\(\int \frac {x^3}{\sqrt {a+b \sqrt {c+d x}}} \, dx\) [132]

Optimal result

Integrand size = 21, antiderivative size = 324 \[ \int \frac {x^3}{\sqrt {a+b \sqrt {c+d x}}} \, dx=-\frac {4 a \left (a^2-b^2 c\right )^3 \sqrt {a+b \sqrt {c+d x}}}{b^8 d^4}+\frac {4 \left (a^2-b^2 c\right )^2 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{3/2}}{3 b^8 d^4}-\frac {12 a \left (7 a^2-3 b^2 c\right ) \left (a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{5/2}}{5 b^8 d^4}+\frac {4 \left (35 a^4-30 a^2 b^2 c+3 b^4 c^2\right ) \left (a+b \sqrt {c+d x}\right )^{7/2}}{7 b^8 d^4}-\frac {20 a \left (7 a^2-3 b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{9/2}}{9 b^8 d^4}+\frac {12 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{11/2}}{11 b^8 d^4}-\frac {28 a \left (a+b \sqrt {c+d x}\right )^{13/2}}{13 b^8 d^4}+\frac {4 \left (a+b \sqrt {c+d x}\right )^{15/2}}{15 b^8 d^4} \] Output:

-4*a*(-b^2*c+a^2)^3*(a+b*(d*x+c)^(1/2))^(1/2)/b^8/d^4+4/3*(-b^2*c+a^2)^2*( 
-b^2*c+7*a^2)*(a+b*(d*x+c)^(1/2))^(3/2)/b^8/d^4-12/5*a*(-3*b^2*c+7*a^2)*(- 
b^2*c+a^2)*(a+b*(d*x+c)^(1/2))^(5/2)/b^8/d^4+4/7*(3*b^4*c^2-30*a^2*b^2*c+3 
5*a^4)*(a+b*(d*x+c)^(1/2))^(7/2)/b^8/d^4-20/9*a*(-3*b^2*c+7*a^2)*(a+b*(d*x 
+c)^(1/2))^(9/2)/b^8/d^4+12/11*(-b^2*c+7*a^2)*(a+b*(d*x+c)^(1/2))^(11/2)/b 
^8/d^4-28/13*a*(a+b*(d*x+c)^(1/2))^(13/2)/b^8/d^4+4/15*(a+b*(d*x+c)^(1/2)) 
^(15/2)/b^8/d^4
 

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.72 \[ \int \frac {x^3}{\sqrt {a+b \sqrt {c+d x}}} \, dx=\frac {4 \sqrt {a+b \sqrt {c+d x}} \left (-14336 a^7+768 a^5 b^2 (58 c-7 d x)+7168 a^6 b \sqrt {c+d x}-640 a^4 b^3 (32 c-7 d x) \sqrt {c+d x}+24 a^2 b^5 \sqrt {c+d x} \left (784 c^2-356 c d x+147 d^2 x^2\right )-16 a^3 b^4 \left (2936 c^2-680 c d x+245 d^2 x^2\right )+6 a b^6 \left (2880 c^3-928 c^2 d x+658 c d^2 x^2-539 d^3 x^3\right )-39 b^7 \sqrt {c+d x} \left (128 c^3-96 c^2 d x+84 c d^2 x^2-77 d^3 x^3\right )\right )}{45045 b^8 d^4} \] Input:

Integrate[x^3/Sqrt[a + b*Sqrt[c + d*x]],x]
 

Output:

(4*Sqrt[a + b*Sqrt[c + d*x]]*(-14336*a^7 + 768*a^5*b^2*(58*c - 7*d*x) + 71 
68*a^6*b*Sqrt[c + d*x] - 640*a^4*b^3*(32*c - 7*d*x)*Sqrt[c + d*x] + 24*a^2 
*b^5*Sqrt[c + d*x]*(784*c^2 - 356*c*d*x + 147*d^2*x^2) - 16*a^3*b^4*(2936* 
c^2 - 680*c*d*x + 245*d^2*x^2) + 6*a*b^6*(2880*c^3 - 928*c^2*d*x + 658*c*d 
^2*x^2 - 539*d^3*x^3) - 39*b^7*Sqrt[c + d*x]*(128*c^3 - 96*c^2*d*x + 84*c* 
d^2*x^2 - 77*d^3*x^3)))/(45045*b^8*d^4)
 

Rubi [A] (verified)

Time = 0.73 (sec) , antiderivative size = 305, 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.

\(\displaystyle \int \frac {x^3}{\sqrt {a+b \sqrt {c+d x}}} \, dx\)

\(\Big \downarrow \) 896

\(\displaystyle \frac {\int \frac {d^3 x^3}{\sqrt {a+b \sqrt {c+d x}}}d(c+d x)}{d^4}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\int -\frac {d^3 x^3}{\sqrt {a+b \sqrt {c+d x}}}d(c+d x)}{d^4}\)

\(\Big \downarrow \) 1732

\(\displaystyle -\frac {2 \int -\frac {d^3 x^3 \sqrt {c+d x}}{\sqrt {a+b \sqrt {c+d x}}}d\sqrt {c+d x}}{d^4}\)

\(\Big \downarrow \) 522

\(\displaystyle -\frac {2 \int \left (-\frac {\left (a+b \sqrt {c+d x}\right )^{13/2}}{b^7}+\frac {7 a \left (a+b \sqrt {c+d x}\right )^{11/2}}{b^7}+\frac {3 \left (b^2 c-7 a^2\right ) \left (a+b \sqrt {c+d x}\right )^{9/2}}{b^7}-\frac {5 \left (3 a b^2 c-7 a^3\right ) \left (a+b \sqrt {c+d x}\right )^{7/2}}{b^7}+\frac {\left (-35 a^4+30 b^2 c a^2-3 b^4 c^2\right ) \left (a+b \sqrt {c+d x}\right )^{5/2}}{b^7}+\frac {3 \left (7 a^5-10 b^2 c a^3+3 b^4 c^2 a\right ) \left (a+b \sqrt {c+d x}\right )^{3/2}}{b^7}+\frac {\left (b^2 c-7 a^2\right ) \left (b^2 c-a^2\right )^2 \sqrt {a+b \sqrt {c+d x}}}{b^7}+\frac {a \left (a^2-b^2 c\right )^3}{b^7 \sqrt {a+b \sqrt {c+d x}}}\right )d\sqrt {c+d x}}{d^4}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 \left (-\frac {6 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{11/2}}{11 b^8}+\frac {10 a \left (7 a^2-3 b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{9/2}}{9 b^8}+\frac {6 a \left (7 a^2-3 b^2 c\right ) \left (a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{5/2}}{5 b^8}-\frac {2 \left (a^2-b^2 c\right )^2 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{3/2}}{3 b^8}+\frac {2 a \left (a^2-b^2 c\right )^3 \sqrt {a+b \sqrt {c+d x}}}{b^8}-\frac {2 \left (35 a^4-30 a^2 b^2 c+3 b^4 c^2\right ) \left (a+b \sqrt {c+d x}\right )^{7/2}}{7 b^8}-\frac {2 \left (a+b \sqrt {c+d x}\right )^{15/2}}{15 b^8}+\frac {14 a \left (a+b \sqrt {c+d x}\right )^{13/2}}{13 b^8}\right )}{d^4}\)

Input:

Int[x^3/Sqrt[a + b*Sqrt[c + d*x]],x]
 

Output:

(-2*((2*a*(a^2 - b^2*c)^3*Sqrt[a + b*Sqrt[c + d*x]])/b^8 - (2*(a^2 - b^2*c 
)^2*(7*a^2 - b^2*c)*(a + b*Sqrt[c + d*x])^(3/2))/(3*b^8) + (6*a*(7*a^2 - 3 
*b^2*c)*(a^2 - b^2*c)*(a + b*Sqrt[c + d*x])^(5/2))/(5*b^8) - (2*(35*a^4 - 
30*a^2*b^2*c + 3*b^4*c^2)*(a + b*Sqrt[c + d*x])^(7/2))/(7*b^8) + (10*a*(7* 
a^2 - 3*b^2*c)*(a + b*Sqrt[c + d*x])^(9/2))/(9*b^8) - (6*(7*a^2 - b^2*c)*( 
a + b*Sqrt[c + d*x])^(11/2))/(11*b^8) + (14*a*(a + b*Sqrt[c + d*x])^(13/2) 
)/(13*b^8) - (2*(a + b*Sqrt[c + d*x])^(15/2))/(15*b^8)))/d^4
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 [A] (verified)

Time = 0.22 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.19

method	result	size
derivativedivides	\(-\frac {4 \left (-\frac {\left (a +b \sqrt {d x +c}\right )^{\frac {15}{2}}}{15}+\frac {7 a \left (a +b \sqrt {d x +c}\right )^{\frac {13}{2}}}{13}+\frac {\left (3 b^{2} c -21 a^{2}\right ) \left (a +b \sqrt {d x +c}\right )^{\frac {11}{2}}}{11}+\frac {\left (8 \left (-b^{2} c +a^{2}\right ) a +2 a \left (-2 b^{2} c +6 a^{2}\right )+\left (-3 b^{2} c +15 a^{2}\right ) a \right ) \left (a +b \sqrt {d x +c}\right )^{\frac {9}{2}}}{9}+\frac {\left (-\left (-b^{2} c +a^{2}\right ) \left (-2 b^{2} c +6 a^{2}\right )-8 a^{2} \left (-b^{2} c +a^{2}\right )-\left (-b^{2} c +a^{2}\right )^{2}+\left (-8 \left (-b^{2} c +a^{2}\right ) a -2 a \left (-2 b^{2} c +6 a^{2}\right )\right ) a \right ) \left (a +b \sqrt {d x +c}\right )^{\frac {7}{2}}}{7}+\frac {\left (6 \left (-b^{2} c +a^{2}\right )^{2} a +\left (\left (-b^{2} c +a^{2}\right ) \left (-2 b^{2} c +6 a^{2}\right )+8 a^{2} \left (-b^{2} c +a^{2}\right )+\left (-b^{2} c +a^{2}\right )^{2}\right ) a \right ) \left (a +b \sqrt {d x +c}\right )^{\frac {5}{2}}}{5}+\frac {\left (-\left (-b^{2} c +a^{2}\right )^{3}-6 \left (-b^{2} c +a^{2}\right )^{2} a^{2}\right ) \left (a +b \sqrt {d x +c}\right )^{\frac {3}{2}}}{3}+\left (-b^{2} c +a^{2}\right )^{3} a \sqrt {a +b \sqrt {d x +c}}\right )}{d^{4} b^{8}}\)	\(384\)
default	\(-\frac {4 \left (-\frac {\left (a +b \sqrt {d x +c}\right )^{\frac {15}{2}}}{15}+\frac {7 a \left (a +b \sqrt {d x +c}\right )^{\frac {13}{2}}}{13}+\frac {\left (3 b^{2} c -21 a^{2}\right ) \left (a +b \sqrt {d x +c}\right )^{\frac {11}{2}}}{11}+\frac {\left (8 \left (-b^{2} c +a^{2}\right ) a +2 a \left (-2 b^{2} c +6 a^{2}\right )+\left (-3 b^{2} c +15 a^{2}\right ) a \right ) \left (a +b \sqrt {d x +c}\right )^{\frac {9}{2}}}{9}+\frac {\left (-\left (-b^{2} c +a^{2}\right ) \left (-2 b^{2} c +6 a^{2}\right )-8 a^{2} \left (-b^{2} c +a^{2}\right )-\left (-b^{2} c +a^{2}\right )^{2}+\left (-8 \left (-b^{2} c +a^{2}\right ) a -2 a \left (-2 b^{2} c +6 a^{2}\right )\right ) a \right ) \left (a +b \sqrt {d x +c}\right )^{\frac {7}{2}}}{7}+\frac {\left (6 \left (-b^{2} c +a^{2}\right )^{2} a +\left (\left (-b^{2} c +a^{2}\right ) \left (-2 b^{2} c +6 a^{2}\right )+8 a^{2} \left (-b^{2} c +a^{2}\right )+\left (-b^{2} c +a^{2}\right )^{2}\right ) a \right ) \left (a +b \sqrt {d x +c}\right )^{\frac {5}{2}}}{5}+\frac {\left (-\left (-b^{2} c +a^{2}\right )^{3}-6 \left (-b^{2} c +a^{2}\right )^{2} a^{2}\right ) \left (a +b \sqrt {d x +c}\right )^{\frac {3}{2}}}{3}+\left (-b^{2} c +a^{2}\right )^{3} a \sqrt {a +b \sqrt {d x +c}}\right )}{d^{4} b^{8}}\)	\(384\)

Input:

int(x^3/(a+b*(d*x+c)^(1/2))^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-4/d^4/b^8*(-1/15*(a+b*(d*x+c)^(1/2))^(15/2)+7/13*a*(a+b*(d*x+c)^(1/2))^(1 
3/2)+1/11*(3*b^2*c-21*a^2)*(a+b*(d*x+c)^(1/2))^(11/2)+1/9*(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))^(9/2)+1/7* 
(-(-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))^(7/2)+1/5*(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))^(5/2)+1/3*(-(-b^2*c+a^2)^3-6*(-b^2*c+a^2)^2*a^2 
)*(a+b*(d*x+c)^(1/2))^(3/2)+(-b^2*c+a^2)^3*a*(a+b*(d*x+c)^(1/2))^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.71 \[ \int \frac {x^3}{\sqrt {a+b \sqrt {c+d x}}} \, dx=-\frac {4 \, {\left (3234 \, a b^{6} d^{3} x^{3} - 17280 \, a b^{6} c^{3} + 46976 \, a^{3} b^{4} c^{2} - 44544 \, a^{5} b^{2} c + 14336 \, a^{7} - 28 \, {\left (141 \, a b^{6} c - 140 \, a^{3} b^{4}\right )} d^{2} x^{2} + 64 \, {\left (87 \, a b^{6} c^{2} - 170 \, a^{3} b^{4} c + 84 \, a^{5} b^{2}\right )} d x - {\left (3003 \, b^{7} d^{3} x^{3} - 4992 \, b^{7} c^{3} + 18816 \, a^{2} b^{5} c^{2} - 20480 \, a^{4} b^{3} c + 7168 \, a^{6} b - 252 \, {\left (13 \, b^{7} c - 14 \, a^{2} b^{5}\right )} d^{2} x^{2} + 32 \, {\left (117 \, b^{7} c^{2} - 267 \, a^{2} b^{5} c + 140 \, a^{4} b^{3}\right )} d x\right )} \sqrt {d x + c}\right )} \sqrt {\sqrt {d x + c} b + a}}{45045 \, b^{8} d^{4}} \] Input:

integrate(x^3/(a+b*(d*x+c)^(1/2))^(1/2),x, algorithm="fricas")
 

Output:

-4/45045*(3234*a*b^6*d^3*x^3 - 17280*a*b^6*c^3 + 46976*a^3*b^4*c^2 - 44544 
*a^5*b^2*c + 14336*a^7 - 28*(141*a*b^6*c - 140*a^3*b^4)*d^2*x^2 + 64*(87*a 
*b^6*c^2 - 170*a^3*b^4*c + 84*a^5*b^2)*d*x - (3003*b^7*d^3*x^3 - 4992*b^7* 
c^3 + 18816*a^2*b^5*c^2 - 20480*a^4*b^3*c + 7168*a^6*b - 252*(13*b^7*c - 1 
4*a^2*b^5)*d^2*x^2 + 32*(117*b^7*c^2 - 267*a^2*b^5*c + 140*a^4*b^3)*d*x)*s 
qrt(d*x + c))*sqrt(sqrt(d*x + c)*b + a)/(b^8*d^4)
 

Sympy [A] (verification not implemented)

Time = 0.91 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.10 \[ \int \frac {x^3}{\sqrt {a+b \sqrt {c+d x}}} \, dx=\begin {cases} \frac {2 \left (\begin {cases} \frac {2 \left (- \frac {7 a \left (a + b \sqrt {c + d x}\right )^{\frac {13}{2}}}{13 b^{6}} + \frac {\left (a + b \sqrt {c + d x}\right )^{\frac {15}{2}}}{15 b^{6}} + \frac {\left (a + b \sqrt {c + d x}\right )^{\frac {11}{2}} \cdot \left (21 a^{2} - 3 b^{2} c\right )}{11 b^{6}} + \frac {\left (a + b \sqrt {c + d x}\right )^{\frac {9}{2}} \left (- 35 a^{3} + 15 a b^{2} c\right )}{9 b^{6}} + \frac {\left (a + b \sqrt {c + d x}\right )^{\frac {7}{2}} \cdot \left (35 a^{4} - 30 a^{2} b^{2} c + 3 b^{4} c^{2}\right )}{7 b^{6}} + \frac {\left (a + b \sqrt {c + d x}\right )^{\frac {5}{2}} \left (- 21 a^{5} + 30 a^{3} b^{2} c - 9 a b^{4} c^{2}\right )}{5 b^{6}} + \frac {\left (a + b \sqrt {c + d x}\right )^{\frac {3}{2}} \cdot \left (7 a^{6} - 15 a^{4} b^{2} c + 9 a^{2} b^{4} c^{2} - b^{6} c^{3}\right )}{3 b^{6}} + \frac {\sqrt {a + b \sqrt {c + d x}} \left (- a^{7} + 3 a^{5} b^{2} c - 3 a^{3} b^{4} c^{2} + a b^{6} c^{3}\right )}{b^{6}}\right )}{b^{2}} & \text {for}\: b \neq 0 \\\frac {d^{4} x^{4}}{8 \sqrt {a}} & \text {otherwise} \end {cases}\right )}{d^{4}} & \text {for}\: d \neq 0 \\\frac {x^{4}}{4 \sqrt {a + b \sqrt {c}}} & \text {otherwise} \end {cases} \] 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))**(13/2)/(13*b**6) + 
(a + b*sqrt(c + d*x))**(15/2)/(15*b**6) + (a + b*sqrt(c + d*x))**(11/2)*(2 
1*a**2 - 3*b**2*c)/(11*b**6) + (a + b*sqrt(c + d*x))**(9/2)*(-35*a**3 + 15 
*a*b**2*c)/(9*b**6) + (a + b*sqrt(c + d*x))**(7/2)*(35*a**4 - 30*a**2*b**2 
*c + 3*b**4*c**2)/(7*b**6) + (a + b*sqrt(c + d*x))**(5/2)*(-21*a**5 + 30*a 
**3*b**2*c - 9*a*b**4*c**2)/(5*b**6) + (a + b*sqrt(c + d*x))**(3/2)*(7*a** 
6 - 15*a**4*b**2*c + 9*a**2*b**4*c**2 - b**6*c**3)/(3*b**6) + sqrt(a + b*s 
qrt(c + d*x))*(-a**7 + 3*a**5*b**2*c - 3*a**3*b**4*c**2 + a*b**6*c**3)/b** 
6)/b**2, Ne(b, 0)), (d**4*x**4/(8*sqrt(a)), True))/d**4, Ne(d, 0)), (x**4/ 
(4*sqrt(a + b*sqrt(c))), True))
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 268, normalized size of antiderivative = 0.83 \[ \int \frac {x^3}{\sqrt {a+b \sqrt {c+d x}}} \, dx=\frac {4 \, {\left (3003 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {15}{2}} - 24255 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {13}{2}} a - 12285 \, {\left (b^{2} c - 7 \, a^{2}\right )} {\left (\sqrt {d x + c} b + a\right )}^{\frac {11}{2}} + 25025 \, {\left (3 \, a b^{2} c - 7 \, a^{3}\right )} {\left (\sqrt {d x + c} b + a\right )}^{\frac {9}{2}} + 6435 \, {\left (3 \, b^{4} c^{2} - 30 \, a^{2} b^{2} c + 35 \, a^{4}\right )} {\left (\sqrt {d x + c} b + a\right )}^{\frac {7}{2}} - 27027 \, {\left (3 \, a b^{4} c^{2} - 10 \, a^{3} b^{2} c + 7 \, a^{5}\right )} {\left (\sqrt {d x + c} b + a\right )}^{\frac {5}{2}} - 15015 \, {\left (b^{6} c^{3} - 9 \, a^{2} b^{4} c^{2} + 15 \, a^{4} b^{2} c - 7 \, a^{6}\right )} {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} + 45045 \, {\left (a b^{6} c^{3} - 3 \, a^{3} b^{4} c^{2} + 3 \, a^{5} b^{2} c - a^{7}\right )} \sqrt {\sqrt {d x + c} b + a}\right )}}{45045 \, b^{8} d^{4}} \] Input:

integrate(x^3/(a+b*(d*x+c)^(1/2))^(1/2),x, algorithm="maxima")
 

Output:

4/45045*(3003*(sqrt(d*x + c)*b + a)^(15/2) - 24255*(sqrt(d*x + c)*b + a)^( 
13/2)*a - 12285*(b^2*c - 7*a^2)*(sqrt(d*x + c)*b + a)^(11/2) + 25025*(3*a* 
b^2*c - 7*a^3)*(sqrt(d*x + c)*b + a)^(9/2) + 6435*(3*b^4*c^2 - 30*a^2*b^2* 
c + 35*a^4)*(sqrt(d*x + c)*b + a)^(7/2) - 27027*(3*a*b^4*c^2 - 10*a^3*b^2* 
c + 7*a^5)*(sqrt(d*x + c)*b + a)^(5/2) - 15015*(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)^(3/2) + 45045*(a*b^6*c^3 - 3*a 
^3*b^4*c^2 + 3*a^5*b^2*c - a^7)*sqrt(sqrt(d*x + c)*b + a))/(b^8*d^4)
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 409, normalized size of antiderivative = 1.26 \[ \int \frac {x^3}{\sqrt {a+b \sqrt {c+d x}}} \, dx=-\frac {4 \, {\left (15015 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} b^{6} c^{3} - 45045 \, \sqrt {\sqrt {d x + c} b + a} a b^{6} c^{3} - 19305 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {7}{2}} b^{4} c^{2} + 81081 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {5}{2}} a b^{4} c^{2} - 135135 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} a^{2} b^{4} c^{2} + 135135 \, \sqrt {\sqrt {d x + c} b + a} a^{3} b^{4} c^{2} + 12285 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {11}{2}} b^{2} c - 75075 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {9}{2}} a b^{2} c + 193050 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {7}{2}} a^{2} b^{2} c - 270270 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {5}{2}} a^{3} b^{2} c + 225225 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} a^{4} b^{2} c - 135135 \, \sqrt {\sqrt {d x + c} b + a} a^{5} b^{2} c - 3003 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {15}{2}} + 24255 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {13}{2}} a - 85995 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {11}{2}} a^{2} + 175175 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {9}{2}} a^{3} - 225225 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {7}{2}} a^{4} + 189189 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {5}{2}} a^{5} - 105105 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} a^{6} + 45045 \, \sqrt {\sqrt {d x + c} b + a} a^{7}\right )}}{45045 \, b^{8} d^{4}} \] Input:

integrate(x^3/(a+b*(d*x+c)^(1/2))^(1/2),x, algorithm="giac")
 

Output:

-4/45045*(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 + 81081* 
(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 - 135135 
*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)/(b^8*d^4)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3}{\sqrt {a+b \sqrt {c+d x}}} \, dx=\int \frac {x^3}{\sqrt {a+b\,\sqrt {c+d\,x}}} \,d x \] 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)
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 287, normalized size of antiderivative = 0.89 \[ \int \frac {x^3}{\sqrt {a+b \sqrt {c+d x}}} \, dx=\frac {4 \sqrt {\sqrt {d x +c}\, b +a}\, \left (7168 \sqrt {d x +c}\, a^{6} b -20480 \sqrt {d x +c}\, a^{4} b^{3} c +4480 \sqrt {d x +c}\, a^{4} b^{3} d x +18816 \sqrt {d x +c}\, a^{2} b^{5} c^{2}-8544 \sqrt {d x +c}\, a^{2} b^{5} c d x +3528 \sqrt {d x +c}\, a^{2} b^{5} d^{2} x^{2}-4992 \sqrt {d x +c}\, b^{7} c^{3}+3744 \sqrt {d x +c}\, b^{7} c^{2} d x -3276 \sqrt {d x +c}\, b^{7} c \,d^{2} x^{2}+3003 \sqrt {d x +c}\, b^{7} d^{3} x^{3}-14336 a^{7}+44544 a^{5} b^{2} c -5376 a^{5} b^{2} d x -46976 a^{3} b^{4} c^{2}+10880 a^{3} b^{4} c d x -3920 a^{3} b^{4} d^{2} x^{2}+17280 a \,b^{6} c^{3}-5568 a \,b^{6} c^{2} d x +3948 a \,b^{6} c \,d^{2} x^{2}-3234 a \,b^{6} d^{3} x^{3}\right )}{45045 b^{8} d^{4}} \] 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**6*b - 20480*sqrt(c + d 
*x)*a**4*b**3*c + 4480*sqrt(c + d*x)*a**4*b**3*d*x + 18816*sqrt(c + d*x)*a 
**2*b**5*c**2 - 8544*sqrt(c + d*x)*a**2*b**5*c*d*x + 3528*sqrt(c + d*x)*a* 
*2*b**5*d**2*x**2 - 4992*sqrt(c + d*x)*b**7*c**3 + 3744*sqrt(c + d*x)*b**7 
*c**2*d*x - 3276*sqrt(c + d*x)*b**7*c*d**2*x**2 + 3003*sqrt(c + d*x)*b**7* 
d**3*x**3 - 14336*a**7 + 44544*a**5*b**2*c - 5376*a**5*b**2*d*x - 46976*a* 
*3*b**4*c**2 + 10880*a**3*b**4*c*d*x - 3920*a**3*b**4*d**2*x**2 + 17280*a* 
b**6*c**3 - 5568*a*b**6*c**2*d*x + 3948*a*b**6*c*d**2*x**2 - 3234*a*b**6*d 
**3*x**3))/(45045*b**8*d**4)