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

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

[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)

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Rubi [A]  time = 0.156336, antiderivative size = 224, normalized size of antiderivative = 1., 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 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]

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]

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*}

Mathematica [A]  time = 0.198368, size = 147, normalized size = 0.66 \[ \frac{4 \left (a+b \sqrt{c+d x}\right )^{3/2} \left (32 a^3 b^2 (68 c-75 d x)+16 a^2 b^3 \sqrt{c+d x} (175 d x-254 c)+1920 a^4 b \sqrt{c+d x}-1280 a^5-6 a b^4 \left (96 c^2-380 c d x+525 d^2 x^2\right )+77 b^5 \sqrt{c+d x} \left (32 c^2-40 c d x+45 d^2 x^2\right )\right )}{45045 b^6 d^3} \]

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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Maple [A]  time = 0.003, size = 183, normalized size = 0.8 \begin{align*} 4\,{\frac{1}{{d}^{3}{b}^{6}} \left ( 1/13\, \left ( a+b\sqrt{dx+c} \right ) ^{13/2}-{\frac{5\,a \left ( a+b\sqrt{dx+c} \right ) ^{11/2}}{11}}+1/9\, \left ( -2\,{b}^{2}c+10\,{a}^{2} \right ) \left ( a+b\sqrt{dx+c} \right ) ^{9/2}+1/7\, \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{dx+c} \right ) ^{7/2}+1/5\, \left ( \left ( -{b}^{2}c+{a}^{2} \right ) ^{2}+4\,{a}^{2} \left ( -{b}^{2}c+{a}^{2} \right ) \right ) \left ( a+b\sqrt{dx+c} \right ) ^{5/2}-1/3\, \left ( -{b}^{2}c+{a}^{2} \right ) ^{2}a \left ( a+b\sqrt{dx+c} \right ) ^{3/2} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.09632, size = 225, normalized size = 1. \begin{align*} \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}} \end{align*}

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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Fricas [A]  time = 2.34869, size = 441, normalized size = 1.97 \begin{align*} \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}} \end{align*}

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \sqrt{a + b \sqrt{c + d x}}\, dx \end{align*}

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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Giac [B]  time = 1.21627, size = 923, normalized size = 4.12 \begin{align*} \text{result too large to display} \end{align*}

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*(9009*sqrt((sqrt(d*x + c)*b + a)*b^2)*(sqrt(d*x + c)*b + a)^2*b^6*c^2*sgn((sqrt(d*x + c)*b + a)*b - a*
b) - 15015*sqrt((sqrt(d*x + c)*b + a)*b^2)*(sqrt(d*x + c)*b + a)*a*b^6*c^2*sgn((sqrt(d*x + c)*b + a)*b - a*b)
- 10010*sqrt((sqrt(d*x + c)*b + a)*b^2)*(sqrt(d*x + c)*b + a)^4*b^4*c*sgn((sqrt(d*x + c)*b + a)*b - a*b) + 386
10*sqrt((sqrt(d*x + c)*b + a)*b^2)*(sqrt(d*x + c)*b + a)^3*a*b^4*c*sgn((sqrt(d*x + c)*b + a)*b - a*b) - 54054*
sqrt((sqrt(d*x + c)*b + a)*b^2)*(sqrt(d*x + c)*b + a)^2*a^2*b^4*c*sgn((sqrt(d*x + c)*b + a)*b - a*b) + 30030*s
qrt((sqrt(d*x + c)*b + a)*b^2)*(sqrt(d*x + c)*b + a)*a^3*b^4*c*sgn((sqrt(d*x + c)*b + a)*b - a*b) + 3465*sqrt(
(sqrt(d*x + c)*b + a)*b^2)*(sqrt(d*x + c)*b + a)^6*b^2*sgn((sqrt(d*x + c)*b + a)*b - a*b) - 20475*sqrt((sqrt(d
*x + c)*b + a)*b^2)*(sqrt(d*x + c)*b + a)^5*a*b^2*sgn((sqrt(d*x + c)*b + a)*b - a*b) + 50050*sqrt((sqrt(d*x +
c)*b + a)*b^2)*(sqrt(d*x + c)*b + a)^4*a^2*b^2*sgn((sqrt(d*x + c)*b + a)*b - a*b) - 64350*sqrt((sqrt(d*x + c)*
b + a)*b^2)*(sqrt(d*x + c)*b + a)^3*a^3*b^2*sgn((sqrt(d*x + c)*b + a)*b - a*b) + 45045*sqrt((sqrt(d*x + c)*b +
a)*b^2)*(sqrt(d*x + c)*b + a)^2*a^4*b^2*sgn((sqrt(d*x + c)*b + a)*b - a*b) - 15015*sqrt((sqrt(d*x + c)*b + a)
*b^2)*(sqrt(d*x + c)*b + a)*a^5*b^2*sgn((sqrt(d*x + c)*b + a)*b - a*b))*abs(b)/(b^10*d^3)