Optimal. Leaf size=206 \[ \frac{c (d+e x)^8 \left (a B e^2-2 A c d e+5 B c d^2\right )}{4 e^6}-\frac{2 c (d+e x)^7 \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{7 e^6}+\frac{(d+e x)^6 \left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{6 e^6}-\frac{(d+e x)^5 \left (a e^2+c d^2\right )^2 (B d-A e)}{5 e^6}-\frac{c^2 (d+e x)^9 (5 B d-A e)}{9 e^6}+\frac{B c^2 (d+e x)^{10}}{10 e^6} \]
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Rubi [A] time = 0.27349, antiderivative size = 206, 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 = {772} \[ \frac{c (d+e x)^8 \left (a B e^2-2 A c d e+5 B c d^2\right )}{4 e^6}-\frac{2 c (d+e x)^7 \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{7 e^6}+\frac{(d+e x)^6 \left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{6 e^6}-\frac{(d+e x)^5 \left (a e^2+c d^2\right )^2 (B d-A e)}{5 e^6}-\frac{c^2 (d+e x)^9 (5 B d-A e)}{9 e^6}+\frac{B c^2 (d+e x)^{10}}{10 e^6} \]
Antiderivative was successfully verified.
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Rule 772
Rubi steps
\begin{align*} \int (A+B x) (d+e x)^4 \left (a+c x^2\right )^2 \, dx &=\int \left (\frac{(-B d+A e) \left (c d^2+a e^2\right )^2 (d+e x)^4}{e^5}+\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 ) (d+e x)^5}{e^5}+\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 ) (d+e x)^6}{e^5}-\frac{2 c \left (-5 B c d^2+2 A c d e-a B e^2\right ) (d+e x)^7}{e^5}+\frac{c^2 (-5 B d+A e) (d+e x)^8}{e^5}+\frac{B c^2 (d+e x)^9}{e^5}\right ) \, dx\\ &=-\frac{(B d-A e) \left (c d^2+a e^2\right )^2 (d+e x)^5}{5 e^6}+\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 ) (d+e x)^6}{6 e^6}-\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 ) (d+e x)^7}{7 e^6}+\frac{c \left (5 B c d^2-2 A c d e+a B e^2\right ) (d+e x)^8}{4 e^6}-\frac{c^2 (5 B d-A e) (d+e x)^9}{9 e^6}+\frac{B c^2 (d+e x)^{10}}{10 e^6}\\ \end{align*}
Mathematica [A] time = 0.0859645, size = 314, normalized size = 1.52 \[ \frac{1}{6} x^6 \left (a^2 B e^4+8 a A c d e^3+12 a B c d^2 e^2+4 A c^2 d^3 e+B c^2 d^4\right )+\frac{1}{5} x^5 \left (a^2 A e^4+4 a^2 B d e^3+12 a A c d^2 e^2+8 a B c d^3 e+A c^2 d^4\right )+\frac{1}{2} a^2 d^3 x^2 (4 A e+B d)+a^2 A d^4 x+\frac{1}{4} c e^2 x^8 \left (a B e^2+2 A c d e+3 B c d^2\right )+\frac{2}{7} c e x^7 \left (a A e^3+4 a B d e^2+3 A c d^2 e+2 B c d^3\right )+\frac{1}{2} a d x^4 \left (2 a A e^3+3 a B d e^2+4 A c d^2 e+B c d^3\right )+\frac{2}{3} a d^2 x^3 \left (3 a A e^2+2 a B d e+A c d^2\right )+\frac{1}{9} c^2 e^3 x^9 (A e+4 B d)+\frac{1}{10} B c^2 e^4 x^{10} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 327, normalized size = 1.6 \begin{align*}{\frac{B{e}^{4}{c}^{2}{x}^{10}}{10}}+{\frac{ \left ( A{e}^{4}+4\,Bd{e}^{3} \right ){c}^{2}{x}^{9}}{9}}+{\frac{ \left ( \left ( 4\,Ad{e}^{3}+6\,B{d}^{2}{e}^{2} \right ){c}^{2}+2\,B{e}^{4}ac \right ){x}^{8}}{8}}+{\frac{ \left ( \left ( 6\,A{d}^{2}{e}^{2}+4\,B{d}^{3}e \right ){c}^{2}+2\, \left ( A{e}^{4}+4\,Bd{e}^{3} \right ) ac \right ){x}^{7}}{7}}+{\frac{ \left ( \left ( 4\,A{d}^{3}e+B{d}^{4} \right ){c}^{2}+2\, \left ( 4\,Ad{e}^{3}+6\,B{d}^{2}{e}^{2} \right ) ac+B{e}^{4}{a}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ( A{d}^{4}{c}^{2}+2\, \left ( 6\,A{d}^{2}{e}^{2}+4\,B{d}^{3}e \right ) ac+ \left ( A{e}^{4}+4\,Bd{e}^{3} \right ){a}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( 2\, \left ( 4\,A{d}^{3}e+B{d}^{4} \right ) ac+ \left ( 4\,Ad{e}^{3}+6\,B{d}^{2}{e}^{2} \right ){a}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ( 2\,A{d}^{4}ac+ \left ( 6\,A{d}^{2}{e}^{2}+4\,B{d}^{3}e \right ){a}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 4\,A{d}^{3}e+B{d}^{4} \right ){a}^{2}{x}^{2}}{2}}+A{d}^{4}{a}^{2}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02937, size = 448, normalized size = 2.17 \begin{align*} \frac{1}{10} \, B c^{2} e^{4} x^{10} + \frac{1}{9} \,{\left (4 \, B c^{2} d e^{3} + A c^{2} e^{4}\right )} x^{9} + \frac{1}{4} \,{\left (3 \, B c^{2} d^{2} e^{2} + 2 \, A c^{2} d e^{3} + B a c e^{4}\right )} x^{8} + A a^{2} d^{4} x + \frac{2}{7} \,{\left (2 \, B c^{2} d^{3} e + 3 \, A c^{2} d^{2} e^{2} + 4 \, B a c d e^{3} + A a c e^{4}\right )} x^{7} + \frac{1}{6} \,{\left (B c^{2} d^{4} + 4 \, A c^{2} d^{3} e + 12 \, B a c d^{2} e^{2} + 8 \, A a c d e^{3} + B a^{2} e^{4}\right )} x^{6} + \frac{1}{5} \,{\left (A c^{2} d^{4} + 8 \, B a c d^{3} e + 12 \, A a c d^{2} e^{2} + 4 \, B a^{2} d e^{3} + A a^{2} e^{4}\right )} x^{5} + \frac{1}{2} \,{\left (B a c d^{4} + 4 \, A a c d^{3} e + 3 \, B a^{2} d^{2} e^{2} + 2 \, A a^{2} d e^{3}\right )} x^{4} + \frac{2}{3} \,{\left (A a c d^{4} + 2 \, B a^{2} d^{3} e + 3 \, A a^{2} d^{2} e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (B a^{2} d^{4} + 4 \, A a^{2} d^{3} e\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64215, size = 853, normalized size = 4.14 \begin{align*} \frac{1}{10} x^{10} e^{4} c^{2} B + \frac{4}{9} x^{9} e^{3} d c^{2} B + \frac{1}{9} x^{9} e^{4} c^{2} A + \frac{3}{4} x^{8} e^{2} d^{2} c^{2} B + \frac{1}{4} x^{8} e^{4} c a B + \frac{1}{2} x^{8} e^{3} d c^{2} A + \frac{4}{7} x^{7} e d^{3} c^{2} B + \frac{8}{7} x^{7} e^{3} d c a B + \frac{6}{7} x^{7} e^{2} d^{2} c^{2} A + \frac{2}{7} x^{7} e^{4} c a A + \frac{1}{6} x^{6} d^{4} c^{2} B + 2 x^{6} e^{2} d^{2} c a B + \frac{1}{6} x^{6} e^{4} a^{2} B + \frac{2}{3} x^{6} e d^{3} c^{2} A + \frac{4}{3} x^{6} e^{3} d c a A + \frac{8}{5} x^{5} e d^{3} c a B + \frac{4}{5} x^{5} e^{3} d a^{2} B + \frac{1}{5} x^{5} d^{4} c^{2} A + \frac{12}{5} x^{5} e^{2} d^{2} c a A + \frac{1}{5} x^{5} e^{4} a^{2} A + \frac{1}{2} x^{4} d^{4} c a B + \frac{3}{2} x^{4} e^{2} d^{2} a^{2} B + 2 x^{4} e d^{3} c a A + x^{4} e^{3} d a^{2} A + \frac{4}{3} x^{3} e d^{3} a^{2} B + \frac{2}{3} x^{3} d^{4} c a A + 2 x^{3} e^{2} d^{2} a^{2} A + \frac{1}{2} x^{2} d^{4} a^{2} B + 2 x^{2} e d^{3} a^{2} A + x d^{4} a^{2} A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.352027, size = 398, normalized size = 1.93 \begin{align*} A a^{2} d^{4} x + \frac{B c^{2} e^{4} x^{10}}{10} + x^{9} \left (\frac{A c^{2} e^{4}}{9} + \frac{4 B c^{2} d e^{3}}{9}\right ) + x^{8} \left (\frac{A c^{2} d e^{3}}{2} + \frac{B a c e^{4}}{4} + \frac{3 B c^{2} d^{2} e^{2}}{4}\right ) + x^{7} \left (\frac{2 A a c e^{4}}{7} + \frac{6 A c^{2} d^{2} e^{2}}{7} + \frac{8 B a c d e^{3}}{7} + \frac{4 B c^{2} d^{3} e}{7}\right ) + x^{6} \left (\frac{4 A a c d e^{3}}{3} + \frac{2 A c^{2} d^{3} e}{3} + \frac{B a^{2} e^{4}}{6} + 2 B a c d^{2} e^{2} + \frac{B c^{2} d^{4}}{6}\right ) + x^{5} \left (\frac{A a^{2} e^{4}}{5} + \frac{12 A a c d^{2} e^{2}}{5} + \frac{A c^{2} d^{4}}{5} + \frac{4 B a^{2} d e^{3}}{5} + \frac{8 B a c d^{3} e}{5}\right ) + x^{4} \left (A a^{2} d e^{3} + 2 A a c d^{3} e + \frac{3 B a^{2} d^{2} e^{2}}{2} + \frac{B a c d^{4}}{2}\right ) + x^{3} \left (2 A a^{2} d^{2} e^{2} + \frac{2 A a c d^{4}}{3} + \frac{4 B a^{2} d^{3} e}{3}\right ) + x^{2} \left (2 A a^{2} d^{3} e + \frac{B a^{2} d^{4}}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2961, size = 493, normalized size = 2.39 \begin{align*} \frac{1}{10} \, B c^{2} x^{10} e^{4} + \frac{4}{9} \, B c^{2} d x^{9} e^{3} + \frac{3}{4} \, B c^{2} d^{2} x^{8} e^{2} + \frac{4}{7} \, B c^{2} d^{3} x^{7} e + \frac{1}{6} \, B c^{2} d^{4} x^{6} + \frac{1}{9} \, A c^{2} x^{9} e^{4} + \frac{1}{2} \, A c^{2} d x^{8} e^{3} + \frac{6}{7} \, A c^{2} d^{2} x^{7} e^{2} + \frac{2}{3} \, A c^{2} d^{3} x^{6} e + \frac{1}{5} \, A c^{2} d^{4} x^{5} + \frac{1}{4} \, B a c x^{8} e^{4} + \frac{8}{7} \, B a c d x^{7} e^{3} + 2 \, B a c d^{2} x^{6} e^{2} + \frac{8}{5} \, B a c d^{3} x^{5} e + \frac{1}{2} \, B a c d^{4} x^{4} + \frac{2}{7} \, A a c x^{7} e^{4} + \frac{4}{3} \, A a c d x^{6} e^{3} + \frac{12}{5} \, A a c d^{2} x^{5} e^{2} + 2 \, A a c d^{3} x^{4} e + \frac{2}{3} \, A a c d^{4} x^{3} + \frac{1}{6} \, B a^{2} x^{6} e^{4} + \frac{4}{5} \, B a^{2} d x^{5} e^{3} + \frac{3}{2} \, B a^{2} d^{2} x^{4} e^{2} + \frac{4}{3} \, B a^{2} d^{3} x^{3} e + \frac{1}{2} \, B a^{2} d^{4} x^{2} + \frac{1}{5} \, A a^{2} x^{5} e^{4} + A a^{2} d x^{4} e^{3} + 2 \, A a^{2} d^{2} x^{3} e^{2} + 2 \, A a^{2} d^{3} x^{2} e + A a^{2} d^{4} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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