Optimal. Leaf size=130 \[ -\frac {5 b^4 (d+e x)^3 (b d-a e)}{3 e^6}+\frac {5 b^3 (d+e x)^2 (b d-a e)^2}{e^6}-\frac {10 b^2 x (b d-a e)^3}{e^5}+\frac {(b d-a e)^5}{e^6 (d+e x)}+\frac {5 b (b d-a e)^4 \log (d+e x)}{e^6}+\frac {b^5 (d+e x)^4}{4 e^6} \]
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Rubi [A] time = 0.15, 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 {5 b^4 (d+e x)^3 (b d-a e)}{3 e^6}+\frac {5 b^3 (d+e x)^2 (b d-a e)^2}{e^6}-\frac {10 b^2 x (b d-a e)^3}{e^5}+\frac {(b d-a e)^5}{e^6 (d+e x)}+\frac {5 b (b d-a e)^4 \log (d+e x)}{e^6}+\frac {b^5 (d+e x)^4}{4 e^6} \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)^2} \, dx &=\int \frac {(a+b x)^5}{(d+e x)^2} \, dx\\ &=\int \left (-\frac {10 b^2 (b d-a e)^3}{e^5}+\frac {(-b d+a e)^5}{e^5 (d+e x)^2}+\frac {5 b (b d-a e)^4}{e^5 (d+e x)}+\frac {10 b^3 (b d-a e)^2 (d+e x)}{e^5}-\frac {5 b^4 (b d-a e) (d+e x)^2}{e^5}+\frac {b^5 (d+e x)^3}{e^5}\right ) \, dx\\ &=-\frac {10 b^2 (b d-a e)^3 x}{e^5}+\frac {(b d-a e)^5}{e^6 (d+e x)}+\frac {5 b^3 (b d-a e)^2 (d+e x)^2}{e^6}-\frac {5 b^4 (b d-a e) (d+e x)^3}{3 e^6}+\frac {b^5 (d+e x)^4}{4 e^6}+\frac {5 b (b d-a e)^4 \log (d+e x)}{e^6}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 228, normalized size = 1.75 \begin {gather*} \frac {-12 a^5 e^5+60 a^4 b d e^4+120 a^3 b^2 e^3 \left (-d^2+d e x+e^2 x^2\right )+60 a^2 b^3 e^2 \left (2 d^3-4 d^2 e x-3 d e^2 x^2+e^3 x^3\right )+20 a b^4 e \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 )+60 b (d+e x) (b d-a e)^4 \log (d+e x)+b^5 \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 )}{12 e^6 (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 )^2}{(d+e x)^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.40, size = 373, normalized size = 2.87 \begin {gather*} \frac {3 \, b^{5} e^{5} x^{5} + 12 \, b^{5} d^{5} - 60 \, a b^{4} d^{4} e + 120 \, a^{2} b^{3} d^{3} e^{2} - 120 \, a^{3} b^{2} d^{2} e^{3} + 60 \, a^{4} b d e^{4} - 12 \, a^{5} e^{5} - 5 \, {\left (b^{5} d e^{4} - 4 \, a b^{4} e^{5}\right )} x^{4} + 10 \, {\left (b^{5} d^{2} e^{3} - 4 \, a b^{4} d e^{4} + 6 \, a^{2} b^{3} e^{5}\right )} x^{3} - 30 \, {\left (b^{5} d^{3} e^{2} - 4 \, a b^{4} d^{2} e^{3} + 6 \, a^{2} b^{3} d e^{4} - 4 \, a^{3} b^{2} e^{5}\right )} x^{2} - 12 \, {\left (4 \, b^{5} d^{4} e - 15 \, a b^{4} d^{3} e^{2} + 20 \, a^{2} b^{3} d^{2} e^{3} - 10 \, a^{3} b^{2} d e^{4}\right )} x + 60 \, {\left (b^{5} d^{5} - 4 \, a b^{4} d^{4} e + 6 \, a^{2} b^{3} d^{3} e^{2} - 4 \, a^{3} b^{2} d^{2} e^{3} + a^{4} b d e^{4} + {\left (b^{5} d^{4} e - 4 \, a b^{4} d^{3} e^{2} + 6 \, a^{2} b^{3} d^{2} e^{3} - 4 \, a^{3} b^{2} d e^{4} + a^{4} b e^{5}\right )} x\right )} \log \left (e x + d\right )}{12 \, {\left (e^{7} x + d e^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 328, normalized size = 2.52 \begin {gather*} \frac {1}{12} \, {\left (3 \, b^{5} - \frac {20 \, {\left (b^{5} d e - a b^{4} e^{2}\right )} e^{\left (-1\right )}}{x e + d} + \frac {60 \, {\left (b^{5} d^{2} e^{2} - 2 \, a b^{4} d e^{3} + a^{2} b^{3} e^{4}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} - \frac {120 \, {\left (b^{5} d^{3} e^{3} - 3 \, a b^{4} d^{2} e^{4} + 3 \, a^{2} b^{3} d e^{5} - a^{3} b^{2} e^{6}\right )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{3}}\right )} {\left (x e + d\right )}^{4} e^{\left (-6\right )} - 5 \, {\left (b^{5} d^{4} - 4 \, a b^{4} d^{3} e + 6 \, a^{2} b^{3} d^{2} e^{2} - 4 \, a^{3} b^{2} d e^{3} + a^{4} b e^{4}\right )} e^{\left (-6\right )} \log \left (\frac {{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) + {\left (\frac {b^{5} d^{5} e^{4}}{x e + d} - \frac {5 \, a b^{4} d^{4} e^{5}}{x e + d} + \frac {10 \, a^{2} b^{3} d^{3} e^{6}}{x e + d} - \frac {10 \, a^{3} b^{2} d^{2} e^{7}}{x e + d} + \frac {5 \, a^{4} b d e^{8}}{x e + d} - \frac {a^{5} e^{9}}{x e + d}\right )} e^{\left (-10\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 326, normalized size = 2.51 \begin {gather*} \frac {b^{5} x^{4}}{4 e^{2}}+\frac {5 a \,b^{4} x^{3}}{3 e^{2}}-\frac {2 b^{5} d \,x^{3}}{3 e^{3}}+\frac {5 a^{2} b^{3} x^{2}}{e^{2}}-\frac {5 a \,b^{4} d \,x^{2}}{e^{3}}+\frac {3 b^{5} d^{2} x^{2}}{2 e^{4}}-\frac {a^{5}}{\left (e x +d \right ) e}+\frac {5 a^{4} b d}{\left (e x +d \right ) e^{2}}+\frac {5 a^{4} b \ln \left (e x +d \right )}{e^{2}}-\frac {10 a^{3} b^{2} d^{2}}{\left (e x +d \right ) e^{3}}-\frac {20 a^{3} b^{2} d \ln \left (e x +d \right )}{e^{3}}+\frac {10 a^{3} b^{2} x}{e^{2}}+\frac {10 a^{2} b^{3} d^{3}}{\left (e x +d \right ) e^{4}}+\frac {30 a^{2} b^{3} d^{2} \ln \left (e x +d \right )}{e^{4}}-\frac {20 a^{2} b^{3} d x}{e^{3}}-\frac {5 a \,b^{4} d^{4}}{\left (e x +d \right ) e^{5}}-\frac {20 a \,b^{4} d^{3} \ln \left (e x +d \right )}{e^{5}}+\frac {15 a \,b^{4} d^{2} x}{e^{4}}+\frac {b^{5} d^{5}}{\left (e x +d \right ) e^{6}}+\frac {5 b^{5} d^{4} \ln \left (e x +d \right )}{e^{6}}-\frac {4 b^{5} d^{3} x}{e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.51, size = 264, normalized size = 2.03 \begin {gather*} \frac {b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}}{e^{7} x + d e^{6}} + \frac {3 \, b^{5} e^{3} x^{4} - 4 \, {\left (2 \, b^{5} d e^{2} - 5 \, a b^{4} e^{3}\right )} x^{3} + 6 \, {\left (3 \, b^{5} d^{2} e - 10 \, a b^{4} d e^{2} + 10 \, a^{2} b^{3} e^{3}\right )} x^{2} - 12 \, {\left (4 \, b^{5} d^{3} - 15 \, a b^{4} d^{2} e + 20 \, a^{2} b^{3} d e^{2} - 10 \, a^{3} b^{2} e^{3}\right )} x}{12 \, e^{5}} + \frac {5 \, {\left (b^{5} d^{4} - 4 \, a b^{4} d^{3} e + 6 \, a^{2} b^{3} d^{2} e^{2} - 4 \, a^{3} b^{2} d e^{3} + a^{4} b e^{4}\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 = 2.02, size = 327, normalized size = 2.52 \begin {gather*} x^3\,\left (\frac {5\,a\,b^4}{3\,e^2}-\frac {2\,b^5\,d}{3\,e^3}\right )-x^2\,\left (\frac {d\,\left (\frac {5\,a\,b^4}{e^2}-\frac {2\,b^5\,d}{e^3}\right )}{e}-\frac {5\,a^2\,b^3}{e^2}+\frac {b^5\,d^2}{2\,e^4}\right )+x\,\left (\frac {10\,a^3\,b^2}{e^2}+\frac {2\,d\,\left (\frac {2\,d\,\left (\frac {5\,a\,b^4}{e^2}-\frac {2\,b^5\,d}{e^3}\right )}{e}-\frac {10\,a^2\,b^3}{e^2}+\frac {b^5\,d^2}{e^4}\right )}{e}-\frac {d^2\,\left (\frac {5\,a\,b^4}{e^2}-\frac {2\,b^5\,d}{e^3}\right )}{e^2}\right )+\frac {\ln \left (d+e\,x\right )\,\left (5\,a^4\,b\,e^4-20\,a^3\,b^2\,d\,e^3+30\,a^2\,b^3\,d^2\,e^2-20\,a\,b^4\,d^3\,e+5\,b^5\,d^4\right )}{e^6}-\frac {a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5}{e\,\left (x\,e^6+d\,e^5\right )}+\frac {b^5\,x^4}{4\,e^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.90, size = 231, normalized size = 1.78 \begin {gather*} \frac {b^{5} x^{4}}{4 e^{2}} + \frac {5 b \left (a e - b d\right )^{4} \log {\left (d + e x \right )}}{e^{6}} + x^{3} \left (\frac {5 a b^{4}}{3 e^{2}} - \frac {2 b^{5} d}{3 e^{3}}\right ) + x^{2} \left (\frac {5 a^{2} b^{3}}{e^{2}} - \frac {5 a b^{4} d}{e^{3}} + \frac {3 b^{5} d^{2}}{2 e^{4}}\right ) + x \left (\frac {10 a^{3} b^{2}}{e^{2}} - \frac {20 a^{2} b^{3} d}{e^{3}} + \frac {15 a b^{4} d^{2}}{e^{4}} - \frac {4 b^{5} d^{3}}{e^{5}}\right ) + \frac {- a^{5} e^{5} + 5 a^{4} b d e^{4} - 10 a^{3} b^{2} d^{2} e^{3} + 10 a^{2} b^{3} d^{3} e^{2} - 5 a b^{4} d^{4} e + b^{5} d^{5}}{d e^{6} + e^{7} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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