Optimal. Leaf size=57 \[ \frac {x^4 (4 a B+A b)}{20 a^2 b (a+b x)^4}+\frac {x^4 (A b-a B)}{5 a b (a+b x)^5} \]
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Rubi [A] time = 0.01, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {27, 78, 37} \begin {gather*} \frac {x^4 (4 a B+A b)}{20 a^2 b (a+b x)^4}+\frac {x^4 (A b-a B)}{5 a b (a+b x)^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 37
Rule 78
Rubi steps
\begin {align*} \int \frac {x^3 (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac {x^3 (A+B x)}{(a+b x)^6} \, dx\\ &=\frac {(A b-a B) x^4}{5 a b (a+b x)^5}+\frac {(A b+4 a B) \int \frac {x^3}{(a+b x)^5} \, dx}{5 a b}\\ &=\frac {(A b-a B) x^4}{5 a b (a+b x)^5}+\frac {(A b+4 a B) x^4}{20 a^2 b (a+b x)^4}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 76, normalized size = 1.33 \begin {gather*} -\frac {4 a^4 B+a^3 b (A+20 B x)+5 a^2 b^2 x (A+8 B x)+10 a b^3 x^2 (A+4 B x)+10 b^4 x^3 (A+2 B x)}{20 b^5 (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^3 (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 = 0.39, size = 139, normalized size = 2.44 \begin {gather*} -\frac {20 \, B b^{4} x^{4} + 4 \, B a^{4} + A a^{3} b + 10 \, {\left (4 \, B a b^{3} + A b^{4}\right )} x^{3} + 10 \, {\left (4 \, B a^{2} b^{2} + A a b^{3}\right )} x^{2} + 5 \, {\left (4 \, B a^{3} b + A a^{2} b^{2}\right )} x}{20 \, {\left (b^{10} x^{5} + 5 \, a b^{9} x^{4} + 10 \, a^{2} b^{8} x^{3} + 10 \, a^{3} b^{7} x^{2} + 5 \, a^{4} b^{6} x + a^{5} b^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 93, normalized size = 1.63 \begin {gather*} -\frac {20 \, B b^{4} x^{4} + 40 \, B a b^{3} x^{3} + 10 \, A b^{4} x^{3} + 40 \, B a^{2} b^{2} x^{2} + 10 \, A a b^{3} x^{2} + 20 \, B a^{3} b x + 5 \, A a^{2} b^{2} x + 4 \, B a^{4} + A a^{3} b}{20 \, {\left (b x + a\right )}^{5} b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 102, normalized size = 1.79 \begin {gather*} \frac {\left (A b -B a \right ) a^{3}}{5 \left (b x +a \right )^{5} b^{5}}-\frac {\left (3 A b -4 B a \right ) a^{2}}{4 \left (b x +a \right )^{4} b^{5}}-\frac {B}{\left (b x +a \right ) b^{5}}+\frac {\left (A b -2 B a \right ) a}{\left (b x +a \right )^{3} b^{5}}-\frac {A b -4 B a}{2 \left (b x +a \right )^{2} b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.65, size = 139, normalized size = 2.44 \begin {gather*} -\frac {20 \, B b^{4} x^{4} + 4 \, B a^{4} + A a^{3} b + 10 \, {\left (4 \, B a b^{3} + A b^{4}\right )} x^{3} + 10 \, {\left (4 \, B a^{2} b^{2} + A a b^{3}\right )} x^{2} + 5 \, {\left (4 \, B a^{3} b + A a^{2} b^{2}\right )} x}{20 \, {\left (b^{10} x^{5} + 5 \, a b^{9} x^{4} + 10 \, a^{2} b^{8} x^{3} + 10 \, a^{3} b^{7} x^{2} + 5 \, a^{4} b^{6} x + a^{5} b^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 128, normalized size = 2.25 \begin {gather*} -\frac {\frac {B\,x^4}{b}+\frac {a^3\,\left (A\,b+4\,B\,a\right )}{20\,b^5}+\frac {x^3\,\left (A\,b+4\,B\,a\right )}{2\,b^2}+\frac {a\,x^2\,\left (A\,b+4\,B\,a\right )}{2\,b^3}+\frac {a^2\,x\,\left (A\,b+4\,B\,a\right )}{4\,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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.47, size = 150, normalized size = 2.63 \begin {gather*} \frac {- A a^{3} b - 4 B a^{4} - 20 B b^{4} x^{4} + x^{3} \left (- 10 A b^{4} - 40 B a b^{3}\right ) + x^{2} \left (- 10 A a b^{3} - 40 B a^{2} b^{2}\right ) + x \left (- 5 A a^{2} b^{2} - 20 B a^{3} b\right )}{20 a^{5} b^{5} + 100 a^{4} b^{6} x + 200 a^{3} b^{7} x^{2} + 200 a^{2} b^{8} x^{3} + 100 a b^{9} x^{4} + 20 b^{10} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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