Optimal. Leaf size=84 \[ \frac{(2 A b e-B (a e+b d)) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{b^{3/2} e^{3/2}}+\frac{B \sqrt{a+b x} \sqrt{d+e x}}{b e} \]
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Rubi [A] time = 0.0598965, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {80, 63, 217, 206} \[ \frac{(2 A b e-B (a e+b d)) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{b^{3/2} e^{3/2}}+\frac{B \sqrt{a+b x} \sqrt{d+e x}}{b e} \]
Antiderivative was successfully verified.
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Rule 80
Rule 63
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{A+B x}{\sqrt{a+b x} \sqrt{d+e x}} \, dx &=\frac{B \sqrt{a+b x} \sqrt{d+e x}}{b e}+\frac{\left (A b e-B \left (\frac{b d}{2}+\frac{a e}{2}\right )\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{d+e x}} \, dx}{b e}\\ &=\frac{B \sqrt{a+b x} \sqrt{d+e x}}{b e}+\frac{\left (2 \left (A b e-B \left (\frac{b d}{2}+\frac{a e}{2}\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{d-\frac{a e}{b}+\frac{e x^2}{b}}} \, dx,x,\sqrt{a+b x}\right )}{b^2 e}\\ &=\frac{B \sqrt{a+b x} \sqrt{d+e x}}{b e}+\frac{\left (2 \left (A b e-B \left (\frac{b d}{2}+\frac{a e}{2}\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{e x^2}{b}} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{d+e x}}\right )}{b^2 e}\\ &=\frac{B \sqrt{a+b x} \sqrt{d+e x}}{b e}+\frac{(2 A b e-B (b d+a e)) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{b^{3/2} e^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.189946, size = 118, normalized size = 1.4 \[ \frac{b B \sqrt{e} \sqrt{a+b x} (d+e x)-\sqrt{b d-a e} \sqrt{\frac{b (d+e x)}{b d-a e}} (a B e-2 A b e+b B d) \sinh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b d-a e}}\right )}{b^2 e^{3/2} \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.019, size = 198, normalized size = 2.4 \begin{align*}{\frac{1}{2\,be} \left ( 2\,A\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ) be-B\ln \left ({\frac{1}{2} \left ( 2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd \right ){\frac{1}{\sqrt{be}}}} \right ) ae-B\ln \left ({\frac{1}{2} \left ( 2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd \right ){\frac{1}{\sqrt{be}}}} \right ) bd+2\,B\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be} \right ) \sqrt{bx+a}\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }}}{\frac{1}{\sqrt{be}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.3878, size = 606, normalized size = 7.21 \begin{align*} \left [\frac{4 \, \sqrt{b x + a} \sqrt{e x + d} B b e -{\left (B b d +{\left (B a - 2 \, A b\right )} e\right )} \sqrt{b e} \log \left (8 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 6 \, a b d e + a^{2} e^{2} + 4 \,{\left (2 \, b e x + b d + a e\right )} \sqrt{b e} \sqrt{b x + a} \sqrt{e x + d} + 8 \,{\left (b^{2} d e + a b e^{2}\right )} x\right )}{4 \, b^{2} e^{2}}, \frac{2 \, \sqrt{b x + a} \sqrt{e x + d} B b e +{\left (B b d +{\left (B a - 2 \, A b\right )} e\right )} \sqrt{-b e} \arctan \left (\frac{{\left (2 \, b e x + b d + a e\right )} \sqrt{-b e} \sqrt{b x + a} \sqrt{e x + d}}{2 \,{\left (b^{2} e^{2} x^{2} + a b d e +{\left (b^{2} d e + a b e^{2}\right )} x\right )}}\right )}{2 \, b^{2} e^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{A + B x}{\sqrt{a + b x} \sqrt{d + e x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.97696, size = 143, normalized size = 1.7 \begin{align*} \frac{{\left (\frac{{\left (B b d + B a e - 2 \, A b e\right )} e^{\left (-\frac{3}{2}\right )} \log \left ({\left | -\sqrt{b x + a} \sqrt{b} e^{\frac{1}{2}} + \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e} \right |}\right )}{b^{\frac{3}{2}}} + \frac{\sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e} \sqrt{b x + a} B e^{\left (-1\right )}}{b^{2}}\right )} b}{{\left | b \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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