Optimal. Leaf size=206 \[ \frac {c (d+e x)^8 \left (a B e^2-2 A c d e+5 B c d^2\right )}{4 e^6}+\frac {(d+e x)^6 \left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{6 e^6}-\frac {(d+e x)^5 \left (a e^2+c d^2\right )^2 (B d-A e)}{5 e^6}-\frac {2 c (d+e x)^7 \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{7 e^6}-\frac {c^2 (d+e x)^9 (5 B d-A e)}{9 e^6}+\frac {B c^2 (d+e x)^{10}}{10 e^6} \]
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Rubi [A] time = 0.27, antiderivative size = 206, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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} \end {gather*}
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*}
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Mathematica [A] time = 0.08, size = 314, normalized size = 1.52 \begin {gather*} \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}{3} a d^2 x^3 \left (3 a A e^2+2 a B d e+A 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 {1}{9} c^2 e^3 x^9 (A e+4 B d)+\frac {1}{10} B c^2 e^4 x^{10} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (A+B x) (d+e x)^4 \left (a+c x^2\right )^2 \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.36, size = 377, normalized size = 1.83 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 365, normalized size = 1.77 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 327, normalized size = 1.59 \begin {gather*} \frac {B \,c^{2} e^{4} x^{10}}{10}+\frac {\left (A \,e^{4}+4 B d \,e^{3}\right ) c^{2} x^{9}}{9}+A \,a^{2} d^{4} x +\frac {\left (2 B a c \,e^{4}+\left (4 A d \,e^{3}+6 B \,d^{2} e^{2}\right ) c^{2}\right ) x^{8}}{8}+\frac {\left (2 \left (A \,e^{4}+4 B d \,e^{3}\right ) a c +\left (6 A \,d^{2} e^{2}+4 B \,d^{3} e \right ) c^{2}\right ) x^{7}}{7}+\frac {\left (B \,a^{2} e^{4}+2 \left (4 A d \,e^{3}+6 B \,d^{2} e^{2}\right ) a c +\left (4 A \,d^{3} e +B \,d^{4}\right ) c^{2}\right ) x^{6}}{6}+\frac {\left (A \,c^{2} d^{4}+\left (A \,e^{4}+4 B d \,e^{3}\right ) a^{2}+2 \left (6 A \,d^{2} e^{2}+4 B \,d^{3} e \right ) a c \right ) x^{5}}{5}+\frac {\left (4 A \,d^{3} e +B \,d^{4}\right ) a^{2} x^{2}}{2}+\frac {\left (\left (4 A d \,e^{3}+6 B \,d^{2} e^{2}\right ) a^{2}+2 \left (4 A \,d^{3} e +B \,d^{4}\right ) a c \right ) x^{4}}{4}+\frac {\left (2 A a c \,d^{4}+\left (6 A \,d^{2} e^{2}+4 B \,d^{3} e \right ) a^{2}\right ) x^{3}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 332, normalized size = 1.61 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 298, normalized size = 1.45 \begin {gather*} x^5\,\left (\frac {4\,B\,a^2\,d\,e^3}{5}+\frac {A\,a^2\,e^4}{5}+\frac {8\,B\,a\,c\,d^3\,e}{5}+\frac {12\,A\,a\,c\,d^2\,e^2}{5}+\frac {A\,c^2\,d^4}{5}\right )+x^6\,\left (\frac {B\,a^2\,e^4}{6}+2\,B\,a\,c\,d^2\,e^2+\frac {4\,A\,a\,c\,d\,e^3}{3}+\frac {B\,c^2\,d^4}{6}+\frac {2\,A\,c^2\,d^3\,e}{3}\right )+\frac {2\,a\,d^2\,x^3\,\left (A\,c\,d^2+2\,B\,a\,d\,e+3\,A\,a\,e^2\right )}{3}+\frac {c\,e^2\,x^8\,\left (3\,B\,c\,d^2+2\,A\,c\,d\,e+B\,a\,e^2\right )}{4}+\frac {a^2\,d^3\,x^2\,\left (4\,A\,e+B\,d\right )}{2}+\frac {c^2\,e^3\,x^9\,\left (A\,e+4\,B\,d\right )}{9}+\frac {a\,d\,x^4\,\left (B\,c\,d^3+4\,A\,c\,d^2\,e+3\,B\,a\,d\,e^2+2\,A\,a\,e^3\right )}{2}+\frac {2\,c\,e\,x^7\,\left (2\,B\,c\,d^3+3\,A\,c\,d^2\,e+4\,B\,a\,d\,e^2+A\,a\,e^3\right )}{7}+A\,a^2\,d^4\,x+\frac {B\,c^2\,e^4\,x^{10}}{10} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 398, normalized size = 1.93 \begin {gather*} 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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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