Integrand size = 27, antiderivative size = 121 \[ \int x (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=-\frac {a (A b-a B) (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{6 b^3}+\frac {(A b-2 a B) (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{7 b^3}+\frac {B (a+b x)^7 \sqrt {a^2+2 a b x+b^2 x^2}}{8 b^3} \] Output:
-1/6*a*(A*b-B*a)*(b*x+a)^5*((b*x+a)^2)^(1/2)/b^3+1/7*(A*b-2*B*a)*(b*x+a)^6 *((b*x+a)^2)^(1/2)/b^3+1/8*B*(b*x+a)^7*((b*x+a)^2)^(1/2)/b^3
Time = 1.06 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.03 \[ \int x (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {x^2 \sqrt {(a+b x)^2} \left (28 a^5 (3 A+2 B x)+70 a^4 b x (4 A+3 B x)+84 a^3 b^2 x^2 (5 A+4 B x)+56 a^2 b^3 x^3 (6 A+5 B x)+20 a b^4 x^4 (7 A+6 B x)+3 b^5 x^5 (8 A+7 B x)\right )}{168 (a+b x)} \] Input:
Integrate[x*(A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]
Output:
(x^2*Sqrt[(a + b*x)^2]*(28*a^5*(3*A + 2*B*x) + 70*a^4*b*x*(4*A + 3*B*x) + 84*a^3*b^2*x^2*(5*A + 4*B*x) + 56*a^2*b^3*x^3*(6*A + 5*B*x) + 20*a*b^4*x^4 *(7*A + 6*B*x) + 3*b^5*x^5*(8*A + 7*B*x)))/(168*(a + b*x))
Time = 0.42 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.74, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {1187, 27, 85, 2009}
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.
\(\displaystyle \int x \left (a^2+2 a b x+b^2 x^2\right )^{5/2} (A+B x) \, dx\) |
\(\Big \downarrow \) 1187 |
\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int b^5 x (a+b x)^5 (A+B x)dx}{b^5 (a+b x)}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int x (a+b x)^5 (A+B x)dx}{a+b x}\) |
\(\Big \downarrow \) 85 |
\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {B (a+b x)^7}{b^2}+\frac {(A b-2 a B) (a+b x)^6}{b^2}+\frac {a (a B-A b) (a+b x)^5}{b^2}\right )dx}{a+b x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {(a+b x)^7 (A b-2 a B)}{7 b^3}-\frac {a (a+b x)^6 (A b-a B)}{6 b^3}+\frac {B (a+b x)^8}{8 b^3}\right )}{a+b x}\) |
Input:
Int[x*(A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]
Output:
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-1/6*(a*(A*b - a*B)*(a + b*x)^6)/b^3 + ((A *b - 2*a*B)*(a + b*x)^7)/(7*b^3) + (B*(a + b*x)^8)/(8*b^3)))/(a + b*x)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_] : > Int[ExpandIntegrand[(a + b*x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && NeQ[b*e + a* f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x + c*x^2)^FracPart[p]/(c^ IntPart[p]*(b/2 + c*x)^(2*FracPart[p])) Int[(d + e*x)^m*(f + g*x)^n*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p]
Time = 1.06 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.16
method | result | size |
gosper | \(\frac {x^{2} \left (21 B \,b^{5} x^{6}+24 A \,b^{5} x^{5}+120 B a \,b^{4} x^{5}+140 A a \,b^{4} x^{4}+280 B \,a^{2} b^{3} x^{4}+336 A \,a^{2} b^{3} x^{3}+336 B \,a^{3} b^{2} x^{3}+420 A \,a^{3} b^{2} x^{2}+210 B \,a^{4} b \,x^{2}+280 A \,a^{4} b x +56 B \,a^{5} x +84 A \,a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{168 \left (b x +a \right )^{5}}\) | \(140\) |
default | \(\frac {x^{2} \left (21 B \,b^{5} x^{6}+24 A \,b^{5} x^{5}+120 B a \,b^{4} x^{5}+140 A a \,b^{4} x^{4}+280 B \,a^{2} b^{3} x^{4}+336 A \,a^{2} b^{3} x^{3}+336 B \,a^{3} b^{2} x^{3}+420 A \,a^{3} b^{2} x^{2}+210 B \,a^{4} b \,x^{2}+280 A \,a^{4} b x +56 B \,a^{5} x +84 A \,a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{168 \left (b x +a \right )^{5}}\) | \(140\) |
orering | \(\frac {x^{2} \left (21 B \,b^{5} x^{6}+24 A \,b^{5} x^{5}+120 B a \,b^{4} x^{5}+140 A a \,b^{4} x^{4}+280 B \,a^{2} b^{3} x^{4}+336 A \,a^{2} b^{3} x^{3}+336 B \,a^{3} b^{2} x^{3}+420 A \,a^{3} b^{2} x^{2}+210 B \,a^{4} b \,x^{2}+280 A \,a^{4} b x +56 B \,a^{5} x +84 A \,a^{5}\right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{\frac {5}{2}}}{168 \left (b x +a \right )^{5}}\) | \(149\) |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, B \,b^{5} x^{8}}{8 b x +8 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (A \,b^{5}+5 B a \,b^{4}\right ) x^{7}}{7 b x +7 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (5 A a \,b^{4}+10 B \,a^{2} b^{3}\right ) x^{6}}{6 b x +6 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (10 A \,a^{2} b^{3}+10 B \,a^{3} b^{2}\right ) x^{5}}{5 b x +5 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (10 a^{3} A \,b^{2}+5 B \,a^{4} b \right ) x^{4}}{4 b x +4 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (5 A \,a^{4} b +B \,a^{5}\right ) x^{3}}{3 b x +3 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, A \,a^{5} x^{2}}{2 b x +2 a}\) | \(236\) |
Input:
int(x*(B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2),x,method=_RETURNVERBOSE)
Output:
1/168*x^2*(21*B*b^5*x^6+24*A*b^5*x^5+120*B*a*b^4*x^5+140*A*a*b^4*x^4+280*B *a^2*b^3*x^4+336*A*a^2*b^3*x^3+336*B*a^3*b^2*x^3+420*A*a^3*b^2*x^2+210*B*a ^4*b*x^2+280*A*a^4*b*x+56*B*a^5*x+84*A*a^5)*((b*x+a)^2)^(5/2)/(b*x+a)^5
Time = 0.07 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.98 \[ \int x (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {1}{8} \, B b^{5} x^{8} + \frac {1}{2} \, A a^{5} x^{2} + \frac {1}{7} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{7} + \frac {5}{6} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{6} + 2 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{5} + \frac {5}{4} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + \frac {1}{3} \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3} \] Input:
integrate(x*(B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")
Output:
1/8*B*b^5*x^8 + 1/2*A*a^5*x^2 + 1/7*(5*B*a*b^4 + A*b^5)*x^7 + 5/6*(2*B*a^2 *b^3 + A*a*b^4)*x^6 + 2*(B*a^3*b^2 + A*a^2*b^3)*x^5 + 5/4*(B*a^4*b + 2*A*a ^3*b^2)*x^4 + 1/3*(B*a^5 + 5*A*a^4*b)*x^3
Leaf count of result is larger than twice the leaf count of optimal. 3563 vs. \(2 (85) = 170\).
Time = 1.12 (sec) , antiderivative size = 3563, normalized size of antiderivative = 29.45 \[ \int x (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\text {Too large to display} \] Input:
integrate(x*(B*x+A)*(b**2*x**2+2*a*b*x+a**2)**(5/2),x)
Output:
Piecewise((sqrt(a**2 + 2*a*b*x + b**2*x**2)*(B*b**4*x**7/8 + x**6*(A*b**6 + 33*B*a*b**5/8)/(7*b**2) + x**5*(6*A*a*b**5 + 113*B*a**2*b**4/8 - 13*a*(A *b**6 + 33*B*a*b**5/8)/(7*b))/(6*b**2) + x**4*(15*A*a**2*b**4 + 20*B*a**3* b**3 - 6*a**2*(A*b**6 + 33*B*a*b**5/8)/(7*b**2) - 11*a*(6*A*a*b**5 + 113*B *a**2*b**4/8 - 13*a*(A*b**6 + 33*B*a*b**5/8)/(7*b))/(6*b))/(5*b**2) + x**3 *(20*A*a**3*b**3 + 15*B*a**4*b**2 - 5*a**2*(6*A*a*b**5 + 113*B*a**2*b**4/8 - 13*a*(A*b**6 + 33*B*a*b**5/8)/(7*b))/(6*b**2) - 9*a*(15*A*a**2*b**4 + 2 0*B*a**3*b**3 - 6*a**2*(A*b**6 + 33*B*a*b**5/8)/(7*b**2) - 11*a*(6*A*a*b** 5 + 113*B*a**2*b**4/8 - 13*a*(A*b**6 + 33*B*a*b**5/8)/(7*b))/(6*b))/(5*b)) /(4*b**2) + x**2*(15*A*a**4*b**2 + 6*B*a**5*b - 4*a**2*(15*A*a**2*b**4 + 2 0*B*a**3*b**3 - 6*a**2*(A*b**6 + 33*B*a*b**5/8)/(7*b**2) - 11*a*(6*A*a*b** 5 + 113*B*a**2*b**4/8 - 13*a*(A*b**6 + 33*B*a*b**5/8)/(7*b))/(6*b))/(5*b** 2) - 7*a*(20*A*a**3*b**3 + 15*B*a**4*b**2 - 5*a**2*(6*A*a*b**5 + 113*B*a** 2*b**4/8 - 13*a*(A*b**6 + 33*B*a*b**5/8)/(7*b))/(6*b**2) - 9*a*(15*A*a**2* b**4 + 20*B*a**3*b**3 - 6*a**2*(A*b**6 + 33*B*a*b**5/8)/(7*b**2) - 11*a*(6 *A*a*b**5 + 113*B*a**2*b**4/8 - 13*a*(A*b**6 + 33*B*a*b**5/8)/(7*b))/(6*b) )/(5*b))/(4*b))/(3*b**2) + x*(6*A*a**5*b + B*a**6 - 3*a**2*(20*A*a**3*b**3 + 15*B*a**4*b**2 - 5*a**2*(6*A*a*b**5 + 113*B*a**2*b**4/8 - 13*a*(A*b**6 + 33*B*a*b**5/8)/(7*b))/(6*b**2) - 9*a*(15*A*a**2*b**4 + 20*B*a**3*b**3 - 6*a**2*(A*b**6 + 33*B*a*b**5/8)/(7*b**2) - 11*a*(6*A*a*b**5 + 113*B*a**...
Leaf count of result is larger than twice the leaf count of optimal. 183 vs. \(2 (83) = 166\).
Time = 0.04 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.51 \[ \int x (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B a^{2} x}{6 \, b^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A a x}{6 \, b} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B a^{3}}{6 \, b^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A a^{2}}{6 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B x}{8 \, b^{2}} - \frac {9 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B a}{56 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A}{7 \, b^{2}} \] Input:
integrate(x*(B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="maxima")
Output:
1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*B*a^2*x/b^2 - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*A*a*x/b + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*B*a^3/b^3 - 1/6* (b^2*x^2 + 2*a*b*x + a^2)^(5/2)*A*a^2/b^2 + 1/8*(b^2*x^2 + 2*a*b*x + a^2)^ (7/2)*B*x/b^2 - 9/56*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*a/b^3 + 1/7*(b^2*x^ 2 + 2*a*b*x + a^2)^(7/2)*A/b^2
Leaf count of result is larger than twice the leaf count of optimal. 221 vs. \(2 (83) = 166\).
Time = 0.24 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.83 \[ \int x (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {1}{8} \, B b^{5} x^{8} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{7} \, B a b^{4} x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{7} \, A b^{5} x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{3} \, B a^{2} b^{3} x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{6} \, A a b^{4} x^{6} \mathrm {sgn}\left (b x + a\right ) + 2 \, B a^{3} b^{2} x^{5} \mathrm {sgn}\left (b x + a\right ) + 2 \, A a^{2} b^{3} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{4} \, B a^{4} b x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, A a^{3} b^{2} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{3} \, B a^{5} x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{3} \, A a^{4} b x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, A a^{5} x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {{\left (B a^{8} - 4 \, A a^{7} b\right )} \mathrm {sgn}\left (b x + a\right )}{168 \, b^{3}} \] Input:
integrate(x*(B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")
Output:
1/8*B*b^5*x^8*sgn(b*x + a) + 5/7*B*a*b^4*x^7*sgn(b*x + a) + 1/7*A*b^5*x^7* sgn(b*x + a) + 5/3*B*a^2*b^3*x^6*sgn(b*x + a) + 5/6*A*a*b^4*x^6*sgn(b*x + a) + 2*B*a^3*b^2*x^5*sgn(b*x + a) + 2*A*a^2*b^3*x^5*sgn(b*x + a) + 5/4*B*a ^4*b*x^4*sgn(b*x + a) + 5/2*A*a^3*b^2*x^4*sgn(b*x + a) + 1/3*B*a^5*x^3*sgn (b*x + a) + 5/3*A*a^4*b*x^3*sgn(b*x + a) + 1/2*A*a^5*x^2*sgn(b*x + a) + 1/ 168*(B*a^8 - 4*A*a^7*b)*sgn(b*x + a)/b^3
Timed out. \[ \int x (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\int x\,\left (A+B\,x\right )\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2} \,d x \] Input:
int(x*(A + B*x)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2),x)
Output:
int(x*(A + B*x)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2), x)
Time = 0.28 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.56 \[ \int x (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {x^{2} \left (7 b^{6} x^{6}+48 a \,b^{5} x^{5}+140 a^{2} b^{4} x^{4}+224 a^{3} b^{3} x^{3}+210 a^{4} b^{2} x^{2}+112 a^{5} b x +28 a^{6}\right )}{56} \] Input:
int(x*(B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2),x)
Output:
(x**2*(28*a**6 + 112*a**5*b*x + 210*a**4*b**2*x**2 + 224*a**3*b**3*x**3 + 140*a**2*b**4*x**4 + 48*a*b**5*x**5 + 7*b**6*x**6))/56