Integrand size = 21, antiderivative size = 224 \[ \int x^2 \sqrt {a+b \sqrt {c+d x}} \, dx=-\frac {4 a \left (a^2-b^2 c\right )^2 \left (a+b \sqrt {c+d x}\right )^{3/2}}{3 b^6 d^3}+\frac {4 \left (5 a^4-6 a^2 b^2 c+b^4 c^2\right ) \left (a+b \sqrt {c+d x}\right )^{5/2}}{5 b^6 d^3}-\frac {8 a \left (5 a^2-3 b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{7/2}}{7 b^6 d^3}+\frac {8 \left (5 a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{9/2}}{9 b^6 d^3}-\frac {20 a \left (a+b \sqrt {c+d x}\right )^{11/2}}{11 b^6 d^3}+\frac {4 \left (a+b \sqrt {c+d x}\right )^{13/2}}{13 b^6 d^3} \]
-4/3*a*(-b^2*c+a^2)^2*(a+b*(d*x+c)^(1/2))^(3/2)/b^6/d^3+4/5*(b^4*c^2-6*a^2 *b^2*c+5*a^4)*(a+b*(d*x+c)^(1/2))^(5/2)/b^6/d^3-8/7*a*(-3*b^2*c+5*a^2)*(a+ b*(d*x+c)^(1/2))^(7/2)/b^6/d^3+8/9*(-b^2*c+5*a^2)*(a+b*(d*x+c)^(1/2))^(9/2 )/b^6/d^3-20/11*a*(a+b*(d*x+c)^(1/2))^(11/2)/b^6/d^3+4/13*(a+b*(d*x+c)^(1/ 2))^(13/2)/b^6/d^3
Time = 0.13 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.66 \[ \int x^2 \sqrt {a+b \sqrt {c+d x}} \, dx=\frac {4 \left (a+b \sqrt {c+d x}\right )^{3/2} \left (-1280 a^5+32 a^3 b^2 (68 c-75 d x)+1920 a^4 b \sqrt {c+d x}+16 a^2 b^3 \sqrt {c+d x} (-254 c+175 d x)+77 b^5 \sqrt {c+d x} \left (32 c^2-40 c d x+45 d^2 x^2\right )-6 a b^4 \left (96 c^2-380 c d x+525 d^2 x^2\right )\right )}{45045 b^6 d^3} \]
(4*(a + b*Sqrt[c + d*x])^(3/2)*(-1280*a^5 + 32*a^3*b^2*(68*c - 75*d*x) + 1 920*a^4*b*Sqrt[c + d*x] + 16*a^2*b^3*Sqrt[c + d*x]*(-254*c + 175*d*x) + 77 *b^5*Sqrt[c + d*x]*(32*c^2 - 40*c*d*x + 45*d^2*x^2) - 6*a*b^4*(96*c^2 - 38 0*c*d*x + 525*d^2*x^2)))/(45045*b^6*d^3)
Time = 0.32 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.94, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, 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.
| \(\displaystyle \int x^2 \sqrt {a+b \sqrt {c+d x}} \, dx\) |
| \(\Big \downarrow \) 896 |
| \(\displaystyle \frac {\int d^2 x^2 \sqrt {a+b \sqrt {c+d x}}d(c+d x)}{d^3}\) |
| \(\Big \downarrow \) 1732 |
| \(\displaystyle \frac {2 \int d^2 x^2 \sqrt {c+d x} \sqrt {a+b \sqrt {c+d x}}d\sqrt {c+d x}}{d^3}\) |
| \(\Big \downarrow \) 522 |
| \(\displaystyle \frac {2 \int \left (\frac {\left (a+b \sqrt {c+d x}\right )^{11/2}}{b^5}-\frac {5 a \left (a+b \sqrt {c+d x}\right )^{9/2}}{b^5}-\frac {2 \left (b^2 c-5 a^2\right ) \left (a+b \sqrt {c+d x}\right )^{7/2}}{b^5}-\frac {2 \left (5 a^3-3 a b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{5/2}}{b^5}+\frac {\left (5 a^4-6 b^2 c a^2+b^4 c^2\right ) \left (a+b \sqrt {c+d x}\right )^{3/2}}{b^5}-\frac {a \left (a^2-b^2 c\right )^2 \sqrt {a+b \sqrt {c+d x}}}{b^5}\right )d\sqrt {c+d x}}{d^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \left (\frac {4 \left (5 a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{9/2}}{9 b^6}-\frac {4 a \left (5 a^2-3 b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{7/2}}{7 b^6}-\frac {2 a \left (a^2-b^2 c\right )^2 \left (a+b \sqrt {c+d x}\right )^{3/2}}{3 b^6}+\frac {2 \left (5 a^4-6 a^2 b^2 c+b^4 c^2\right ) \left (a+b \sqrt {c+d x}\right )^{5/2}}{5 b^6}+\frac {2 \left (a+b \sqrt {c+d x}\right )^{13/2}}{13 b^6}-\frac {10 a \left (a+b \sqrt {c+d x}\right )^{11/2}}{11 b^6}\right )}{d^3}\) |
(2*((-2*a*(a^2 - b^2*c)^2*(a + b*Sqrt[c + d*x])^(3/2))/(3*b^6) + (2*(5*a^4 - 6*a^2*b^2*c + b^4*c^2)*(a + b*Sqrt[c + d*x])^(5/2))/(5*b^6) - (4*a*(5*a ^2 - 3*b^2*c)*(a + b*Sqrt[c + d*x])^(7/2))/(7*b^6) + (4*(5*a^2 - b^2*c)*(a + b*Sqrt[c + d*x])^(9/2))/(9*b^6) - (10*a*(a + b*Sqrt[c + d*x])^(11/2))/( 11*b^6) + (2*(a + b*Sqrt[c + d*x])^(13/2))/(13*b^6)))/d^3
3.7.26.3.1 Defintions of rubi rules used
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]
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]
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]
Time = 0.15 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.82
| method | result | size |
| derivativedivides | \(\frac {\frac {4 \left (a +b \sqrt {d x +c}\right )^{\frac {13}{2}}}{13}-\frac {20 a \left (a +b \sqrt {d x +c}\right )^{\frac {11}{2}}}{11}-\frac {4 \left (2 b^{2} c -10 a^{2}\right ) \left (a +b \sqrt {d x +c}\right )^{\frac {9}{2}}}{9}-\frac {4 \left (4 \left (-b^{2} c +a^{2}\right ) a +a \left (-2 b^{2} c +6 a^{2}\right )\right ) \left (a +b \sqrt {d x +c}\right )^{\frac {7}{2}}}{7}-\frac {4 \left (-\left (-b^{2} c +a^{2}\right )^{2}-4 a^{2} \left (-b^{2} c +a^{2}\right )\right ) \left (a +b \sqrt {d x +c}\right )^{\frac {5}{2}}}{5}-\frac {4 \left (-b^{2} c +a^{2}\right )^{2} a \left (a +b \sqrt {d x +c}\right )^{\frac {3}{2}}}{3}}{d^{3} b^{6}}\) | \(184\) |
| default | \(\frac {\frac {4 \left (a +b \sqrt {d x +c}\right )^{\frac {13}{2}}}{13}-\frac {20 a \left (a +b \sqrt {d x +c}\right )^{\frac {11}{2}}}{11}-\frac {4 \left (2 b^{2} c -10 a^{2}\right ) \left (a +b \sqrt {d x +c}\right )^{\frac {9}{2}}}{9}-\frac {4 \left (4 \left (-b^{2} c +a^{2}\right ) a +a \left (-2 b^{2} c +6 a^{2}\right )\right ) \left (a +b \sqrt {d x +c}\right )^{\frac {7}{2}}}{7}-\frac {4 \left (-\left (-b^{2} c +a^{2}\right )^{2}-4 a^{2} \left (-b^{2} c +a^{2}\right )\right ) \left (a +b \sqrt {d x +c}\right )^{\frac {5}{2}}}{5}-\frac {4 \left (-b^{2} c +a^{2}\right )^{2} a \left (a +b \sqrt {d x +c}\right )^{\frac {3}{2}}}{3}}{d^{3} b^{6}}\) | \(184\) |
4/d^3/b^6*(1/13*(a+b*(d*x+c)^(1/2))^(13/2)-5/11*a*(a+b*(d*x+c)^(1/2))^(11/ 2)-1/9*(2*b^2*c-10*a^2)*(a+b*(d*x+c)^(1/2))^(9/2)-1/7*(4*(-b^2*c+a^2)*a+a* (-2*b^2*c+6*a^2))*(a+b*(d*x+c)^(1/2))^(7/2)-1/5*(-(-b^2*c+a^2)^2-4*a^2*(-b ^2*c+a^2))*(a+b*(d*x+c)^(1/2))^(5/2)-1/3*(-b^2*c+a^2)^2*a*(a+b*(d*x+c)^(1/ 2))^(3/2))
Time = 0.31 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.82 \[ \int x^2 \sqrt {a+b \sqrt {c+d x}} \, dx=\frac {4 \, {\left (3465 \, b^{6} d^{3} x^{3} + 2464 \, b^{6} c^{3} - 4640 \, a^{2} b^{4} c^{2} + 4096 \, a^{4} b^{2} c - 1280 \, a^{6} + 35 \, {\left (11 \, b^{6} c - 10 \, a^{2} b^{4}\right )} d^{2} x^{2} - 8 \, {\left (77 \, b^{6} c^{2} - 127 \, a^{2} b^{4} c + 60 \, a^{4} b^{2}\right )} d x + {\left (315 \, a b^{5} d^{2} x^{2} + 1888 \, a b^{5} c^{2} - 1888 \, a^{3} b^{3} c + 640 \, a^{5} b - 400 \, {\left (2 \, a b^{5} c - a^{3} b^{3}\right )} d x\right )} \sqrt {d x + c}\right )} \sqrt {\sqrt {d x + c} b + a}}{45045 \, b^{6} d^{3}} \]
4/45045*(3465*b^6*d^3*x^3 + 2464*b^6*c^3 - 4640*a^2*b^4*c^2 + 4096*a^4*b^2 *c - 1280*a^6 + 35*(11*b^6*c - 10*a^2*b^4)*d^2*x^2 - 8*(77*b^6*c^2 - 127*a ^2*b^4*c + 60*a^4*b^2)*d*x + (315*a*b^5*d^2*x^2 + 1888*a*b^5*c^2 - 1888*a^ 3*b^3*c + 640*a^5*b - 400*(2*a*b^5*c - a^3*b^3)*d*x)*sqrt(d*x + c))*sqrt(s qrt(d*x + c)*b + a)/(b^6*d^3)
Time = 0.75 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.08 \[ \int x^2 \sqrt {a+b \sqrt {c+d x}} \, dx=\begin {cases} \frac {2 \left (\begin {cases} \frac {2 \left (- \frac {5 a \left (a + b \sqrt {c + d x}\right )^{\frac {11}{2}}}{11 b^{4}} + \frac {\left (a + b \sqrt {c + d x}\right )^{\frac {13}{2}}}{13 b^{4}} + \frac {\left (a + b \sqrt {c + d x}\right )^{\frac {9}{2}} \cdot \left (10 a^{2} - 2 b^{2} c\right )}{9 b^{4}} + \frac {\left (a + b \sqrt {c + d x}\right )^{\frac {7}{2}} \left (- 10 a^{3} + 6 a b^{2} c\right )}{7 b^{4}} + \frac {\left (a + b \sqrt {c + d x}\right )^{\frac {5}{2}} \cdot \left (5 a^{4} - 6 a^{2} b^{2} c + b^{4} c^{2}\right )}{5 b^{4}} + \frac {\left (a + b \sqrt {c + d x}\right )^{\frac {3}{2}} \left (- a^{5} + 2 a^{3} b^{2} c - a b^{4} c^{2}\right )}{3 b^{4}}\right )}{b^{2}} & \text {for}\: b \neq 0 \\\frac {\sqrt {a} d^{3} x^{3}}{6} & \text {otherwise} \end {cases}\right )}{d^{3}} & \text {for}\: d \neq 0 \\\frac {x^{3} \sqrt {a + b \sqrt {c}}}{3} & \text {otherwise} \end {cases} \]
Piecewise((2*Piecewise((2*(-5*a*(a + b*sqrt(c + d*x))**(11/2)/(11*b**4) + (a + b*sqrt(c + d*x))**(13/2)/(13*b**4) + (a + b*sqrt(c + d*x))**(9/2)*(10 *a**2 - 2*b**2*c)/(9*b**4) + (a + b*sqrt(c + d*x))**(7/2)*(-10*a**3 + 6*a* b**2*c)/(7*b**4) + (a + b*sqrt(c + d*x))**(5/2)*(5*a**4 - 6*a**2*b**2*c + b**4*c**2)/(5*b**4) + (a + b*sqrt(c + d*x))**(3/2)*(-a**5 + 2*a**3*b**2*c - a*b**4*c**2)/(3*b**4))/b**2, Ne(b, 0)), (sqrt(a)*d**3*x**3/6, True))/d** 3, Ne(d, 0)), (x**3*sqrt(a + b*sqrt(c))/3, True))
Time = 0.19 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.75 \[ \int x^2 \sqrt {a+b \sqrt {c+d x}} \, dx=\frac {4 \, {\left (3465 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {13}{2}} - 20475 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {11}{2}} a - 10010 \, {\left (b^{2} c - 5 \, a^{2}\right )} {\left (\sqrt {d x + c} b + a\right )}^{\frac {9}{2}} + 12870 \, {\left (3 \, a b^{2} c - 5 \, a^{3}\right )} {\left (\sqrt {d x + c} b + a\right )}^{\frac {7}{2}} + 9009 \, {\left (b^{4} c^{2} - 6 \, a^{2} b^{2} c + 5 \, a^{4}\right )} {\left (\sqrt {d x + c} b + a\right )}^{\frac {5}{2}} - 15015 \, {\left (a b^{4} c^{2} - 2 \, a^{3} b^{2} c + a^{5}\right )} {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}}\right )}}{45045 \, b^{6} d^{3}} \]
4/45045*(3465*(sqrt(d*x + c)*b + a)^(13/2) - 20475*(sqrt(d*x + c)*b + a)^( 11/2)*a - 10010*(b^2*c - 5*a^2)*(sqrt(d*x + c)*b + a)^(9/2) + 12870*(3*a*b ^2*c - 5*a^3)*(sqrt(d*x + c)*b + a)^(7/2) + 9009*(b^4*c^2 - 6*a^2*b^2*c + 5*a^4)*(sqrt(d*x + c)*b + a)^(5/2) - 15015*(a*b^4*c^2 - 2*a^3*b^2*c + a^5) *(sqrt(d*x + c)*b + a)^(3/2))/(b^6*d^3)
Leaf count of result is larger than twice the leaf count of optimal. 549 vs. \(2 (188) = 376\).
Time = 0.39 (sec) , antiderivative size = 549, normalized size of antiderivative = 2.45 \[ \int x^2 \sqrt {a+b \sqrt {c+d x}} \, dx=\frac {4 \, {\left (\frac {13 \, {\left (1155 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} b^{4} c^{2} - 3465 \, \sqrt {\sqrt {d x + c} b + a} a b^{4} c^{2} - 990 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {7}{2}} b^{2} c + 4158 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {5}{2}} a b^{2} c - 6930 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} a^{2} b^{2} c + 6930 \, \sqrt {\sqrt {d x + c} b + a} a^{3} b^{2} c + 315 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {11}{2}} - 1925 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {9}{2}} a + 4950 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {7}{2}} a^{2} - 6930 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {5}{2}} a^{3} + 5775 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} a^{4} - 3465 \, \sqrt {\sqrt {d x + c} b + a} a^{5}\right )} a}{b^{5} d^{2}} + \frac {9009 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {5}{2}} b^{4} c^{2} - 30030 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} a b^{4} c^{2} + 45045 \, \sqrt {\sqrt {d x + c} b + a} a^{2} b^{4} c^{2} - 10010 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {9}{2}} b^{2} c + 51480 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {7}{2}} a b^{2} c - 108108 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {5}{2}} a^{2} b^{2} c + 120120 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} a^{3} b^{2} c - 90090 \, \sqrt {\sqrt {d x + c} b + a} a^{4} b^{2} c + 3465 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {13}{2}} - 24570 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {11}{2}} a + 75075 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {9}{2}} a^{2} - 128700 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {7}{2}} a^{3} + 135135 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {5}{2}} a^{4} - 90090 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} a^{5} + 45045 \, \sqrt {\sqrt {d x + c} b + a} a^{6}}{b^{5} d^{2}}\right )}}{45045 \, b d} \]
4/45045*(13*(1155*(sqrt(d*x + c)*b + a)^(3/2)*b^4*c^2 - 3465*sqrt(sqrt(d*x + c)*b + a)*a*b^4*c^2 - 990*(sqrt(d*x + c)*b + a)^(7/2)*b^2*c + 4158*(sqr t(d*x + c)*b + a)^(5/2)*a*b^2*c - 6930*(sqrt(d*x + c)*b + a)^(3/2)*a^2*b^2 *c + 6930*sqrt(sqrt(d*x + c)*b + a)*a^3*b^2*c + 315*(sqrt(d*x + c)*b + a)^ (11/2) - 1925*(sqrt(d*x + c)*b + a)^(9/2)*a + 4950*(sqrt(d*x + c)*b + a)^( 7/2)*a^2 - 6930*(sqrt(d*x + c)*b + a)^(5/2)*a^3 + 5775*(sqrt(d*x + c)*b + a)^(3/2)*a^4 - 3465*sqrt(sqrt(d*x + c)*b + a)*a^5)*a/(b^5*d^2) + (9009*(sq rt(d*x + c)*b + a)^(5/2)*b^4*c^2 - 30030*(sqrt(d*x + c)*b + a)^(3/2)*a*b^4 *c^2 + 45045*sqrt(sqrt(d*x + c)*b + a)*a^2*b^4*c^2 - 10010*(sqrt(d*x + c)* b + a)^(9/2)*b^2*c + 51480*(sqrt(d*x + c)*b + a)^(7/2)*a*b^2*c - 108108*(s qrt(d*x + c)*b + a)^(5/2)*a^2*b^2*c + 120120*(sqrt(d*x + c)*b + a)^(3/2)*a ^3*b^2*c - 90090*sqrt(sqrt(d*x + c)*b + a)*a^4*b^2*c + 3465*(sqrt(d*x + c) *b + a)^(13/2) - 24570*(sqrt(d*x + c)*b + a)^(11/2)*a + 75075*(sqrt(d*x + c)*b + a)^(9/2)*a^2 - 128700*(sqrt(d*x + c)*b + a)^(7/2)*a^3 + 135135*(sqr t(d*x + c)*b + a)^(5/2)*a^4 - 90090*(sqrt(d*x + c)*b + a)^(3/2)*a^5 + 4504 5*sqrt(sqrt(d*x + c)*b + a)*a^6)/(b^5*d^2))/(b*d)
Timed out. \[ \int x^2 \sqrt {a+b \sqrt {c+d x}} \, dx=\int x^2\,\sqrt {a+b\,\sqrt {c+d\,x}} \,d x \]