Optimal. Leaf size=134 \[ 5 a^2 b^3 x (4 a B+3 A b)+5 a^3 b^2 \log (x) (3 a B+4 A b)-\frac{a^5 (a B+6 A b)}{2 x^2}-\frac{3 a^4 b (2 a B+5 A b)}{x}-\frac{a^6 A}{3 x^3}+\frac{3}{2} a b^4 x^2 (5 a B+2 A b)+\frac{1}{3} b^5 x^3 (6 a B+A b)+\frac{1}{4} b^6 B x^4 \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0799055, antiderivative size = 134, normalized size of antiderivative = 1., 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, 76} \[ 5 a^2 b^3 x (4 a B+3 A b)+5 a^3 b^2 \log (x) (3 a B+4 A b)-\frac{a^5 (a B+6 A b)}{2 x^2}-\frac{3 a^4 b (2 a B+5 A b)}{x}-\frac{a^6 A}{3 x^3}+\frac{3}{2} a b^4 x^2 (5 a B+2 A b)+\frac{1}{3} b^5 x^3 (6 a B+A b)+\frac{1}{4} b^6 B x^4 \]
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 )^3}{x^4} \, dx &=\int \frac{(a+b x)^6 (A+B x)}{x^4} \, dx\\ &=\int \left (5 a^2 b^3 (3 A b+4 a B)+\frac{a^6 A}{x^4}+\frac{a^5 (6 A b+a B)}{x^3}+\frac{3 a^4 b (5 A b+2 a B)}{x^2}+\frac{5 a^3 b^2 (4 A b+3 a B)}{x}+3 a b^4 (2 A b+5 a B) x+b^5 (A b+6 a B) x^2+b^6 B x^3\right ) \, dx\\ &=-\frac{a^6 A}{3 x^3}-\frac{a^5 (6 A b+a B)}{2 x^2}-\frac{3 a^4 b (5 A b+2 a B)}{x}+5 a^2 b^3 (3 A b+4 a B) x+\frac{3}{2} a b^4 (2 A b+5 a B) x^2+\frac{1}{3} b^5 (A b+6 a B) x^3+\frac{1}{4} b^6 B x^4+5 a^3 b^2 (4 A b+3 a B) \log (x)\\ \end{align*}
Mathematica [A] time = 0.0486854, size = 127, normalized size = 0.95 \[ \frac{15}{2} a^2 b^4 x (2 A+B x)+5 a^3 b^2 \log (x) (3 a B+4 A b)-\frac{15 a^4 A b^2}{x}-\frac{3 a^5 b (A+2 B x)}{x^2}-\frac{a^6 (2 A+3 B x)}{6 x^3}+20 a^3 b^3 B x+a b^5 x^2 (3 A+2 B x)+\frac{1}{12} b^6 x^3 (4 A+3 B x) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.009, size = 144, normalized size = 1.1 \begin{align*}{\frac{{b}^{6}B{x}^{4}}{4}}+{\frac{A{x}^{3}{b}^{6}}{3}}+2\,B{x}^{3}a{b}^{5}+3\,A{x}^{2}a{b}^{5}+{\frac{15\,B{x}^{2}{a}^{2}{b}^{4}}{2}}+15\,A{a}^{2}{b}^{4}x+20\,B{a}^{3}{b}^{3}x+20\,A\ln \left ( x \right ){a}^{3}{b}^{3}+15\,B\ln \left ( x \right ){a}^{4}{b}^{2}-{\frac{A{a}^{6}}{3\,{x}^{3}}}-3\,{\frac{A{a}^{5}b}{{x}^{2}}}-{\frac{B{a}^{6}}{2\,{x}^{2}}}-15\,{\frac{A{a}^{4}{b}^{2}}{x}}-6\,{\frac{B{a}^{5}b}{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.11835, size = 196, normalized size = 1.46 \begin{align*} \frac{1}{4} \, B b^{6} x^{4} + \frac{1}{3} \,{\left (6 \, B a b^{5} + A b^{6}\right )} x^{3} + \frac{3}{2} \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{2} + 5 \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x + 5 \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} \log \left (x\right ) - \frac{2 \, A a^{6} + 18 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 3 \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x}{6 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.54442, size = 323, normalized size = 2.41 \begin{align*} \frac{3 \, B b^{6} x^{7} - 4 \, A a^{6} + 4 \,{\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 18 \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 60 \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 60 \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} \log \left (x\right ) - 36 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} - 6 \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x}{12 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 1.12624, size = 146, normalized size = 1.09 \begin{align*} \frac{B b^{6} x^{4}}{4} + 5 a^{3} b^{2} \left (4 A b + 3 B a\right ) \log{\left (x \right )} + x^{3} \left (\frac{A b^{6}}{3} + 2 B a b^{5}\right ) + x^{2} \left (3 A a b^{5} + \frac{15 B a^{2} b^{4}}{2}\right ) + x \left (15 A a^{2} b^{4} + 20 B a^{3} b^{3}\right ) - \frac{2 A a^{6} + x^{2} \left (90 A a^{4} b^{2} + 36 B a^{5} b\right ) + x \left (18 A a^{5} b + 3 B a^{6}\right )}{6 x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.13823, size = 196, normalized size = 1.46 \begin{align*} \frac{1}{4} \, B b^{6} x^{4} + 2 \, B a b^{5} x^{3} + \frac{1}{3} \, A b^{6} x^{3} + \frac{15}{2} \, B a^{2} b^{4} x^{2} + 3 \, A a b^{5} x^{2} + 20 \, B a^{3} b^{3} x + 15 \, A a^{2} b^{4} x + 5 \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} \log \left ({\left | x \right |}\right ) - \frac{2 \, A a^{6} + 18 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 3 \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x}{6 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]