Optimal. Leaf size=128 \[ -\frac{2 b (d+e x)^{7/2} (-2 a B e-A b e+3 b B d)}{7 e^4}+\frac{2 (d+e x)^{5/2} (b d-a e) (-a B e-2 A b e+3 b B d)}{5 e^4}-\frac{2 (d+e x)^{3/2} (b d-a e)^2 (B d-A e)}{3 e^4}+\frac{2 b^2 B (d+e x)^{9/2}}{9 e^4} \]
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Rubi [A] time = 0.0489692, antiderivative size = 128, 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} \[ -\frac{2 b (d+e x)^{7/2} (-2 a B e-A b e+3 b B d)}{7 e^4}+\frac{2 (d+e x)^{5/2} (b d-a e) (-a B e-2 A b e+3 b B d)}{5 e^4}-\frac{2 (d+e x)^{3/2} (b d-a e)^2 (B d-A e)}{3 e^4}+\frac{2 b^2 B (d+e x)^{9/2}}{9 e^4} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin{align*} \int (a+b x)^2 (A+B x) \sqrt{d+e x} \, dx &=\int \left (\frac{(-b d+a e)^2 (-B d+A e) \sqrt{d+e x}}{e^3}+\frac{(-b d+a e) (-3 b B d+2 A b e+a B e) (d+e x)^{3/2}}{e^3}+\frac{b (-3 b B d+A b e+2 a B e) (d+e x)^{5/2}}{e^3}+\frac{b^2 B (d+e x)^{7/2}}{e^3}\right ) \, dx\\ &=-\frac{2 (b d-a e)^2 (B d-A e) (d+e x)^{3/2}}{3 e^4}+\frac{2 (b d-a e) (3 b B d-2 A b e-a B e) (d+e x)^{5/2}}{5 e^4}-\frac{2 b (3 b B d-A b e-2 a B e) (d+e x)^{7/2}}{7 e^4}+\frac{2 b^2 B (d+e x)^{9/2}}{9 e^4}\\ \end{align*}
Mathematica [A] time = 0.0994933, size = 107, normalized size = 0.84 \[ \frac{2 (d+e x)^{3/2} \left (-45 b (d+e x)^2 (-2 a B e-A b e+3 b B d)+63 (d+e x) (b d-a e) (-a B e-2 A b e+3 b B d)-105 (b d-a e)^2 (B d-A e)+35 b^2 B (d+e x)^3\right )}{315 e^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 169, normalized size = 1.3 \begin{align*}{\frac{70\,B{b}^{2}{x}^{3}{e}^{3}+90\,A{b}^{2}{e}^{3}{x}^{2}+180\,Bab{e}^{3}{x}^{2}-60\,B{b}^{2}d{e}^{2}{x}^{2}+252\,Aab{e}^{3}x-72\,A{b}^{2}d{e}^{2}x+126\,B{a}^{2}{e}^{3}x-144\,Babd{e}^{2}x+48\,B{b}^{2}{d}^{2}ex+210\,{a}^{2}A{e}^{3}-168\,Aabd{e}^{2}+48\,A{b}^{2}{d}^{2}e-84\,B{a}^{2}d{e}^{2}+96\,Bab{d}^{2}e-32\,B{b}^{2}{d}^{3}}{315\,{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 = 1.0523, size = 215, normalized size = 1.68 \begin{align*} \frac{2 \,{\left (35 \,{\left (e x + d\right )}^{\frac{9}{2}} B b^{2} - 45 \,{\left (3 \, B b^{2} d -{\left (2 \, B a b + A b^{2}\right )} e\right )}{\left (e x + d\right )}^{\frac{7}{2}} + 63 \,{\left (3 \, B b^{2} d^{2} - 2 \,{\left (2 \, B a b + A b^{2}\right )} d e +{\left (B a^{2} + 2 \, A a b\right )} e^{2}\right )}{\left (e x + d\right )}^{\frac{5}{2}} - 105 \,{\left (B b^{2} d^{3} - A a^{2} e^{3} -{\left (2 \, B a b + A b^{2}\right )} d^{2} e +{\left (B a^{2} + 2 \, A a b\right )} d e^{2}\right )}{\left (e x + d\right )}^{\frac{3}{2}}\right )}}{315 \, e^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.80282, size = 489, normalized size = 3.82 \begin{align*} \frac{2 \,{\left (35 \, B b^{2} e^{4} x^{4} - 16 \, B b^{2} d^{4} + 105 \, A a^{2} d e^{3} + 24 \,{\left (2 \, B a b + A b^{2}\right )} d^{3} e - 42 \,{\left (B a^{2} + 2 \, A a b\right )} d^{2} e^{2} + 5 \,{\left (B b^{2} d e^{3} + 9 \,{\left (2 \, B a b + A b^{2}\right )} e^{4}\right )} x^{3} - 3 \,{\left (2 \, B b^{2} d^{2} e^{2} - 3 \,{\left (2 \, B a b + A b^{2}\right )} d e^{3} - 21 \,{\left (B a^{2} + 2 \, A a b\right )} e^{4}\right )} x^{2} +{\left (8 \, B b^{2} d^{3} e + 105 \, A a^{2} e^{4} - 12 \,{\left (2 \, B a b + A b^{2}\right )} d^{2} e^{2} + 21 \,{\left (B a^{2} + 2 \, A a b\right )} d e^{3}\right )} x\right )} \sqrt{e x + d}}{315 \, e^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.49169, size = 201, normalized size = 1.57 \begin{align*} \frac{2 \left (\frac{B b^{2} \left (d + e x\right )^{\frac{9}{2}}}{9 e^{3}} + \frac{\left (d + e x\right )^{\frac{7}{2}} \left (A b^{2} e + 2 B a b e - 3 B b^{2} d\right )}{7 e^{3}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (2 A a b e^{2} - 2 A b^{2} d e + B a^{2} e^{2} - 4 B a b d e + 3 B b^{2} d^{2}\right )}{5 e^{3}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (A a^{2} e^{3} - 2 A a b d e^{2} + A b^{2} d^{2} e - B a^{2} d e^{2} + 2 B a b d^{2} e - B b^{2} d^{3}\right )}{3 e^{3}}\right )}{e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.40587, size = 294, normalized size = 2.3 \begin{align*} \frac{2}{315} \,{\left (21 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} B a^{2} e^{\left (-1\right )} + 42 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} A a b e^{\left (-1\right )} + 6 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2}\right )} B a b e^{\left (-2\right )} + 3 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2}\right )} A b^{2} e^{\left (-2\right )} +{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} d + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3}\right )} B b^{2} e^{\left (-3\right )} + 105 \,{\left (x e + d\right )}^{\frac{3}{2}} A a^{2}\right )} e^{\left (-1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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