Integrand size = 21, antiderivative size = 326 \[ \int x^3 \sqrt {a+b \sqrt {c+d x}} \, dx=-\frac {4 a \left (a^2-b^2 c\right )^3 \left (a+b \sqrt {c+d x}\right )^{3/2}}{3 b^8 d^4}+\frac {4 \left (a^2-b^2 c\right )^2 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{5/2}}{5 b^8 d^4}-\frac {12 a \left (7 a^2-3 b^2 c\right ) \left (a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{7/2}}{7 b^8 d^4}+\frac {4 \left (35 a^4-30 a^2 b^2 c+3 b^4 c^2\right ) \left (a+b \sqrt {c+d x}\right )^{9/2}}{9 b^8 d^4}-\frac {20 a \left (7 a^2-3 b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{11/2}}{11 b^8 d^4}+\frac {12 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{13/2}}{13 b^8 d^4}-\frac {28 a \left (a+b \sqrt {c+d x}\right )^{15/2}}{15 b^8 d^4}+\frac {4 \left (a+b \sqrt {c+d x}\right )^{17/2}}{17 b^8 d^4} \] Output:
-4/3*a*(-b^2*c+a^2)^3*(a+b*(d*x+c)^(1/2))^(3/2)/b^8/d^4+4/5*(-b^2*c+a^2)^2 *(-b^2*c+7*a^2)*(a+b*(d*x+c)^(1/2))^(5/2)/b^8/d^4-12/7*a*(-3*b^2*c+7*a^2)* (-b^2*c+a^2)*(a+b*(d*x+c)^(1/2))^(7/2)/b^8/d^4+4/9*(3*b^4*c^2-30*a^2*b^2*c +35*a^4)*(a+b*(d*x+c)^(1/2))^(9/2)/b^8/d^4-20/11*a*(-3*b^2*c+7*a^2)*(a+b*( d*x+c)^(1/2))^(11/2)/b^8/d^4+12/13*(-b^2*c+7*a^2)*(a+b*(d*x+c)^(1/2))^(13/ 2)/b^8/d^4-28/15*a*(a+b*(d*x+c)^(1/2))^(15/2)/b^8/d^4+4/17*(a+b*(d*x+c)^(1 /2))^(17/2)/b^8/d^4
Time = 0.26 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.71 \[ \int x^3 \sqrt {a+b \sqrt {c+d x}} \, dx=\frac {4 \left (a+b \sqrt {c+d x}\right )^{3/2} \left (-14336 a^7+3840 a^5 b^2 (10 c-7 d x)+21504 a^6 b \sqrt {c+d x}-640 a^4 b^3 (104 c-49 d x) \sqrt {c+d x}-48 a^3 b^4 \left (616 c^2-1080 c d x+735 d^2 x^2\right )+24 a^2 b^5 \sqrt {c+d x} \left (2960 c^2-2716 c d x+1617 d^2 x^2\right )+6 a b^6 \left (320 c^3-3936 c^2 d x+5754 c d^2 x^2-7007 d^3 x^3\right )-231 b^7 \sqrt {c+d x} \left (128 c^3-160 c^2 d x+180 c d^2 x^2-195 d^3 x^3\right )\right )}{765765 b^8 d^4} \] Input:
Integrate[x^3*Sqrt[a + b*Sqrt[c + d*x]],x]
Output:
(4*(a + b*Sqrt[c + d*x])^(3/2)*(-14336*a^7 + 3840*a^5*b^2*(10*c - 7*d*x) + 21504*a^6*b*Sqrt[c + d*x] - 640*a^4*b^3*(104*c - 49*d*x)*Sqrt[c + d*x] - 48*a^3*b^4*(616*c^2 - 1080*c*d*x + 735*d^2*x^2) + 24*a^2*b^5*Sqrt[c + d*x] *(2960*c^2 - 2716*c*d*x + 1617*d^2*x^2) + 6*a*b^6*(320*c^3 - 3936*c^2*d*x + 5754*c*d^2*x^2 - 7007*d^3*x^3) - 231*b^7*Sqrt[c + d*x]*(128*c^3 - 160*c^ 2*d*x + 180*c*d^2*x^2 - 195*d^3*x^3)))/(765765*b^8*d^4)
Time = 0.73 (sec) , antiderivative size = 307, normalized size of antiderivative = 0.94, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {896, 25, 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.
| \(\displaystyle \int x^3 \sqrt {a+b \sqrt {c+d x}} \, dx\) |
| \(\Big \downarrow \) 896 |
| \(\displaystyle \frac {\int d^3 x^3 \sqrt {a+b \sqrt {c+d x}}d(c+d x)}{d^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -d^3 x^3 \sqrt {a+b \sqrt {c+d x}}d(c+d x)}{d^4}\) |
| \(\Big \downarrow \) 1732 |
| \(\displaystyle -\frac {2 \int -d^3 x^3 \sqrt {c+d x} \sqrt {a+b \sqrt {c+d x}}d\sqrt {c+d x}}{d^4}\) |
| \(\Big \downarrow \) 522 |
| \(\displaystyle -\frac {2 \int \left (-\frac {\left (a+b \sqrt {c+d x}\right )^{15/2}}{b^7}+\frac {7 a \left (a+b \sqrt {c+d x}\right )^{13/2}}{b^7}+\frac {3 \left (b^2 c-7 a^2\right ) \left (a+b \sqrt {c+d x}\right )^{11/2}}{b^7}-\frac {5 \left (3 a b^2 c-7 a^3\right ) \left (a+b \sqrt {c+d x}\right )^{9/2}}{b^7}+\frac {\left (-35 a^4+30 b^2 c a^2-3 b^4 c^2\right ) \left (a+b \sqrt {c+d x}\right )^{7/2}}{b^7}+\frac {3 \left (7 a^5-10 b^2 c a^3+3 b^4 c^2 a\right ) \left (a+b \sqrt {c+d x}\right )^{5/2}}{b^7}+\frac {\left (b^2 c-7 a^2\right ) \left (b^2 c-a^2\right )^2 \left (a+b \sqrt {c+d x}\right )^{3/2}}{b^7}+\frac {a \left (a^2-b^2 c\right )^3 \sqrt {a+b \sqrt {c+d x}}}{b^7}\right )d\sqrt {c+d x}}{d^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \left (-\frac {6 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{13/2}}{13 b^8}+\frac {10 a \left (7 a^2-3 b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{11/2}}{11 b^8}+\frac {6 a \left (7 a^2-3 b^2 c\right ) \left (a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{7/2}}{7 b^8}-\frac {2 \left (a^2-b^2 c\right )^2 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{5/2}}{5 b^8}+\frac {2 a \left (a^2-b^2 c\right )^3 \left (a+b \sqrt {c+d x}\right )^{3/2}}{3 b^8}-\frac {2 \left (35 a^4-30 a^2 b^2 c+3 b^4 c^2\right ) \left (a+b \sqrt {c+d x}\right )^{9/2}}{9 b^8}-\frac {2 \left (a+b \sqrt {c+d x}\right )^{17/2}}{17 b^8}+\frac {14 a \left (a+b \sqrt {c+d x}\right )^{15/2}}{15 b^8}\right )}{d^4}\) |
Input:
Int[x^3*Sqrt[a + b*Sqrt[c + d*x]],x]
Output:
(-2*((2*a*(a^2 - b^2*c)^3*(a + b*Sqrt[c + d*x])^(3/2))/(3*b^8) - (2*(a^2 - b^2*c)^2*(7*a^2 - b^2*c)*(a + b*Sqrt[c + d*x])^(5/2))/(5*b^8) + (6*a*(7*a ^2 - 3*b^2*c)*(a^2 - b^2*c)*(a + b*Sqrt[c + d*x])^(7/2))/(7*b^8) - (2*(35* a^4 - 30*a^2*b^2*c + 3*b^4*c^2)*(a + b*Sqrt[c + d*x])^(9/2))/(9*b^8) + (10 *a*(7*a^2 - 3*b^2*c)*(a + b*Sqrt[c + d*x])^(11/2))/(11*b^8) - (6*(7*a^2 - b^2*c)*(a + b*Sqrt[c + d*x])^(13/2))/(13*b^8) + (14*a*(a + b*Sqrt[c + d*x] )^(15/2))/(15*b^8) - (2*(a + b*Sqrt[c + d*x])^(17/2))/(17*b^8)))/d^4
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 = 0.20 (sec) , antiderivative size = 385, normalized size of antiderivative = 1.18
| method | result | size |
| derivativedivides | \(-\frac {4 \left (-\frac {\left (a +b \sqrt {d x +c}\right )^{\frac {17}{2}}}{17}+\frac {7 a \left (a +b \sqrt {d x +c}\right )^{\frac {15}{2}}}{15}+\frac {\left (3 b^{2} c -21 a^{2}\right ) \left (a +b \sqrt {d x +c}\right )^{\frac {13}{2}}}{13}+\frac {\left (8 \left (-b^{2} c +a^{2}\right ) a +2 a \left (-2 b^{2} c +6 a^{2}\right )+\left (-3 b^{2} c +15 a^{2}\right ) a \right ) \left (a +b \sqrt {d x +c}\right )^{\frac {11}{2}}}{11}+\frac {\left (-\left (-b^{2} c +a^{2}\right ) \left (-2 b^{2} c +6 a^{2}\right )-8 a^{2} \left (-b^{2} c +a^{2}\right )-\left (-b^{2} c +a^{2}\right )^{2}+\left (-8 \left (-b^{2} c +a^{2}\right ) a -2 a \left (-2 b^{2} c +6 a^{2}\right )\right ) a \right ) \left (a +b \sqrt {d x +c}\right )^{\frac {9}{2}}}{9}+\frac {\left (6 \left (-b^{2} c +a^{2}\right )^{2} a +\left (\left (-b^{2} c +a^{2}\right ) \left (-2 b^{2} c +6 a^{2}\right )+8 a^{2} \left (-b^{2} c +a^{2}\right )+\left (-b^{2} c +a^{2}\right )^{2}\right ) a \right ) \left (a +b \sqrt {d x +c}\right )^{\frac {7}{2}}}{7}+\frac {\left (-\left (-b^{2} c +a^{2}\right )^{3}-6 \left (-b^{2} c +a^{2}\right )^{2} a^{2}\right ) \left (a +b \sqrt {d x +c}\right )^{\frac {5}{2}}}{5}+\frac {\left (-b^{2} c +a^{2}\right )^{3} a \left (a +b \sqrt {d x +c}\right )^{\frac {3}{2}}}{3}\right )}{d^{4} b^{8}}\) | \(385\) |
| default | \(-\frac {4 \left (-\frac {\left (a +b \sqrt {d x +c}\right )^{\frac {17}{2}}}{17}+\frac {7 a \left (a +b \sqrt {d x +c}\right )^{\frac {15}{2}}}{15}+\frac {\left (3 b^{2} c -21 a^{2}\right ) \left (a +b \sqrt {d x +c}\right )^{\frac {13}{2}}}{13}+\frac {\left (8 \left (-b^{2} c +a^{2}\right ) a +2 a \left (-2 b^{2} c +6 a^{2}\right )+\left (-3 b^{2} c +15 a^{2}\right ) a \right ) \left (a +b \sqrt {d x +c}\right )^{\frac {11}{2}}}{11}+\frac {\left (-\left (-b^{2} c +a^{2}\right ) \left (-2 b^{2} c +6 a^{2}\right )-8 a^{2} \left (-b^{2} c +a^{2}\right )-\left (-b^{2} c +a^{2}\right )^{2}+\left (-8 \left (-b^{2} c +a^{2}\right ) a -2 a \left (-2 b^{2} c +6 a^{2}\right )\right ) a \right ) \left (a +b \sqrt {d x +c}\right )^{\frac {9}{2}}}{9}+\frac {\left (6 \left (-b^{2} c +a^{2}\right )^{2} a +\left (\left (-b^{2} c +a^{2}\right ) \left (-2 b^{2} c +6 a^{2}\right )+8 a^{2} \left (-b^{2} c +a^{2}\right )+\left (-b^{2} c +a^{2}\right )^{2}\right ) a \right ) \left (a +b \sqrt {d x +c}\right )^{\frac {7}{2}}}{7}+\frac {\left (-\left (-b^{2} c +a^{2}\right )^{3}-6 \left (-b^{2} c +a^{2}\right )^{2} a^{2}\right ) \left (a +b \sqrt {d x +c}\right )^{\frac {5}{2}}}{5}+\frac {\left (-b^{2} c +a^{2}\right )^{3} a \left (a +b \sqrt {d x +c}\right )^{\frac {3}{2}}}{3}\right )}{d^{4} b^{8}}\) | \(385\) |
Input:
int(x^3*(a+b*(d*x+c)^(1/2))^(1/2),x,method=_RETURNVERBOSE)
Output:
-4/d^4/b^8*(-1/17*(a+b*(d*x+c)^(1/2))^(17/2)+7/15*a*(a+b*(d*x+c)^(1/2))^(1 5/2)+1/13*(3*b^2*c-21*a^2)*(a+b*(d*x+c)^(1/2))^(13/2)+1/11*(8*(-b^2*c+a^2) *a+2*a*(-2*b^2*c+6*a^2)+(-3*b^2*c+15*a^2)*a)*(a+b*(d*x+c)^(1/2))^(11/2)+1/ 9*(-(-b^2*c+a^2)*(-2*b^2*c+6*a^2)-8*a^2*(-b^2*c+a^2)-(-b^2*c+a^2)^2+(-8*(- b^2*c+a^2)*a-2*a*(-2*b^2*c+6*a^2))*a)*(a+b*(d*x+c)^(1/2))^(9/2)+1/7*(6*(-b ^2*c+a^2)^2*a+((-b^2*c+a^2)*(-2*b^2*c+6*a^2)+8*a^2*(-b^2*c+a^2)+(-b^2*c+a^ 2)^2)*a)*(a+b*(d*x+c)^(1/2))^(7/2)+1/5*(-(-b^2*c+a^2)^3-6*(-b^2*c+a^2)^2*a ^2)*(a+b*(d*x+c)^(1/2))^(5/2)+1/3*(-b^2*c+a^2)^3*a*(a+b*(d*x+c)^(1/2))^(3/ 2))
Time = 0.12 (sec) , antiderivative size = 286, normalized size of antiderivative = 0.88 \[ \int x^3 \sqrt {a+b \sqrt {c+d x}} \, dx=\frac {4 \, {\left (45045 \, b^{8} d^{4} x^{4} - 29568 \, b^{8} c^{4} + 72960 \, a^{2} b^{6} c^{3} - 96128 \, a^{4} b^{4} c^{2} + 59904 \, a^{6} b^{2} c - 14336 \, a^{8} + 231 \, {\left (15 \, b^{8} c - 14 \, a^{2} b^{6}\right )} d^{3} x^{3} - 28 \, {\left (165 \, b^{8} c^{2} - 291 \, a^{2} b^{6} c + 140 \, a^{4} b^{4}\right )} d^{2} x^{2} + 32 \, {\left (231 \, b^{8} c^{3} - 555 \, a^{2} b^{6} c^{2} + 520 \, a^{4} b^{4} c - 168 \, a^{6} b^{2}\right )} d x + {\left (3003 \, a b^{7} d^{3} x^{3} - 27648 \, a b^{7} c^{3} + 41472 \, a^{3} b^{5} c^{2} - 28160 \, a^{5} b^{3} c + 7168 \, a^{7} b - 3528 \, {\left (2 \, a b^{7} c - a^{3} b^{5}\right )} d^{2} x^{2} + 32 \, {\left (417 \, a b^{7} c^{2} - 417 \, a^{3} b^{5} c + 140 \, a^{5} b^{3}\right )} d x\right )} \sqrt {d x + c}\right )} \sqrt {\sqrt {d x + c} b + a}}{765765 \, b^{8} d^{4}} \] Input:
integrate(x^3*(a+b*(d*x+c)^(1/2))^(1/2),x, algorithm="fricas")
Output:
4/765765*(45045*b^8*d^4*x^4 - 29568*b^8*c^4 + 72960*a^2*b^6*c^3 - 96128*a^ 4*b^4*c^2 + 59904*a^6*b^2*c - 14336*a^8 + 231*(15*b^8*c - 14*a^2*b^6)*d^3* x^3 - 28*(165*b^8*c^2 - 291*a^2*b^6*c + 140*a^4*b^4)*d^2*x^2 + 32*(231*b^8 *c^3 - 555*a^2*b^6*c^2 + 520*a^4*b^4*c - 168*a^6*b^2)*d*x + (3003*a*b^7*d^ 3*x^3 - 27648*a*b^7*c^3 + 41472*a^3*b^5*c^2 - 28160*a^5*b^3*c + 7168*a^7*b - 3528*(2*a*b^7*c - a^3*b^5)*d^2*x^2 + 32*(417*a*b^7*c^2 - 417*a^3*b^5*c + 140*a^5*b^3)*d*x)*sqrt(d*x + c))*sqrt(sqrt(d*x + c)*b + a)/(b^8*d^4)
Time = 0.88 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.10 \[ \int x^3 \sqrt {a+b \sqrt {c+d x}} \, dx=\begin {cases} \frac {2 \left (\begin {cases} \frac {2 \left (- \frac {7 a \left (a + b \sqrt {c + d x}\right )^{\frac {15}{2}}}{15 b^{6}} + \frac {\left (a + b \sqrt {c + d x}\right )^{\frac {17}{2}}}{17 b^{6}} + \frac {\left (a + b \sqrt {c + d x}\right )^{\frac {13}{2}} \cdot \left (21 a^{2} - 3 b^{2} c\right )}{13 b^{6}} + \frac {\left (a + b \sqrt {c + d x}\right )^{\frac {11}{2}} \left (- 35 a^{3} + 15 a b^{2} c\right )}{11 b^{6}} + \frac {\left (a + b \sqrt {c + d x}\right )^{\frac {9}{2}} \cdot \left (35 a^{4} - 30 a^{2} b^{2} c + 3 b^{4} c^{2}\right )}{9 b^{6}} + \frac {\left (a + b \sqrt {c + d x}\right )^{\frac {7}{2}} \left (- 21 a^{5} + 30 a^{3} b^{2} c - 9 a b^{4} c^{2}\right )}{7 b^{6}} + \frac {\left (a + b \sqrt {c + d x}\right )^{\frac {5}{2}} \cdot \left (7 a^{6} - 15 a^{4} b^{2} c + 9 a^{2} b^{4} c^{2} - b^{6} c^{3}\right )}{5 b^{6}} + \frac {\left (a + b \sqrt {c + d x}\right )^{\frac {3}{2}} \left (- a^{7} + 3 a^{5} b^{2} c - 3 a^{3} b^{4} c^{2} + a b^{6} c^{3}\right )}{3 b^{6}}\right )}{b^{2}} & \text {for}\: b \neq 0 \\\frac {\sqrt {a} d^{4} x^{4}}{8} & \text {otherwise} \end {cases}\right )}{d^{4}} & \text {for}\: d \neq 0 \\\frac {x^{4} \sqrt {a + b \sqrt {c}}}{4} & \text {otherwise} \end {cases} \] Input:
integrate(x**3*(a+b*(d*x+c)**(1/2))**(1/2),x)
Output:
Piecewise((2*Piecewise((2*(-7*a*(a + b*sqrt(c + d*x))**(15/2)/(15*b**6) + (a + b*sqrt(c + d*x))**(17/2)/(17*b**6) + (a + b*sqrt(c + d*x))**(13/2)*(2 1*a**2 - 3*b**2*c)/(13*b**6) + (a + b*sqrt(c + d*x))**(11/2)*(-35*a**3 + 1 5*a*b**2*c)/(11*b**6) + (a + b*sqrt(c + d*x))**(9/2)*(35*a**4 - 30*a**2*b* *2*c + 3*b**4*c**2)/(9*b**6) + (a + b*sqrt(c + d*x))**(7/2)*(-21*a**5 + 30 *a**3*b**2*c - 9*a*b**4*c**2)/(7*b**6) + (a + b*sqrt(c + d*x))**(5/2)*(7*a **6 - 15*a**4*b**2*c + 9*a**2*b**4*c**2 - b**6*c**3)/(5*b**6) + (a + b*sqr t(c + d*x))**(3/2)*(-a**7 + 3*a**5*b**2*c - 3*a**3*b**4*c**2 + a*b**6*c**3 )/(3*b**6))/b**2, Ne(b, 0)), (sqrt(a)*d**4*x**4/8, True))/d**4, Ne(d, 0)), (x**4*sqrt(a + b*sqrt(c))/4, True))
Time = 0.04 (sec) , antiderivative size = 268, normalized size of antiderivative = 0.82 \[ \int x^3 \sqrt {a+b \sqrt {c+d x}} \, dx=\frac {4 \, {\left (45045 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {17}{2}} - 357357 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {15}{2}} a - 176715 \, {\left (b^{2} c - 7 \, a^{2}\right )} {\left (\sqrt {d x + c} b + a\right )}^{\frac {13}{2}} + 348075 \, {\left (3 \, a b^{2} c - 7 \, a^{3}\right )} {\left (\sqrt {d x + c} b + a\right )}^{\frac {11}{2}} + 85085 \, {\left (3 \, b^{4} c^{2} - 30 \, a^{2} b^{2} c + 35 \, a^{4}\right )} {\left (\sqrt {d x + c} b + a\right )}^{\frac {9}{2}} - 328185 \, {\left (3 \, a b^{4} c^{2} - 10 \, a^{3} b^{2} c + 7 \, a^{5}\right )} {\left (\sqrt {d x + c} b + a\right )}^{\frac {7}{2}} - 153153 \, {\left (b^{6} c^{3} - 9 \, a^{2} b^{4} c^{2} + 15 \, a^{4} b^{2} c - 7 \, a^{6}\right )} {\left (\sqrt {d x + c} b + a\right )}^{\frac {5}{2}} + 255255 \, {\left (a b^{6} c^{3} - 3 \, a^{3} b^{4} c^{2} + 3 \, a^{5} b^{2} c - a^{7}\right )} {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}}\right )}}{765765 \, b^{8} d^{4}} \] Input:
integrate(x^3*(a+b*(d*x+c)^(1/2))^(1/2),x, algorithm="maxima")
Output:
4/765765*(45045*(sqrt(d*x + c)*b + a)^(17/2) - 357357*(sqrt(d*x + c)*b + a )^(15/2)*a - 176715*(b^2*c - 7*a^2)*(sqrt(d*x + c)*b + a)^(13/2) + 348075* (3*a*b^2*c - 7*a^3)*(sqrt(d*x + c)*b + a)^(11/2) + 85085*(3*b^4*c^2 - 30*a ^2*b^2*c + 35*a^4)*(sqrt(d*x + c)*b + a)^(9/2) - 328185*(3*a*b^4*c^2 - 10* a^3*b^2*c + 7*a^5)*(sqrt(d*x + c)*b + a)^(7/2) - 153153*(b^6*c^3 - 9*a^2*b ^4*c^2 + 15*a^4*b^2*c - 7*a^6)*(sqrt(d*x + c)*b + a)^(5/2) + 255255*(a*b^6 *c^3 - 3*a^3*b^4*c^2 + 3*a^5*b^2*c - a^7)*(sqrt(d*x + c)*b + a)^(3/2))/(b^ 8*d^4)
Leaf count of result is larger than twice the leaf count of optimal. 915 vs. \(2 (279) = 558\).
Time = 0.16 (sec) , antiderivative size = 915, normalized size of antiderivative = 2.81 \[ \int x^3 \sqrt {a+b \sqrt {c+d x}} \, dx=\text {Too large to display} \] Input:
integrate(x^3*(a+b*(d*x+c)^(1/2))^(1/2),x, algorithm="giac")
Output:
-4/765765*(17*(15015*(sqrt(d*x + c)*b + a)^(3/2)*b^6*c^3 - 45045*sqrt(sqrt (d*x + c)*b + a)*a*b^6*c^3 - 19305*(sqrt(d*x + c)*b + a)^(7/2)*b^4*c^2 + 8 1081*(sqrt(d*x + c)*b + a)^(5/2)*a*b^4*c^2 - 135135*(sqrt(d*x + c)*b + a)^ (3/2)*a^2*b^4*c^2 + 135135*sqrt(sqrt(d*x + c)*b + a)*a^3*b^4*c^2 + 12285*( sqrt(d*x + c)*b + a)^(11/2)*b^2*c - 75075*(sqrt(d*x + c)*b + a)^(9/2)*a*b^ 2*c + 193050*(sqrt(d*x + c)*b + a)^(7/2)*a^2*b^2*c - 270270*(sqrt(d*x + c) *b + a)^(5/2)*a^3*b^2*c + 225225*(sqrt(d*x + c)*b + a)^(3/2)*a^4*b^2*c - 1 35135*sqrt(sqrt(d*x + c)*b + a)*a^5*b^2*c - 3003*(sqrt(d*x + c)*b + a)^(15 /2) + 24255*(sqrt(d*x + c)*b + a)^(13/2)*a - 85995*(sqrt(d*x + c)*b + a)^( 11/2)*a^2 + 175175*(sqrt(d*x + c)*b + a)^(9/2)*a^3 - 225225*(sqrt(d*x + c) *b + a)^(7/2)*a^4 + 189189*(sqrt(d*x + c)*b + a)^(5/2)*a^5 - 105105*(sqrt( d*x + c)*b + a)^(3/2)*a^6 + 45045*sqrt(sqrt(d*x + c)*b + a)*a^7)*a/(b^7*d^ 3) + (153153*(sqrt(d*x + c)*b + a)^(5/2)*b^6*c^3 - 510510*(sqrt(d*x + c)*b + a)^(3/2)*a*b^6*c^3 + 765765*sqrt(sqrt(d*x + c)*b + a)*a^2*b^6*c^3 - 255 255*(sqrt(d*x + c)*b + a)^(9/2)*b^4*c^2 + 1312740*(sqrt(d*x + c)*b + a)^(7 /2)*a*b^4*c^2 - 2756754*(sqrt(d*x + c)*b + a)^(5/2)*a^2*b^4*c^2 + 3063060* (sqrt(d*x + c)*b + a)^(3/2)*a^3*b^4*c^2 - 2297295*sqrt(sqrt(d*x + c)*b + a )*a^4*b^4*c^2 + 176715*(sqrt(d*x + c)*b + a)^(13/2)*b^2*c - 1253070*(sqrt( d*x + c)*b + a)^(11/2)*a*b^2*c + 3828825*(sqrt(d*x + c)*b + a)^(9/2)*a^2*b ^2*c - 6563700*(sqrt(d*x + c)*b + a)^(7/2)*a^3*b^2*c + 6891885*(sqrt(d*...
Timed out. \[ \int x^3 \sqrt {a+b \sqrt {c+d x}} \, dx=\int x^3\,\sqrt {a+b\,\sqrt {c+d\,x}} \,d x \] Input:
int(x^3*(a + b*(c + d*x)^(1/2))^(1/2),x)
Output:
int(x^3*(a + b*(c + d*x)^(1/2))^(1/2), x)
Time = 0.23 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.09 \[ \int x^3 \sqrt {a+b \sqrt {c+d x}} \, dx=\frac {4 \sqrt {\sqrt {d x +c}\, b +a}\, \left (-28160 \sqrt {d x +c}\, a^{5} b^{3} c +41472 \sqrt {d x +c}\, a^{3} b^{5} c^{2}-27648 \sqrt {d x +c}\, a \,b^{7} c^{3}-5376 a^{6} b^{2} d x -3920 a^{4} b^{4} d^{2} x^{2}-3234 a^{2} b^{6} d^{3} x^{3}+7392 b^{8} c^{3} d x -4620 b^{8} c^{2} d^{2} x^{2}+3465 b^{8} c \,d^{3} x^{3}-29568 b^{8} c^{4}-13344 \sqrt {d x +c}\, a^{3} b^{5} c d x +13344 \sqrt {d x +c}\, a \,b^{7} c^{2} d x -7056 \sqrt {d x +c}\, a \,b^{7} c \,d^{2} x^{2}+59904 a^{6} b^{2} c -96128 a^{4} b^{4} c^{2}+72960 a^{2} b^{6} c^{3}+45045 b^{8} d^{4} x^{4}+4480 \sqrt {d x +c}\, a^{5} b^{3} d x +3528 \sqrt {d x +c}\, a^{3} b^{5} d^{2} x^{2}+3003 \sqrt {d x +c}\, a \,b^{7} d^{3} x^{3}+16640 a^{4} b^{4} c d x -17760 a^{2} b^{6} c^{2} d x +8148 a^{2} b^{6} c \,d^{2} x^{2}+7168 \sqrt {d x +c}\, a^{7} b -14336 a^{8}\right )}{765765 b^{8} d^{4}} \] Input:
int(x^3*(a+b*(d*x+c)^(1/2))^(1/2),x)
Output:
(4*sqrt(sqrt(c + d*x)*b + a)*(7168*sqrt(c + d*x)*a**7*b - 28160*sqrt(c + d *x)*a**5*b**3*c + 4480*sqrt(c + d*x)*a**5*b**3*d*x + 41472*sqrt(c + d*x)*a **3*b**5*c**2 - 13344*sqrt(c + d*x)*a**3*b**5*c*d*x + 3528*sqrt(c + d*x)*a **3*b**5*d**2*x**2 - 27648*sqrt(c + d*x)*a*b**7*c**3 + 13344*sqrt(c + d*x) *a*b**7*c**2*d*x - 7056*sqrt(c + d*x)*a*b**7*c*d**2*x**2 + 3003*sqrt(c + d *x)*a*b**7*d**3*x**3 - 14336*a**8 + 59904*a**6*b**2*c - 5376*a**6*b**2*d*x - 96128*a**4*b**4*c**2 + 16640*a**4*b**4*c*d*x - 3920*a**4*b**4*d**2*x**2 + 72960*a**2*b**6*c**3 - 17760*a**2*b**6*c**2*d*x + 8148*a**2*b**6*c*d**2 *x**2 - 3234*a**2*b**6*d**3*x**3 - 29568*b**8*c**4 + 7392*b**8*c**3*d*x - 4620*b**8*c**2*d**2*x**2 + 3465*b**8*c*d**3*x**3 + 45045*b**8*d**4*x**4))/ (765765*b**8*d**4)