3.2967 \(\int x^3 \sqrt{a+b \sqrt{c x^3}} \, dx\)

Optimal. Leaf size=843 \[ \frac{4}{19} \sqrt{a+b \sqrt{c x^3}} x^4+\frac{12 a \sqrt{c x^3} \sqrt{a+b \sqrt{c x^3}} x}{247 b c}-\frac{120 a^2 \sqrt{a+b \sqrt{c x^3}} x}{1729 b^2 c}-\frac{240 \sqrt [4]{3} \sqrt{2-\sqrt{3}} a^{10/3} \left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\sqrt [3]{a}\right ) \sqrt{\frac{-\frac{\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+b^{2/3} \sqrt [3]{c} x+a^{2/3}}{\left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}\right )^2}} E\left (\sin ^{-1}\left (\frac{\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{1729 b^{8/3} c^{4/3} \sqrt{\frac{\sqrt [3]{a} \left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\sqrt [3]{a}\right )}{\left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}\right )^2}} \sqrt{a+b \sqrt{c x^3}}}+\frac{160 \sqrt{2} 3^{3/4} a^{10/3} \left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\sqrt [3]{a}\right ) \sqrt{\frac{-\frac{\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+b^{2/3} \sqrt [3]{c} x+a^{2/3}}{\left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt{3}\right )}{1729 b^{8/3} c^{4/3} \sqrt{\frac{\sqrt [3]{a} \left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\sqrt [3]{a}\right )}{\left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}\right )^2}} \sqrt{a+b \sqrt{c x^3}}}+\frac{480 a^3 \sqrt{a+b \sqrt{c x^3}}}{1729 b^{8/3} c^{4/3} \left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}\right )} \]

[Out]

(-120*a^2*x*Sqrt[a + b*Sqrt[c*x^3]])/(1729*b^2*c) + (4*x^4*Sqrt[a + b*Sqrt[c*x^3]])/19 + (12*a*x*Sqrt[c*x^3]*S
qrt[a + b*Sqrt[c*x^3]])/(247*b*c) + (480*a^3*Sqrt[a + b*Sqrt[c*x^3]])/(1729*b^(8/3)*c^(4/3)*((1 + Sqrt[3])*a^(
1/3) + (b^(1/3)*c^(2/3)*x^2)/Sqrt[c*x^3])) - (240*3^(1/4)*Sqrt[2 - Sqrt[3]]*a^(10/3)*(a^(1/3) + (b^(1/3)*c^(2/
3)*x^2)/Sqrt[c*x^3])*Sqrt[(a^(2/3) + b^(2/3)*c^(1/3)*x - (a^(1/3)*b^(1/3)*c^(2/3)*x^2)/Sqrt[c*x^3])/((1 + Sqrt
[3])*a^(1/3) + (b^(1/3)*c^(2/3)*x^2)/Sqrt[c*x^3])^2]*EllipticE[ArcSin[((1 - Sqrt[3])*a^(1/3) + (b^(1/3)*c^(2/3
)*x^2)/Sqrt[c*x^3])/((1 + Sqrt[3])*a^(1/3) + (b^(1/3)*c^(2/3)*x^2)/Sqrt[c*x^3])], -7 - 4*Sqrt[3]])/(1729*b^(8/
3)*c^(4/3)*Sqrt[(a^(1/3)*(a^(1/3) + (b^(1/3)*c^(2/3)*x^2)/Sqrt[c*x^3]))/((1 + Sqrt[3])*a^(1/3) + (b^(1/3)*c^(2
/3)*x^2)/Sqrt[c*x^3])^2]*Sqrt[a + b*Sqrt[c*x^3]]) + (160*Sqrt[2]*3^(3/4)*a^(10/3)*(a^(1/3) + (b^(1/3)*c^(2/3)*
x^2)/Sqrt[c*x^3])*Sqrt[(a^(2/3) + b^(2/3)*c^(1/3)*x - (a^(1/3)*b^(1/3)*c^(2/3)*x^2)/Sqrt[c*x^3])/((1 + Sqrt[3]
)*a^(1/3) + (b^(1/3)*c^(2/3)*x^2)/Sqrt[c*x^3])^2]*EllipticF[ArcSin[((1 - Sqrt[3])*a^(1/3) + (b^(1/3)*c^(2/3)*x
^2)/Sqrt[c*x^3])/((1 + Sqrt[3])*a^(1/3) + (b^(1/3)*c^(2/3)*x^2)/Sqrt[c*x^3])], -7 - 4*Sqrt[3]])/(1729*b^(8/3)*
c^(4/3)*Sqrt[(a^(1/3)*(a^(1/3) + (b^(1/3)*c^(2/3)*x^2)/Sqrt[c*x^3]))/((1 + Sqrt[3])*a^(1/3) + (b^(1/3)*c^(2/3)
*x^2)/Sqrt[c*x^3])^2]*Sqrt[a + b*Sqrt[c*x^3]])

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Rubi [A]  time = 0.636343, antiderivative size = 843, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {369, 341, 279, 321, 303, 218, 1877} \[ \frac{4}{19} \sqrt{a+b \sqrt{c x^3}} x^4+\frac{12 a \sqrt{c x^3} \sqrt{a+b \sqrt{c x^3}} x}{247 b c}-\frac{120 a^2 \sqrt{a+b \sqrt{c x^3}} x}{1729 b^2 c}-\frac{240 \sqrt [4]{3} \sqrt{2-\sqrt{3}} a^{10/3} \left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\sqrt [3]{a}\right ) \sqrt{\frac{-\frac{\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+b^{2/3} \sqrt [3]{c} x+a^{2/3}}{\left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}\right )^2}} E\left (\sin ^{-1}\left (\frac{\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{1729 b^{8/3} c^{4/3} \sqrt{\frac{\sqrt [3]{a} \left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\sqrt [3]{a}\right )}{\left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}\right )^2}} \sqrt{a+b \sqrt{c x^3}}}+\frac{160 \sqrt{2} 3^{3/4} a^{10/3} \left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\sqrt [3]{a}\right ) \sqrt{\frac{-\frac{\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+b^{2/3} \sqrt [3]{c} x+a^{2/3}}{\left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}\right )^2}} F\left (\sin ^{-1}\left (\frac{\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{1729 b^{8/3} c^{4/3} \sqrt{\frac{\sqrt [3]{a} \left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\sqrt [3]{a}\right )}{\left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}\right )^2}} \sqrt{a+b \sqrt{c x^3}}}+\frac{480 a^3 \sqrt{a+b \sqrt{c x^3}}}{1729 b^{8/3} c^{4/3} \left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}\right )} \]

Antiderivative was successfully verified.

[In]

Int[x^3*Sqrt[a + b*Sqrt[c*x^3]],x]

[Out]

(-120*a^2*x*Sqrt[a + b*Sqrt[c*x^3]])/(1729*b^2*c) + (4*x^4*Sqrt[a + b*Sqrt[c*x^3]])/19 + (12*a*x*Sqrt[c*x^3]*S
qrt[a + b*Sqrt[c*x^3]])/(247*b*c) + (480*a^3*Sqrt[a + b*Sqrt[c*x^3]])/(1729*b^(8/3)*c^(4/3)*((1 + Sqrt[3])*a^(
1/3) + (b^(1/3)*c^(2/3)*x^2)/Sqrt[c*x^3])) - (240*3^(1/4)*Sqrt[2 - Sqrt[3]]*a^(10/3)*(a^(1/3) + (b^(1/3)*c^(2/
3)*x^2)/Sqrt[c*x^3])*Sqrt[(a^(2/3) + b^(2/3)*c^(1/3)*x - (a^(1/3)*b^(1/3)*c^(2/3)*x^2)/Sqrt[c*x^3])/((1 + Sqrt
[3])*a^(1/3) + (b^(1/3)*c^(2/3)*x^2)/Sqrt[c*x^3])^2]*EllipticE[ArcSin[((1 - Sqrt[3])*a^(1/3) + (b^(1/3)*c^(2/3
)*x^2)/Sqrt[c*x^3])/((1 + Sqrt[3])*a^(1/3) + (b^(1/3)*c^(2/3)*x^2)/Sqrt[c*x^3])], -7 - 4*Sqrt[3]])/(1729*b^(8/
3)*c^(4/3)*Sqrt[(a^(1/3)*(a^(1/3) + (b^(1/3)*c^(2/3)*x^2)/Sqrt[c*x^3]))/((1 + Sqrt[3])*a^(1/3) + (b^(1/3)*c^(2
/3)*x^2)/Sqrt[c*x^3])^2]*Sqrt[a + b*Sqrt[c*x^3]]) + (160*Sqrt[2]*3^(3/4)*a^(10/3)*(a^(1/3) + (b^(1/3)*c^(2/3)*
x^2)/Sqrt[c*x^3])*Sqrt[(a^(2/3) + b^(2/3)*c^(1/3)*x - (a^(1/3)*b^(1/3)*c^(2/3)*x^2)/Sqrt[c*x^3])/((1 + Sqrt[3]
)*a^(1/3) + (b^(1/3)*c^(2/3)*x^2)/Sqrt[c*x^3])^2]*EllipticF[ArcSin[((1 - Sqrt[3])*a^(1/3) + (b^(1/3)*c^(2/3)*x
^2)/Sqrt[c*x^3])/((1 + Sqrt[3])*a^(1/3) + (b^(1/3)*c^(2/3)*x^2)/Sqrt[c*x^3])], -7 - 4*Sqrt[3]])/(1729*b^(8/3)*
c^(4/3)*Sqrt[(a^(1/3)*(a^(1/3) + (b^(1/3)*c^(2/3)*x^2)/Sqrt[c*x^3]))/((1 + Sqrt[3])*a^(1/3) + (b^(1/3)*c^(2/3)
*x^2)/Sqrt[c*x^3])^2]*Sqrt[a + b*Sqrt[c*x^3]])

Rule 369

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*((c_.)*(x_)^(q_))^(n_))^(p_.), x_Symbol] :> With[{k = Denominator[n]}, Su
bst[Int[(d*x)^m*(a + b*c^n*x^(n*q))^p, x], x^(1/k), (c*x^q)^(1/k)/(c^(1/k)*(x^(1/k))^(q - 1))]] /; FreeQ[{a, b
, c, d, m, p, q}, x] && FractionQ[n]

Rule 341

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[n]}, Dist[k, Subst[Int[x^(k*(
m + 1) - 1)*(a + b*x^(k*n))^p, x], x, x^(1/k)], x]] /; FreeQ[{a, b, m, p}, x] && FractionQ[n]

Rule 279

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m +
n*p + 1)), x] + Dist[(a*n*p)/(m + n*p + 1), Int[(c*x)^m*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x]
&& IGtQ[n, 0] && GtQ[p, 0] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 303

Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Dist[(Sq
rt[2]*s)/(Sqrt[2 + Sqrt[3]]*r), Int[1/Sqrt[a + b*x^3], x], x] + Dist[1/r, Int[((1 - Sqrt[3])*s + r*x)/Sqrt[a +
b*x^3], x], x]] /; FreeQ[{a, b}, x] && PosQ[a]

Rule 218

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(2*Sqr
t[2 + Sqrt[3]]*(s + r*x)*Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]*EllipticF[ArcSin[((1 - Sqrt[3
])*s + r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]])/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[(s*(s + r*x))/((1 + Sqr
t[3])*s + r*x)^2]), x]] /; FreeQ[{a, b}, x] && PosQ[a]

Rule 1877

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Simplify[((1 - Sqrt[3])*d)/c]]
, s = Denom[Simplify[((1 - Sqrt[3])*d)/c]]}, Simp[(2*d*s^3*Sqrt[a + b*x^3])/(a*r^2*((1 + Sqrt[3])*s + r*x)), x
] - Simp[(3^(1/4)*Sqrt[2 - Sqrt[3]]*d*s*(s + r*x)*Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]*Elli
pticE[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]])/(r^2*Sqrt[a + b*x^3]*Sqrt[(s*(
s + r*x))/((1 + Sqrt[3])*s + r*x)^2]), x]] /; FreeQ[{a, b, c, d}, x] && PosQ[a] && EqQ[b*c^3 - 2*(5 - 3*Sqrt[3
])*a*d^3, 0]

Rubi steps

\begin{align*} \int x^3 \sqrt{a+b \sqrt{c x^3}} \, dx &=\operatorname{Subst}\left (\int x^3 \sqrt{a+b \sqrt{c} x^{3/2}} \, dx,\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=\operatorname{Subst}\left (2 \operatorname{Subst}\left (\int x^7 \sqrt{a+b \sqrt{c} x^3} \, dx,x,\sqrt{x}\right ),\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=\frac{4}{19} x^4 \sqrt{a+b \sqrt{c x^3}}+\operatorname{Subst}\left (\frac{1}{19} (6 a) \operatorname{Subst}\left (\int \frac{x^7}{\sqrt{a+b \sqrt{c} x^3}} \, dx,x,\sqrt{x}\right ),\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=\frac{4}{19} x^4 \sqrt{a+b \sqrt{c x^3}}+\frac{12 a x \sqrt{c x^3} \sqrt{a+b \sqrt{c x^3}}}{247 b c}-\operatorname{Subst}\left (\frac{\left (60 a^2\right ) \operatorname{Subst}\left (\int \frac{x^4}{\sqrt{a+b \sqrt{c} x^3}} \, dx,x,\sqrt{x}\right )}{247 b \sqrt{c}},\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=-\frac{120 a^2 x \sqrt{a+b \sqrt{c x^3}}}{1729 b^2 c}+\frac{4}{19} x^4 \sqrt{a+b \sqrt{c x^3}}+\frac{12 a x \sqrt{c x^3} \sqrt{a+b \sqrt{c x^3}}}{247 b c}+\operatorname{Subst}\left (\frac{\left (240 a^3\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{a+b \sqrt{c} x^3}} \, dx,x,\sqrt{x}\right )}{1729 b^2 c},\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=-\frac{120 a^2 x \sqrt{a+b \sqrt{c x^3}}}{1729 b^2 c}+\frac{4}{19} x^4 \sqrt{a+b \sqrt{c x^3}}+\frac{12 a x \sqrt{c x^3} \sqrt{a+b \sqrt{c x^3}}}{247 b c}+\operatorname{Subst}\left (\frac{\left (240 a^3\right ) \operatorname{Subst}\left (\int \frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt [6]{c} x}{\sqrt{a+b \sqrt{c} x^3}} \, dx,x,\sqrt{x}\right )}{1729 b^{7/3} c^{7/6}},\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )+\operatorname{Subst}\left (\frac{\left (240 \sqrt{2 \left (2-\sqrt{3}\right )} a^{10/3}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b \sqrt{c} x^3}} \, dx,x,\sqrt{x}\right )}{1729 b^{7/3} c^{7/6}},\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=-\frac{120 a^2 x \sqrt{a+b \sqrt{c x^3}}}{1729 b^2 c}+\frac{4}{19} x^4 \sqrt{a+b \sqrt{c x^3}}+\frac{12 a x \sqrt{c x^3} \sqrt{a+b \sqrt{c x^3}}}{247 b c}+\frac{480 a^3 \sqrt{a+b \sqrt{c x^3}}}{1729 b^{8/3} c^{4/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}\right )}-\frac{240 \sqrt [4]{3} \sqrt{2-\sqrt{3}} a^{10/3} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}\right ) \sqrt{\frac{a^{2/3}+b^{2/3} \sqrt [3]{c} x-\frac{\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}}\right )|-7-4 \sqrt{3}\right )}{1729 b^{8/3} c^{4/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}\right )^2}} \sqrt{a+b \sqrt{c x^3}}}+\frac{160 \sqrt{2} 3^{3/4} a^{10/3} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}\right ) \sqrt{\frac{a^{2/3}+b^{2/3} \sqrt [3]{c} x-\frac{\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}}\right )|-7-4 \sqrt{3}\right )}{1729 b^{8/3} c^{4/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}\right )^2}} \sqrt{a+b \sqrt{c x^3}}}\\ \end{align*}

Mathematica [F]  time = 0.0441376, size = 0, normalized size = 0. \[ \int x^3 \sqrt{a+b \sqrt{c x^3}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[x^3*Sqrt[a + b*Sqrt[c*x^3]],x]

[Out]

Integrate[x^3*Sqrt[a + b*Sqrt[c*x^3]], x]

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Maple [A]  time = 0.176, size = 932, normalized size = 1.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a+b*(c*x^3)^(1/2))^(1/2),x)

[Out]

4/1729/x^2/c^2*(30*I*3^(1/2)*x^2*2^(1/2)*(-I*(I*3^(1/2)*x*(-a*c*b^2)^(1/3)-2*b*(c*x^3)^(1/2)-(-a*c*b^2)^(1/3)*
x)*3^(1/2)/(-a*c*b^2)^(1/3)/x)^(1/2)*(-a*c*b^2)^(2/3)*((b*(c*x^3)^(1/2)-(-a*c*b^2)^(1/3)*x)/x/(-a*c*b^2)^(1/3)
/(I*3^(1/2)-3))^(1/2)*(-I*(I*3^(1/2)*x*(-a*c*b^2)^(1/3)+2*b*(c*x^3)^(1/2)+(-a*c*b^2)^(1/3)*x)*3^(1/2)/(-a*c*b^
2)^(1/3)/x)^(1/2)*EllipticE(1/6*3^(1/2)*2^(1/2)*(-I*(I*3^(1/2)*x*(-a*c*b^2)^(1/3)-2*b*(c*x^3)^(1/2)-(-a*c*b^2)
^(1/3)*x)*3^(1/2)/(-a*c*b^2)^(1/3)/x)^(1/2),2^(1/2)*(I*3^(1/2)/(I*3^(1/2)-3))^(1/2))*a^3-20*I*3^(1/2)*x^2*2^(1
/2)*(-I*(I*3^(1/2)*x*(-a*c*b^2)^(1/3)-2*b*(c*x^3)^(1/2)-(-a*c*b^2)^(1/3)*x)*3^(1/2)/(-a*c*b^2)^(1/3)/x)^(1/2)*
(-a*c*b^2)^(2/3)*((b*(c*x^3)^(1/2)-(-a*c*b^2)^(1/3)*x)/x/(-a*c*b^2)^(1/3)/(I*3^(1/2)-3))^(1/2)*(-I*(I*3^(1/2)*
x*(-a*c*b^2)^(1/3)+2*b*(c*x^3)^(1/2)+(-a*c*b^2)^(1/3)*x)*3^(1/2)/(-a*c*b^2)^(1/3)/x)^(1/2)*EllipticF(1/6*3^(1/
2)*2^(1/2)*(-I*(I*3^(1/2)*x*(-a*c*b^2)^(1/3)-2*b*(c*x^3)^(1/2)-(-a*c*b^2)^(1/3)*x)*3^(1/2)/(-a*c*b^2)^(1/3)/x)
^(1/2),2^(1/2)*(I*3^(1/2)/(I*3^(1/2)-3))^(1/2))*a^3+91*(c*x^3)^(1/2)*x^6*b^5*c^2+112*x^6*a*b^4*c^2+30*x^2*2^(1
/2)*(-I*(I*3^(1/2)*x*(-a*c*b^2)^(1/3)-2*b*(c*x^3)^(1/2)-(-a*c*b^2)^(1/3)*x)*3^(1/2)/(-a*c*b^2)^(1/3)/x)^(1/2)*
(-a*c*b^2)^(2/3)*((b*(c*x^3)^(1/2)-(-a*c*b^2)^(1/3)*x)/x/(-a*c*b^2)^(1/3)/(I*3^(1/2)-3))^(1/2)*(-I*(I*3^(1/2)*
x*(-a*c*b^2)^(1/3)+2*b*(c*x^3)^(1/2)+(-a*c*b^2)^(1/3)*x)*3^(1/2)/(-a*c*b^2)^(1/3)/x)^(1/2)*EllipticE(1/6*3^(1/
2)*2^(1/2)*(-I*(I*3^(1/2)*x*(-a*c*b^2)^(1/3)-2*b*(c*x^3)^(1/2)-(-a*c*b^2)^(1/3)*x)*3^(1/2)/(-a*c*b^2)^(1/3)/x)
^(1/2),2^(1/2)*(I*3^(1/2)/(I*3^(1/2)-3))^(1/2))*a^3-30*(c*x^3)^(1/2)*x^3*a^2*b^3*c-30*x^3*a^3*b^2*c+21*(c*x^3)
^(3/2)*a^2*b^3)/b^4/(a+b*(c*x^3)^(1/2))^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\sqrt{c x^{3}} b + a} x^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*(c*x^3)^(1/2))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(sqrt(c*x^3)*b + a)*x^3, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{\sqrt{c x^{3}} b + a} x^{3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*(c*x^3)^(1/2))^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(sqrt(c*x^3)*b + a)*x^3, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \sqrt{a + b \sqrt{c x^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(a+b*(c*x**3)**(1/2))**(1/2),x)

[Out]

Integral(x**3*sqrt(a + b*sqrt(c*x**3)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\sqrt{c x^{3}} b + a} x^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*(c*x^3)^(1/2))^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(sqrt(c*x^3)*b + a)*x^3, x)