\(\int x \sqrt {a+b \sqrt {c x^3}} \, dx\) [2964]

Optimal result

Integrand size = 19, antiderivative size = 400 \[ \int x \sqrt {a+b \sqrt {c x^3}} \, dx=\frac {4}{11} x^2 \sqrt {a+b \sqrt {c x^3}}+\frac {12 a x^2 \sqrt {a+b \sqrt {c x^3}}}{55 b \sqrt {c x^3}}-\frac {8\ 3^{3/4} \sqrt {2+\sqrt {3}} a^2 \left (\sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right ) \sqrt {\frac {a^{2/3}+b^{2/3} \sqrt [3]{c} x-\frac {\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}}\right ),-7-4 \sqrt {3}\right )}{55 b^{4/3} c^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )^2}} \sqrt {a+b \sqrt {c x^3}}} \]

[Out]

-8/55*3^(3/4)*a^2*EllipticF((a^(1/3)*(1-3^(1/2))+b^(1/3)*c^(2/3)*x^2/(c*x^3)^(1/2))/(a^(1/3)*(1+3^(1/2))+b^(1/
3)*c^(2/3)*x^2/(c*x^3)^(1/2)),I*3^(1/2)+2*I)*(a^(1/3)+b^(1/3)*c^(2/3)*x^2/(c*x^3)^(1/2))*(1/2*6^(1/2)+1/2*2^(1
/2))*((a^(2/3)+b^(2/3)*c^(1/3)*x-a^(1/3)*b^(1/3)*c^(2/3)*x^2/(c*x^3)^(1/2))/(a^(1/3)*(1+3^(1/2))+b^(1/3)*c^(2/
3)*x^2/(c*x^3)^(1/2))^2)^(1/2)/b^(4/3)/c^(2/3)/(a^(1/3)*(a^(1/3)+b^(1/3)*c^(2/3)*x^2/(c*x^3)^(1/2))/(a^(1/3)*(
1+3^(1/2))+b^(1/3)*c^(2/3)*x^2/(c*x^3)^(1/2))^2)^(1/2)/(a+b*(c*x^3)^(1/2))^(1/2)+4/11*x^2*(a+b*(c*x^3)^(1/2))^
(1/2)+12/55*a*x^2*(a+b*(c*x^3)^(1/2))^(1/2)/b/(c*x^3)^(1/2)

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 400, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {376, 348, 285, 327, 224} \[ \int x \sqrt {a+b \sqrt {c x^3}} \, dx=-\frac {8\ 3^{3/4} \sqrt {2+\sqrt {3}} a^2 \left (\sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right ) \sqrt {\frac {a^{2/3}-\frac {\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}+b^{2/3} \sqrt [3]{c} x}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right )}{55 b^{4/3} c^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )^2}} \sqrt {a+b \sqrt {c x^3}}}+\frac {12 a x^2 \sqrt {a+b \sqrt {c x^3}}}{55 b \sqrt {c x^3}}+\frac {4}{11} x^2 \sqrt {a+b \sqrt {c x^3}} \]

[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 224

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

Rule 285

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 327

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 348

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 376

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]

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int x \sqrt {a+b \sqrt {c} x^{3/2}} \, dx,\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right ) \\ & = \text {Subst}\left (2 \text {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}}+\text {Subst}\left (\frac {1}{11} (6 a) \text {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}}-\text {Subst}\left (\frac {\left (12 a^2\right ) \text {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]

\[ \int x \sqrt {a+b \sqrt {c x^3}} \, dx=\int x \sqrt {a+b \sqrt {c x^3}} \, dx \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 5.47 (sec) , antiderivative size = 350, normalized size of antiderivative = 0.88

method	result	size
default	\(\frac {\frac {4 i a^{2} \sqrt {3}\, \left (-a \,b^{2} c \right )^{\frac {1}{3}} \sqrt {2}\, \sqrt {-\frac {i \left (i \sqrt {3}\, x \left (-a \,b^{2} c \right )^{\frac {1}{3}}-2 b \sqrt {c \,x^{3}}-x \left (-a \,b^{2} c \right )^{\frac {1}{3}}\right ) \sqrt {3}}{x \left (-a \,b^{2} c \right )^{\frac {1}{3}}}}\, \sqrt {\frac {b \sqrt {c \,x^{3}}-x \left (-a \,b^{2} c \right )^{\frac {1}{3}}}{x \left (-a \,b^{2} c \right )^{\frac {1}{3}} \left (i \sqrt {3}-3\right )}}\, \sqrt {-\frac {i \left (i \sqrt {3}\, x \left (-a \,b^{2} c \right )^{\frac {1}{3}}+2 b \sqrt {c \,x^{3}}+x \left (-a \,b^{2} c \right )^{\frac {1}{3}}\right ) \sqrt {3}}{x \left (-a \,b^{2} c \right )^{\frac {1}{3}}}}\, F\left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {-\frac {i \left (i \sqrt {3}\, x \left (-a \,b^{2} c \right )^{\frac {1}{3}}-2 b \sqrt {c \,x^{3}}-x \left (-a \,b^{2} c \right )^{\frac {1}{3}}\right ) \sqrt {3}}{x \left (-a \,b^{2} c \right )^{\frac {1}{3}}}}}{6}, \sqrt {2}\, \sqrt {\frac {i \sqrt {3}}{i \sqrt {3}-3}}\right ) \sqrt {c \,x^{3}}}{55}+\frac {4 c^{2} x^{5} b^{3}}{11}+\frac {32 \sqrt {c \,x^{3}}\, a \,b^{2} c \,x^{2}}{55}+\frac {12 a^{2} b c \,x^{2}}{55}}{c \,b^{2} \sqrt {c \,x^{3}}\, \sqrt {a +b \sqrt {c \,x^{3}}}}\)	\(350\)

[In]

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

[Out]

4/55*(I*a^2*3^(1/2)*(-a*b^2*c)^(1/3)*2^(1/2)*(-I*(I*3^(1/2)*x*(-a*b^2*c)^(1/3)-2*b*(c*x^3)^(1/2)-x*(-a*b^2*c)^
(1/3))*3^(1/2)/x/(-a*b^2*c)^(1/3))^(1/2)*((b*(c*x^3)^(1/2)-x*(-a*b^2*c)^(1/3))/x/(-a*b^2*c)^(1/3)/(I*3^(1/2)-3
))^(1/2)*(-I*(I*3^(1/2)*x*(-a*b^2*c)^(1/3)+2*b*(c*x^3)^(1/2)+x*(-a*b^2*c)^(1/3))*3^(1/2)/x/(-a*b^2*c)^(1/3))^(
1/2)*EllipticF(1/6*3^(1/2)*2^(1/2)*(-I*(I*3^(1/2)*x*(-a*b^2*c)^(1/3)-2*b*(c*x^3)^(1/2)-x*(-a*b^2*c)^(1/3))*3^(
1/2)/x/(-a*b^2*c)^(1/3))^(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)*a*b^2*c*x^2+3*a^2*b*c*x^2)/c/b^2/(c*x^3)^(1/2)/(a+b*(c*x^3)^(1/2))^(1/2)

Fricas [F]

\[ \int x \sqrt {a+b \sqrt {c x^3}} \, dx=\int { \sqrt {\sqrt {c x^{3}} b + a} x \,d x } \]

[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)

Sympy [F]

\[ \int x \sqrt {a+b \sqrt {c x^3}} \, dx=\int x \sqrt {a + b \sqrt {c x^{3}}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int x \sqrt {a+b \sqrt {c x^3}} \, dx=\int { \sqrt {\sqrt {c x^{3}} b + a} x \,d x } \]

[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)

Giac [F]

\[ \int x \sqrt {a+b \sqrt {c x^3}} \, dx=\int { \sqrt {\sqrt {c x^{3}} b + a} x \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int x \sqrt {a+b \sqrt {c x^3}} \, dx=\int x\,\sqrt {a+b\,\sqrt {c\,x^3}} \,d x \]

[In]

int(x*(a + b*(c*x^3)^(1/2))^(1/2),x)

[Out]

int(x*(a + b*(c*x^3)^(1/2))^(1/2), x)