\(\int x^2 (A+B x) (a^2+2 a b x+b^2 x^2)^{5/2} \, dx\) [316]

Optimal result

Integrand size = 29, antiderivative size = 167 \[ \int x^2 (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {a^2 (A b-a B) (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{6 b^4}-\frac {a (2 A b-3 a B) (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{7 b^4}+\frac {(A b-3 a B) (a+b x)^7 \sqrt {a^2+2 a b x+b^2 x^2}}{8 b^4}+\frac {B (a+b x)^8 \sqrt {a^2+2 a b x+b^2 x^2}}{9 b^4} \] Output:

1/6*a^2*(A*b-B*a)*(b*x+a)^5*((b*x+a)^2)^(1/2)/b^4-1/7*a*(2*A*b-3*B*a)*(b*x 
+a)^6*((b*x+a)^2)^(1/2)/b^4+1/8*(A*b-3*B*a)*(b*x+a)^7*((b*x+a)^2)^(1/2)/b^ 
4+1/9*B*(b*x+a)^8*((b*x+a)^2)^(1/2)/b^4
 

Mathematica [A] (verified)

Time = 1.06 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.75 \[ \int x^2 (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {x^3 \sqrt {(a+b x)^2} \left (42 a^5 (4 A+3 B x)+126 a^4 b x (5 A+4 B x)+168 a^3 b^2 x^2 (6 A+5 B x)+120 a^2 b^3 x^3 (7 A+6 B x)+45 a b^4 x^4 (8 A+7 B x)+7 b^5 x^5 (9 A+8 B x)\right )}{504 (a+b x)} \] Input:

Integrate[x^2*(A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]
 

Output:

(x^3*Sqrt[(a + b*x)^2]*(42*a^5*(4*A + 3*B*x) + 126*a^4*b*x*(5*A + 4*B*x) + 
 168*a^3*b^2*x^2*(6*A + 5*B*x) + 120*a^2*b^3*x^3*(7*A + 6*B*x) + 45*a*b^4* 
x^4*(8*A + 7*B*x) + 7*b^5*x^5*(9*A + 8*B*x)))/(504*(a + b*x))
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.69, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, 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.

\(\displaystyle \int x^2 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} (A+B x) \, dx\)

\(\Big \downarrow \) 1187

\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int b^5 x^2 (a+b x)^5 (A+B x)dx}{b^5 (a+b x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int x^2 (a+b x)^5 (A+B x)dx}{a+b x}\)

\(\Big \downarrow \) 85

\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {B (a+b x)^8}{b^3}+\frac {(A b-3 a B) (a+b x)^7}{b^3}+\frac {a (3 a B-2 A b) (a+b x)^6}{b^3}-\frac {a^2 (a B-A b) (a+b x)^5}{b^3}\right )dx}{a+b x}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {a^2 (a+b x)^6 (A b-a B)}{6 b^4}+\frac {(a+b x)^8 (A b-3 a B)}{8 b^4}-\frac {a (a+b x)^7 (2 A b-3 a B)}{7 b^4}+\frac {B (a+b x)^9}{9 b^4}\right )}{a+b x}\)

Input:

Int[x^2*(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]*((a^2*(A*b - a*B)*(a + b*x)^6)/(6*b^4) - (a 
*(2*A*b - 3*a*B)*(a + b*x)^7)/(7*b^4) + ((A*b - 3*a*B)*(a + b*x)^8)/(8*b^4 
) + (B*(a + b*x)^9)/(9*b^4)))/(a + b*x)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 85

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])
 

rule 1187

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]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 1.05 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.84

method	result	size
gosper	\(\frac {x^{3} \left (56 B \,b^{5} x^{6}+63 A \,b^{5} x^{5}+315 B a \,b^{4} x^{5}+360 A a \,b^{4} x^{4}+720 B \,a^{2} b^{3} x^{4}+840 A \,a^{2} b^{3} x^{3}+840 B \,a^{3} b^{2} x^{3}+1008 A \,a^{3} b^{2} x^{2}+504 B \,a^{4} b \,x^{2}+630 A \,a^{4} b x +126 B \,a^{5} x +168 A \,a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{504 \left (b x +a \right )^{5}}\)	\(140\)
default	\(\frac {x^{3} \left (56 B \,b^{5} x^{6}+63 A \,b^{5} x^{5}+315 B a \,b^{4} x^{5}+360 A a \,b^{4} x^{4}+720 B \,a^{2} b^{3} x^{4}+840 A \,a^{2} b^{3} x^{3}+840 B \,a^{3} b^{2} x^{3}+1008 A \,a^{3} b^{2} x^{2}+504 B \,a^{4} b \,x^{2}+630 A \,a^{4} b x +126 B \,a^{5} x +168 A \,a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{504 \left (b x +a \right )^{5}}\)	\(140\)
orering	\(\frac {x^{3} \left (56 B \,b^{5} x^{6}+63 A \,b^{5} x^{5}+315 B a \,b^{4} x^{5}+360 A a \,b^{4} x^{4}+720 B \,a^{2} b^{3} x^{4}+840 A \,a^{2} b^{3} x^{3}+840 B \,a^{3} b^{2} x^{3}+1008 A \,a^{3} b^{2} x^{2}+504 B \,a^{4} b \,x^{2}+630 A \,a^{4} b x +126 B \,a^{5} x +168 A \,a^{5}\right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{\frac {5}{2}}}{504 \left (b x +a \right )^{5}}\)	\(149\)
risch	\(\frac {\sqrt {\left (b x +a \right )^{2}}\, B \,b^{5} x^{9}}{9 b x +9 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (A \,b^{5}+5 B a \,b^{4}\right ) x^{8}}{8 b x +8 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (5 A a \,b^{4}+10 B \,a^{2} b^{3}\right ) x^{7}}{7 b x +7 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (10 A \,a^{2} b^{3}+10 B \,a^{3} b^{2}\right ) x^{6}}{6 b x +6 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (10 a^{3} A \,b^{2}+5 B \,a^{4} b \right ) x^{5}}{5 b x +5 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (5 A \,a^{4} b +B \,a^{5}\right ) x^{4}}{4 b x +4 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, A \,a^{5} x^{3}}{3 b x +3 a}\)	\(236\)

Input:

int(x^2*(B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

1/504*x^3*(56*B*b^5*x^6+63*A*b^5*x^5+315*B*a*b^4*x^5+360*A*a*b^4*x^4+720*B 
*a^2*b^3*x^4+840*A*a^2*b^3*x^3+840*B*a^3*b^2*x^3+1008*A*a^3*b^2*x^2+504*B* 
a^4*b*x^2+630*A*a^4*b*x+126*B*a^5*x+168*A*a^5)*((b*x+a)^2)^(5/2)/(b*x+a)^5
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.71 \[ \int x^2 (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {1}{9} \, B b^{5} x^{9} + \frac {1}{3} \, A a^{5} x^{3} + \frac {1}{8} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{8} + \frac {5}{7} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{7} + \frac {5}{3} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{5} + \frac {1}{4} \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{4} \] Input:

integrate(x^2*(B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")
 

Output:

1/9*B*b^5*x^9 + 1/3*A*a^5*x^3 + 1/8*(5*B*a*b^4 + A*b^5)*x^8 + 5/7*(2*B*a^2 
*b^3 + A*a*b^4)*x^7 + 5/3*(B*a^3*b^2 + A*a^2*b^3)*x^6 + (B*a^4*b + 2*A*a^3 
*b^2)*x^5 + 1/4*(B*a^5 + 5*A*a^4*b)*x^4
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5836 vs. \(2 (121) = 242\).

Time = 0.91 (sec) , antiderivative size = 5836, normalized size of antiderivative = 34.95 \[ \int x^2 (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\text {Too large to display} \] Input:

integrate(x**2*(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**8/9 + x**7*(A*b**6 
+ 37*B*a*b**5/9)/(8*b**2) + x**6*(6*A*a*b**5 + 127*B*a**2*b**4/9 - 15*a*(A 
*b**6 + 37*B*a*b**5/9)/(8*b))/(7*b**2) + x**5*(15*A*a**2*b**4 + 20*B*a**3* 
b**3 - 7*a**2*(A*b**6 + 37*B*a*b**5/9)/(8*b**2) - 13*a*(6*A*a*b**5 + 127*B 
*a**2*b**4/9 - 15*a*(A*b**6 + 37*B*a*b**5/9)/(8*b))/(7*b))/(6*b**2) + x**4 
*(20*A*a**3*b**3 + 15*B*a**4*b**2 - 6*a**2*(6*A*a*b**5 + 127*B*a**2*b**4/9 
 - 15*a*(A*b**6 + 37*B*a*b**5/9)/(8*b))/(7*b**2) - 11*a*(15*A*a**2*b**4 + 
20*B*a**3*b**3 - 7*a**2*(A*b**6 + 37*B*a*b**5/9)/(8*b**2) - 13*a*(6*A*a*b* 
*5 + 127*B*a**2*b**4/9 - 15*a*(A*b**6 + 37*B*a*b**5/9)/(8*b))/(7*b))/(6*b) 
)/(5*b**2) + x**3*(15*A*a**4*b**2 + 6*B*a**5*b - 5*a**2*(15*A*a**2*b**4 + 
20*B*a**3*b**3 - 7*a**2*(A*b**6 + 37*B*a*b**5/9)/(8*b**2) - 13*a*(6*A*a*b* 
*5 + 127*B*a**2*b**4/9 - 15*a*(A*b**6 + 37*B*a*b**5/9)/(8*b))/(7*b))/(6*b* 
*2) - 9*a*(20*A*a**3*b**3 + 15*B*a**4*b**2 - 6*a**2*(6*A*a*b**5 + 127*B*a* 
*2*b**4/9 - 15*a*(A*b**6 + 37*B*a*b**5/9)/(8*b))/(7*b**2) - 11*a*(15*A*a** 
2*b**4 + 20*B*a**3*b**3 - 7*a**2*(A*b**6 + 37*B*a*b**5/9)/(8*b**2) - 13*a* 
(6*A*a*b**5 + 127*B*a**2*b**4/9 - 15*a*(A*b**6 + 37*B*a*b**5/9)/(8*b))/(7* 
b))/(6*b))/(5*b))/(4*b**2) + x**2*(6*A*a**5*b + B*a**6 - 4*a**2*(20*A*a**3 
*b**3 + 15*B*a**4*b**2 - 6*a**2*(6*A*a*b**5 + 127*B*a**2*b**4/9 - 15*a*(A* 
b**6 + 37*B*a*b**5/9)/(8*b))/(7*b**2) - 11*a*(15*A*a**2*b**4 + 20*B*a**3*b 
**3 - 7*a**2*(A*b**6 + 37*B*a*b**5/9)/(8*b**2) - 13*a*(6*A*a*b**5 + 127...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 241 vs. \(2 (116) = 232\).

Time = 0.03 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.44 \[ \int x^2 (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=-\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B a^{3} x}{6 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A a^{2} x}{6 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B x^{2}}{9 \, b^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B a^{4}}{6 \, b^{4}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A a^{3}}{6 \, b^{3}} - \frac {11 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B a x}{72 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A x}{8 \, b^{2}} + \frac {83 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B a^{2}}{504 \, b^{4}} - \frac {9 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A a}{56 \, b^{3}} \] Input:

integrate(x^2*(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^3*x/b^3 + 1/6*(b^2*x^2 + 2*a*b*x 
+ a^2)^(5/2)*A*a^2*x/b^2 + 1/9*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*x^2/b^2 - 
 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*B*a^4/b^4 + 1/6*(b^2*x^2 + 2*a*b*x + 
a^2)^(5/2)*A*a^3/b^3 - 11/72*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*a*x/b^3 + 1 
/8*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*x/b^2 + 83/504*(b^2*x^2 + 2*a*b*x + a 
^2)^(7/2)*B*a^2/b^4 - 9/56*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*a/b^3
 

Giac [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.32 \[ \int x^2 (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {1}{9} \, B b^{5} x^{9} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{8} \, B a b^{4} x^{8} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{8} \, A b^{5} x^{8} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{7} \, B a^{2} b^{3} x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{7} \, A a b^{4} x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{3} \, B a^{3} b^{2} x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{3} \, A a^{2} b^{3} x^{6} \mathrm {sgn}\left (b x + a\right ) + B a^{4} b x^{5} \mathrm {sgn}\left (b x + a\right ) + 2 \, A a^{3} b^{2} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{4} \, B a^{5} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{4} \, A a^{4} b x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{3} \, A a^{5} x^{3} \mathrm {sgn}\left (b x + a\right ) - \frac {{\left (B a^{9} - 3 \, A a^{8} b\right )} \mathrm {sgn}\left (b x + a\right )}{504 \, b^{4}} \] Input:

integrate(x^2*(B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")
 

Output:

1/9*B*b^5*x^9*sgn(b*x + a) + 5/8*B*a*b^4*x^8*sgn(b*x + a) + 1/8*A*b^5*x^8* 
sgn(b*x + a) + 10/7*B*a^2*b^3*x^7*sgn(b*x + a) + 5/7*A*a*b^4*x^7*sgn(b*x + 
 a) + 5/3*B*a^3*b^2*x^6*sgn(b*x + a) + 5/3*A*a^2*b^3*x^6*sgn(b*x + a) + B* 
a^4*b*x^5*sgn(b*x + a) + 2*A*a^3*b^2*x^5*sgn(b*x + a) + 1/4*B*a^5*x^4*sgn( 
b*x + a) + 5/4*A*a^4*b*x^4*sgn(b*x + a) + 1/3*A*a^5*x^3*sgn(b*x + a) - 1/5 
04*(B*a^9 - 3*A*a^8*b)*sgn(b*x + a)/b^4
 

Mupad [F(-1)]

Timed out. \[ \int x^2 (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\int x^2\,\left (A+B\,x\right )\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2} \,d x \] Input:

int(x^2*(A + B*x)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2),x)
 

Output:

int(x^2*(A + B*x)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.41 \[ \int x^2 (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {x^{3} \left (28 b^{6} x^{6}+189 a \,b^{5} x^{5}+540 a^{2} b^{4} x^{4}+840 a^{3} b^{3} x^{3}+756 a^{4} b^{2} x^{2}+378 a^{5} b x +84 a^{6}\right )}{252} \] Input:

int(x^2*(B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2),x)
 

Output:

(x**3*(84*a**6 + 378*a**5*b*x + 756*a**4*b**2*x**2 + 840*a**3*b**3*x**3 + 
540*a**2*b**4*x**4 + 189*a*b**5*x**5 + 28*b**6*x**6))/252