Optimal. Leaf size=146 \[ \frac {(b d-a e)^6 \log (d+e x)}{e^7}-\frac {b x (b d-a e)^5}{e^6}+\frac {(a+b x)^2 (b d-a e)^4}{2 e^5}-\frac {(a+b x)^3 (b d-a e)^3}{3 e^4}+\frac {(a+b x)^4 (b d-a e)^2}{4 e^3}-\frac {(a+b x)^5 (b d-a e)}{5 e^2}+\frac {(a+b x)^6}{6 e} \]
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Rubi [A] time = 0.06, antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {27, 43} \[ -\frac {b x (b d-a e)^5}{e^6}+\frac {(a+b x)^2 (b d-a e)^4}{2 e^5}-\frac {(a+b x)^3 (b d-a e)^3}{3 e^4}+\frac {(a+b x)^4 (b d-a e)^2}{4 e^3}-\frac {(a+b x)^5 (b d-a e)}{5 e^2}+\frac {(b d-a e)^6 \log (d+e x)}{e^7}+\frac {(a+b x)^6}{6 e} \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin {align*} \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{d+e x} \, dx &=\int \frac {(a+b x)^6}{d+e x} \, dx\\ &=\int \left (-\frac {b (b d-a e)^5}{e^6}+\frac {b (b d-a e)^4 (a+b x)}{e^5}-\frac {b (b d-a e)^3 (a+b x)^2}{e^4}+\frac {b (b d-a e)^2 (a+b x)^3}{e^3}-\frac {b (b d-a e) (a+b x)^4}{e^2}+\frac {b (a+b x)^5}{e}+\frac {(-b d+a e)^6}{e^6 (d+e x)}\right ) \, dx\\ &=-\frac {b (b d-a e)^5 x}{e^6}+\frac {(b d-a e)^4 (a+b x)^2}{2 e^5}-\frac {(b d-a e)^3 (a+b x)^3}{3 e^4}+\frac {(b d-a e)^2 (a+b x)^4}{4 e^3}-\frac {(b d-a e) (a+b x)^5}{5 e^2}+\frac {(a+b x)^6}{6 e}+\frac {(b d-a e)^6 \log (d+e x)}{e^7}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 230, normalized size = 1.58 \[ \frac {b e x \left (360 a^5 e^5+450 a^4 b e^4 (e x-2 d)+200 a^3 b^2 e^3 \left (6 d^2-3 d e x+2 e^2 x^2\right )+75 a^2 b^3 e^2 \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )+6 a b^4 e \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )+b^5 \left (-60 d^5+30 d^4 e x-20 d^3 e^2 x^2+15 d^2 e^3 x^3-12 d e^4 x^4+10 e^5 x^5\right )\right )+60 (b d-a e)^6 \log (d+e x)}{60 e^7} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.88, size = 351, normalized size = 2.40 \[ \frac {10 \, b^{6} e^{6} x^{6} - 12 \, {\left (b^{6} d e^{5} - 6 \, a b^{5} e^{6}\right )} x^{5} + 15 \, {\left (b^{6} d^{2} e^{4} - 6 \, a b^{5} d e^{5} + 15 \, a^{2} b^{4} e^{6}\right )} x^{4} - 20 \, {\left (b^{6} d^{3} e^{3} - 6 \, a b^{5} d^{2} e^{4} + 15 \, a^{2} b^{4} d e^{5} - 20 \, a^{3} b^{3} e^{6}\right )} x^{3} + 30 \, {\left (b^{6} d^{4} e^{2} - 6 \, a b^{5} d^{3} e^{3} + 15 \, a^{2} b^{4} d^{2} e^{4} - 20 \, a^{3} b^{3} d e^{5} + 15 \, a^{4} b^{2} e^{6}\right )} x^{2} - 60 \, {\left (b^{6} d^{5} e - 6 \, a b^{5} d^{4} e^{2} + 15 \, a^{2} b^{4} d^{3} e^{3} - 20 \, a^{3} b^{3} d^{2} e^{4} + 15 \, a^{4} b^{2} d e^{5} - 6 \, a^{5} b e^{6}\right )} x + 60 \, {\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} \log \left (e x + d\right )}{60 \, e^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 354, normalized size = 2.42 \[ {\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{60} \, {\left (10 \, b^{6} x^{6} e^{5} - 12 \, b^{6} d x^{5} e^{4} + 15 \, b^{6} d^{2} x^{4} e^{3} - 20 \, b^{6} d^{3} x^{3} e^{2} + 30 \, b^{6} d^{4} x^{2} e - 60 \, b^{6} d^{5} x + 72 \, a b^{5} x^{5} e^{5} - 90 \, a b^{5} d x^{4} e^{4} + 120 \, a b^{5} d^{2} x^{3} e^{3} - 180 \, a b^{5} d^{3} x^{2} e^{2} + 360 \, a b^{5} d^{4} x e + 225 \, a^{2} b^{4} x^{4} e^{5} - 300 \, a^{2} b^{4} d x^{3} e^{4} + 450 \, a^{2} b^{4} d^{2} x^{2} e^{3} - 900 \, a^{2} b^{4} d^{3} x e^{2} + 400 \, a^{3} b^{3} x^{3} e^{5} - 600 \, a^{3} b^{3} d x^{2} e^{4} + 1200 \, a^{3} b^{3} d^{2} x e^{3} + 450 \, a^{4} b^{2} x^{2} e^{5} - 900 \, a^{4} b^{2} d x e^{4} + 360 \, a^{5} b x e^{5}\right )} e^{\left (-6\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 412, normalized size = 2.82 \[ \frac {b^{6} x^{6}}{6 e}+\frac {6 a \,b^{5} x^{5}}{5 e}-\frac {b^{6} d \,x^{5}}{5 e^{2}}+\frac {15 a^{2} b^{4} x^{4}}{4 e}-\frac {3 a \,b^{5} d \,x^{4}}{2 e^{2}}+\frac {b^{6} d^{2} x^{4}}{4 e^{3}}+\frac {20 a^{3} b^{3} x^{3}}{3 e}-\frac {5 a^{2} b^{4} d \,x^{3}}{e^{2}}+\frac {2 a \,b^{5} d^{2} x^{3}}{e^{3}}-\frac {b^{6} d^{3} x^{3}}{3 e^{4}}+\frac {15 a^{4} b^{2} x^{2}}{2 e}-\frac {10 a^{3} b^{3} d \,x^{2}}{e^{2}}+\frac {15 a^{2} b^{4} d^{2} x^{2}}{2 e^{3}}-\frac {3 a \,b^{5} d^{3} x^{2}}{e^{4}}+\frac {b^{6} d^{4} x^{2}}{2 e^{5}}+\frac {a^{6} \ln \left (e x +d \right )}{e}-\frac {6 a^{5} b d \ln \left (e x +d \right )}{e^{2}}+\frac {6 a^{5} b x}{e}+\frac {15 a^{4} b^{2} d^{2} \ln \left (e x +d \right )}{e^{3}}-\frac {15 a^{4} b^{2} d x}{e^{2}}-\frac {20 a^{3} b^{3} d^{3} \ln \left (e x +d \right )}{e^{4}}+\frac {20 a^{3} b^{3} d^{2} x}{e^{3}}+\frac {15 a^{2} b^{4} d^{4} \ln \left (e x +d \right )}{e^{5}}-\frac {15 a^{2} b^{4} d^{3} x}{e^{4}}-\frac {6 a \,b^{5} d^{5} \ln \left (e x +d \right )}{e^{6}}+\frac {6 a \,b^{5} d^{4} x}{e^{5}}+\frac {b^{6} d^{6} \ln \left (e x +d \right )}{e^{7}}-\frac {b^{6} d^{5} x}{e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.46, size = 349, normalized size = 2.39 \[ \frac {10 \, b^{6} e^{5} x^{6} - 12 \, {\left (b^{6} d e^{4} - 6 \, a b^{5} e^{5}\right )} x^{5} + 15 \, {\left (b^{6} d^{2} e^{3} - 6 \, a b^{5} d e^{4} + 15 \, a^{2} b^{4} e^{5}\right )} x^{4} - 20 \, {\left (b^{6} d^{3} e^{2} - 6 \, a b^{5} d^{2} e^{3} + 15 \, a^{2} b^{4} d e^{4} - 20 \, a^{3} b^{3} e^{5}\right )} x^{3} + 30 \, {\left (b^{6} d^{4} e - 6 \, a b^{5} d^{3} e^{2} + 15 \, a^{2} b^{4} d^{2} e^{3} - 20 \, a^{3} b^{3} d e^{4} + 15 \, a^{4} b^{2} e^{5}\right )} x^{2} - 60 \, {\left (b^{6} d^{5} - 6 \, a b^{5} d^{4} e + 15 \, a^{2} b^{4} d^{3} e^{2} - 20 \, a^{3} b^{3} d^{2} e^{3} + 15 \, a^{4} b^{2} d e^{4} - 6 \, a^{5} b e^{5}\right )} x}{60 \, e^{6}} + \frac {{\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} \log \left (e x + d\right )}{e^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.52, size = 385, normalized size = 2.64 \[ x^5\,\left (\frac {6\,a\,b^5}{5\,e}-\frac {b^6\,d}{5\,e^2}\right )+x^3\,\left (\frac {d\,\left (\frac {d\,\left (\frac {6\,a\,b^5}{e}-\frac {b^6\,d}{e^2}\right )}{e}-\frac {15\,a^2\,b^4}{e}\right )}{3\,e}+\frac {20\,a^3\,b^3}{3\,e}\right )+x\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {6\,a\,b^5}{e}-\frac {b^6\,d}{e^2}\right )}{e}-\frac {15\,a^2\,b^4}{e}\right )}{e}+\frac {20\,a^3\,b^3}{e}\right )}{e}-\frac {15\,a^4\,b^2}{e}\right )}{e}+\frac {6\,a^5\,b}{e}\right )-x^4\,\left (\frac {d\,\left (\frac {6\,a\,b^5}{e}-\frac {b^6\,d}{e^2}\right )}{4\,e}-\frac {15\,a^2\,b^4}{4\,e}\right )-x^2\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {6\,a\,b^5}{e}-\frac {b^6\,d}{e^2}\right )}{e}-\frac {15\,a^2\,b^4}{e}\right )}{e}+\frac {20\,a^3\,b^3}{e}\right )}{2\,e}-\frac {15\,a^4\,b^2}{2\,e}\right )+\frac {\ln \left (d+e\,x\right )\,\left (a^6\,e^6-6\,a^5\,b\,d\,e^5+15\,a^4\,b^2\,d^2\,e^4-20\,a^3\,b^3\,d^3\,e^3+15\,a^2\,b^4\,d^4\,e^2-6\,a\,b^5\,d^5\,e+b^6\,d^6\right )}{e^7}+\frac {b^6\,x^6}{6\,e} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.65, size = 296, normalized size = 2.03 \[ \frac {b^{6} x^{6}}{6 e} + x^{5} \left (\frac {6 a b^{5}}{5 e} - \frac {b^{6} d}{5 e^{2}}\right ) + x^{4} \left (\frac {15 a^{2} b^{4}}{4 e} - \frac {3 a b^{5} d}{2 e^{2}} + \frac {b^{6} d^{2}}{4 e^{3}}\right ) + x^{3} \left (\frac {20 a^{3} b^{3}}{3 e} - \frac {5 a^{2} b^{4} d}{e^{2}} + \frac {2 a b^{5} d^{2}}{e^{3}} - \frac {b^{6} d^{3}}{3 e^{4}}\right ) + x^{2} \left (\frac {15 a^{4} b^{2}}{2 e} - \frac {10 a^{3} b^{3} d}{e^{2}} + \frac {15 a^{2} b^{4} d^{2}}{2 e^{3}} - \frac {3 a b^{5} d^{3}}{e^{4}} + \frac {b^{6} d^{4}}{2 e^{5}}\right ) + x \left (\frac {6 a^{5} b}{e} - \frac {15 a^{4} b^{2} d}{e^{2}} + \frac {20 a^{3} b^{3} d^{2}}{e^{3}} - \frac {15 a^{2} b^{4} d^{3}}{e^{4}} + \frac {6 a b^{5} d^{4}}{e^{5}} - \frac {b^{6} d^{5}}{e^{6}}\right ) + \frac {\left (a e - b d\right )^{6} \log {\left (d + e x \right )}}{e^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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