Optimal. Leaf size=87 \[ -\frac {a^2 (A b-a B)}{5 b^4 (a+b x)^5}+\frac {a (2 A b-3 a B)}{4 b^4 (a+b x)^4}-\frac {A b-3 a B}{3 b^4 (a+b x)^3}-\frac {B}{2 b^4 (a+b x)^2} \]
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Rubi [A] time = 0.07, antiderivative size = 87, 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, 77} \begin {gather*} -\frac {a^2 (A b-a B)}{5 b^4 (a+b x)^5}+\frac {a (2 A b-3 a B)}{4 b^4 (a+b x)^4}-\frac {A b-3 a B}{3 b^4 (a+b x)^3}-\frac {B}{2 b^4 (a+b x)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 77
Rubi steps
\begin {align*} \int \frac {x^2 (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac {x^2 (A+B x)}{(a+b x)^6} \, dx\\ &=\int \left (-\frac {a^2 (-A b+a B)}{b^3 (a+b x)^6}+\frac {a (-2 A b+3 a B)}{b^3 (a+b x)^5}+\frac {A b-3 a B}{b^3 (a+b x)^4}+\frac {B}{b^3 (a+b x)^3}\right ) \, dx\\ &=-\frac {a^2 (A b-a B)}{5 b^4 (a+b x)^5}+\frac {a (2 A b-3 a B)}{4 b^4 (a+b x)^4}-\frac {A b-3 a B}{3 b^4 (a+b x)^3}-\frac {B}{2 b^4 (a+b x)^2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 63, normalized size = 0.72 \begin {gather*} -\frac {3 a^3 B+a^2 b (2 A+15 B x)+10 a b^2 x (A+3 B x)+10 b^3 x^2 (2 A+3 B x)}{60 b^4 (a+b x)^5} \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 {x^2 (A+B x)}{\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 [A] time = 0.39, size = 119, normalized size = 1.37 \begin {gather*} -\frac {30 \, B b^{3} x^{3} + 3 \, B a^{3} + 2 \, A a^{2} b + 10 \, {\left (3 \, B a b^{2} + 2 \, A b^{3}\right )} x^{2} + 5 \, {\left (3 \, B a^{2} b + 2 \, A a b^{2}\right )} x}{60 \, {\left (b^{9} x^{5} + 5 \, a b^{8} x^{4} + 10 \, a^{2} b^{7} x^{3} + 10 \, a^{3} b^{6} x^{2} + 5 \, a^{4} b^{5} x + a^{5} b^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 70, normalized size = 0.80 \begin {gather*} -\frac {30 \, B b^{3} x^{3} + 30 \, B a b^{2} x^{2} + 20 \, A b^{3} x^{2} + 15 \, B a^{2} b x + 10 \, A a b^{2} x + 3 \, B a^{3} + 2 \, A a^{2} b}{60 \, {\left (b x + a\right )}^{5} b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 80, normalized size = 0.92 \begin {gather*} -\frac {\left (A b -B a \right ) a^{2}}{5 \left (b x +a \right )^{5} b^{4}}-\frac {B}{2 \left (b x +a \right )^{2} b^{4}}+\frac {\left (2 A b -3 B a \right ) a}{4 \left (b x +a \right )^{4} b^{4}}-\frac {A b -3 B a}{3 \left (b x +a \right )^{3} b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.61, size = 119, normalized size = 1.37 \begin {gather*} -\frac {30 \, B b^{3} x^{3} + 3 \, B a^{3} + 2 \, A a^{2} b + 10 \, {\left (3 \, B a b^{2} + 2 \, A b^{3}\right )} x^{2} + 5 \, {\left (3 \, B a^{2} b + 2 \, A a b^{2}\right )} x}{60 \, {\left (b^{9} x^{5} + 5 \, a b^{8} x^{4} + 10 \, a^{2} b^{7} x^{3} + 10 \, a^{3} b^{6} x^{2} + 5 \, a^{4} b^{5} x + a^{5} b^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 113, normalized size = 1.30 \begin {gather*} -\frac {\frac {B\,x^3}{2\,b}+\frac {a^2\,\left (2\,A\,b+3\,B\,a\right )}{60\,b^4}+\frac {x^2\,\left (2\,A\,b+3\,B\,a\right )}{6\,b^2}+\frac {a\,x\,\left (2\,A\,b+3\,B\,a\right )}{12\,b^3}}{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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.01, size = 126, normalized size = 1.45 \begin {gather*} \frac {- 2 A a^{2} b - 3 B a^{3} - 30 B b^{3} x^{3} + x^{2} \left (- 20 A b^{3} - 30 B a b^{2}\right ) + x \left (- 10 A a b^{2} - 15 B a^{2} b\right )}{60 a^{5} b^{4} + 300 a^{4} b^{5} x + 600 a^{3} b^{6} x^{2} + 600 a^{2} b^{7} x^{3} + 300 a b^{8} x^{4} + 60 b^{9} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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