Optimal. Leaf size=114 \[ \frac{2 (d+e x)^{3/2} \left (a B e^2-2 A c d e+3 B c d^2\right )}{3 e^4}-\frac{2 \sqrt{d+e x} \left (a e^2+c d^2\right ) (B d-A e)}{e^4}-\frac{2 c (d+e x)^{5/2} (3 B d-A e)}{5 e^4}+\frac{2 B c (d+e x)^{7/2}}{7 e^4} \]
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Rubi [A] time = 0.0487892, antiderivative size = 114, 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} \[ \frac{2 (d+e x)^{3/2} \left (a B e^2-2 A c d e+3 B c d^2\right )}{3 e^4}-\frac{2 \sqrt{d+e x} \left (a e^2+c d^2\right ) (B d-A e)}{e^4}-\frac{2 c (d+e x)^{5/2} (3 B d-A e)}{5 e^4}+\frac{2 B c (d+e x)^{7/2}}{7 e^4} \]
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 )}{\sqrt{d+e x}} \, dx &=\int \left (\frac{(-B d+A e) \left (c d^2+a e^2\right )}{e^3 \sqrt{d+e x}}+\frac{\left (3 B c d^2-2 A c d e+a B e^2\right ) \sqrt{d+e x}}{e^3}+\frac{c (-3 B d+A e) (d+e x)^{3/2}}{e^3}+\frac{B c (d+e x)^{5/2}}{e^3}\right ) \, dx\\ &=-\frac{2 (B d-A e) \left (c d^2+a e^2\right ) \sqrt{d+e x}}{e^4}+\frac{2 \left (3 B c d^2-2 A c d e+a B e^2\right ) (d+e x)^{3/2}}{3 e^4}-\frac{2 c (3 B d-A 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 = 0.0843641, size = 96, normalized size = 0.84 \[ \frac{2 \sqrt{d+e x} \left (105 a A e^3+35 a B e^2 (e x-2 d)+7 A c e \left (8 d^2-4 d e x+3 e^2 x^2\right )-3 B c \left (-8 d^2 e x+16 d^3+6 d e^2 x^2-5 e^3 x^3\right )\right )}{105 e^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 101, normalized size = 0.9 \begin{align*}{\frac{30\,Bc{x}^{3}{e}^{3}+42\,Ac{e}^{3}{x}^{2}-36\,Bcd{e}^{2}{x}^{2}-56\,Acd{e}^{2}x+70\,Ba{e}^{3}x+48\,Bc{d}^{2}ex+210\,aA{e}^{3}+112\,Ac{d}^{2}e-140\,aBd{e}^{2}-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 = 1.029, size = 140, normalized size = 1.23 \begin{align*} \frac{2 \,{\left (15 \,{\left (e x + d\right )}^{\frac{7}{2}} B c - 21 \,{\left (3 \, B c d - A c e\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 35 \,{\left (3 \, B c d^{2} - 2 \, A c d e + B a e^{2}\right )}{\left (e x + d\right )}^{\frac{3}{2}} - 105 \,{\left (B c d^{3} - A c d^{2} e + B a d e^{2} - A a e^{3}\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 = 1.6927, size = 243, normalized size = 2.13 \begin{align*} \frac{2 \,{\left (15 \, B c e^{3} x^{3} - 48 \, B c d^{3} + 56 \, A c d^{2} e - 70 \, B a d e^{2} + 105 \, A a e^{3} - 3 \,{\left (6 \, B c d e^{2} - 7 \, A c e^{3}\right )} x^{2} +{\left (24 \, B c d^{2} e - 28 \, A c d e^{2} + 35 \, B a e^{3}\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 = 28.0998, size = 374, normalized size = 3.28 \begin{align*} \begin{cases} - \frac{\frac{2 A a d}{\sqrt{d + e x}} + 2 A a \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right ) + \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 a d \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right )}{e} + \frac{2 B a \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 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{A a x + \frac{A c x^{3}}{3} + \frac{B a x^{2}}{2} + \frac{B c x^{4}}{4}}{\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 = 1.14139, size = 186, normalized size = 1.63 \begin{align*} \frac{2}{105} \,{\left (35 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{x e + d} d\right )} B a 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 )} 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 )} + 105 \, \sqrt{x e + d} A a\right )} e^{\left (-1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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