Optimal. Leaf size=232 \[ \frac {\log (d+e x) \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{e^6}-\frac {x \left (A c e (3 c d-2 b e)-B \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )\right )}{e^5}+\frac {d^2 (B d-A e) (c d-b e)^2}{2 e^6 (d+e x)^2}-\frac {d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{e^6 (d+e x)}-\frac {c x^2 (-A c e-2 b B e+3 B c d)}{2 e^4}+\frac {B c^2 x^3}{3 e^3} \]
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Rubi [A] time = 0.32, antiderivative size = 232, 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 {x \left (A c e (3 c d-2 b e)-B \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )\right )}{e^5}+\frac {\log (d+e x) \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{e^6}+\frac {d^2 (B d-A e) (c d-b e)^2}{2 e^6 (d+e x)^2}-\frac {c x^2 (-A c e-2 b B e+3 B c d)}{2 e^4}-\frac {d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{e^6 (d+e x)}+\frac {B c^2 x^3}{3 e^3} \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 )^2}{(d+e x)^3} \, dx &=\int \left (\frac {-A c e (3 c d-2 b e)+B \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )}{e^5}+\frac {c (-3 B c d+2 b B e+A c e) x}{e^4}+\frac {B c^2 x^2}{e^3}-\frac {d^2 (B d-A e) (c d-b e)^2}{e^5 (d+e x)^3}+\frac {d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{e^5 (d+e x)^2}+\frac {A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )}{e^5 (d+e x)}\right ) \, dx\\ &=-\frac {\left (A c e (3 c d-2 b e)-B \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )\right ) x}{e^5}-\frac {c (3 B c d-2 b B e-A c e) x^2}{2 e^4}+\frac {B c^2 x^3}{3 e^3}+\frac {d^2 (B d-A e) (c d-b e)^2}{2 e^6 (d+e x)^2}-\frac {d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{e^6 (d+e x)}+\frac {\left (A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )\right ) \log (d+e x)}{e^6}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 219, normalized size = 0.94 \begin {gather*} \frac {6 e x \left (A c e (2 b e-3 c d)+B \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )\right )+6 \log (d+e x) \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )+B d \left (-3 b^2 e^2+12 b c d e-10 c^2 d^2\right )\right )+\frac {3 d^2 (B d-A e) (c d-b e)^2}{(d+e x)^2}+3 c e^2 x^2 (A c e+2 b B e-3 B c d)-\frac {6 d (c d-b e) (2 A e (b e-2 c d)+B d (5 c d-3 b e))}{d+e x}+2 B c^2 e^3 x^3}{6 e^6} \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 )^2}{(d+e x)^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.41, size = 483, normalized size = 2.08 \begin {gather*} \frac {2 \, B c^{2} e^{5} x^{5} - 27 \, B c^{2} d^{5} + 9 \, A b^{2} d^{2} e^{3} + 21 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e - 15 \, {\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2} - {\left (5 \, B c^{2} d e^{4} - 3 \, {\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + 2 \, {\left (10 \, B c^{2} d^{2} e^{3} - 6 \, {\left (2 \, B b c + A c^{2}\right )} d e^{4} + 3 \, {\left (B b^{2} + 2 \, A b c\right )} e^{5}\right )} x^{3} + 3 \, {\left (21 \, B c^{2} d^{3} e^{2} - 11 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 4 \, {\left (B b^{2} + 2 \, A b c\right )} d e^{4}\right )} x^{2} + 6 \, {\left (B c^{2} d^{4} e + 2 \, A b^{2} d e^{4} + {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} - 2 \, {\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{3}\right )} x - 6 \, {\left (10 \, B c^{2} d^{5} - A b^{2} d^{2} e^{3} - 6 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2} + {\left (10 \, B c^{2} d^{3} e^{2} - A b^{2} e^{5} - 6 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d e^{4}\right )} x^{2} + 2 \, {\left (10 \, B c^{2} d^{4} e - A b^{2} d e^{4} - 6 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{3}\right )} x\right )} \log \left (e x + d\right )}{6 \, {\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} e^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 307, normalized size = 1.32 \begin {gather*} -{\left (10 \, B c^{2} d^{3} - 12 \, B b c d^{2} e - 6 \, A c^{2} d^{2} e + 3 \, B b^{2} d e^{2} + 6 \, A b c d e^{2} - A b^{2} e^{3}\right )} e^{\left (-6\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{6} \, {\left (2 \, B c^{2} x^{3} e^{6} - 9 \, B c^{2} d x^{2} e^{5} + 36 \, B c^{2} d^{2} x e^{4} + 6 \, B b c x^{2} e^{6} + 3 \, A c^{2} x^{2} e^{6} - 36 \, B b c d x e^{5} - 18 \, A c^{2} d x e^{5} + 6 \, B b^{2} x e^{6} + 12 \, A b c x e^{6}\right )} e^{\left (-9\right )} - \frac {{\left (9 \, B c^{2} d^{5} - 14 \, B b c d^{4} e - 7 \, A c^{2} d^{4} e + 5 \, B b^{2} d^{3} e^{2} + 10 \, A b c d^{3} e^{2} - 3 \, A b^{2} d^{2} e^{3} + 2 \, {\left (5 \, B c^{2} d^{4} e - 8 \, B b c d^{3} e^{2} - 4 \, A c^{2} d^{3} e^{2} + 3 \, B b^{2} d^{2} e^{3} + 6 \, A b c d^{2} e^{3} - 2 \, A b^{2} d e^{4}\right )} x\right )} e^{\left (-6\right )}}{2 \, {\left (x e + d\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 420, normalized size = 1.81 \begin {gather*} \frac {B \,c^{2} x^{3}}{3 e^{3}}-\frac {A \,b^{2} d^{2}}{2 \left (e x +d \right )^{2} e^{3}}+\frac {A b c \,d^{3}}{\left (e x +d \right )^{2} e^{4}}-\frac {A \,c^{2} d^{4}}{2 \left (e x +d \right )^{2} e^{5}}+\frac {A \,c^{2} x^{2}}{2 e^{3}}+\frac {B \,b^{2} d^{3}}{2 \left (e x +d \right )^{2} e^{4}}-\frac {B b c \,d^{4}}{\left (e x +d \right )^{2} e^{5}}+\frac {B b c \,x^{2}}{e^{3}}+\frac {B \,c^{2} d^{5}}{2 \left (e x +d \right )^{2} e^{6}}-\frac {3 B \,c^{2} d \,x^{2}}{2 e^{4}}+\frac {2 A \,b^{2} d}{\left (e x +d \right ) e^{3}}+\frac {A \,b^{2} \ln \left (e x +d \right )}{e^{3}}-\frac {6 A b c \,d^{2}}{\left (e x +d \right ) e^{4}}-\frac {6 A b c d \ln \left (e x +d \right )}{e^{4}}+\frac {2 A b c x}{e^{3}}+\frac {4 A \,c^{2} d^{3}}{\left (e x +d \right ) e^{5}}+\frac {6 A \,c^{2} d^{2} \ln \left (e x +d \right )}{e^{5}}-\frac {3 A \,c^{2} d x}{e^{4}}-\frac {3 B \,b^{2} d^{2}}{\left (e x +d \right ) e^{4}}-\frac {3 B \,b^{2} d \ln \left (e x +d \right )}{e^{4}}+\frac {B \,b^{2} x}{e^{3}}+\frac {8 B b c \,d^{3}}{\left (e x +d \right ) e^{5}}+\frac {12 B b c \,d^{2} \ln \left (e x +d \right )}{e^{5}}-\frac {6 B b c d x}{e^{4}}-\frac {5 B \,c^{2} d^{4}}{\left (e x +d \right ) e^{6}}-\frac {10 B \,c^{2} d^{3} \ln \left (e x +d \right )}{e^{6}}+\frac {6 B \,c^{2} d^{2} x}{e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.62, size = 302, normalized size = 1.30 \begin {gather*} -\frac {9 \, B c^{2} d^{5} - 3 \, A b^{2} d^{2} e^{3} - 7 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e + 5 \, {\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2} + 2 \, {\left (5 \, B c^{2} d^{4} e - 2 \, A b^{2} d e^{4} - 4 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{3}\right )} x}{2 \, {\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} e^{6}\right )}} + \frac {2 \, B c^{2} e^{2} x^{3} - 3 \, {\left (3 \, B c^{2} d e - {\left (2 \, B b c + A c^{2}\right )} e^{2}\right )} x^{2} + 6 \, {\left (6 \, B c^{2} d^{2} - 3 \, {\left (2 \, B b c + A c^{2}\right )} d e + {\left (B b^{2} + 2 \, A b c\right )} e^{2}\right )} x}{6 \, e^{5}} - \frac {{\left (10 \, B c^{2} d^{3} - A b^{2} e^{3} - 6 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d e^{2}\right )} \log \left (e x + d\right )}{e^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 334, normalized size = 1.44 \begin {gather*} x^2\,\left (\frac {A\,c^2+2\,B\,b\,c}{2\,e^3}-\frac {3\,B\,c^2\,d}{2\,e^4}\right )-x\,\left (\frac {3\,d\,\left (\frac {A\,c^2+2\,B\,b\,c}{e^3}-\frac {3\,B\,c^2\,d}{e^4}\right )}{e}-\frac {B\,b^2+2\,A\,c\,b}{e^3}+\frac {3\,B\,c^2\,d^2}{e^5}\right )-\frac {\frac {5\,B\,b^2\,d^3\,e^2-3\,A\,b^2\,d^2\,e^3-14\,B\,b\,c\,d^4\,e+10\,A\,b\,c\,d^3\,e^2+9\,B\,c^2\,d^5-7\,A\,c^2\,d^4\,e}{2\,e}+x\,\left (3\,B\,b^2\,d^2\,e^2-2\,A\,b^2\,d\,e^3-8\,B\,b\,c\,d^3\,e+6\,A\,b\,c\,d^2\,e^2+5\,B\,c^2\,d^4-4\,A\,c^2\,d^3\,e\right )}{d^2\,e^5+2\,d\,e^6\,x+e^7\,x^2}+\frac {\ln \left (d+e\,x\right )\,\left (-3\,B\,b^2\,d\,e^2+A\,b^2\,e^3+12\,B\,b\,c\,d^2\,e-6\,A\,b\,c\,d\,e^2-10\,B\,c^2\,d^3+6\,A\,c^2\,d^2\,e\right )}{e^6}+\frac {B\,c^2\,x^3}{3\,e^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.07, size = 362, normalized size = 1.56 \begin {gather*} \frac {B c^{2} x^{3}}{3 e^{3}} + x^{2} \left (\frac {A c^{2}}{2 e^{3}} + \frac {B b c}{e^{3}} - \frac {3 B c^{2} d}{2 e^{4}}\right ) + x \left (\frac {2 A b c}{e^{3}} - \frac {3 A c^{2} d}{e^{4}} + \frac {B b^{2}}{e^{3}} - \frac {6 B b c d}{e^{4}} + \frac {6 B c^{2} d^{2}}{e^{5}}\right ) + \frac {3 A b^{2} d^{2} e^{3} - 10 A b c d^{3} e^{2} + 7 A c^{2} d^{4} e - 5 B b^{2} d^{3} e^{2} + 14 B b c d^{4} e - 9 B c^{2} d^{5} + x \left (4 A b^{2} d e^{4} - 12 A b c d^{2} e^{3} + 8 A c^{2} d^{3} e^{2} - 6 B b^{2} d^{2} e^{3} + 16 B b c d^{3} e^{2} - 10 B c^{2} d^{4} e\right )}{2 d^{2} e^{6} + 4 d e^{7} x + 2 e^{8} x^{2}} - \frac {\left (- A b^{2} e^{3} + 6 A b c d e^{2} - 6 A c^{2} d^{2} e + 3 B b^{2} d e^{2} - 12 B b c d^{2} e + 10 B c^{2} d^{3}\right ) \log {\left (d + e x \right )}}{e^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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