Optimal. Leaf size=49 \[ -\frac{a^2 A}{3 x^3}-\frac{a (a B+2 A b)}{2 x^2}-\frac{b (2 a B+A b)}{x}+b^2 B \log (x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0240986, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {27, 76} \[ -\frac{a^2 A}{3 x^3}-\frac{a (a B+2 A b)}{2 x^2}-\frac{b (2 a B+A b)}{x}+b^2 B \log (x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 27
Rule 76
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )}{x^4} \, dx &=\int \frac{(a+b x)^2 (A+B x)}{x^4} \, dx\\ &=\int \left (\frac{a^2 A}{x^4}+\frac{a (2 A b+a B)}{x^3}+\frac{b (A b+2 a B)}{x^2}+\frac{b^2 B}{x}\right ) \, dx\\ &=-\frac{a^2 A}{3 x^3}-\frac{a (2 A b+a B)}{2 x^2}-\frac{b (A b+2 a B)}{x}+b^2 B \log (x)\\ \end{align*}
Mathematica [A] time = 0.0254893, size = 48, normalized size = 0.98 \[ b^2 B \log (x)-\frac{a^2 (2 A+3 B x)+6 a b x (A+2 B x)+6 A b^2 x^2}{6 x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.006, size = 52, normalized size = 1.1 \begin{align*}{b}^{2}B\ln \left ( x \right ) -{\frac{A{a}^{2}}{3\,{x}^{3}}}-{\frac{Aab}{{x}^{2}}}-{\frac{B{a}^{2}}{2\,{x}^{2}}}-{\frac{A{b}^{2}}{x}}-2\,{\frac{abB}{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.991728, size = 68, normalized size = 1.39 \begin{align*} B b^{2} \log \left (x\right ) - \frac{2 \, A a^{2} + 6 \,{\left (2 \, B a b + A b^{2}\right )} x^{2} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} x}{6 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.23697, size = 122, normalized size = 2.49 \begin{align*} \frac{6 \, B b^{2} x^{3} \log \left (x\right ) - 2 \, A a^{2} - 6 \,{\left (2 \, B a b + A b^{2}\right )} x^{2} - 3 \,{\left (B a^{2} + 2 \, A a b\right )} x}{6 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.739027, size = 51, normalized size = 1.04 \begin{align*} B b^{2} \log{\left (x \right )} - \frac{2 A a^{2} + x^{2} \left (6 A b^{2} + 12 B a b\right ) + x \left (6 A a b + 3 B a^{2}\right )}{6 x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.13386, size = 69, normalized size = 1.41 \begin{align*} B b^{2} \log \left ({\left | x \right |}\right ) - \frac{2 \, A a^{2} + 6 \,{\left (2 \, B a b + A b^{2}\right )} x^{2} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} x}{6 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]