Optimal. Leaf size=93 \[ -\frac {d (c d-b e)^2 x}{e^4}+\frac {(c d-b e)^2 x^2}{2 e^3}-\frac {c (c d-2 b e) x^3}{3 e^2}+\frac {c^2 x^4}{4 e}+\frac {d^2 (c d-b e)^2 \log (d+e x)}{e^5} \]
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Rubi [A]
time = 0.06, antiderivative size = 93, 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 = {712}
\begin {gather*} \frac {d^2 (c d-b e)^2 \log (d+e x)}{e^5}-\frac {d x (c d-b e)^2}{e^4}+\frac {x^2 (c d-b e)^2}{2 e^3}-\frac {c x^3 (c d-2 b e)}{3 e^2}+\frac {c^2 x^4}{4 e} \end {gather*}
Antiderivative was successfully verified.
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Rule 712
Rubi steps
\begin {align*} \int \frac {\left (b x+c x^2\right )^2}{d+e x} \, dx &=\int \left (-\frac {d (c d-b e)^2}{e^4}+\frac {(-c d+b e)^2 x}{e^3}-\frac {c (c d-2 b e) x^2}{e^2}+\frac {c^2 x^3}{e}+\frac {d^2 (c d-b e)^2}{e^4 (d+e x)}\right ) \, dx\\ &=-\frac {d (c d-b e)^2 x}{e^4}+\frac {(c d-b e)^2 x^2}{2 e^3}-\frac {c (c d-2 b e) x^3}{3 e^2}+\frac {c^2 x^4}{4 e}+\frac {d^2 (c d-b e)^2 \log (d+e x)}{e^5}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 106, normalized size = 1.14 \begin {gather*} -\frac {d (c d-b e)^2 x}{e^4}+\frac {(-c d+b e)^2 x^2}{2 e^3}-\frac {c (c d-2 b e) x^3}{3 e^2}+\frac {c^2 x^4}{4 e}+\frac {\left (c^2 d^4-2 b c d^3 e+b^2 d^2 e^2\right ) \log (d+e x)}{e^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.50, size = 135, normalized size = 1.45
method | result | size |
norman | \(\frac {c^{2} x^{4}}{4 e}+\frac {\left (b^{2} e^{2}-2 b c d e +d^{2} c^{2}\right ) x^{2}}{2 e^{3}}-\frac {d \left (b^{2} e^{2}-2 b c d e +d^{2} c^{2}\right ) x}{e^{4}}+\frac {c \left (2 b e -c d \right ) x^{3}}{3 e^{2}}+\frac {d^{2} \left (b^{2} e^{2}-2 b c d e +d^{2} c^{2}\right ) \ln \left (e x +d \right )}{e^{5}}\) | \(122\) |
default | \(-\frac {-\frac {c^{2} x^{4} e^{3}}{4}+\frac {\left (-\left (b e -c d \right ) e^{2} c -e^{3} b c \right ) x^{3}}{3}+\frac {\left (-\left (b e -c d \right ) b \,e^{2}+c e \left (b d e -c \,d^{2}\right )\right ) x^{2}}{2}+\left (b e -c d \right ) \left (b d e -c \,d^{2}\right ) x}{e^{4}}+\frac {d^{2} \left (b^{2} e^{2}-2 b c d e +d^{2} c^{2}\right ) \ln \left (e x +d \right )}{e^{5}}\) | \(135\) |
risch | \(\frac {c^{2} x^{4}}{4 e}+\frac {2 x^{3} b c}{3 e}-\frac {c^{2} d \,x^{3}}{3 e^{2}}+\frac {x^{2} b^{2}}{2 e}-\frac {x^{2} d b c}{e^{2}}+\frac {x^{2} d^{2} c^{2}}{2 e^{3}}-\frac {b^{2} d x}{e^{2}}+\frac {2 b c \,d^{2} x}{e^{3}}-\frac {c^{2} d^{3} x}{e^{4}}+\frac {d^{2} \ln \left (e x +d \right ) b^{2}}{e^{3}}-\frac {2 d^{3} \ln \left (e x +d \right ) b c}{e^{4}}+\frac {d^{4} \ln \left (e x +d \right ) c^{2}}{e^{5}}\) | \(152\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 126, normalized size = 1.35 \begin {gather*} {\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} e^{\left (-5\right )} \log \left (x e + d\right ) + \frac {1}{12} \, {\left (3 \, c^{2} x^{4} e^{3} - 4 \, {\left (c^{2} d e^{2} - 2 \, b c e^{3}\right )} x^{3} + 6 \, {\left (c^{2} d^{2} e - 2 \, b c d e^{2} + b^{2} e^{3}\right )} x^{2} - 12 \, {\left (c^{2} d^{3} - 2 \, b c d^{2} e + b^{2} d e^{2}\right )} x\right )} e^{\left (-4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.66, size = 130, normalized size = 1.40 \begin {gather*} -\frac {1}{12} \, {\left (12 \, c^{2} d^{3} x e - {\left (3 \, c^{2} x^{4} + 8 \, b c x^{3} + 6 \, b^{2} x^{2}\right )} e^{4} + 4 \, {\left (c^{2} d x^{3} + 3 \, b c d x^{2} + 3 \, b^{2} d x\right )} e^{3} - 6 \, {\left (c^{2} d^{2} x^{2} + 4 \, b c d^{2} x\right )} e^{2} - 12 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} \log \left (x e + d\right )\right )} e^{\left (-5\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.14, size = 116, normalized size = 1.25 \begin {gather*} \frac {c^{2} x^{4}}{4 e} + \frac {d^{2} \left (b e - c d\right )^{2} \log {\left (d + e x \right )}}{e^{5}} + x^{3} \cdot \left (\frac {2 b c}{3 e} - \frac {c^{2} d}{3 e^{2}}\right ) + x^{2} \left (\frac {b^{2}}{2 e} - \frac {b c d}{e^{2}} + \frac {c^{2} d^{2}}{2 e^{3}}\right ) + x \left (- \frac {b^{2} d}{e^{2}} + \frac {2 b c d^{2}}{e^{3}} - \frac {c^{2} d^{3}}{e^{4}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.18, size = 134, normalized size = 1.44 \begin {gather*} {\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} e^{\left (-5\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{12} \, {\left (3 \, c^{2} x^{4} e^{3} - 4 \, c^{2} d x^{3} e^{2} + 6 \, c^{2} d^{2} x^{2} e - 12 \, c^{2} d^{3} x + 8 \, b c x^{3} e^{3} - 12 \, b c d x^{2} e^{2} + 24 \, b c d^{2} x e + 6 \, b^{2} x^{2} e^{3} - 12 \, b^{2} d x e^{2}\right )} e^{\left (-4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 141, normalized size = 1.52 \begin {gather*} x^2\,\left (\frac {b^2}{2\,e}+\frac {d\,\left (\frac {c^2\,d}{e^2}-\frac {2\,b\,c}{e}\right )}{2\,e}\right )-x^3\,\left (\frac {c^2\,d}{3\,e^2}-\frac {2\,b\,c}{3\,e}\right )+\frac {c^2\,x^4}{4\,e}+\frac {\ln \left (d+e\,x\right )\,\left (b^2\,d^2\,e^2-2\,b\,c\,d^3\,e+c^2\,d^4\right )}{e^5}-\frac {d\,x\,\left (\frac {b^2}{e}+\frac {d\,\left (\frac {c^2\,d}{e^2}-\frac {2\,b\,c}{e}\right )}{e}\right )}{e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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