Optimal. Leaf size=186 \[ -\frac {7 b^6 (d+e x)^5 (b d-a e)}{5 e^8}+\frac {21 b^5 (d+e x)^4 (b d-a e)^2}{4 e^8}-\frac {35 b^4 (d+e x)^3 (b d-a e)^3}{3 e^8}+\frac {35 b^3 (d+e x)^2 (b d-a e)^4}{2 e^8}-\frac {21 b^2 x (b d-a e)^5}{e^7}+\frac {(b d-a e)^7}{e^8 (d+e x)}+\frac {7 b (b d-a e)^6 \log (d+e x)}{e^8}+\frac {b^7 (d+e x)^6}{6 e^8} \]
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Rubi [A] time = 0.26, antiderivative size = 186, 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)^5 (b d-a e)}{5 e^8}+\frac {21 b^5 (d+e x)^4 (b d-a e)^2}{4 e^8}-\frac {35 b^4 (d+e x)^3 (b d-a e)^3}{3 e^8}+\frac {35 b^3 (d+e x)^2 (b d-a e)^4}{2 e^8}-\frac {21 b^2 x (b d-a e)^5}{e^7}+\frac {(b d-a e)^7}{e^8 (d+e x)}+\frac {7 b (b d-a e)^6 \log (d+e x)}{e^8}+\frac {b^7 (d+e x)^6}{6 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)^2} \, dx &=\int \frac {(a+b x)^7}{(d+e x)^2} \, dx\\ &=\int \left (-\frac {21 b^2 (b d-a e)^5}{e^7}+\frac {(-b d+a e)^7}{e^7 (d+e x)^2}+\frac {7 b (b d-a e)^6}{e^7 (d+e x)}+\frac {35 b^3 (b d-a e)^4 (d+e x)}{e^7}-\frac {35 b^4 (b d-a e)^3 (d+e x)^2}{e^7}+\frac {21 b^5 (b d-a e)^2 (d+e x)^3}{e^7}-\frac {7 b^6 (b d-a e) (d+e x)^4}{e^7}+\frac {b^7 (d+e x)^5}{e^7}\right ) \, dx\\ &=-\frac {21 b^2 (b d-a e)^5 x}{e^7}+\frac {(b d-a e)^7}{e^8 (d+e x)}+\frac {35 b^3 (b d-a e)^4 (d+e x)^2}{2 e^8}-\frac {35 b^4 (b d-a e)^3 (d+e x)^3}{3 e^8}+\frac {21 b^5 (b d-a e)^2 (d+e x)^4}{4 e^8}-\frac {7 b^6 (b d-a e) (d+e x)^5}{5 e^8}+\frac {b^7 (d+e x)^6}{6 e^8}+\frac {7 b (b d-a e)^6 \log (d+e x)}{e^8}\\ \end {align*}
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Mathematica [B] time = 0.11, size = 387, normalized size = 2.08 \begin {gather*} \frac {-60 a^7 e^7+420 a^6 b d e^6+1260 a^5 b^2 e^5 \left (-d^2+d e x+e^2 x^2\right )+1050 a^4 b^3 e^4 \left (2 d^3-4 d^2 e x-3 d e^2 x^2+e^3 x^3\right )+700 a^3 b^4 e^3 \left (-3 d^4+9 d^3 e x+6 d^2 e^2 x^2-2 d e^3 x^3+e^4 x^4\right )+105 a^2 b^5 e^2 \left (12 d^5-48 d^4 e x-30 d^3 e^2 x^2+10 d^2 e^3 x^3-5 d e^4 x^4+3 e^5 x^5\right )+42 a b^6 e \left (-10 d^6+50 d^5 e x+30 d^4 e^2 x^2-10 d^3 e^3 x^3+5 d^2 e^4 x^4-3 d e^5 x^5+2 e^6 x^6\right )+420 b (d+e x) (b d-a e)^6 \log (d+e x)+b^7 \left (60 d^7-360 d^6 e x-210 d^5 e^2 x^2+70 d^4 e^3 x^3-35 d^3 e^4 x^4+21 d^2 e^5 x^5-14 d e^6 x^6+10 e^7 x^7\right )}{60 e^8 (d+e x)} \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)^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.44, size = 629, normalized size = 3.38 \begin {gather*} \frac {10 \, b^{7} e^{7} x^{7} + 60 \, b^{7} d^{7} - 420 \, a b^{6} d^{6} e + 1260 \, a^{2} b^{5} d^{5} e^{2} - 2100 \, a^{3} b^{4} d^{4} e^{3} + 2100 \, a^{4} b^{3} d^{3} e^{4} - 1260 \, a^{5} b^{2} d^{2} e^{5} + 420 \, a^{6} b d e^{6} - 60 \, a^{7} e^{7} - 14 \, {\left (b^{7} d e^{6} - 6 \, a b^{6} e^{7}\right )} x^{6} + 21 \, {\left (b^{7} d^{2} e^{5} - 6 \, a b^{6} d e^{6} + 15 \, a^{2} b^{5} e^{7}\right )} x^{5} - 35 \, {\left (b^{7} d^{3} e^{4} - 6 \, a b^{6} d^{2} e^{5} + 15 \, a^{2} b^{5} d e^{6} - 20 \, a^{3} b^{4} e^{7}\right )} x^{4} + 70 \, {\left (b^{7} d^{4} e^{3} - 6 \, a b^{6} d^{3} e^{4} + 15 \, a^{2} b^{5} d^{2} e^{5} - 20 \, a^{3} b^{4} d e^{6} + 15 \, a^{4} b^{3} e^{7}\right )} x^{3} - 210 \, {\left (b^{7} d^{5} e^{2} - 6 \, a b^{6} d^{4} e^{3} + 15 \, a^{2} b^{5} d^{3} e^{4} - 20 \, a^{3} b^{4} d^{2} e^{5} + 15 \, a^{4} b^{3} d e^{6} - 6 \, a^{5} b^{2} e^{7}\right )} x^{2} - 60 \, {\left (6 \, b^{7} d^{6} e - 35 \, a b^{6} d^{5} e^{2} + 84 \, a^{2} b^{5} d^{4} e^{3} - 105 \, a^{3} b^{4} d^{3} e^{4} + 70 \, a^{4} b^{3} d^{2} e^{5} - 21 \, a^{5} b^{2} d e^{6}\right )} x + 420 \, {\left (b^{7} d^{7} - 6 \, a b^{6} d^{6} e + 15 \, a^{2} b^{5} d^{5} e^{2} - 20 \, a^{3} b^{4} d^{4} e^{3} + 15 \, a^{4} b^{3} d^{3} e^{4} - 6 \, a^{5} b^{2} d^{2} e^{5} + a^{6} b d e^{6} + {\left (b^{7} d^{6} e - 6 \, a b^{6} d^{5} e^{2} + 15 \, a^{2} b^{5} d^{4} e^{3} - 20 \, a^{3} b^{4} d^{3} e^{4} + 15 \, a^{4} b^{3} d^{2} e^{5} - 6 \, a^{5} b^{2} d e^{6} + a^{6} b e^{7}\right )} x\right )} \log \left (e x + d\right )}{60 \, {\left (e^{9} x + d e^{8}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 542, normalized size = 2.91 \begin {gather*} \frac {1}{60} \, {\left (10 \, b^{7} - \frac {84 \, {\left (b^{7} d e - a b^{6} e^{2}\right )} e^{\left (-1\right )}}{x e + d} + \frac {315 \, {\left (b^{7} d^{2} e^{2} - 2 \, a b^{6} d e^{3} + a^{2} b^{5} e^{4}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} - \frac {700 \, {\left (b^{7} d^{3} e^{3} - 3 \, a b^{6} d^{2} e^{4} + 3 \, a^{2} b^{5} d e^{5} - a^{3} b^{4} e^{6}\right )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{3}} + \frac {1050 \, {\left (b^{7} d^{4} e^{4} - 4 \, a b^{6} d^{3} e^{5} + 6 \, a^{2} b^{5} d^{2} e^{6} - 4 \, a^{3} b^{4} d e^{7} + a^{4} b^{3} e^{8}\right )} e^{\left (-4\right )}}{{\left (x e + d\right )}^{4}} - \frac {1260 \, {\left (b^{7} d^{5} e^{5} - 5 \, a b^{6} d^{4} e^{6} + 10 \, a^{2} b^{5} d^{3} e^{7} - 10 \, a^{3} b^{4} d^{2} e^{8} + 5 \, a^{4} b^{3} d e^{9} - a^{5} b^{2} e^{10}\right )} e^{\left (-5\right )}}{{\left (x e + d\right )}^{5}}\right )} {\left (x e + d\right )}^{6} e^{\left (-8\right )} - 7 \, {\left (b^{7} d^{6} - 6 \, a b^{6} d^{5} e + 15 \, a^{2} b^{5} d^{4} e^{2} - 20 \, a^{3} b^{4} d^{3} e^{3} + 15 \, a^{4} b^{3} d^{2} e^{4} - 6 \, a^{5} b^{2} d e^{5} + a^{6} b e^{6}\right )} e^{\left (-8\right )} \log \left (\frac {{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) + {\left (\frac {b^{7} d^{7} e^{6}}{x e + d} - \frac {7 \, a b^{6} d^{6} e^{7}}{x e + d} + \frac {21 \, a^{2} b^{5} d^{5} e^{8}}{x e + d} - \frac {35 \, a^{3} b^{4} d^{4} e^{9}}{x e + d} + \frac {35 \, a^{4} b^{3} d^{3} e^{10}}{x e + d} - \frac {21 \, a^{5} b^{2} d^{2} e^{11}}{x e + d} + \frac {7 \, a^{6} b d e^{12}}{x e + d} - \frac {a^{7} e^{13}}{x e + d}\right )} e^{\left (-14\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 571, normalized size = 3.07 \begin {gather*} \frac {b^{7} x^{6}}{6 e^{2}}+\frac {7 a \,b^{6} x^{5}}{5 e^{2}}-\frac {2 b^{7} d \,x^{5}}{5 e^{3}}+\frac {21 a^{2} b^{5} x^{4}}{4 e^{2}}-\frac {7 a \,b^{6} d \,x^{4}}{2 e^{3}}+\frac {3 b^{7} d^{2} x^{4}}{4 e^{4}}+\frac {35 a^{3} b^{4} x^{3}}{3 e^{2}}-\frac {14 a^{2} b^{5} d \,x^{3}}{e^{3}}+\frac {7 a \,b^{6} d^{2} x^{3}}{e^{4}}-\frac {4 b^{7} d^{3} x^{3}}{3 e^{5}}+\frac {35 a^{4} b^{3} x^{2}}{2 e^{2}}-\frac {35 a^{3} b^{4} d \,x^{2}}{e^{3}}+\frac {63 a^{2} b^{5} d^{2} x^{2}}{2 e^{4}}-\frac {14 a \,b^{6} d^{3} x^{2}}{e^{5}}+\frac {5 b^{7} d^{4} x^{2}}{2 e^{6}}-\frac {a^{7}}{\left (e x +d \right ) e}+\frac {7 a^{6} b d}{\left (e x +d \right ) e^{2}}+\frac {7 a^{6} b \ln \left (e x +d \right )}{e^{2}}-\frac {21 a^{5} b^{2} d^{2}}{\left (e x +d \right ) e^{3}}-\frac {42 a^{5} b^{2} d \ln \left (e x +d \right )}{e^{3}}+\frac {21 a^{5} b^{2} x}{e^{2}}+\frac {35 a^{4} b^{3} d^{3}}{\left (e x +d \right ) e^{4}}+\frac {105 a^{4} b^{3} d^{2} \ln \left (e x +d \right )}{e^{4}}-\frac {70 a^{4} b^{3} d x}{e^{3}}-\frac {35 a^{3} b^{4} d^{4}}{\left (e x +d \right ) e^{5}}-\frac {140 a^{3} b^{4} d^{3} \ln \left (e x +d \right )}{e^{5}}+\frac {105 a^{3} b^{4} d^{2} x}{e^{4}}+\frac {21 a^{2} b^{5} d^{5}}{\left (e x +d \right ) e^{6}}+\frac {105 a^{2} b^{5} d^{4} \ln \left (e x +d \right )}{e^{6}}-\frac {84 a^{2} b^{5} d^{3} x}{e^{5}}-\frac {7 a \,b^{6} d^{6}}{\left (e x +d \right ) e^{7}}-\frac {42 a \,b^{6} d^{5} \ln \left (e x +d \right )}{e^{7}}+\frac {35 a \,b^{6} d^{4} x}{e^{6}}+\frac {b^{7} d^{7}}{\left (e x +d \right ) e^{8}}+\frac {7 b^{7} d^{6} \ln \left (e x +d \right )}{e^{8}}-\frac {6 b^{7} d^{5} x}{e^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.79, size = 466, normalized size = 2.51 \begin {gather*} \frac {b^{7} d^{7} - 7 \, a b^{6} d^{6} e + 21 \, a^{2} b^{5} d^{5} e^{2} - 35 \, a^{3} b^{4} d^{4} e^{3} + 35 \, a^{4} b^{3} d^{3} e^{4} - 21 \, a^{5} b^{2} d^{2} e^{5} + 7 \, a^{6} b d e^{6} - a^{7} e^{7}}{e^{9} x + d e^{8}} + \frac {10 \, b^{7} e^{5} x^{6} - 12 \, {\left (2 \, b^{7} d e^{4} - 7 \, a b^{6} e^{5}\right )} x^{5} + 15 \, {\left (3 \, b^{7} d^{2} e^{3} - 14 \, a b^{6} d e^{4} + 21 \, a^{2} b^{5} e^{5}\right )} x^{4} - 20 \, {\left (4 \, b^{7} d^{3} e^{2} - 21 \, a b^{6} d^{2} e^{3} + 42 \, a^{2} b^{5} d e^{4} - 35 \, a^{3} b^{4} e^{5}\right )} x^{3} + 30 \, {\left (5 \, b^{7} d^{4} e - 28 \, a b^{6} d^{3} e^{2} + 63 \, a^{2} b^{5} d^{2} e^{3} - 70 \, a^{3} b^{4} d e^{4} + 35 \, a^{4} b^{3} e^{5}\right )} x^{2} - 60 \, {\left (6 \, b^{7} d^{5} - 35 \, a b^{6} d^{4} e + 84 \, a^{2} b^{5} d^{3} e^{2} - 105 \, a^{3} b^{4} d^{2} e^{3} + 70 \, a^{4} b^{3} d e^{4} - 21 \, a^{5} b^{2} e^{5}\right )} x}{60 \, e^{7}} + \frac {7 \, {\left (b^{7} d^{6} - 6 \, a b^{6} d^{5} e + 15 \, a^{2} b^{5} d^{4} e^{2} - 20 \, a^{3} b^{4} d^{3} e^{3} + 15 \, a^{4} b^{3} d^{2} e^{4} - 6 \, a^{5} b^{2} d e^{5} + a^{6} b e^{6}\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 = 2.00, size = 839, normalized size = 4.51 \begin {gather*} x^5\,\left (\frac {7\,a\,b^6}{5\,e^2}-\frac {2\,b^7\,d}{5\,e^3}\right )+x^2\,\left (\frac {d^2\,\left (\frac {2\,d\,\left (\frac {7\,a\,b^6}{e^2}-\frac {2\,b^7\,d}{e^3}\right )}{e}-\frac {21\,a^2\,b^5}{e^2}+\frac {b^7\,d^2}{e^4}\right )}{2\,e^2}-\frac {d\,\left (\frac {35\,a^3\,b^4}{e^2}+\frac {2\,d\,\left (\frac {2\,d\,\left (\frac {7\,a\,b^6}{e^2}-\frac {2\,b^7\,d}{e^3}\right )}{e}-\frac {21\,a^2\,b^5}{e^2}+\frac {b^7\,d^2}{e^4}\right )}{e}-\frac {d^2\,\left (\frac {7\,a\,b^6}{e^2}-\frac {2\,b^7\,d}{e^3}\right )}{e^2}\right )}{e}+\frac {35\,a^4\,b^3}{2\,e^2}\right )-x^4\,\left (\frac {d\,\left (\frac {7\,a\,b^6}{e^2}-\frac {2\,b^7\,d}{e^3}\right )}{2\,e}-\frac {21\,a^2\,b^5}{4\,e^2}+\frac {b^7\,d^2}{4\,e^4}\right )+x^3\,\left (\frac {35\,a^3\,b^4}{3\,e^2}+\frac {2\,d\,\left (\frac {2\,d\,\left (\frac {7\,a\,b^6}{e^2}-\frac {2\,b^7\,d}{e^3}\right )}{e}-\frac {21\,a^2\,b^5}{e^2}+\frac {b^7\,d^2}{e^4}\right )}{3\,e}-\frac {d^2\,\left (\frac {7\,a\,b^6}{e^2}-\frac {2\,b^7\,d}{e^3}\right )}{3\,e^2}\right )-x\,\left (\frac {d^2\,\left (\frac {35\,a^3\,b^4}{e^2}+\frac {2\,d\,\left (\frac {2\,d\,\left (\frac {7\,a\,b^6}{e^2}-\frac {2\,b^7\,d}{e^3}\right )}{e}-\frac {21\,a^2\,b^5}{e^2}+\frac {b^7\,d^2}{e^4}\right )}{e}-\frac {d^2\,\left (\frac {7\,a\,b^6}{e^2}-\frac {2\,b^7\,d}{e^3}\right )}{e^2}\right )}{e^2}-\frac {21\,a^5\,b^2}{e^2}+\frac {2\,d\,\left (\frac {d^2\,\left (\frac {2\,d\,\left (\frac {7\,a\,b^6}{e^2}-\frac {2\,b^7\,d}{e^3}\right )}{e}-\frac {21\,a^2\,b^5}{e^2}+\frac {b^7\,d^2}{e^4}\right )}{e^2}-\frac {2\,d\,\left (\frac {35\,a^3\,b^4}{e^2}+\frac {2\,d\,\left (\frac {2\,d\,\left (\frac {7\,a\,b^6}{e^2}-\frac {2\,b^7\,d}{e^3}\right )}{e}-\frac {21\,a^2\,b^5}{e^2}+\frac {b^7\,d^2}{e^4}\right )}{e}-\frac {d^2\,\left (\frac {7\,a\,b^6}{e^2}-\frac {2\,b^7\,d}{e^3}\right )}{e^2}\right )}{e}+\frac {35\,a^4\,b^3}{e^2}\right )}{e}\right )+\frac {\ln \left (d+e\,x\right )\,\left (7\,a^6\,b\,e^6-42\,a^5\,b^2\,d\,e^5+105\,a^4\,b^3\,d^2\,e^4-140\,a^3\,b^4\,d^3\,e^3+105\,a^2\,b^5\,d^4\,e^2-42\,a\,b^6\,d^5\,e+7\,b^7\,d^6\right )}{e^8}-\frac {a^7\,e^7-7\,a^6\,b\,d\,e^6+21\,a^5\,b^2\,d^2\,e^5-35\,a^4\,b^3\,d^3\,e^4+35\,a^3\,b^4\,d^4\,e^3-21\,a^2\,b^5\,d^5\,e^2+7\,a\,b^6\,d^6\,e-b^7\,d^7}{e\,\left (x\,e^8+d\,e^7\right )}+\frac {b^7\,x^6}{6\,e^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.50, size = 428, normalized size = 2.30 \begin {gather*} \frac {b^{7} x^{6}}{6 e^{2}} + \frac {7 b \left (a e - b d\right )^{6} \log {\left (d + e x \right )}}{e^{8}} + x^{5} \left (\frac {7 a b^{6}}{5 e^{2}} - \frac {2 b^{7} d}{5 e^{3}}\right ) + x^{4} \left (\frac {21 a^{2} b^{5}}{4 e^{2}} - \frac {7 a b^{6} d}{2 e^{3}} + \frac {3 b^{7} d^{2}}{4 e^{4}}\right ) + x^{3} \left (\frac {35 a^{3} b^{4}}{3 e^{2}} - \frac {14 a^{2} b^{5} d}{e^{3}} + \frac {7 a b^{6} d^{2}}{e^{4}} - \frac {4 b^{7} d^{3}}{3 e^{5}}\right ) + x^{2} \left (\frac {35 a^{4} b^{3}}{2 e^{2}} - \frac {35 a^{3} b^{4} d}{e^{3}} + \frac {63 a^{2} b^{5} d^{2}}{2 e^{4}} - \frac {14 a b^{6} d^{3}}{e^{5}} + \frac {5 b^{7} d^{4}}{2 e^{6}}\right ) + x \left (\frac {21 a^{5} b^{2}}{e^{2}} - \frac {70 a^{4} b^{3} d}{e^{3}} + \frac {105 a^{3} b^{4} d^{2}}{e^{4}} - \frac {84 a^{2} b^{5} d^{3}}{e^{5}} + \frac {35 a b^{6} d^{4}}{e^{6}} - \frac {6 b^{7} d^{5}}{e^{7}}\right ) + \frac {- a^{7} e^{7} + 7 a^{6} b d e^{6} - 21 a^{5} b^{2} d^{2} e^{5} + 35 a^{4} b^{3} d^{3} e^{4} - 35 a^{3} b^{4} d^{4} e^{3} + 21 a^{2} b^{5} d^{5} e^{2} - 7 a b^{6} d^{6} e + b^{7} d^{7}}{d e^{8} + e^{9} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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