\(\int x^2 (a+b \sqrt {c+d x})^p \, dx\) [140]

Optimal result

Integrand size = 19, antiderivative size = 242 \[ \int x^2 \left (a+b \sqrt {c+d x}\right )^p \, dx=-\frac {2 a \left (a^2-b^2 c\right )^2 \left (a+b \sqrt {c+d x}\right )^{1+p}}{b^6 d^3 (1+p)}+\frac {2 \left (5 a^4-6 a^2 b^2 c+b^4 c^2\right ) \left (a+b \sqrt {c+d x}\right )^{2+p}}{b^6 d^3 (2+p)}-\frac {4 a \left (5 a^2-3 b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{3+p}}{b^6 d^3 (3+p)}+\frac {4 \left (5 a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{4+p}}{b^6 d^3 (4+p)}-\frac {10 a \left (a+b \sqrt {c+d x}\right )^{5+p}}{b^6 d^3 (5+p)}+\frac {2 \left (a+b \sqrt {c+d x}\right )^{6+p}}{b^6 d^3 (6+p)} \] Output:

-2*a*(-b^2*c+a^2)^2*(a+b*(d*x+c)^(1/2))^(p+1)/b^6/d^3/(p+1)+2*(b^4*c^2-6*a 
^2*b^2*c+5*a^4)*(a+b*(d*x+c)^(1/2))^(2+p)/b^6/d^3/(2+p)-4*a*(-3*b^2*c+5*a^ 
2)*(a+b*(d*x+c)^(1/2))^(3+p)/b^6/d^3/(3+p)+4*(-b^2*c+5*a^2)*(a+b*(d*x+c)^( 
1/2))^(4+p)/b^6/d^3/(4+p)-10*a*(a+b*(d*x+c)^(1/2))^(5+p)/b^6/d^3/(5+p)+2*( 
a+b*(d*x+c)^(1/2))^(6+p)/b^6/d^3/(6+p)
 

Mathematica [A] (verified)

Time = 0.55 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.17 \[ \int x^2 \left (a+b \sqrt {c+d x}\right )^p \, dx=-\frac {2 \left (a+b \sqrt {c+d x}\right )^{1+p} \left (120 a^5-120 a^4 b (1+p) \sqrt {c+d x}+12 a^3 b^2 \left (4 c \left (-5+p+p^2\right )+5 d \left (2+3 p+p^2\right ) x\right )-4 a^2 b^3 (1+p) \sqrt {c+d x} \left (2 c \left (-30-4 p+p^2\right )+5 d \left (6+5 p+p^2\right ) x\right )-b^5 \left (15+23 p+9 p^2+p^3\right ) \sqrt {c+d x} \left (8 c^2-4 c d (2+p) x+d^2 \left (8+6 p+p^2\right ) x^2\right )+a b^4 \left (-8 c^2 \left (-15+10 p+12 p^2+2 p^3\right )+4 c d \left (-30-43 p-10 p^2+4 p^3+p^4\right ) x+5 d^2 \left (24+50 p+35 p^2+10 p^3+p^4\right ) x^2\right )\right )}{b^6 d^3 (1+p) (2+p) (3+p) (4+p) (5+p) (6+p)} \] Input:

Integrate[x^2*(a + b*Sqrt[c + d*x])^p,x]
 

Output:

(-2*(a + b*Sqrt[c + d*x])^(1 + p)*(120*a^5 - 120*a^4*b*(1 + p)*Sqrt[c + d* 
x] + 12*a^3*b^2*(4*c*(-5 + p + p^2) + 5*d*(2 + 3*p + p^2)*x) - 4*a^2*b^3*( 
1 + p)*Sqrt[c + d*x]*(2*c*(-30 - 4*p + p^2) + 5*d*(6 + 5*p + p^2)*x) - b^5 
*(15 + 23*p + 9*p^2 + p^3)*Sqrt[c + d*x]*(8*c^2 - 4*c*d*(2 + p)*x + d^2*(8 
 + 6*p + p^2)*x^2) + a*b^4*(-8*c^2*(-15 + 10*p + 12*p^2 + 2*p^3) + 4*c*d*( 
-30 - 43*p - 10*p^2 + 4*p^3 + p^4)*x + 5*d^2*(24 + 50*p + 35*p^2 + 10*p^3 
+ p^4)*x^2)))/(b^6*d^3*(1 + p)*(2 + p)*(3 + p)*(4 + p)*(5 + p)*(6 + p))
 

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.94, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {896, 1732, 522, 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.

\(\displaystyle \int x^2 \left (a+b \sqrt {c+d x}\right )^p \, dx\)

\(\Big \downarrow \) 896

\(\displaystyle \frac {\int d^2 x^2 \left (a+b \sqrt {c+d x}\right )^pd(c+d x)}{d^3}\)

\(\Big \downarrow \) 1732

\(\displaystyle \frac {2 \int d^2 x^2 \sqrt {c+d x} \left (a+b \sqrt {c+d x}\right )^pd\sqrt {c+d x}}{d^3}\)

\(\Big \downarrow \) 522

\(\displaystyle \frac {2 \int \left (-\frac {a \left (a^2-b^2 c\right )^2 \left (a+b \sqrt {c+d x}\right )^p}{b^5}+\frac {\left (5 a^4-6 b^2 c a^2+b^4 c^2\right ) \left (a+b \sqrt {c+d x}\right )^{p+1}}{b^5}-\frac {2 \left (5 a^3-3 a b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{p+2}}{b^5}-\frac {2 \left (b^2 c-5 a^2\right ) \left (a+b \sqrt {c+d x}\right )^{p+3}}{b^5}-\frac {5 a \left (a+b \sqrt {c+d x}\right )^{p+4}}{b^5}+\frac {\left (a+b \sqrt {c+d x}\right )^{p+5}}{b^5}\right )d\sqrt {c+d x}}{d^3}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 \left (-\frac {a \left (a^2-b^2 c\right )^2 \left (a+b \sqrt {c+d x}\right )^{p+1}}{b^6 (p+1)}-\frac {2 a \left (5 a^2-3 b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{p+3}}{b^6 (p+3)}+\frac {2 \left (5 a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{p+4}}{b^6 (p+4)}+\frac {\left (5 a^4-6 a^2 b^2 c+b^4 c^2\right ) \left (a+b \sqrt {c+d x}\right )^{p+2}}{b^6 (p+2)}-\frac {5 a \left (a+b \sqrt {c+d x}\right )^{p+5}}{b^6 (p+5)}+\frac {\left (a+b \sqrt {c+d x}\right )^{p+6}}{b^6 (p+6)}\right )}{d^3}\)

Input:

Int[x^2*(a + b*Sqrt[c + d*x])^p,x]
 

Output:

(2*(-((a*(a^2 - b^2*c)^2*(a + b*Sqrt[c + d*x])^(1 + p))/(b^6*(1 + p))) + ( 
(5*a^4 - 6*a^2*b^2*c + b^4*c^2)*(a + b*Sqrt[c + d*x])^(2 + p))/(b^6*(2 + p 
)) - (2*a*(5*a^2 - 3*b^2*c)*(a + b*Sqrt[c + d*x])^(3 + p))/(b^6*(3 + p)) + 
 (2*(5*a^2 - b^2*c)*(a + b*Sqrt[c + d*x])^(4 + p))/(b^6*(4 + p)) - (5*a*(a 
 + b*Sqrt[c + d*x])^(5 + p))/(b^6*(5 + p)) + (a + b*Sqrt[c + d*x])^(6 + p) 
/(b^6*(6 + p))))/d^3
 

Defintions of rubi rules used

rule 522

Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. 
), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], 
x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
 

rule 896

Int[((a_) + (b_.)*(v_)^(n_))^(p_.)*(x_)^(m_.), x_Symbol] :> With[{c = Coeff 
icient[v, x, 0], d = Coefficient[v, x, 1]}, Simp[1/d^(m + 1)   Subst[Int[Si 
mplifyIntegrand[(x - c)^m*(a + b*x^n)^p, x], x], x, v], x] /; NeQ[c, 0]] /; 
 FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && IntegerQ[m]
 

rule 1732

Int[((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symb 
ol] :> With[{g = Denominator[n]}, Simp[g   Subst[Int[x^(g - 1)*(d + e*x^(g* 
n))^q*(a + c*x^(2*g*n))^p, x], x, x^(1/g)], x]] /; FreeQ[{a, c, d, e, p, q} 
, x] && EqQ[n2, 2*n] && FractionQ[n]
 

rule 2009

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

Maple [F]

\[\int x^{2} \left (a +b \sqrt {d x +c}\right )^{p}d x\]

Input:

int(x^2*(a+b*(d*x+c)^(1/2))^p,x)
 

Output:

int(x^2*(a+b*(d*x+c)^(1/2))^p,x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 712 vs. \(2 (230) = 460\).

Time = 0.22 (sec) , antiderivative size = 712, normalized size of antiderivative = 2.94 \[ \int x^2 \left (a+b \sqrt {c+d x}\right )^p \, dx=\frac {2 \, {\left (120 \, b^{6} c^{3} - 360 \, a^{2} b^{4} c^{2} + 360 \, a^{4} b^{2} c - 120 \, a^{6} + 8 \, {\left (b^{6} c^{3} + 3 \, a^{2} b^{4} c^{2}\right )} p^{3} + {\left (b^{6} d^{3} p^{5} + 15 \, b^{6} d^{3} p^{4} + 85 \, b^{6} d^{3} p^{3} + 225 \, b^{6} d^{3} p^{2} + 274 \, b^{6} d^{3} p + 120 \, b^{6} d^{3}\right )} x^{3} + 24 \, {\left (3 \, b^{6} c^{3} + 3 \, a^{2} b^{4} c^{2} - 2 \, a^{4} b^{2} c\right )} p^{2} + {\left (b^{6} c d^{2} p^{5} + {\left (11 \, b^{6} c - 5 \, a^{2} b^{4}\right )} d^{2} p^{4} + {\left (41 \, b^{6} c - 30 \, a^{2} b^{4}\right )} d^{2} p^{3} + {\left (61 \, b^{6} c - 55 \, a^{2} b^{4}\right )} d^{2} p^{2} + 30 \, {\left (b^{6} c - a^{2} b^{4}\right )} d^{2} p\right )} x^{2} + 8 \, {\left (23 \, b^{6} c^{3} - 24 \, a^{2} b^{4} c^{2} + 9 \, a^{4} b^{2} c\right )} p - 4 \, {\left ({\left (b^{6} c^{2} + a^{2} b^{4} c\right )} d p^{4} + 3 \, {\left (3 \, b^{6} c^{2} - a^{2} b^{4} c\right )} d p^{3} + {\left (23 \, b^{6} c^{2} - 34 \, a^{2} b^{4} c + 15 \, a^{4} b^{2}\right )} d p^{2} + 15 \, {\left (b^{6} c^{2} - 2 \, a^{2} b^{4} c + a^{4} b^{2}\right )} d p\right )} x + {\left (8 \, {\left (3 \, a b^{5} c^{2} + a^{3} b^{3} c\right )} p^{3} + 24 \, {\left (7 \, a b^{5} c^{2} - 3 \, a^{3} b^{3} c\right )} p^{2} + {\left (a b^{5} d^{2} p^{5} + 10 \, a b^{5} d^{2} p^{4} + 35 \, a b^{5} d^{2} p^{3} + 50 \, a b^{5} d^{2} p^{2} + 24 \, a b^{5} d^{2} p\right )} x^{2} + 8 \, {\left (33 \, a b^{5} c^{2} - 40 \, a^{3} b^{3} c + 15 \, a^{5} b\right )} p - 4 \, {\left (2 \, a b^{5} c d p^{4} + 5 \, {\left (3 \, a b^{5} c - a^{3} b^{3}\right )} d p^{3} + {\left (31 \, a b^{5} c - 15 \, a^{3} b^{3}\right )} d p^{2} + 2 \, {\left (9 \, a b^{5} c - 5 \, a^{3} b^{3}\right )} d p\right )} x\right )} \sqrt {d x + c}\right )} {\left (\sqrt {d x + c} b + a\right )}^{p}}{b^{6} d^{3} p^{6} + 21 \, b^{6} d^{3} p^{5} + 175 \, b^{6} d^{3} p^{4} + 735 \, b^{6} d^{3} p^{3} + 1624 \, b^{6} d^{3} p^{2} + 1764 \, b^{6} d^{3} p + 720 \, b^{6} d^{3}} \] Input:

integrate(x^2*(a+b*(d*x+c)^(1/2))^p,x, algorithm="fricas")
 

Output:

2*(120*b^6*c^3 - 360*a^2*b^4*c^2 + 360*a^4*b^2*c - 120*a^6 + 8*(b^6*c^3 + 
3*a^2*b^4*c^2)*p^3 + (b^6*d^3*p^5 + 15*b^6*d^3*p^4 + 85*b^6*d^3*p^3 + 225* 
b^6*d^3*p^2 + 274*b^6*d^3*p + 120*b^6*d^3)*x^3 + 24*(3*b^6*c^3 + 3*a^2*b^4 
*c^2 - 2*a^4*b^2*c)*p^2 + (b^6*c*d^2*p^5 + (11*b^6*c - 5*a^2*b^4)*d^2*p^4 
+ (41*b^6*c - 30*a^2*b^4)*d^2*p^3 + (61*b^6*c - 55*a^2*b^4)*d^2*p^2 + 30*( 
b^6*c - a^2*b^4)*d^2*p)*x^2 + 8*(23*b^6*c^3 - 24*a^2*b^4*c^2 + 9*a^4*b^2*c 
)*p - 4*((b^6*c^2 + a^2*b^4*c)*d*p^4 + 3*(3*b^6*c^2 - a^2*b^4*c)*d*p^3 + ( 
23*b^6*c^2 - 34*a^2*b^4*c + 15*a^4*b^2)*d*p^2 + 15*(b^6*c^2 - 2*a^2*b^4*c 
+ a^4*b^2)*d*p)*x + (8*(3*a*b^5*c^2 + a^3*b^3*c)*p^3 + 24*(7*a*b^5*c^2 - 3 
*a^3*b^3*c)*p^2 + (a*b^5*d^2*p^5 + 10*a*b^5*d^2*p^4 + 35*a*b^5*d^2*p^3 + 5 
0*a*b^5*d^2*p^2 + 24*a*b^5*d^2*p)*x^2 + 8*(33*a*b^5*c^2 - 40*a^3*b^3*c + 1 
5*a^5*b)*p - 4*(2*a*b^5*c*d*p^4 + 5*(3*a*b^5*c - a^3*b^3)*d*p^3 + (31*a*b^ 
5*c - 15*a^3*b^3)*d*p^2 + 2*(9*a*b^5*c - 5*a^3*b^3)*d*p)*x)*sqrt(d*x + c)) 
*(sqrt(d*x + c)*b + a)^p/(b^6*d^3*p^6 + 21*b^6*d^3*p^5 + 175*b^6*d^3*p^4 + 
 735*b^6*d^3*p^3 + 1624*b^6*d^3*p^2 + 1764*b^6*d^3*p + 720*b^6*d^3)
 

Sympy [F]

\[ \int x^2 \left (a+b \sqrt {c+d x}\right )^p \, dx=\int x^{2} \left (a + b \sqrt {c + d x}\right )^{p}\, dx \] Input:

integrate(x**2*(a+b*(d*x+c)**(1/2))**p,x)
 

Output:

Integral(x**2*(a + b*sqrt(c + d*x))**p, x)
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.66 \[ \int x^2 \left (a+b \sqrt {c+d x}\right )^p \, dx=\frac {2 \, {\left (\frac {{\left ({\left (d x + c\right )} b^{2} {\left (p + 1\right )} + \sqrt {d x + c} a b p - a^{2}\right )} {\left (\sqrt {d x + c} b + a\right )}^{p} c^{2}}{{\left (p^{2} + 3 \, p + 2\right )} b^{2}} - \frac {2 \, {\left ({\left (p^{3} + 6 \, p^{2} + 11 \, p + 6\right )} {\left (d x + c\right )}^{2} b^{4} + {\left (p^{3} + 3 \, p^{2} + 2 \, p\right )} {\left (d x + c\right )}^{\frac {3}{2}} a b^{3} - 3 \, {\left (p^{2} + p\right )} {\left (d x + c\right )} a^{2} b^{2} + 6 \, \sqrt {d x + c} a^{3} b p - 6 \, a^{4}\right )} {\left (\sqrt {d x + c} b + a\right )}^{p} c}{{\left (p^{4} + 10 \, p^{3} + 35 \, p^{2} + 50 \, p + 24\right )} b^{4}} + \frac {{\left ({\left (p^{5} + 15 \, p^{4} + 85 \, p^{3} + 225 \, p^{2} + 274 \, p + 120\right )} {\left (d x + c\right )}^{3} b^{6} + {\left (p^{5} + 10 \, p^{4} + 35 \, p^{3} + 50 \, p^{2} + 24 \, p\right )} {\left (d x + c\right )}^{\frac {5}{2}} a b^{5} - 5 \, {\left (p^{4} + 6 \, p^{3} + 11 \, p^{2} + 6 \, p\right )} {\left (d x + c\right )}^{2} a^{2} b^{4} + 20 \, {\left (p^{3} + 3 \, p^{2} + 2 \, p\right )} {\left (d x + c\right )}^{\frac {3}{2}} a^{3} b^{3} - 60 \, {\left (p^{2} + p\right )} {\left (d x + c\right )} a^{4} b^{2} + 120 \, \sqrt {d x + c} a^{5} b p - 120 \, a^{6}\right )} {\left (\sqrt {d x + c} b + a\right )}^{p}}{{\left (p^{6} + 21 \, p^{5} + 175 \, p^{4} + 735 \, p^{3} + 1624 \, p^{2} + 1764 \, p + 720\right )} b^{6}}\right )}}{d^{3}} \] Input:

integrate(x^2*(a+b*(d*x+c)^(1/2))^p,x, algorithm="maxima")
 

Output:

2*(((d*x + c)*b^2*(p + 1) + sqrt(d*x + c)*a*b*p - a^2)*(sqrt(d*x + c)*b + 
a)^p*c^2/((p^2 + 3*p + 2)*b^2) - 2*((p^3 + 6*p^2 + 11*p + 6)*(d*x + c)^2*b 
^4 + (p^3 + 3*p^2 + 2*p)*(d*x + c)^(3/2)*a*b^3 - 3*(p^2 + p)*(d*x + c)*a^2 
*b^2 + 6*sqrt(d*x + c)*a^3*b*p - 6*a^4)*(sqrt(d*x + c)*b + a)^p*c/((p^4 + 
10*p^3 + 35*p^2 + 50*p + 24)*b^4) + ((p^5 + 15*p^4 + 85*p^3 + 225*p^2 + 27 
4*p + 120)*(d*x + c)^3*b^6 + (p^5 + 10*p^4 + 35*p^3 + 50*p^2 + 24*p)*(d*x 
+ c)^(5/2)*a*b^5 - 5*(p^4 + 6*p^3 + 11*p^2 + 6*p)*(d*x + c)^2*a^2*b^4 + 20 
*(p^3 + 3*p^2 + 2*p)*(d*x + c)^(3/2)*a^3*b^3 - 60*(p^2 + p)*(d*x + c)*a^4* 
b^2 + 120*sqrt(d*x + c)*a^5*b*p - 120*a^6)*(sqrt(d*x + c)*b + a)^p/((p^6 + 
 21*p^5 + 175*p^4 + 735*p^3 + 1624*p^2 + 1764*p + 720)*b^6))/d^3
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2490 vs. \(2 (230) = 460\).

Time = 0.22 (sec) , antiderivative size = 2490, normalized size of antiderivative = 10.29 \[ \int x^2 \left (a+b \sqrt {c+d x}\right )^p \, dx=\text {Too large to display} \] Input:

integrate(x^2*(a+b*(d*x+c)^(1/2))^p,x, algorithm="giac")
 

Output:

2*((sqrt(d*x + c)*b + a)^2*(sqrt(d*x + c)*b + a)^p*b^4*c^2*p^5 - (sqrt(d*x 
 + c)*b + a)*(sqrt(d*x + c)*b + a)^p*a*b^4*c^2*p^5 + 19*(sqrt(d*x + c)*b + 
 a)^2*(sqrt(d*x + c)*b + a)^p*b^4*c^2*p^4 - 20*(sqrt(d*x + c)*b + a)*(sqrt 
(d*x + c)*b + a)^p*a*b^4*c^2*p^4 - 2*(sqrt(d*x + c)*b + a)^4*(sqrt(d*x + c 
)*b + a)^p*b^2*c*p^5 + 6*(sqrt(d*x + c)*b + a)^3*(sqrt(d*x + c)*b + a)^p*a 
*b^2*c*p^5 - 6*(sqrt(d*x + c)*b + a)^2*(sqrt(d*x + c)*b + a)^p*a^2*b^2*c*p 
^5 + 2*(sqrt(d*x + c)*b + a)*(sqrt(d*x + c)*b + a)^p*a^3*b^2*c*p^5 + 137*( 
sqrt(d*x + c)*b + a)^2*(sqrt(d*x + c)*b + a)^p*b^4*c^2*p^3 - 155*(sqrt(d*x 
 + c)*b + a)*(sqrt(d*x + c)*b + a)^p*a*b^4*c^2*p^3 - 34*(sqrt(d*x + c)*b + 
 a)^4*(sqrt(d*x + c)*b + a)^p*b^2*c*p^4 + 108*(sqrt(d*x + c)*b + a)^3*(sqr 
t(d*x + c)*b + a)^p*a*b^2*c*p^4 - 114*(sqrt(d*x + c)*b + a)^2*(sqrt(d*x + 
c)*b + a)^p*a^2*b^2*c*p^4 + 40*(sqrt(d*x + c)*b + a)*(sqrt(d*x + c)*b + a) 
^p*a^3*b^2*c*p^4 + (sqrt(d*x + c)*b + a)^6*(sqrt(d*x + c)*b + a)^p*p^5 - 5 
*(sqrt(d*x + c)*b + a)^5*(sqrt(d*x + c)*b + a)^p*a*p^5 + 10*(sqrt(d*x + c) 
*b + a)^4*(sqrt(d*x + c)*b + a)^p*a^2*p^5 - 10*(sqrt(d*x + c)*b + a)^3*(sq 
rt(d*x + c)*b + a)^p*a^3*p^5 + 5*(sqrt(d*x + c)*b + a)^2*(sqrt(d*x + c)*b 
+ a)^p*a^4*p^5 - (sqrt(d*x + c)*b + a)*(sqrt(d*x + c)*b + a)^p*a^5*p^5 + 4 
61*(sqrt(d*x + c)*b + a)^2*(sqrt(d*x + c)*b + a)^p*b^4*c^2*p^2 - 580*(sqrt 
(d*x + c)*b + a)*(sqrt(d*x + c)*b + a)^p*a*b^4*c^2*p^2 - 214*(sqrt(d*x + c 
)*b + a)^4*(sqrt(d*x + c)*b + a)^p*b^2*c*p^3 + 726*(sqrt(d*x + c)*b + a...
 

Mupad [F(-1)]

Timed out. \[ \int x^2 \left (a+b \sqrt {c+d x}\right )^p \, dx=\int x^2\,{\left (a+b\,\sqrt {c+d\,x}\right )}^p \,d x \] Input:

int(x^2*(a + b*(c + d*x)^(1/2))^p,x)
 

Output:

int(x^2*(a + b*(c + d*x)^(1/2))^p, x)
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 865, normalized size of antiderivative = 3.57 \[ \int x^2 \left (a+b \sqrt {c+d x}\right )^p \, dx=\frac {2 \left (\sqrt {d x +c}\, b +a \right )^{p} \left (-360 a^{2} b^{4} c^{2}+8 b^{6} c^{3} p^{3}+72 b^{6} c^{3} p^{2}+184 b^{6} c^{3} p -120 a^{6}+8 \sqrt {d x +c}\, a^{3} b^{3} c \,p^{3}-72 \sqrt {d x +c}\, a^{3} b^{3} c \,p^{2}-320 \sqrt {d x +c}\, a^{3} b^{3} c p +24 \sqrt {d x +c}\, a \,b^{5} c^{2} p^{3}+168 \sqrt {d x +c}\, a \,b^{5} c^{2} p^{2}+264 \sqrt {d x +c}\, a \,b^{5} c^{2} p -60 a^{4} b^{2} d \,p^{2} x -60 a^{4} b^{2} d p x +360 a^{4} b^{2} c +b^{6} c \,d^{2} p^{5} x^{2}+20 \sqrt {d x +c}\, a^{3} b^{3} d \,p^{3} x +60 \sqrt {d x +c}\, a^{3} b^{3} d \,p^{2} x +40 \sqrt {d x +c}\, a^{3} b^{3} d p x +10 \sqrt {d x +c}\, a \,b^{5} d^{2} p^{4} x^{2}+35 \sqrt {d x +c}\, a \,b^{5} d^{2} p^{3} x^{2}-5 a^{2} b^{4} d^{2} p^{4} x^{2}-30 a^{2} b^{4} d^{2} p^{3} x^{2}-55 a^{2} b^{4} d^{2} p^{2} x^{2}-30 a^{2} b^{4} d^{2} p \,x^{2}-4 b^{6} c^{2} d \,p^{4} x -36 b^{6} c^{2} d \,p^{3} x -92 b^{6} c^{2} d \,p^{2} x -60 b^{6} c^{2} d p x +11 b^{6} c \,d^{2} p^{4} x^{2}+41 b^{6} c \,d^{2} p^{3} x^{2}+61 b^{6} c \,d^{2} p^{2} x^{2}+30 b^{6} c \,d^{2} p \,x^{2}+\sqrt {d x +c}\, a \,b^{5} d^{2} p^{5} x^{2}+120 b^{6} d^{3} x^{3}+50 \sqrt {d x +c}\, a \,b^{5} d^{2} p^{2} x^{2}+24 \sqrt {d x +c}\, a \,b^{5} d^{2} p \,x^{2}-4 a^{2} b^{4} c d \,p^{4} x +12 a^{2} b^{4} c d \,p^{3} x +136 a^{2} b^{4} c d \,p^{2} x +120 a^{2} b^{4} c d p x +b^{6} d^{3} p^{5} x^{3}+120 \sqrt {d x +c}\, a^{5} b p -48 a^{4} b^{2} c \,p^{2}+72 a^{4} b^{2} c p +24 a^{2} b^{4} c^{2} p^{3}+72 a^{2} b^{4} c^{2} p^{2}-192 a^{2} b^{4} c^{2} p +15 b^{6} d^{3} p^{4} x^{3}+85 b^{6} d^{3} p^{3} x^{3}+225 b^{6} d^{3} p^{2} x^{3}+274 b^{6} d^{3} p \,x^{3}-8 \sqrt {d x +c}\, a \,b^{5} c d \,p^{4} x -60 \sqrt {d x +c}\, a \,b^{5} c d \,p^{3} x -124 \sqrt {d x +c}\, a \,b^{5} c d \,p^{2} x -72 \sqrt {d x +c}\, a \,b^{5} c d p x +120 b^{6} c^{3}\right )}{b^{6} d^{3} \left (p^{6}+21 p^{5}+175 p^{4}+735 p^{3}+1624 p^{2}+1764 p +720\right )} \] Input:

int(x^2*(a+b*(d*x+c)^(1/2))^p,x)
 

Output:

(2*(sqrt(c + d*x)*b + a)**p*(120*sqrt(c + d*x)*a**5*b*p + 8*sqrt(c + d*x)* 
a**3*b**3*c*p**3 - 72*sqrt(c + d*x)*a**3*b**3*c*p**2 - 320*sqrt(c + d*x)*a 
**3*b**3*c*p + 20*sqrt(c + d*x)*a**3*b**3*d*p**3*x + 60*sqrt(c + d*x)*a**3 
*b**3*d*p**2*x + 40*sqrt(c + d*x)*a**3*b**3*d*p*x + 24*sqrt(c + d*x)*a*b** 
5*c**2*p**3 + 168*sqrt(c + d*x)*a*b**5*c**2*p**2 + 264*sqrt(c + d*x)*a*b** 
5*c**2*p - 8*sqrt(c + d*x)*a*b**5*c*d*p**4*x - 60*sqrt(c + d*x)*a*b**5*c*d 
*p**3*x - 124*sqrt(c + d*x)*a*b**5*c*d*p**2*x - 72*sqrt(c + d*x)*a*b**5*c* 
d*p*x + sqrt(c + d*x)*a*b**5*d**2*p**5*x**2 + 10*sqrt(c + d*x)*a*b**5*d**2 
*p**4*x**2 + 35*sqrt(c + d*x)*a*b**5*d**2*p**3*x**2 + 50*sqrt(c + d*x)*a*b 
**5*d**2*p**2*x**2 + 24*sqrt(c + d*x)*a*b**5*d**2*p*x**2 - 120*a**6 - 48*a 
**4*b**2*c*p**2 + 72*a**4*b**2*c*p + 360*a**4*b**2*c - 60*a**4*b**2*d*p**2 
*x - 60*a**4*b**2*d*p*x + 24*a**2*b**4*c**2*p**3 + 72*a**2*b**4*c**2*p**2 
- 192*a**2*b**4*c**2*p - 360*a**2*b**4*c**2 - 4*a**2*b**4*c*d*p**4*x + 12* 
a**2*b**4*c*d*p**3*x + 136*a**2*b**4*c*d*p**2*x + 120*a**2*b**4*c*d*p*x - 
5*a**2*b**4*d**2*p**4*x**2 - 30*a**2*b**4*d**2*p**3*x**2 - 55*a**2*b**4*d* 
*2*p**2*x**2 - 30*a**2*b**4*d**2*p*x**2 + 8*b**6*c**3*p**3 + 72*b**6*c**3* 
p**2 + 184*b**6*c**3*p + 120*b**6*c**3 - 4*b**6*c**2*d*p**4*x - 36*b**6*c* 
*2*d*p**3*x - 92*b**6*c**2*d*p**2*x - 60*b**6*c**2*d*p*x + b**6*c*d**2*p** 
5*x**2 + 11*b**6*c*d**2*p**4*x**2 + 41*b**6*c*d**2*p**3*x**2 + 61*b**6*c*d 
**2*p**2*x**2 + 30*b**6*c*d**2*p*x**2 + b**6*d**3*p**5*x**3 + 15*b**6*d...