Integrand size = 19, antiderivative size = 133 \[ \int x \sqrt {a+b \sqrt {c+d x}} \, dx=-\frac {4 a \left (a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{3/2}}{3 b^4 d^2}+\frac {4 \left (3 a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{5/2}}{5 b^4 d^2}-\frac {12 a \left (a+b \sqrt {c+d x}\right )^{7/2}}{7 b^4 d^2}+\frac {4 \left (a+b \sqrt {c+d x}\right )^{9/2}}{9 b^4 d^2} \] Output:
-4/3*a*(-b^2*c+a^2)*(a+b*(d*x+c)^(1/2))^(3/2)/b^4/d^2+4/5*(-b^2*c+3*a^2)*( a+b*(d*x+c)^(1/2))^(5/2)/b^4/d^2-12/7*a*(a+b*(d*x+c)^(1/2))^(7/2)/b^4/d^2+ 4/9*(a+b*(d*x+c)^(1/2))^(9/2)/b^4/d^2
Time = 0.09 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.63 \[ \int x \sqrt {a+b \sqrt {c+d x}} \, dx=\frac {4 \left (a+b \sqrt {c+d x}\right )^{3/2} \left (-16 a^3+6 a b^2 (2 c-5 d x)+24 a^2 b \sqrt {c+d x}+7 b^3 \sqrt {c+d x} (-4 c+5 d x)\right )}{315 b^4 d^2} \] Input:
Integrate[x*Sqrt[a + b*Sqrt[c + d*x]],x]
Output:
(4*(a + b*Sqrt[c + d*x])^(3/2)*(-16*a^3 + 6*a*b^2*(2*c - 5*d*x) + 24*a^2*b *Sqrt[c + d*x] + 7*b^3*Sqrt[c + d*x]*(-4*c + 5*d*x)))/(315*b^4*d^2)
Time = 0.46 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.95, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, 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.
\(\displaystyle \int x \sqrt {a+b \sqrt {c+d x}} \, dx\) |
\(\Big \downarrow \) 896 |
\(\displaystyle \frac {\int d x \sqrt {a+b \sqrt {c+d x}}d(c+d x)}{d^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\int -d x \sqrt {a+b \sqrt {c+d x}}d(c+d x)}{d^2}\) |
\(\Big \downarrow \) 1732 |
\(\displaystyle -\frac {2 \int -d x \sqrt {c+d x} \sqrt {a+b \sqrt {c+d x}}d\sqrt {c+d x}}{d^2}\) |
\(\Big \downarrow \) 522 |
\(\displaystyle -\frac {2 \int \left (-\frac {\left (a+b \sqrt {c+d x}\right )^{7/2}}{b^3}+\frac {3 a \left (a+b \sqrt {c+d x}\right )^{5/2}}{b^3}+\frac {\left (b^2 c-3 a^2\right ) \left (a+b \sqrt {c+d x}\right )^{3/2}}{b^3}+\frac {\left (a^3-a b^2 c\right ) \sqrt {a+b \sqrt {c+d x}}}{b^3}\right )d\sqrt {c+d x}}{d^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 \left (-\frac {2 \left (3 a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{5/2}}{5 b^4}+\frac {2 a \left (a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{3/2}}{3 b^4}-\frac {2 \left (a+b \sqrt {c+d x}\right )^{9/2}}{9 b^4}+\frac {6 a \left (a+b \sqrt {c+d x}\right )^{7/2}}{7 b^4}\right )}{d^2}\) |
Input:
Int[x*Sqrt[a + b*Sqrt[c + d*x]],x]
Output:
(-2*((2*a*(a^2 - b^2*c)*(a + b*Sqrt[c + d*x])^(3/2))/(3*b^4) - (2*(3*a^2 - b^2*c)*(a + b*Sqrt[c + d*x])^(5/2))/(5*b^4) + (6*a*(a + b*Sqrt[c + d*x])^ (7/2))/(7*b^4) - (2*(a + b*Sqrt[c + d*x])^(9/2))/(9*b^4)))/d^2
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 = 93, normalized size of antiderivative = 0.70
method | result | size |
derivativedivides | \(-\frac {4 \left (-\frac {\left (a +b \sqrt {d x +c}\right )^{\frac {9}{2}}}{9}+\frac {3 a \left (a +b \sqrt {d x +c}\right )^{\frac {7}{2}}}{7}+\frac {\left (b^{2} c -3 a^{2}\right ) \left (a +b \sqrt {d x +c}\right )^{\frac {5}{2}}}{5}+\frac {\left (-b^{2} c +a^{2}\right ) a \left (a +b \sqrt {d x +c}\right )^{\frac {3}{2}}}{3}\right )}{d^{2} b^{4}}\) | \(93\) |
default | \(-\frac {4 \left (-\frac {\left (a +b \sqrt {d x +c}\right )^{\frac {9}{2}}}{9}+\frac {3 a \left (a +b \sqrt {d x +c}\right )^{\frac {7}{2}}}{7}+\frac {\left (b^{2} c -3 a^{2}\right ) \left (a +b \sqrt {d x +c}\right )^{\frac {5}{2}}}{5}+\frac {\left (-b^{2} c +a^{2}\right ) a \left (a +b \sqrt {d x +c}\right )^{\frac {3}{2}}}{3}\right )}{d^{2} b^{4}}\) | \(93\) |
Input:
int(x*(a+b*(d*x+c)^(1/2))^(1/2),x,method=_RETURNVERBOSE)
Output:
-4/d^2/b^4*(-1/9*(a+b*(d*x+c)^(1/2))^(9/2)+3/7*a*(a+b*(d*x+c)^(1/2))^(7/2) +1/5*(b^2*c-3*a^2)*(a+b*(d*x+c)^(1/2))^(5/2)+1/3*(-b^2*c+a^2)*a*(a+b*(d*x+ c)^(1/2))^(3/2))
Time = 0.12 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.77 \[ \int x \sqrt {a+b \sqrt {c+d x}} \, dx=\frac {4 \, {\left (35 \, b^{4} d^{2} x^{2} - 28 \, b^{4} c^{2} + 36 \, a^{2} b^{2} c - 16 \, a^{4} + {\left (7 \, b^{4} c - 6 \, a^{2} b^{2}\right )} d x + {\left (5 \, a b^{3} d x - 16 \, a b^{3} c + 8 \, a^{3} b\right )} \sqrt {d x + c}\right )} \sqrt {\sqrt {d x + c} b + a}}{315 \, b^{4} d^{2}} \] Input:
integrate(x*(a+b*(d*x+c)^(1/2))^(1/2),x, algorithm="fricas")
Output:
4/315*(35*b^4*d^2*x^2 - 28*b^4*c^2 + 36*a^2*b^2*c - 16*a^4 + (7*b^4*c - 6* a^2*b^2)*d*x + (5*a*b^3*d*x - 16*a*b^3*c + 8*a^3*b)*sqrt(d*x + c))*sqrt(sq rt(d*x + c)*b + a)/(b^4*d^2)
Time = 0.66 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.17 \[ \int x \sqrt {a+b \sqrt {c+d x}} \, dx=\begin {cases} \frac {2 \left (\begin {cases} \frac {2 \left (- \frac {3 a \left (a + b \sqrt {c + d x}\right )^{\frac {7}{2}}}{7 b^{2}} + \frac {\left (a + b \sqrt {c + d x}\right )^{\frac {9}{2}}}{9 b^{2}} + \frac {\left (a + b \sqrt {c + d x}\right )^{\frac {5}{2}} \cdot \left (3 a^{2} - b^{2} c\right )}{5 b^{2}} + \frac {\left (a + b \sqrt {c + d x}\right )^{\frac {3}{2}} \left (- a^{3} + a b^{2} c\right )}{3 b^{2}}\right )}{b^{2}} & \text {for}\: b \neq 0 \\\sqrt {a} \left (- \frac {c \left (c + d x\right )}{2} + \frac {\left (c + d x\right )^{2}}{4}\right ) & \text {otherwise} \end {cases}\right )}{d^{2}} & \text {for}\: d \neq 0 \\\frac {x^{2} \sqrt {a + b \sqrt {c}}}{2} & \text {otherwise} \end {cases} \] Input:
integrate(x*(a+b*(d*x+c)**(1/2))**(1/2),x)
Output:
Piecewise((2*Piecewise((2*(-3*a*(a + b*sqrt(c + d*x))**(7/2)/(7*b**2) + (a + b*sqrt(c + d*x))**(9/2)/(9*b**2) + (a + b*sqrt(c + d*x))**(5/2)*(3*a**2 - b**2*c)/(5*b**2) + (a + b*sqrt(c + d*x))**(3/2)*(-a**3 + a*b**2*c)/(3*b **2))/b**2, Ne(b, 0)), (sqrt(a)*(-c*(c + d*x)/2 + (c + d*x)**2/4), True))/ d**2, Ne(d, 0)), (x**2*sqrt(a + b*sqrt(c))/2, True))
Time = 0.03 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.70 \[ \int x \sqrt {a+b \sqrt {c+d x}} \, dx=\frac {4 \, {\left (35 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {9}{2}} - 135 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {7}{2}} a - 63 \, {\left (b^{2} c - 3 \, a^{2}\right )} {\left (\sqrt {d x + c} b + a\right )}^{\frac {5}{2}} + 105 \, {\left (a b^{2} c - a^{3}\right )} {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}}\right )}}{315 \, b^{4} d^{2}} \] Input:
integrate(x*(a+b*(d*x+c)^(1/2))^(1/2),x, algorithm="maxima")
Output:
4/315*(35*(sqrt(d*x + c)*b + a)^(9/2) - 135*(sqrt(d*x + c)*b + a)^(7/2)*a - 63*(b^2*c - 3*a^2)*(sqrt(d*x + c)*b + a)^(5/2) + 105*(a*b^2*c - a^3)*(sq rt(d*x + c)*b + a)^(3/2))/(b^4*d^2)
Leaf count of result is larger than twice the leaf count of optimal. 279 vs. \(2 (109) = 218\).
Time = 0.17 (sec) , antiderivative size = 279, normalized size of antiderivative = 2.10 \[ \int x \sqrt {a+b \sqrt {c+d x}} \, dx=-\frac {4 \, {\left (\frac {3 \, {\left (35 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} b^{2} c - 105 \, \sqrt {\sqrt {d x + c} b + a} a b^{2} c - 15 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {7}{2}} + 63 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {5}{2}} a - 105 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} a^{2} + 105 \, \sqrt {\sqrt {d x + c} b + a} a^{3}\right )} a}{b^{3} d} + \frac {63 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {5}{2}} b^{2} c - 210 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} a b^{2} c + 315 \, \sqrt {\sqrt {d x + c} b + a} a^{2} b^{2} c - 35 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {9}{2}} + 180 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {7}{2}} a - 378 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {5}{2}} a^{2} + 420 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} a^{3} - 315 \, \sqrt {\sqrt {d x + c} b + a} a^{4}}{b^{3} d}\right )}}{315 \, b d} \] Input:
integrate(x*(a+b*(d*x+c)^(1/2))^(1/2),x, algorithm="giac")
Output:
-4/315*(3*(35*(sqrt(d*x + c)*b + a)^(3/2)*b^2*c - 105*sqrt(sqrt(d*x + c)*b + a)*a*b^2*c - 15*(sqrt(d*x + c)*b + a)^(7/2) + 63*(sqrt(d*x + c)*b + a)^ (5/2)*a - 105*(sqrt(d*x + c)*b + a)^(3/2)*a^2 + 105*sqrt(sqrt(d*x + c)*b + a)*a^3)*a/(b^3*d) + (63*(sqrt(d*x + c)*b + a)^(5/2)*b^2*c - 210*(sqrt(d*x + c)*b + a)^(3/2)*a*b^2*c + 315*sqrt(sqrt(d*x + c)*b + a)*a^2*b^2*c - 35* (sqrt(d*x + c)*b + a)^(9/2) + 180*(sqrt(d*x + c)*b + a)^(7/2)*a - 378*(sqr t(d*x + c)*b + a)^(5/2)*a^2 + 420*(sqrt(d*x + c)*b + a)^(3/2)*a^3 - 315*sq rt(sqrt(d*x + c)*b + a)*a^4)/(b^3*d))/(b*d)
Timed out. \[ \int x \sqrt {a+b \sqrt {c+d x}} \, dx=\int x\,\sqrt {a+b\,\sqrt {c+d\,x}} \,d x \] Input:
int(x*(a + b*(c + d*x)^(1/2))^(1/2),x)
Output:
int(x*(a + b*(c + d*x)^(1/2))^(1/2), x)
Time = 0.22 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.83 \[ \int x \sqrt {a+b \sqrt {c+d x}} \, dx=\frac {4 \sqrt {\sqrt {d x +c}\, b +a}\, \left (8 \sqrt {d x +c}\, a^{3} b -16 \sqrt {d x +c}\, a \,b^{3} c +5 \sqrt {d x +c}\, a \,b^{3} d x -16 a^{4}+36 a^{2} b^{2} c -6 a^{2} b^{2} d x -28 b^{4} c^{2}+7 b^{4} c d x +35 b^{4} d^{2} x^{2}\right )}{315 b^{4} d^{2}} \] Input:
int(x*(a+b*(d*x+c)^(1/2))^(1/2),x)
Output:
(4*sqrt(sqrt(c + d*x)*b + a)*(8*sqrt(c + d*x)*a**3*b - 16*sqrt(c + d*x)*a* b**3*c + 5*sqrt(c + d*x)*a*b**3*d*x - 16*a**4 + 36*a**2*b**2*c - 6*a**2*b* *2*d*x - 28*b**4*c**2 + 7*b**4*c*d*x + 35*b**4*d**2*x**2))/(315*b**4*d**2)