Optimal. Leaf size=130 \[ -\frac {b^4 x (4 b d-5 a e)}{e^5}+\frac {10 b^3 (b d-a e)^2 \log (d+e x)}{e^6}+\frac {10 b^2 (b d-a e)^3}{e^6 (d+e x)}-\frac {5 b (b d-a e)^4}{2 e^6 (d+e x)^2}+\frac {(b d-a e)^5}{3 e^6 (d+e x)^3}+\frac {b^5 x^2}{2 e^4} \]
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Rubi [A] time = 0.12, antiderivative size = 130, 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 {b^4 x (4 b d-5 a e)}{e^5}+\frac {10 b^2 (b d-a e)^3}{e^6 (d+e x)}+\frac {10 b^3 (b d-a e)^2 \log (d+e x)}{e^6}-\frac {5 b (b d-a e)^4}{2 e^6 (d+e x)^2}+\frac {(b d-a e)^5}{3 e^6 (d+e x)^3}+\frac {b^5 x^2}{2 e^4} \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 )^2}{(d+e x)^4} \, dx &=\int \frac {(a+b x)^5}{(d+e x)^4} \, dx\\ &=\int \left (-\frac {b^4 (4 b d-5 a e)}{e^5}+\frac {b^5 x}{e^4}+\frac {(-b d+a e)^5}{e^5 (d+e x)^4}+\frac {5 b (b d-a e)^4}{e^5 (d+e x)^3}-\frac {10 b^2 (b d-a e)^3}{e^5 (d+e x)^2}+\frac {10 b^3 (b d-a e)^2}{e^5 (d+e x)}\right ) \, dx\\ &=-\frac {b^4 (4 b d-5 a e) x}{e^5}+\frac {b^5 x^2}{2 e^4}+\frac {(b d-a e)^5}{3 e^6 (d+e x)^3}-\frac {5 b (b d-a e)^4}{2 e^6 (d+e x)^2}+\frac {10 b^2 (b d-a e)^3}{e^6 (d+e x)}+\frac {10 b^3 (b d-a e)^2 \log (d+e x)}{e^6}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 229, normalized size = 1.76 \begin {gather*} \frac {-2 a^5 e^5-5 a^4 b e^4 (d+3 e x)-20 a^3 b^2 e^3 \left (d^2+3 d e x+3 e^2 x^2\right )+10 a^2 b^3 d e^2 \left (11 d^2+27 d e x+18 e^2 x^2\right )+10 a b^4 e \left (-13 d^4-27 d^3 e x-9 d^2 e^2 x^2+9 d e^3 x^3+3 e^4 x^4\right )+60 b^3 (d+e x)^3 (b d-a e)^2 \log (d+e x)+b^5 \left (47 d^5+81 d^4 e x-9 d^3 e^2 x^2-63 d^2 e^3 x^3-15 d e^4 x^4+3 e^5 x^5\right )}{6 e^6 (d+e x)^3} \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 )^2}{(d+e x)^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.40, size = 425, normalized size = 3.27 \begin {gather*} \frac {3 \, b^{5} e^{5} x^{5} + 47 \, b^{5} d^{5} - 130 \, a b^{4} d^{4} e + 110 \, a^{2} b^{3} d^{3} e^{2} - 20 \, a^{3} b^{2} d^{2} e^{3} - 5 \, a^{4} b d e^{4} - 2 \, a^{5} e^{5} - 15 \, {\left (b^{5} d e^{4} - 2 \, a b^{4} e^{5}\right )} x^{4} - 9 \, {\left (7 \, b^{5} d^{2} e^{3} - 10 \, a b^{4} d e^{4}\right )} x^{3} - 3 \, {\left (3 \, b^{5} d^{3} e^{2} + 30 \, a b^{4} d^{2} e^{3} - 60 \, a^{2} b^{3} d e^{4} + 20 \, a^{3} b^{2} e^{5}\right )} x^{2} + 3 \, {\left (27 \, b^{5} d^{4} e - 90 \, a b^{4} d^{3} e^{2} + 90 \, a^{2} b^{3} d^{2} e^{3} - 20 \, a^{3} b^{2} d e^{4} - 5 \, a^{4} b e^{5}\right )} x + 60 \, {\left (b^{5} d^{5} - 2 \, a b^{4} d^{4} e + a^{2} b^{3} d^{3} e^{2} + {\left (b^{5} d^{2} e^{3} - 2 \, a b^{4} d e^{4} + a^{2} b^{3} e^{5}\right )} x^{3} + 3 \, {\left (b^{5} d^{3} e^{2} - 2 \, a b^{4} d^{2} e^{3} + a^{2} b^{3} d e^{4}\right )} x^{2} + 3 \, {\left (b^{5} d^{4} e - 2 \, a b^{4} d^{3} e^{2} + a^{2} b^{3} d^{2} e^{3}\right )} x\right )} \log \left (e x + d\right )}{6 \, {\left (e^{9} x^{3} + 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x + d^{3} e^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 249, normalized size = 1.92 \begin {gather*} 10 \, {\left (b^{5} d^{2} - 2 \, a b^{4} d e + a^{2} b^{3} e^{2}\right )} e^{\left (-6\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{2} \, {\left (b^{5} x^{2} e^{4} - 8 \, b^{5} d x e^{3} + 10 \, a b^{4} x e^{4}\right )} e^{\left (-8\right )} + \frac {{\left (47 \, b^{5} d^{5} - 130 \, a b^{4} d^{4} e + 110 \, a^{2} b^{3} d^{3} e^{2} - 20 \, a^{3} b^{2} d^{2} e^{3} - 5 \, a^{4} b d e^{4} - 2 \, a^{5} e^{5} + 60 \, {\left (b^{5} d^{3} e^{2} - 3 \, a b^{4} d^{2} e^{3} + 3 \, a^{2} b^{3} d e^{4} - a^{3} b^{2} e^{5}\right )} x^{2} + 15 \, {\left (7 \, b^{5} d^{4} e - 20 \, a b^{4} d^{3} e^{2} + 18 \, a^{2} b^{3} d^{2} e^{3} - 4 \, a^{3} b^{2} d e^{4} - a^{4} b e^{5}\right )} x\right )} e^{\left (-6\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 = 361, normalized size = 2.78 \begin {gather*} -\frac {a^{5}}{3 \left (e x +d \right )^{3} e}+\frac {5 a^{4} b d}{3 \left (e x +d \right )^{3} e^{2}}-\frac {10 a^{3} b^{2} d^{2}}{3 \left (e x +d \right )^{3} e^{3}}+\frac {10 a^{2} b^{3} d^{3}}{3 \left (e x +d \right )^{3} e^{4}}-\frac {5 a \,b^{4} d^{4}}{3 \left (e x +d \right )^{3} e^{5}}+\frac {b^{5} d^{5}}{3 \left (e x +d \right )^{3} e^{6}}-\frac {5 a^{4} b}{2 \left (e x +d \right )^{2} e^{2}}+\frac {10 a^{3} b^{2} d}{\left (e x +d \right )^{2} e^{3}}-\frac {15 a^{2} b^{3} d^{2}}{\left (e x +d \right )^{2} e^{4}}+\frac {10 a \,b^{4} d^{3}}{\left (e x +d \right )^{2} e^{5}}-\frac {5 b^{5} d^{4}}{2 \left (e x +d \right )^{2} e^{6}}+\frac {b^{5} x^{2}}{2 e^{4}}-\frac {10 a^{3} b^{2}}{\left (e x +d \right ) e^{3}}+\frac {30 a^{2} b^{3} d}{\left (e x +d \right ) e^{4}}+\frac {10 a^{2} b^{3} \ln \left (e x +d \right )}{e^{4}}-\frac {30 a \,b^{4} d^{2}}{\left (e x +d \right ) e^{5}}-\frac {20 a \,b^{4} d \ln \left (e x +d \right )}{e^{5}}+\frac {5 a \,b^{4} x}{e^{4}}+\frac {10 b^{5} d^{3}}{\left (e x +d \right ) e^{6}}+\frac {10 b^{5} d^{2} \ln \left (e x +d \right )}{e^{6}}-\frac {4 b^{5} d x}{e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.76, size = 282, normalized size = 2.17 \begin {gather*} \frac {47 \, b^{5} d^{5} - 130 \, a b^{4} d^{4} e + 110 \, a^{2} b^{3} d^{3} e^{2} - 20 \, a^{3} b^{2} d^{2} e^{3} - 5 \, a^{4} b d e^{4} - 2 \, a^{5} e^{5} + 60 \, {\left (b^{5} d^{3} e^{2} - 3 \, a b^{4} d^{2} e^{3} + 3 \, a^{2} b^{3} d e^{4} - a^{3} b^{2} e^{5}\right )} x^{2} + 15 \, {\left (7 \, b^{5} d^{4} e - 20 \, a b^{4} d^{3} e^{2} + 18 \, a^{2} b^{3} d^{2} e^{3} - 4 \, a^{3} b^{2} d e^{4} - a^{4} b e^{5}\right )} x}{6 \, {\left (e^{9} x^{3} + 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x + d^{3} e^{6}\right )}} + \frac {b^{5} e x^{2} - 2 \, {\left (4 \, b^{5} d - 5 \, a b^{4} e\right )} x}{2 \, e^{5}} + \frac {10 \, {\left (b^{5} d^{2} - 2 \, a b^{4} d e + a^{2} b^{3} e^{2}\right )} \log \left (e x + d\right )}{e^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 285, normalized size = 2.19 \begin {gather*} x\,\left (\frac {5\,a\,b^4}{e^4}-\frac {4\,b^5\,d}{e^5}\right )-\frac {\frac {2\,a^5\,e^5+5\,a^4\,b\,d\,e^4+20\,a^3\,b^2\,d^2\,e^3-110\,a^2\,b^3\,d^3\,e^2+130\,a\,b^4\,d^4\,e-47\,b^5\,d^5}{6\,e}+x\,\left (\frac {5\,a^4\,b\,e^4}{2}+10\,a^3\,b^2\,d\,e^3-45\,a^2\,b^3\,d^2\,e^2+50\,a\,b^4\,d^3\,e-\frac {35\,b^5\,d^4}{2}\right )-x^2\,\left (-10\,a^3\,b^2\,e^4+30\,a^2\,b^3\,d\,e^3-30\,a\,b^4\,d^2\,e^2+10\,b^5\,d^3\,e\right )}{d^3\,e^5+3\,d^2\,e^6\,x+3\,d\,e^7\,x^2+e^8\,x^3}+\frac {b^5\,x^2}{2\,e^4}+\frac {\ln \left (d+e\,x\right )\,\left (10\,a^2\,b^3\,e^2-20\,a\,b^4\,d\,e+10\,b^5\,d^2\right )}{e^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 3.06, size = 284, normalized size = 2.18 \begin {gather*} \frac {b^{5} x^{2}}{2 e^{4}} + \frac {10 b^{3} \left (a e - b d\right )^{2} \log {\left (d + e x \right )}}{e^{6}} + x \left (\frac {5 a b^{4}}{e^{4}} - \frac {4 b^{5} d}{e^{5}}\right ) + \frac {- 2 a^{5} e^{5} - 5 a^{4} b d e^{4} - 20 a^{3} b^{2} d^{2} e^{3} + 110 a^{2} b^{3} d^{3} e^{2} - 130 a b^{4} d^{4} e + 47 b^{5} d^{5} + x^{2} \left (- 60 a^{3} b^{2} e^{5} + 180 a^{2} b^{3} d e^{4} - 180 a b^{4} d^{2} e^{3} + 60 b^{5} d^{3} e^{2}\right ) + x \left (- 15 a^{4} b e^{5} - 60 a^{3} b^{2} d e^{4} + 270 a^{2} b^{3} d^{2} e^{3} - 300 a b^{4} d^{3} e^{2} + 105 b^{5} d^{4} e\right )}{6 d^{3} e^{6} + 18 d^{2} e^{7} x + 18 d e^{8} x^{2} + 6 e^{9} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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