Optimal. Leaf size=112 \[ -\frac{2 \left (a B e^2-2 A c d e+3 B c d^2\right )}{e^4 \sqrt{d+e x}}+\frac{2 \left (a e^2+c d^2\right ) (B d-A e)}{3 e^4 (d+e x)^{3/2}}-\frac{2 c \sqrt{d+e x} (3 B d-A e)}{e^4}+\frac{2 B c (d+e x)^{3/2}}{3 e^4} \]
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Rubi [A] time = 0.0496668, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {772} \[ -\frac{2 \left (a B e^2-2 A c d e+3 B c d^2\right )}{e^4 \sqrt{d+e x}}+\frac{2 \left (a e^2+c d^2\right ) (B d-A e)}{3 e^4 (d+e x)^{3/2}}-\frac{2 c \sqrt{d+e x} (3 B d-A e)}{e^4}+\frac{2 B c (d+e x)^{3/2}}{3 e^4} \]
Antiderivative was successfully verified.
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Rule 772
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (a+c x^2\right )}{(d+e x)^{5/2}} \, dx &=\int \left (\frac{(-B d+A e) \left (c d^2+a e^2\right )}{e^3 (d+e x)^{5/2}}+\frac{3 B c d^2-2 A c d e+a B e^2}{e^3 (d+e x)^{3/2}}+\frac{c (-3 B d+A e)}{e^3 \sqrt{d+e x}}+\frac{B c \sqrt{d+e x}}{e^3}\right ) \, dx\\ &=\frac{2 (B d-A e) \left (c d^2+a e^2\right )}{3 e^4 (d+e x)^{3/2}}-\frac{2 \left (3 B c d^2-2 A c d e+a B e^2\right )}{e^4 \sqrt{d+e x}}-\frac{2 c (3 B d-A e) \sqrt{d+e x}}{e^4}+\frac{2 B c (d+e x)^{3/2}}{3 e^4}\\ \end{align*}
Mathematica [A] time = 0.0779698, size = 94, normalized size = 0.84 \[ -\frac{2 \left (a A e^3+a B e^2 (2 d+3 e x)-A c e \left (8 d^2+12 d e x+3 e^2 x^2\right )+B c \left (24 d^2 e x+16 d^3+6 d e^2 x^2-e^3 x^3\right )\right )}{3 e^4 (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 100, normalized size = 0.9 \begin{align*} -{\frac{-2\,Bc{x}^{3}{e}^{3}-6\,Ac{e}^{3}{x}^{2}+12\,Bcd{e}^{2}{x}^{2}-24\,Acd{e}^{2}x+6\,Ba{e}^{3}x+48\,Bc{d}^{2}ex+2\,aA{e}^{3}-16\,Ac{d}^{2}e+4\,aBd{e}^{2}+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 = 1.02886, size = 146, normalized size = 1.3 \begin{align*} \frac{2 \,{\left (\frac{{\left (e x + d\right )}^{\frac{3}{2}} B c - 3 \,{\left (3 \, B c d - A c e\right )} \sqrt{e x + d}}{e^{3}} + \frac{B c d^{3} - A c d^{2} e + B a d e^{2} - A a e^{3} - 3 \,{\left (3 \, B c d^{2} - 2 \, A c d e + B a e^{2}\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 = 1.47714, size = 259, normalized size = 2.31 \begin{align*} \frac{2 \,{\left (B c e^{3} x^{3} - 16 \, B c d^{3} + 8 \, A c d^{2} e - 2 \, B a d e^{2} - A a e^{3} - 3 \,{\left (2 \, B c d e^{2} - A c e^{3}\right )} x^{2} - 3 \,{\left (8 \, B c d^{2} e - 4 \, A c d e^{2} + B a e^{3}\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 = 1.51504, size = 449, normalized size = 4.01 \begin{align*} \begin{cases} - \frac{2 A a e^{3}}{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{4 B a d e^{2}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} - \frac{6 B a e^{3} x}{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{A a x + \frac{A c x^{3}}{3} + \frac{B a x^{2}}{2} + \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 = 1.16531, size = 170, normalized size = 1.52 \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} 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 )} A c d e + A c d^{2} e + 3 \,{\left (x e + d\right )} B a e^{2} - B a d e^{2} + A a e^{3}\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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