Optimal. Leaf size=38 \[ \frac {a B-A b}{5 b^2 (a+b x)^5}-\frac {B}{4 b^2 (a+b x)^4} \]
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Rubi [A] time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {27, 43} \begin {gather*} -\frac {A b-a B}{5 b^2 (a+b x)^5}-\frac {B}{4 b^2 (a+b x)^4} \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 )^3} \, dx &=\int \frac {A+B x}{(a+b x)^6} \, dx\\ &=\int \left (\frac {A b-a B}{b (a+b x)^6}+\frac {B}{b (a+b x)^5}\right ) \, dx\\ &=-\frac {A b-a B}{5 b^2 (a+b x)^5}-\frac {B}{4 b^2 (a+b x)^4}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 27, normalized size = 0.71 \begin {gather*} -\frac {B (a+5 b x)+4 A b}{20 b^2 (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 {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 [B] time = 0.40, size = 72, normalized size = 1.89 \begin {gather*} -\frac {5 \, B b x + B a + 4 \, A b}{20 \, {\left (b^{7} x^{5} + 5 \, a b^{6} x^{4} + 10 \, a^{2} b^{5} x^{3} + 10 \, a^{3} b^{4} x^{2} + 5 \, a^{4} b^{3} x + a^{5} b^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 25, normalized size = 0.66 \begin {gather*} -\frac {5 \, B b x + B a + 4 \, A b}{20 \, {\left (b x + a\right )}^{5} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 35, normalized size = 0.92 \begin {gather*} -\frac {B}{4 \left (b x +a \right )^{4} b^{2}}-\frac {A b -B a}{5 \left (b x +a \right )^{5} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.69, size = 72, normalized size = 1.89 \begin {gather*} -\frac {5 \, B b x + B a + 4 \, A b}{20 \, {\left (b^{7} x^{5} + 5 \, a b^{6} x^{4} + 10 \, a^{2} b^{5} x^{3} + 10 \, a^{3} b^{4} x^{2} + 5 \, a^{4} b^{3} x + a^{5} b^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.08, size = 74, normalized size = 1.95 \begin {gather*} -\frac {\frac {4\,A\,b+B\,a}{20\,b^2}+\frac {B\,x}{4\,b}}{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 [B] time = 0.54, size = 76, normalized size = 2.00 \begin {gather*} \frac {- 4 A b - B a - 5 B b x}{20 a^{5} b^{2} + 100 a^{4} b^{3} x + 200 a^{3} b^{4} x^{2} + 200 a^{2} b^{5} x^{3} + 100 a b^{6} x^{4} + 20 b^{7} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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