Optimal. Leaf size=257 \[ -\frac {x^4 \left (B (c d-b e)^3-A c e \left (3 b^2 e^2-3 b c d e+c^2 d^2\right )\right )}{4 e^4}-\frac {c x^5 \left (A c e (c d-3 b e)-B \left (3 b^2 e^2-3 b c d e+c^2 d^2\right )\right )}{5 e^3}-\frac {c^2 x^6 (-A c e-3 b B e+B c d)}{6 e^2}-\frac {d^3 (B d-A e) (c d-b e)^3 \log (d+e x)}{e^8}+\frac {d^2 x (B d-A e) (c d-b e)^3}{e^7}-\frac {d x^2 (B d-A e) (c d-b e)^3}{2 e^6}+\frac {x^3 (B d-A e) (c d-b e)^3}{3 e^5}+\frac {B c^3 x^7}{7 e} \]
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Rubi [A] time = 0.45, antiderivative size = 257, normalized size of antiderivative = 1.00, 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 = {771} \begin {gather*} -\frac {c x^5 \left (A c e (c d-3 b e)-B \left (3 b^2 e^2-3 b c d e+c^2 d^2\right )\right )}{5 e^3}-\frac {x^4 \left (B (c d-b e)^3-A c e \left (3 b^2 e^2-3 b c d e+c^2 d^2\right )\right )}{4 e^4}-\frac {c^2 x^6 (-A c e-3 b B e+B c d)}{6 e^2}+\frac {d^2 x (B d-A e) (c d-b e)^3}{e^7}-\frac {d^3 (B d-A e) (c d-b e)^3 \log (d+e x)}{e^8}+\frac {x^3 (B d-A e) (c d-b e)^3}{3 e^5}-\frac {d x^2 (B d-A e) (c d-b e)^3}{2 e^6}+\frac {B c^3 x^7}{7 e} \end {gather*}
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (b x+c x^2\right )^3}{d+e x} \, dx &=\int \left (\frac {d^2 (B d-A e) (c d-b e)^3}{e^7}-\frac {d (B d-A e) (c d-b e)^3 x}{e^6}+\frac {(-B d+A e) (-c d+b e)^3 x^2}{e^5}+\frac {\left (-B (c d-b e)^3+A c e \left (c^2 d^2-3 b c d e+3 b^2 e^2\right )\right ) x^3}{e^4}+\frac {c \left (-A c e (c d-3 b e)+B \left (c^2 d^2-3 b c d e+3 b^2 e^2\right )\right ) x^4}{e^3}+\frac {c^2 (-B c d+3 b B e+A c e) x^5}{e^2}+\frac {B c^3 x^6}{e}-\frac {d^3 (B d-A e) (c d-b e)^3}{e^7 (d+e x)}\right ) \, dx\\ &=\frac {d^2 (B d-A e) (c d-b e)^3 x}{e^7}-\frac {d (B d-A e) (c d-b e)^3 x^2}{2 e^6}+\frac {(B d-A e) (c d-b e)^3 x^3}{3 e^5}-\frac {\left (B (c d-b e)^3-A c e \left (c^2 d^2-3 b c d e+3 b^2 e^2\right )\right ) x^4}{4 e^4}-\frac {c \left (A c e (c d-3 b e)-B \left (c^2 d^2-3 b c d e+3 b^2 e^2\right )\right ) x^5}{5 e^3}-\frac {c^2 (B c d-3 b B e-A c e) x^6}{6 e^2}+\frac {B c^3 x^7}{7 e}-\frac {d^3 (B d-A e) (c d-b e)^3 \log (d+e x)}{e^8}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 248, normalized size = 0.96 \begin {gather*} \frac {84 c e^5 x^5 \left (A c e (3 b e-c d)+B \left (3 b^2 e^2-3 b c d e+c^2 d^2\right )\right )+105 e^4 x^4 \left (A c e \left (3 b^2 e^2-3 b c d e+c^2 d^2\right )-B (c d-b e)^3\right )+70 c^2 e^6 x^6 (A c e+3 b B e-B c d)-420 d^3 (B d-A e) (c d-b e)^3 \log (d+e x)+420 d^2 e x (B d-A e) (c d-b e)^3+140 e^3 x^3 (A e-B d) (b e-c d)^3-210 d e^2 x^2 (B d-A e) (c d-b e)^3+60 B c^3 e^7 x^7}{420 e^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 \frac {(A+B x) \left (b x+c x^2\right )^3}{d+e x} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.43, size = 531, normalized size = 2.07 \begin {gather*} \frac {60 \, B c^{3} e^{7} x^{7} - 70 \, {\left (B c^{3} d e^{6} - {\left (3 \, B b c^{2} + A c^{3}\right )} e^{7}\right )} x^{6} + 84 \, {\left (B c^{3} d^{2} e^{5} - {\left (3 \, B b c^{2} + A c^{3}\right )} d e^{6} + 3 \, {\left (B b^{2} c + A b c^{2}\right )} e^{7}\right )} x^{5} - 105 \, {\left (B c^{3} d^{3} e^{4} - {\left (3 \, B b c^{2} + A c^{3}\right )} d^{2} e^{5} + 3 \, {\left (B b^{2} c + A b c^{2}\right )} d e^{6} - {\left (B b^{3} + 3 \, A b^{2} c\right )} e^{7}\right )} x^{4} + 140 \, {\left (B c^{3} d^{4} e^{3} + A b^{3} e^{7} - {\left (3 \, B b c^{2} + A c^{3}\right )} d^{3} e^{4} + 3 \, {\left (B b^{2} c + A b c^{2}\right )} d^{2} e^{5} - {\left (B b^{3} + 3 \, A b^{2} c\right )} d e^{6}\right )} x^{3} - 210 \, {\left (B c^{3} d^{5} e^{2} + A b^{3} d e^{6} - {\left (3 \, B b c^{2} + A c^{3}\right )} d^{4} e^{3} + 3 \, {\left (B b^{2} c + A b c^{2}\right )} d^{3} e^{4} - {\left (B b^{3} + 3 \, A b^{2} c\right )} d^{2} e^{5}\right )} x^{2} + 420 \, {\left (B c^{3} d^{6} e + A b^{3} d^{2} e^{5} - {\left (3 \, B b c^{2} + A c^{3}\right )} d^{5} e^{2} + 3 \, {\left (B b^{2} c + A b c^{2}\right )} d^{4} e^{3} - {\left (B b^{3} + 3 \, A b^{2} c\right )} d^{3} e^{4}\right )} x - 420 \, {\left (B c^{3} d^{7} + A b^{3} d^{3} e^{4} - {\left (3 \, B b c^{2} + A c^{3}\right )} d^{6} e + 3 \, {\left (B b^{2} c + A b c^{2}\right )} d^{5} e^{2} - {\left (B b^{3} + 3 \, A b^{2} c\right )} d^{4} e^{3}\right )} \log \left (e x + d\right )}{420 \, e^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.18, size = 629, normalized size = 2.45 \begin {gather*} -{\left (B c^{3} d^{7} - 3 \, B b c^{2} d^{6} e - A c^{3} d^{6} e + 3 \, B b^{2} c d^{5} e^{2} + 3 \, A b c^{2} d^{5} e^{2} - B b^{3} d^{4} e^{3} - 3 \, A b^{2} c d^{4} e^{3} + A b^{3} d^{3} e^{4}\right )} e^{\left (-8\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{420} \, {\left (60 \, B c^{3} x^{7} e^{6} - 70 \, B c^{3} d x^{6} e^{5} + 84 \, B c^{3} d^{2} x^{5} e^{4} - 105 \, B c^{3} d^{3} x^{4} e^{3} + 140 \, B c^{3} d^{4} x^{3} e^{2} - 210 \, B c^{3} d^{5} x^{2} e + 420 \, B c^{3} d^{6} x + 210 \, B b c^{2} x^{6} e^{6} + 70 \, A c^{3} x^{6} e^{6} - 252 \, B b c^{2} d x^{5} e^{5} - 84 \, A c^{3} d x^{5} e^{5} + 315 \, B b c^{2} d^{2} x^{4} e^{4} + 105 \, A c^{3} d^{2} x^{4} e^{4} - 420 \, B b c^{2} d^{3} x^{3} e^{3} - 140 \, A c^{3} d^{3} x^{3} e^{3} + 630 \, B b c^{2} d^{4} x^{2} e^{2} + 210 \, A c^{3} d^{4} x^{2} e^{2} - 1260 \, B b c^{2} d^{5} x e - 420 \, A c^{3} d^{5} x e + 252 \, B b^{2} c x^{5} e^{6} + 252 \, A b c^{2} x^{5} e^{6} - 315 \, B b^{2} c d x^{4} e^{5} - 315 \, A b c^{2} d x^{4} e^{5} + 420 \, B b^{2} c d^{2} x^{3} e^{4} + 420 \, A b c^{2} d^{2} x^{3} e^{4} - 630 \, B b^{2} c d^{3} x^{2} e^{3} - 630 \, A b c^{2} d^{3} x^{2} e^{3} + 1260 \, B b^{2} c d^{4} x e^{2} + 1260 \, A b c^{2} d^{4} x e^{2} + 105 \, B b^{3} x^{4} e^{6} + 315 \, A b^{2} c x^{4} e^{6} - 140 \, B b^{3} d x^{3} e^{5} - 420 \, A b^{2} c d x^{3} e^{5} + 210 \, B b^{3} d^{2} x^{2} e^{4} + 630 \, A b^{2} c d^{2} x^{2} e^{4} - 420 \, B b^{3} d^{3} x e^{3} - 1260 \, A b^{2} c d^{3} x e^{3} + 140 \, A b^{3} x^{3} e^{6} - 210 \, A b^{3} d x^{2} e^{5} + 420 \, A b^{3} d^{2} x e^{4}\right )} e^{\left (-7\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 708, normalized size = 2.75 \begin {gather*} \frac {B \,c^{3} x^{7}}{7 e}+\frac {A \,c^{3} x^{6}}{6 e}+\frac {B b \,c^{2} x^{6}}{2 e}-\frac {B \,c^{3} d \,x^{6}}{6 e^{2}}+\frac {3 A b \,c^{2} x^{5}}{5 e}-\frac {A \,c^{3} d \,x^{5}}{5 e^{2}}+\frac {3 B \,b^{2} c \,x^{5}}{5 e}-\frac {3 B b \,c^{2} d \,x^{5}}{5 e^{2}}+\frac {B \,c^{3} d^{2} x^{5}}{5 e^{3}}+\frac {3 A \,b^{2} c \,x^{4}}{4 e}-\frac {3 A b \,c^{2} d \,x^{4}}{4 e^{2}}+\frac {A \,c^{3} d^{2} x^{4}}{4 e^{3}}+\frac {B \,b^{3} x^{4}}{4 e}-\frac {3 B \,b^{2} c d \,x^{4}}{4 e^{2}}+\frac {3 B b \,c^{2} d^{2} x^{4}}{4 e^{3}}-\frac {B \,c^{3} d^{3} x^{4}}{4 e^{4}}+\frac {A \,b^{3} x^{3}}{3 e}-\frac {A \,b^{2} c d \,x^{3}}{e^{2}}+\frac {A b \,c^{2} d^{2} x^{3}}{e^{3}}-\frac {A \,c^{3} d^{3} x^{3}}{3 e^{4}}-\frac {B \,b^{3} d \,x^{3}}{3 e^{2}}+\frac {B \,b^{2} c \,d^{2} x^{3}}{e^{3}}-\frac {B b \,c^{2} d^{3} x^{3}}{e^{4}}+\frac {B \,c^{3} d^{4} x^{3}}{3 e^{5}}-\frac {A \,b^{3} d \,x^{2}}{2 e^{2}}+\frac {3 A \,b^{2} c \,d^{2} x^{2}}{2 e^{3}}-\frac {3 A b \,c^{2} d^{3} x^{2}}{2 e^{4}}+\frac {A \,c^{3} d^{4} x^{2}}{2 e^{5}}+\frac {B \,b^{3} d^{2} x^{2}}{2 e^{3}}-\frac {3 B \,b^{2} c \,d^{3} x^{2}}{2 e^{4}}+\frac {3 B b \,c^{2} d^{4} x^{2}}{2 e^{5}}-\frac {B \,c^{3} d^{5} x^{2}}{2 e^{6}}-\frac {A \,b^{3} d^{3} \ln \left (e x +d \right )}{e^{4}}+\frac {A \,b^{3} d^{2} x}{e^{3}}+\frac {3 A \,b^{2} c \,d^{4} \ln \left (e x +d \right )}{e^{5}}-\frac {3 A \,b^{2} c \,d^{3} x}{e^{4}}-\frac {3 A b \,c^{2} d^{5} \ln \left (e x +d \right )}{e^{6}}+\frac {3 A b \,c^{2} d^{4} x}{e^{5}}+\frac {A \,c^{3} d^{6} \ln \left (e x +d \right )}{e^{7}}-\frac {A \,c^{3} d^{5} x}{e^{6}}+\frac {B \,b^{3} d^{4} \ln \left (e x +d \right )}{e^{5}}-\frac {B \,b^{3} d^{3} x}{e^{4}}-\frac {3 B \,b^{2} c \,d^{5} \ln \left (e x +d \right )}{e^{6}}+\frac {3 B \,b^{2} c \,d^{4} x}{e^{5}}+\frac {3 B b \,c^{2} d^{6} \ln \left (e x +d \right )}{e^{7}}-\frac {3 B b \,c^{2} d^{5} x}{e^{6}}-\frac {B \,c^{3} d^{7} \ln \left (e x +d \right )}{e^{8}}+\frac {B \,c^{3} d^{6} x}{e^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.55, size = 530, normalized size = 2.06 \begin {gather*} \frac {60 \, B c^{3} e^{6} x^{7} - 70 \, {\left (B c^{3} d e^{5} - {\left (3 \, B b c^{2} + A c^{3}\right )} e^{6}\right )} x^{6} + 84 \, {\left (B c^{3} d^{2} e^{4} - {\left (3 \, B b c^{2} + A c^{3}\right )} d e^{5} + 3 \, {\left (B b^{2} c + A b c^{2}\right )} e^{6}\right )} x^{5} - 105 \, {\left (B c^{3} d^{3} e^{3} - {\left (3 \, B b c^{2} + A c^{3}\right )} d^{2} e^{4} + 3 \, {\left (B b^{2} c + A b c^{2}\right )} d e^{5} - {\left (B b^{3} + 3 \, A b^{2} c\right )} e^{6}\right )} x^{4} + 140 \, {\left (B c^{3} d^{4} e^{2} + A b^{3} e^{6} - {\left (3 \, B b c^{2} + A c^{3}\right )} d^{3} e^{3} + 3 \, {\left (B b^{2} c + A b c^{2}\right )} d^{2} e^{4} - {\left (B b^{3} + 3 \, A b^{2} c\right )} d e^{5}\right )} x^{3} - 210 \, {\left (B c^{3} d^{5} e + A b^{3} d e^{5} - {\left (3 \, B b c^{2} + A c^{3}\right )} d^{4} e^{2} + 3 \, {\left (B b^{2} c + A b c^{2}\right )} d^{3} e^{3} - {\left (B b^{3} + 3 \, A b^{2} c\right )} d^{2} e^{4}\right )} x^{2} + 420 \, {\left (B c^{3} d^{6} + A b^{3} d^{2} e^{4} - {\left (3 \, B b c^{2} + A c^{3}\right )} d^{5} e + 3 \, {\left (B b^{2} c + A b c^{2}\right )} d^{4} e^{2} - {\left (B b^{3} + 3 \, A b^{2} c\right )} d^{3} e^{3}\right )} x}{420 \, e^{7}} - \frac {{\left (B c^{3} d^{7} + A b^{3} d^{3} e^{4} - {\left (3 \, B b c^{2} + A c^{3}\right )} d^{6} e + 3 \, {\left (B b^{2} c + A b c^{2}\right )} d^{5} e^{2} - {\left (B b^{3} + 3 \, A b^{2} c\right )} d^{4} e^{3}\right )} \log \left (e x + d\right )}{e^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.39, size = 560, normalized size = 2.18 \begin {gather*} x^4\,\left (\frac {B\,b^3+3\,A\,c\,b^2}{4\,e}+\frac {d\,\left (\frac {d\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{e}-\frac {B\,c^3\,d}{e^2}\right )}{e}-\frac {3\,b\,c\,\left (A\,c+B\,b\right )}{e}\right )}{4\,e}\right )-x^5\,\left (\frac {d\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{e}-\frac {B\,c^3\,d}{e^2}\right )}{5\,e}-\frac {3\,b\,c\,\left (A\,c+B\,b\right )}{5\,e}\right )+x^3\,\left (\frac {A\,b^3}{3\,e}-\frac {d\,\left (\frac {B\,b^3+3\,A\,c\,b^2}{e}+\frac {d\,\left (\frac {d\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{e}-\frac {B\,c^3\,d}{e^2}\right )}{e}-\frac {3\,b\,c\,\left (A\,c+B\,b\right )}{e}\right )}{e}\right )}{3\,e}\right )+x^6\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{6\,e}-\frac {B\,c^3\,d}{6\,e^2}\right )-\frac {\ln \left (d+e\,x\right )\,\left (-B\,b^3\,d^4\,e^3+A\,b^3\,d^3\,e^4+3\,B\,b^2\,c\,d^5\,e^2-3\,A\,b^2\,c\,d^4\,e^3-3\,B\,b\,c^2\,d^6\,e+3\,A\,b\,c^2\,d^5\,e^2+B\,c^3\,d^7-A\,c^3\,d^6\,e\right )}{e^8}-\frac {d\,x^2\,\left (\frac {A\,b^3}{e}-\frac {d\,\left (\frac {B\,b^3+3\,A\,c\,b^2}{e}+\frac {d\,\left (\frac {d\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{e}-\frac {B\,c^3\,d}{e^2}\right )}{e}-\frac {3\,b\,c\,\left (A\,c+B\,b\right )}{e}\right )}{e}\right )}{e}\right )}{2\,e}+\frac {d^2\,x\,\left (\frac {A\,b^3}{e}-\frac {d\,\left (\frac {B\,b^3+3\,A\,c\,b^2}{e}+\frac {d\,\left (\frac {d\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{e}-\frac {B\,c^3\,d}{e^2}\right )}{e}-\frac {3\,b\,c\,\left (A\,c+B\,b\right )}{e}\right )}{e}\right )}{e}\right )}{e^2}+\frac {B\,c^3\,x^7}{7\,e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.01, size = 578, normalized size = 2.25 \begin {gather*} \frac {B c^{3} x^{7}}{7 e} + \frac {d^{3} \left (- A e + B d\right ) \left (b e - c d\right )^{3} \log {\left (d + e x \right )}}{e^{8}} + x^{6} \left (\frac {A c^{3}}{6 e} + \frac {B b c^{2}}{2 e} - \frac {B c^{3} d}{6 e^{2}}\right ) + x^{5} \left (\frac {3 A b c^{2}}{5 e} - \frac {A c^{3} d}{5 e^{2}} + \frac {3 B b^{2} c}{5 e} - \frac {3 B b c^{2} d}{5 e^{2}} + \frac {B c^{3} d^{2}}{5 e^{3}}\right ) + x^{4} \left (\frac {3 A b^{2} c}{4 e} - \frac {3 A b c^{2} d}{4 e^{2}} + \frac {A c^{3} d^{2}}{4 e^{3}} + \frac {B b^{3}}{4 e} - \frac {3 B b^{2} c d}{4 e^{2}} + \frac {3 B b c^{2} d^{2}}{4 e^{3}} - \frac {B c^{3} d^{3}}{4 e^{4}}\right ) + x^{3} \left (\frac {A b^{3}}{3 e} - \frac {A b^{2} c d}{e^{2}} + \frac {A b c^{2} d^{2}}{e^{3}} - \frac {A c^{3} d^{3}}{3 e^{4}} - \frac {B b^{3} d}{3 e^{2}} + \frac {B b^{2} c d^{2}}{e^{3}} - \frac {B b c^{2} d^{3}}{e^{4}} + \frac {B c^{3} d^{4}}{3 e^{5}}\right ) + x^{2} \left (- \frac {A b^{3} d}{2 e^{2}} + \frac {3 A b^{2} c d^{2}}{2 e^{3}} - \frac {3 A b c^{2} d^{3}}{2 e^{4}} + \frac {A c^{3} d^{4}}{2 e^{5}} + \frac {B b^{3} d^{2}}{2 e^{3}} - \frac {3 B b^{2} c d^{3}}{2 e^{4}} + \frac {3 B b c^{2} d^{4}}{2 e^{5}} - \frac {B c^{3} d^{5}}{2 e^{6}}\right ) + x \left (\frac {A b^{3} d^{2}}{e^{3}} - \frac {3 A b^{2} c d^{3}}{e^{4}} + \frac {3 A b c^{2} d^{4}}{e^{5}} - \frac {A c^{3} d^{5}}{e^{6}} - \frac {B b^{3} d^{3}}{e^{4}} + \frac {3 B b^{2} c d^{4}}{e^{5}} - \frac {3 B b c^{2} d^{5}}{e^{6}} + \frac {B c^{3} d^{6}}{e^{7}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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