Optimal. Leaf size=177 \[ -\frac {A}{2 a^6 x^2}+\frac {6 A b-a B}{a^7 x}+\frac {b (A b-a B)}{5 a^3 (a+b x)^5}+\frac {b (3 A b-2 a B)}{4 a^4 (a+b x)^4}+\frac {b (2 A b-a B)}{a^5 (a+b x)^3}+\frac {b (5 A b-2 a B)}{a^6 (a+b x)^2}+\frac {5 b (3 A b-a B)}{a^7 (a+b x)}+\frac {3 b (7 A b-2 a B) \log (x)}{a^8}-\frac {3 b (7 A b-2 a B) \log (a+b x)}{a^8} \]
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Rubi [A]
time = 0.14, antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {27, 78}
\begin {gather*} \frac {3 b \log (x) (7 A b-2 a B)}{a^8}-\frac {3 b (7 A b-2 a B) \log (a+b x)}{a^8}+\frac {6 A b-a B}{a^7 x}+\frac {5 b (3 A b-a B)}{a^7 (a+b x)}+\frac {b (5 A b-2 a B)}{a^6 (a+b x)^2}-\frac {A}{2 a^6 x^2}+\frac {b (2 A b-a B)}{a^5 (a+b x)^3}+\frac {b (3 A b-2 a B)}{4 a^4 (a+b x)^4}+\frac {b (A b-a B)}{5 a^3 (a+b x)^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 78
Rubi steps
\begin {align*} \int \frac {A+B x}{x^3 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac {A+B x}{x^3 (a+b x)^6} \, dx\\ &=\int \left (\frac {A}{a^6 x^3}+\frac {-6 A b+a B}{a^7 x^2}-\frac {3 b (-7 A b+2 a B)}{a^8 x}+\frac {b^2 (-A b+a B)}{a^3 (a+b x)^6}+\frac {b^2 (-3 A b+2 a B)}{a^4 (a+b x)^5}+\frac {3 b^2 (-2 A b+a B)}{a^5 (a+b x)^4}+\frac {2 b^2 (-5 A b+2 a B)}{a^6 (a+b x)^3}+\frac {5 b^2 (-3 A b+a B)}{a^7 (a+b x)^2}+\frac {3 b^2 (-7 A b+2 a B)}{a^8 (a+b x)}\right ) \, dx\\ &=-\frac {A}{2 a^6 x^2}+\frac {6 A b-a B}{a^7 x}+\frac {b (A b-a B)}{5 a^3 (a+b x)^5}+\frac {b (3 A b-2 a B)}{4 a^4 (a+b x)^4}+\frac {b (2 A b-a B)}{a^5 (a+b x)^3}+\frac {b (5 A b-2 a B)}{a^6 (a+b x)^2}+\frac {5 b (3 A b-a B)}{a^7 (a+b x)}+\frac {3 b (7 A b-2 a B) \log (x)}{a^8}-\frac {3 b (7 A b-2 a B) \log (a+b x)}{a^8}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 162, normalized size = 0.92 \begin {gather*} \frac {\frac {a \left (420 A b^6 x^6+5 a^3 b^3 x^3 (539 A-188 B x)+2 a^5 b x (35 A-137 B x)+7 a^4 b^2 x^2 (137 A-110 B x)+10 a^2 b^4 x^4 (329 A-54 B x)+30 a b^5 x^5 (63 A-4 B x)-10 a^6 (A+2 B x)\right )}{x^2 (a+b x)^5}+60 b (7 A b-2 a B) \log (x)+60 b (-7 A b+2 a B) \log (a+b x)}{20 a^8} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.76, size = 172, normalized size = 0.97
method | result | size |
default | \(\frac {b \left (2 A b -B a \right )}{a^{5} \left (b x +a \right )^{3}}+\frac {b \left (3 A b -2 B a \right )}{4 a^{4} \left (b x +a \right )^{4}}+\frac {5 b \left (3 A b -B a \right )}{a^{7} \left (b x +a \right )}+\frac {b \left (5 A b -2 B a \right )}{a^{6} \left (b x +a \right )^{2}}-\frac {3 b \left (7 A b -2 B a \right ) \ln \left (b x +a \right )}{a^{8}}+\frac {b \left (A b -B a \right )}{5 a^{3} \left (b x +a \right )^{5}}-\frac {A}{2 a^{6} x^{2}}-\frac {-6 A b +B a}{a^{7} x}+\frac {3 b \left (7 A b -2 B a \right ) \ln \left (x \right )}{a^{8}}\) | \(172\) |
norman | \(\frac {-\frac {A}{2 a}+\frac {\left (7 A b -2 B a \right ) x}{2 a^{2}}-\frac {5 b \left (21 b^{2} A -6 a b B \right ) x^{3}}{a^{4}}-\frac {5 b^{2} \left (63 b^{2} A -18 a b B \right ) x^{4}}{a^{5}}-\frac {5 b^{3} \left (77 b^{2} A -22 a b B \right ) x^{5}}{a^{6}}-\frac {5 b^{4} \left (175 b^{2} A -50 a b B \right ) x^{6}}{4 a^{7}}-\frac {b^{5} \left (959 b^{2} A -274 a b B \right ) x^{7}}{20 a^{8}}}{x^{2} \left (b x +a \right )^{5}}+\frac {3 b \left (7 A b -2 B a \right ) \ln \left (x \right )}{a^{8}}-\frac {3 b \left (7 A b -2 B a \right ) \ln \left (b x +a \right )}{a^{8}}\) | \(186\) |
risch | \(\frac {\frac {3 b^{5} \left (7 A b -2 B a \right ) x^{6}}{a^{7}}+\frac {27 \left (7 A b -2 B a \right ) b^{4} x^{5}}{2 a^{6}}+\frac {47 b^{3} \left (7 A b -2 B a \right ) x^{4}}{2 a^{5}}+\frac {77 b^{2} \left (7 A b -2 B a \right ) x^{3}}{4 a^{4}}+\frac {137 b \left (7 A b -2 B a \right ) x^{2}}{20 a^{3}}+\frac {\left (7 A b -2 B a \right ) x}{2 a^{2}}-\frac {A}{2 a}}{x^{2} \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{2} \left (b x +a \right )}-\frac {21 b^{2} \ln \left (b x +a \right ) A}{a^{8}}+\frac {6 b \ln \left (b x +a \right ) B}{a^{7}}+\frac {21 b^{2} \ln \left (-x \right ) A}{a^{8}}-\frac {6 b \ln \left (-x \right ) B}{a^{7}}\) | \(203\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 242, normalized size = 1.37 \begin {gather*} -\frac {10 \, A a^{6} + 60 \, {\left (2 \, B a b^{5} - 7 \, A b^{6}\right )} x^{6} + 270 \, {\left (2 \, B a^{2} b^{4} - 7 \, A a b^{5}\right )} x^{5} + 470 \, {\left (2 \, B a^{3} b^{3} - 7 \, A a^{2} b^{4}\right )} x^{4} + 385 \, {\left (2 \, B a^{4} b^{2} - 7 \, A a^{3} b^{3}\right )} x^{3} + 137 \, {\left (2 \, B a^{5} b - 7 \, A a^{4} b^{2}\right )} x^{2} + 10 \, {\left (2 \, B a^{6} - 7 \, A a^{5} b\right )} x}{20 \, {\left (a^{7} b^{5} x^{7} + 5 \, a^{8} b^{4} x^{6} + 10 \, a^{9} b^{3} x^{5} + 10 \, a^{10} b^{2} x^{4} + 5 \, a^{11} b x^{3} + a^{12} x^{2}\right )}} + \frac {3 \, {\left (2 \, B a b - 7 \, A b^{2}\right )} \log \left (b x + a\right )}{a^{8}} - \frac {3 \, {\left (2 \, B a b - 7 \, A b^{2}\right )} \log \left (x\right )}{a^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 484 vs.
\(2 (171) = 342\).
time = 2.74, size = 484, normalized size = 2.73 \begin {gather*} -\frac {10 \, A a^{7} + 60 \, {\left (2 \, B a^{2} b^{5} - 7 \, A a b^{6}\right )} x^{6} + 270 \, {\left (2 \, B a^{3} b^{4} - 7 \, A a^{2} b^{5}\right )} x^{5} + 470 \, {\left (2 \, B a^{4} b^{3} - 7 \, A a^{3} b^{4}\right )} x^{4} + 385 \, {\left (2 \, B a^{5} b^{2} - 7 \, A a^{4} b^{3}\right )} x^{3} + 137 \, {\left (2 \, B a^{6} b - 7 \, A a^{5} b^{2}\right )} x^{2} + 10 \, {\left (2 \, B a^{7} - 7 \, A a^{6} b\right )} x - 60 \, {\left ({\left (2 \, B a b^{6} - 7 \, A b^{7}\right )} x^{7} + 5 \, {\left (2 \, B a^{2} b^{5} - 7 \, A a b^{6}\right )} x^{6} + 10 \, {\left (2 \, B a^{3} b^{4} - 7 \, A a^{2} b^{5}\right )} x^{5} + 10 \, {\left (2 \, B a^{4} b^{3} - 7 \, A a^{3} b^{4}\right )} x^{4} + 5 \, {\left (2 \, B a^{5} b^{2} - 7 \, A a^{4} b^{3}\right )} x^{3} + {\left (2 \, B a^{6} b - 7 \, A a^{5} b^{2}\right )} x^{2}\right )} \log \left (b x + a\right ) + 60 \, {\left ({\left (2 \, B a b^{6} - 7 \, A b^{7}\right )} x^{7} + 5 \, {\left (2 \, B a^{2} b^{5} - 7 \, A a b^{6}\right )} x^{6} + 10 \, {\left (2 \, B a^{3} b^{4} - 7 \, A a^{2} b^{5}\right )} x^{5} + 10 \, {\left (2 \, B a^{4} b^{3} - 7 \, A a^{3} b^{4}\right )} x^{4} + 5 \, {\left (2 \, B a^{5} b^{2} - 7 \, A a^{4} b^{3}\right )} x^{3} + {\left (2 \, B a^{6} b - 7 \, A a^{5} b^{2}\right )} x^{2}\right )} \log \left (x\right )}{20 \, {\left (a^{8} b^{5} x^{7} + 5 \, a^{9} b^{4} x^{6} + 10 \, a^{10} b^{3} x^{5} + 10 \, a^{11} b^{2} x^{4} + 5 \, a^{12} b x^{3} + a^{13} x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.61, size = 335, normalized size = 1.89 \begin {gather*} \frac {- 10 A a^{6} + x^{6} \cdot \left (420 A b^{6} - 120 B a b^{5}\right ) + x^{5} \cdot \left (1890 A a b^{5} - 540 B a^{2} b^{4}\right ) + x^{4} \cdot \left (3290 A a^{2} b^{4} - 940 B a^{3} b^{3}\right ) + x^{3} \cdot \left (2695 A a^{3} b^{3} - 770 B a^{4} b^{2}\right ) + x^{2} \cdot \left (959 A a^{4} b^{2} - 274 B a^{5} b\right ) + x \left (70 A a^{5} b - 20 B a^{6}\right )}{20 a^{12} x^{2} + 100 a^{11} b x^{3} + 200 a^{10} b^{2} x^{4} + 200 a^{9} b^{3} x^{5} + 100 a^{8} b^{4} x^{6} + 20 a^{7} b^{5} x^{7}} - \frac {3 b \left (- 7 A b + 2 B a\right ) \log {\left (x + \frac {- 21 A a b^{2} + 6 B a^{2} b - 3 a b \left (- 7 A b + 2 B a\right )}{- 42 A b^{3} + 12 B a b^{2}} \right )}}{a^{8}} + \frac {3 b \left (- 7 A b + 2 B a\right ) \log {\left (x + \frac {- 21 A a b^{2} + 6 B a^{2} b + 3 a b \left (- 7 A b + 2 B a\right )}{- 42 A b^{3} + 12 B a b^{2}} \right )}}{a^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.12, size = 205, normalized size = 1.16 \begin {gather*} -\frac {3 \, {\left (2 \, B a b - 7 \, A b^{2}\right )} \log \left ({\left | x \right |}\right )}{a^{8}} + \frac {3 \, {\left (2 \, B a b^{2} - 7 \, A b^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{8} b} - \frac {10 \, A a^{7} + 60 \, {\left (2 \, B a^{2} b^{5} - 7 \, A a b^{6}\right )} x^{6} + 270 \, {\left (2 \, B a^{3} b^{4} - 7 \, A a^{2} b^{5}\right )} x^{5} + 470 \, {\left (2 \, B a^{4} b^{3} - 7 \, A a^{3} b^{4}\right )} x^{4} + 385 \, {\left (2 \, B a^{5} b^{2} - 7 \, A a^{4} b^{3}\right )} x^{3} + 137 \, {\left (2 \, B a^{6} b - 7 \, A a^{5} b^{2}\right )} x^{2} + 10 \, {\left (2 \, B a^{7} - 7 \, A a^{6} b\right )} x}{20 \, {\left (b x + a\right )}^{5} a^{8} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.21, size = 230, normalized size = 1.30 \begin {gather*} \frac {\frac {x\,\left (7\,A\,b-2\,B\,a\right )}{2\,a^2}-\frac {A}{2\,a}+\frac {77\,b^2\,x^3\,\left (7\,A\,b-2\,B\,a\right )}{4\,a^4}+\frac {47\,b^3\,x^4\,\left (7\,A\,b-2\,B\,a\right )}{2\,a^5}+\frac {27\,b^4\,x^5\,\left (7\,A\,b-2\,B\,a\right )}{2\,a^6}+\frac {3\,b^5\,x^6\,\left (7\,A\,b-2\,B\,a\right )}{a^7}+\frac {137\,b\,x^2\,\left (7\,A\,b-2\,B\,a\right )}{20\,a^3}}{a^5\,x^2+5\,a^4\,b\,x^3+10\,a^3\,b^2\,x^4+10\,a^2\,b^3\,x^5+5\,a\,b^4\,x^6+b^5\,x^7}-\frac {6\,b\,\mathrm {atanh}\left (\frac {3\,b\,\left (7\,A\,b-2\,B\,a\right )\,\left (a+2\,b\,x\right )}{a\,\left (21\,A\,b^2-6\,B\,a\,b\right )}\right )\,\left (7\,A\,b-2\,B\,a\right )}{a^8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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