Optimal. Leaf size=106 \[ -\frac {a^4 B}{4 b^6 (a+b x)^4}+\frac {4 a^3 B}{3 b^6 (a+b x)^3}-\frac {3 a^2 B}{b^6 (a+b x)^2}+\frac {x^5 (A b-a B)}{5 a b (a+b x)^5}+\frac {4 a B}{b^6 (a+b x)}+\frac {B \log (a+b x)}{b^6} \]
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Rubi [A] time = 0.06, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {27, 78, 43} \[ -\frac {3 a^2 B}{b^6 (a+b x)^2}+\frac {4 a^3 B}{3 b^6 (a+b x)^3}-\frac {a^4 B}{4 b^6 (a+b x)^4}+\frac {x^5 (A b-a B)}{5 a b (a+b x)^5}+\frac {4 a B}{b^6 (a+b x)}+\frac {B \log (a+b x)}{b^6} \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rule 78
Rubi steps
\begin {align*} \int \frac {x^4 (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac {x^4 (A+B x)}{(a+b x)^6} \, dx\\ &=\frac {(A b-a B) x^5}{5 a b (a+b x)^5}+\frac {B \int \frac {x^4}{(a+b x)^5} \, dx}{b}\\ &=\frac {(A b-a B) x^5}{5 a b (a+b x)^5}+\frac {B \int \left (\frac {a^4}{b^4 (a+b x)^5}-\frac {4 a^3}{b^4 (a+b x)^4}+\frac {6 a^2}{b^4 (a+b x)^3}-\frac {4 a}{b^4 (a+b x)^2}+\frac {1}{b^4 (a+b x)}\right ) \, dx}{b}\\ &=\frac {(A b-a B) x^5}{5 a b (a+b x)^5}-\frac {a^4 B}{4 b^6 (a+b x)^4}+\frac {4 a^3 B}{3 b^6 (a+b x)^3}-\frac {3 a^2 B}{b^6 (a+b x)^2}+\frac {4 a B}{b^6 (a+b x)}+\frac {B \log (a+b x)}{b^6}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 113, normalized size = 1.07 \[ \frac {137 a^5 B+a^4 (625 b B x-12 A b)+20 a^3 b^2 x (55 B x-3 A)+60 a^2 b^3 x^2 (15 B x-2 A)+60 a b^4 x^3 (5 B x-2 A)+60 B (a+b x)^5 \log (a+b x)-60 A b^5 x^4}{60 b^6 (a+b x)^5} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.72, size = 222, normalized size = 2.09 \[ \frac {137 \, B a^{5} - 12 \, A a^{4} b + 60 \, {\left (5 \, B a b^{4} - A b^{5}\right )} x^{4} + 60 \, {\left (15 \, B a^{2} b^{3} - 2 \, A a b^{4}\right )} x^{3} + 20 \, {\left (55 \, B a^{3} b^{2} - 6 \, A a^{2} b^{3}\right )} x^{2} + 5 \, {\left (125 \, B a^{4} b - 12 \, A a^{3} b^{2}\right )} x + 60 \, {\left (B b^{5} x^{5} + 5 \, B a b^{4} x^{4} + 10 \, B a^{2} b^{3} x^{3} + 10 \, B a^{3} b^{2} x^{2} + 5 \, B a^{4} b x + B a^{5}\right )} \log \left (b x + a\right )}{60 \, {\left (b^{11} x^{5} + 5 \, a b^{10} x^{4} + 10 \, a^{2} b^{9} x^{3} + 10 \, a^{3} b^{8} x^{2} + 5 \, a^{4} b^{7} x + a^{5} b^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 124, normalized size = 1.17 \[ \frac {B \log \left ({\left | b x + a \right |}\right )}{b^{6}} + \frac {60 \, {\left (5 \, B a b^{3} - A b^{4}\right )} x^{4} + 60 \, {\left (15 \, B a^{2} b^{2} - 2 \, A a b^{3}\right )} x^{3} + 20 \, {\left (55 \, B a^{3} b - 6 \, A a^{2} b^{2}\right )} x^{2} + 5 \, {\left (125 \, B a^{4} - 12 \, A a^{3} b\right )} x + \frac {137 \, B a^{5} - 12 \, A a^{4} b}{b}}{60 \, {\left (b x + a\right )}^{5} b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 165, normalized size = 1.56 \[ -\frac {A \,a^{4}}{5 \left (b x +a \right )^{5} b^{5}}+\frac {B \,a^{5}}{5 \left (b x +a \right )^{5} b^{6}}+\frac {A \,a^{3}}{\left (b x +a \right )^{4} b^{5}}-\frac {5 B \,a^{4}}{4 \left (b x +a \right )^{4} b^{6}}-\frac {2 A \,a^{2}}{\left (b x +a \right )^{3} b^{5}}+\frac {10 B \,a^{3}}{3 \left (b x +a \right )^{3} b^{6}}+\frac {2 A a}{\left (b x +a \right )^{2} b^{5}}-\frac {5 B \,a^{2}}{\left (b x +a \right )^{2} b^{6}}-\frac {A}{\left (b x +a \right ) b^{5}}+\frac {5 B a}{\left (b x +a \right ) b^{6}}+\frac {B \ln \left (b x +a \right )}{b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.60, size = 170, normalized size = 1.60 \[ \frac {137 \, B a^{5} - 12 \, A a^{4} b + 60 \, {\left (5 \, B a b^{4} - A b^{5}\right )} x^{4} + 60 \, {\left (15 \, B a^{2} b^{3} - 2 \, A a b^{4}\right )} x^{3} + 20 \, {\left (55 \, B a^{3} b^{2} - 6 \, A a^{2} b^{3}\right )} x^{2} + 5 \, {\left (125 \, B a^{4} b - 12 \, A a^{3} b^{2}\right )} x}{60 \, {\left (b^{11} x^{5} + 5 \, a b^{10} x^{4} + 10 \, a^{2} b^{9} x^{3} + 10 \, a^{3} b^{8} x^{2} + 5 \, a^{4} b^{7} x + a^{5} b^{6}\right )}} + \frac {B \log \left (b x + a\right )}{b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.17, size = 161, normalized size = 1.52 \[ \frac {\frac {137\,B\,a^5-12\,A\,a^4\,b}{60\,b^6}+\frac {x^3\,\left (15\,B\,a^2-2\,A\,a\,b\right )}{b^3}+\frac {x\,\left (125\,B\,a^4-12\,A\,a^3\,b\right )}{12\,b^5}-\frac {x^4\,\left (A\,b-5\,B\,a\right )}{b^2}+\frac {x^2\,\left (55\,B\,a^3-6\,A\,a^2\,b\right )}{3\,b^4}}{a^5+5\,a^4\,b\,x+10\,a^3\,b^2\,x^2+10\,a^2\,b^3\,x^3+5\,a\,b^4\,x^4+b^5\,x^5}+\frac {B\,\ln \left (a+b\,x\right )}{b^6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.95, size = 172, normalized size = 1.62 \[ \frac {B \log {\left (a + b x \right )}}{b^{6}} + \frac {- 12 A a^{4} b + 137 B a^{5} + x^{4} \left (- 60 A b^{5} + 300 B a b^{4}\right ) + x^{3} \left (- 120 A a b^{4} + 900 B a^{2} b^{3}\right ) + x^{2} \left (- 120 A a^{2} b^{3} + 1100 B a^{3} b^{2}\right ) + x \left (- 60 A a^{3} b^{2} + 625 B a^{4} b\right )}{60 a^{5} b^{6} + 300 a^{4} b^{7} x + 600 a^{3} b^{8} x^{2} + 600 a^{2} b^{9} x^{3} + 300 a b^{10} x^{4} + 60 b^{11} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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