Optimal. Leaf size=209 \[ -\frac{b^2 (b+2 c x) \sqrt{b x+c x^2} \left (-12 b c (A e+B d)+24 A c^2 d+7 b^2 B e\right )}{512 c^4}+\frac{(b+2 c x) \left (b x+c x^2\right )^{3/2} \left (-12 b c (A e+B d)+24 A c^2 d+7 b^2 B e\right )}{192 c^3}+\frac{b^4 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \left (-12 b c (A e+B d)+24 A c^2 d+7 b^2 B e\right )}{512 c^{9/2}}-\frac{\left (b x+c x^2\right )^{5/2} (-12 c (A e+B d)+7 b B e-10 B c e x)}{60 c^2} \]
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Rubi [A] time = 0.190785, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {779, 612, 620, 206} \[ -\frac{b^2 (b+2 c x) \sqrt{b x+c x^2} \left (-12 b c (A e+B d)+24 A c^2 d+7 b^2 B e\right )}{512 c^4}+\frac{(b+2 c x) \left (b x+c x^2\right )^{3/2} \left (-12 b c (A e+B d)+24 A c^2 d+7 b^2 B e\right )}{192 c^3}+\frac{b^4 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \left (-12 b c (A e+B d)+24 A c^2 d+7 b^2 B e\right )}{512 c^{9/2}}-\frac{\left (b x+c x^2\right )^{5/2} (-12 c (A e+B d)+7 b B e-10 B c e x)}{60 c^2} \]
Antiderivative was successfully verified.
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Rule 779
Rule 612
Rule 620
Rule 206
Rubi steps
\begin{align*} \int (A+B x) (d+e x) \left (b x+c x^2\right )^{3/2} \, dx &=-\frac{(7 b B e-12 c (B d+A e)-10 B c e x) \left (b x+c x^2\right )^{5/2}}{60 c^2}+\frac{\left (\frac{7}{2} b^2 B e+6 c (2 A c d-b (B d+A e))\right ) \int \left (b x+c x^2\right )^{3/2} \, dx}{12 c^2}\\ &=\frac{\left (24 A c^2 d+7 b^2 B e-12 b c (B d+A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{192 c^3}-\frac{(7 b B e-12 c (B d+A e)-10 B c e x) \left (b x+c x^2\right )^{5/2}}{60 c^2}-\frac{\left (b^2 \left (24 A c^2 d+7 b^2 B e-12 b c (B d+A e)\right )\right ) \int \sqrt{b x+c x^2} \, dx}{128 c^3}\\ &=-\frac{b^2 \left (24 A c^2 d+7 b^2 B e-12 b c (B d+A e)\right ) (b+2 c x) \sqrt{b x+c x^2}}{512 c^4}+\frac{\left (24 A c^2 d+7 b^2 B e-12 b c (B d+A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{192 c^3}-\frac{(7 b B e-12 c (B d+A e)-10 B c e x) \left (b x+c x^2\right )^{5/2}}{60 c^2}+\frac{\left (b^4 \left (24 A c^2 d+7 b^2 B e-12 b c (B d+A e)\right )\right ) \int \frac{1}{\sqrt{b x+c x^2}} \, dx}{1024 c^4}\\ &=-\frac{b^2 \left (24 A c^2 d+7 b^2 B e-12 b c (B d+A e)\right ) (b+2 c x) \sqrt{b x+c x^2}}{512 c^4}+\frac{\left (24 A c^2 d+7 b^2 B e-12 b c (B d+A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{192 c^3}-\frac{(7 b B e-12 c (B d+A e)-10 B c e x) \left (b x+c x^2\right )^{5/2}}{60 c^2}+\frac{\left (b^4 \left (24 A c^2 d+7 b^2 B e-12 b c (B d+A e)\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{b x+c x^2}}\right )}{512 c^4}\\ &=-\frac{b^2 \left (24 A c^2 d+7 b^2 B e-12 b c (B d+A e)\right ) (b+2 c x) \sqrt{b x+c x^2}}{512 c^4}+\frac{\left (24 A c^2 d+7 b^2 B e-12 b c (B d+A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{192 c^3}-\frac{(7 b B e-12 c (B d+A e)-10 B c e x) \left (b x+c x^2\right )^{5/2}}{60 c^2}+\frac{b^4 \left (24 A c^2 d+7 b^2 B e-12 b c (B d+A e)\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{512 c^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.630616, size = 245, normalized size = 1.17 \[ \frac{\sqrt{x (b+c x)} \left (\sqrt{c} \left (-8 b^3 c^2 (15 A (3 d+e x)+B x (15 d+7 e x))+48 b^2 c^3 x (A (5 d+2 e x)+B x (2 d+e x))+10 b^4 c (18 A e+18 B d+7 B e x)+64 b c^4 x^2 (A (45 d+33 e x)+B x (33 d+26 e x))+128 c^5 x^3 (3 A (5 d+4 e x)+2 B x (6 d+5 e x))-105 b^5 B e\right )+\frac{15 b^{7/2} \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right ) \left (-12 b c (A e+B d)+24 A c^2 d+7 b^2 B e\right )}{\sqrt{x} \sqrt{\frac{c x}{b}+1}}\right )}{7680 c^{9/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.007, size = 544, normalized size = 2.6 \begin{align*}{\frac{Bex}{6\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}-{\frac{7\,bBe}{60\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}+{\frac{7\,{b}^{2}Bex}{96\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{7\,{b}^{3}Be}{192\,{c}^{3}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{7\,Be{b}^{4}x}{256\,{c}^{3}}\sqrt{c{x}^{2}+bx}}-{\frac{7\,Be{b}^{5}}{512\,{c}^{4}}\sqrt{c{x}^{2}+bx}}+{\frac{7\,Be{b}^{6}}{1024}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{9}{2}}}}+{\frac{Ae}{5\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}+{\frac{Bd}{5\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}-{\frac{Abex}{8\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{Bbdx}{8\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{A{b}^{2}e}{16\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{{b}^{2}Bd}{16\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{b}^{3}xAe}{64\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{3\,{b}^{3}Bxd}{64\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{3\,A{b}^{4}e}{128\,{c}^{3}}\sqrt{c{x}^{2}+bx}}+{\frac{3\,{b}^{4}Bd}{128\,{c}^{3}}\sqrt{c{x}^{2}+bx}}-{\frac{3\,A{b}^{5}e}{256}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{7}{2}}}}-{\frac{3\,B{b}^{5}d}{256}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{7}{2}}}}+{\frac{dAx}{4} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{Abd}{8\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{3\,dA{b}^{2}x}{32\,c}\sqrt{c{x}^{2}+bx}}-{\frac{3\,dA{b}^{3}}{64\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{3\,dA{b}^{4}}{128}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{5}{2}}}} \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 = 1.05214, size = 1355, normalized size = 6.48 \begin{align*} \left [\frac{15 \,{\left (12 \,{\left (B b^{5} c - 2 \, A b^{4} c^{2}\right )} d -{\left (7 \, B b^{6} - 12 \, A b^{5} c\right )} e\right )} \sqrt{c} \log \left (2 \, c x + b - 2 \, \sqrt{c x^{2} + b x} \sqrt{c}\right ) + 2 \,{\left (1280 \, B c^{6} e x^{5} + 128 \,{\left (12 \, B c^{6} d +{\left (13 \, B b c^{5} + 12 \, A c^{6}\right )} e\right )} x^{4} + 48 \,{\left (4 \,{\left (11 \, B b c^{5} + 10 \, A c^{6}\right )} d +{\left (B b^{2} c^{4} + 44 \, A b c^{5}\right )} e\right )} x^{3} + 8 \,{\left (12 \,{\left (B b^{2} c^{4} + 30 \, A b c^{5}\right )} d -{\left (7 \, B b^{3} c^{3} - 12 \, A b^{2} c^{4}\right )} e\right )} x^{2} + 180 \,{\left (B b^{4} c^{2} - 2 \, A b^{3} c^{3}\right )} d - 15 \,{\left (7 \, B b^{5} c - 12 \, A b^{4} c^{2}\right )} e - 10 \,{\left (12 \,{\left (B b^{3} c^{3} - 2 \, A b^{2} c^{4}\right )} d -{\left (7 \, B b^{4} c^{2} - 12 \, A b^{3} c^{3}\right )} e\right )} x\right )} \sqrt{c x^{2} + b x}}{15360 \, c^{5}}, \frac{15 \,{\left (12 \,{\left (B b^{5} c - 2 \, A b^{4} c^{2}\right )} d -{\left (7 \, B b^{6} - 12 \, A b^{5} c\right )} e\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) +{\left (1280 \, B c^{6} e x^{5} + 128 \,{\left (12 \, B c^{6} d +{\left (13 \, B b c^{5} + 12 \, A c^{6}\right )} e\right )} x^{4} + 48 \,{\left (4 \,{\left (11 \, B b c^{5} + 10 \, A c^{6}\right )} d +{\left (B b^{2} c^{4} + 44 \, A b c^{5}\right )} e\right )} x^{3} + 8 \,{\left (12 \,{\left (B b^{2} c^{4} + 30 \, A b c^{5}\right )} d -{\left (7 \, B b^{3} c^{3} - 12 \, A b^{2} c^{4}\right )} e\right )} x^{2} + 180 \,{\left (B b^{4} c^{2} - 2 \, A b^{3} c^{3}\right )} d - 15 \,{\left (7 \, B b^{5} c - 12 \, A b^{4} c^{2}\right )} e - 10 \,{\left (12 \,{\left (B b^{3} c^{3} - 2 \, A b^{2} c^{4}\right )} d -{\left (7 \, B b^{4} c^{2} - 12 \, A b^{3} c^{3}\right )} e\right )} x\right )} \sqrt{c x^{2} + b x}}{7680 \, c^{5}}\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 \left (x \left (b + c x\right )\right )^{\frac{3}{2}} \left (A + B x\right ) \left (d + e x\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16881, size = 428, normalized size = 2.05 \begin{align*} \frac{1}{7680} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (10 \, B c x e + \frac{12 \, B c^{6} d + 13 \, B b c^{5} e + 12 \, A c^{6} e}{c^{5}}\right )} x + \frac{3 \,{\left (44 \, B b c^{5} d + 40 \, A c^{6} d + B b^{2} c^{4} e + 44 \, A b c^{5} e\right )}}{c^{5}}\right )} x + \frac{12 \, B b^{2} c^{4} d + 360 \, A b c^{5} d - 7 \, B b^{3} c^{3} e + 12 \, A b^{2} c^{4} e}{c^{5}}\right )} x - \frac{5 \,{\left (12 \, B b^{3} c^{3} d - 24 \, A b^{2} c^{4} d - 7 \, B b^{4} c^{2} e + 12 \, A b^{3} c^{3} e\right )}}{c^{5}}\right )} x + \frac{15 \,{\left (12 \, B b^{4} c^{2} d - 24 \, A b^{3} c^{3} d - 7 \, B b^{5} c e + 12 \, A b^{4} c^{2} e\right )}}{c^{5}}\right )} + \frac{{\left (12 \, B b^{5} c d - 24 \, A b^{4} c^{2} d - 7 \, B b^{6} e + 12 \, A b^{5} c e\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{1024 \, c^{\frac{9}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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