Optimal. Leaf size=107 \[ -\frac {d^2 (c d-b e)^2}{e^5 (d+e x)}-\frac {2 d (c d-b e) (2 c d-b e) \log (d+e x)}{e^5}+\frac {x (c d-b e) (3 c d-b e)}{e^4}-\frac {c x^2 (c d-b e)}{e^3}+\frac {c^2 x^3}{3 e^2} \]
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Rubi [A] time = 0.10, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {698} \[ -\frac {d^2 (c d-b e)^2}{e^5 (d+e x)}-\frac {c x^2 (c d-b e)}{e^3}+\frac {x (c d-b e) (3 c d-b e)}{e^4}-\frac {2 d (c d-b e) (2 c d-b e) \log (d+e x)}{e^5}+\frac {c^2 x^3}{3 e^2} \]
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin {align*} \int \frac {\left (b x+c x^2\right )^2}{(d+e x)^2} \, dx &=\int \left (\frac {(c d-b e) (3 c d-b e)}{e^4}-\frac {2 c (c d-b e) x}{e^3}+\frac {c^2 x^2}{e^2}+\frac {d^2 (c d-b e)^2}{e^4 (d+e x)^2}+\frac {2 d (c d-b e) (-2 c d+b e)}{e^4 (d+e x)}\right ) \, dx\\ &=\frac {(c d-b e) (3 c d-b e) x}{e^4}-\frac {c (c d-b e) x^2}{e^3}+\frac {c^2 x^3}{3 e^2}-\frac {d^2 (c d-b e)^2}{e^5 (d+e x)}-\frac {2 d (c d-b e) (2 c d-b e) \log (d+e x)}{e^5}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 114, normalized size = 1.07 \[ \frac {3 e x \left (b^2 e^2-4 b c d e+3 c^2 d^2\right )-6 d \left (b^2 e^2-3 b c d e+2 c^2 d^2\right ) \log (d+e x)-\frac {3 d^2 (c d-b e)^2}{d+e x}-3 c e^2 x^2 (c d-b e)+c^2 e^3 x^3}{3 e^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 203, normalized size = 1.90 \[ \frac {c^{2} e^{4} x^{4} - 3 \, c^{2} d^{4} + 6 \, b c d^{3} e - 3 \, b^{2} d^{2} e^{2} - {\left (2 \, c^{2} d e^{3} - 3 \, b c e^{4}\right )} x^{3} + 3 \, {\left (2 \, c^{2} d^{2} e^{2} - 3 \, b c d e^{3} + b^{2} e^{4}\right )} x^{2} + 3 \, {\left (3 \, c^{2} d^{3} e - 4 \, b c d^{2} e^{2} + b^{2} d e^{3}\right )} x - 6 \, {\left (2 \, c^{2} d^{4} - 3 \, b c d^{3} e + b^{2} d^{2} e^{2} + {\left (2 \, c^{2} d^{3} e - 3 \, b c d^{2} e^{2} + b^{2} d e^{3}\right )} x\right )} \log \left (e x + d\right )}{3 \, {\left (e^{6} x + d e^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 184, normalized size = 1.72 \[ \frac {1}{3} \, {\left (c^{2} - \frac {3 \, {\left (2 \, c^{2} d e - b c e^{2}\right )} e^{\left (-1\right )}}{x e + d} + \frac {3 \, {\left (6 \, c^{2} d^{2} e^{2} - 6 \, b c d e^{3} + b^{2} e^{4}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}}\right )} {\left (x e + d\right )}^{3} e^{\left (-5\right )} + 2 \, {\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2}\right )} e^{\left (-5\right )} \log \left (\frac {{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) - {\left (\frac {c^{2} d^{4} e^{3}}{x e + d} - \frac {2 \, b c d^{3} e^{4}}{x e + d} + \frac {b^{2} d^{2} e^{5}}{x e + d}\right )} e^{\left (-8\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 164, normalized size = 1.53 \[ \frac {c^{2} x^{3}}{3 e^{2}}+\frac {b c \,x^{2}}{e^{2}}-\frac {c^{2} d \,x^{2}}{e^{3}}-\frac {b^{2} d^{2}}{\left (e x +d \right ) e^{3}}-\frac {2 b^{2} d \ln \left (e x +d \right )}{e^{3}}+\frac {b^{2} x}{e^{2}}+\frac {2 b c \,d^{3}}{\left (e x +d \right ) e^{4}}+\frac {6 b c \,d^{2} \ln \left (e x +d \right )}{e^{4}}-\frac {4 b c d x}{e^{3}}-\frac {c^{2} d^{4}}{\left (e x +d \right ) e^{5}}-\frac {4 c^{2} d^{3} \ln \left (e x +d \right )}{e^{5}}+\frac {3 c^{2} d^{2} x}{e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.37, size = 138, normalized size = 1.29 \[ -\frac {c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}}{e^{6} x + d e^{5}} + \frac {c^{2} e^{2} x^{3} - 3 \, {\left (c^{2} d e - b c e^{2}\right )} x^{2} + 3 \, {\left (3 \, c^{2} d^{2} - 4 \, b c d e + b^{2} e^{2}\right )} x}{3 \, e^{4}} - \frac {2 \, {\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2}\right )} \log \left (e x + d\right )}{e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.18, size = 158, normalized size = 1.48 \[ x\,\left (\frac {b^2}{e^2}+\frac {2\,d\,\left (\frac {2\,c^2\,d}{e^3}-\frac {2\,b\,c}{e^2}\right )}{e}-\frac {c^2\,d^2}{e^4}\right )-x^2\,\left (\frac {c^2\,d}{e^3}-\frac {b\,c}{e^2}\right )-\frac {b^2\,d^2\,e^2-2\,b\,c\,d^3\,e+c^2\,d^4}{e\,\left (x\,e^5+d\,e^4\right )}-\frac {\ln \left (d+e\,x\right )\,\left (2\,b^2\,d\,e^2-6\,b\,c\,d^2\,e+4\,c^2\,d^3\right )}{e^5}+\frac {c^2\,x^3}{3\,e^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.58, size = 126, normalized size = 1.18 \[ \frac {c^{2} x^{3}}{3 e^{2}} - \frac {2 d \left (b e - 2 c d\right ) \left (b e - c d\right ) \log {\left (d + e x \right )}}{e^{5}} + x^{2} \left (\frac {b c}{e^{2}} - \frac {c^{2} d}{e^{3}}\right ) + x \left (\frac {b^{2}}{e^{2}} - \frac {4 b c d}{e^{3}} + \frac {3 c^{2} d^{2}}{e^{4}}\right ) + \frac {- b^{2} d^{2} e^{2} + 2 b c d^{3} e - c^{2} d^{4}}{d e^{5} + e^{6} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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