Optimal. Leaf size=240 \[ \frac{\sqrt{a+b x} (d+e x)^{3/2} \left (35 a^2 e^2-90 a b d e+73 b^2 d^2\right )}{12 b^3}+\frac{\sqrt{a+b x} \sqrt{d+e x} (b d-a e) \left (35 a^2 e^2-90 a b d e+73 b^2 d^2\right )}{8 b^4}+\frac{(b d-a e)^2 \left (35 a^2 e^2-90 a b d e+73 b^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{8 b^{9/2} \sqrt{e}}+\frac{2 e (a+b x)^{3/2} (d+e x)^{5/2}}{b^2}+\frac{\sqrt{a+b x} (d+e x)^{5/2} (17 b d-13 a e)}{3 b^2} \]
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Rubi [A] time = 0.225081, antiderivative size = 240, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {951, 80, 50, 63, 217, 206} \[ \frac{\sqrt{a+b x} (d+e x)^{3/2} \left (35 a^2 e^2-90 a b d e+73 b^2 d^2\right )}{12 b^3}+\frac{\sqrt{a+b x} \sqrt{d+e x} (b d-a e) \left (35 a^2 e^2-90 a b d e+73 b^2 d^2\right )}{8 b^4}+\frac{(b d-a e)^2 \left (35 a^2 e^2-90 a b d e+73 b^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{8 b^{9/2} \sqrt{e}}+\frac{2 e (a+b x)^{3/2} (d+e x)^{5/2}}{b^2}+\frac{\sqrt{a+b x} (d+e x)^{5/2} (17 b d-13 a e)}{3 b^2} \]
Antiderivative was successfully verified.
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Rule 951
Rule 80
Rule 50
Rule 63
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{(d+e x)^{3/2} \left (15 d^2+20 d e x+8 e^2 x^2\right )}{\sqrt{a+b x}} \, dx &=\frac{2 e (a+b x)^{3/2} (d+e x)^{5/2}}{b^2}+\frac{\int \frac{(d+e x)^{3/2} \left (4 e \left (15 b^2 d^2-3 a b d e-5 a^2 e^2\right )+4 b e^2 (17 b d-13 a e) x\right )}{\sqrt{a+b x}} \, dx}{4 b^2 e}\\ &=\frac{(17 b d-13 a e) \sqrt{a+b x} (d+e x)^{5/2}}{3 b^2}+\frac{2 e (a+b x)^{3/2} (d+e x)^{5/2}}{b^2}+\frac{\left (73 b^2 d^2-90 a b d e+35 a^2 e^2\right ) \int \frac{(d+e x)^{3/2}}{\sqrt{a+b x}} \, dx}{6 b^2}\\ &=\frac{\left (73 b^2 d^2-90 a b d e+35 a^2 e^2\right ) \sqrt{a+b x} (d+e x)^{3/2}}{12 b^3}+\frac{(17 b d-13 a e) \sqrt{a+b x} (d+e x)^{5/2}}{3 b^2}+\frac{2 e (a+b x)^{3/2} (d+e x)^{5/2}}{b^2}+\frac{\left ((b d-a e) \left (73 b^2 d^2-90 a b d e+35 a^2 e^2\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{a+b x}} \, dx}{8 b^3}\\ &=\frac{(b d-a e) \left (73 b^2 d^2-90 a b d e+35 a^2 e^2\right ) \sqrt{a+b x} \sqrt{d+e x}}{8 b^4}+\frac{\left (73 b^2 d^2-90 a b d e+35 a^2 e^2\right ) \sqrt{a+b x} (d+e x)^{3/2}}{12 b^3}+\frac{(17 b d-13 a e) \sqrt{a+b x} (d+e x)^{5/2}}{3 b^2}+\frac{2 e (a+b x)^{3/2} (d+e x)^{5/2}}{b^2}+\frac{\left ((b d-a e)^2 \left (73 b^2 d^2-90 a b d e+35 a^2 e^2\right )\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{d+e x}} \, dx}{16 b^4}\\ &=\frac{(b d-a e) \left (73 b^2 d^2-90 a b d e+35 a^2 e^2\right ) \sqrt{a+b x} \sqrt{d+e x}}{8 b^4}+\frac{\left (73 b^2 d^2-90 a b d e+35 a^2 e^2\right ) \sqrt{a+b x} (d+e x)^{3/2}}{12 b^3}+\frac{(17 b d-13 a e) \sqrt{a+b x} (d+e x)^{5/2}}{3 b^2}+\frac{2 e (a+b x)^{3/2} (d+e x)^{5/2}}{b^2}+\frac{\left ((b d-a e)^2 \left (73 b^2 d^2-90 a b d e+35 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 )}{8 b^5}\\ &=\frac{(b d-a e) \left (73 b^2 d^2-90 a b d e+35 a^2 e^2\right ) \sqrt{a+b x} \sqrt{d+e x}}{8 b^4}+\frac{\left (73 b^2 d^2-90 a b d e+35 a^2 e^2\right ) \sqrt{a+b x} (d+e x)^{3/2}}{12 b^3}+\frac{(17 b d-13 a e) \sqrt{a+b x} (d+e x)^{5/2}}{3 b^2}+\frac{2 e (a+b x)^{3/2} (d+e x)^{5/2}}{b^2}+\frac{\left ((b d-a e)^2 \left (73 b^2 d^2-90 a b d e+35 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 )}{8 b^5}\\ &=\frac{(b d-a e) \left (73 b^2 d^2-90 a b d e+35 a^2 e^2\right ) \sqrt{a+b x} \sqrt{d+e x}}{8 b^4}+\frac{\left (73 b^2 d^2-90 a b d e+35 a^2 e^2\right ) \sqrt{a+b x} (d+e x)^{3/2}}{12 b^3}+\frac{(17 b d-13 a e) \sqrt{a+b x} (d+e x)^{5/2}}{3 b^2}+\frac{2 e (a+b x)^{3/2} (d+e x)^{5/2}}{b^2}+\frac{(b d-a e)^2 \left (73 b^2 d^2-90 a b d e+35 a^2 e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{8 b^{9/2} \sqrt{e}}\\ \end{align*}
Mathematica [A] time = 0.909865, size = 204, normalized size = 0.85 \[ \frac{\sqrt{d+e x} \left (\sqrt{a+b x} \left (5 a^2 b e^2 (89 d+14 e x)-105 a^3 e^3-a b^2 e \left (725 d^2+292 d e x+56 e^2 x^2\right )+b^3 \left (466 d^2 e x+501 d^3+232 d e^2 x^2+48 e^3 x^3\right )\right )+\frac{3 \left (35 a^2 e^2-90 a b d e+73 b^2 d^2\right ) (b d-a e)^{3/2} \sinh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b d-a e}}\right )}{\sqrt{e} \sqrt{\frac{b (d+e x)}{b d-a e}}}\right )}{24 b^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.375, size = 571, normalized size = 2.4 \begin{align*}{\frac{1}{48\,{b}^{4}}\sqrt{ex+d}\sqrt{bx+a} \left ( 96\,{x}^{3}{b}^{3}{e}^{3}\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}-112\,{x}^{2}a{b}^{2}{e}^{3}\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+464\,{x}^{2}{b}^{3}d{e}^{2}\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+105\,\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}^{4}{e}^{4}-480\,\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}^{3}bd{e}^{3}+864\,\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}{b}^{2}{d}^{2}{e}^{2}-708\,\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{b}^{3}{d}^{3}e+219\,\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}^{4}{d}^{4}+140\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}x{a}^{2}b{e}^{3}-584\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}xa{b}^{2}d{e}^{2}+932\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}x{b}^{3}{d}^{2}e-210\,\sqrt{be}\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }{a}^{3}{e}^{3}+890\,\sqrt{be}\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }{a}^{2}bd{e}^{2}-1450\,\sqrt{be}\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }a{b}^{2}{d}^{2}e+1002\,\sqrt{be}\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }{b}^{3}{d}^{3} \right ){\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 = 2.12577, size = 1250, normalized size = 5.21 \begin{align*} \left [\frac{3 \,{\left (73 \, b^{4} d^{4} - 236 \, a b^{3} d^{3} e + 288 \, a^{2} b^{2} d^{2} e^{2} - 160 \, a^{3} b d e^{3} + 35 \, a^{4} e^{4}\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 (48 \, b^{4} e^{4} x^{3} + 501 \, b^{4} d^{3} e - 725 \, a b^{3} d^{2} e^{2} + 445 \, a^{2} b^{2} d e^{3} - 105 \, a^{3} b e^{4} + 8 \,{\left (29 \, b^{4} d e^{3} - 7 \, a b^{3} e^{4}\right )} x^{2} + 2 \,{\left (233 \, b^{4} d^{2} e^{2} - 146 \, a b^{3} d e^{3} + 35 \, a^{2} b^{2} e^{4}\right )} x\right )} \sqrt{b x + a} \sqrt{e x + d}}{96 \, b^{5} e}, -\frac{3 \,{\left (73 \, b^{4} d^{4} - 236 \, a b^{3} d^{3} e + 288 \, a^{2} b^{2} d^{2} e^{2} - 160 \, a^{3} b d e^{3} + 35 \, a^{4} e^{4}\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 (48 \, b^{4} e^{4} x^{3} + 501 \, b^{4} d^{3} e - 725 \, a b^{3} d^{2} e^{2} + 445 \, a^{2} b^{2} d e^{3} - 105 \, a^{3} b e^{4} + 8 \,{\left (29 \, b^{4} d e^{3} - 7 \, a b^{3} e^{4}\right )} x^{2} + 2 \,{\left (233 \, b^{4} d^{2} e^{2} - 146 \, a b^{3} d e^{3} + 35 \, a^{2} b^{2} e^{4}\right )} x\right )} \sqrt{b x + a} \sqrt{e x + d}}{48 \, b^{5} e}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.46333, size = 975, normalized size = 4.06 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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