\(\int \frac {x^2}{\sqrt {a+b \sqrt {c+d x}}} \, dx\) [133]

Optimal result

Integrand size = 21, antiderivative size = 222 \[ \int \frac {x^2}{\sqrt {a+b \sqrt {c+d x}}} \, dx=-\frac {4 a \left (a^2-b^2 c\right )^2 \sqrt {a+b \sqrt {c+d x}}}{b^6 d^3}+\frac {4 \left (5 a^4-6 a^2 b^2 c+b^4 c^2\right ) \left (a+b \sqrt {c+d x}\right )^{3/2}}{3 b^6 d^3}-\frac {8 a \left (5 a^2-3 b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{5/2}}{5 b^6 d^3}+\frac {8 \left (5 a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{7/2}}{7 b^6 d^3}-\frac {20 a \left (a+b \sqrt {c+d x}\right )^{9/2}}{9 b^6 d^3}+\frac {4 \left (a+b \sqrt {c+d x}\right )^{11/2}}{11 b^6 d^3} \] Output:

-4*a*(-b^2*c+a^2)^2*(a+b*(d*x+c)^(1/2))^(1/2)/b^6/d^3+4/3*(b^4*c^2-6*a^2*b 
^2*c+5*a^4)*(a+b*(d*x+c)^(1/2))^(3/2)/b^6/d^3-8/5*a*(-3*b^2*c+5*a^2)*(a+b* 
(d*x+c)^(1/2))^(5/2)/b^6/d^3+8/7*(-b^2*c+5*a^2)*(a+b*(d*x+c)^(1/2))^(7/2)/ 
b^6/d^3-20/9*a*(a+b*(d*x+c)^(1/2))^(9/2)/b^6/d^3+4/11*(a+b*(d*x+c)^(1/2))^ 
(11/2)/b^6/d^3
 

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.66 \[ \int \frac {x^2}{\sqrt {a+b \sqrt {c+d x}}} \, dx=\frac {4 \sqrt {a+b \sqrt {c+d x}} \left (-1280 a^5+96 a^3 b^2 (28 c-5 d x)+640 a^4 b \sqrt {c+d x}-16 a^2 b^3 (74 c-25 d x) \sqrt {c+d x}+15 b^5 \sqrt {c+d x} \left (32 c^2-24 c d x+21 d^2 x^2\right )-2 a b^4 \left (736 c^2-244 c d x+175 d^2 x^2\right )\right )}{3465 b^6 d^3} \] Input:

Integrate[x^2/Sqrt[a + b*Sqrt[c + d*x]],x]
 

Output:

(4*Sqrt[a + b*Sqrt[c + d*x]]*(-1280*a^5 + 96*a^3*b^2*(28*c - 5*d*x) + 640* 
a^4*b*Sqrt[c + d*x] - 16*a^2*b^3*(74*c - 25*d*x)*Sqrt[c + d*x] + 15*b^5*Sq 
rt[c + d*x]*(32*c^2 - 24*c*d*x + 21*d^2*x^2) - 2*a*b^4*(736*c^2 - 244*c*d* 
x + 175*d^2*x^2)))/(3465*b^6*d^3)
 

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 209, 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.

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

\(\Big \downarrow \) 896

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

\(\Big \downarrow \) 1732

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

\(\Big \downarrow \) 522

\(\displaystyle \frac {2 \int \left (\frac {\left (a+b \sqrt {c+d x}\right )^{9/2}}{b^5}-\frac {5 a \left (a+b \sqrt {c+d x}\right )^{7/2}}{b^5}-\frac {2 \left (b^2 c-5 a^2\right ) \left (a+b \sqrt {c+d x}\right )^{5/2}}{b^5}-\frac {2 \left (5 a^3-3 a b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{3/2}}{b^5}+\frac {\left (5 a^4-6 b^2 c a^2+b^4 c^2\right ) \sqrt {a+b \sqrt {c+d x}}}{b^5}-\frac {a \left (a^2-b^2 c\right )^2}{b^5 \sqrt {a+b \sqrt {c+d x}}}\right )d\sqrt {c+d x}}{d^3}\)

\(\Big \downarrow \) 2009

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

Input:

Int[x^2/Sqrt[a + b*Sqrt[c + d*x]],x]
 

Output:

(2*((-2*a*(a^2 - b^2*c)^2*Sqrt[a + b*Sqrt[c + d*x]])/b^6 + (2*(5*a^4 - 6*a 
^2*b^2*c + b^4*c^2)*(a + b*Sqrt[c + d*x])^(3/2))/(3*b^6) - (4*a*(5*a^2 - 3 
*b^2*c)*(a + b*Sqrt[c + d*x])^(5/2))/(5*b^6) + (4*(5*a^2 - b^2*c)*(a + b*S 
qrt[c + d*x])^(7/2))/(7*b^6) - (10*a*(a + b*Sqrt[c + d*x])^(9/2))/(9*b^6) 
+ (2*(a + b*Sqrt[c + d*x])^(11/2))/(11*b^6)))/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 [A] (verified)

Time = 0.22 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.83

method	result	size
derivativedivides	\(\frac {\frac {4 \left (a +b \sqrt {d x +c}\right )^{\frac {11}{2}}}{11}-\frac {20 a \left (a +b \sqrt {d x +c}\right )^{\frac {9}{2}}}{9}-\frac {4 \left (2 b^{2} c -10 a^{2}\right ) \left (a +b \sqrt {d x +c}\right )^{\frac {7}{2}}}{7}-\frac {4 \left (4 \left (-b^{2} c +a^{2}\right ) a +a \left (-2 b^{2} c +6 a^{2}\right )\right ) \left (a +b \sqrt {d x +c}\right )^{\frac {5}{2}}}{5}-\frac {4 \left (-\left (-b^{2} c +a^{2}\right )^{2}-4 a^{2} \left (-b^{2} c +a^{2}\right )\right ) \left (a +b \sqrt {d x +c}\right )^{\frac {3}{2}}}{3}-4 \left (-b^{2} c +a^{2}\right )^{2} a \sqrt {a +b \sqrt {d x +c}}}{d^{3} b^{6}}\)	\(184\)
default	\(\frac {\frac {4 \left (a +b \sqrt {d x +c}\right )^{\frac {11}{2}}}{11}-\frac {20 a \left (a +b \sqrt {d x +c}\right )^{\frac {9}{2}}}{9}-\frac {4 \left (2 b^{2} c -10 a^{2}\right ) \left (a +b \sqrt {d x +c}\right )^{\frac {7}{2}}}{7}-\frac {4 \left (4 \left (-b^{2} c +a^{2}\right ) a +a \left (-2 b^{2} c +6 a^{2}\right )\right ) \left (a +b \sqrt {d x +c}\right )^{\frac {5}{2}}}{5}-\frac {4 \left (-\left (-b^{2} c +a^{2}\right )^{2}-4 a^{2} \left (-b^{2} c +a^{2}\right )\right ) \left (a +b \sqrt {d x +c}\right )^{\frac {3}{2}}}{3}-4 \left (-b^{2} c +a^{2}\right )^{2} a \sqrt {a +b \sqrt {d x +c}}}{d^{3} b^{6}}\)	\(184\)

Input:

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

Output:

4/d^3/b^6*(1/11*(a+b*(d*x+c)^(1/2))^(11/2)-5/9*a*(a+b*(d*x+c)^(1/2))^(9/2) 
-1/7*(2*b^2*c-10*a^2)*(a+b*(d*x+c)^(1/2))^(7/2)-1/5*(4*(-b^2*c+a^2)*a+a*(- 
2*b^2*c+6*a^2))*(a+b*(d*x+c)^(1/2))^(5/2)-1/3*(-(-b^2*c+a^2)^2-4*a^2*(-b^2 
*c+a^2))*(a+b*(d*x+c)^(1/2))^(3/2)-(-b^2*c+a^2)^2*a*(a+b*(d*x+c)^(1/2))^(1 
/2))
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.63 \[ \int \frac {x^2}{\sqrt {a+b \sqrt {c+d x}}} \, dx=-\frac {4 \, {\left (350 \, a b^{4} d^{2} x^{2} + 1472 \, a b^{4} c^{2} - 2688 \, a^{3} b^{2} c + 1280 \, a^{5} - 8 \, {\left (61 \, a b^{4} c - 60 \, a^{3} b^{2}\right )} d x - {\left (315 \, b^{5} d^{2} x^{2} + 480 \, b^{5} c^{2} - 1184 \, a^{2} b^{3} c + 640 \, a^{4} b - 40 \, {\left (9 \, b^{5} c - 10 \, a^{2} b^{3}\right )} d x\right )} \sqrt {d x + c}\right )} \sqrt {\sqrt {d x + c} b + a}}{3465 \, b^{6} d^{3}} \] Input:

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

Output:

-4/3465*(350*a*b^4*d^2*x^2 + 1472*a*b^4*c^2 - 2688*a^3*b^2*c + 1280*a^5 - 
8*(61*a*b^4*c - 60*a^3*b^2)*d*x - (315*b^5*d^2*x^2 + 480*b^5*c^2 - 1184*a^ 
2*b^3*c + 640*a^4*b - 40*(9*b^5*c - 10*a^2*b^3)*d*x)*sqrt(d*x + c))*sqrt(s 
qrt(d*x + c)*b + a)/(b^6*d^3)
 

Sympy [A] (verification not implemented)

Time = 0.95 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.08 \[ \int \frac {x^2}{\sqrt {a+b \sqrt {c+d x}}} \, dx=\begin {cases} \frac {2 \left (\begin {cases} \frac {2 \left (- \frac {5 a \left (a + b \sqrt {c + d x}\right )^{\frac {9}{2}}}{9 b^{4}} + \frac {\left (a + b \sqrt {c + d x}\right )^{\frac {11}{2}}}{11 b^{4}} + \frac {\left (a + b \sqrt {c + d x}\right )^{\frac {7}{2}} \cdot \left (10 a^{2} - 2 b^{2} c\right )}{7 b^{4}} + \frac {\left (a + b \sqrt {c + d x}\right )^{\frac {5}{2}} \left (- 10 a^{3} + 6 a b^{2} c\right )}{5 b^{4}} + \frac {\left (a + b \sqrt {c + d x}\right )^{\frac {3}{2}} \cdot \left (5 a^{4} - 6 a^{2} b^{2} c + b^{4} c^{2}\right )}{3 b^{4}} + \frac {\sqrt {a + b \sqrt {c + d x}} \left (- a^{5} + 2 a^{3} b^{2} c - a b^{4} c^{2}\right )}{b^{4}}\right )}{b^{2}} & \text {for}\: b \neq 0 \\\frac {d^{3} x^{3}}{6 \sqrt {a}} & \text {otherwise} \end {cases}\right )}{d^{3}} & \text {for}\: d \neq 0 \\\frac {x^{3}}{3 \sqrt {a + b \sqrt {c}}} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((2*Piecewise((2*(-5*a*(a + b*sqrt(c + d*x))**(9/2)/(9*b**4) + (a 
 + b*sqrt(c + d*x))**(11/2)/(11*b**4) + (a + b*sqrt(c + d*x))**(7/2)*(10*a 
**2 - 2*b**2*c)/(7*b**4) + (a + b*sqrt(c + d*x))**(5/2)*(-10*a**3 + 6*a*b* 
*2*c)/(5*b**4) + (a + b*sqrt(c + d*x))**(3/2)*(5*a**4 - 6*a**2*b**2*c + b* 
*4*c**2)/(3*b**4) + sqrt(a + b*sqrt(c + d*x))*(-a**5 + 2*a**3*b**2*c - a*b 
**4*c**2)/b**4)/b**2, Ne(b, 0)), (d**3*x**3/(6*sqrt(a)), True))/d**3, Ne(d 
, 0)), (x**3/(3*sqrt(a + b*sqrt(c))), True))
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.75 \[ \int \frac {x^2}{\sqrt {a+b \sqrt {c+d x}}} \, dx=\frac {4 \, {\left (315 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {11}{2}} - 1925 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {9}{2}} a - 990 \, {\left (b^{2} c - 5 \, a^{2}\right )} {\left (\sqrt {d x + c} b + a\right )}^{\frac {7}{2}} + 1386 \, {\left (3 \, a b^{2} c - 5 \, a^{3}\right )} {\left (\sqrt {d x + c} b + a\right )}^{\frac {5}{2}} + 1155 \, {\left (b^{4} c^{2} - 6 \, a^{2} b^{2} c + 5 \, a^{4}\right )} {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} - 3465 \, {\left (a b^{4} c^{2} - 2 \, a^{3} b^{2} c + a^{5}\right )} \sqrt {\sqrt {d x + c} b + a}\right )}}{3465 \, b^{6} d^{3}} \] Input:

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

Output:

4/3465*(315*(sqrt(d*x + c)*b + a)^(11/2) - 1925*(sqrt(d*x + c)*b + a)^(9/2 
)*a - 990*(b^2*c - 5*a^2)*(sqrt(d*x + c)*b + a)^(7/2) + 1386*(3*a*b^2*c - 
5*a^3)*(sqrt(d*x + c)*b + a)^(5/2) + 1155*(b^4*c^2 - 6*a^2*b^2*c + 5*a^4)* 
(sqrt(d*x + c)*b + a)^(3/2) - 3465*(a*b^4*c^2 - 2*a^3*b^2*c + a^5)*sqrt(sq 
rt(d*x + c)*b + a))/(b^6*d^3)
 

Giac [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.07 \[ \int \frac {x^2}{\sqrt {a+b \sqrt {c+d x}}} \, dx=\frac {4 \, {\left (1155 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} b^{4} c^{2} - 3465 \, \sqrt {\sqrt {d x + c} b + a} a b^{4} c^{2} - 990 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {7}{2}} b^{2} c + 4158 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {5}{2}} a b^{2} c - 6930 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} a^{2} b^{2} c + 6930 \, \sqrt {\sqrt {d x + c} b + a} a^{3} b^{2} c + 315 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {11}{2}} - 1925 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {9}{2}} a + 4950 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {7}{2}} a^{2} - 6930 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {5}{2}} a^{3} + 5775 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} a^{4} - 3465 \, \sqrt {\sqrt {d x + c} b + a} a^{5}\right )}}{3465 \, b^{6} d^{3}} \] Input:

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

Output:

4/3465*(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*(sqrt(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)/(b^6*d^3)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2}{\sqrt {a+b \sqrt {c+d x}}} \, dx=\int \frac {x^2}{\sqrt {a+b\,\sqrt {c+d\,x}}} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.73 \[ \int \frac {x^2}{\sqrt {a+b \sqrt {c+d x}}} \, dx=\frac {4 \sqrt {\sqrt {d x +c}\, b +a}\, \left (640 \sqrt {d x +c}\, a^{4} b -1184 \sqrt {d x +c}\, a^{2} b^{3} c +400 \sqrt {d x +c}\, a^{2} b^{3} d x +480 \sqrt {d x +c}\, b^{5} c^{2}-360 \sqrt {d x +c}\, b^{5} c d x +315 \sqrt {d x +c}\, b^{5} d^{2} x^{2}-1280 a^{5}+2688 a^{3} b^{2} c -480 a^{3} b^{2} d x -1472 a \,b^{4} c^{2}+488 a \,b^{4} c d x -350 a \,b^{4} d^{2} x^{2}\right )}{3465 b^{6} d^{3}} \] Input:

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

Output:

(4*sqrt(sqrt(c + d*x)*b + a)*(640*sqrt(c + d*x)*a**4*b - 1184*sqrt(c + d*x 
)*a**2*b**3*c + 400*sqrt(c + d*x)*a**2*b**3*d*x + 480*sqrt(c + d*x)*b**5*c 
**2 - 360*sqrt(c + d*x)*b**5*c*d*x + 315*sqrt(c + d*x)*b**5*d**2*x**2 - 12 
80*a**5 + 2688*a**3*b**2*c - 480*a**3*b**2*d*x - 1472*a*b**4*c**2 + 488*a* 
b**4*c*d*x - 350*a*b**4*d**2*x**2))/(3465*b**6*d**3)