Optimal. Leaf size=251 \[ -\frac {3 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right ) \left (a e^2-b d e+c d^2\right )}{e^7 (d+e x)}-\frac {(2 c d-b e) \log (d+e x) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{e^7}+\frac {c x \left (-3 c e (4 b d-a e)+3 b^2 e^2+10 c^2 d^2\right )}{e^6}-\frac {\left (a e^2-b d e+c d^2\right )^3}{3 e^7 (d+e x)^3}+\frac {3 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{2 e^7 (d+e x)^2}-\frac {c^2 x^2 (4 c d-3 b e)}{2 e^5}+\frac {c^3 x^3}{3 e^4} \]
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Rubi [A] time = 0.30, antiderivative size = 251, 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 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right ) \left (a e^2-b d e+c d^2\right )}{e^7 (d+e x)}+\frac {c x \left (-3 c e (4 b d-a e)+3 b^2 e^2+10 c^2 d^2\right )}{e^6}-\frac {(2 c d-b e) \log (d+e x) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{e^7}-\frac {\left (a e^2-b d e+c d^2\right )^3}{3 e^7 (d+e x)^3}+\frac {3 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{2 e^7 (d+e x)^2}-\frac {c^2 x^2 (4 c d-3 b e)}{2 e^5}+\frac {c^3 x^3}{3 e^4} \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)^4} \, dx &=\int \left (\frac {c \left (10 c^2 d^2+3 b^2 e^2-3 c e (4 b d-a e)\right )}{e^6}-\frac {c^2 (4 c d-3 b e) x}{e^5}+\frac {c^3 x^2}{e^4}+\frac {\left (c d^2-b d e+a e^2\right )^3}{e^6 (d+e x)^4}+\frac {3 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )^2}{e^6 (d+e x)^3}+\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)^2}+\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)}\right ) \, dx\\ &=\frac {c \left (10 c^2 d^2+3 b^2 e^2-3 c e (4 b d-a e)\right ) x}{e^6}-\frac {c^2 (4 c d-3 b e) x^2}{2 e^5}+\frac {c^3 x^3}{3 e^4}-\frac {\left (c d^2-b d e+a e^2\right )^3}{3 e^7 (d+e x)^3}+\frac {3 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2}{2 e^7 (d+e x)^2}-\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 )}{e^7 (d+e x)}-\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 ) \log (d+e x)}{e^7}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 260, normalized size = 1.04 \begin {gather*} \frac {-\frac {18 \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 )}{d+e x}+6 c e x \left (3 c e (a e-4 b d)+3 b^2 e^2+10 c^2 d^2\right )-6 (2 c d-b e) \log (d+e x) \left (2 c e (3 a e-5 b d)+b^2 e^2+10 c^2 d^2\right )-\frac {2 \left (e (a e-b d)+c d^2\right )^3}{(d+e x)^3}+\frac {9 (2 c d-b e) \left (e (a e-b d)+c d^2\right )^2}{(d+e x)^2}+3 c^2 e^2 x^2 (3 b e-4 c d)+2 c^3 e^3 x^3}{6 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)^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.41, size = 664, normalized size = 2.65 \begin {gather*} \frac {2 \, c^{3} e^{6} x^{6} - 74 \, c^{3} d^{6} + 141 \, b c^{2} d^{5} e - 3 \, a^{2} b d e^{5} - 2 \, a^{3} e^{6} - 78 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} + 11 \, {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} - 6 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} - 3 \, {\left (2 \, c^{3} d e^{5} - 3 \, b c^{2} e^{6}\right )} x^{5} + 3 \, {\left (10 \, c^{3} d^{2} e^{4} - 15 \, b c^{2} d e^{5} + 6 \, {\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} + {\left (146 \, c^{3} d^{3} e^{3} - 189 \, b c^{2} d^{2} e^{4} + 54 \, {\left (b^{2} c + a c^{2}\right )} d e^{5}\right )} x^{3} + 3 \, {\left (26 \, c^{3} d^{4} e^{2} - 9 \, b c^{2} d^{3} e^{3} - 18 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} + 6 \, {\left (b^{3} + 6 \, a b c\right )} d e^{5} - 6 \, {\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} - 3 \, {\left (34 \, c^{3} d^{5} e - 81 \, b c^{2} d^{4} e^{2} + 3 \, a^{2} b e^{6} + 54 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} - 9 \, {\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 - 6 \, {\left (20 \, c^{3} d^{6} - 30 \, b c^{2} d^{5} e + 12 \, {\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 (20 \, c^{3} d^{3} e^{3} - 30 \, b c^{2} d^{2} e^{4} + 12 \, {\left (b^{2} c + a c^{2}\right )} d e^{5} - {\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} + 3 \, {\left (20 \, c^{3} d^{4} e^{2} - 30 \, b c^{2} d^{3} e^{3} + 12 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} - {\left (b^{3} + 6 \, a b c\right )} d e^{5}\right )} x^{2} + 3 \, {\left (20 \, c^{3} d^{5} e - 30 \, b c^{2} d^{4} e^{2} + 12 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} - {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4}\right )} x\right )} \log \left (e x + d\right )}{6 \, {\left (e^{10} x^{3} + 3 \, d e^{9} x^{2} + 3 \, d^{2} e^{8} x + d^{3} 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 = 424, normalized size = 1.69 \begin {gather*} -{\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \, b^{2} c d e^{2} + 12 \, a c^{2} d e^{2} - b^{3} e^{3} - 6 \, a b c e^{3}\right )} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{6} \, {\left (2 \, c^{3} x^{3} e^{8} - 12 \, c^{3} d x^{2} e^{7} + 60 \, c^{3} d^{2} x e^{6} + 9 \, b c^{2} x^{2} e^{8} - 72 \, b c^{2} d x e^{7} + 18 \, b^{2} c x e^{8} + 18 \, a c^{2} x e^{8}\right )} e^{\left (-12\right )} - \frac {{\left (74 \, c^{3} d^{6} - 141 \, b c^{2} d^{5} e + 78 \, b^{2} c d^{4} e^{2} + 78 \, a c^{2} d^{4} e^{2} - 11 \, b^{3} d^{3} e^{3} - 66 \, a b c d^{3} e^{3} + 6 \, a b^{2} d^{2} e^{4} + 6 \, a^{2} c d^{2} e^{4} + 3 \, a^{2} b d e^{5} + 2 \, a^{3} e^{6} + 18 \, {\left (5 \, c^{3} d^{4} e^{2} - 10 \, b c^{2} d^{3} e^{3} + 6 \, b^{2} c d^{2} e^{4} + 6 \, a c^{2} d^{2} e^{4} - b^{3} d e^{5} - 6 \, a b c d e^{5} + a b^{2} e^{6} + a^{2} c e^{6}\right )} x^{2} + 9 \, {\left (18 \, c^{3} d^{5} e - 35 \, b c^{2} d^{4} e^{2} + 20 \, b^{2} c d^{3} e^{3} + 20 \, a c^{2} d^{3} e^{3} - 3 \, b^{3} d^{2} e^{4} - 18 \, 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 )}}{6 \, {\left (x e + d\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 653, normalized size = 2.60 \begin {gather*} -\frac {a^{3}}{3 \left (e x +d \right )^{3} e}+\frac {a^{2} b d}{\left (e x +d \right )^{3} e^{2}}-\frac {a^{2} c \,d^{2}}{\left (e x +d \right )^{3} e^{3}}-\frac {a \,b^{2} d^{2}}{\left (e x +d \right )^{3} e^{3}}+\frac {2 a b c \,d^{3}}{\left (e x +d \right )^{3} e^{4}}-\frac {a \,c^{2} d^{4}}{\left (e x +d \right )^{3} e^{5}}+\frac {b^{3} d^{3}}{3 \left (e x +d \right )^{3} e^{4}}-\frac {b^{2} c \,d^{4}}{\left (e x +d \right )^{3} e^{5}}+\frac {b \,c^{2} d^{5}}{\left (e x +d \right )^{3} e^{6}}-\frac {c^{3} d^{6}}{3 \left (e x +d \right )^{3} e^{7}}+\frac {c^{3} x^{3}}{3 e^{4}}-\frac {3 a^{2} b}{2 \left (e x +d \right )^{2} e^{2}}+\frac {3 a^{2} c d}{\left (e x +d \right )^{2} e^{3}}+\frac {3 a \,b^{2} d}{\left (e x +d \right )^{2} e^{3}}-\frac {9 a b c \,d^{2}}{\left (e x +d \right )^{2} e^{4}}+\frac {6 a \,c^{2} d^{3}}{\left (e x +d \right )^{2} e^{5}}-\frac {3 b^{3} d^{2}}{2 \left (e x +d \right )^{2} e^{4}}+\frac {6 b^{2} c \,d^{3}}{\left (e x +d \right )^{2} e^{5}}-\frac {15 b \,c^{2} d^{4}}{2 \left (e x +d \right )^{2} e^{6}}+\frac {3 b \,c^{2} x^{2}}{2 e^{4}}+\frac {3 c^{3} d^{5}}{\left (e x +d \right )^{2} e^{7}}-\frac {2 c^{3} d \,x^{2}}{e^{5}}-\frac {3 a^{2} c}{\left (e x +d \right ) e^{3}}-\frac {3 a \,b^{2}}{\left (e x +d \right ) e^{3}}+\frac {18 a b c d}{\left (e x +d \right ) e^{4}}+\frac {6 a b c \ln \left (e x +d \right )}{e^{4}}-\frac {18 a \,c^{2} d^{2}}{\left (e x +d \right ) e^{5}}-\frac {12 a \,c^{2} d \ln \left (e x +d \right )}{e^{5}}+\frac {3 a \,c^{2} x}{e^{4}}+\frac {3 b^{3} d}{\left (e x +d \right ) e^{4}}+\frac {b^{3} \ln \left (e x +d \right )}{e^{4}}-\frac {18 b^{2} c \,d^{2}}{\left (e x +d \right ) e^{5}}-\frac {12 b^{2} c d \ln \left (e x +d \right )}{e^{5}}+\frac {3 b^{2} c x}{e^{4}}+\frac {30 b \,c^{2} d^{3}}{\left (e x +d \right ) e^{6}}+\frac {30 b \,c^{2} d^{2} \ln \left (e x +d \right )}{e^{6}}-\frac {12 b \,c^{2} d x}{e^{5}}-\frac {15 c^{3} d^{4}}{\left (e x +d \right ) e^{7}}-\frac {20 c^{3} d^{3} \ln \left (e x +d \right )}{e^{7}}+\frac {10 c^{3} d^{2} x}{e^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.22, size = 430, normalized size = 1.71 \begin {gather*} -\frac {74 \, c^{3} d^{6} - 141 \, b c^{2} d^{5} e + 3 \, a^{2} b d e^{5} + 2 \, a^{3} e^{6} + 78 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} - 11 \, {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 6 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} + 18 \, {\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} + 9 \, {\left (18 \, c^{3} d^{5} e - 35 \, b c^{2} d^{4} e^{2} + a^{2} b e^{6} + 20 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} - 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}{6 \, {\left (e^{10} x^{3} + 3 \, d e^{9} x^{2} + 3 \, d^{2} e^{8} x + d^{3} e^{7}\right )}} + \frac {2 \, c^{3} e^{2} x^{3} - 3 \, {\left (4 \, c^{3} d e - 3 \, b c^{2} e^{2}\right )} x^{2} + 6 \, {\left (10 \, c^{3} d^{2} - 12 \, b c^{2} d e + 3 \, {\left (b^{2} c + a c^{2}\right )} e^{2}\right )} x}{6 \, e^{6}} - \frac {{\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \, {\left (b^{2} c + a c^{2}\right )} d e^{2} - {\left (b^{3} + 6 \, a b c\right )} e^{3}\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 = 0.14, size = 484, normalized size = 1.93 \begin {gather*} x^2\,\left (\frac {3\,b\,c^2}{2\,e^4}-\frac {2\,c^3\,d}{e^5}\right )-x\,\left (\frac {4\,d\,\left (\frac {3\,b\,c^2}{e^4}-\frac {4\,c^3\,d}{e^5}\right )}{e}+\frac {6\,c^3\,d^2}{e^6}-\frac {3\,c\,\left (b^2+a\,c\right )}{e^4}\right )-\frac {x\,\left (\frac {3\,a^2\,b\,e^5}{2}+3\,a^2\,c\,d\,e^4+3\,a\,b^2\,d\,e^4-27\,a\,b\,c\,d^2\,e^3+30\,a\,c^2\,d^3\,e^2-\frac {9\,b^3\,d^2\,e^3}{2}+30\,b^2\,c\,d^3\,e^2-\frac {105\,b\,c^2\,d^4\,e}{2}+27\,c^3\,d^5\right )+\frac {2\,a^3\,e^6+3\,a^2\,b\,d\,e^5+6\,a^2\,c\,d^2\,e^4+6\,a\,b^2\,d^2\,e^4-66\,a\,b\,c\,d^3\,e^3+78\,a\,c^2\,d^4\,e^2-11\,b^3\,d^3\,e^3+78\,b^2\,c\,d^4\,e^2-141\,b\,c^2\,d^5\,e+74\,c^3\,d^6}{6\,e}+x^2\,\left (3\,a^2\,c\,e^5+3\,a\,b^2\,e^5-18\,a\,b\,c\,d\,e^4+18\,a\,c^2\,d^2\,e^3-3\,b^3\,d\,e^4+18\,b^2\,c\,d^2\,e^3-30\,b\,c^2\,d^3\,e^2+15\,c^3\,d^4\,e\right )}{d^3\,e^6+3\,d^2\,e^7\,x+3\,d\,e^8\,x^2+e^9\,x^3}+\frac {\ln \left (d+e\,x\right )\,\left (b^3\,e^3-12\,b^2\,c\,d\,e^2+30\,b\,c^2\,d^2\,e+6\,a\,b\,c\,e^3-20\,c^3\,d^3-12\,a\,c^2\,d\,e^2\right )}{e^7}+\frac {c^3\,x^3}{3\,e^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 30.61, size = 502, normalized size = 2.00 \begin {gather*} \frac {c^{3} x^{3}}{3 e^{4}} + x^{2} \left (\frac {3 b c^{2}}{2 e^{4}} - \frac {2 c^{3} d}{e^{5}}\right ) + x \left (\frac {3 a c^{2}}{e^{4}} + \frac {3 b^{2} c}{e^{4}} - \frac {12 b c^{2} d}{e^{5}} + \frac {10 c^{3} d^{2}}{e^{6}}\right ) + \frac {- 2 a^{3} e^{6} - 3 a^{2} b d e^{5} - 6 a^{2} c d^{2} e^{4} - 6 a b^{2} d^{2} e^{4} + 66 a b c d^{3} e^{3} - 78 a c^{2} d^{4} e^{2} + 11 b^{3} d^{3} e^{3} - 78 b^{2} c d^{4} e^{2} + 141 b c^{2} d^{5} e - 74 c^{3} d^{6} + x^{2} \left (- 18 a^{2} c e^{6} - 18 a b^{2} e^{6} + 108 a b c d e^{5} - 108 a c^{2} d^{2} e^{4} + 18 b^{3} d e^{5} - 108 b^{2} c d^{2} e^{4} + 180 b c^{2} d^{3} e^{3} - 90 c^{3} d^{4} e^{2}\right ) + x \left (- 9 a^{2} b e^{6} - 18 a^{2} c d e^{5} - 18 a b^{2} d e^{5} + 162 a b c d^{2} e^{4} - 180 a c^{2} d^{3} e^{3} + 27 b^{3} d^{2} e^{4} - 180 b^{2} c d^{3} e^{3} + 315 b c^{2} d^{4} e^{2} - 162 c^{3} d^{5} e\right )}{6 d^{3} e^{7} + 18 d^{2} e^{8} x + 18 d e^{9} x^{2} + 6 e^{10} x^{3}} + \frac {\left (b e - 2 c d\right ) \left (6 a c e^{2} + b^{2} e^{2} - 10 b c d e + 10 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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