Optimal. Leaf size=126 \[ -\frac{2 (d+e x)^{13/2} (-A c e-b B e+3 B c d)}{13 e^4}+\frac{2 (d+e x)^{11/2} (B d (3 c d-2 b e)-A e (2 c d-b e))}{11 e^4}-\frac{2 d (d+e x)^{9/2} (B d-A e) (c d-b e)}{9 e^4}+\frac{2 B c (d+e x)^{15/2}}{15 e^4} \]
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Rubi [A] time = 0.0885321, 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} \[ -\frac{2 (d+e x)^{13/2} (-A c e-b B e+3 B c d)}{13 e^4}+\frac{2 (d+e x)^{11/2} (B d (3 c d-2 b e)-A e (2 c d-b e))}{11 e^4}-\frac{2 d (d+e x)^{9/2} (B d-A e) (c d-b e)}{9 e^4}+\frac{2 B c (d+e x)^{15/2}}{15 e^4} \]
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin{align*} \int (A+B x) (d+e x)^{7/2} \left (b x+c x^2\right ) \, dx &=\int \left (-\frac{d (B d-A e) (c d-b e) (d+e x)^{7/2}}{e^3}+\frac{(B d (3 c d-2 b e)-A e (2 c d-b e)) (d+e x)^{9/2}}{e^3}+\frac{(-3 B c d+b B e+A c e) (d+e x)^{11/2}}{e^3}+\frac{B c (d+e x)^{13/2}}{e^3}\right ) \, dx\\ &=-\frac{2 d (B d-A e) (c d-b e) (d+e x)^{9/2}}{9 e^4}+\frac{2 (B d (3 c d-2 b e)-A e (2 c d-b e)) (d+e x)^{11/2}}{11 e^4}-\frac{2 (3 B c d-b B e-A c e) (d+e x)^{13/2}}{13 e^4}+\frac{2 B c (d+e x)^{15/2}}{15 e^4}\\ \end{align*}
Mathematica [A] time = 0.148778, size = 113, normalized size = 0.9 \[ \frac{2 (d+e x)^{9/2} \left (5 A e \left (13 b e (9 e x-2 d)+c \left (8 d^2-36 d e x+99 e^2 x^2\right )\right )+B \left (5 b e \left (8 d^2-36 d e x+99 e^2 x^2\right )+c \left (72 d^2 e x-16 d^3-198 d e^2 x^2+429 e^3 x^3\right )\right )\right )}{6435 e^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 121, normalized size = 1. \begin{align*} -{\frac{-858\,Bc{x}^{3}{e}^{3}-990\,Ac{e}^{3}{x}^{2}-990\,Bb{e}^{3}{x}^{2}+396\,Bcd{e}^{2}{x}^{2}-1170\,Ab{e}^{3}x+360\,Acd{e}^{2}x+360\,Bbd{e}^{2}x-144\,Bc{d}^{2}ex+260\,Abd{e}^{2}-80\,Ac{d}^{2}e-80\,Bb{d}^{2}e+32\,Bc{d}^{3}}{6435\,{e}^{4}} \left ( ex+d \right ) ^{{\frac{9}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08202, size = 151, normalized size = 1.2 \begin{align*} \frac{2 \,{\left (429 \,{\left (e x + d\right )}^{\frac{15}{2}} B c - 495 \,{\left (3 \, B c d -{\left (B b + A c\right )} e\right )}{\left (e x + d\right )}^{\frac{13}{2}} + 585 \,{\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{11}{2}} - 715 \,{\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{9}{2}}\right )}}{6435 \, e^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.71046, size = 629, normalized size = 4.99 \begin{align*} \frac{2 \,{\left (429 \, B c e^{7} x^{7} - 16 \, B c d^{7} - 130 \, A b d^{5} e^{2} + 40 \,{\left (B b + A c\right )} d^{6} e + 33 \,{\left (46 \, B c d e^{6} + 15 \,{\left (B b + A c\right )} e^{7}\right )} x^{6} + 9 \,{\left (206 \, B c d^{2} e^{5} + 65 \, A b e^{7} + 200 \,{\left (B b + A c\right )} d e^{6}\right )} x^{5} + 10 \,{\left (80 \, B c d^{3} e^{4} + 221 \, A b d e^{6} + 229 \,{\left (B b + A c\right )} d^{2} e^{5}\right )} x^{4} + 5 \,{\left (B c d^{4} e^{3} + 598 \, A b d^{2} e^{5} + 212 \,{\left (B b + A c\right )} d^{3} e^{4}\right )} x^{3} - 3 \,{\left (2 \, B c d^{5} e^{2} - 520 \, A b d^{3} e^{4} - 5 \,{\left (B b + A c\right )} d^{4} e^{3}\right )} x^{2} +{\left (8 \, B c d^{6} e + 65 \, A b d^{4} e^{3} - 20 \,{\left (B b + A c\right )} d^{5} e^{2}\right )} x\right )} \sqrt{e x + d}}{6435 \, e^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 21.1923, size = 683, normalized size = 5.42 \begin{align*} \begin{cases} - \frac{4 A b d^{5} \sqrt{d + e x}}{99 e^{2}} + \frac{2 A b d^{4} x \sqrt{d + e x}}{99 e} + \frac{16 A b d^{3} x^{2} \sqrt{d + e x}}{33} + \frac{92 A b d^{2} e x^{3} \sqrt{d + e x}}{99} + \frac{68 A b d e^{2} x^{4} \sqrt{d + e x}}{99} + \frac{2 A b e^{3} x^{5} \sqrt{d + e x}}{11} + \frac{16 A c d^{6} \sqrt{d + e x}}{1287 e^{3}} - \frac{8 A c d^{5} x \sqrt{d + e x}}{1287 e^{2}} + \frac{2 A c d^{4} x^{2} \sqrt{d + e x}}{429 e} + \frac{424 A c d^{3} x^{3} \sqrt{d + e x}}{1287} + \frac{916 A c d^{2} e x^{4} \sqrt{d + e x}}{1287} + \frac{80 A c d e^{2} x^{5} \sqrt{d + e x}}{143} + \frac{2 A c e^{3} x^{6} \sqrt{d + e x}}{13} + \frac{16 B b d^{6} \sqrt{d + e x}}{1287 e^{3}} - \frac{8 B b d^{5} x \sqrt{d + e x}}{1287 e^{2}} + \frac{2 B b d^{4} x^{2} \sqrt{d + e x}}{429 e} + \frac{424 B b d^{3} x^{3} \sqrt{d + e x}}{1287} + \frac{916 B b d^{2} e x^{4} \sqrt{d + e x}}{1287} + \frac{80 B b d e^{2} x^{5} \sqrt{d + e x}}{143} + \frac{2 B b e^{3} x^{6} \sqrt{d + e x}}{13} - \frac{32 B c d^{7} \sqrt{d + e x}}{6435 e^{4}} + \frac{16 B c d^{6} x \sqrt{d + e x}}{6435 e^{3}} - \frac{4 B c d^{5} x^{2} \sqrt{d + e x}}{2145 e^{2}} + \frac{2 B c d^{4} x^{3} \sqrt{d + e x}}{1287 e} + \frac{320 B c d^{3} x^{4} \sqrt{d + e x}}{1287} + \frac{412 B c d^{2} e x^{5} \sqrt{d + e x}}{715} + \frac{92 B c d e^{2} x^{6} \sqrt{d + e x}}{195} + \frac{2 B c e^{3} x^{7} \sqrt{d + e x}}{15} & \text{for}\: e \neq 0 \\d^{\frac{7}{2}} \left (\frac{A b x^{2}}{2} + \frac{A c x^{3}}{3} + \frac{B b x^{3}}{3} + \frac{B c x^{4}}{4}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.43297, size = 1350, normalized size = 10.71 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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