Optimal. Leaf size=255 \[ -\frac {x \left (-9 c^2 d e (2 b d-a e)+3 b c e^2 (3 b d-2 a e)-b^3 e^3+10 c^3 d^3\right )}{e^6}+\frac {3 \log (d+e x) \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7}+\frac {3 c x^2 \left (-c e (3 b d-a e)+b^2 e^2+2 c^2 d^2\right )}{2 e^5}+\frac {3 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^7 (d+e x)}-\frac {\left (a e^2-b d e+c d^2\right )^3}{2 e^7 (d+e x)^2}-\frac {c^2 x^3 (c d-b e)}{e^4}+\frac {c^3 x^4}{4 e^3} \]
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Rubi [A] time = 0.34, antiderivative size = 255, 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} \begin {gather*} \frac {3 c x^2 \left (-c e (3 b d-a e)+b^2 e^2+2 c^2 d^2\right )}{2 e^5}-\frac {x \left (-9 c^2 d e (2 b d-a e)+3 b c e^2 (3 b d-2 a e)-b^3 e^3+10 c^3 d^3\right )}{e^6}+\frac {3 \log (d+e x) \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 )}{e^7}+\frac {3 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^7 (d+e x)}-\frac {\left (a e^2-b d e+c d^2\right )^3}{2 e^7 (d+e x)^2}-\frac {c^2 x^3 (c d-b e)}{e^4}+\frac {c^3 x^4}{4 e^3} \end {gather*}
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)^3} \, dx &=\int \left (\frac {-10 c^3 d^3+b^3 e^3-3 b c e^2 (3 b d-2 a e)+9 c^2 d e (2 b d-a e)}{e^6}+\frac {3 c \left (2 c^2 d^2+b^2 e^2-c e (3 b d-a e)\right ) x}{e^5}-\frac {3 c^2 (c d-b e) x^2}{e^4}+\frac {c^3 x^3}{e^3}+\frac {\left (c d^2-b d e+a e^2\right )^3}{e^6 (d+e x)^3}+\frac {3 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )^2}{e^6 (d+e x)^2}+\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)}\right ) \, dx\\ &=-\frac {\left (10 c^3 d^3-b^3 e^3+3 b c e^2 (3 b d-2 a e)-9 c^2 d e (2 b d-a e)\right ) x}{e^6}+\frac {3 c \left (2 c^2 d^2+b^2 e^2-c e (3 b d-a e)\right ) x^2}{2 e^5}-\frac {c^2 (c d-b e) x^3}{e^4}+\frac {c^3 x^4}{4 e^3}-\frac {\left (c d^2-b d e+a e^2\right )^3}{2 e^7 (d+e x)^2}+\frac {3 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2}{e^7 (d+e x)}+\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 ) \log (d+e x)}{e^7}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 265, normalized size = 1.04 \begin {gather*} \frac {12 \log (d+e x) \left (c e^2 \left (a^2 e^2-6 a b d e+6 b^2 d^2\right )+b^2 e^3 (a e-b d)+2 c^2 d^2 e (3 a e-5 b d)+5 c^3 d^4\right )+4 e x \left (9 c^2 d e (2 b d-a e)+3 b c e^2 (2 a e-3 b d)+b^3 e^3-10 c^3 d^3\right )+6 c e^2 x^2 \left (c e (a e-3 b d)+b^2 e^2+2 c^2 d^2\right )+\frac {12 (2 c d-b e) \left (e (a e-b d)+c d^2\right )^2}{d+e x}-\frac {2 \left (e (a e-b d)+c d^2\right )^3}{(d+e x)^2}+4 c^2 e^3 x^3 (b e-c d)+c^3 e^4 x^4}{4 e^7} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.40, size = 638, normalized size = 2.50 \begin {gather*} \frac {c^{3} e^{6} x^{6} + 22 \, c^{3} d^{6} - 54 \, b c^{2} d^{5} e - 6 \, a^{2} b d e^{5} - 2 \, a^{3} e^{6} + 42 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} - 10 \, {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 18 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} - 2 \, {\left (c^{3} d e^{5} - 2 \, b c^{2} e^{6}\right )} x^{5} + {\left (5 \, c^{3} d^{2} e^{4} - 10 \, b c^{2} d e^{5} + 6 \, {\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} - 4 \, {\left (5 \, c^{3} d^{3} e^{3} - 10 \, 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} - 2 \, {\left (34 \, c^{3} d^{4} e^{2} - 63 \, b c^{2} d^{3} e^{3} + 33 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} - 4 \, {\left (b^{3} + 6 \, a b c\right )} d e^{5}\right )} x^{2} - 4 \, {\left (4 \, c^{3} d^{5} e - 3 \, b c^{2} d^{4} e^{2} + 3 \, a^{2} b e^{6} - 3 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} + 2 \, {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} - 6 \, {\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x + 12 \, {\left (5 \, c^{3} d^{6} - 10 \, b c^{2} d^{5} e + 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} + {\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} + {\left (5 \, c^{3} d^{4} e^{2} - 10 \, 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} + {\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} + 2 \, {\left (5 \, c^{3} d^{5} e - 10 \, b c^{2} d^{4} e^{2} + 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} + {\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x\right )} \log \left (e x + d\right )}{4 \, {\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} e^{7}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 432, normalized size = 1.69 \begin {gather*} 3 \, {\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \, b^{2} c d^{2} e^{2} + 6 \, a c^{2} d^{2} e^{2} - b^{3} d e^{3} - 6 \, a b c d e^{3} + a b^{2} e^{4} + a^{2} c e^{4}\right )} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{4} \, {\left (c^{3} x^{4} e^{9} - 4 \, c^{3} d x^{3} e^{8} + 12 \, c^{3} d^{2} x^{2} e^{7} - 40 \, c^{3} d^{3} x e^{6} + 4 \, b c^{2} x^{3} e^{9} - 18 \, b c^{2} d x^{2} e^{8} + 72 \, b c^{2} d^{2} x e^{7} + 6 \, b^{2} c x^{2} e^{9} + 6 \, a c^{2} x^{2} e^{9} - 36 \, b^{2} c d x e^{8} - 36 \, a c^{2} d x e^{8} + 4 \, b^{3} x e^{9} + 24 \, a b c x e^{9}\right )} e^{\left (-12\right )} + \frac {{\left (11 \, c^{3} d^{6} - 27 \, b c^{2} d^{5} e + 21 \, b^{2} c d^{4} e^{2} + 21 \, a c^{2} d^{4} e^{2} - 5 \, b^{3} d^{3} e^{3} - 30 \, a b c d^{3} e^{3} + 9 \, a b^{2} d^{2} e^{4} + 9 \, a^{2} c d^{2} e^{4} - 3 \, a^{2} b d e^{5} - a^{3} e^{6} + 6 \, {\left (2 \, c^{3} d^{5} e - 5 \, b c^{2} d^{4} e^{2} + 4 \, b^{2} c d^{3} e^{3} + 4 \, a c^{2} d^{3} e^{3} - b^{3} d^{2} e^{4} - 6 \, a b c d^{2} e^{4} + 2 \, a b^{2} d e^{5} + 2 \, a^{2} c d e^{5} - a^{2} b e^{6}\right )} x\right )} e^{\left (-7\right )}}{2 \, {\left (x e + d\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 624, normalized size = 2.45 \begin {gather*} \frac {c^{3} x^{4}}{4 e^{3}}+\frac {b \,c^{2} x^{3}}{e^{3}}-\frac {c^{3} d \,x^{3}}{e^{4}}-\frac {a^{3}}{2 \left (e x +d \right )^{2} e}+\frac {3 a^{2} b d}{2 \left (e x +d \right )^{2} e^{2}}-\frac {3 a^{2} c \,d^{2}}{2 \left (e x +d \right )^{2} e^{3}}-\frac {3 a \,b^{2} d^{2}}{2 \left (e x +d \right )^{2} e^{3}}+\frac {3 a b c \,d^{3}}{\left (e x +d \right )^{2} e^{4}}-\frac {3 a \,c^{2} d^{4}}{2 \left (e x +d \right )^{2} e^{5}}+\frac {3 a \,c^{2} x^{2}}{2 e^{3}}+\frac {b^{3} d^{3}}{2 \left (e x +d \right )^{2} e^{4}}-\frac {3 b^{2} c \,d^{4}}{2 \left (e x +d \right )^{2} e^{5}}+\frac {3 b^{2} c \,x^{2}}{2 e^{3}}+\frac {3 b \,c^{2} d^{5}}{2 \left (e x +d \right )^{2} e^{6}}-\frac {9 b \,c^{2} d \,x^{2}}{2 e^{4}}-\frac {c^{3} d^{6}}{2 \left (e x +d \right )^{2} e^{7}}+\frac {3 c^{3} d^{2} x^{2}}{e^{5}}-\frac {3 a^{2} b}{\left (e x +d \right ) e^{2}}+\frac {6 a^{2} c d}{\left (e x +d \right ) e^{3}}+\frac {3 a^{2} c \ln \left (e x +d \right )}{e^{3}}+\frac {6 a \,b^{2} d}{\left (e x +d \right ) e^{3}}+\frac {3 a \,b^{2} \ln \left (e x +d \right )}{e^{3}}-\frac {18 a b c \,d^{2}}{\left (e x +d \right ) e^{4}}-\frac {18 a b c d \ln \left (e x +d \right )}{e^{4}}+\frac {6 a b c x}{e^{3}}+\frac {12 a \,c^{2} d^{3}}{\left (e x +d \right ) e^{5}}+\frac {18 a \,c^{2} d^{2} \ln \left (e x +d \right )}{e^{5}}-\frac {9 a \,c^{2} d x}{e^{4}}-\frac {3 b^{3} d^{2}}{\left (e x +d \right ) e^{4}}-\frac {3 b^{3} d \ln \left (e x +d \right )}{e^{4}}+\frac {b^{3} x}{e^{3}}+\frac {12 b^{2} c \,d^{3}}{\left (e x +d \right ) e^{5}}+\frac {18 b^{2} c \,d^{2} \ln \left (e x +d \right )}{e^{5}}-\frac {9 b^{2} c d x}{e^{4}}-\frac {15 b \,c^{2} d^{4}}{\left (e x +d \right ) e^{6}}-\frac {30 b \,c^{2} d^{3} \ln \left (e x +d \right )}{e^{6}}+\frac {18 b \,c^{2} d^{2} x}{e^{5}}+\frac {6 c^{3} d^{5}}{\left (e x +d \right ) e^{7}}+\frac {15 c^{3} d^{4} \ln \left (e x +d \right )}{e^{7}}-\frac {10 c^{3} d^{3} x}{e^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.21, size = 417, normalized size = 1.64 \begin {gather*} \frac {11 \, c^{3} d^{6} - 27 \, b c^{2} d^{5} e - 3 \, a^{2} b d e^{5} - a^{3} e^{6} + 21 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} - 5 \, {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 9 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} + 6 \, {\left (2 \, c^{3} d^{5} e - 5 \, b c^{2} d^{4} e^{2} - a^{2} b e^{6} + 4 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} - {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} + 2 \, {\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x}{2 \, {\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} e^{7}\right )}} + \frac {c^{3} e^{3} x^{4} - 4 \, {\left (c^{3} d e^{2} - b c^{2} e^{3}\right )} x^{3} + 6 \, {\left (2 \, c^{3} d^{2} e - 3 \, b c^{2} d e^{2} + {\left (b^{2} c + a c^{2}\right )} e^{3}\right )} x^{2} - 4 \, {\left (10 \, c^{3} d^{3} - 18 \, b c^{2} d^{2} e + 9 \, {\left (b^{2} c + a c^{2}\right )} d e^{2} - {\left (b^{3} + 6 \, a b c\right )} e^{3}\right )} x}{4 \, e^{6}} + \frac {3 \, {\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d e^{3} + {\left (a b^{2} + a^{2} c\right )} e^{4}\right )} \log \left (e x + d\right )}{e^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.76, size = 521, normalized size = 2.04 \begin {gather*} \frac {x\,\left (-3\,a^2\,b\,e^5+6\,a^2\,c\,d\,e^4+6\,a\,b^2\,d\,e^4-18\,a\,b\,c\,d^2\,e^3+12\,a\,c^2\,d^3\,e^2-3\,b^3\,d^2\,e^3+12\,b^2\,c\,d^3\,e^2-15\,b\,c^2\,d^4\,e+6\,c^3\,d^5\right )-\frac {a^3\,e^6+3\,a^2\,b\,d\,e^5-9\,a^2\,c\,d^2\,e^4-9\,a\,b^2\,d^2\,e^4+30\,a\,b\,c\,d^3\,e^3-21\,a\,c^2\,d^4\,e^2+5\,b^3\,d^3\,e^3-21\,b^2\,c\,d^4\,e^2+27\,b\,c^2\,d^5\,e-11\,c^3\,d^6}{2\,e}}{d^2\,e^6+2\,d\,e^7\,x+e^8\,x^2}+x^3\,\left (\frac {b\,c^2}{e^3}-\frac {c^3\,d}{e^4}\right )+x\,\left (\frac {b^3+6\,a\,c\,b}{e^3}-\frac {c^3\,d^3}{e^6}+\frac {3\,d\,\left (\frac {3\,d\,\left (\frac {3\,b\,c^2}{e^3}-\frac {3\,c^3\,d}{e^4}\right )}{e}+\frac {3\,c^3\,d^2}{e^5}-\frac {3\,c\,\left (b^2+a\,c\right )}{e^3}\right )}{e}-\frac {3\,d^2\,\left (\frac {3\,b\,c^2}{e^3}-\frac {3\,c^3\,d}{e^4}\right )}{e^2}\right )-x^2\,\left (\frac {3\,d\,\left (\frac {3\,b\,c^2}{e^3}-\frac {3\,c^3\,d}{e^4}\right )}{2\,e}+\frac {3\,c^3\,d^2}{2\,e^5}-\frac {3\,c\,\left (b^2+a\,c\right )}{2\,e^3}\right )+\frac {\ln \left (d+e\,x\right )\,\left (3\,a^2\,c\,e^4+3\,a\,b^2\,e^4-18\,a\,b\,c\,d\,e^3+18\,a\,c^2\,d^2\,e^2-3\,b^3\,d\,e^3+18\,b^2\,c\,d^2\,e^2-30\,b\,c^2\,d^3\,e+15\,c^3\,d^4\right )}{e^7}+\frac {c^3\,x^4}{4\,e^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 8.60, size = 466, normalized size = 1.83 \begin {gather*} \frac {c^{3} x^{4}}{4 e^{3}} + x^{3} \left (\frac {b c^{2}}{e^{3}} - \frac {c^{3} d}{e^{4}}\right ) + x^{2} \left (\frac {3 a c^{2}}{2 e^{3}} + \frac {3 b^{2} c}{2 e^{3}} - \frac {9 b c^{2} d}{2 e^{4}} + \frac {3 c^{3} d^{2}}{e^{5}}\right ) + x \left (\frac {6 a b c}{e^{3}} - \frac {9 a c^{2} d}{e^{4}} + \frac {b^{3}}{e^{3}} - \frac {9 b^{2} c d}{e^{4}} + \frac {18 b c^{2} d^{2}}{e^{5}} - \frac {10 c^{3} d^{3}}{e^{6}}\right ) + \frac {- a^{3} e^{6} - 3 a^{2} b d e^{5} + 9 a^{2} c d^{2} e^{4} + 9 a b^{2} d^{2} e^{4} - 30 a b c d^{3} e^{3} + 21 a c^{2} d^{4} e^{2} - 5 b^{3} d^{3} e^{3} + 21 b^{2} c d^{4} e^{2} - 27 b c^{2} d^{5} e + 11 c^{3} d^{6} + x \left (- 6 a^{2} b e^{6} + 12 a^{2} c d e^{5} + 12 a b^{2} d e^{5} - 36 a b c d^{2} e^{4} + 24 a c^{2} d^{3} e^{3} - 6 b^{3} d^{2} e^{4} + 24 b^{2} c d^{3} e^{3} - 30 b c^{2} d^{4} e^{2} + 12 c^{3} d^{5} e\right )}{2 d^{2} e^{7} + 4 d e^{8} x + 2 e^{9} x^{2}} + \frac {3 \left (a e^{2} - b d e + c d^{2}\right ) \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right ) \log {\left (d + e x \right )}}{e^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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