Optimal. Leaf size=44 \[ \frac{(a+b x)^3 (A b-4 a B)}{12 a^2 x^3}-\frac{A (a+b x)^3}{4 a x^4} \]
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Rubi [A] time = 0.0113715, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {27, 78, 37} \[ \frac{(a+b x)^3 (A b-4 a B)}{12 a^2 x^3}-\frac{A (a+b x)^3}{4 a x^4} \]
Antiderivative was successfully verified.
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Rule 27
Rule 78
Rule 37
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )}{x^5} \, dx &=\int \frac{(a+b x)^2 (A+B x)}{x^5} \, dx\\ &=-\frac{A (a+b x)^3}{4 a x^4}+\frac{(-A b+4 a B) \int \frac{(a+b x)^2}{x^4} \, dx}{4 a}\\ &=-\frac{A (a+b x)^3}{4 a x^4}+\frac{(A b-4 a B) (a+b x)^3}{12 a^2 x^3}\\ \end{align*}
Mathematica [A] time = 0.0152344, size = 47, normalized size = 1.07 \[ -\frac{a^2 (3 A+4 B x)+4 a b x (2 A+3 B x)+6 b^2 x^2 (A+2 B x)}{12 x^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 48, normalized size = 1.1 \begin{align*} -{\frac{a \left ( 2\,Ab+aB \right ) }{3\,{x}^{3}}}-{\frac{b \left ( Ab+2\,aB \right ) }{2\,{x}^{2}}}-{\frac{{b}^{2}B}{x}}-{\frac{A{a}^{2}}{4\,{x}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.972463, size = 69, normalized size = 1.57 \begin{align*} -\frac{12 \, B b^{2} x^{3} + 3 \, A a^{2} + 6 \,{\left (2 \, B a b + A b^{2}\right )} x^{2} + 4 \,{\left (B a^{2} + 2 \, A a b\right )} x}{12 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.24892, size = 116, normalized size = 2.64 \begin{align*} -\frac{12 \, B b^{2} x^{3} + 3 \, A a^{2} + 6 \,{\left (2 \, B a b + A b^{2}\right )} x^{2} + 4 \,{\left (B a^{2} + 2 \, A a b\right )} x}{12 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.940999, size = 54, normalized size = 1.23 \begin{align*} - \frac{3 A a^{2} + 12 B b^{2} x^{3} + x^{2} \left (6 A b^{2} + 12 B a b\right ) + x \left (8 A a b + 4 B a^{2}\right )}{12 x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11764, size = 69, normalized size = 1.57 \begin{align*} -\frac{12 \, B b^{2} x^{3} + 12 \, B a b x^{2} + 6 \, A b^{2} x^{2} + 4 \, B a^{2} x + 8 \, A a b x + 3 \, A a^{2}}{12 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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