Optimal. Leaf size=166 \[ -\frac {c^2 x^4 (2 c d-3 b e)}{4 e^3}-\frac {d^3 (c d-b e)^3}{e^7 (d+e x)}-\frac {3 d^2 (c d-b e)^2 (2 c d-b e) \log (d+e x)}{e^7}+\frac {d x (5 c d-2 b e) (c d-b e)^2}{e^6}-\frac {x^2 (c d-b e)^2 (4 c d-b e)}{2 e^5}+\frac {c x^3 (c d-b e)^2}{e^4}+\frac {c^3 x^5}{5 e^2} \]
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Rubi [A] time = 0.19, antiderivative size = 166, 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} \begin {gather*} -\frac {c^2 x^4 (2 c d-3 b e)}{4 e^3}-\frac {d^3 (c d-b e)^3}{e^7 (d+e x)}-\frac {3 d^2 (c d-b e)^2 (2 c d-b e) \log (d+e x)}{e^7}+\frac {c x^3 (c d-b e)^2}{e^4}-\frac {x^2 (c d-b e)^2 (4 c d-b e)}{2 e^5}+\frac {d x (5 c d-2 b e) (c d-b e)^2}{e^6}+\frac {c^3 x^5}{5 e^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin {align*} \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^2} \, dx &=\int \left (\frac {d (5 c d-2 b e) (c d-b e)^2}{e^6}+\frac {(-4 c d+b e) (-c d+b e)^2 x}{e^5}+\frac {3 c (c d-b e)^2 x^2}{e^4}-\frac {c^2 (2 c d-3 b e) x^3}{e^3}+\frac {c^3 x^4}{e^2}+\frac {d^3 (c d-b e)^3}{e^6 (d+e x)^2}-\frac {3 d^2 (c d-b e)^2 (2 c d-b e)}{e^6 (d+e x)}\right ) \, dx\\ &=\frac {d (5 c d-2 b e) (c d-b e)^2 x}{e^6}-\frac {(c d-b e)^2 (4 c d-b e) x^2}{2 e^5}+\frac {c (c d-b e)^2 x^3}{e^4}-\frac {c^2 (2 c d-3 b e) x^4}{4 e^3}+\frac {c^3 x^5}{5 e^2}-\frac {d^3 (c d-b e)^3}{e^7 (d+e x)}-\frac {3 d^2 (c d-b e)^2 (2 c d-b e) \log (d+e x)}{e^7}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 160, normalized size = 0.96 \begin {gather*} \frac {-5 c^2 e^4 x^4 (2 c d-3 b e)-\frac {20 d^3 (c d-b e)^3}{d+e x}-60 d^2 (c d-b e)^2 (2 c d-b e) \log (d+e x)+20 c e^3 x^3 (c d-b e)^2+10 e^2 x^2 (c d-b e)^2 (b e-4 c d)+20 d e x (5 c d-2 b e) (c d-b e)^2+4 c^3 e^5 x^5}{20 e^7} \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 {\left (b x+c x^2\right )^3}{(d+e x)^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.41, size = 370, normalized size = 2.23 \begin {gather*} \frac {4 \, c^{3} e^{6} x^{6} - 20 \, c^{3} d^{6} + 60 \, b c^{2} d^{5} e - 60 \, b^{2} c d^{4} e^{2} + 20 \, b^{3} d^{3} e^{3} - 3 \, {\left (2 \, c^{3} d e^{5} - 5 \, b c^{2} e^{6}\right )} x^{5} + 5 \, {\left (2 \, c^{3} d^{2} e^{4} - 5 \, b c^{2} d e^{5} + 4 \, b^{2} c e^{6}\right )} x^{4} - 10 \, {\left (2 \, c^{3} d^{3} e^{3} - 5 \, b c^{2} d^{2} e^{4} + 4 \, b^{2} c d e^{5} - b^{3} e^{6}\right )} x^{3} + 30 \, {\left (2 \, c^{3} d^{4} e^{2} - 5 \, b c^{2} d^{3} e^{3} + 4 \, b^{2} c d^{2} e^{4} - b^{3} d e^{5}\right )} x^{2} + 20 \, {\left (5 \, c^{3} d^{5} e - 12 \, b c^{2} d^{4} e^{2} + 9 \, b^{2} c d^{3} e^{3} - 2 \, b^{3} d^{2} e^{4}\right )} x - 60 \, {\left (2 \, c^{3} d^{6} - 5 \, b c^{2} d^{5} e + 4 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3} + {\left (2 \, c^{3} d^{5} e - 5 \, b c^{2} d^{4} e^{2} + 4 \, b^{2} c d^{3} e^{3} - b^{3} d^{2} e^{4}\right )} x\right )} \log \left (e x + d\right )}{20 \, {\left (e^{8} x + d e^{7}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.20, size = 332, normalized size = 2.00 \begin {gather*} \frac {1}{20} \, {\left (4 \, c^{3} - \frac {15 \, {\left (2 \, c^{3} d e - b c^{2} e^{2}\right )} e^{\left (-1\right )}}{x e + d} + \frac {20 \, {\left (5 \, c^{3} d^{2} e^{2} - 5 \, b c^{2} d e^{3} + b^{2} c e^{4}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} - \frac {10 \, {\left (20 \, c^{3} d^{3} e^{3} - 30 \, b c^{2} d^{2} e^{4} + 12 \, b^{2} c d e^{5} - b^{3} e^{6}\right )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{3}} + \frac {60 \, {\left (5 \, c^{3} d^{4} e^{4} - 10 \, b c^{2} d^{3} e^{5} + 6 \, b^{2} c d^{2} e^{6} - b^{3} d e^{7}\right )} e^{\left (-4\right )}}{{\left (x e + d\right )}^{4}}\right )} {\left (x e + d\right )}^{5} e^{\left (-7\right )} + 3 \, {\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e + 4 \, b^{2} c d^{3} e^{2} - b^{3} d^{2} e^{3}\right )} e^{\left (-7\right )} \log \left (\frac {{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) - {\left (\frac {c^{3} d^{6} e^{5}}{x e + d} - \frac {3 \, b c^{2} d^{5} e^{6}}{x e + d} + \frac {3 \, b^{2} c d^{4} e^{7}}{x e + d} - \frac {b^{3} d^{3} e^{8}}{x e + d}\right )} e^{\left (-12\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 318, normalized size = 1.92 \begin {gather*} \frac {c^{3} x^{5}}{5 e^{2}}+\frac {3 b \,c^{2} x^{4}}{4 e^{2}}-\frac {c^{3} d \,x^{4}}{2 e^{3}}+\frac {b^{2} c \,x^{3}}{e^{2}}-\frac {2 b \,c^{2} d \,x^{3}}{e^{3}}+\frac {c^{3} d^{2} x^{3}}{e^{4}}+\frac {b^{3} x^{2}}{2 e^{2}}-\frac {3 b^{2} c d \,x^{2}}{e^{3}}+\frac {9 b \,c^{2} d^{2} x^{2}}{2 e^{4}}-\frac {2 c^{3} d^{3} x^{2}}{e^{5}}+\frac {b^{3} d^{3}}{\left (e x +d \right ) e^{4}}+\frac {3 b^{3} d^{2} \ln \left (e x +d \right )}{e^{4}}-\frac {2 b^{3} d x}{e^{3}}-\frac {3 b^{2} c \,d^{4}}{\left (e x +d \right ) e^{5}}-\frac {12 b^{2} c \,d^{3} \ln \left (e x +d \right )}{e^{5}}+\frac {9 b^{2} c \,d^{2} x}{e^{4}}+\frac {3 b \,c^{2} d^{5}}{\left (e x +d \right ) e^{6}}+\frac {15 b \,c^{2} d^{4} \ln \left (e x +d \right )}{e^{6}}-\frac {12 b \,c^{2} d^{3} x}{e^{5}}-\frac {c^{3} d^{6}}{\left (e x +d \right ) e^{7}}-\frac {6 c^{3} d^{5} \ln \left (e x +d \right )}{e^{7}}+\frac {5 c^{3} d^{4} x}{e^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.39, size = 273, normalized size = 1.64 \begin {gather*} -\frac {c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}}{e^{8} x + d e^{7}} + \frac {4 \, c^{3} e^{4} x^{5} - 5 \, {\left (2 \, c^{3} d e^{3} - 3 \, b c^{2} e^{4}\right )} x^{4} + 20 \, {\left (c^{3} d^{2} e^{2} - 2 \, b c^{2} d e^{3} + b^{2} c e^{4}\right )} x^{3} - 10 \, {\left (4 \, c^{3} d^{3} e - 9 \, b c^{2} d^{2} e^{2} + 6 \, b^{2} c d e^{3} - b^{3} e^{4}\right )} x^{2} + 20 \, {\left (5 \, c^{3} d^{4} - 12 \, b c^{2} d^{3} e + 9 \, b^{2} c d^{2} e^{2} - 2 \, b^{3} d e^{3}\right )} x}{20 \, e^{6}} - \frac {3 \, {\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e + 4 \, b^{2} c d^{3} e^{2} - b^{3} d^{2} e^{3}\right )} \log \left (e x + d\right )}{e^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.21, size = 435, normalized size = 2.62 \begin {gather*} x^4\,\left (\frac {3\,b\,c^2}{4\,e^2}-\frac {c^3\,d}{2\,e^3}\right )-x^3\,\left (\frac {2\,d\,\left (\frac {3\,b\,c^2}{e^2}-\frac {2\,c^3\,d}{e^3}\right )}{3\,e}-\frac {b^2\,c}{e^2}+\frac {c^3\,d^2}{3\,e^4}\right )+x^2\,\left (\frac {b^3}{2\,e^2}+\frac {d\,\left (\frac {2\,d\,\left (\frac {3\,b\,c^2}{e^2}-\frac {2\,c^3\,d}{e^3}\right )}{e}-\frac {3\,b^2\,c}{e^2}+\frac {c^3\,d^2}{e^4}\right )}{e}-\frac {d^2\,\left (\frac {3\,b\,c^2}{e^2}-\frac {2\,c^3\,d}{e^3}\right )}{2\,e^2}\right )+x\,\left (\frac {d^2\,\left (\frac {2\,d\,\left (\frac {3\,b\,c^2}{e^2}-\frac {2\,c^3\,d}{e^3}\right )}{e}-\frac {3\,b^2\,c}{e^2}+\frac {c^3\,d^2}{e^4}\right )}{e^2}-\frac {2\,d\,\left (\frac {b^3}{e^2}+\frac {2\,d\,\left (\frac {2\,d\,\left (\frac {3\,b\,c^2}{e^2}-\frac {2\,c^3\,d}{e^3}\right )}{e}-\frac {3\,b^2\,c}{e^2}+\frac {c^3\,d^2}{e^4}\right )}{e}-\frac {d^2\,\left (\frac {3\,b\,c^2}{e^2}-\frac {2\,c^3\,d}{e^3}\right )}{e^2}\right )}{e}\right )-\frac {\ln \left (d+e\,x\right )\,\left (-3\,b^3\,d^2\,e^3+12\,b^2\,c\,d^3\,e^2-15\,b\,c^2\,d^4\,e+6\,c^3\,d^5\right )}{e^7}-\frac {-b^3\,d^3\,e^3+3\,b^2\,c\,d^4\,e^2-3\,b\,c^2\,d^5\,e+c^3\,d^6}{e\,\left (x\,e^7+d\,e^6\right )}+\frac {c^3\,x^5}{5\,e^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.92, size = 257, normalized size = 1.55 \begin {gather*} \frac {c^{3} x^{5}}{5 e^{2}} + \frac {3 d^{2} \left (b e - 2 c d\right ) \left (b e - c d\right )^{2} \log {\left (d + e x \right )}}{e^{7}} + x^{4} \left (\frac {3 b c^{2}}{4 e^{2}} - \frac {c^{3} d}{2 e^{3}}\right ) + x^{3} \left (\frac {b^{2} c}{e^{2}} - \frac {2 b c^{2} d}{e^{3}} + \frac {c^{3} d^{2}}{e^{4}}\right ) + x^{2} \left (\frac {b^{3}}{2 e^{2}} - \frac {3 b^{2} c d}{e^{3}} + \frac {9 b c^{2} d^{2}}{2 e^{4}} - \frac {2 c^{3} d^{3}}{e^{5}}\right ) + x \left (- \frac {2 b^{3} d}{e^{3}} + \frac {9 b^{2} c d^{2}}{e^{4}} - \frac {12 b c^{2} d^{3}}{e^{5}} + \frac {5 c^{3} d^{4}}{e^{6}}\right ) + \frac {b^{3} d^{3} e^{3} - 3 b^{2} c d^{4} e^{2} + 3 b c^{2} d^{5} e - c^{3} d^{6}}{d e^{7} + e^{8} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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