Optimal. Leaf size=107 \[ \frac {(c d-b e) (3 c d-b e) x}{e^4}-\frac {c (c d-b e) x^2}{e^3}+\frac {c^2 x^3}{3 e^2}-\frac {d^2 (c d-b e)^2}{e^5 (d+e x)}-\frac {2 d (c d-b e) (2 c d-b e) \log (d+e x)}{e^5} \]
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Rubi [A]
time = 0.07, 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 = {712}
\begin {gather*} -\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} \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)^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 = 0.07, size = 114, normalized size = 1.07 \begin {gather*} \frac {3 e \left (3 c^2 d^2-4 b c d e+b^2 e^2\right ) x-3 c e^2 (c d-b e) x^2+c^2 e^3 x^3-\frac {3 d^2 (c d-b e)^2}{d+e x}-6 d \left (2 c^2 d^2-3 b c d e+b^2 e^2\right ) \log (d+e x)}{3 e^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.45, size = 131, normalized size = 1.22
method | result | size |
default | \(\frac {\frac {1}{3} c^{2} e^{2} x^{3}+b c \,e^{2} x^{2}-c^{2} d e \,x^{2}+b^{2} e^{2} x -4 b c d e x +3 d^{2} c^{2} x}{e^{4}}-\frac {d^{2} \left (b^{2} e^{2}-2 b c d e +d^{2} c^{2}\right )}{e^{5} \left (e x +d \right )}-\frac {2 d \left (b^{2} e^{2}-3 b c d e +2 d^{2} c^{2}\right ) \ln \left (e x +d \right )}{e^{5}}\) | \(131\) |
norman | \(\frac {\frac {\left (b^{2} e^{2}-3 b c d e +2 d^{2} c^{2}\right ) x^{2}}{e^{3}}+\frac {\left (2 d^{2} e^{2} b^{2}-6 d^{3} e b c +4 c^{2} d^{4}\right ) x}{d \,e^{4}}+\frac {c^{2} x^{4}}{3 e}+\frac {c \left (3 b e -2 c d \right ) x^{3}}{3 e^{2}}}{e x +d}-\frac {2 d \left (b^{2} e^{2}-3 b c d e +2 d^{2} c^{2}\right ) \ln \left (e x +d \right )}{e^{5}}\) | \(139\) |
risch | \(\frac {c^{2} x^{3}}{3 e^{2}}+\frac {b c \,x^{2}}{e^{2}}-\frac {c^{2} d \,x^{2}}{e^{3}}+\frac {b^{2} x}{e^{2}}-\frac {4 b c d x}{e^{3}}+\frac {3 d^{2} c^{2} x}{e^{4}}-\frac {d^{2} b^{2}}{e^{3} \left (e x +d \right )}+\frac {2 d^{3} b c}{e^{4} \left (e x +d \right )}-\frac {d^{4} c^{2}}{e^{5} \left (e x +d \right )}-\frac {2 d \ln \left (e x +d \right ) b^{2}}{e^{3}}+\frac {6 d^{2} \ln \left (e x +d \right ) b c}{e^{4}}-\frac {4 d^{3} \ln \left (e x +d \right ) c^{2}}{e^{5}}\) | \(164\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 134, normalized size = 1.25 \begin {gather*} -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 (x e + d\right ) + \frac {1}{3} \, {\left (c^{2} x^{3} e^{2} - 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\right )} e^{\left (-4\right )} - \frac {c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}}{x e^{6} + d e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.05, size = 197, normalized size = 1.84 \begin {gather*} -\frac {3 \, c^{2} d^{4} - {\left (c^{2} x^{4} + 3 \, b c x^{3} + 3 \, b^{2} x^{2}\right )} e^{4} + {\left (2 \, c^{2} d x^{3} + 9 \, b c d x^{2} - 3 \, b^{2} d x\right )} e^{3} - 3 \, {\left (2 \, c^{2} d^{2} x^{2} - 4 \, b c d^{2} x - b^{2} d^{2}\right )} e^{2} - 3 \, {\left (3 \, c^{2} d^{3} x + 2 \, b c d^{3}\right )} e + 6 \, {\left (2 \, c^{2} d^{4} + b^{2} d x e^{3} - {\left (3 \, b c d^{2} x - b^{2} d^{2}\right )} e^{2} + {\left (2 \, c^{2} d^{3} x - 3 \, b c d^{3}\right )} e\right )} \log \left (x e + d\right )}{3 \, {\left (x e^{6} + d e^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.27, size = 126, normalized size = 1.18 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.89, size = 184, normalized size = 1.72 \begin {gather*} \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 )} \end {gather*}
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 \begin {gather*} 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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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