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

Optimal. Leaf size=224 \[ \frac {8 \left (5 a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{9/2}}{9 b^6 d^3}-\frac {8 a \left (5 a^2-3 b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{7/2}}{7 b^6 d^3}-\frac {4 a \left (a^2-b^2 c\right )^2 \left (a+b \sqrt {c+d x}\right )^{3/2}}{3 b^6 d^3}+\frac {4 \left (5 a^4-6 a^2 b^2 c+b^4 c^2\right ) \left (a+b \sqrt {c+d x}\right )^{5/2}}{5 b^6 d^3}+\frac {4 \left (a+b \sqrt {c+d x}\right )^{13/2}}{13 b^6 d^3}-\frac {20 a \left (a+b \sqrt {c+d x}\right )^{11/2}}{11 b^6 d^3} \]

[Out]

-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

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Rubi [A]  time = 0.16, antiderivative size = 224, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {371, 1398, 772} \[ \frac {4 \left (-6 a^2 b^2 c+5 a^4+b^4 c^2\right ) \left (a+b \sqrt {c+d x}\right )^{5/2}}{5 b^6 d^3}+\frac {8 \left (5 a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{9/2}}{9 b^6 d^3}-\frac {8 a \left (5 a^2-3 b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{7/2}}{7 b^6 d^3}-\frac {4 a \left (a^2-b^2 c\right )^2 \left (a+b \sqrt {c+d x}\right )^{3/2}}{3 b^6 d^3}+\frac {4 \left (a+b \sqrt {c+d x}\right )^{13/2}}{13 b^6 d^3}-\frac {20 a \left (a+b \sqrt {c+d x}\right )^{11/2}}{11 b^6 d^3} \]

Antiderivative was successfully verified.

[In]

Int[x^2*Sqrt[a + b*Sqrt[c + d*x]],x]

[Out]

(-4*a*(a^2 - b^2*c)^2*(a + b*Sqrt[c + d*x])^(3/2))/(3*b^6*d^3) + (4*(5*a^4 - 6*a^2*b^2*c + b^4*c^2)*(a + b*Sqr
t[c + d*x])^(5/2))/(5*b^6*d^3) - (8*a*(5*a^2 - 3*b^2*c)*(a + b*Sqrt[c + d*x])^(7/2))/(7*b^6*d^3) + (8*(5*a^2 -
b^2*c)*(a + b*Sqrt[c + d*x])^(9/2))/(9*b^6*d^3) - (20*a*(a + b*Sqrt[c + d*x])^(11/2))/(11*b^6*d^3) + (4*(a +
b*Sqrt[c + d*x])^(13/2))/(13*b^6*d^3)

Rule 371

Int[((a_) + (b_.)*(v_)^(n_))^(p_.)*(x_)^(m_.), x_Symbol] :> With[{c = Coefficient[v, x, 0], d = Coefficient[v,
x, 1]}, Dist[1/d^(m + 1), Subst[Int[SimplifyIntegrand[(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 772

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegr
and[(d + e*x)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IGtQ[p, 0]

Rule 1398

Int[((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> With[{g = Denominator[n]}, D
ist[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]

Rubi steps

\begin {align*} \int x^2 \sqrt {a+b \sqrt {c+d x}} \, dx &=\frac {\operatorname {Subst}\left (\int \sqrt {a+b \sqrt {x}} (-c+x)^2 \, dx,x,c+d x\right )}{d^3}\\ &=\frac {2 \operatorname {Subst}\left (\int x \sqrt {a+b x} \left (-c+x^2\right )^2 \, dx,x,\sqrt {c+d x}\right )}{d^3}\\ &=\frac {2 \operatorname {Subst}\left (\int \left (-\frac {a \left (a^2-b^2 c\right )^2 \sqrt {a+b x}}{b^5}+\frac {\left (5 a^4-6 a^2 b^2 c+b^4 c^2\right ) (a+b x)^{3/2}}{b^5}-\frac {2 \left (5 a^3-3 a b^2 c\right ) (a+b x)^{5/2}}{b^5}-\frac {2 \left (-5 a^2+b^2 c\right ) (a+b x)^{7/2}}{b^5}-\frac {5 a (a+b x)^{9/2}}{b^5}+\frac {(a+b x)^{11/2}}{b^5}\right ) \, dx,x,\sqrt {c+d x}\right )}{d^3}\\ &=-\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}\\ \end {align*}

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Mathematica [A]  time = 0.24, size = 147, normalized size = 0.66 \[ \frac {4 \left (a+b \sqrt {c+d x}\right )^{3/2} \left (-1280 a^5+1920 a^4 b \sqrt {c+d x}+32 a^3 b^2 (68 c-75 d x)+16 a^2 b^3 \sqrt {c+d x} (175 d x-254 c)-6 a b^4 \left (96 c^2-380 c d x+525 d^2 x^2\right )+77 b^5 \sqrt {c+d x} \left (32 c^2-40 c d x+45 d^2 x^2\right )\right )}{45045 b^6 d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*(a + b*Sqrt[c + d*x])^(3/2)*(-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]*(32*c^2 - 40*c*d*x + 45*d^2*x^2) - 6*a*b^4*(96*c^2 -
380*c*d*x + 525*d^2*x^2)))/(45045*b^6*d^3)

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fricas [A]  time = 0.52, size = 184, normalized size = 0.82 \[ \frac {4 \, {\left (3465 \, b^{6} d^{3} x^{3} + 2464 \, b^{6} c^{3} - 4640 \, a^{2} b^{4} c^{2} + 4096 \, a^{4} b^{2} c - 1280 \, a^{6} + 35 \, {\left (11 \, b^{6} c - 10 \, a^{2} b^{4}\right )} d^{2} x^{2} - 8 \, {\left (77 \, b^{6} c^{2} - 127 \, a^{2} b^{4} c + 60 \, a^{4} b^{2}\right )} d x + {\left (315 \, a b^{5} d^{2} x^{2} + 1888 \, a b^{5} c^{2} - 1888 \, a^{3} b^{3} c + 640 \, a^{5} b - 400 \, {\left (2 \, a b^{5} c - a^{3} b^{3}\right )} d x\right )} \sqrt {d x + c}\right )} \sqrt {\sqrt {d x + c} b + a}}{45045 \, b^{6} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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(sqrt(d*x + c)*b + a)/(b^6*d^3)

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giac [B]  time = 0.46, size = 549, normalized size = 2.45 \[ \frac {4 \, {\left (\frac {13 \, {\left (1155 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} b^{4} c^{2} - 3465 \, \sqrt {\sqrt {d x + c} b + a} a b^{4} c^{2} - 990 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {7}{2}} b^{2} c + 4158 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {5}{2}} a b^{2} c - 6930 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} a^{2} b^{2} c + 6930 \, \sqrt {\sqrt {d x + c} b + a} a^{3} b^{2} c + 315 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {11}{2}} - 1925 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {9}{2}} a + 4950 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {7}{2}} a^{2} - 6930 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {5}{2}} a^{3} + 5775 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} a^{4} - 3465 \, \sqrt {\sqrt {d x + c} b + a} a^{5}\right )} a}{b^{5} d^{2}} + \frac {9009 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {5}{2}} b^{4} c^{2} - 30030 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} a b^{4} c^{2} + 45045 \, \sqrt {\sqrt {d x + c} b + a} a^{2} b^{4} c^{2} - 10010 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {9}{2}} b^{2} c + 51480 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {7}{2}} a b^{2} c - 108108 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {5}{2}} a^{2} b^{2} c + 120120 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} a^{3} b^{2} c - 90090 \, \sqrt {\sqrt {d x + c} b + a} a^{4} b^{2} c + 3465 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {13}{2}} - 24570 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {11}{2}} a + 75075 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {9}{2}} a^{2} - 128700 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {7}{2}} a^{3} + 135135 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {5}{2}} a^{4} - 90090 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} a^{5} + 45045 \, \sqrt {\sqrt {d x + c} b + a} a^{6}}{b^{5} d^{2}}\right )}}{45045 \, b d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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*(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)*a/(b^5*d^2) + (9009*(sqrt(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*(sqrt(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*(sqrt(d*x + c)*b + a)^(5/2)*a^4 - 90090*(sqrt(d*x + c)*b + a)^(3/2)*
a^5 + 45045*sqrt(sqrt(d*x + c)*b + a)*a^6)/(b^5*d^2))/(b*d)

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maple [A]  time = 0.00, size = 183, normalized size = 0.82 \[ \frac {-\frac {20 \left (a +\sqrt {d x +c}\, b \right )^{\frac {11}{2}} a}{11}-\frac {4 \left (-b^{2} c +a^{2}\right )^{2} \left (a +\sqrt {d x +c}\, b \right )^{\frac {3}{2}} a}{3}+\frac {4 \left (a +\sqrt {d x +c}\, b \right )^{\frac {13}{2}}}{13}+\frac {4 \left (-2 b^{2} c +10 a^{2}\right ) \left (a +\sqrt {d x +c}\, b \right )^{\frac {9}{2}}}{9}+\frac {4 \left (-4 \left (-b^{2} c +a^{2}\right ) a -\left (-2 b^{2} c +6 a^{2}\right ) a \right ) \left (a +\sqrt {d x +c}\, b \right )^{\frac {7}{2}}}{7}+\frac {4 \left (4 \left (-b^{2} c +a^{2}\right ) a^{2}+\left (-b^{2} c +a^{2}\right )^{2}\right ) \left (a +\sqrt {d x +c}\, b \right )^{\frac {5}{2}}}{5}}{b^{6} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/d^3/b^6*(1/13*(a+(d*x+c)^(1/2)*b)^(13/2)-5/11*a*(a+(d*x+c)^(1/2)*b)^(11/2)+1/9*(-2*b^2*c+10*a^2)*(a+(d*x+c)^
(1/2)*b)^(9/2)+1/7*(-4*(-b^2*c+a^2)*a-(-2*b^2*c+6*a^2)*a)*(a+(d*x+c)^(1/2)*b)^(7/2)+1/5*((-b^2*c+a^2)^2+4*(-b^
2*c+a^2)*a^2)*(a+(d*x+c)^(1/2)*b)^(5/2)-1/3*(-b^2*c+a^2)^2*a*(a+(d*x+c)^(1/2)*b)^(3/2))

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maxima [A]  time = 0.94, size = 167, normalized size = 0.75 \[ \frac {4 \, {\left (3465 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {13}{2}} - 20475 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {11}{2}} a - 10010 \, {\left (b^{2} c - 5 \, a^{2}\right )} {\left (\sqrt {d x + c} b + a\right )}^{\frac {9}{2}} + 12870 \, {\left (3 \, a b^{2} c - 5 \, a^{3}\right )} {\left (\sqrt {d x + c} b + a\right )}^{\frac {7}{2}} + 9009 \, {\left (b^{4} c^{2} - 6 \, a^{2} b^{2} c + 5 \, a^{4}\right )} {\left (\sqrt {d x + c} b + a\right )}^{\frac {5}{2}} - 15015 \, {\left (a b^{4} c^{2} - 2 \, a^{3} b^{2} c + a^{5}\right )} {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}}\right )}}{45045 \, b^{6} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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)*(sqr
t(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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int x^2\,\sqrt {a+b\,\sqrt {c+d\,x}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \sqrt {a + b \sqrt {c + d x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2*sqrt(a + b*sqrt(c + d*x)), x)

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