Optimal. Leaf size=187 \[ -\frac {7 b^6 (d+e x)^3 (b d-a e)}{3 e^8}+\frac {21 b^5 (d+e x)^2 (b d-a e)^2}{2 e^8}-\frac {35 b^4 x (b d-a e)^3}{e^7}+\frac {35 b^3 (b d-a e)^4 \log (d+e x)}{e^8}+\frac {21 b^2 (b d-a e)^5}{e^8 (d+e x)}-\frac {7 b (b d-a e)^6}{2 e^8 (d+e x)^2}+\frac {(b d-a e)^7}{3 e^8 (d+e x)^3}+\frac {b^7 (d+e x)^4}{4 e^8} \]
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Rubi [A] time = 0.21, antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {27, 43} \begin {gather*} -\frac {7 b^6 (d+e x)^3 (b d-a e)}{3 e^8}+\frac {21 b^5 (d+e x)^2 (b d-a e)^2}{2 e^8}-\frac {35 b^4 x (b d-a e)^3}{e^7}+\frac {21 b^2 (b d-a e)^5}{e^8 (d+e x)}+\frac {35 b^3 (b d-a e)^4 \log (d+e x)}{e^8}-\frac {7 b (b d-a e)^6}{2 e^8 (d+e x)^2}+\frac {(b d-a e)^7}{3 e^8 (d+e x)^3}+\frac {b^7 (d+e x)^4}{4 e^8} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin {align*} \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^4} \, dx &=\int \frac {(a+b x)^7}{(d+e x)^4} \, dx\\ &=\int \left (-\frac {35 b^4 (b d-a e)^3}{e^7}+\frac {(-b d+a e)^7}{e^7 (d+e x)^4}+\frac {7 b (b d-a e)^6}{e^7 (d+e x)^3}-\frac {21 b^2 (b d-a e)^5}{e^7 (d+e x)^2}+\frac {35 b^3 (b d-a e)^4}{e^7 (d+e x)}+\frac {21 b^5 (b d-a e)^2 (d+e x)}{e^7}-\frac {7 b^6 (b d-a e) (d+e x)^2}{e^7}+\frac {b^7 (d+e x)^3}{e^7}\right ) \, dx\\ &=-\frac {35 b^4 (b d-a e)^3 x}{e^7}+\frac {(b d-a e)^7}{3 e^8 (d+e x)^3}-\frac {7 b (b d-a e)^6}{2 e^8 (d+e x)^2}+\frac {21 b^2 (b d-a e)^5}{e^8 (d+e x)}+\frac {21 b^5 (b d-a e)^2 (d+e x)^2}{2 e^8}-\frac {7 b^6 (b d-a e) (d+e x)^3}{3 e^8}+\frac {b^7 (d+e x)^4}{4 e^8}+\frac {35 b^3 (b d-a e)^4 \log (d+e x)}{e^8}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 199, normalized size = 1.06 \begin {gather*} \frac {6 b^5 e^2 x^2 \left (21 a^2 e^2-28 a b d e+10 b^2 d^2\right )-12 b^4 e x \left (-35 a^3 e^3+84 a^2 b d e^2-70 a b^2 d^2 e+20 b^3 d^3\right )-4 b^6 e^3 x^3 (4 b d-7 a e)+420 b^3 (b d-a e)^4 \log (d+e x)+\frac {252 b^2 (b d-a e)^5}{d+e x}-\frac {42 b (b d-a e)^6}{(d+e x)^2}+\frac {4 (b d-a e)^7}{(d+e x)^3}+3 b^7 e^4 x^4}{12 e^8} \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 {(a+b x) \left (a^2+2 a b x+b^2 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 = 0.43, size = 737, normalized size = 3.94 \begin {gather*} \frac {3 \, b^{7} e^{7} x^{7} + 214 \, b^{7} d^{7} - 1036 \, a b^{6} d^{6} e + 1974 \, a^{2} b^{5} d^{5} e^{2} - 1820 \, a^{3} b^{4} d^{4} e^{3} + 770 \, a^{4} b^{3} d^{3} e^{4} - 84 \, a^{5} b^{2} d^{2} e^{5} - 14 \, a^{6} b d e^{6} - 4 \, a^{7} e^{7} - 7 \, {\left (b^{7} d e^{6} - 4 \, a b^{6} e^{7}\right )} x^{6} + 21 \, {\left (b^{7} d^{2} e^{5} - 4 \, a b^{6} d e^{6} + 6 \, a^{2} b^{5} e^{7}\right )} x^{5} - 105 \, {\left (b^{7} d^{3} e^{4} - 4 \, a b^{6} d^{2} e^{5} + 6 \, a^{2} b^{5} d e^{6} - 4 \, a^{3} b^{4} e^{7}\right )} x^{4} - 2 \, {\left (278 \, b^{7} d^{4} e^{3} - 1022 \, a b^{6} d^{3} e^{4} + 1323 \, a^{2} b^{5} d^{2} e^{5} - 630 \, a^{3} b^{4} d e^{6}\right )} x^{3} - 6 \, {\left (68 \, b^{7} d^{5} e^{2} - 182 \, a b^{6} d^{4} e^{3} + 63 \, a^{2} b^{5} d^{3} e^{4} + 210 \, a^{3} b^{4} d^{2} e^{5} - 210 \, a^{4} b^{3} d e^{6} + 42 \, a^{5} b^{2} e^{7}\right )} x^{2} + 6 \, {\left (37 \, b^{7} d^{6} e - 238 \, a b^{6} d^{5} e^{2} + 567 \, a^{2} b^{5} d^{4} e^{3} - 630 \, a^{3} b^{4} d^{3} e^{4} + 315 \, a^{4} b^{3} d^{2} e^{5} - 42 \, a^{5} b^{2} d e^{6} - 7 \, a^{6} b e^{7}\right )} x + 420 \, {\left (b^{7} d^{7} - 4 \, a b^{6} d^{6} e + 6 \, a^{2} b^{5} d^{5} e^{2} - 4 \, a^{3} b^{4} d^{4} e^{3} + a^{4} b^{3} d^{3} e^{4} + {\left (b^{7} d^{4} e^{3} - 4 \, a b^{6} d^{3} e^{4} + 6 \, a^{2} b^{5} d^{2} e^{5} - 4 \, a^{3} b^{4} d e^{6} + a^{4} b^{3} e^{7}\right )} x^{3} + 3 \, {\left (b^{7} d^{5} e^{2} - 4 \, a b^{6} d^{4} e^{3} + 6 \, a^{2} b^{5} d^{3} e^{4} - 4 \, a^{3} b^{4} d^{2} e^{5} + a^{4} b^{3} d e^{6}\right )} x^{2} + 3 \, {\left (b^{7} d^{6} e - 4 \, a b^{6} d^{5} e^{2} + 6 \, a^{2} b^{5} d^{4} e^{3} - 4 \, a^{3} b^{4} d^{3} e^{4} + a^{4} b^{3} d^{2} e^{5}\right )} x\right )} \log \left (e x + d\right )}{12 \, {\left (e^{11} x^{3} + 3 \, d e^{10} x^{2} + 3 \, d^{2} e^{9} x + d^{3} e^{8}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.18, size = 442, normalized size = 2.36 \begin {gather*} 35 \, {\left (b^{7} d^{4} - 4 \, a b^{6} d^{3} e + 6 \, a^{2} b^{5} d^{2} e^{2} - 4 \, a^{3} b^{4} d e^{3} + a^{4} b^{3} e^{4}\right )} e^{\left (-8\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{12} \, {\left (3 \, b^{7} x^{4} e^{12} - 16 \, b^{7} d x^{3} e^{11} + 60 \, b^{7} d^{2} x^{2} e^{10} - 240 \, b^{7} d^{3} x e^{9} + 28 \, a b^{6} x^{3} e^{12} - 168 \, a b^{6} d x^{2} e^{11} + 840 \, a b^{6} d^{2} x e^{10} + 126 \, a^{2} b^{5} x^{2} e^{12} - 1008 \, a^{2} b^{5} d x e^{11} + 420 \, a^{3} b^{4} x e^{12}\right )} e^{\left (-16\right )} + \frac {{\left (107 \, b^{7} d^{7} - 518 \, a b^{6} d^{6} e + 987 \, a^{2} b^{5} d^{5} e^{2} - 910 \, a^{3} b^{4} d^{4} e^{3} + 385 \, a^{4} b^{3} d^{3} e^{4} - 42 \, a^{5} b^{2} d^{2} e^{5} - 7 \, a^{6} b d e^{6} - 2 \, a^{7} e^{7} + 126 \, {\left (b^{7} d^{5} e^{2} - 5 \, a b^{6} d^{4} e^{3} + 10 \, a^{2} b^{5} d^{3} e^{4} - 10 \, a^{3} b^{4} d^{2} e^{5} + 5 \, a^{4} b^{3} d e^{6} - a^{5} b^{2} e^{7}\right )} x^{2} + 21 \, {\left (11 \, b^{7} d^{6} e - 54 \, a b^{6} d^{5} e^{2} + 105 \, a^{2} b^{5} d^{4} e^{3} - 100 \, a^{3} b^{4} d^{3} e^{4} + 45 \, a^{4} b^{3} d^{2} e^{5} - 6 \, a^{5} b^{2} d e^{6} - a^{6} b e^{7}\right )} x\right )} e^{\left (-8\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 = 0.07, size = 622, normalized size = 3.33 \begin {gather*} \frac {b^{7} x^{4}}{4 e^{4}}-\frac {a^{7}}{3 \left (e x +d \right )^{3} e}+\frac {7 a^{6} b d}{3 \left (e x +d \right )^{3} e^{2}}-\frac {7 a^{5} b^{2} d^{2}}{\left (e x +d \right )^{3} e^{3}}+\frac {35 a^{4} b^{3} d^{3}}{3 \left (e x +d \right )^{3} e^{4}}-\frac {35 a^{3} b^{4} d^{4}}{3 \left (e x +d \right )^{3} e^{5}}+\frac {7 a^{2} b^{5} d^{5}}{\left (e x +d \right )^{3} e^{6}}-\frac {7 a \,b^{6} d^{6}}{3 \left (e x +d \right )^{3} e^{7}}+\frac {7 a \,b^{6} x^{3}}{3 e^{4}}+\frac {b^{7} d^{7}}{3 \left (e x +d \right )^{3} e^{8}}-\frac {4 b^{7} d \,x^{3}}{3 e^{5}}-\frac {7 a^{6} b}{2 \left (e x +d \right )^{2} e^{2}}+\frac {21 a^{5} b^{2} d}{\left (e x +d \right )^{2} e^{3}}-\frac {105 a^{4} b^{3} d^{2}}{2 \left (e x +d \right )^{2} e^{4}}+\frac {70 a^{3} b^{4} d^{3}}{\left (e x +d \right )^{2} e^{5}}-\frac {105 a^{2} b^{5} d^{4}}{2 \left (e x +d \right )^{2} e^{6}}+\frac {21 a^{2} b^{5} x^{2}}{2 e^{4}}+\frac {21 a \,b^{6} d^{5}}{\left (e x +d \right )^{2} e^{7}}-\frac {14 a \,b^{6} d \,x^{2}}{e^{5}}-\frac {7 b^{7} d^{6}}{2 \left (e x +d \right )^{2} e^{8}}+\frac {5 b^{7} d^{2} x^{2}}{e^{6}}-\frac {21 a^{5} b^{2}}{\left (e x +d \right ) e^{3}}+\frac {105 a^{4} b^{3} d}{\left (e x +d \right ) e^{4}}+\frac {35 a^{4} b^{3} \ln \left (e x +d \right )}{e^{4}}-\frac {210 a^{3} b^{4} d^{2}}{\left (e x +d \right ) e^{5}}-\frac {140 a^{3} b^{4} d \ln \left (e x +d \right )}{e^{5}}+\frac {35 a^{3} b^{4} x}{e^{4}}+\frac {210 a^{2} b^{5} d^{3}}{\left (e x +d \right ) e^{6}}+\frac {210 a^{2} b^{5} d^{2} \ln \left (e x +d \right )}{e^{6}}-\frac {84 a^{2} b^{5} d x}{e^{5}}-\frac {105 a \,b^{6} d^{4}}{\left (e x +d \right ) e^{7}}-\frac {140 a \,b^{6} d^{3} \ln \left (e x +d \right )}{e^{7}}+\frac {70 a \,b^{6} d^{2} x}{e^{6}}+\frac {21 b^{7} d^{5}}{\left (e x +d \right ) e^{8}}+\frac {35 b^{7} d^{4} \ln \left (e x +d \right )}{e^{8}}-\frac {20 b^{7} d^{3} x}{e^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.67, size = 485, normalized size = 2.59 \begin {gather*} \frac {107 \, b^{7} d^{7} - 518 \, a b^{6} d^{6} e + 987 \, a^{2} b^{5} d^{5} e^{2} - 910 \, a^{3} b^{4} d^{4} e^{3} + 385 \, a^{4} b^{3} d^{3} e^{4} - 42 \, a^{5} b^{2} d^{2} e^{5} - 7 \, a^{6} b d e^{6} - 2 \, a^{7} e^{7} + 126 \, {\left (b^{7} d^{5} e^{2} - 5 \, a b^{6} d^{4} e^{3} + 10 \, a^{2} b^{5} d^{3} e^{4} - 10 \, a^{3} b^{4} d^{2} e^{5} + 5 \, a^{4} b^{3} d e^{6} - a^{5} b^{2} e^{7}\right )} x^{2} + 21 \, {\left (11 \, b^{7} d^{6} e - 54 \, a b^{6} d^{5} e^{2} + 105 \, a^{2} b^{5} d^{4} e^{3} - 100 \, a^{3} b^{4} d^{3} e^{4} + 45 \, a^{4} b^{3} d^{2} e^{5} - 6 \, a^{5} b^{2} d e^{6} - a^{6} b e^{7}\right )} x}{6 \, {\left (e^{11} x^{3} + 3 \, d e^{10} x^{2} + 3 \, d^{2} e^{9} x + d^{3} e^{8}\right )}} + \frac {3 \, b^{7} e^{3} x^{4} - 4 \, {\left (4 \, b^{7} d e^{2} - 7 \, a b^{6} e^{3}\right )} x^{3} + 6 \, {\left (10 \, b^{7} d^{2} e - 28 \, a b^{6} d e^{2} + 21 \, a^{2} b^{5} e^{3}\right )} x^{2} - 12 \, {\left (20 \, b^{7} d^{3} - 70 \, a b^{6} d^{2} e + 84 \, a^{2} b^{5} d e^{2} - 35 \, a^{3} b^{4} e^{3}\right )} x}{12 \, e^{7}} + \frac {35 \, {\left (b^{7} d^{4} - 4 \, a b^{6} d^{3} e + 6 \, a^{2} b^{5} d^{2} e^{2} - 4 \, a^{3} b^{4} d e^{3} + a^{4} b^{3} e^{4}\right )} \log \left (e x + d\right )}{e^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.12, size = 558, normalized size = 2.98 \begin {gather*} x\,\left (\frac {35\,a^3\,b^4}{e^4}-\frac {4\,b^7\,d^3}{e^7}+\frac {4\,d\,\left (\frac {4\,d\,\left (\frac {7\,a\,b^6}{e^4}-\frac {4\,b^7\,d}{e^5}\right )}{e}-\frac {21\,a^2\,b^5}{e^4}+\frac {6\,b^7\,d^2}{e^6}\right )}{e}-\frac {6\,d^2\,\left (\frac {7\,a\,b^6}{e^4}-\frac {4\,b^7\,d}{e^5}\right )}{e^2}\right )-\frac {\frac {2\,a^7\,e^7+7\,a^6\,b\,d\,e^6+42\,a^5\,b^2\,d^2\,e^5-385\,a^4\,b^3\,d^3\,e^4+910\,a^3\,b^4\,d^4\,e^3-987\,a^2\,b^5\,d^5\,e^2+518\,a\,b^6\,d^6\,e-107\,b^7\,d^7}{6\,e}+x\,\left (\frac {7\,a^6\,b\,e^6}{2}+21\,a^5\,b^2\,d\,e^5-\frac {315\,a^4\,b^3\,d^2\,e^4}{2}+350\,a^3\,b^4\,d^3\,e^3-\frac {735\,a^2\,b^5\,d^4\,e^2}{2}+189\,a\,b^6\,d^5\,e-\frac {77\,b^7\,d^6}{2}\right )-x^2\,\left (-21\,a^5\,b^2\,e^6+105\,a^4\,b^3\,d\,e^5-210\,a^3\,b^4\,d^2\,e^4+210\,a^2\,b^5\,d^3\,e^3-105\,a\,b^6\,d^4\,e^2+21\,b^7\,d^5\,e\right )}{d^3\,e^7+3\,d^2\,e^8\,x+3\,d\,e^9\,x^2+e^{10}\,x^3}+x^3\,\left (\frac {7\,a\,b^6}{3\,e^4}-\frac {4\,b^7\,d}{3\,e^5}\right )-x^2\,\left (\frac {2\,d\,\left (\frac {7\,a\,b^6}{e^4}-\frac {4\,b^7\,d}{e^5}\right )}{e}-\frac {21\,a^2\,b^5}{2\,e^4}+\frac {3\,b^7\,d^2}{e^6}\right )+\frac {\ln \left (d+e\,x\right )\,\left (35\,a^4\,b^3\,e^4-140\,a^3\,b^4\,d\,e^3+210\,a^2\,b^5\,d^2\,e^2-140\,a\,b^6\,d^3\,e+35\,b^7\,d^4\right )}{e^8}+\frac {b^7\,x^4}{4\,e^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 6.27, size = 474, normalized size = 2.53 \begin {gather*} \frac {b^{7} x^{4}}{4 e^{4}} + \frac {35 b^{3} \left (a e - b d\right )^{4} \log {\left (d + e x \right )}}{e^{8}} + x^{3} \left (\frac {7 a b^{6}}{3 e^{4}} - \frac {4 b^{7} d}{3 e^{5}}\right ) + x^{2} \left (\frac {21 a^{2} b^{5}}{2 e^{4}} - \frac {14 a b^{6} d}{e^{5}} + \frac {5 b^{7} d^{2}}{e^{6}}\right ) + x \left (\frac {35 a^{3} b^{4}}{e^{4}} - \frac {84 a^{2} b^{5} d}{e^{5}} + \frac {70 a b^{6} d^{2}}{e^{6}} - \frac {20 b^{7} d^{3}}{e^{7}}\right ) + \frac {- 2 a^{7} e^{7} - 7 a^{6} b d e^{6} - 42 a^{5} b^{2} d^{2} e^{5} + 385 a^{4} b^{3} d^{3} e^{4} - 910 a^{3} b^{4} d^{4} e^{3} + 987 a^{2} b^{5} d^{5} e^{2} - 518 a b^{6} d^{6} e + 107 b^{7} d^{7} + x^{2} \left (- 126 a^{5} b^{2} e^{7} + 630 a^{4} b^{3} d e^{6} - 1260 a^{3} b^{4} d^{2} e^{5} + 1260 a^{2} b^{5} d^{3} e^{4} - 630 a b^{6} d^{4} e^{3} + 126 b^{7} d^{5} e^{2}\right ) + x \left (- 21 a^{6} b e^{7} - 126 a^{5} b^{2} d e^{6} + 945 a^{4} b^{3} d^{2} e^{5} - 2100 a^{3} b^{4} d^{3} e^{4} + 2205 a^{2} b^{5} d^{4} e^{3} - 1134 a b^{6} d^{5} e^{2} + 231 b^{7} d^{6} e\right )}{6 d^{3} e^{8} + 18 d^{2} e^{9} x + 18 d e^{10} x^{2} + 6 e^{11} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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