Optimal. Leaf size=287 \[ -\frac{\sqrt [4]{a} e \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (\sqrt{a} B-3 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{3 c^{5/4} \sqrt{e x} \sqrt{a+c x^2}}-\frac{2 \sqrt [4]{a} A e \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{c^{3/4} \sqrt{e x} \sqrt{a+c x^2}}+\frac{2 A e x \sqrt{a+c x^2}}{\sqrt{c} \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}+\frac{2 B \sqrt{e x} \sqrt{a+c x^2}}{3 c} \]
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Rubi [A] time = 0.60994, antiderivative size = 287, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{\sqrt [4]{a} e \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (\sqrt{a} B-3 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{3 c^{5/4} \sqrt{e x} \sqrt{a+c x^2}}-\frac{2 \sqrt [4]{a} A e \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{c^{3/4} \sqrt{e x} \sqrt{a+c x^2}}+\frac{2 A e x \sqrt{a+c x^2}}{\sqrt{c} \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}+\frac{2 B \sqrt{e x} \sqrt{a+c x^2}}{3 c} \]
Antiderivative was successfully verified.
[In] Int[(Sqrt[e*x]*(A + B*x))/Sqrt[a + c*x^2],x]
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Rubi in Sympy [A] time = 38.2631, size = 267, normalized size = 0.93 \[ - \frac{2 A \sqrt [4]{a} e \sqrt{x} \sqrt{\frac{a + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{c^{\frac{3}{4}} \sqrt{e x} \sqrt{a + c x^{2}}} + \frac{2 A e x \sqrt{a + c x^{2}}}{\sqrt{c} \sqrt{e x} \left (\sqrt{a} + \sqrt{c} x\right )} + \frac{2 B \sqrt{e x} \sqrt{a + c x^{2}}}{3 c} + \frac{\sqrt [4]{a} e \sqrt{x} \sqrt{\frac{a + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) \left (3 A \sqrt{c} - B \sqrt{a}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{3 c^{\frac{5}{4}} \sqrt{e x} \sqrt{a + c x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x)**(1/2)*(B*x+A)/(c*x**2+a)**(1/2),x)
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Mathematica [C] time = 0.902064, size = 216, normalized size = 0.75 \[ \frac{2 e \left (\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} \left (a+c x^2\right ) (3 A+B x)+\sqrt{a} x^{3/2} \sqrt{\frac{a}{c x^2}+1} \left (3 A \sqrt{c}-i \sqrt{a} B\right ) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )-3 \sqrt{a} A \sqrt{c} x^{3/2} \sqrt{\frac{a}{c x^2}+1} E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{3 c \sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} \sqrt{e x} \sqrt{a+c x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(Sqrt[e*x]*(A + B*x))/Sqrt[a + c*x^2],x]
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Maple [A] time = 0.016, size = 295, normalized size = 1. \[ -{\frac{1}{3\,{c}^{2}x}\sqrt{ex} \left ( 3\,A\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ) ac-6\,A\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ) ac+B\sqrt{{1 \left ( cx+\sqrt{-ac} \right ){\frac{1}{\sqrt{-ac}}}}}\sqrt{2}\sqrt{{1 \left ( -cx+\sqrt{-ac} \right ){\frac{1}{\sqrt{-ac}}}}}\sqrt{-{cx{\frac{1}{\sqrt{-ac}}}}}{\it EllipticF} \left ( \sqrt{{1 \left ( cx+\sqrt{-ac} \right ){\frac{1}{\sqrt{-ac}}}}},{\frac{\sqrt{2}}{2}} \right ) \sqrt{-ac}a-2\,B{c}^{2}{x}^{3}-2\,aBcx \right ){\frac{1}{\sqrt{c{x}^{2}+a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x)^(1/2)*(B*x+A)/(c*x^2+a)^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x + A\right )} \sqrt{e x}}{\sqrt{c x^{2} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*sqrt(e*x)/sqrt(c*x^2 + a),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (B x + A\right )} \sqrt{e x}}{\sqrt{c x^{2} + a}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*sqrt(e*x)/sqrt(c*x^2 + a),x, algorithm="fricas")
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Sympy [A] time = 7.14877, size = 92, normalized size = 0.32 \[ \frac{A \left (e x\right )^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt{a} e \Gamma \left (\frac{7}{4}\right )} + \frac{B \left (e x\right )^{\frac{5}{2}} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt{a} e^{2} \Gamma \left (\frac{9}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x)**(1/2)*(B*x+A)/(c*x**2+a)**(1/2),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x + A\right )} \sqrt{e x}}{\sqrt{c x^{2} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*sqrt(e*x)/sqrt(c*x^2 + a),x, algorithm="giac")
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