Optimal. Leaf size=126 \[ -\frac{2 (d+e x)^{9/2} (-A c e-b B e+3 B c d)}{9 e^4}+\frac{2 (d+e x)^{7/2} (B d (3 c d-2 b e)-A e (2 c d-b e))}{7 e^4}-\frac{2 d (d+e x)^{5/2} (B d-A e) (c d-b e)}{5 e^4}+\frac{2 B c (d+e x)^{11/2}}{11 e^4} \]
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Rubi [A] time = 0.0732722, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {771} \[ -\frac{2 (d+e x)^{9/2} (-A c e-b B e+3 B c d)}{9 e^4}+\frac{2 (d+e x)^{7/2} (B d (3 c d-2 b e)-A e (2 c d-b e))}{7 e^4}-\frac{2 d (d+e x)^{5/2} (B d-A e) (c d-b e)}{5 e^4}+\frac{2 B c (d+e x)^{11/2}}{11 e^4} \]
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin{align*} \int (A+B x) (d+e x)^{3/2} \left (b x+c x^2\right ) \, dx &=\int \left (-\frac{d (B d-A e) (c d-b e) (d+e x)^{3/2}}{e^3}+\frac{(B d (3 c d-2 b e)-A e (2 c d-b e)) (d+e x)^{5/2}}{e^3}+\frac{(-3 B c d+b B e+A c e) (d+e x)^{7/2}}{e^3}+\frac{B c (d+e x)^{9/2}}{e^3}\right ) \, dx\\ &=-\frac{2 d (B d-A e) (c d-b e) (d+e x)^{5/2}}{5 e^4}+\frac{2 (B d (3 c d-2 b e)-A e (2 c d-b e)) (d+e x)^{7/2}}{7 e^4}-\frac{2 (3 B c d-b B e-A c e) (d+e x)^{9/2}}{9 e^4}+\frac{2 B c (d+e x)^{11/2}}{11 e^4}\\ \end{align*}
Mathematica [A] time = 0.120248, size = 114, normalized size = 0.9 \[ \frac{2 (d+e x)^{5/2} \left (11 A e \left (9 b e (5 e x-2 d)+c \left (8 d^2-20 d e x+35 e^2 x^2\right )\right )+B \left (11 b e \left (8 d^2-20 d e x+35 e^2 x^2\right )-3 c \left (-40 d^2 e x+16 d^3+70 d e^2 x^2-105 e^3 x^3\right )\right )\right )}{3465 e^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 121, normalized size = 1. \begin{align*} -{\frac{-630\,Bc{x}^{3}{e}^{3}-770\,Ac{e}^{3}{x}^{2}-770\,Bb{e}^{3}{x}^{2}+420\,Bcd{e}^{2}{x}^{2}-990\,Ab{e}^{3}x+440\,Acd{e}^{2}x+440\,Bbd{e}^{2}x-240\,Bc{d}^{2}ex+396\,Abd{e}^{2}-176\,Ac{d}^{2}e-176\,Bb{d}^{2}e+96\,Bc{d}^{3}}{3465\,{e}^{4}} \left ( ex+d \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03735, size = 151, normalized size = 1.2 \begin{align*} \frac{2 \,{\left (315 \,{\left (e x + d\right )}^{\frac{11}{2}} B c - 385 \,{\left (3 \, B c d -{\left (B b + A c\right )} e\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 495 \,{\left (3 \, B c d^{2} + A b e^{2} - 2 \,{\left (B b + A c\right )} d e\right )}{\left (e x + d\right )}^{\frac{7}{2}} - 693 \,{\left (B c d^{3} + A b d e^{2} -{\left (B b + A c\right )} d^{2} e\right )}{\left (e x + d\right )}^{\frac{5}{2}}\right )}}{3465 \, e^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.73702, size = 446, normalized size = 3.54 \begin{align*} \frac{2 \,{\left (315 \, B c e^{5} x^{5} - 48 \, B c d^{5} - 198 \, A b d^{3} e^{2} + 88 \,{\left (B b + A c\right )} d^{4} e + 35 \,{\left (12 \, B c d e^{4} + 11 \,{\left (B b + A c\right )} e^{5}\right )} x^{4} + 5 \,{\left (3 \, B c d^{2} e^{3} + 99 \, A b e^{5} + 110 \,{\left (B b + A c\right )} d e^{4}\right )} x^{3} - 3 \,{\left (6 \, B c d^{3} e^{2} - 264 \, A b d e^{4} - 11 \,{\left (B b + A c\right )} d^{2} e^{3}\right )} x^{2} +{\left (24 \, B c d^{4} e + 99 \, A b d^{2} e^{3} - 44 \,{\left (B b + A c\right )} d^{3} e^{2}\right )} x\right )} \sqrt{e x + d}}{3465 \, e^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 18.9717, size = 434, normalized size = 3.44 \begin{align*} \frac{2 A b d \left (- \frac{d \left (d + e x\right )^{\frac{3}{2}}}{3} + \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e^{2}} + \frac{2 A b \left (\frac{d^{2} \left (d + e x\right )^{\frac{3}{2}}}{3} - \frac{2 d \left (d + e x\right )^{\frac{5}{2}}}{5} + \frac{\left (d + e x\right )^{\frac{7}{2}}}{7}\right )}{e^{2}} + \frac{2 A c d \left (\frac{d^{2} \left (d + e x\right )^{\frac{3}{2}}}{3} - \frac{2 d \left (d + e x\right )^{\frac{5}{2}}}{5} + \frac{\left (d + e x\right )^{\frac{7}{2}}}{7}\right )}{e^{3}} + \frac{2 A c \left (- \frac{d^{3} \left (d + e x\right )^{\frac{3}{2}}}{3} + \frac{3 d^{2} \left (d + e x\right )^{\frac{5}{2}}}{5} - \frac{3 d \left (d + e x\right )^{\frac{7}{2}}}{7} + \frac{\left (d + e x\right )^{\frac{9}{2}}}{9}\right )}{e^{3}} + \frac{2 B b d \left (\frac{d^{2} \left (d + e x\right )^{\frac{3}{2}}}{3} - \frac{2 d \left (d + e x\right )^{\frac{5}{2}}}{5} + \frac{\left (d + e x\right )^{\frac{7}{2}}}{7}\right )}{e^{3}} + \frac{2 B b \left (- \frac{d^{3} \left (d + e x\right )^{\frac{3}{2}}}{3} + \frac{3 d^{2} \left (d + e x\right )^{\frac{5}{2}}}{5} - \frac{3 d \left (d + e x\right )^{\frac{7}{2}}}{7} + \frac{\left (d + e x\right )^{\frac{9}{2}}}{9}\right )}{e^{3}} + \frac{2 B c d \left (- \frac{d^{3} \left (d + e x\right )^{\frac{3}{2}}}{3} + \frac{3 d^{2} \left (d + e x\right )^{\frac{5}{2}}}{5} - \frac{3 d \left (d + e x\right )^{\frac{7}{2}}}{7} + \frac{\left (d + e x\right )^{\frac{9}{2}}}{9}\right )}{e^{4}} + \frac{2 B c \left (\frac{d^{4} \left (d + e x\right )^{\frac{3}{2}}}{3} - \frac{4 d^{3} \left (d + e x\right )^{\frac{5}{2}}}{5} + \frac{6 d^{2} \left (d + e x\right )^{\frac{7}{2}}}{7} - \frac{4 d \left (d + e x\right )^{\frac{9}{2}}}{9} + \frac{\left (d + e x\right )^{\frac{11}{2}}}{11}\right )}{e^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.49063, size = 524, normalized size = 4.16 \begin{align*} \frac{2}{3465} \,{\left (231 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} A b d e^{\left (-1\right )} + 33 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2}\right )} B b d e^{\left (-2\right )} + 33 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2}\right )} A c d e^{\left (-2\right )} + 11 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} d + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3}\right )} B c d e^{\left (-3\right )} + 33 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2}\right )} A b e^{\left (-1\right )} + 11 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} d + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3}\right )} B b e^{\left (-2\right )} + 11 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} d + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3}\right )} A c e^{\left (-2\right )} +{\left (315 \,{\left (x e + d\right )}^{\frac{11}{2}} - 1540 \,{\left (x e + d\right )}^{\frac{9}{2}} d + 2970 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{2} - 2772 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{3} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{4}\right )} B c e^{\left (-3\right )}\right )} e^{\left (-1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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