Optimal. Leaf size=122 \[ \frac{2 \left (3 a^2 e^2-8 a b d e+8 b^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{b^{5/2} \sqrt{e}}+\frac{4 e (a+b x)^{3/2} \sqrt{d+e x}}{b^2}+\frac{2 \sqrt{a+b x} \sqrt{d+e x} (7 b d-5 a e)}{b^2} \]
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Rubi [A] time = 0.123517, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.132, Rules used = {951, 80, 63, 217, 206} \[ \frac{2 \left (3 a^2 e^2-8 a b d e+8 b^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{b^{5/2} \sqrt{e}}+\frac{4 e (a+b x)^{3/2} \sqrt{d+e x}}{b^2}+\frac{2 \sqrt{a+b x} \sqrt{d+e x} (7 b d-5 a e)}{b^2} \]
Antiderivative was successfully verified.
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Rule 951
Rule 80
Rule 63
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{15 d^2+20 d e x+8 e^2 x^2}{\sqrt{a+b x} \sqrt{d+e x}} \, dx &=\frac{4 e (a+b x)^{3/2} \sqrt{d+e x}}{b^2}+\frac{\int \frac{2 e \left (15 b^2 d^2-6 a b d e-2 a^2 e^2\right )+4 b e^2 (7 b d-5 a e) x}{\sqrt{a+b x} \sqrt{d+e x}} \, dx}{2 b^2 e}\\ &=\frac{2 (7 b d-5 a e) \sqrt{a+b x} \sqrt{d+e x}}{b^2}+\frac{4 e (a+b x)^{3/2} \sqrt{d+e x}}{b^2}+\frac{\left (8 b^2 d^2-8 a b d e+3 a^2 e^2\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{d+e x}} \, dx}{b^2}\\ &=\frac{2 (7 b d-5 a e) \sqrt{a+b x} \sqrt{d+e x}}{b^2}+\frac{4 e (a+b x)^{3/2} \sqrt{d+e x}}{b^2}+\frac{\left (2 \left (8 b^2 d^2-8 a b d e+3 a^2 e^2\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^3}\\ &=\frac{2 (7 b d-5 a e) \sqrt{a+b x} \sqrt{d+e x}}{b^2}+\frac{4 e (a+b x)^{3/2} \sqrt{d+e x}}{b^2}+\frac{\left (2 \left (8 b^2 d^2-8 a b d e+3 a^2 e^2\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^3}\\ &=\frac{2 (7 b d-5 a e) \sqrt{a+b x} \sqrt{d+e x}}{b^2}+\frac{4 e (a+b x)^{3/2} \sqrt{d+e x}}{b^2}+\frac{2 \left (8 b^2 d^2-8 a b d e+3 a^2 e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{b^{5/2} \sqrt{e}}\\ \end{align*}
Mathematica [A] time = 0.421291, size = 135, normalized size = 1.11 \[ \frac{2 \left (\frac{\sqrt{b d-a e} \left (3 a^2 e^2-8 a b d e+8 b^2 d^2\right ) \sqrt{\frac{b (d+e x)}{b d-a e}} \sinh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b d-a e}}\right )}{\sqrt{e}}+b \sqrt{a+b x} (d+e x) (-3 a e+7 b d+2 b e x)\right )}{b^3 \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.318, size = 247, normalized size = 2. \begin{align*}{\frac{1}{{b}^{2}} \left ( 3\,\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ){a}^{2}{e}^{2}-8\,\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ) abde+8\,\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ){b}^{2}{d}^{2}+4\,\sqrt{be}\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }xbe-6\,\sqrt{be}\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }ae+14\,\sqrt{be}\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }bd \right ) \sqrt{ex+d}\sqrt{bx+a}{\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 = 1.97896, size = 710, normalized size = 5.82 \begin{align*} \left [\frac{{\left (8 \, b^{2} d^{2} - 8 \, a b d e + 3 \, a^{2} e^{2}\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 (2 \, b^{2} e^{2} x + 7 \, b^{2} d e - 3 \, a b e^{2}\right )} \sqrt{b x + a} \sqrt{e x + d}}{2 \, b^{3} e}, -\frac{{\left (8 \, b^{2} d^{2} - 8 \, a b d e + 3 \, a^{2} e^{2}\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 (2 \, b^{2} e^{2} x + 7 \, b^{2} d e - 3 \, a b e^{2}\right )} \sqrt{b x + a} \sqrt{e x + d}}{b^{3} e}\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{15 d^{2} + 20 d e x + 8 e^{2} x^{2}}{\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 = 1.19239, size = 196, normalized size = 1.61 \begin{align*} \frac{2 \,{\left (\sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e} \sqrt{b x + a}{\left (\frac{2 \,{\left (b x + a\right )} e}{b^{3}} + \frac{{\left (7 \, b^{6} d e^{2} - 5 \, a b^{5} e^{3}\right )} e^{\left (-2\right )}}{b^{8}}\right )} - \frac{{\left (8 \, b^{2} d^{2} - 8 \, a b d e + 3 \, a^{2} e^{2}\right )} e^{\left (-\frac{1}{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{5}{2}}}\right )} b}{{\left | b \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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