Optimal. Leaf size=266 \[ -\frac {3 c \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{2 e^7 (d+e x)^2}+\frac {(2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{3 e^7 (d+e x)^3}-\frac {3 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{4 e^7 (d+e x)^4}+\frac {3 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{5 e^7 (d+e x)^5}-\frac {\left (a e^2-b d e+c d^2\right )^3}{6 e^7 (d+e x)^6}+\frac {3 c^2 (2 c d-b e)}{e^7 (d+e x)}+\frac {c^3 \log (d+e x)}{e^7} \]
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Rubi [A] time = 0.23, antiderivative size = 266, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {698} \[ -\frac {3 c \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{2 e^7 (d+e x)^2}+\frac {(2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{3 e^7 (d+e x)^3}-\frac {3 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{4 e^7 (d+e x)^4}+\frac {3 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{5 e^7 (d+e x)^5}-\frac {\left (a e^2-b d e+c d^2\right )^3}{6 e^7 (d+e x)^6}+\frac {3 c^2 (2 c d-b e)}{e^7 (d+e x)}+\frac {c^3 \log (d+e x)}{e^7} \]
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin {align*} \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^7} \, dx &=\int \left (\frac {\left (c d^2-b d e+a e^2\right )^3}{e^6 (d+e x)^7}+\frac {3 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )^2}{e^6 (d+e x)^6}+\frac {3 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2-5 b c d e+b^2 e^2+a c e^2\right )}{e^6 (d+e x)^5}+\frac {(2 c d-b e) \left (-10 c^2 d^2-b^2 e^2+2 c e (5 b d-3 a e)\right )}{e^6 (d+e x)^4}+\frac {3 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\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 {\left (c d^2-b d e+a e^2\right )^3}{6 e^7 (d+e x)^6}+\frac {3 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2}{5 e^7 (d+e x)^5}-\frac {3 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{4 e^7 (d+e x)^4}+\frac {(2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right )}{3 e^7 (d+e x)^3}-\frac {3 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\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 = 0.16, size = 385, normalized size = 1.45 \[ \frac {-3 c e^2 \left (a^2 e^2 \left (d^2+6 d e x+15 e^2 x^2\right )+2 a b e \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )+2 b^2 \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )\right )-e^3 \left (10 a^3 e^3+6 a^2 b e^2 (d+6 e x)+3 a b^2 e \left (d^2+6 d e x+15 e^2 x^2\right )+b^3 \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )\right )-6 c^2 e \left (a e \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )+5 b \left (d^5+6 d^4 e x+15 d^3 e^2 x^2+20 d^2 e^3 x^3+15 d e^4 x^4+6 e^5 x^5\right )\right )+c^3 d \left (147 d^5+822 d^4 e x+1875 d^3 e^2 x^2+2200 d^2 e^3 x^3+1350 d e^4 x^4+360 e^5 x^5\right )+60 c^3 (d+e x)^6 \log (d+e x)}{60 e^7 (d+e x)^6} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.93, size = 545, normalized size = 2.05 \[ \frac {147 \, c^{3} d^{6} - 30 \, b c^{2} d^{5} e - 6 \, a^{2} b d e^{5} - 10 \, a^{3} e^{6} - 6 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} - 3 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} + 180 \, {\left (2 \, c^{3} d e^{5} - b c^{2} e^{6}\right )} x^{5} + 90 \, {\left (15 \, c^{3} d^{2} e^{4} - 5 \, b c^{2} d e^{5} - {\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} + 20 \, {\left (110 \, c^{3} d^{3} e^{3} - 30 \, b c^{2} d^{2} e^{4} - 6 \, {\left (b^{2} c + a c^{2}\right )} d e^{5} - {\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} + 15 \, {\left (125 \, c^{3} d^{4} e^{2} - 30 \, b c^{2} d^{3} e^{3} - 6 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} - {\left (b^{3} + 6 \, a b c\right )} d e^{5} - 3 \, {\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} + 6 \, {\left (137 \, c^{3} d^{5} e - 30 \, b c^{2} d^{4} e^{2} - 6 \, a^{2} b e^{6} - 6 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} - {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} - 3 \, {\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x + 60 \, {\left (c^{3} e^{6} x^{6} + 6 \, c^{3} d e^{5} x^{5} + 15 \, c^{3} d^{2} e^{4} x^{4} + 20 \, c^{3} d^{3} e^{3} x^{3} + 15 \, c^{3} d^{4} e^{2} x^{2} + 6 \, c^{3} d^{5} e x + c^{3} d^{6}\right )} \log \left (e x + d\right )}{60 \, {\left (e^{13} x^{6} + 6 \, d e^{12} x^{5} + 15 \, d^{2} e^{11} x^{4} + 20 \, d^{3} e^{10} x^{3} + 15 \, d^{4} e^{9} x^{2} + 6 \, d^{5} e^{8} x + d^{6} e^{7}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 425, normalized size = 1.60 \[ c^{3} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {{\left (180 \, {\left (2 \, c^{3} d e^{4} - b c^{2} e^{5}\right )} x^{5} + 90 \, {\left (15 \, c^{3} d^{2} e^{3} - 5 \, b c^{2} d e^{4} - b^{2} c e^{5} - a c^{2} e^{5}\right )} x^{4} + 20 \, {\left (110 \, c^{3} d^{3} e^{2} - 30 \, b c^{2} d^{2} e^{3} - 6 \, b^{2} c d e^{4} - 6 \, a c^{2} d e^{4} - b^{3} e^{5} - 6 \, a b c e^{5}\right )} x^{3} + 15 \, {\left (125 \, c^{3} d^{4} e - 30 \, b c^{2} d^{3} e^{2} - 6 \, b^{2} c d^{2} e^{3} - 6 \, a c^{2} d^{2} e^{3} - b^{3} d e^{4} - 6 \, a b c d e^{4} - 3 \, a b^{2} e^{5} - 3 \, a^{2} c e^{5}\right )} x^{2} + 6 \, {\left (137 \, c^{3} d^{5} - 30 \, b c^{2} d^{4} e - 6 \, b^{2} c d^{3} e^{2} - 6 \, a c^{2} d^{3} e^{2} - b^{3} d^{2} e^{3} - 6 \, a b c d^{2} e^{3} - 3 \, a b^{2} d e^{4} - 3 \, a^{2} c d e^{4} - 6 \, a^{2} b e^{5}\right )} x + {\left (147 \, c^{3} d^{6} - 30 \, b c^{2} d^{5} e - 6 \, b^{2} c d^{4} e^{2} - 6 \, a c^{2} d^{4} e^{2} - b^{3} d^{3} e^{3} - 6 \, a b c d^{3} e^{3} - 3 \, a b^{2} d^{2} e^{4} - 3 \, a^{2} c d^{2} e^{4} - 6 \, a^{2} b d e^{5} - 10 \, a^{3} e^{6}\right )} e^{\left (-1\right )}\right )} e^{\left (-6\right )}}{60 \, {\left (x e + d\right )}^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 695, normalized size = 2.61 \[ -\frac {a^{3}}{6 \left (e x +d \right )^{6} e}+\frac {a^{2} b d}{2 \left (e x +d \right )^{6} e^{2}}-\frac {a^{2} c \,d^{2}}{2 \left (e x +d \right )^{6} e^{3}}-\frac {a \,b^{2} d^{2}}{2 \left (e x +d \right )^{6} e^{3}}+\frac {a b c \,d^{3}}{\left (e x +d \right )^{6} e^{4}}-\frac {a \,c^{2} d^{4}}{2 \left (e x +d \right )^{6} e^{5}}+\frac {b^{3} d^{3}}{6 \left (e x +d \right )^{6} e^{4}}-\frac {b^{2} c \,d^{4}}{2 \left (e x +d \right )^{6} e^{5}}+\frac {b \,c^{2} d^{5}}{2 \left (e x +d \right )^{6} e^{6}}-\frac {c^{3} d^{6}}{6 \left (e x +d \right )^{6} e^{7}}-\frac {3 a^{2} b}{5 \left (e x +d \right )^{5} e^{2}}+\frac {6 a^{2} c d}{5 \left (e x +d \right )^{5} e^{3}}+\frac {6 a \,b^{2} d}{5 \left (e x +d \right )^{5} e^{3}}-\frac {18 a b c \,d^{2}}{5 \left (e x +d \right )^{5} e^{4}}+\frac {12 a \,c^{2} d^{3}}{5 \left (e x +d \right )^{5} e^{5}}-\frac {3 b^{3} d^{2}}{5 \left (e x +d \right )^{5} e^{4}}+\frac {12 b^{2} c \,d^{3}}{5 \left (e x +d \right )^{5} e^{5}}-\frac {3 b \,c^{2} d^{4}}{\left (e x +d \right )^{5} e^{6}}+\frac {6 c^{3} d^{5}}{5 \left (e x +d \right )^{5} e^{7}}-\frac {3 a^{2} c}{4 \left (e x +d \right )^{4} e^{3}}-\frac {3 a \,b^{2}}{4 \left (e x +d \right )^{4} e^{3}}+\frac {9 a b c d}{2 \left (e x +d \right )^{4} e^{4}}-\frac {9 a \,c^{2} d^{2}}{2 \left (e x +d \right )^{4} e^{5}}+\frac {3 b^{3} d}{4 \left (e x +d \right )^{4} e^{4}}-\frac {9 b^{2} c \,d^{2}}{2 \left (e x +d \right )^{4} e^{5}}+\frac {15 b \,c^{2} d^{3}}{2 \left (e x +d \right )^{4} e^{6}}-\frac {15 c^{3} d^{4}}{4 \left (e x +d \right )^{4} e^{7}}-\frac {2 a b c}{\left (e x +d \right )^{3} e^{4}}+\frac {4 a \,c^{2} d}{\left (e x +d \right )^{3} e^{5}}-\frac {b^{3}}{3 \left (e x +d \right )^{3} e^{4}}+\frac {4 b^{2} c d}{\left (e x +d \right )^{3} e^{5}}-\frac {10 b \,c^{2} d^{2}}{\left (e x +d \right )^{3} e^{6}}+\frac {20 c^{3} d^{3}}{3 \left (e x +d \right )^{3} e^{7}}-\frac {3 a \,c^{2}}{2 \left (e x +d \right )^{2} e^{5}}-\frac {3 b^{2} c}{2 \left (e x +d \right )^{2} e^{5}}+\frac {15 b \,c^{2} d}{2 \left (e x +d \right )^{2} e^{6}}-\frac {15 c^{3} d^{2}}{2 \left (e x +d \right )^{2} e^{7}}-\frac {3 b \,c^{2}}{\left (e x +d \right ) e^{6}}+\frac {6 c^{3} d}{\left (e x +d \right ) e^{7}}+\frac {c^{3} \ln \left (e x +d \right )}{e^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.26, size = 469, normalized size = 1.76 \[ \frac {147 \, c^{3} d^{6} - 30 \, b c^{2} d^{5} e - 6 \, a^{2} b d e^{5} - 10 \, a^{3} e^{6} - 6 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} - 3 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} + 180 \, {\left (2 \, c^{3} d e^{5} - b c^{2} e^{6}\right )} x^{5} + 90 \, {\left (15 \, c^{3} d^{2} e^{4} - 5 \, b c^{2} d e^{5} - {\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} + 20 \, {\left (110 \, c^{3} d^{3} e^{3} - 30 \, b c^{2} d^{2} e^{4} - 6 \, {\left (b^{2} c + a c^{2}\right )} d e^{5} - {\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} + 15 \, {\left (125 \, c^{3} d^{4} e^{2} - 30 \, b c^{2} d^{3} e^{3} - 6 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} - {\left (b^{3} + 6 \, a b c\right )} d e^{5} - 3 \, {\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} + 6 \, {\left (137 \, c^{3} d^{5} e - 30 \, b c^{2} d^{4} e^{2} - 6 \, a^{2} b e^{6} - 6 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} - {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} - 3 \, {\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x}{60 \, {\left (e^{13} x^{6} + 6 \, d e^{12} x^{5} + 15 \, d^{2} e^{11} x^{4} + 20 \, d^{3} e^{10} x^{3} + 15 \, d^{4} e^{9} x^{2} + 6 \, d^{5} e^{8} x + d^{6} e^{7}\right )}} + \frac {c^{3} \log \left (e x + d\right )}{e^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.79, size = 497, normalized size = 1.87 \[ \frac {c^3\,\ln \left (d+e\,x\right )}{e^7}-\frac {\frac {10\,a^3\,e^6+6\,a^2\,b\,d\,e^5+3\,a^2\,c\,d^2\,e^4+3\,a\,b^2\,d^2\,e^4+6\,a\,b\,c\,d^3\,e^3+6\,a\,c^2\,d^4\,e^2+b^3\,d^3\,e^3+6\,b^2\,c\,d^4\,e^2+30\,b\,c^2\,d^5\,e-147\,c^3\,d^6}{60\,e^7}+\frac {3\,x^4\,\left (b^2\,c\,e^2+5\,b\,c^2\,d\,e-15\,c^3\,d^2+a\,c^2\,e^2\right )}{2\,e^3}+\frac {x^3\,\left (b^3\,e^3+6\,b^2\,c\,d\,e^2+30\,b\,c^2\,d^2\,e+6\,a\,b\,c\,e^3-110\,c^3\,d^3+6\,a\,c^2\,d\,e^2\right )}{3\,e^4}+\frac {x^2\,\left (3\,a^2\,c\,e^4+3\,a\,b^2\,e^4+6\,a\,b\,c\,d\,e^3+6\,a\,c^2\,d^2\,e^2+b^3\,d\,e^3+6\,b^2\,c\,d^2\,e^2+30\,b\,c^2\,d^3\,e-125\,c^3\,d^4\right )}{4\,e^5}+\frac {x\,\left (6\,a^2\,b\,e^5+3\,a^2\,c\,d\,e^4+3\,a\,b^2\,d\,e^4+6\,a\,b\,c\,d^2\,e^3+6\,a\,c^2\,d^3\,e^2+b^3\,d^2\,e^3+6\,b^2\,c\,d^3\,e^2+30\,b\,c^2\,d^4\,e-137\,c^3\,d^5\right )}{10\,e^6}+\frac {3\,c^2\,x^5\,\left (b\,e-2\,c\,d\right )}{e^2}}{d^6+6\,d^5\,e\,x+15\,d^4\,e^2\,x^2+20\,d^3\,e^3\,x^3+15\,d^2\,e^4\,x^4+6\,d\,e^5\,x^5+e^6\,x^6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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