Optimal. Leaf size=108 \[ \frac {(d+e x)^5 \left (a B e^2-2 A c d e+3 B c d^2\right )}{5 e^4}-\frac {(d+e x)^4 \left (a e^2+c d^2\right ) (B d-A e)}{4 e^4}-\frac {c (d+e x)^6 (3 B d-A e)}{6 e^4}+\frac {B c (d+e x)^7}{7 e^4} \]
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Rubi [A] time = 0.11, 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} \[ \frac {(d+e x)^5 \left (a B e^2-2 A c d e+3 B c d^2\right )}{5 e^4}-\frac {(d+e x)^4 \left (a e^2+c d^2\right ) (B d-A e)}{4 e^4}-\frac {c (d+e x)^6 (3 B d-A e)}{6 e^4}+\frac {B c (d+e x)^7}{7 e^4} \]
Antiderivative was successfully verified.
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Rule 772
Rubi steps
\begin {align*} \int (A+B x) (d+e x)^3 \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)^3}{e^3}+\frac {\left (3 B c d^2-2 A c d e+a B e^2\right ) (d+e x)^4}{e^3}+\frac {c (-3 B d+A e) (d+e x)^5}{e^3}+\frac {B c (d+e x)^6}{e^3}\right ) \, dx\\ &=-\frac {(B d-A e) \left (c d^2+a e^2\right ) (d+e x)^4}{4 e^4}+\frac {\left (3 B c d^2-2 A c d e+a B e^2\right ) (d+e x)^5}{5 e^4}-\frac {c (3 B d-A e) (d+e x)^6}{6 e^4}+\frac {B c (d+e x)^7}{7 e^4}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 150, normalized size = 1.39 \[ \frac {1}{5} e x^5 \left (a B e^2+3 A c d e+3 B c d^2\right )+\frac {1}{3} d x^3 \left (3 a A e^2+3 a B d e+A c d^2\right )+\frac {1}{4} x^4 \left (a A e^3+3 a B d e^2+3 A c d^2 e+B c d^3\right )+\frac {1}{2} a d^2 x^2 (3 A e+B d)+a A d^3 x+\frac {1}{6} c e^2 x^6 (A e+3 B d)+\frac {1}{7} B c e^3 x^7 \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 164, normalized size = 1.52 \[ \frac {1}{7} x^{7} e^{3} c B + \frac {1}{2} x^{6} e^{2} d c B + \frac {1}{6} x^{6} e^{3} c A + \frac {3}{5} x^{5} e d^{2} c B + \frac {1}{5} x^{5} e^{3} a B + \frac {3}{5} x^{5} e^{2} d c A + \frac {1}{4} x^{4} d^{3} c B + \frac {3}{4} x^{4} e^{2} d a B + \frac {3}{4} x^{4} e d^{2} c A + \frac {1}{4} x^{4} e^{3} a A + x^{3} e d^{2} a B + \frac {1}{3} x^{3} d^{3} c A + x^{3} e^{2} d a A + \frac {1}{2} x^{2} d^{3} a B + \frac {3}{2} x^{2} e d^{2} a A + x d^{3} a A \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 160, normalized size = 1.48 \[ \frac {1}{7} \, B c x^{7} e^{3} + \frac {1}{2} \, B c d x^{6} e^{2} + \frac {3}{5} \, B c d^{2} x^{5} e + \frac {1}{4} \, B c d^{3} x^{4} + \frac {1}{6} \, A c x^{6} e^{3} + \frac {3}{5} \, A c d x^{5} e^{2} + \frac {3}{4} \, A c d^{2} x^{4} e + \frac {1}{3} \, A c d^{3} x^{3} + \frac {1}{5} \, B a x^{5} e^{3} + \frac {3}{4} \, B a d x^{4} e^{2} + B a d^{2} x^{3} e + \frac {1}{2} \, B a d^{3} x^{2} + \frac {1}{4} \, A a x^{4} e^{3} + A a d x^{3} e^{2} + \frac {3}{2} \, A a d^{2} x^{2} e + A a d^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 151, normalized size = 1.40 \[ \frac {B c \,e^{3} x^{7}}{7}+\frac {\left (A \,e^{3}+3 B d \,e^{2}\right ) c \,x^{6}}{6}+A a \,d^{3} x +\frac {\left (B a \,e^{3}+\left (3 A d \,e^{2}+3 B \,d^{2} e \right ) c \right ) x^{5}}{5}+\frac {\left (\left (A \,e^{3}+3 B d \,e^{2}\right ) a +\left (3 A \,d^{2} e +B \,d^{3}\right ) c \right ) x^{4}}{4}+\frac {\left (3 A \,d^{2} e +B \,d^{3}\right ) a \,x^{2}}{2}+\frac {\left (A c \,d^{3}+\left (3 A d \,e^{2}+3 B \,d^{2} e \right ) a \right ) x^{3}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.64, size = 148, normalized size = 1.37 \[ \frac {1}{7} \, B c e^{3} x^{7} + \frac {1}{6} \, {\left (3 \, B c d e^{2} + A c e^{3}\right )} x^{6} + A a d^{3} x + \frac {1}{5} \, {\left (3 \, B c d^{2} e + 3 \, A c d e^{2} + B a e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (B c d^{3} + 3 \, A c d^{2} e + 3 \, B a d e^{2} + A a e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (A c d^{3} + 3 \, B a d^{2} e + 3 \, A a d e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B a d^{3} + 3 \, A a d^{2} e\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 141, normalized size = 1.31 \[ x^4\,\left (\frac {B\,c\,d^3}{4}+\frac {3\,A\,c\,d^2\,e}{4}+\frac {3\,B\,a\,d\,e^2}{4}+\frac {A\,a\,e^3}{4}\right )+x^3\,\left (\frac {A\,c\,d^3}{3}+B\,a\,d^2\,e+A\,a\,d\,e^2\right )+x^5\,\left (\frac {3\,B\,c\,d^2\,e}{5}+\frac {3\,A\,c\,d\,e^2}{5}+\frac {B\,a\,e^3}{5}\right )+A\,a\,d^3\,x+\frac {B\,c\,e^3\,x^7}{7}+\frac {a\,d^2\,x^2\,\left (3\,A\,e+B\,d\right )}{2}+\frac {c\,e^2\,x^6\,\left (A\,e+3\,B\,d\right )}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 173, normalized size = 1.60 \[ A a d^{3} x + \frac {B c e^{3} x^{7}}{7} + x^{6} \left (\frac {A c e^{3}}{6} + \frac {B c d e^{2}}{2}\right ) + x^{5} \left (\frac {3 A c d e^{2}}{5} + \frac {B a e^{3}}{5} + \frac {3 B c d^{2} e}{5}\right ) + x^{4} \left (\frac {A a e^{3}}{4} + \frac {3 A c d^{2} e}{4} + \frac {3 B a d e^{2}}{4} + \frac {B c d^{3}}{4}\right ) + x^{3} \left (A a d e^{2} + \frac {A c d^{3}}{3} + B a d^{2} e\right ) + x^{2} \left (\frac {3 A a d^{2} e}{2} + \frac {B a d^{3}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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