Optimal. Leaf size=214 \[ \frac{4 c \sqrt{d+e x} \left (a B e^2-2 A c d e+5 B c d^2\right )}{e^6}+\frac{4 c \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{e^6 \sqrt{d+e x}}-\frac{2 \left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{3 e^6 (d+e x)^{3/2}}+\frac{2 \left (a e^2+c d^2\right )^2 (B d-A e)}{5 e^6 (d+e x)^{5/2}}-\frac{2 c^2 (d+e x)^{3/2} (5 B d-A e)}{3 e^6}+\frac{2 B c^2 (d+e x)^{5/2}}{5 e^6} \]
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Rubi [A] time = 0.0916629, antiderivative size = 214, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {772} \[ \frac{4 c \sqrt{d+e x} \left (a B e^2-2 A c d e+5 B c d^2\right )}{e^6}+\frac{4 c \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{e^6 \sqrt{d+e x}}-\frac{2 \left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{3 e^6 (d+e x)^{3/2}}+\frac{2 \left (a e^2+c d^2\right )^2 (B d-A e)}{5 e^6 (d+e x)^{5/2}}-\frac{2 c^2 (d+e x)^{3/2} (5 B d-A e)}{3 e^6}+\frac{2 B c^2 (d+e x)^{5/2}}{5 e^6} \]
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 )^2}{(d+e x)^{7/2}} \, dx &=\int \left (\frac{(-B d+A e) \left (c d^2+a e^2\right )^2}{e^5 (d+e x)^{7/2}}+\frac{\left (c d^2+a e^2\right ) \left (5 B c d^2-4 A c d e+a B e^2\right )}{e^5 (d+e x)^{5/2}}+\frac{2 c \left (-5 B c d^3+3 A c d^2 e-3 a B d e^2+a A e^3\right )}{e^5 (d+e x)^{3/2}}-\frac{2 c \left (-5 B c d^2+2 A c d e-a B e^2\right )}{e^5 \sqrt{d+e x}}+\frac{c^2 (-5 B d+A e) \sqrt{d+e x}}{e^5}+\frac{B c^2 (d+e x)^{3/2}}{e^5}\right ) \, dx\\ &=\frac{2 (B d-A e) \left (c d^2+a e^2\right )^2}{5 e^6 (d+e x)^{5/2}}-\frac{2 \left (c d^2+a e^2\right ) \left (5 B c d^2-4 A c d e+a B e^2\right )}{3 e^6 (d+e x)^{3/2}}+\frac{4 c \left (5 B c d^3-3 A c d^2 e+3 a B d e^2-a A e^3\right )}{e^6 \sqrt{d+e x}}+\frac{4 c \left (5 B c d^2-2 A c d e+a B e^2\right ) \sqrt{d+e x}}{e^6}-\frac{2 c^2 (5 B d-A e) (d+e x)^{3/2}}{3 e^6}+\frac{2 B c^2 (d+e x)^{5/2}}{5 e^6}\\ \end{align*}
Mathematica [A] time = 0.152679, size = 212, normalized size = 0.99 \[ -\frac{2 \left (3 a^2 A e^5+a^2 B e^4 (2 d+5 e x)+2 a A c e^3 \left (8 d^2+20 d e x+15 e^2 x^2\right )-6 a B c e^2 \left (40 d^2 e x+16 d^3+30 d e^2 x^2+5 e^3 x^3\right )+A c^2 e \left (240 d^2 e^2 x^2+320 d^3 e x+128 d^4+40 d e^3 x^3-5 e^4 x^4\right )-B c^2 \left (480 d^3 e^2 x^2+80 d^2 e^3 x^3+640 d^4 e x+256 d^5-10 d e^4 x^4+3 e^5 x^5\right )\right )}{15 e^6 (d+e x)^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 259, normalized size = 1.2 \begin{align*} -{\frac{-6\,B{c}^{2}{x}^{5}{e}^{5}-10\,A{c}^{2}{e}^{5}{x}^{4}+20\,B{c}^{2}d{e}^{4}{x}^{4}+80\,A{c}^{2}d{e}^{4}{x}^{3}-60\,Bac{e}^{5}{x}^{3}-160\,B{c}^{2}{d}^{2}{e}^{3}{x}^{3}+60\,Aac{e}^{5}{x}^{2}+480\,A{c}^{2}{d}^{2}{e}^{3}{x}^{2}-360\,Bacd{e}^{4}{x}^{2}-960\,B{c}^{2}{d}^{3}{e}^{2}{x}^{2}+80\,Aacd{e}^{4}x+640\,A{c}^{2}{d}^{3}{e}^{2}x+10\,B{a}^{2}{e}^{5}x-480\,Bac{d}^{2}{e}^{3}x-1280\,B{c}^{2}{d}^{4}ex+6\,A{a}^{2}{e}^{5}+32\,A{d}^{2}ac{e}^{3}+256\,A{d}^{4}{c}^{2}e+4\,B{a}^{2}d{e}^{4}-192\,aBc{d}^{3}{e}^{2}-512\,B{c}^{2}{d}^{5}}{15\,{e}^{6}} \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 = 1.00067, size = 344, normalized size = 1.61 \begin{align*} \frac{2 \,{\left (\frac{3 \,{\left (e x + d\right )}^{\frac{5}{2}} B c^{2} - 5 \,{\left (5 \, B c^{2} d - A c^{2} e\right )}{\left (e x + d\right )}^{\frac{3}{2}} + 30 \,{\left (5 \, B c^{2} d^{2} - 2 \, A c^{2} d e + B a c e^{2}\right )} \sqrt{e x + d}}{e^{5}} + \frac{3 \, B c^{2} d^{5} - 3 \, A c^{2} d^{4} e + 6 \, B a c d^{3} e^{2} - 6 \, A a c d^{2} e^{3} + 3 \, B a^{2} d e^{4} - 3 \, A a^{2} e^{5} + 30 \,{\left (5 \, B c^{2} d^{3} - 3 \, A c^{2} d^{2} e + 3 \, B a c d e^{2} - A a c e^{3}\right )}{\left (e x + d\right )}^{2} - 5 \,{\left (5 \, B c^{2} d^{4} - 4 \, A c^{2} d^{3} e + 6 \, B a c d^{2} e^{2} - 4 \, A a c d e^{3} + B a^{2} e^{4}\right )}{\left (e x + d\right )}}{{\left (e x + d\right )}^{\frac{5}{2}} e^{5}}\right )}}{15 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.49272, size = 601, normalized size = 2.81 \begin{align*} \frac{2 \,{\left (3 \, B c^{2} e^{5} x^{5} + 256 \, B c^{2} d^{5} - 128 \, A c^{2} d^{4} e + 96 \, B a c d^{3} e^{2} - 16 \, A a c d^{2} e^{3} - 2 \, B a^{2} d e^{4} - 3 \, A a^{2} e^{5} - 5 \,{\left (2 \, B c^{2} d e^{4} - A c^{2} e^{5}\right )} x^{4} + 10 \,{\left (8 \, B c^{2} d^{2} e^{3} - 4 \, A c^{2} d e^{4} + 3 \, B a c e^{5}\right )} x^{3} + 30 \,{\left (16 \, B c^{2} d^{3} e^{2} - 8 \, A c^{2} d^{2} e^{3} + 6 \, B a c d e^{4} - A a c e^{5}\right )} x^{2} + 5 \,{\left (128 \, B c^{2} d^{4} e - 64 \, A c^{2} d^{3} e^{2} + 48 \, B a c d^{2} e^{3} - 8 \, A a c d e^{4} - B a^{2} e^{5}\right )} x\right )} \sqrt{e x + d}}{15 \,{\left (e^{9} x^{3} + 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x + d^{3} e^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.50323, size = 1426, normalized size = 6.66 \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 = 1.23115, size = 431, normalized size = 2.01 \begin{align*} \frac{2}{15} \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} B c^{2} e^{24} - 25 \,{\left (x e + d\right )}^{\frac{3}{2}} B c^{2} d e^{24} + 150 \, \sqrt{x e + d} B c^{2} d^{2} e^{24} + 5 \,{\left (x e + d\right )}^{\frac{3}{2}} A c^{2} e^{25} - 60 \, \sqrt{x e + d} A c^{2} d e^{25} + 30 \, \sqrt{x e + d} B a c e^{26}\right )} e^{\left (-30\right )} + \frac{2 \,{\left (150 \,{\left (x e + d\right )}^{2} B c^{2} d^{3} - 25 \,{\left (x e + d\right )} B c^{2} d^{4} + 3 \, B c^{2} d^{5} - 90 \,{\left (x e + d\right )}^{2} A c^{2} d^{2} e + 20 \,{\left (x e + d\right )} A c^{2} d^{3} e - 3 \, A c^{2} d^{4} e + 90 \,{\left (x e + d\right )}^{2} B a c d e^{2} - 30 \,{\left (x e + d\right )} B a c d^{2} e^{2} + 6 \, B a c d^{3} e^{2} - 30 \,{\left (x e + d\right )}^{2} A a c e^{3} + 20 \,{\left (x e + d\right )} A a c d e^{3} - 6 \, A a c d^{2} e^{3} - 5 \,{\left (x e + d\right )} B a^{2} e^{4} + 3 \, B a^{2} d e^{4} - 3 \, A a^{2} e^{5}\right )} e^{\left (-6\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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