Optimal. Leaf size=81 \[ -\frac{2 (d+e x)^{3/2} (-a B e-A b e+2 b B d)}{3 e^3}+\frac{2 \sqrt{d+e x} (b d-a e) (B d-A e)}{e^3}+\frac{2 b B (d+e x)^{5/2}}{5 e^3} \]
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Rubi [A] time = 0.0344749, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {77} \[ -\frac{2 (d+e x)^{3/2} (-a B e-A b e+2 b B d)}{3 e^3}+\frac{2 \sqrt{d+e x} (b d-a e) (B d-A e)}{e^3}+\frac{2 b B (d+e x)^{5/2}}{5 e^3} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin{align*} \int \frac{(a+b x) (A+B x)}{\sqrt{d+e x}} \, dx &=\int \left (\frac{(-b d+a e) (-B d+A e)}{e^2 \sqrt{d+e x}}+\frac{(-2 b B d+A b e+a B e) \sqrt{d+e x}}{e^2}+\frac{b B (d+e x)^{3/2}}{e^2}\right ) \, dx\\ &=\frac{2 (b d-a e) (B d-A e) \sqrt{d+e x}}{e^3}-\frac{2 (2 b B d-A b e-a B e) (d+e x)^{3/2}}{3 e^3}+\frac{2 b B (d+e x)^{5/2}}{5 e^3}\\ \end{align*}
Mathematica [A] time = 0.0533718, size = 68, normalized size = 0.84 \[ \frac{2 \sqrt{d+e x} \left (5 a e (3 A e-2 B d+B e x)+5 A b e (e x-2 d)+b B \left (8 d^2-4 d e x+3 e^2 x^2\right )\right )}{15 e^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 73, normalized size = 0.9 \begin{align*}{\frac{6\,bB{x}^{2}{e}^{2}+10\,Ab{e}^{2}x+10\,Ba{e}^{2}x-8\,Bbdex+30\,aA{e}^{2}-20\,Abde-20\,Bade+16\,bB{d}^{2}}{15\,{e}^{3}}\sqrt{ex+d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.1957, size = 101, normalized size = 1.25 \begin{align*} \frac{2 \,{\left (3 \,{\left (e x + d\right )}^{\frac{5}{2}} B b - 5 \,{\left (2 \, B b d -{\left (B a + A b\right )} e\right )}{\left (e x + d\right )}^{\frac{3}{2}} + 15 \,{\left (B b d^{2} + A a e^{2} -{\left (B a + A b\right )} d e\right )} \sqrt{e x + d}\right )}}{15 \, e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.56279, size = 165, normalized size = 2.04 \begin{align*} \frac{2 \,{\left (3 \, B b e^{2} x^{2} + 8 \, B b d^{2} + 15 \, A a e^{2} - 10 \,{\left (B a + A b\right )} d e -{\left (4 \, B b d e - 5 \,{\left (B a + A b\right )} e^{2}\right )} x\right )} \sqrt{e x + d}}{15 \, e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 19.3251, size = 311, normalized size = 3.84 \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 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 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 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}}}{e} & \text{for}\: e \neq 0 \\\frac{A a x + \frac{B b x^{3}}{3} + \frac{x^{2} \left (A b + B a\right )}{2}}{\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 = 2.08091, size = 147, normalized size = 1.81 \begin{align*} \frac{2}{15} \,{\left (5 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{x e + d} d\right )} B a e^{\left (-1\right )} + 5 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{x e + d} d\right )} A b e^{\left (-1\right )} +{\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 )} + 15 \, \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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