Optimal. Leaf size=122 \[ -\frac{2 \sqrt{d+e x} (-A c e-b B e+3 B c d)}{e^4}-\frac{2 (B d (3 c d-2 b e)-A e (2 c d-b e))}{e^4 \sqrt{d+e x}}+\frac{2 d (B d-A e) (c d-b e)}{3 e^4 (d+e x)^{3/2}}+\frac{2 B c (d+e x)^{3/2}}{3 e^4} \]
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Rubi [A] time = 0.0744253, 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} \[ -\frac{2 \sqrt{d+e x} (-A c e-b B e+3 B c d)}{e^4}-\frac{2 (B d (3 c d-2 b e)-A e (2 c d-b e))}{e^4 \sqrt{d+e x}}+\frac{2 d (B d-A e) (c d-b e)}{3 e^4 (d+e x)^{3/2}}+\frac{2 B c (d+e x)^{3/2}}{3 e^4} \]
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)^{5/2}} \, dx &=\int \left (-\frac{d (B d-A e) (c d-b e)}{e^3 (d+e x)^{5/2}}+\frac{B d (3 c d-2 b e)-A e (2 c d-b e)}{e^3 (d+e x)^{3/2}}+\frac{-3 B c d+b B e+A c e}{e^3 \sqrt{d+e x}}+\frac{B c \sqrt{d+e x}}{e^3}\right ) \, dx\\ &=\frac{2 d (B d-A e) (c d-b e)}{3 e^4 (d+e x)^{3/2}}-\frac{2 (B d (3 c d-2 b e)-A e (2 c d-b e))}{e^4 \sqrt{d+e x}}-\frac{2 (3 B c d-b B e-A c e) \sqrt{d+e x}}{e^4}+\frac{2 B c (d+e x)^{3/2}}{3 e^4}\\ \end{align*}
Mathematica [A] time = 0.0517101, size = 110, normalized size = 0.9 \[ \frac{2 \left (A e \left (c \left (8 d^2+12 d e x+3 e^2 x^2\right )-b e (2 d+3 e x)\right )+B \left (b e \left (8 d^2+12 d e x+3 e^2 x^2\right )+c \left (-24 d^2 e x-16 d^3-6 d e^2 x^2+e^3 x^3\right )\right )\right )}{3 e^4 (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 121, normalized size = 1. \begin{align*} -{\frac{-2\,Bc{x}^{3}{e}^{3}-6\,Ac{e}^{3}{x}^{2}-6\,Bb{e}^{3}{x}^{2}+12\,Bcd{e}^{2}{x}^{2}+6\,Ab{e}^{3}x-24\,Acd{e}^{2}x-24\,Bbd{e}^{2}x+48\,Bc{d}^{2}ex+4\,Abd{e}^{2}-16\,Ac{d}^{2}e-16\,Bb{d}^{2}e+32\,Bc{d}^{3}}{3\,{e}^{4}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05466, size = 157, normalized size = 1.29 \begin{align*} \frac{2 \,{\left (\frac{{\left (e x + d\right )}^{\frac{3}{2}} B c - 3 \,{\left (3 \, B c d -{\left (B b + A c\right )} e\right )} \sqrt{e x + d}}{e^{3}} + \frac{B c d^{3} + A b d e^{2} -{\left (B b + A c\right )} d^{2} e - 3 \,{\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 )}}{{\left (e x + d\right )}^{\frac{3}{2}} e^{3}}\right )}}{3 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.69689, size = 278, normalized size = 2.28 \begin{align*} \frac{2 \,{\left (B c e^{3} x^{3} - 16 \, B c d^{3} - 2 \, A b d e^{2} + 8 \,{\left (B b + A c\right )} d^{2} e - 3 \,{\left (2 \, B c d e^{2} -{\left (B b + A c\right )} e^{3}\right )} x^{2} - 3 \,{\left (8 \, B c d^{2} e + A b e^{3} - 4 \,{\left (B b + A c\right )} d e^{2}\right )} x\right )} \sqrt{e x + d}}{3 \,{\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.81655, size = 539, normalized size = 4.42 \begin{align*} \begin{cases} - \frac{4 A b d e^{2}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} - \frac{6 A b e^{3} x}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} + \frac{16 A c d^{2} e}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} + \frac{24 A c d e^{2} x}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} + \frac{6 A c e^{3} x^{2}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} + \frac{16 B b d^{2} e}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} + \frac{24 B b d e^{2} x}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} + \frac{6 B b e^{3} x^{2}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} - \frac{32 B c d^{3}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} - \frac{48 B c d^{2} e x}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} - \frac{12 B c d e^{2} x^{2}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} + \frac{2 B c e^{3} x^{3}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} & \text{for}\: e \neq 0 \\\frac{\frac{A b x^{2}}{2} + \frac{A c x^{3}}{3} + \frac{B b x^{3}}{3} + \frac{B c x^{4}}{4}}{d^{\frac{5}{2}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23992, size = 211, normalized size = 1.73 \begin{align*} \frac{2}{3} \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} B c e^{8} - 9 \, \sqrt{x e + d} B c d e^{8} + 3 \, \sqrt{x e + d} B b e^{9} + 3 \, \sqrt{x e + d} A c e^{9}\right )} e^{\left (-12\right )} - \frac{2 \,{\left (9 \,{\left (x e + d\right )} B c d^{2} - B c d^{3} - 6 \,{\left (x e + d\right )} B b d e - 6 \,{\left (x e + d\right )} A c d e + B b d^{2} e + A c d^{2} e + 3 \,{\left (x e + d\right )} A b e^{2} - A b d e^{2}\right )} e^{\left (-4\right )}}{3 \,{\left (x e + d\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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