Optimal. Leaf size=298 \[ \frac{e \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (\sqrt{a} B-A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 a^{3/4} c^{5/4} \sqrt{e x} \sqrt{a+c x^2}}+\frac{A e \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{a^{3/4} c^{3/4} \sqrt{e x} \sqrt{a+c x^2}}-\frac{\sqrt{e x} (a B-A c x)}{a c \sqrt{a+c x^2}}-\frac{A e x \sqrt{a+c x^2}}{a \sqrt{c} \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )} \]
[Out]
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Rubi [A] time = 0.601901, antiderivative size = 298, 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 \[ \frac{e \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (\sqrt{a} B-A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 a^{3/4} c^{5/4} \sqrt{e x} \sqrt{a+c x^2}}+\frac{A e \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{a^{3/4} c^{3/4} \sqrt{e x} \sqrt{a+c x^2}}-\frac{\sqrt{e x} (a B-A c x)}{a c \sqrt{a+c x^2}}-\frac{A e x \sqrt{a+c x^2}}{a \sqrt{c} \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )} \]
Antiderivative was successfully verified.
[In] Int[(Sqrt[e*x]*(A + B*x))/(a + c*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 59.6342, size = 269, normalized size = 0.9 \[ - \frac{A e x \sqrt{a + c x^{2}}}{a \sqrt{c} \sqrt{e x} \left (\sqrt{a} + \sqrt{c} x\right )} + \frac{A e \sqrt{x} \sqrt{\frac{a + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{a^{\frac{3}{4}} c^{\frac{3}{4}} \sqrt{e x} \sqrt{a + c x^{2}}} - \frac{\sqrt{e x} \left (- A c x + B a\right )}{a c \sqrt{a + c x^{2}}} - \frac{e \sqrt{x} \sqrt{\frac{a + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) \left (A \sqrt{c} - B \sqrt{a}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{2 a^{\frac{3}{4}} c^{\frac{5}{4}} \sqrt{e x} \sqrt{a + c x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x)**(1/2)*(B*x+A)/(c*x**2+a)**(3/2),x)
[Out]
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Mathematica [C] time = 1.0937, size = 204, normalized size = 0.68 \[ \frac{i e \left (-x^{3/2} \sqrt{\frac{a}{c x^2}+1} \left (A \sqrt{c}-i \sqrt{a} B\right ) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )-\sqrt{a} \sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} (A+B x)+A \sqrt{c} x^{3/2} \sqrt{\frac{a}{c x^2}+1} E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{c^{3/2} \left (\frac{i \sqrt{a}}{\sqrt{c}}\right )^{3/2} \sqrt{e x} \sqrt{a+c x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(Sqrt[e*x]*(A + B*x))/(a + c*x^2)^(3/2),x]
[Out]
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Maple [A] time = 0.02, size = 297, normalized size = 1. \[{\frac{1}{2\,ax{c}^{2}}\sqrt{ex} \left ( A\sqrt{{1 \left ( cx+\sqrt{-ac} \right ){\frac{1}{\sqrt{-ac}}}}}\sqrt{2}\sqrt{{1 \left ( -cx+\sqrt{-ac} \right ){\frac{1}{\sqrt{-ac}}}}}\sqrt{-{cx{\frac{1}{\sqrt{-ac}}}}}{\it EllipticF} \left ( \sqrt{{1 \left ( cx+\sqrt{-ac} \right ){\frac{1}{\sqrt{-ac}}}}},{\frac{\sqrt{2}}{2}} \right ) ac-2\,A\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ) ac+B\sqrt{{1 \left ( cx+\sqrt{-ac} \right ){\frac{1}{\sqrt{-ac}}}}}\sqrt{2}\sqrt{{1 \left ( -cx+\sqrt{-ac} \right ){\frac{1}{\sqrt{-ac}}}}}\sqrt{-{cx{\frac{1}{\sqrt{-ac}}}}}{\it EllipticF} \left ( \sqrt{{1 \left ( cx+\sqrt{-ac} \right ){\frac{1}{\sqrt{-ac}}}}},{\frac{\sqrt{2}}{2}} \right ) \sqrt{-ac}a+2\,A{c}^{2}{x}^{2}-2\,aBcx \right ){\frac{1}{\sqrt{c{x}^{2}+a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x)^(1/2)*(B*x+A)/(c*x^2+a)^(3/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x + A\right )} \sqrt{e x}}{{\left (c x^{2} + a\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*sqrt(e*x)/(c*x^2 + a)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (B x + A\right )} \sqrt{e x}}{{\left (c x^{2} + a\right )}^{\frac{3}{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*sqrt(e*x)/(c*x^2 + a)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 41.6011, size = 94, normalized size = 0.32 \[ \frac{A \sqrt{e} x^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{3}{2} \\ \frac{7}{4} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac{3}{2}} \Gamma \left (\frac{7}{4}\right )} + \frac{B \sqrt{e} x^{\frac{5}{2}} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{5}{4}, \frac{3}{2} \\ \frac{9}{4} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac{3}{2}} \Gamma \left (\frac{9}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x)**(1/2)*(B*x+A)/(c*x**2+a)**(3/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x + A\right )} \sqrt{e x}}{{\left (c x^{2} + a\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*sqrt(e*x)/(c*x^2 + a)^(3/2),x, algorithm="giac")
[Out]