Optimal. Leaf size=124 \[ -\frac{2 (d+e x)^{5/2} (-A c e-b B e+3 B c d)}{5 e^4}+\frac{2 (d+e x)^{3/2} (B d (3 c d-2 b e)-A e (2 c d-b e))}{3 e^4}-\frac{2 d \sqrt{d+e x} (B d-A e) (c d-b e)}{e^4}+\frac{2 B c (d+e x)^{7/2}}{7 e^4} \]
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Rubi [A] time = 0.0764481, antiderivative size = 124, 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} \[ -\frac{2 (d+e x)^{5/2} (-A c e-b B e+3 B c d)}{5 e^4}+\frac{2 (d+e x)^{3/2} (B d (3 c d-2 b e)-A e (2 c d-b e))}{3 e^4}-\frac{2 d \sqrt{d+e x} (B d-A e) (c d-b e)}{e^4}+\frac{2 B c (d+e x)^{7/2}}{7 e^4} \]
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 )}{\sqrt{d+e x}} \, dx &=\int \left (-\frac{d (B d-A e) (c d-b e)}{e^3 \sqrt{d+e x}}+\frac{(B d (3 c d-2 b e)-A e (2 c d-b e)) \sqrt{d+e x}}{e^3}+\frac{(-3 B c d+b B e+A c e) (d+e x)^{3/2}}{e^3}+\frac{B c (d+e x)^{5/2}}{e^3}\right ) \, dx\\ &=-\frac{2 d (B d-A e) (c d-b e) \sqrt{d+e x}}{e^4}+\frac{2 (B d (3 c d-2 b e)-A e (2 c d-b e)) (d+e x)^{3/2}}{3 e^4}-\frac{2 (3 B c d-b B e-A c e) (d+e x)^{5/2}}{5 e^4}+\frac{2 B c (d+e x)^{7/2}}{7 e^4}\\ \end{align*}
Mathematica [A] time = 0.0880854, size = 113, normalized size = 0.91 \[ \frac{2 \sqrt{d+e x} \left (7 A e \left (5 b e (e x-2 d)+c \left (8 d^2-4 d e x+3 e^2 x^2\right )\right )+B \left (7 b e \left (8 d^2-4 d e x+3 e^2 x^2\right )-3 c \left (-8 d^2 e x+16 d^3+6 d e^2 x^2-5 e^3 x^3\right )\right )\right )}{105 e^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 121, normalized size = 1. \begin{align*} -{\frac{-30\,Bc{x}^{3}{e}^{3}-42\,Ac{e}^{3}{x}^{2}-42\,Bb{e}^{3}{x}^{2}+36\,Bcd{e}^{2}{x}^{2}-70\,Ab{e}^{3}x+56\,Acd{e}^{2}x+56\,Bbd{e}^{2}x-48\,Bc{d}^{2}ex+140\,Abd{e}^{2}-112\,Ac{d}^{2}e-112\,Bb{d}^{2}e+96\,Bc{d}^{3}}{105\,{e}^{4}}\sqrt{ex+d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.09575, size = 151, normalized size = 1.22 \begin{align*} \frac{2 \,{\left (15 \,{\left (e x + d\right )}^{\frac{7}{2}} B c - 21 \,{\left (3 \, B c d -{\left (B b + A c\right )} e\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 35 \,{\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{3}{2}} - 105 \,{\left (B c d^{3} + A b d e^{2} -{\left (B b + A c\right )} d^{2} e\right )} \sqrt{e x + d}\right )}}{105 \, e^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.74753, size = 257, normalized size = 2.07 \begin{align*} \frac{2 \,{\left (15 \, B c e^{3} x^{3} - 48 \, B c d^{3} - 70 \, A b d e^{2} + 56 \,{\left (B b + A c\right )} d^{2} e - 3 \,{\left (6 \, B c d e^{2} - 7 \,{\left (B b + A c\right )} e^{3}\right )} x^{2} +{\left (24 \, B c d^{2} e + 35 \, A b e^{3} - 28 \,{\left (B b + A c\right )} d e^{2}\right )} x\right )} \sqrt{e x + d}}{105 \, e^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 45.1931, size = 430, normalized size = 3.47 \begin{align*} \begin{cases} - \frac{\frac{2 A b d \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right )}{e} + \frac{2 A b \left (\frac{d^{2}}{\sqrt{d + e x}} + 2 d \sqrt{d + e x} - \frac{\left (d + e x\right )^{\frac{3}{2}}}{3}\right )}{e} + \frac{2 A c d \left (\frac{d^{2}}{\sqrt{d + e x}} + 2 d \sqrt{d + e x} - \frac{\left (d + e x\right )^{\frac{3}{2}}}{3}\right )}{e^{2}} + \frac{2 A c \left (- \frac{d^{3}}{\sqrt{d + e x}} - 3 d^{2} \sqrt{d + e x} + d \left (d + e x\right )^{\frac{3}{2}} - \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e^{2}} + \frac{2 B b d \left (\frac{d^{2}}{\sqrt{d + e x}} + 2 d \sqrt{d + e x} - \frac{\left (d + e x\right )^{\frac{3}{2}}}{3}\right )}{e^{2}} + \frac{2 B b \left (- \frac{d^{3}}{\sqrt{d + e x}} - 3 d^{2} \sqrt{d + e x} + d \left (d + e x\right )^{\frac{3}{2}} - \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e^{2}} + \frac{2 B c d \left (- \frac{d^{3}}{\sqrt{d + e x}} - 3 d^{2} \sqrt{d + e x} + d \left (d + e x\right )^{\frac{3}{2}} - \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e^{3}} + \frac{2 B c \left (\frac{d^{4}}{\sqrt{d + e x}} + 4 d^{3} \sqrt{d + e x} - 2 d^{2} \left (d + e x\right )^{\frac{3}{2}} + \frac{4 d \left (d + e x\right )^{\frac{5}{2}}}{5} - \frac{\left (d + e x\right )^{\frac{7}{2}}}{7}\right )}{e^{3}}}{e} & \text{for}\: e \neq 0 \\\frac{\frac{A b x^{2}}{2} + \frac{B c x^{4}}{4} + \frac{x^{3} \left (A c + B b\right )}{3}}{\sqrt{d}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25621, size = 225, normalized size = 1.81 \begin{align*} \frac{2}{105} \,{\left (35 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{x e + d} d\right )} A b e^{\left (-1\right )} + 7 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} d + 15 \, \sqrt{x e + d} d^{2}\right )} B b e^{\left (-2\right )} + 7 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} d + 15 \, \sqrt{x e + d} d^{2}\right )} A c e^{\left (-2\right )} + 3 \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} - 21 \,{\left (x e + d\right )}^{\frac{5}{2}} d + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} - 35 \, \sqrt{x e + d} d^{3}\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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