Optimal. Leaf size=44 \[ \frac{(a+b x)^6 (A b-7 a B)}{42 a^2 x^6}-\frac{A (a+b x)^6}{7 a x^7} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0131699, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {78, 37} \[ \frac{(a+b x)^6 (A b-7 a B)}{42 a^2 x^6}-\frac{A (a+b x)^6}{7 a x^7} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 78
Rule 37
Rubi steps
\begin{align*} \int \frac{(a+b x)^5 (A+B x)}{x^8} \, dx &=-\frac{A (a+b x)^6}{7 a x^7}+\frac{(-A b+7 a B) \int \frac{(a+b x)^5}{x^7} \, dx}{7 a}\\ &=-\frac{A (a+b x)^6}{7 a x^7}+\frac{(A b-7 a B) (a+b x)^6}{42 a^2 x^6}\\ \end{align*}
Mathematica [B] time = 0.0273834, size = 104, normalized size = 2.36 \[ -\frac{21 a^3 b^2 x^2 (4 A+5 B x)+35 a^2 b^3 x^3 (3 A+4 B x)+7 a^4 b x (5 A+6 B x)+a^5 (6 A+7 B x)+35 a b^4 x^4 (2 A+3 B x)+21 b^5 x^5 (A+2 B x)}{42 x^7} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.006, size = 104, normalized size = 2.4 \begin{align*} -{\frac{5\,a{b}^{3} \left ( Ab+2\,Ba \right ) }{3\,{x}^{3}}}-{\frac{{a}^{3}b \left ( 2\,Ab+Ba \right ) }{{x}^{5}}}-{\frac{5\,{b}^{2}{a}^{2} \left ( Ab+Ba \right ) }{2\,{x}^{4}}}-{\frac{{b}^{4} \left ( Ab+5\,Ba \right ) }{2\,{x}^{2}}}-{\frac{{a}^{4} \left ( 5\,Ab+Ba \right ) }{6\,{x}^{6}}}-{\frac{A{a}^{5}}{7\,{x}^{7}}}-{\frac{B{b}^{5}}{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 0.997001, size = 161, normalized size = 3.66 \begin{align*} -\frac{42 \, B b^{5} x^{6} + 6 \, A a^{5} + 21 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 70 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 105 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 42 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 7 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x}{42 \, x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.55259, size = 258, normalized size = 5.86 \begin{align*} -\frac{42 \, B b^{5} x^{6} + 6 \, A a^{5} + 21 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 70 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 105 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 42 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 7 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x}{42 \, x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 4.52484, size = 126, normalized size = 2.86 \begin{align*} - \frac{6 A a^{5} + 42 B b^{5} x^{6} + x^{5} \left (21 A b^{5} + 105 B a b^{4}\right ) + x^{4} \left (70 A a b^{4} + 140 B a^{2} b^{3}\right ) + x^{3} \left (105 A a^{2} b^{3} + 105 B a^{3} b^{2}\right ) + x^{2} \left (84 A a^{3} b^{2} + 42 B a^{4} b\right ) + x \left (35 A a^{4} b + 7 B a^{5}\right )}{42 x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.19975, size = 166, normalized size = 3.77 \begin{align*} -\frac{42 \, B b^{5} x^{6} + 105 \, B a b^{4} x^{5} + 21 \, A b^{5} x^{5} + 140 \, B a^{2} b^{3} x^{4} + 70 \, A a b^{4} x^{4} + 105 \, B a^{3} b^{2} x^{3} + 105 \, A a^{2} b^{3} x^{3} + 42 \, B a^{4} b x^{2} + 84 \, A a^{3} b^{2} x^{2} + 7 \, B a^{5} x + 35 \, A a^{4} b x + 6 \, A a^{5}}{42 \, x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]