Integrand size = 29, antiderivative size = 75 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^8} \, dx=\frac {(A b-a B) (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{6 a^2 x^6}-\frac {A \left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{7 a^2 x^7} \] Output:
1/6*(A*b-B*a)*(b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/a^2/x^6-1/7*A*(b^2*x^2+2 *a*b*x+a^2)^(7/2)/a^2/x^7
Time = 1.05 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.63 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^8} \, dx=-\frac {\sqrt {(a+b x)^2} \left (21 b^5 x^5 (A+2 B x)+35 a b^4 x^4 (2 A+3 B x)+35 a^2 b^3 x^3 (3 A+4 B x)+21 a^3 b^2 x^2 (4 A+5 B x)+7 a^4 b x (5 A+6 B x)+a^5 (6 A+7 B x)\right )}{42 x^7 (a+b x)} \] Input:
Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/x^8,x]
Output:
-1/42*(Sqrt[(a + b*x)^2]*(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*(a + b*x))
Time = 0.38 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.03, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {1186, 1102, 27, 48}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2} (A+B x)}{x^8} \, dx\) |
\(\Big \downarrow \) 1186 |
\(\displaystyle -\frac {(A b-a B) \int \frac {\left (a^2+2 b x a+b^2 x^2\right )^{5/2}}{x^7}dx}{a}-\frac {A \left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{7 a^2 x^7}\) |
\(\Big \downarrow \) 1102 |
\(\displaystyle -\frac {\sqrt {a^2+2 a b x+b^2 x^2} (A b-a B) \int \frac {b^5 (a+b x)^5}{x^7}dx}{a b^5 (a+b x)}-\frac {A \left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{7 a^2 x^7}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\sqrt {a^2+2 a b x+b^2 x^2} (A b-a B) \int \frac {(a+b x)^5}{x^7}dx}{a (a+b x)}-\frac {A \left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{7 a^2 x^7}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle \frac {(a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2} (A b-a B)}{6 a^2 x^6}-\frac {A \left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{7 a^2 x^7}\) |
Input:
Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/x^8,x]
Output:
((A*b - a*B)*(a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*a^2*x^6) - (A*( a^2 + 2*a*b*x + b^2*x^2)^(7/2))/(7*a^2*x^7)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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] && EqQ[m + n + 2, 0] && NeQ[m, -1]
Int[((d_.) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*F racPart[p])) Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0]
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_), x_Symbol] :> Simp[-2*c*(e*f - d*g)*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(2*c*d - b*e)^2)), x] + Simp[(2*c*f - b*g)/ (2*c*d - b*e) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[ {a, b, c, d, e, f, g, m, p}, x] && EqQ[b^2 - 4*a*c, 0] && EqQ[m + 2*p + 3, 0] && NeQ[2*c*f - b*g, 0] && NeQ[2*c*d - b*e, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(135\) vs. \(2(67)=134\).
Time = 1.37 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.81
method | result | size |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-B \,b^{5} x^{6}+\left (-\frac {1}{2} A \,b^{5}-\frac {5}{2} B a \,b^{4}\right ) x^{5}+\left (-\frac {5}{3} A a \,b^{4}-\frac {10}{3} B \,a^{2} b^{3}\right ) x^{4}+\left (-\frac {5}{2} A \,a^{2} b^{3}-\frac {5}{2} B \,a^{3} b^{2}\right ) x^{3}+\left (-2 a^{3} A \,b^{2}-B \,a^{4} b \right ) x^{2}+\left (-\frac {5}{6} A \,a^{4} b -\frac {1}{6} B \,a^{5}\right ) x -\frac {A \,a^{5}}{7}\right )}{\left (b x +a \right ) x^{7}}\) | \(136\) |
gosper | \(-\frac {\left (42 B \,b^{5} x^{6}+21 A \,b^{5} x^{5}+105 B a \,b^{4} x^{5}+70 A a \,b^{4} x^{4}+140 B \,a^{2} b^{3} x^{4}+105 A \,a^{2} b^{3} x^{3}+105 B \,a^{3} b^{2} x^{3}+84 A \,a^{3} b^{2} x^{2}+42 B \,a^{4} b \,x^{2}+35 A \,a^{4} b x +7 B \,a^{5} x +6 A \,a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{42 x^{7} \left (b x +a \right )^{5}}\) | \(140\) |
default | \(-\frac {\left (42 B \,b^{5} x^{6}+21 A \,b^{5} x^{5}+105 B a \,b^{4} x^{5}+70 A a \,b^{4} x^{4}+140 B \,a^{2} b^{3} x^{4}+105 A \,a^{2} b^{3} x^{3}+105 B \,a^{3} b^{2} x^{3}+84 A \,a^{3} b^{2} x^{2}+42 B \,a^{4} b \,x^{2}+35 A \,a^{4} b x +7 B \,a^{5} x +6 A \,a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{42 x^{7} \left (b x +a \right )^{5}}\) | \(140\) |
orering | \(-\frac {\left (42 B \,b^{5} x^{6}+21 A \,b^{5} x^{5}+105 B a \,b^{4} x^{5}+70 A a \,b^{4} x^{4}+140 B \,a^{2} b^{3} x^{4}+105 A \,a^{2} b^{3} x^{3}+105 B \,a^{3} b^{2} x^{3}+84 A \,a^{3} b^{2} x^{2}+42 B \,a^{4} b \,x^{2}+35 A \,a^{4} b x +7 B \,a^{5} x +6 A \,a^{5}\right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{\frac {5}{2}}}{42 x^{7} \left (b x +a \right )^{5}}\) | \(149\) |
Input:
int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/x^8,x,method=_RETURNVERBOSE)
Output:
((b*x+a)^2)^(1/2)/(b*x+a)*(-B*b^5*x^6+(-1/2*A*b^5-5/2*B*a*b^4)*x^5+(-5/3*A *a*b^4-10/3*B*a^2*b^3)*x^4+(-5/2*A*a^2*b^3-5/2*B*a^3*b^2)*x^3+(-2*A*a^3*b^ 2-B*a^4*b)*x^2+(-5/6*A*a^4*b-1/6*B*a^5)*x-1/7*A*a^5)/x^7
Time = 0.07 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.59 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^8} \, dx=-\frac {42 \, B b^{5} x^{6} + 6 \, A a^{5} + 21 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 70 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 105 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 42 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 7 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x}{42 \, x^{7}} \] Input:
integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/x^8,x, algorithm="fricas")
Output:
-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
\[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^8} \, dx=\int \frac {\left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}{x^{8}}\, dx \] Input:
integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**(5/2)/x**8,x)
Output:
Integral((A + B*x)*((a + b*x)**2)**(5/2)/x**8, x)
Leaf count of result is larger than twice the leaf count of optimal. 435 vs. \(2 (67) = 134\).
Time = 0.04 (sec) , antiderivative size = 435, normalized size of antiderivative = 5.80 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^8} \, dx=\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b^{6}}{6 \, a^{6}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{7}}{6 \, a^{7}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b^{5}}{6 \, a^{5} x} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{6}}{6 \, a^{6} x} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{4}}{6 \, a^{6} x^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{5}}{6 \, a^{7} x^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{3}}{6 \, a^{5} x^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{4}}{6 \, a^{6} x^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{2}}{6 \, a^{4} x^{4}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{3}}{6 \, a^{5} x^{4}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b}{6 \, a^{3} x^{5}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{2}}{6 \, a^{4} x^{5}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B}{6 \, a^{2} x^{6}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b}{6 \, a^{3} x^{6}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A}{7 \, a^{2} x^{7}} \] Input:
integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/x^8,x, algorithm="maxima")
Output:
1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*B*b^6/a^6 - 1/6*(b^2*x^2 + 2*a*b*x + a ^2)^(5/2)*A*b^7/a^7 + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*B*b^5/(a^5*x) - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*A*b^6/(a^6*x) - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*b^4/(a^6*x^2) + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b^5/ (a^7*x^2) + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*b^3/(a^5*x^3) - 1/6*(b^2 *x^2 + 2*a*b*x + a^2)^(7/2)*A*b^4/(a^6*x^3) - 1/6*(b^2*x^2 + 2*a*b*x + a^2 )^(7/2)*B*b^2/(a^4*x^4) + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b^3/(a^5*x ^4) + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*b/(a^3*x^5) - 1/6*(b^2*x^2 + 2 *a*b*x + a^2)^(7/2)*A*b^2/(a^4*x^5) - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)* B/(a^2*x^6) + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b/(a^3*x^6) - 1/7*(b^2 *x^2 + 2*a*b*x + a^2)^(7/2)*A/(a^2*x^7)
Leaf count of result is larger than twice the leaf count of optimal. 221 vs. \(2 (67) = 134\).
Time = 0.22 (sec) , antiderivative size = 221, normalized size of antiderivative = 2.95 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^8} \, dx=-\frac {{\left (7 \, B a b^{6} - A b^{7}\right )} \mathrm {sgn}\left (b x + a\right )}{42 \, a^{2}} - \frac {42 \, B b^{5} x^{6} \mathrm {sgn}\left (b x + a\right ) + 105 \, B a b^{4} x^{5} \mathrm {sgn}\left (b x + a\right ) + 21 \, A b^{5} x^{5} \mathrm {sgn}\left (b x + a\right ) + 140 \, B a^{2} b^{3} x^{4} \mathrm {sgn}\left (b x + a\right ) + 70 \, A a b^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + 105 \, B a^{3} b^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + 105 \, A a^{2} b^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 42 \, B a^{4} b x^{2} \mathrm {sgn}\left (b x + a\right ) + 84 \, A a^{3} b^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 7 \, B a^{5} x \mathrm {sgn}\left (b x + a\right ) + 35 \, A a^{4} b x \mathrm {sgn}\left (b x + a\right ) + 6 \, A a^{5} \mathrm {sgn}\left (b x + a\right )}{42 \, x^{7}} \] Input:
integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/x^8,x, algorithm="giac")
Output:
-1/42*(7*B*a*b^6 - A*b^7)*sgn(b*x + a)/a^2 - 1/42*(42*B*b^5*x^6*sgn(b*x + a) + 105*B*a*b^4*x^5*sgn(b*x + a) + 21*A*b^5*x^5*sgn(b*x + a) + 140*B*a^2* b^3*x^4*sgn(b*x + a) + 70*A*a*b^4*x^4*sgn(b*x + a) + 105*B*a^3*b^2*x^3*sgn (b*x + a) + 105*A*a^2*b^3*x^3*sgn(b*x + a) + 42*B*a^4*b*x^2*sgn(b*x + a) + 84*A*a^3*b^2*x^2*sgn(b*x + a) + 7*B*a^5*x*sgn(b*x + a) + 35*A*a^4*b*x*sgn (b*x + a) + 6*A*a^5*sgn(b*x + a))/x^7
Time = 10.80 (sec) , antiderivative size = 284, normalized size of antiderivative = 3.79 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^8} \, dx=-\frac {\left (\frac {B\,a^5}{6}+\frac {5\,A\,b\,a^4}{6}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^6\,\left (a+b\,x\right )}-\frac {\left (\frac {A\,b^5}{2}+\frac {5\,B\,a\,b^4}{2}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^2\,\left (a+b\,x\right )}-\frac {A\,a^5\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{7\,x^7\,\left (a+b\,x\right )}-\frac {B\,b^5\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x\,\left (a+b\,x\right )}-\frac {5\,a\,b^3\,\left (A\,b+2\,B\,a\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{3\,x^3\,\left (a+b\,x\right )}-\frac {a^3\,b\,\left (2\,A\,b+B\,a\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^5\,\left (a+b\,x\right )}-\frac {5\,a^2\,b^2\,\left (A\,b+B\,a\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{2\,x^4\,\left (a+b\,x\right )} \] Input:
int(((A + B*x)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2))/x^8,x)
Output:
- (((B*a^5)/6 + (5*A*a^4*b)/6)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(x^6*(a + b*x)) - (((A*b^5)/2 + (5*B*a*b^4)/2)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(x^2 *(a + b*x)) - (A*a^5*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(7*x^7*(a + b*x)) - (B*b^5*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(x*(a + b*x)) - (5*a*b^3*(A*b + 2* B*a)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(3*x^3*(a + b*x)) - (a^3*b*(2*A*b + B*a)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(x^5*(a + b*x)) - (5*a^2*b^2*(A*b + B*a)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(2*x^4*(a + b*x))
Time = 0.29 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.91 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^8} \, dx=\frac {-7 b^{6} x^{6}-21 a \,b^{5} x^{5}-35 a^{2} b^{4} x^{4}-35 a^{3} b^{3} x^{3}-21 a^{4} b^{2} x^{2}-7 a^{5} b x -a^{6}}{7 x^{7}} \] Input:
int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/x^8,x)
Output:
( - a**6 - 7*a**5*b*x - 21*a**4*b**2*x**2 - 35*a**3*b**3*x**3 - 35*a**2*b* *4*x**4 - 21*a*b**5*x**5 - 7*b**6*x**6)/(7*x**7)