Optimal. Leaf size=192 \[ \frac {15 b^2 e^4 \log (a+b x)}{(b d-a e)^7}-\frac {15 b^2 e^4 \log (d+e x)}{(b d-a e)^7}+\frac {10 b^2 e^3}{(a+b x) (b d-a e)^6}-\frac {3 b^2 e^2}{(a+b x)^2 (b d-a e)^5}+\frac {b^2 e}{(a+b x)^3 (b d-a e)^4}-\frac {b^2}{4 (a+b x)^4 (b d-a e)^3}+\frac {5 b e^4}{(d+e x) (b d-a e)^6}+\frac {e^4}{2 (d+e x)^2 (b d-a e)^5} \]
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Rubi [A] time = 0.19, antiderivative size = 192, 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, 44} \begin {gather*} \frac {10 b^2 e^3}{(a+b x) (b d-a e)^6}-\frac {3 b^2 e^2}{(a+b x)^2 (b d-a e)^5}+\frac {15 b^2 e^4 \log (a+b x)}{(b d-a e)^7}-\frac {15 b^2 e^4 \log (d+e x)}{(b d-a e)^7}+\frac {b^2 e}{(a+b x)^3 (b d-a e)^4}-\frac {b^2}{4 (a+b x)^4 (b d-a e)^3}+\frac {5 b e^4}{(d+e x) (b d-a e)^6}+\frac {e^4}{2 (d+e x)^2 (b d-a e)^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 44
Rubi steps
\begin {align*} \int \frac {a+b x}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac {1}{(a+b x)^5 (d+e x)^3} \, dx\\ &=\int \left (\frac {b^3}{(b d-a e)^3 (a+b x)^5}-\frac {3 b^3 e}{(b d-a e)^4 (a+b x)^4}+\frac {6 b^3 e^2}{(b d-a e)^5 (a+b x)^3}-\frac {10 b^3 e^3}{(b d-a e)^6 (a+b x)^2}+\frac {15 b^3 e^4}{(b d-a e)^7 (a+b x)}-\frac {e^5}{(b d-a e)^5 (d+e x)^3}-\frac {5 b e^5}{(b d-a e)^6 (d+e x)^2}-\frac {15 b^2 e^5}{(b d-a e)^7 (d+e x)}\right ) \, dx\\ &=-\frac {b^2}{4 (b d-a e)^3 (a+b x)^4}+\frac {b^2 e}{(b d-a e)^4 (a+b x)^3}-\frac {3 b^2 e^2}{(b d-a e)^5 (a+b x)^2}+\frac {10 b^2 e^3}{(b d-a e)^6 (a+b x)}+\frac {e^4}{2 (b d-a e)^5 (d+e x)^2}+\frac {5 b e^4}{(b d-a e)^6 (d+e x)}+\frac {15 b^2 e^4 \log (a+b x)}{(b d-a e)^7}-\frac {15 b^2 e^4 \log (d+e x)}{(b d-a e)^7}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 179, normalized size = 0.93 \begin {gather*} \frac {\frac {40 b^2 e^3 (b d-a e)}{a+b x}-\frac {12 b^2 e^2 (b d-a e)^2}{(a+b x)^2}+\frac {4 b^2 e (b d-a e)^3}{(a+b x)^3}-\frac {b^2 (b d-a e)^4}{(a+b x)^4}+60 b^2 e^4 \log (a+b x)+\frac {20 b e^4 (b d-a e)}{d+e x}+\frac {2 e^4 (b d-a e)^2}{(d+e x)^2}-60 b^2 e^4 \log (d+e x)}{4 (b d-a 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 {a+b x}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.45, size = 1565, normalized size = 8.15
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 547, normalized size = 2.85 \begin {gather*} \frac {15 \, b^{3} e^{4} \log \left ({\left | b x + a \right |}\right )}{b^{8} d^{7} - 7 \, a b^{7} d^{6} e + 21 \, a^{2} b^{6} d^{5} e^{2} - 35 \, a^{3} b^{5} d^{4} e^{3} + 35 \, a^{4} b^{4} d^{3} e^{4} - 21 \, a^{5} b^{3} d^{2} e^{5} + 7 \, a^{6} b^{2} d e^{6} - a^{7} b e^{7}} - \frac {15 \, b^{2} e^{5} \log \left ({\left | x e + d \right |}\right )}{b^{7} d^{7} e - 7 \, a b^{6} d^{6} e^{2} + 21 \, a^{2} b^{5} d^{5} e^{3} - 35 \, a^{3} b^{4} d^{4} e^{4} + 35 \, a^{4} b^{3} d^{3} e^{5} - 21 \, a^{5} b^{2} d^{2} e^{6} + 7 \, a^{6} b d e^{7} - a^{7} e^{8}} - \frac {b^{6} d^{6} - 8 \, a b^{5} d^{5} e + 30 \, a^{2} b^{4} d^{4} e^{2} - 80 \, a^{3} b^{3} d^{3} e^{3} + 35 \, a^{4} b^{2} d^{2} e^{4} + 24 \, a^{5} b d e^{5} - 2 \, a^{6} e^{6} - 60 \, {\left (b^{6} d e^{5} - a b^{5} e^{6}\right )} x^{5} - 30 \, {\left (3 \, b^{6} d^{2} e^{4} + 4 \, a b^{5} d e^{5} - 7 \, a^{2} b^{4} e^{6}\right )} x^{4} - 20 \, {\left (b^{6} d^{3} e^{3} + 15 \, a b^{5} d^{2} e^{4} - 3 \, a^{2} b^{4} d e^{5} - 13 \, a^{3} b^{3} e^{6}\right )} x^{3} + 5 \, {\left (b^{6} d^{4} e^{2} - 16 \, a b^{5} d^{3} e^{3} - 66 \, a^{2} b^{4} d^{2} e^{4} + 56 \, a^{3} b^{3} d e^{5} + 25 \, a^{4} b^{2} e^{6}\right )} x^{2} - 2 \, {\left (b^{6} d^{5} e - 10 \, a b^{5} d^{4} e^{2} + 60 \, a^{2} b^{4} d^{3} e^{3} + 50 \, a^{3} b^{3} d^{2} e^{4} - 95 \, a^{4} b^{2} d e^{5} - 6 \, a^{5} b e^{6}\right )} x}{4 \, {\left (b d - a e\right )}^{7} {\left (b x + a\right )}^{4} {\left (x e + d\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 189, normalized size = 0.98 \begin {gather*} -\frac {15 b^{2} e^{4} \ln \left (b x +a \right )}{\left (a e -b d \right )^{7}}+\frac {15 b^{2} e^{4} \ln \left (e x +d \right )}{\left (a e -b d \right )^{7}}+\frac {10 b^{2} e^{3}}{\left (a e -b d \right )^{6} \left (b x +a \right )}+\frac {5 b \,e^{4}}{\left (a e -b d \right )^{6} \left (e x +d \right )}+\frac {3 b^{2} e^{2}}{\left (a e -b d \right )^{5} \left (b x +a \right )^{2}}-\frac {e^{4}}{2 \left (a e -b d \right )^{5} \left (e x +d \right )^{2}}+\frac {b^{2} e}{\left (a e -b d \right )^{4} \left (b x +a \right )^{3}}+\frac {b^{2}}{4 \left (a e -b d \right )^{3} \left (b x +a \right )^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.27, size = 1200, normalized size = 6.25 \begin {gather*} \frac {15 \, b^{2} e^{4} \log \left (b x + a\right )}{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}} - \frac {15 \, b^{2} e^{4} \log \left (e x + d\right )}{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}} + \frac {60 \, b^{5} e^{5} x^{5} - b^{5} d^{5} + 7 \, a b^{4} d^{4} e - 23 \, a^{2} b^{3} d^{3} e^{2} + 57 \, a^{3} b^{2} d^{2} e^{3} + 22 \, a^{4} b d e^{4} - 2 \, a^{5} e^{5} + 30 \, {\left (3 \, b^{5} d e^{4} + 7 \, a b^{4} e^{5}\right )} x^{4} + 20 \, {\left (b^{5} d^{2} e^{3} + 16 \, a b^{4} d e^{4} + 13 \, a^{2} b^{3} e^{5}\right )} x^{3} - 5 \, {\left (b^{5} d^{3} e^{2} - 15 \, a b^{4} d^{2} e^{3} - 81 \, a^{2} b^{3} d e^{4} - 25 \, a^{3} b^{2} e^{5}\right )} x^{2} + 2 \, {\left (b^{5} d^{4} e - 9 \, a b^{4} d^{3} e^{2} + 51 \, a^{2} b^{3} d^{2} e^{3} + 101 \, a^{3} b^{2} d e^{4} + 6 \, a^{4} b e^{5}\right )} x}{4 \, {\left (a^{4} b^{6} d^{8} - 6 \, a^{5} b^{5} d^{7} e + 15 \, a^{6} b^{4} d^{6} e^{2} - 20 \, a^{7} b^{3} d^{5} e^{3} + 15 \, a^{8} b^{2} d^{4} e^{4} - 6 \, a^{9} b d^{3} e^{5} + a^{10} d^{2} e^{6} + {\left (b^{10} d^{6} e^{2} - 6 \, a b^{9} d^{5} e^{3} + 15 \, a^{2} b^{8} d^{4} e^{4} - 20 \, a^{3} b^{7} d^{3} e^{5} + 15 \, a^{4} b^{6} d^{2} e^{6} - 6 \, a^{5} b^{5} d e^{7} + a^{6} b^{4} e^{8}\right )} x^{6} + 2 \, {\left (b^{10} d^{7} e - 4 \, a b^{9} d^{6} e^{2} + 3 \, a^{2} b^{8} d^{5} e^{3} + 10 \, a^{3} b^{7} d^{4} e^{4} - 25 \, a^{4} b^{6} d^{3} e^{5} + 24 \, a^{5} b^{5} d^{2} e^{6} - 11 \, a^{6} b^{4} d e^{7} + 2 \, a^{7} b^{3} e^{8}\right )} x^{5} + {\left (b^{10} d^{8} + 2 \, a b^{9} d^{7} e - 27 \, a^{2} b^{8} d^{6} e^{2} + 64 \, a^{3} b^{7} d^{5} e^{3} - 55 \, a^{4} b^{6} d^{4} e^{4} - 6 \, a^{5} b^{5} d^{3} e^{5} + 43 \, a^{6} b^{4} d^{2} e^{6} - 28 \, a^{7} b^{3} d e^{7} + 6 \, a^{8} b^{2} e^{8}\right )} x^{4} + 4 \, {\left (a b^{9} d^{8} - 3 \, a^{2} b^{8} d^{7} e - 2 \, a^{3} b^{7} d^{6} e^{2} + 19 \, a^{4} b^{6} d^{5} e^{3} - 30 \, a^{5} b^{5} d^{4} e^{4} + 19 \, a^{6} b^{4} d^{3} e^{5} - 2 \, a^{7} b^{3} d^{2} e^{6} - 3 \, a^{8} b^{2} d e^{7} + a^{9} b e^{8}\right )} x^{3} + {\left (6 \, a^{2} b^{8} d^{8} - 28 \, a^{3} b^{7} d^{7} e + 43 \, a^{4} b^{6} d^{6} e^{2} - 6 \, a^{5} b^{5} d^{5} e^{3} - 55 \, a^{6} b^{4} d^{4} e^{4} + 64 \, a^{7} b^{3} d^{3} e^{5} - 27 \, a^{8} b^{2} d^{2} e^{6} + 2 \, a^{9} b d e^{7} + a^{10} e^{8}\right )} x^{2} + 2 \, {\left (2 \, a^{3} b^{7} d^{8} - 11 \, a^{4} b^{6} d^{7} e + 24 \, a^{5} b^{5} d^{6} e^{2} - 25 \, a^{6} b^{4} d^{5} e^{3} + 10 \, a^{7} b^{3} d^{4} e^{4} + 3 \, a^{8} b^{2} d^{3} e^{5} - 4 \, a^{9} b d^{2} e^{6} + a^{10} d e^{7}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.69, size = 1098, normalized size = 5.72 \begin {gather*} \frac {\frac {5\,e^3\,x^3\,\left (13\,a^2\,b^3\,e^2+16\,a\,b^4\,d\,e+b^5\,d^2\right )}{a^6\,e^6-6\,a^5\,b\,d\,e^5+15\,a^4\,b^2\,d^2\,e^4-20\,a^3\,b^3\,d^3\,e^3+15\,a^2\,b^4\,d^4\,e^2-6\,a\,b^5\,d^5\,e+b^6\,d^6}-\frac {2\,a^5\,e^5-22\,a^4\,b\,d\,e^4-57\,a^3\,b^2\,d^2\,e^3+23\,a^2\,b^3\,d^3\,e^2-7\,a\,b^4\,d^4\,e+b^5\,d^5}{4\,\left (a^6\,e^6-6\,a^5\,b\,d\,e^5+15\,a^4\,b^2\,d^2\,e^4-20\,a^3\,b^3\,d^3\,e^3+15\,a^2\,b^4\,d^4\,e^2-6\,a\,b^5\,d^5\,e+b^6\,d^6\right )}+\frac {5\,e^2\,x^2\,\left (25\,a^3\,b^2\,e^3+81\,a^2\,b^3\,d\,e^2+15\,a\,b^4\,d^2\,e-b^5\,d^3\right )}{4\,\left (a^6\,e^6-6\,a^5\,b\,d\,e^5+15\,a^4\,b^2\,d^2\,e^4-20\,a^3\,b^3\,d^3\,e^3+15\,a^2\,b^4\,d^4\,e^2-6\,a\,b^5\,d^5\,e+b^6\,d^6\right )}+\frac {15\,b^5\,e^5\,x^5}{a^6\,e^6-6\,a^5\,b\,d\,e^5+15\,a^4\,b^2\,d^2\,e^4-20\,a^3\,b^3\,d^3\,e^3+15\,a^2\,b^4\,d^4\,e^2-6\,a\,b^5\,d^5\,e+b^6\,d^6}+\frac {e\,x\,\left (6\,a^4\,b\,e^4+101\,a^3\,b^2\,d\,e^3+51\,a^2\,b^3\,d^2\,e^2-9\,a\,b^4\,d^3\,e+b^5\,d^4\right )}{2\,\left (a^6\,e^6-6\,a^5\,b\,d\,e^5+15\,a^4\,b^2\,d^2\,e^4-20\,a^3\,b^3\,d^3\,e^3+15\,a^2\,b^4\,d^4\,e^2-6\,a\,b^5\,d^5\,e+b^6\,d^6\right )}+\frac {15\,b\,e^3\,x^4\,\left (3\,d\,b^4\,e+7\,a\,b^3\,e^2\right )}{2\,\left (a^6\,e^6-6\,a^5\,b\,d\,e^5+15\,a^4\,b^2\,d^2\,e^4-20\,a^3\,b^3\,d^3\,e^3+15\,a^2\,b^4\,d^4\,e^2-6\,a\,b^5\,d^5\,e+b^6\,d^6\right )}}{x\,\left (2\,e\,a^4\,d+4\,b\,a^3\,d^2\right )+x^2\,\left (a^4\,e^2+8\,a^3\,b\,d\,e+6\,a^2\,b^2\,d^2\right )+x^4\,\left (6\,a^2\,b^2\,e^2+8\,a\,b^3\,d\,e+b^4\,d^2\right )+x^5\,\left (2\,d\,b^4\,e+4\,a\,b^3\,e^2\right )+x^3\,\left (4\,a^3\,b\,e^2+12\,a^2\,b^2\,d\,e+4\,a\,b^3\,d^2\right )+a^4\,d^2+b^4\,e^2\,x^6}-\frac {30\,b^2\,e^4\,\mathrm {atanh}\left (\frac {a^7\,e^7-5\,a^6\,b\,d\,e^6+9\,a^5\,b^2\,d^2\,e^5-5\,a^4\,b^3\,d^3\,e^4-5\,a^3\,b^4\,d^4\,e^3+9\,a^2\,b^5\,d^5\,e^2-5\,a\,b^6\,d^6\,e+b^7\,d^7}{{\left (a\,e-b\,d\right )}^7}+\frac {2\,b\,e\,x\,\left (a^6\,e^6-6\,a^5\,b\,d\,e^5+15\,a^4\,b^2\,d^2\,e^4-20\,a^3\,b^3\,d^3\,e^3+15\,a^2\,b^4\,d^4\,e^2-6\,a\,b^5\,d^5\,e+b^6\,d^6\right )}{{\left (a\,e-b\,d\right )}^7}\right )}{{\left (a\,e-b\,d\right )}^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 5.06, size = 1571, normalized size = 8.18
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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