Optimal. Leaf size=290 \[ \frac {x \left (B \left (a e^2+c d^2\right )^3-A c d e \left (3 a^2 e^4+3 a c d^2 e^2+c^2 d^4\right )\right )}{e^7}-\frac {c x^2 \left (3 a^2 e^4+3 a c d^2 e^2+c^2 d^4\right ) (B d-A e)}{2 e^6}-\frac {c x^3 \left (A c d e \left (3 a e^2+c d^2\right )-B \left (3 a^2 e^4+3 a c d^2 e^2+c^2 d^4\right )\right )}{3 e^5}-\frac {c^2 x^4 \left (3 a e^2+c d^2\right ) (B d-A e)}{4 e^4}+\frac {c^2 x^5 \left (3 a B e^2-A c d e+B c d^2\right )}{5 e^3}-\frac {\left (a e^2+c d^2\right )^3 (B d-A e) \log (d+e x)}{e^8}-\frac {c^3 x^6 (B d-A e)}{6 e^2}+\frac {B c^3 x^7}{7 e} \]
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Rubi [A] time = 0.40, antiderivative size = 290, 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} \[ -\frac {c x^3 \left (A c d e \left (3 a e^2+c d^2\right )-B \left (3 a^2 e^4+3 a c d^2 e^2+c^2 d^4\right )\right )}{3 e^5}-\frac {c x^2 \left (3 a^2 e^4+3 a c d^2 e^2+c^2 d^4\right ) (B d-A e)}{2 e^6}+\frac {x \left (B \left (a e^2+c d^2\right )^3-A c d e \left (3 a^2 e^4+3 a c d^2 e^2+c^2 d^4\right )\right )}{e^7}+\frac {c^2 x^5 \left (3 a B e^2-A c d e+B c d^2\right )}{5 e^3}-\frac {c^2 x^4 \left (3 a e^2+c d^2\right ) (B d-A e)}{4 e^4}-\frac {\left (a e^2+c d^2\right )^3 (B d-A e) \log (d+e x)}{e^8}-\frac {c^3 x^6 (B d-A e)}{6 e^2}+\frac {B c^3 x^7}{7 e} \]
Antiderivative was successfully verified.
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Rule 772
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a+c x^2\right )^3}{d+e x} \, dx &=\int \left (\frac {B \left (c d^2+a e^2\right )^3-A c d e \left (c^2 d^4+3 a c d^2 e^2+3 a^2 e^4\right )}{e^7}+\frac {c (-B d+A e) \left (c^2 d^4+3 a c d^2 e^2+3 a^2 e^4\right ) x}{e^6}+\frac {c \left (-A c d e \left (c d^2+3 a e^2\right )+B \left (c^2 d^4+3 a c d^2 e^2+3 a^2 e^4\right )\right ) x^2}{e^5}+\frac {c^2 (-B d+A e) \left (c d^2+3 a e^2\right ) x^3}{e^4}-\frac {c^2 \left (-B c d^2+A c d e-3 a B e^2\right ) x^4}{e^3}+\frac {c^3 (-B d+A e) x^5}{e^2}+\frac {B c^3 x^6}{e}+\frac {(-B d+A e) \left (c d^2+a e^2\right )^3}{e^7 (d+e x)}\right ) \, dx\\ &=\frac {\left (B \left (c d^2+a e^2\right )^3-A c d e \left (c^2 d^4+3 a c d^2 e^2+3 a^2 e^4\right )\right ) x}{e^7}-\frac {c (B d-A e) \left (c^2 d^4+3 a c d^2 e^2+3 a^2 e^4\right ) x^2}{2 e^6}-\frac {c \left (A c d e \left (c d^2+3 a e^2\right )-B \left (c^2 d^4+3 a c d^2 e^2+3 a^2 e^4\right )\right ) x^3}{3 e^5}-\frac {c^2 (B d-A e) \left (c d^2+3 a e^2\right ) x^4}{4 e^4}+\frac {c^2 \left (B c d^2-A c d e+3 a B e^2\right ) x^5}{5 e^3}-\frac {c^3 (B d-A e) x^6}{6 e^2}+\frac {B c^3 x^7}{7 e}-\frac {(B d-A e) \left (c d^2+a e^2\right )^3 \log (d+e x)}{e^8}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 311, normalized size = 1.07 \[ \frac {e x \left (7 A c e \left (90 a^2 e^4 (e x-2 d)+15 a c e^2 \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )+c^2 \left (-60 d^5+30 d^4 e x-20 d^3 e^2 x^2+15 d^2 e^3 x^3-12 d e^4 x^4+10 e^5 x^5\right )\right )+B \left (420 a^3 e^6+210 a^2 c e^4 \left (6 d^2-3 d e x+2 e^2 x^2\right )+21 a c^2 e^2 \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )+c^3 \left (420 d^6-210 d^5 e x+140 d^4 e^2 x^2-105 d^3 e^3 x^3+84 d^2 e^4 x^4-70 d e^5 x^5+60 e^6 x^6\right )\right )\right )-420 \left (a e^2+c d^2\right )^3 (B d-A e) \log (d+e x)}{420 e^8} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 448, normalized size = 1.54 \[ \frac {60 \, B c^{3} e^{7} x^{7} - 70 \, {\left (B c^{3} d e^{6} - A c^{3} e^{7}\right )} x^{6} + 84 \, {\left (B c^{3} d^{2} e^{5} - A c^{3} d e^{6} + 3 \, B a c^{2} e^{7}\right )} x^{5} - 105 \, {\left (B c^{3} d^{3} e^{4} - A c^{3} d^{2} e^{5} + 3 \, B a c^{2} d e^{6} - 3 \, A a c^{2} e^{7}\right )} x^{4} + 140 \, {\left (B c^{3} d^{4} e^{3} - A c^{3} d^{3} e^{4} + 3 \, B a c^{2} d^{2} e^{5} - 3 \, A a c^{2} d e^{6} + 3 \, B a^{2} c e^{7}\right )} x^{3} - 210 \, {\left (B c^{3} d^{5} e^{2} - A c^{3} d^{4} e^{3} + 3 \, B a c^{2} d^{3} e^{4} - 3 \, A a c^{2} d^{2} e^{5} + 3 \, B a^{2} c d e^{6} - 3 \, A a^{2} c e^{7}\right )} x^{2} + 420 \, {\left (B c^{3} d^{6} e - A c^{3} d^{5} e^{2} + 3 \, B a c^{2} d^{4} e^{3} - 3 \, A a c^{2} d^{3} e^{4} + 3 \, B a^{2} c d^{2} e^{5} - 3 \, A a^{2} c d e^{6} + B a^{3} e^{7}\right )} x - 420 \, {\left (B c^{3} d^{7} - A c^{3} d^{6} e + 3 \, B a c^{2} d^{5} e^{2} - 3 \, A a c^{2} d^{4} e^{3} + 3 \, B a^{2} c d^{3} e^{4} - 3 \, A a^{2} c d^{2} e^{5} + B a^{3} d e^{6} - A a^{3} e^{7}\right )} \log \left (e x + d\right )}{420 \, e^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 459, normalized size = 1.58 \[ -{\left (B c^{3} d^{7} - A c^{3} d^{6} e + 3 \, B a c^{2} d^{5} e^{2} - 3 \, A a c^{2} d^{4} e^{3} + 3 \, B a^{2} c d^{3} e^{4} - 3 \, A a^{2} c d^{2} e^{5} + B a^{3} d e^{6} - A a^{3} e^{7}\right )} e^{\left (-8\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{420} \, {\left (60 \, B c^{3} x^{7} e^{6} - 70 \, B c^{3} d x^{6} e^{5} + 84 \, B c^{3} d^{2} x^{5} e^{4} - 105 \, B c^{3} d^{3} x^{4} e^{3} + 140 \, B c^{3} d^{4} x^{3} e^{2} - 210 \, B c^{3} d^{5} x^{2} e + 420 \, B c^{3} d^{6} x + 70 \, A c^{3} x^{6} e^{6} - 84 \, A c^{3} d x^{5} e^{5} + 105 \, A c^{3} d^{2} x^{4} e^{4} - 140 \, A c^{3} d^{3} x^{3} e^{3} + 210 \, A c^{3} d^{4} x^{2} e^{2} - 420 \, A c^{3} d^{5} x e + 252 \, B a c^{2} x^{5} e^{6} - 315 \, B a c^{2} d x^{4} e^{5} + 420 \, B a c^{2} d^{2} x^{3} e^{4} - 630 \, B a c^{2} d^{3} x^{2} e^{3} + 1260 \, B a c^{2} d^{4} x e^{2} + 315 \, A a c^{2} x^{4} e^{6} - 420 \, A a c^{2} d x^{3} e^{5} + 630 \, A a c^{2} d^{2} x^{2} e^{4} - 1260 \, A a c^{2} d^{3} x e^{3} + 420 \, B a^{2} c x^{3} e^{6} - 630 \, B a^{2} c d x^{2} e^{5} + 1260 \, B a^{2} c d^{2} x e^{4} + 630 \, A a^{2} c x^{2} e^{6} - 1260 \, A a^{2} c d x e^{5} + 420 \, B a^{3} x e^{6}\right )} e^{\left (-7\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 526, normalized size = 1.81 \[ \frac {B \,c^{3} x^{7}}{7 e}+\frac {A \,c^{3} x^{6}}{6 e}-\frac {B \,c^{3} d \,x^{6}}{6 e^{2}}-\frac {A \,c^{3} d \,x^{5}}{5 e^{2}}+\frac {3 B a \,c^{2} x^{5}}{5 e}+\frac {B \,c^{3} d^{2} x^{5}}{5 e^{3}}+\frac {3 A a \,c^{2} x^{4}}{4 e}+\frac {A \,c^{3} d^{2} x^{4}}{4 e^{3}}-\frac {3 B a \,c^{2} d \,x^{4}}{4 e^{2}}-\frac {B \,c^{3} d^{3} x^{4}}{4 e^{4}}-\frac {A a \,c^{2} d \,x^{3}}{e^{2}}-\frac {A \,c^{3} d^{3} x^{3}}{3 e^{4}}+\frac {B \,a^{2} c \,x^{3}}{e}+\frac {B a \,c^{2} d^{2} x^{3}}{e^{3}}+\frac {B \,c^{3} d^{4} x^{3}}{3 e^{5}}+\frac {3 A \,a^{2} c \,x^{2}}{2 e}+\frac {3 A a \,c^{2} d^{2} x^{2}}{2 e^{3}}+\frac {A \,c^{3} d^{4} x^{2}}{2 e^{5}}-\frac {3 B \,a^{2} c d \,x^{2}}{2 e^{2}}-\frac {3 B a \,c^{2} d^{3} x^{2}}{2 e^{4}}-\frac {B \,c^{3} d^{5} x^{2}}{2 e^{6}}+\frac {A \,a^{3} \ln \left (e x +d \right )}{e}+\frac {3 A \,a^{2} c \,d^{2} \ln \left (e x +d \right )}{e^{3}}-\frac {3 A \,a^{2} c d x}{e^{2}}+\frac {3 A a \,c^{2} d^{4} \ln \left (e x +d \right )}{e^{5}}-\frac {3 A a \,c^{2} d^{3} x}{e^{4}}+\frac {A \,c^{3} d^{6} \ln \left (e x +d \right )}{e^{7}}-\frac {A \,c^{3} d^{5} x}{e^{6}}-\frac {B \,a^{3} d \ln \left (e x +d \right )}{e^{2}}+\frac {B \,a^{3} x}{e}-\frac {3 B \,a^{2} c \,d^{3} \ln \left (e x +d \right )}{e^{4}}+\frac {3 B \,a^{2} c \,d^{2} x}{e^{3}}-\frac {3 B a \,c^{2} d^{5} \ln \left (e x +d \right )}{e^{6}}+\frac {3 B a \,c^{2} d^{4} x}{e^{5}}-\frac {B \,c^{3} d^{7} \ln \left (e x +d \right )}{e^{8}}+\frac {B \,c^{3} d^{6} x}{e^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.56, size = 447, normalized size = 1.54 \[ \frac {60 \, B c^{3} e^{6} x^{7} - 70 \, {\left (B c^{3} d e^{5} - A c^{3} e^{6}\right )} x^{6} + 84 \, {\left (B c^{3} d^{2} e^{4} - A c^{3} d e^{5} + 3 \, B a c^{2} e^{6}\right )} x^{5} - 105 \, {\left (B c^{3} d^{3} e^{3} - A c^{3} d^{2} e^{4} + 3 \, B a c^{2} d e^{5} - 3 \, A a c^{2} e^{6}\right )} x^{4} + 140 \, {\left (B c^{3} d^{4} e^{2} - A c^{3} d^{3} e^{3} + 3 \, B a c^{2} d^{2} e^{4} - 3 \, A a c^{2} d e^{5} + 3 \, B a^{2} c e^{6}\right )} x^{3} - 210 \, {\left (B c^{3} d^{5} e - A c^{3} d^{4} e^{2} + 3 \, B a c^{2} d^{3} e^{3} - 3 \, A a c^{2} d^{2} e^{4} + 3 \, B a^{2} c d e^{5} - 3 \, A a^{2} c e^{6}\right )} x^{2} + 420 \, {\left (B c^{3} d^{6} - A c^{3} d^{5} e + 3 \, B a c^{2} d^{4} e^{2} - 3 \, A a c^{2} d^{3} e^{3} + 3 \, B a^{2} c d^{2} e^{4} - 3 \, A a^{2} c d e^{5} + B a^{3} e^{6}\right )} x}{420 \, e^{7}} - \frac {{\left (B c^{3} d^{7} - A c^{3} d^{6} e + 3 \, B a c^{2} d^{5} e^{2} - 3 \, A a c^{2} d^{4} e^{3} + 3 \, B a^{2} c d^{3} e^{4} - 3 \, A a^{2} c d^{2} e^{5} + B a^{3} d e^{6} - A a^{3} e^{7}\right )} \log \left (e x + d\right )}{e^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 494, normalized size = 1.70 \[ x^2\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {A\,c^3}{e}-\frac {B\,c^3\,d}{e^2}\right )}{e}-\frac {3\,B\,a\,c^2}{e}\right )}{e}+\frac {3\,A\,a\,c^2}{e}\right )}{e}-\frac {3\,B\,a^2\,c}{e}\right )}{2\,e}+\frac {3\,A\,a^2\,c}{2\,e}\right )+x^4\,\left (\frac {d\,\left (\frac {d\,\left (\frac {A\,c^3}{e}-\frac {B\,c^3\,d}{e^2}\right )}{e}-\frac {3\,B\,a\,c^2}{e}\right )}{4\,e}+\frac {3\,A\,a\,c^2}{4\,e}\right )+x^6\,\left (\frac {A\,c^3}{6\,e}-\frac {B\,c^3\,d}{6\,e^2}\right )-x^3\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {A\,c^3}{e}-\frac {B\,c^3\,d}{e^2}\right )}{e}-\frac {3\,B\,a\,c^2}{e}\right )}{e}+\frac {3\,A\,a\,c^2}{e}\right )}{3\,e}-\frac {B\,a^2\,c}{e}\right )+x\,\left (\frac {B\,a^3}{e}-\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {A\,c^3}{e}-\frac {B\,c^3\,d}{e^2}\right )}{e}-\frac {3\,B\,a\,c^2}{e}\right )}{e}+\frac {3\,A\,a\,c^2}{e}\right )}{e}-\frac {3\,B\,a^2\,c}{e}\right )}{e}+\frac {3\,A\,a^2\,c}{e}\right )}{e}\right )-x^5\,\left (\frac {d\,\left (\frac {A\,c^3}{e}-\frac {B\,c^3\,d}{e^2}\right )}{5\,e}-\frac {3\,B\,a\,c^2}{5\,e}\right )+\frac {\ln \left (d+e\,x\right )\,\left (-B\,a^3\,d\,e^6+A\,a^3\,e^7-3\,B\,a^2\,c\,d^3\,e^4+3\,A\,a^2\,c\,d^2\,e^5-3\,B\,a\,c^2\,d^5\,e^2+3\,A\,a\,c^2\,d^4\,e^3-B\,c^3\,d^7+A\,c^3\,d^6\,e\right )}{e^8}+\frac {B\,c^3\,x^7}{7\,e} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.92, size = 410, normalized size = 1.41 \[ \frac {B c^{3} x^{7}}{7 e} + x^{6} \left (\frac {A c^{3}}{6 e} - \frac {B c^{3} d}{6 e^{2}}\right ) + x^{5} \left (- \frac {A c^{3} d}{5 e^{2}} + \frac {3 B a c^{2}}{5 e} + \frac {B c^{3} d^{2}}{5 e^{3}}\right ) + x^{4} \left (\frac {3 A a c^{2}}{4 e} + \frac {A c^{3} d^{2}}{4 e^{3}} - \frac {3 B a c^{2} d}{4 e^{2}} - \frac {B c^{3} d^{3}}{4 e^{4}}\right ) + x^{3} \left (- \frac {A a c^{2} d}{e^{2}} - \frac {A c^{3} d^{3}}{3 e^{4}} + \frac {B a^{2} c}{e} + \frac {B a c^{2} d^{2}}{e^{3}} + \frac {B c^{3} d^{4}}{3 e^{5}}\right ) + x^{2} \left (\frac {3 A a^{2} c}{2 e} + \frac {3 A a c^{2} d^{2}}{2 e^{3}} + \frac {A c^{3} d^{4}}{2 e^{5}} - \frac {3 B a^{2} c d}{2 e^{2}} - \frac {3 B a c^{2} d^{3}}{2 e^{4}} - \frac {B c^{3} d^{5}}{2 e^{6}}\right ) + x \left (- \frac {3 A a^{2} c d}{e^{2}} - \frac {3 A a c^{2} d^{3}}{e^{4}} - \frac {A c^{3} d^{5}}{e^{6}} + \frac {B a^{3}}{e} + \frac {3 B a^{2} c d^{2}}{e^{3}} + \frac {3 B a c^{2} d^{4}}{e^{5}} + \frac {B c^{3} d^{6}}{e^{7}}\right ) - \frac {\left (- A e + B d\right ) \left (a e^{2} + c d^{2}\right )^{3} \log {\left (d + e x \right )}}{e^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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