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

Optimal. Leaf size=222 \[ \frac{4 \left (-6 a^2 b^2 c+5 a^4+b^4 c^2\right ) \left (a+b \sqrt{c+d x}\right )^{3/2}}{3 b^6 d^3}+\frac{8 \left (5 a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{7/2}}{7 b^6 d^3}-\frac{8 a \left (5 a^2-3 b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{5/2}}{5 b^6 d^3}-\frac{4 a \left (a^2-b^2 c\right )^2 \sqrt{a+b \sqrt{c+d x}}}{b^6 d^3}+\frac{4 \left (a+b \sqrt{c+d x}\right )^{11/2}}{11 b^6 d^3}-\frac{20 a \left (a+b \sqrt{c+d x}\right )^{9/2}}{9 b^6 d^3} \]

[Out]

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

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Rubi [A]  time = 0.158127, antiderivative size = 222, 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} \[ \frac{4 \left (-6 a^2 b^2 c+5 a^4+b^4 c^2\right ) \left (a+b \sqrt{c+d x}\right )^{3/2}}{3 b^6 d^3}+\frac{8 \left (5 a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{7/2}}{7 b^6 d^3}-\frac{8 a \left (5 a^2-3 b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{5/2}}{5 b^6 d^3}-\frac{4 a \left (a^2-b^2 c\right )^2 \sqrt{a+b \sqrt{c+d x}}}{b^6 d^3}+\frac{4 \left (a+b \sqrt{c+d x}\right )^{11/2}}{11 b^6 d^3}-\frac{20 a \left (a+b \sqrt{c+d x}\right )^{9/2}}{9 b^6 d^3} \]

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*Sqrt[a + b*Sqrt[c + d*x]])/(b^6*d^3) + (4*(5*a^4 - 6*a^2*b^2*c + b^4*c^2)*(a + b*Sqrt[c
+ d*x])^(3/2))/(3*b^6*d^3) - (8*a*(5*a^2 - 3*b^2*c)*(a + b*Sqrt[c + d*x])^(5/2))/(5*b^6*d^3) + (8*(5*a^2 - b^2
*c)*(a + b*Sqrt[c + d*x])^(7/2))/(7*b^6*d^3) - (20*a*(a + b*Sqrt[c + d*x])^(9/2))/(9*b^6*d^3) + (4*(a + b*Sqrt
[c + d*x])^(11/2))/(11*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 \frac{x^2}{\sqrt{a+b \sqrt{c+d x}}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(-c+x)^2}{\sqrt{a+b \sqrt{x}}} \, dx,x,c+d x\right )}{d^3}\\ &=\frac{2 \operatorname{Subst}\left (\int \frac{x \left (-c+x^2\right )^2}{\sqrt{a+b x}} \, 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}{b^5 \sqrt{a+b x}}+\frac{\left (5 a^4-6 a^2 b^2 c+b^4 c^2\right ) \sqrt{a+b x}}{b^5}-\frac{2 \left (5 a^3-3 a b^2 c\right ) (a+b x)^{3/2}}{b^5}-\frac{2 \left (-5 a^2+b^2 c\right ) (a+b x)^{5/2}}{b^5}-\frac{5 a (a+b x)^{7/2}}{b^5}+\frac{(a+b x)^{9/2}}{b^5}\right ) \, dx,x,\sqrt{c+d x}\right )}{d^3}\\ &=-\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}\\ \end{align*}

Mathematica [A]  time = 0.173945, size = 147, normalized size = 0.66 \[ \frac{4 \sqrt{a+b \sqrt{c+d x}} \left (96 a^3 b^2 (28 c-5 d x)-16 a^2 b^3 (74 c-25 d x) \sqrt{c+d x}+640 a^4 b \sqrt{c+d x}-1280 a^5-2 a b^4 \left (736 c^2-244 c d x+175 d^2 x^2\right )+15 b^5 \sqrt{c+d x} \left (32 c^2-24 c d x+21 d^2 x^2\right )\right )}{3465 b^6 d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(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*Sqrt[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)

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

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Maxima [A]  time = 1.07706, size = 225, normalized size = 1.01 \begin{align*} \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}} \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]

4/3465*(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(sqrt(d*x + c)*b + a))/(b^6*d^3)

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Fricas [A]  time = 2.24425, size = 342, normalized size = 1.54 \begin{align*} -\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}} \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]

-4/3465*(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 - (3
15*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
(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 \frac{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 = 1.29198, size = 849, normalized size = 3.82 \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]

4/3465*(1155*sqrt((sqrt(d*x + c)*b + a)*b^2)*(sqrt(d*x + c)*b + a)*b^4*c^2*sgn((sqrt(d*x + c)*b + a)*b - a*b)
- 3465*sqrt((sqrt(d*x + c)*b + a)*b^2)*a*b^4*c^2*sgn((sqrt(d*x + c)*b + a)*b - a*b) - 990*sqrt((sqrt(d*x + c)*
b + a)*b^2)*(sqrt(d*x + c)*b + a)^3*b^2*c*sgn((sqrt(d*x + c)*b + a)*b - a*b) + 4158*sqrt((sqrt(d*x + c)*b + a)
*b^2)*(sqrt(d*x + c)*b + a)^2*a*b^2*c*sgn((sqrt(d*x + c)*b + a)*b - a*b) - 6930*sqrt((sqrt(d*x + c)*b + a)*b^2
)*(sqrt(d*x + c)*b + a)*a^2*b^2*c*sgn((sqrt(d*x + c)*b + a)*b - a*b) + 6930*sqrt((sqrt(d*x + c)*b + a)*b^2)*a^
3*b^2*c*sgn((sqrt(d*x + c)*b + a)*b - a*b) + 315*sqrt((sqrt(d*x + c)*b + a)*b^2)*(sqrt(d*x + c)*b + a)^5*sgn((
sqrt(d*x + c)*b + a)*b - a*b) - 1925*sqrt((sqrt(d*x + c)*b + a)*b^2)*(sqrt(d*x + c)*b + a)^4*a*sgn((sqrt(d*x +
c)*b + a)*b - a*b) + 4950*sqrt((sqrt(d*x + c)*b + a)*b^2)*(sqrt(d*x + c)*b + a)^3*a^2*sgn((sqrt(d*x + c)*b +
a)*b - a*b) - 6930*sqrt((sqrt(d*x + c)*b + a)*b^2)*(sqrt(d*x + c)*b + a)^2*a^3*sgn((sqrt(d*x + c)*b + a)*b - a
*b) + 5775*sqrt((sqrt(d*x + c)*b + a)*b^2)*(sqrt(d*x + c)*b + a)*a^4*sgn((sqrt(d*x + c)*b + a)*b - a*b) - 3465
*sqrt((sqrt(d*x + c)*b + a)*b^2)*a^5*sgn((sqrt(d*x + c)*b + a)*b - a*b))/(b^6*d^3*abs(b))