\(\int \frac {(a+b x)^5 (A+B x)}{x^8} \, dx\) [132]

Optimal result

Integrand size = 16, antiderivative size = 44 \[ \int \frac {(a+b x)^5 (A+B x)}{x^8} \, dx=-\frac {A (a+b x)^6}{7 a x^7}+\frac {(A b-7 a B) (a+b x)^6}{42 a^2 x^6} \]

[Out]

-1/7*A*(b*x+a)^6/a/x^7+1/42*(A*b-7*B*a)*(b*x+a)^6/a^2/x^6

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {79, 37} \[ \int \frac {(a+b x)^5 (A+B x)}{x^8} \, dx=\frac {(a+b x)^6 (A b-7 a B)}{42 a^2 x^6}-\frac {A (a+b x)^6}{7 a x^7} \]

[In]

Int[((a + b*x)^5*(A + B*x))/x^8,x]

[Out]

-1/7*(A*(a + b*x)^6)/(a*x^7) + ((A*b - 7*a*B)*(a + b*x)^6)/(42*a^2*x^6)

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1
) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

Rubi steps \begin{align*} \text {integral}& = -\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] (verified)

Leaf count is larger than twice the leaf count of optimal. \(104\) vs. \(2(44)=88\).

Time = 0.02 (sec) , antiderivative size = 104, normalized size of antiderivative = 2.36 \[ \int \frac {(a+b x)^5 (A+B x)}{x^8} \, dx=-\frac {21 b^5 x^5 (A+2 B x)+35 a b^4 x^4 (2 A+3 B x)+35 a^2 b^3 x^3 (3 A+4 B x)+21 a^3 b^2 x^2 (4 A+5 B x)+7 a^4 b x (5 A+6 B x)+a^5 (6 A+7 B x)}{42 x^7} \]

[In]

Integrate[((a + b*x)^5*(A + B*x))/x^8,x]

[Out]

-1/42*(21*b^5*x^5*(A + 2*B*x) + 35*a*b^4*x^4*(2*A + 3*B*x) + 35*a^2*b^3*x^3*(3*A + 4*B*x) + 21*a^3*b^2*x^2*(4*
A + 5*B*x) + 7*a^4*b*x*(5*A + 6*B*x) + a^5*(6*A + 7*B*x))/x^7

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(103\) vs. \(2(40)=80\).

Time = 0.40 (sec) , antiderivative size = 104, normalized size of antiderivative = 2.36

method	result	size
default	\(-\frac {a^{4} \left (5 A b +B a \right )}{6 x^{6}}-\frac {a^{5} A}{7 x^{7}}-\frac {5 a \,b^{3} \left (A b +2 B a \right )}{3 x^{3}}-\frac {b^{5} B}{x}-\frac {b^{4} \left (A b +5 B a \right )}{2 x^{2}}-\frac {5 a^{2} b^{2} \left (A b +B a \right )}{2 x^{4}}-\frac {a^{3} b \left (2 A b +B a \right )}{x^{5}}\)	\(104\)
norman	\(\frac {-b^{5} B \,x^{6}+\left (-\frac {1}{2} b^{5} A -\frac {5}{2} a \,b^{4} B \right ) x^{5}+\left (-\frac {5}{3} a \,b^{4} A -\frac {10}{3} a^{2} b^{3} B \right ) x^{4}+\left (-\frac {5}{2} a^{2} b^{3} A -\frac {5}{2} a^{3} b^{2} B \right ) x^{3}+\left (-2 a^{3} b^{2} A -a^{4} b B \right ) x^{2}+\left (-\frac {5}{6} a^{4} b A -\frac {1}{6} a^{5} B \right ) x -\frac {a^{5} A}{7}}{x^{7}}\)	\(120\)
risch	\(\frac {-b^{5} B \,x^{6}+\left (-\frac {1}{2} b^{5} A -\frac {5}{2} a \,b^{4} B \right ) x^{5}+\left (-\frac {5}{3} a \,b^{4} A -\frac {10}{3} a^{2} b^{3} B \right ) x^{4}+\left (-\frac {5}{2} a^{2} b^{3} A -\frac {5}{2} a^{3} b^{2} B \right ) x^{3}+\left (-2 a^{3} b^{2} A -a^{4} b B \right ) x^{2}+\left (-\frac {5}{6} a^{4} b A -\frac {1}{6} a^{5} B \right ) x -\frac {a^{5} A}{7}}{x^{7}}\)	\(120\)
gosper	\(-\frac {42 b^{5} B \,x^{6}+21 A \,b^{5} x^{5}+105 B a \,b^{4} x^{5}+70 a A \,b^{4} x^{4}+140 B \,a^{2} b^{3} x^{4}+105 a^{2} A \,b^{3} x^{3}+105 B \,a^{3} b^{2} x^{3}+84 a^{3} A \,b^{2} x^{2}+42 B \,a^{4} b \,x^{2}+35 a^{4} A b x +7 a^{5} B x +6 a^{5} A}{42 x^{7}}\)	\(124\)
parallelrisch	\(-\frac {42 b^{5} B \,x^{6}+21 A \,b^{5} x^{5}+105 B a \,b^{4} x^{5}+70 a A \,b^{4} x^{4}+140 B \,a^{2} b^{3} x^{4}+105 a^{2} A \,b^{3} x^{3}+105 B \,a^{3} b^{2} x^{3}+84 a^{3} A \,b^{2} x^{2}+42 B \,a^{4} b \,x^{2}+35 a^{4} A b x +7 a^{5} B x +6 a^{5} A}{42 x^{7}}\)	\(124\)

[In]

int((b*x+a)^5*(B*x+A)/x^8,x,method=_RETURNVERBOSE)

[Out]

-1/6*a^4*(5*A*b+B*a)/x^6-1/7*a^5*A/x^7-5/3*a*b^3*(A*b+2*B*a)/x^3-b^5*B/x-1/2*b^4*(A*b+5*B*a)/x^2-5/2*a^2*b^2*(
A*b+B*a)/x^4-a^3*b*(2*A*b+B*a)/x^5

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (41) = 82\).

Time = 0.22 (sec) , antiderivative size = 119, normalized size of antiderivative = 2.70 \[ \int \frac {(a+b x)^5 (A+B x)}{x^8} \, dx=-\frac {42 \, B b^{5} x^{6} + 6 \, A a^{5} + 21 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 70 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 105 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 42 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 7 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x}{42 \, x^{7}} \]

[In]

integrate((b*x+a)^5*(B*x+A)/x^8,x, algorithm="fricas")

[Out]

-1/42*(42*B*b^5*x^6 + 6*A*a^5 + 21*(5*B*a*b^4 + A*b^5)*x^5 + 70*(2*B*a^2*b^3 + A*a*b^4)*x^4 + 105*(B*a^3*b^2 +
A*a^2*b^3)*x^3 + 42*(B*a^4*b + 2*A*a^3*b^2)*x^2 + 7*(B*a^5 + 5*A*a^4*b)*x)/x^7

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 133 vs. \(2 (37) = 74\).

Time = 2.02 (sec) , antiderivative size = 133, normalized size of antiderivative = 3.02 \[ \int \frac {(a+b x)^5 (A+B x)}{x^8} \, dx=\frac {- 6 A a^{5} - 42 B b^{5} x^{6} + x^{5} \left (- 21 A b^{5} - 105 B a b^{4}\right ) + x^{4} \left (- 70 A a b^{4} - 140 B a^{2} b^{3}\right ) + x^{3} \left (- 105 A a^{2} b^{3} - 105 B a^{3} b^{2}\right ) + x^{2} \left (- 84 A a^{3} b^{2} - 42 B a^{4} b\right ) + x \left (- 35 A a^{4} b - 7 B a^{5}\right )}{42 x^{7}} \]

[In]

integrate((b*x+a)**5*(B*x+A)/x**8,x)

[Out]

(-6*A*a**5 - 42*B*b**5*x**6 + x**5*(-21*A*b**5 - 105*B*a*b**4) + x**4*(-70*A*a*b**4 - 140*B*a**2*b**3) + x**3*
(-105*A*a**2*b**3 - 105*B*a**3*b**2) + x**2*(-84*A*a**3*b**2 - 42*B*a**4*b) + x*(-35*A*a**4*b - 7*B*a**5))/(42
*x**7)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (41) = 82\).

Time = 0.21 (sec) , antiderivative size = 119, normalized size of antiderivative = 2.70 \[ \int \frac {(a+b x)^5 (A+B x)}{x^8} \, dx=-\frac {42 \, B b^{5} x^{6} + 6 \, A a^{5} + 21 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 70 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 105 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 42 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 7 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x}{42 \, x^{7}} \]

[In]

integrate((b*x+a)^5*(B*x+A)/x^8,x, algorithm="maxima")

[Out]

-1/42*(42*B*b^5*x^6 + 6*A*a^5 + 21*(5*B*a*b^4 + A*b^5)*x^5 + 70*(2*B*a^2*b^3 + A*a*b^4)*x^4 + 105*(B*a^3*b^2 +
A*a^2*b^3)*x^3 + 42*(B*a^4*b + 2*A*a^3*b^2)*x^2 + 7*(B*a^5 + 5*A*a^4*b)*x)/x^7

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 123 vs. \(2 (41) = 82\).

Time = 0.27 (sec) , antiderivative size = 123, normalized size of antiderivative = 2.80 \[ \int \frac {(a+b x)^5 (A+B x)}{x^8} \, dx=-\frac {42 \, B b^{5} x^{6} + 105 \, B a b^{4} x^{5} + 21 \, A b^{5} x^{5} + 140 \, B a^{2} b^{3} x^{4} + 70 \, A a b^{4} x^{4} + 105 \, B a^{3} b^{2} x^{3} + 105 \, A a^{2} b^{3} x^{3} + 42 \, B a^{4} b x^{2} + 84 \, A a^{3} b^{2} x^{2} + 7 \, B a^{5} x + 35 \, A a^{4} b x + 6 \, A a^{5}}{42 \, x^{7}} \]

[In]

integrate((b*x+a)^5*(B*x+A)/x^8,x, algorithm="giac")

[Out]

-1/42*(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)/x^7

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 118, normalized size of antiderivative = 2.68 \[ \int \frac {(a+b x)^5 (A+B x)}{x^8} \, dx=-\frac {x\,\left (\frac {B\,a^5}{6}+\frac {5\,A\,b\,a^4}{6}\right )+\frac {A\,a^5}{7}+x^2\,\left (B\,a^4\,b+2\,A\,a^3\,b^2\right )+x^4\,\left (\frac {10\,B\,a^2\,b^3}{3}+\frac {5\,A\,a\,b^4}{3}\right )+x^5\,\left (\frac {A\,b^5}{2}+\frac {5\,B\,a\,b^4}{2}\right )+x^3\,\left (\frac {5\,B\,a^3\,b^2}{2}+\frac {5\,A\,a^2\,b^3}{2}\right )+B\,b^5\,x^6}{x^7} \]

[In]

int(((A + B*x)*(a + b*x)^5)/x^8,x)

[Out]

-(x*((B*a^5)/6 + (5*A*a^4*b)/6) + (A*a^5)/7 + x^2*(2*A*a^3*b^2 + B*a^4*b) + x^4*((10*B*a^2*b^3)/3 + (5*A*a*b^4
)/3) + x^5*((A*b^5)/2 + (5*B*a*b^4)/2) + x^3*((5*A*a^2*b^3)/2 + (5*B*a^3*b^2)/2) + B*b^5*x^6)/x^7