Optimal. Leaf size=85 \[ \frac{2 B \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{\sqrt{b} e^{3/2}}-\frac{2 \sqrt{a+b x} (B d-A e)}{e \sqrt{d+e x} (b d-a e)} \]
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Rubi [A] time = 0.0466726, antiderivative size = 85, 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 = {78, 63, 217, 206} \[ \frac{2 B \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{\sqrt{b} e^{3/2}}-\frac{2 \sqrt{a+b x} (B d-A e)}{e \sqrt{d+e x} (b d-a e)} \]
Antiderivative was successfully verified.
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Rule 78
Rule 63
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{A+B x}{\sqrt{a+b x} (d+e x)^{3/2}} \, dx &=-\frac{2 (B d-A e) \sqrt{a+b x}}{e (b d-a e) \sqrt{d+e x}}+\frac{B \int \frac{1}{\sqrt{a+b x} \sqrt{d+e x}} \, dx}{e}\\ &=-\frac{2 (B d-A e) \sqrt{a+b x}}{e (b d-a e) \sqrt{d+e x}}+\frac{(2 B) \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 e}\\ &=-\frac{2 (B d-A e) \sqrt{a+b x}}{e (b d-a e) \sqrt{d+e x}}+\frac{(2 B) \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 e}\\ &=-\frac{2 (B d-A e) \sqrt{a+b x}}{e (b d-a e) \sqrt{d+e x}}+\frac{2 B \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{\sqrt{b} e^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.23828, size = 118, normalized size = 1.39 \[ \frac{2 b \sqrt{e} \sqrt{a+b x} (B d-A e)-2 B (b d-a e)^{3/2} \sqrt{\frac{b (d+e x)}{b d-a e}} \sinh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b d-a e}}\right )}{b e^{3/2} \sqrt{d+e x} (a e-b d)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.019, size = 278, normalized size = 3.3 \begin{align*}{\frac{1}{e \left ( ae-bd \right ) } \left ( 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 ) xa{e}^{2}-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 ) xbde+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 ) ade-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 ) b{d}^{2}-2\,Ae\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+2\,Bd\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be} \right ) \sqrt{bx+a}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }}}{\frac{1}{\sqrt{be}}}{\frac{1}{\sqrt{ex+d}}}} \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 [B] time = 3.08678, size = 802, normalized size = 9.44 \begin{align*} \left [\frac{{\left (B b d^{2} - B a d e +{\left (B b d e - B a e^{2}\right )} x\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 \,{\left (B b d e - A b e^{2}\right )} \sqrt{b x + a} \sqrt{e x + d}}{2 \,{\left (b^{2} d^{2} e^{2} - a b d e^{3} +{\left (b^{2} d e^{3} - a b e^{4}\right )} x\right )}}, -\frac{{\left (B b d^{2} - B a d e +{\left (B b d e - B a e^{2}\right )} x\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 \,{\left (B b d e - A b e^{2}\right )} \sqrt{b x + a} \sqrt{e x + d}}{b^{2} d^{2} e^{2} - a b d e^{3} +{\left (b^{2} d e^{3} - a b e^{4}\right )} x}\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} \left (d + e x\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.48526, size = 162, normalized size = 1.91 \begin{align*} -\frac{2 \, B{\left | b \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{2 \,{\left (B b^{2} d{\left | b \right |} - A b^{2}{\left | b \right |} e\right )} \sqrt{b x + a}}{{\left (b^{3} d e - a b^{2} e^{2}\right )} \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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