Optimal. Leaf size=194 \[ \frac {d^2 (B d-A e) (c d-b e)^2}{e^6 (d+e x)}+\frac {d (c d-b e) \log (d+e x) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{e^6}-\frac {x (c d-b e) (2 B d (2 c d-b e)-A e (3 c d-b e))}{e^5}+\frac {x^2 (c d-b e) (-2 A c e-b B e+3 B c d)}{2 e^4}-\frac {c x^3 (-A c e-2 b B e+2 B c d)}{3 e^3}+\frac {B c^2 x^4}{4 e^2} \]
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Rubi [A] time = 0.28, antiderivative size = 194, 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 {d^2 (B d-A e) (c d-b e)^2}{e^6 (d+e x)}-\frac {c x^3 (-A c e-2 b B e+2 B c d)}{3 e^3}+\frac {x^2 (c d-b e) (-2 A c e-b B e+3 B c d)}{2 e^4}-\frac {x (c d-b e) (2 B d (2 c d-b e)-A e (3 c d-b e))}{e^5}+\frac {d (c d-b e) \log (d+e x) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{e^6}+\frac {B c^2 x^4}{4 e^2} \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)^2} \, dx &=\int \left (\frac {(c d-b e) (-2 B d (2 c d-b e)+A e (3 c d-b e))}{e^5}+\frac {(-c d+b e) (-3 B c d+b B e+2 A c e) x}{e^4}+\frac {c (-2 B c d+2 b B e+A c e) x^2}{e^3}+\frac {B c^2 x^3}{e^2}-\frac {d^2 (B d-A e) (c d-b e)^2}{e^5 (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^5 (d+e x)}\right ) \, dx\\ &=-\frac {(c d-b e) (2 B d (2 c d-b e)-A e (3 c d-b e)) x}{e^5}+\frac {(c d-b e) (3 B c d-b B e-2 A c e) x^2}{2 e^4}-\frac {c (2 B c d-2 b B e-A c e) x^3}{3 e^3}+\frac {B c^2 x^4}{4 e^2}+\frac {d^2 (B d-A e) (c d-b e)^2}{e^6 (d+e x)}+\frac {d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e)) \log (d+e x)}{e^6}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 184, normalized size = 0.95 \begin {gather*} \frac {\frac {12 d^2 (B d-A e) (c d-b e)^2}{d+e x}+4 c e^3 x^3 (A c e+2 b B e-2 B c d)+6 e^2 x^2 (b e-c d) (2 A c e+b B e-3 B c d)+12 e x (b e-c d) (A e (b e-3 c d)+2 B d (2 c d-b e))+12 d (c d-b e) \log (d+e x) (2 A e (b e-2 c d)+B d (5 c d-3 b e))+3 B c^2 e^4 x^4}{12 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)^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.41, size = 420, normalized size = 2.16 \begin {gather*} \frac {3 \, B c^{2} e^{5} x^{5} + 12 \, B c^{2} d^{5} - 12 \, A b^{2} d^{2} e^{3} - 12 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e + 12 \, {\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2} - {\left (5 \, B c^{2} d e^{4} - 4 \, {\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + 2 \, {\left (5 \, B c^{2} d^{2} e^{3} - 4 \, {\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} - 6 \, {\left (5 \, B c^{2} d^{3} e^{2} - 2 \, A b^{2} e^{5} - 4 \, {\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} - 12 \, {\left (4 \, B c^{2} d^{4} e - A b^{2} d e^{4} - 3 \, {\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 + 12 \, {\left (5 \, B c^{2} d^{5} - 2 \, A b^{2} d^{2} e^{3} - 4 \, {\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 (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\right )} \log \left (e x + d\right )}{12 \, {\left (e^{7} x + d e^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 380, normalized size = 1.96 \begin {gather*} \frac {1}{12} \, {\left (3 \, B c^{2} - \frac {4 \, {\left (5 \, B c^{2} d e - 2 \, B b c e^{2} - A c^{2} e^{2}\right )} e^{\left (-1\right )}}{x e + d} + \frac {6 \, {\left (10 \, B c^{2} d^{2} e^{2} - 8 \, B b c d e^{3} - 4 \, A c^{2} d e^{3} + B b^{2} e^{4} + 2 \, A b c e^{4}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} - \frac {12 \, {\left (10 \, B c^{2} d^{3} e^{3} - 12 \, B b c d^{2} e^{4} - 6 \, A c^{2} d^{2} e^{4} + 3 \, B b^{2} d e^{5} + 6 \, A b c d e^{5} - A b^{2} e^{6}\right )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{3}}\right )} {\left (x e + d\right )}^{4} e^{\left (-6\right )} - {\left (5 \, B c^{2} d^{4} - 8 \, B b c d^{3} e - 4 \, A c^{2} d^{3} e + 3 \, B b^{2} d^{2} e^{2} + 6 \, A b c d^{2} e^{2} - 2 \, A b^{2} d e^{3}\right )} e^{\left (-6\right )} \log \left (\frac {{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) + {\left (\frac {B c^{2} d^{5} e^{4}}{x e + d} - \frac {2 \, B b c d^{4} e^{5}}{x e + d} - \frac {A c^{2} d^{4} e^{5}}{x e + d} + \frac {B b^{2} d^{3} e^{6}}{x e + d} + \frac {2 \, A b c d^{3} e^{6}}{x e + d} - \frac {A b^{2} d^{2} e^{7}}{x e + d}\right )} e^{\left (-10\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 394, normalized size = 2.03 \begin {gather*} \frac {B \,c^{2} x^{4}}{4 e^{2}}+\frac {A \,c^{2} x^{3}}{3 e^{2}}+\frac {2 B b c \,x^{3}}{3 e^{2}}-\frac {2 B \,c^{2} d \,x^{3}}{3 e^{3}}+\frac {A b c \,x^{2}}{e^{2}}-\frac {A \,c^{2} d \,x^{2}}{e^{3}}+\frac {B \,b^{2} x^{2}}{2 e^{2}}-\frac {2 B b c d \,x^{2}}{e^{3}}+\frac {3 B \,c^{2} d^{2} x^{2}}{2 e^{4}}-\frac {A \,b^{2} d^{2}}{\left (e x +d \right ) e^{3}}-\frac {2 A \,b^{2} d \ln \left (e x +d \right )}{e^{3}}+\frac {A \,b^{2} x}{e^{2}}+\frac {2 A b c \,d^{3}}{\left (e x +d \right ) e^{4}}+\frac {6 A b c \,d^{2} \ln \left (e x +d \right )}{e^{4}}-\frac {4 A b c d x}{e^{3}}-\frac {A \,c^{2} d^{4}}{\left (e x +d \right ) e^{5}}-\frac {4 A \,c^{2} d^{3} \ln \left (e x +d \right )}{e^{5}}+\frac {3 A \,c^{2} d^{2} x}{e^{4}}+\frac {B \,b^{2} d^{3}}{\left (e x +d \right ) e^{4}}+\frac {3 B \,b^{2} d^{2} \ln \left (e x +d \right )}{e^{4}}-\frac {2 B \,b^{2} d x}{e^{3}}-\frac {2 B b c \,d^{4}}{\left (e x +d \right ) e^{5}}-\frac {8 B b c \,d^{3} \ln \left (e x +d \right )}{e^{5}}+\frac {6 B b c \,d^{2} x}{e^{4}}+\frac {B \,c^{2} d^{5}}{\left (e x +d \right ) e^{6}}+\frac {5 B \,c^{2} d^{4} \ln \left (e x +d \right )}{e^{6}}-\frac {4 B \,c^{2} d^{3} x}{e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 291, normalized size = 1.50 \begin {gather*} \frac {B c^{2} d^{5} - A b^{2} d^{2} e^{3} - {\left (2 \, B b c + A c^{2}\right )} d^{4} e + {\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2}}{e^{7} x + d e^{6}} + \frac {3 \, B c^{2} e^{3} x^{4} - 4 \, {\left (2 \, B c^{2} d e^{2} - {\left (2 \, B b c + A c^{2}\right )} e^{3}\right )} x^{3} + 6 \, {\left (3 \, B c^{2} d^{2} e - 2 \, {\left (2 \, B b c + A c^{2}\right )} d e^{2} + {\left (B b^{2} + 2 \, A b c\right )} e^{3}\right )} x^{2} - 12 \, {\left (4 \, B c^{2} d^{3} - A b^{2} e^{3} - 3 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e + 2 \, {\left (B b^{2} + 2 \, A b c\right )} d e^{2}\right )} x}{12 \, e^{5}} + \frac {{\left (5 \, B c^{2} d^{4} - 2 \, A b^{2} d e^{3} - 4 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d^{2} 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 = 1.43, size = 371, normalized size = 1.91 \begin {gather*} x\,\left (\frac {A\,b^2}{e^2}-\frac {d^2\,\left (\frac {A\,c^2+2\,B\,b\,c}{e^2}-\frac {2\,B\,c^2\,d}{e^3}\right )}{e^2}+\frac {2\,d\,\left (\frac {2\,d\,\left (\frac {A\,c^2+2\,B\,b\,c}{e^2}-\frac {2\,B\,c^2\,d}{e^3}\right )}{e}-\frac {B\,b^2+2\,A\,c\,b}{e^2}+\frac {B\,c^2\,d^2}{e^4}\right )}{e}\right )+x^3\,\left (\frac {A\,c^2+2\,B\,b\,c}{3\,e^2}-\frac {2\,B\,c^2\,d}{3\,e^3}\right )-x^2\,\left (\frac {d\,\left (\frac {A\,c^2+2\,B\,b\,c}{e^2}-\frac {2\,B\,c^2\,d}{e^3}\right )}{e}-\frac {B\,b^2+2\,A\,c\,b}{2\,e^2}+\frac {B\,c^2\,d^2}{2\,e^4}\right )+\frac {B\,b^2\,d^3\,e^2-A\,b^2\,d^2\,e^3-2\,B\,b\,c\,d^4\,e+2\,A\,b\,c\,d^3\,e^2+B\,c^2\,d^5-A\,c^2\,d^4\,e}{e\,\left (x\,e^6+d\,e^5\right )}+\frac {\ln \left (d+e\,x\right )\,\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 )}{e^6}+\frac {B\,c^2\,x^4}{4\,e^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.39, size = 316, normalized size = 1.63 \begin {gather*} \frac {B c^{2} x^{4}}{4 e^{2}} + \frac {d \left (b e - c d\right ) \left (- 2 A b e^{2} + 4 A c d e + 3 B b d e - 5 B c d^{2}\right ) \log {\left (d + e x \right )}}{e^{6}} + x^{3} \left (\frac {A c^{2}}{3 e^{2}} + \frac {2 B b c}{3 e^{2}} - \frac {2 B c^{2} d}{3 e^{3}}\right ) + x^{2} \left (\frac {A b c}{e^{2}} - \frac {A c^{2} d}{e^{3}} + \frac {B b^{2}}{2 e^{2}} - \frac {2 B b c d}{e^{3}} + \frac {3 B c^{2} d^{2}}{2 e^{4}}\right ) + x \left (\frac {A b^{2}}{e^{2}} - \frac {4 A b c d}{e^{3}} + \frac {3 A c^{2} d^{2}}{e^{4}} - \frac {2 B b^{2} d}{e^{3}} + \frac {6 B b c d^{2}}{e^{4}} - \frac {4 B c^{2} d^{3}}{e^{5}}\right ) + \frac {- A b^{2} d^{2} e^{3} + 2 A b c d^{3} e^{2} - A c^{2} d^{4} e + B b^{2} d^{3} e^{2} - 2 B b c d^{4} e + B c^{2} d^{5}}{d e^{6} + e^{7} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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