Optimal. Leaf size=124 \[ \frac{2 b (-2 a B e-A b e+3 b B d)}{e^4 \sqrt{d+e x}}-\frac{2 (b d-a e) (-a B e-2 A b e+3 b B d)}{3 e^4 (d+e x)^{3/2}}+\frac{2 (b d-a e)^2 (B d-A e)}{5 e^4 (d+e x)^{5/2}}+\frac{2 b^2 B \sqrt{d+e x}}{e^4} \]
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Rubi [A] time = 0.0517708, antiderivative size = 124, 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} \[ \frac{2 b (-2 a B e-A b e+3 b B d)}{e^4 \sqrt{d+e x}}-\frac{2 (b d-a e) (-a B e-2 A b e+3 b B d)}{3 e^4 (d+e x)^{3/2}}+\frac{2 (b d-a e)^2 (B d-A e)}{5 e^4 (d+e x)^{5/2}}+\frac{2 b^2 B \sqrt{d+e x}}{e^4} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin{align*} \int \frac{(a+b x)^2 (A+B x)}{(d+e x)^{7/2}} \, dx &=\int \left (\frac{(-b d+a e)^2 (-B d+A e)}{e^3 (d+e x)^{7/2}}+\frac{(-b d+a e) (-3 b B d+2 A b e+a B e)}{e^3 (d+e x)^{5/2}}+\frac{b (-3 b B d+A b e+2 a B e)}{e^3 (d+e x)^{3/2}}+\frac{b^2 B}{e^3 \sqrt{d+e x}}\right ) \, dx\\ &=\frac{2 (b d-a e)^2 (B d-A e)}{5 e^4 (d+e x)^{5/2}}-\frac{2 (b d-a e) (3 b B d-2 A b e-a B e)}{3 e^4 (d+e x)^{3/2}}+\frac{2 b (3 b B d-A b e-2 a B e)}{e^4 \sqrt{d+e x}}+\frac{2 b^2 B \sqrt{d+e x}}{e^4}\\ \end{align*}
Mathematica [A] time = 0.0898908, size = 107, normalized size = 0.86 \[ \frac{2 \left (15 b (d+e x)^2 (-2 a B e-A b e+3 b B d)-5 (d+e x) (b d-a e) (-a B e-2 A b e+3 b B d)+3 (b d-a e)^2 (B d-A e)+15 b^2 B (d+e x)^3\right )}{15 e^4 (d+e x)^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 169, normalized size = 1.4 \begin{align*} -{\frac{-30\,B{b}^{2}{x}^{3}{e}^{3}+30\,A{b}^{2}{e}^{3}{x}^{2}+60\,Bab{e}^{3}{x}^{2}-180\,B{b}^{2}d{e}^{2}{x}^{2}+20\,Aab{e}^{3}x+40\,A{b}^{2}d{e}^{2}x+10\,B{a}^{2}{e}^{3}x+80\,Babd{e}^{2}x-240\,B{b}^{2}{d}^{2}ex+6\,{a}^{2}A{e}^{3}+8\,Aabd{e}^{2}+16\,A{b}^{2}{d}^{2}e+4\,B{a}^{2}d{e}^{2}+32\,Bab{d}^{2}e-96\,B{b}^{2}{d}^{3}}{15\,{e}^{4}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07722, size = 221, normalized size = 1.78 \begin{align*} \frac{2 \,{\left (\frac{15 \, \sqrt{e x + d} B b^{2}}{e^{3}} + \frac{3 \, B b^{2} d^{3} - 3 \, A a^{2} e^{3} - 3 \,{\left (2 \, B a b + A b^{2}\right )} d^{2} e + 3 \,{\left (B a^{2} + 2 \, A a b\right )} d e^{2} + 15 \,{\left (3 \, B b^{2} d -{\left (2 \, B a b + A b^{2}\right )} e\right )}{\left (e x + d\right )}^{2} - 5 \,{\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 )}}{{\left (e x + d\right )}^{\frac{5}{2}} e^{3}}\right )}}{15 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.38872, size = 400, normalized size = 3.23 \begin{align*} \frac{2 \,{\left (15 \, B b^{2} e^{3} x^{3} + 48 \, B b^{2} d^{3} - 3 \, A a^{2} e^{3} - 8 \,{\left (2 \, B a b + A b^{2}\right )} d^{2} e - 2 \,{\left (B a^{2} + 2 \, A a b\right )} d e^{2} + 15 \,{\left (6 \, B b^{2} d e^{2} -{\left (2 \, B a b + A b^{2}\right )} e^{3}\right )} x^{2} + 5 \,{\left (24 \, B b^{2} d^{2} e - 4 \,{\left (2 \, B a b + A b^{2}\right )} d e^{2} -{\left (B a^{2} + 2 \, A a b\right )} e^{3}\right )} x\right )} \sqrt{e x + d}}{15 \,{\left (e^{7} x^{3} + 3 \, d e^{6} x^{2} + 3 \, d^{2} e^{5} x + d^{3} e^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.59072, size = 1015, normalized size = 8.19 \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 [A] time = 1.81845, size = 273, normalized size = 2.2 \begin{align*} 2 \, \sqrt{x e + d} B b^{2} e^{\left (-4\right )} + \frac{2 \,{\left (45 \,{\left (x e + d\right )}^{2} B b^{2} d - 15 \,{\left (x e + d\right )} B b^{2} d^{2} + 3 \, B b^{2} d^{3} - 30 \,{\left (x e + d\right )}^{2} B a b e - 15 \,{\left (x e + d\right )}^{2} A b^{2} e + 20 \,{\left (x e + d\right )} B a b d e + 10 \,{\left (x e + d\right )} A b^{2} d e - 6 \, B a b d^{2} e - 3 \, A b^{2} d^{2} e - 5 \,{\left (x e + d\right )} B a^{2} e^{2} - 10 \,{\left (x e + d\right )} A a b e^{2} + 3 \, B a^{2} d e^{2} + 6 \, A a b d e^{2} - 3 \, A a^{2} e^{3}\right )} e^{\left (-4\right )}}{15 \,{\left (x e + d\right )}^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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