Optimal. Leaf size=122 \[ -\frac{2 (d+e x)^{3/2} (-A c e-b B e+3 B c d)}{3 e^4}+\frac{2 \sqrt{d+e x} (B d (3 c d-2 b e)-A e (2 c d-b e))}{e^4}+\frac{2 d (B d-A e) (c d-b e)}{e^4 \sqrt{d+e x}}+\frac{2 B c (d+e x)^{5/2}}{5 e^4} \]
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Rubi [A] time = 0.0759785, antiderivative size = 122, 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} \[ -\frac{2 (d+e x)^{3/2} (-A c e-b B e+3 B c d)}{3 e^4}+\frac{2 \sqrt{d+e x} (B d (3 c d-2 b e)-A e (2 c d-b e))}{e^4}+\frac{2 d (B d-A e) (c d-b e)}{e^4 \sqrt{d+e x}}+\frac{2 B c (d+e x)^{5/2}}{5 e^4} \]
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (b x+c x^2\right )}{(d+e x)^{3/2}} \, dx &=\int \left (-\frac{d (B d-A e) (c d-b e)}{e^3 (d+e x)^{3/2}}+\frac{B d (3 c d-2 b e)-A e (2 c d-b e)}{e^3 \sqrt{d+e x}}+\frac{(-3 B c d+b B e+A c e) \sqrt{d+e x}}{e^3}+\frac{B c (d+e x)^{3/2}}{e^3}\right ) \, dx\\ &=\frac{2 d (B d-A e) (c d-b e)}{e^4 \sqrt{d+e x}}+\frac{2 (B d (3 c d-2 b e)-A e (2 c d-b e)) \sqrt{d+e x}}{e^4}-\frac{2 (3 B c d-b B e-A c e) (d+e x)^{3/2}}{3 e^4}+\frac{2 B c (d+e x)^{5/2}}{5 e^4}\\ \end{align*}
Mathematica [A] time = 0.0470064, size = 110, normalized size = 0.9 \[ \frac{2 \left (5 A e \left (3 b e (2 d+e x)+c \left (-8 d^2-4 d e x+e^2 x^2\right )\right )+B \left (5 b e \left (-8 d^2-4 d e x+e^2 x^2\right )+3 c \left (8 d^2 e x+16 d^3-2 d e^2 x^2+e^3 x^3\right )\right )\right )}{15 e^4 \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 121, normalized size = 1. \begin{align*}{\frac{6\,Bc{x}^{3}{e}^{3}+10\,Ac{e}^{3}{x}^{2}+10\,Bb{e}^{3}{x}^{2}-12\,Bcd{e}^{2}{x}^{2}+30\,Ab{e}^{3}x-40\,Acd{e}^{2}x-40\,Bbd{e}^{2}x+48\,Bc{d}^{2}ex+60\,Abd{e}^{2}-80\,Ac{d}^{2}e-80\,Bb{d}^{2}e+96\,Bc{d}^{3}}{15\,{e}^{4}}{\frac{1}{\sqrt{ex+d}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06913, size = 162, normalized size = 1.33 \begin{align*} \frac{2 \,{\left (\frac{3 \,{\left (e x + d\right )}^{\frac{5}{2}} B c - 5 \,{\left (3 \, B c d -{\left (B b + A c\right )} e\right )}{\left (e x + d\right )}^{\frac{3}{2}} + 15 \,{\left (3 \, B c d^{2} + A b e^{2} - 2 \,{\left (B b + A c\right )} d e\right )} \sqrt{e x + d}}{e^{3}} + \frac{15 \,{\left (B c d^{3} + A b d e^{2} -{\left (B b + A c\right )} d^{2} e\right )}}{\sqrt{e x + d} e^{3}}\right )}}{15 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.74158, size = 267, normalized size = 2.19 \begin{align*} \frac{2 \,{\left (3 \, B c e^{3} x^{3} + 48 \, B c d^{3} + 30 \, A b d e^{2} - 40 \,{\left (B b + A c\right )} d^{2} e -{\left (6 \, B c d e^{2} - 5 \,{\left (B b + A c\right )} e^{3}\right )} x^{2} +{\left (24 \, B c d^{2} e + 15 \, A b e^{3} - 20 \,{\left (B b + A c\right )} d e^{2}\right )} x\right )} \sqrt{e x + d}}{15 \,{\left (e^{5} x + d e^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 20.3608, size = 126, normalized size = 1.03 \begin{align*} \frac{2 B c \left (d + e x\right )^{\frac{5}{2}}}{5 e^{4}} - \frac{2 d \left (- A e + B d\right ) \left (b e - c d\right )}{e^{4} \sqrt{d + e x}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (2 A c e + 2 B b e - 6 B c d\right )}{3 e^{4}} + \frac{\sqrt{d + e x} \left (2 A b e^{2} - 4 A c d e - 4 B b d e + 6 B c d^{2}\right )}{e^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25082, size = 225, normalized size = 1.84 \begin{align*} \frac{2}{15} \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} B c e^{16} - 15 \,{\left (x e + d\right )}^{\frac{3}{2}} B c d e^{16} + 45 \, \sqrt{x e + d} B c d^{2} e^{16} + 5 \,{\left (x e + d\right )}^{\frac{3}{2}} B b e^{17} + 5 \,{\left (x e + d\right )}^{\frac{3}{2}} A c e^{17} - 30 \, \sqrt{x e + d} B b d e^{17} - 30 \, \sqrt{x e + d} A c d e^{17} + 15 \, \sqrt{x e + d} A b e^{18}\right )} e^{\left (-20\right )} + \frac{2 \,{\left (B c d^{3} - B b d^{2} e - A c d^{2} e + A b d e^{2}\right )} e^{\left (-4\right )}}{\sqrt{x e + d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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