Optimal. Leaf size=228 \[ -\frac {d^3 (c d-b e)^3}{6 e^7 (d+e x)^6}+\frac {3 d^2 (c d-b e)^2 (2 c d-b e)}{5 e^7 (d+e x)^5}-\frac {3 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )}{4 e^7 (d+e x)^4}+\frac {(2 c d-b e) \left (10 c^2 d^2-10 b c d e+b^2 e^2\right )}{3 e^7 (d+e x)^3}-\frac {3 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )}{2 e^7 (d+e x)^2}+\frac {3 c^2 (2 c d-b e)}{e^7 (d+e x)}+\frac {c^3 \log (d+e x)}{e^7} \]
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Rubi [A]
time = 0.12, antiderivative size = 228, 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 {3 c \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{2 e^7 (d+e x)^2}+\frac {(2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{3 e^7 (d+e x)^3}-\frac {3 d (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{4 e^7 (d+e x)^4}+\frac {3 c^2 (2 c d-b e)}{e^7 (d+e x)}-\frac {d^3 (c d-b e)^3}{6 e^7 (d+e x)^6}+\frac {3 d^2 (c d-b e)^2 (2 c d-b e)}{5 e^7 (d+e x)^5}+\frac {c^3 \log (d+e x)}{e^7} \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 )^3}{(d+e x)^7} \, dx &=\int \left (\frac {d^3 (c d-b e)^3}{e^6 (d+e x)^7}-\frac {3 d^2 (c d-b e)^2 (2 c d-b e)}{e^6 (d+e x)^6}+\frac {3 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )}{e^6 (d+e x)^5}+\frac {(2 c d-b e) \left (-10 c^2 d^2+10 b c d e-b^2 e^2\right )}{e^6 (d+e x)^4}+\frac {3 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )}{e^6 (d+e x)^3}-\frac {3 c^2 (2 c d-b e)}{e^6 (d+e x)^2}+\frac {c^3}{e^6 (d+e x)}\right ) \, dx\\ &=-\frac {d^3 (c d-b e)^3}{6 e^7 (d+e x)^6}+\frac {3 d^2 (c d-b e)^2 (2 c d-b e)}{5 e^7 (d+e x)^5}-\frac {3 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )}{4 e^7 (d+e x)^4}+\frac {(2 c d-b e) \left (10 c^2 d^2-10 b c d e+b^2 e^2\right )}{3 e^7 (d+e x)^3}-\frac {3 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )}{2 e^7 (d+e x)^2}+\frac {3 c^2 (2 c d-b e)}{e^7 (d+e x)}+\frac {c^3 \log (d+e x)}{e^7}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 231, normalized size = 1.01 \begin {gather*} \frac {-b^3 e^3 \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )-6 b^2 c e^2 \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )-30 b c^2 e \left (d^5+6 d^4 e x+15 d^3 e^2 x^2+20 d^2 e^3 x^3+15 d e^4 x^4+6 e^5 x^5\right )+c^3 d \left (147 d^5+822 d^4 e x+1875 d^3 e^2 x^2+2200 d^2 e^3 x^3+1350 d e^4 x^4+360 e^5 x^5\right )+60 c^3 (d+e x)^6 \log (d+e x)}{60 e^7 (d+e x)^6} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.43, size = 272, normalized size = 1.19
method | result | size |
risch | \(\frac {-\frac {3 c^{2} \left (b e -2 c d \right ) x^{5}}{e^{2}}-\frac {3 c \left (b^{2} e^{2}+5 b c d e -15 d^{2} c^{2}\right ) x^{4}}{2 e^{3}}-\frac {\left (b^{3} e^{3}+6 b^{2} d \,e^{2} c +30 b \,c^{2} d^{2} e -110 c^{3} d^{3}\right ) x^{3}}{3 e^{4}}-\frac {d \left (b^{3} e^{3}+6 b^{2} d \,e^{2} c +30 b \,c^{2} d^{2} e -125 c^{3} d^{3}\right ) x^{2}}{4 e^{5}}-\frac {d^{2} \left (b^{3} e^{3}+6 b^{2} d \,e^{2} c +30 b \,c^{2} d^{2} e -137 c^{3} d^{3}\right ) x}{10 e^{6}}-\frac {d^{3} \left (b^{3} e^{3}+6 b^{2} d \,e^{2} c +30 b \,c^{2} d^{2} e -147 c^{3} d^{3}\right )}{60 e^{7}}}{\left (e x +d \right )^{6}}+\frac {c^{3} \ln \left (e x +d \right )}{e^{7}}\) | \(252\) |
norman | \(\frac {-\frac {d^{3} \left (b^{3} e^{3}+6 b^{2} d \,e^{2} c +30 b \,c^{2} d^{2} e -147 c^{3} d^{3}\right )}{60 e^{7}}-\frac {3 \left (b \,c^{2} e -2 c^{3} d \right ) x^{5}}{e^{2}}-\frac {3 \left (b^{2} e^{2} c +5 d e b \,c^{2}-15 c^{3} d^{2}\right ) x^{4}}{2 e^{3}}-\frac {\left (b^{3} e^{3}+6 b^{2} d \,e^{2} c +30 b \,c^{2} d^{2} e -110 c^{3} d^{3}\right ) x^{3}}{3 e^{4}}-\frac {d \left (b^{3} e^{3}+6 b^{2} d \,e^{2} c +30 b \,c^{2} d^{2} e -125 c^{3} d^{3}\right ) x^{2}}{4 e^{5}}-\frac {d^{2} \left (b^{3} e^{3}+6 b^{2} d \,e^{2} c +30 b \,c^{2} d^{2} e -137 c^{3} d^{3}\right ) x}{10 e^{6}}}{\left (e x +d \right )^{6}}+\frac {c^{3} \ln \left (e x +d \right )}{e^{7}}\) | \(256\) |
default | \(-\frac {3 c \left (b^{2} e^{2}-5 b c d e +5 d^{2} c^{2}\right )}{2 e^{7} \left (e x +d \right )^{2}}+\frac {3 d \left (b^{3} e^{3}-6 b^{2} d \,e^{2} c +10 b \,c^{2} d^{2} e -5 c^{3} d^{3}\right )}{4 e^{7} \left (e x +d \right )^{4}}+\frac {d^{3} \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}{6 e^{7} \left (e x +d \right )^{6}}-\frac {3 c^{2} \left (b e -2 c d \right )}{e^{7} \left (e x +d \right )}-\frac {b^{3} e^{3}-12 b^{2} d \,e^{2} c +30 b \,c^{2} d^{2} e -20 c^{3} d^{3}}{3 e^{7} \left (e x +d \right )^{3}}+\frac {c^{3} \ln \left (e x +d \right )}{e^{7}}-\frac {3 d^{2} \left (b^{3} e^{3}-4 b^{2} d \,e^{2} c +5 b \,c^{2} d^{2} e -2 c^{3} d^{3}\right )}{5 e^{7} \left (e x +d \right )^{5}}\) | \(272\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 308, normalized size = 1.35 \begin {gather*} c^{3} e^{\left (-7\right )} \log \left (x e + d\right ) + \frac {147 \, c^{3} d^{6} - 30 \, b c^{2} d^{5} e - 6 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3} + 180 \, {\left (2 \, c^{3} d e^{5} - b c^{2} e^{6}\right )} x^{5} + 90 \, {\left (15 \, c^{3} d^{2} e^{4} - 5 \, b c^{2} d e^{5} - b^{2} c e^{6}\right )} x^{4} + 20 \, {\left (110 \, c^{3} d^{3} e^{3} - 30 \, b c^{2} d^{2} e^{4} - 6 \, b^{2} c d e^{5} - b^{3} e^{6}\right )} x^{3} + 15 \, {\left (125 \, c^{3} d^{4} e^{2} - 30 \, b c^{2} d^{3} e^{3} - 6 \, b^{2} c d^{2} e^{4} - b^{3} d e^{5}\right )} x^{2} + 6 \, {\left (137 \, c^{3} d^{5} e - 30 \, b c^{2} d^{4} e^{2} - 6 \, b^{2} c d^{3} e^{3} - b^{3} d^{2} e^{4}\right )} x}{60 \, {\left (x^{6} e^{13} + 6 \, d x^{5} e^{12} + 15 \, d^{2} x^{4} e^{11} + 20 \, d^{3} x^{3} e^{10} + 15 \, d^{4} x^{2} e^{9} + 6 \, d^{5} x e^{8} + d^{6} e^{7}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.13, size = 385, normalized size = 1.69 \begin {gather*} \frac {147 \, c^{3} d^{6} - 10 \, {\left (18 \, b c^{2} x^{5} + 9 \, b^{2} c x^{4} + 2 \, b^{3} x^{3}\right )} e^{6} + 15 \, {\left (24 \, c^{3} d x^{5} - 30 \, b c^{2} d x^{4} - 8 \, b^{2} c d x^{3} - b^{3} d x^{2}\right )} e^{5} + 6 \, {\left (225 \, c^{3} d^{2} x^{4} - 100 \, b c^{2} d^{2} x^{3} - 15 \, b^{2} c d^{2} x^{2} - b^{3} d^{2} x\right )} e^{4} + {\left (2200 \, c^{3} d^{3} x^{3} - 450 \, b c^{2} d^{3} x^{2} - 36 \, b^{2} c d^{3} x - b^{3} d^{3}\right )} e^{3} + 3 \, {\left (625 \, c^{3} d^{4} x^{2} - 60 \, b c^{2} d^{4} x - 2 \, b^{2} c d^{4}\right )} e^{2} + 6 \, {\left (137 \, c^{3} d^{5} x - 5 \, b c^{2} d^{5}\right )} e + 60 \, {\left (c^{3} x^{6} e^{6} + 6 \, c^{3} d x^{5} e^{5} + 15 \, c^{3} d^{2} x^{4} e^{4} + 20 \, c^{3} d^{3} x^{3} e^{3} + 15 \, c^{3} d^{4} x^{2} e^{2} + 6 \, c^{3} d^{5} x e + c^{3} d^{6}\right )} \log \left (x e + d\right )}{60 \, {\left (x^{6} e^{13} + 6 \, d x^{5} e^{12} + 15 \, d^{2} x^{4} e^{11} + 20 \, d^{3} x^{3} e^{10} + 15 \, d^{4} x^{2} e^{9} + 6 \, d^{5} x e^{8} + d^{6} e^{7}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.76, size = 260, normalized size = 1.14 \begin {gather*} c^{3} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {{\left (180 \, {\left (2 \, c^{3} d e^{4} - b c^{2} e^{5}\right )} x^{5} + 90 \, {\left (15 \, c^{3} d^{2} e^{3} - 5 \, b c^{2} d e^{4} - b^{2} c e^{5}\right )} x^{4} + 20 \, {\left (110 \, c^{3} d^{3} e^{2} - 30 \, b c^{2} d^{2} e^{3} - 6 \, b^{2} c d e^{4} - b^{3} e^{5}\right )} x^{3} + 15 \, {\left (125 \, c^{3} d^{4} e - 30 \, b c^{2} d^{3} e^{2} - 6 \, b^{2} c d^{2} e^{3} - b^{3} d e^{4}\right )} x^{2} + 6 \, {\left (137 \, c^{3} d^{5} - 30 \, b c^{2} d^{4} e - 6 \, b^{2} c d^{3} e^{2} - b^{3} d^{2} e^{3}\right )} x + {\left (147 \, c^{3} d^{6} - 30 \, b c^{2} d^{5} e - 6 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}\right )} e^{\left (-1\right )}\right )} e^{\left (-6\right )}}{60 \, {\left (x e + d\right )}^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.16, size = 268, normalized size = 1.18 \begin {gather*} \frac {c^3\,\ln \left (d+e\,x\right )}{e^7}-\frac {x^5\,\left (3\,b\,c^2\,e^6-6\,c^3\,d\,e^5\right )+x^4\,\left (\frac {3\,b^2\,c\,e^6}{2}+\frac {15\,b\,c^2\,d\,e^5}{2}-\frac {45\,c^3\,d^2\,e^4}{2}\right )+x\,\left (\frac {b^3\,d^2\,e^4}{10}+\frac {3\,b^2\,c\,d^3\,e^3}{5}+3\,b\,c^2\,d^4\,e^2-\frac {137\,c^3\,d^5\,e}{10}\right )+x^2\,\left (\frac {b^3\,d\,e^5}{4}+\frac {3\,b^2\,c\,d^2\,e^4}{2}+\frac {15\,b\,c^2\,d^3\,e^3}{2}-\frac {125\,c^3\,d^4\,e^2}{4}\right )+x^3\,\left (\frac {b^3\,e^6}{3}+2\,b^2\,c\,d\,e^5+10\,b\,c^2\,d^2\,e^4-\frac {110\,c^3\,d^3\,e^3}{3}\right )-\frac {49\,c^3\,d^6}{20}+\frac {b^3\,d^3\,e^3}{60}+\frac {b^2\,c\,d^4\,e^2}{10}+\frac {b\,c^2\,d^5\,e}{2}}{e^7\,{\left (d+e\,x\right )}^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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