac{2 (c x)^{3/2} \sqrt{a+b x^2}}{5 c}+\frac{4 a \sqrt{c x} \sqrt{a+b x^2}}{5 \sqrt{b} \left (\sqrt{a}+\sqrt{b} x\right )} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[c*x]*Sqrt[a + b*x^2],x]
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Rubi in Sympy [A] time = 47.4581, size = 246, normalized size = 0.91 \[ - \frac{4 a^{\frac{5}{4}} \sqrt{c} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}} \right )}\middle | \frac{1}{2}\right )}{5 b^{\frac{3}{4}} \sqrt{a + b x^{2}}} + \frac{2 a^{\frac{5}{4}} \sqrt{c} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}} \right )}\middle | \frac{1}{2}\right )}{5 b^{\frac{3}{4}} \sqrt{a + b x^{2}}} + \frac{4 a \sqrt{c x} \sqrt{a + b x^{2}}}{5 \sqrt{b} \left (\sqrt{a} + \sqrt{b} x\right )} + \frac{2 \left (c x\right )^{\frac{3}{2}} \sqrt{a + b x^{2}}}{5 c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x)**(1/2)*(b*x**2+a)**(1/2),x)
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Mathematica [C] time = 0.209423, size = 174, normalized size = 0.65 \[ \frac{2 \sqrt{c x} \left (-2 a^{3/2} \sqrt{\frac{b x^2}{a}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}}\right )\right |-1\right )+2 a^{3/2} \sqrt{\frac{b x^2}{a}+1} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}}\right )\right |-1\right )+\sqrt{b} x \sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}} \left (a+b x^2\right )\right )}{5 \sqrt{b} \sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}} \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[c*x]*Sqrt[a + b*x^2],x]
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Maple [A] time = 0.03, size = 205, normalized size = 0.8 \[{\frac{2}{5\,bx}\sqrt{cx} \left ( 2\,\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticE} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) \sqrt{2}{a}^{2}-\sqrt{{1 \left ( bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{{1 \left ( -bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{-{bx{\frac{1}{\sqrt{-ab}}}}}{\it EllipticF} \left ( \sqrt{{1 \left ( bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}},{\frac{\sqrt{2}}{2}} \right ) \sqrt{2}{a}^{2}+{b}^{2}{x}^{4}+ab{x}^{2} \right ){\frac{1}{\sqrt{b{x}^{2}+a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x)^(1/2)*(b*x^2+a)^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{b x^{2} + a} \sqrt{c x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x^2 + a)*sqrt(c*x),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\sqrt{b x^{2} + a} \sqrt{c x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x^2 + a)*sqrt(c*x),x, algorithm="fricas")
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Sympy [A] time = 3.3459, size = 46, normalized size = 0.17 \[ \frac{\sqrt{a} \sqrt{c} x^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac{7}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x)**(1/2)*(b*x**2+a)**(1/2),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{b x^{2} + a} \sqrt{c x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x^2 + a)*sqrt(c*x),x, algorithm="giac")
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