3.2964 \(\int x \sqrt{a+b \sqrt{c x^3}} \, dx\)

Optimal. Leaf size=400 \[ -\frac{8\ 3^{3/4} \sqrt{2+\sqrt{3}} a^2 \left (\sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}\right ) \sqrt{\frac{a^{2/3}-\frac{\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+b^{2/3} \sqrt [3]{c} x}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}}\right ),-7-4 \sqrt{3}\right )}{55 b^{4/3} c^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}\right )^2}} \sqrt{a+b \sqrt{c x^3}}}+\frac{12 a x^2 \sqrt{a+b \sqrt{c x^3}}}{55 b \sqrt{c x^3}}+\frac{4}{11} x^2 \sqrt{a+b \sqrt{c x^3}} \]

[Out]

(4*x^2*Sqrt[a + b*Sqrt[c*x^3]])/11 + (12*a*x^2*Sqrt[a + b*Sqrt[c*x^3]])/(55*b*Sqrt[c*x^3]) - (8*3^(3/4)*Sqrt[2
+ Sqrt[3]]*a^2*(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)/Sq
rt[c*x^3])], -7 - 4*Sqrt[3]])/(55*b^(4/3)*c^(2/3)*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 = 0.248281, antiderivative size = 400, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {369, 341, 279, 321, 218} \[ -\frac{8\ 3^{3/4} \sqrt{2+\sqrt{3}} a^2 \left (\sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}\right ) \sqrt{\frac{a^{2/3}-\frac{\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+b^{2/3} \sqrt [3]{c} x}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}\right )^2}} F\left (\sin ^{-1}\left (\frac{\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{55 b^{4/3} c^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}\right )^2}} \sqrt{a+b \sqrt{c x^3}}}+\frac{12 a x^2 \sqrt{a+b \sqrt{c x^3}}}{55 b \sqrt{c x^3}}+\frac{4}{11} x^2 \sqrt{a+b \sqrt{c x^3}} \]

Antiderivative was successfully verified.

[In]

Int[x*Sqrt[a + b*Sqrt[c*x^3]],x]

[Out]

(4*x^2*Sqrt[a + b*Sqrt[c*x^3]])/11 + (12*a*x^2*Sqrt[a + b*Sqrt[c*x^3]])/(55*b*Sqrt[c*x^3]) - (8*3^(3/4)*Sqrt[2
+ Sqrt[3]]*a^2*(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)/Sq
rt[c*x^3])], -7 - 4*Sqrt[3]])/(55*b^(4/3)*c^(2/3)*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 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]

Rubi steps

\begin{align*} \int x \sqrt{a+b \sqrt{c x^3}} \, dx &=\operatorname{Subst}\left (\int x \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^3 \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}{11} x^2 \sqrt{a+b \sqrt{c x^3}}+\operatorname{Subst}\left (\frac{1}{11} (6 a) \operatorname{Subst}\left (\int \frac{x^3}{\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}{11} x^2 \sqrt{a+b \sqrt{c x^3}}+\frac{12 a x^2 \sqrt{a+b \sqrt{c x^3}}}{55 b \sqrt{c x^3}}-\operatorname{Subst}\left (\frac{\left (12 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b \sqrt{c} x^3}} \, dx,x,\sqrt{x}\right )}{55 b \sqrt{c}},\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=\frac{4}{11} x^2 \sqrt{a+b \sqrt{c x^3}}+\frac{12 a x^2 \sqrt{a+b \sqrt{c x^3}}}{55 b \sqrt{c x^3}}-\frac{8\ 3^{3/4} \sqrt{2+\sqrt{3}} a^2 \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 )}{55 b^{4/3} c^{2/3} \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 = 0.0394844, size = 0, normalized size = 0. \[ \int x \sqrt{a+b \sqrt{c x^3}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.241, size = 350, normalized size = 0.9 \begin{align*}{\frac{4}{55\,{b}^{2}c} \left ( i{a}^{2}\sqrt{3}\sqrt [3]{-ac{b}^{2}}\sqrt{2}\sqrt{{\frac{-i\sqrt{3}}{x} \left ( i\sqrt{3}x\sqrt [3]{-ac{b}^{2}}-2\,b\sqrt{c{x}^{3}}-\sqrt [3]{-ac{b}^{2}}x \right ){\frac{1}{\sqrt [3]{-ac{b}^{2}}}}}}\sqrt{{\frac{1}{x \left ( i\sqrt{3}-3 \right ) } \left ( b\sqrt{c{x}^{3}}-\sqrt [3]{-ac{b}^{2}}x \right ){\frac{1}{\sqrt [3]{-ac{b}^{2}}}}}}\sqrt{{\frac{-i\sqrt{3}}{x} \left ( i\sqrt{3}x\sqrt [3]{-ac{b}^{2}}+2\,b\sqrt{c{x}^{3}}+\sqrt [3]{-ac{b}^{2}}x \right ){\frac{1}{\sqrt [3]{-ac{b}^{2}}}}}}{\it EllipticF} \left ({\frac{\sqrt{2}\sqrt{3}}{6}\sqrt{{\frac{-i\sqrt{3}}{x} \left ( i\sqrt{3}x\sqrt [3]{-ac{b}^{2}}-2\,b\sqrt{c{x}^{3}}-\sqrt [3]{-ac{b}^{2}}x \right ){\frac{1}{\sqrt [3]{-ac{b}^{2}}}}}}},\sqrt{2}\sqrt{{\frac{i\sqrt{3}}{i\sqrt{3}-3}}} \right ) \sqrt{c{x}^{3}}+5\,{c}^{2}{x}^{5}{b}^{3}+8\,\sqrt{c{x}^{3}}{x}^{2}a{b}^{2}c+3\,{x}^{2}{a}^{2}bc \right ){\frac{1}{\sqrt{c{x}^{3}}}}{\frac{1}{\sqrt{a+b\sqrt{c{x}^{3}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/55*(I*a^2*3^(1/2)*(-a*c*b^2)^(1/3)*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)*((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))*(c*x^3)^(1/2)+5*c^2*x^5*b^3+8*(c*x^3)^
(1/2)*x^2*a*b^2*c+3*x^2*a^2*b*c)/c/(c*x^3)^(1/2)/b^2/(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\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(sqrt(c*x^3)*b + a)*x, 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, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x*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\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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