Optimal. Leaf size=171 \[ -\frac{2 b^2 (d+e x)^{7/2} (-3 a B e-A b e+4 b B d)}{7 e^5}+\frac{6 b (d+e x)^{5/2} (b d-a e) (-a B e-A b e+2 b B d)}{5 e^5}-\frac{2 (d+e x)^{3/2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{3 e^5}+\frac{2 \sqrt{d+e x} (b d-a e)^3 (B d-A e)}{e^5}+\frac{2 b^3 B (d+e x)^{9/2}}{9 e^5} \]
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Rubi [A] time = 0.0682577, antiderivative size = 171, 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 = {77} \[ -\frac{2 b^2 (d+e x)^{7/2} (-3 a B e-A b e+4 b B d)}{7 e^5}+\frac{6 b (d+e x)^{5/2} (b d-a e) (-a B e-A b e+2 b B d)}{5 e^5}-\frac{2 (d+e x)^{3/2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{3 e^5}+\frac{2 \sqrt{d+e x} (b d-a e)^3 (B d-A e)}{e^5}+\frac{2 b^3 B (d+e x)^{9/2}}{9 e^5} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin{align*} \int \frac{(a+b x)^3 (A+B x)}{\sqrt{d+e x}} \, dx &=\int \left (\frac{(-b d+a e)^3 (-B d+A e)}{e^4 \sqrt{d+e x}}+\frac{(-b d+a e)^2 (-4 b B d+3 A b e+a B e) \sqrt{d+e x}}{e^4}-\frac{3 b (b d-a e) (-2 b B d+A b e+a B e) (d+e x)^{3/2}}{e^4}+\frac{b^2 (-4 b B d+A b e+3 a B e) (d+e x)^{5/2}}{e^4}+\frac{b^3 B (d+e x)^{7/2}}{e^4}\right ) \, dx\\ &=\frac{2 (b d-a e)^3 (B d-A e) \sqrt{d+e x}}{e^5}-\frac{2 (b d-a e)^2 (4 b B d-3 A b e-a B e) (d+e x)^{3/2}}{3 e^5}+\frac{6 b (b d-a e) (2 b B d-A b e-a B e) (d+e x)^{5/2}}{5 e^5}-\frac{2 b^2 (4 b B d-A b e-3 a B e) (d+e x)^{7/2}}{7 e^5}+\frac{2 b^3 B (d+e x)^{9/2}}{9 e^5}\\ \end{align*}
Mathematica [A] time = 0.0783036, size = 145, normalized size = 0.85 \[ \frac{2 \sqrt{d+e x} \left (-45 b^2 (d+e x)^3 (-3 a B e-A b e+4 b B d)+189 b (d+e x)^2 (b d-a e) (-a B e-A b e+2 b B d)-105 (d+e x) (b d-a e)^2 (-a B e-3 A b e+4 b B d)+315 (b d-a e)^3 (B d-A e)+35 b^3 B (d+e x)^4\right )}{315 e^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 301, normalized size = 1.8 \begin{align*}{\frac{70\,{b}^{3}B{x}^{4}{e}^{4}+90\,A{b}^{3}{e}^{4}{x}^{3}+270\,Ba{b}^{2}{e}^{4}{x}^{3}-80\,B{b}^{3}d{e}^{3}{x}^{3}+378\,Aa{b}^{2}{e}^{4}{x}^{2}-108\,A{b}^{3}d{e}^{3}{x}^{2}+378\,B{a}^{2}b{e}^{4}{x}^{2}-324\,Ba{b}^{2}d{e}^{3}{x}^{2}+96\,B{b}^{3}{d}^{2}{e}^{2}{x}^{2}+630\,A{a}^{2}b{e}^{4}x-504\,Aa{b}^{2}d{e}^{3}x+144\,A{b}^{3}{d}^{2}{e}^{2}x+210\,B{a}^{3}{e}^{4}x-504\,B{a}^{2}bd{e}^{3}x+432\,Ba{b}^{2}{d}^{2}{e}^{2}x-128\,B{b}^{3}{d}^{3}ex+630\,{a}^{3}A{e}^{4}-1260\,A{a}^{2}bd{e}^{3}+1008\,Aa{b}^{2}{d}^{2}{e}^{2}-288\,A{b}^{3}{d}^{3}e-420\,B{a}^{3}d{e}^{3}+1008\,B{a}^{2}b{d}^{2}{e}^{2}-864\,Ba{b}^{2}{d}^{3}e+256\,B{b}^{3}{d}^{4}}{315\,{e}^{5}}\sqrt{ex+d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04201, size = 358, normalized size = 2.09 \begin{align*} \frac{2 \,{\left (35 \,{\left (e x + d\right )}^{\frac{9}{2}} B b^{3} - 45 \,{\left (4 \, B b^{3} d -{\left (3 \, B a b^{2} + A b^{3}\right )} e\right )}{\left (e x + d\right )}^{\frac{7}{2}} + 189 \,{\left (2 \, B b^{3} d^{2} -{\left (3 \, B a b^{2} + A b^{3}\right )} d e +{\left (B a^{2} b + A a b^{2}\right )} e^{2}\right )}{\left (e x + d\right )}^{\frac{5}{2}} - 105 \,{\left (4 \, B b^{3} d^{3} - 3 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e + 6 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{2} -{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{3}\right )}{\left (e x + d\right )}^{\frac{3}{2}} + 315 \,{\left (B b^{3} d^{4} + A a^{3} e^{4} -{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 3 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} -{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3}\right )} \sqrt{e x + d}\right )}}{315 \, e^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.35619, size = 579, normalized size = 3.39 \begin{align*} \frac{2 \,{\left (35 \, B b^{3} e^{4} x^{4} + 128 \, B b^{3} d^{4} + 315 \, A a^{3} e^{4} - 144 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 504 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} - 210 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} - 5 \,{\left (8 \, B b^{3} d e^{3} - 9 \,{\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 3 \,{\left (16 \, B b^{3} d^{2} e^{2} - 18 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 63 \,{\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} -{\left (64 \, B b^{3} d^{3} e - 72 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 252 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{3} - 105 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x\right )} \sqrt{e x + d}}{315 \, e^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 66.7763, size = 916, normalized size = 5.36 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.37284, size = 466, normalized size = 2.73 \begin{align*} \frac{2}{315} \,{\left (105 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{x e + d} d\right )} B a^{3} e^{\left (-1\right )} + 315 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{x e + d} d\right )} A a^{2} b e^{\left (-1\right )} + 63 \,{\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 a^{2} b e^{\left (-2\right )} + 63 \,{\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 a b^{2} e^{\left (-2\right )} + 27 \,{\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 a b^{2} e^{\left (-3\right )} + 9 \,{\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 )} A b^{3} e^{\left (-3\right )} +{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} - 180 \,{\left (x e + d\right )}^{\frac{7}{2}} d + 378 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} - 420 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} + 315 \, \sqrt{x e + d} d^{4}\right )} B b^{3} e^{\left (-4\right )} + 315 \, \sqrt{x e + d} A a^{3}\right )} e^{\left (-1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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