Optimal. Leaf size=108 \[ \frac {(d+e x)^6 \left (a B e^2-2 A c d e+3 B c d^2\right )}{6 e^4}-\frac {(d+e x)^5 \left (a e^2+c d^2\right ) (B d-A e)}{5 e^4}-\frac {c (d+e x)^7 (3 B d-A e)}{7 e^4}+\frac {B c (d+e x)^8}{8 e^4} \]
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Rubi [A] time = 0.14, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {772} \begin {gather*} \frac {(d+e x)^6 \left (a B e^2-2 A c d e+3 B c d^2\right )}{6 e^4}-\frac {(d+e x)^5 \left (a e^2+c d^2\right ) (B d-A e)}{5 e^4}-\frac {c (d+e x)^7 (3 B d-A e)}{7 e^4}+\frac {B c (d+e x)^8}{8 e^4} \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 ) \, dx &=\int \left (\frac {(-B d+A e) \left (c d^2+a e^2\right ) (d+e x)^4}{e^3}+\frac {\left (3 B c d^2-2 A c d e+a B e^2\right ) (d+e x)^5}{e^3}+\frac {c (-3 B d+A e) (d+e x)^6}{e^3}+\frac {B c (d+e x)^7}{e^3}\right ) \, dx\\ &=-\frac {(B d-A e) \left (c d^2+a e^2\right ) (d+e x)^5}{5 e^4}+\frac {\left (3 B c d^2-2 A c d e+a B e^2\right ) (d+e x)^6}{6 e^4}-\frac {c (3 B d-A e) (d+e x)^7}{7 e^4}+\frac {B c (d+e x)^8}{8 e^4}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 194, normalized size = 1.80 \begin {gather*} \frac {1}{6} e^2 x^6 \left (a B e^2+4 A c d e+6 B c d^2\right )+\frac {1}{3} d^2 x^3 \left (6 a A e^2+4 a B d e+A c d^2\right )+\frac {1}{5} e x^5 \left (a A e^3+4 a B d e^2+6 A c d^2 e+4 B c d^3\right )+\frac {1}{4} d x^4 \left (4 a A e^3+6 a B d e^2+4 A c d^2 e+B c d^3\right )+\frac {1}{2} a d^3 x^2 (4 A e+B d)+a A d^4 x+\frac {1}{7} c e^3 x^7 (A e+4 B d)+\frac {1}{8} B c e^4 x^8 \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 ) \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.36, size = 215, normalized size = 1.99 \begin {gather*} \frac {1}{8} x^{8} e^{4} c B + \frac {4}{7} x^{7} e^{3} d c B + \frac {1}{7} x^{7} e^{4} c A + x^{6} e^{2} d^{2} c B + \frac {1}{6} x^{6} e^{4} a B + \frac {2}{3} x^{6} e^{3} d c A + \frac {4}{5} x^{5} e d^{3} c B + \frac {4}{5} x^{5} e^{3} d a B + \frac {6}{5} x^{5} e^{2} d^{2} c A + \frac {1}{5} x^{5} e^{4} a A + \frac {1}{4} x^{4} d^{4} c B + \frac {3}{2} x^{4} e^{2} d^{2} a B + x^{4} e d^{3} c A + x^{4} e^{3} d a A + \frac {4}{3} x^{3} e d^{3} a B + \frac {1}{3} x^{3} d^{4} c A + 2 x^{3} e^{2} d^{2} a A + \frac {1}{2} x^{2} d^{4} a B + 2 x^{2} e d^{3} a A + x d^{4} a A \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 207, normalized size = 1.92 \begin {gather*} \frac {1}{8} \, B c x^{8} e^{4} + \frac {4}{7} \, B c d x^{7} e^{3} + B c d^{2} x^{6} e^{2} + \frac {4}{5} \, B c d^{3} x^{5} e + \frac {1}{4} \, B c d^{4} x^{4} + \frac {1}{7} \, A c x^{7} e^{4} + \frac {2}{3} \, A c d x^{6} e^{3} + \frac {6}{5} \, A c d^{2} x^{5} e^{2} + A c d^{3} x^{4} e + \frac {1}{3} \, A c d^{4} x^{3} + \frac {1}{6} \, B a x^{6} e^{4} + \frac {4}{5} \, B a d x^{5} e^{3} + \frac {3}{2} \, B a d^{2} x^{4} e^{2} + \frac {4}{3} \, B a d^{3} x^{3} e + \frac {1}{2} \, B a d^{4} x^{2} + \frac {1}{5} \, A a x^{5} e^{4} + A a d x^{4} e^{3} + 2 \, A a d^{2} x^{3} e^{2} + 2 \, A a d^{3} x^{2} e + A a 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 = 199, normalized size = 1.84 \begin {gather*} \frac {B c \,e^{4} x^{8}}{8}+\frac {\left (A \,e^{4}+4 B d \,e^{3}\right ) c \,x^{7}}{7}+A a \,d^{4} x +\frac {\left (B a \,e^{4}+\left (4 A d \,e^{3}+6 B \,d^{2} e^{2}\right ) c \right ) x^{6}}{6}+\frac {\left (\left (A \,e^{4}+4 B d \,e^{3}\right ) a +\left (6 A \,d^{2} e^{2}+4 B \,d^{3} e \right ) c \right ) x^{5}}{5}+\frac {\left (\left (4 A d \,e^{3}+6 B \,d^{2} e^{2}\right ) a +\left (4 A \,d^{3} e +B \,d^{4}\right ) c \right ) x^{4}}{4}+\frac {\left (4 A \,d^{3} e +B \,d^{4}\right ) a \,x^{2}}{2}+\frac {\left (A c \,d^{4}+\left (6 A \,d^{2} e^{2}+4 B \,d^{3} e \right ) a \right ) x^{3}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 194, normalized size = 1.80 \begin {gather*} \frac {1}{8} \, B c e^{4} x^{8} + \frac {1}{7} \, {\left (4 \, B c d e^{3} + A c e^{4}\right )} x^{7} + A a d^{4} x + \frac {1}{6} \, {\left (6 \, B c d^{2} e^{2} + 4 \, A c d e^{3} + B a e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (4 \, B c d^{3} e + 6 \, A c d^{2} e^{2} + 4 \, B a d e^{3} + A a e^{4}\right )} x^{5} + \frac {1}{4} \, {\left (B c d^{4} + 4 \, A c d^{3} e + 6 \, B a d^{2} e^{2} + 4 \, A a d e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (A c d^{4} + 4 \, B a d^{3} e + 6 \, A a d^{2} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B a d^{4} + 4 \, A a 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 = 0.09, size = 185, normalized size = 1.71 \begin {gather*} x^3\,\left (\frac {A\,c\,d^4}{3}+\frac {4\,B\,a\,d^3\,e}{3}+2\,A\,a\,d^2\,e^2\right )+x^6\,\left (B\,c\,d^2\,e^2+\frac {2\,A\,c\,d\,e^3}{3}+\frac {B\,a\,e^4}{6}\right )+x^4\,\left (\frac {B\,c\,d^4}{4}+A\,c\,d^3\,e+\frac {3\,B\,a\,d^2\,e^2}{2}+A\,a\,d\,e^3\right )+x^5\,\left (\frac {4\,B\,c\,d^3\,e}{5}+\frac {6\,A\,c\,d^2\,e^2}{5}+\frac {4\,B\,a\,d\,e^3}{5}+\frac {A\,a\,e^4}{5}\right )+A\,a\,d^4\,x+\frac {B\,c\,e^4\,x^8}{8}+\frac {a\,d^3\,x^2\,\left (4\,A\,e+B\,d\right )}{2}+\frac {c\,e^3\,x^7\,\left (A\,e+4\,B\,d\right )}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.10, size = 226, normalized size = 2.09 \begin {gather*} A a d^{4} x + \frac {B c e^{4} x^{8}}{8} + x^{7} \left (\frac {A c e^{4}}{7} + \frac {4 B c d e^{3}}{7}\right ) + x^{6} \left (\frac {2 A c d e^{3}}{3} + \frac {B a e^{4}}{6} + B c d^{2} e^{2}\right ) + x^{5} \left (\frac {A a e^{4}}{5} + \frac {6 A c d^{2} e^{2}}{5} + \frac {4 B a d e^{3}}{5} + \frac {4 B c d^{3} e}{5}\right ) + x^{4} \left (A a d e^{3} + A c d^{3} e + \frac {3 B a d^{2} e^{2}}{2} + \frac {B c d^{4}}{4}\right ) + x^{3} \left (2 A a d^{2} e^{2} + \frac {A c d^{4}}{3} + \frac {4 B a d^{3} e}{3}\right ) + x^{2} \left (2 A a d^{3} e + \frac {B a d^{4}}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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