\(\int \frac {\sqrt {a+b \sqrt {c x^3}}}{x^2} \, dx\) [2965]

Optimal result

Integrand size = 21, antiderivative size = 355 \[ \int \frac {\sqrt {a+b \sqrt {c x^3}}}{x^2} \, dx=-\frac {\sqrt {a+b \sqrt {c x^3}}}{x}+\frac {3^{3/4} \sqrt {2+\sqrt {3}} b^{2/3} \sqrt [3]{c} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right ) \sqrt {\frac {a^{2/3}+b^{2/3} \sqrt [3]{c} x-\frac {\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}}\right ),-7-4 \sqrt {3}\right )}{\sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )^2}} \sqrt {a+b \sqrt {c x^3}}} \]

[Out]

3^(3/4)*b^(2/3)*c^(1/3)*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)/(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)-(a+b*(c*x^3)^(1/2))^(1/2)/x

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {376, 348, 283, 224} \[ \int \frac {\sqrt {a+b \sqrt {c x^3}}}{x^2} \, dx=\frac {3^{3/4} \sqrt {2+\sqrt {3}} b^{2/3} \sqrt [3]{c} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right ) \sqrt {\frac {a^{2/3}-\frac {\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}+b^{2/3} \sqrt [3]{c} x}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right )}{\sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )^2}} \sqrt {a+b \sqrt {c x^3}}}-\frac {\sqrt {a+b \sqrt {c x^3}}}{x} \]

[In]

Int[Sqrt[a + b*Sqrt[c*x^3]]/x^2,x]

[Out]

-(Sqrt[a + b*Sqrt[c*x^3]]/x) + (3^(3/4)*Sqrt[2 + Sqrt[3]]*b^(2/3)*c^(1/3)*(a^(1/3) + (b^(1/3)*c^(2/3)*x^2)/Sqr
t[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]])/(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 283

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^p/(c*(m + 1
))), x] - Dist[b*n*(p/(c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 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 \frac {\sqrt {a+b \sqrt {c} x^{3/2}}}{x^2} \, dx,\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right ) \\ & = \text {Subst}\left (2 \text {Subst}\left (\int \frac {\sqrt {a+b \sqrt {c} x^3}}{x^3} \, dx,x,\sqrt {x}\right ),\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right ) \\ & = -\frac {\sqrt {a+b \sqrt {c x^3}}}{x}+\text {Subst}\left (\frac {1}{2} \left (3 b \sqrt {c}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b \sqrt {c} x^3}} \, dx,x,\sqrt {x}\right ),\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right ) \\ & = -\frac {\sqrt {a+b \sqrt {c x^3}}}{x}+\frac {3^{3/4} \sqrt {2+\sqrt {3}} b^{2/3} \sqrt [3]{c} \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 )}{\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]

\[ \int \frac {\sqrt {a+b \sqrt {c x^3}}}{x^2} \, dx=\int \frac {\sqrt {a+b \sqrt {c x^3}}}{x^2} \, dx \]

[In]

Integrate[Sqrt[a + b*Sqrt[c*x^3]]/x^2,x]

[Out]

Integrate[Sqrt[a + b*Sqrt[c*x^3]]/x^2, x]

Maple [A] (verified)

Time = 5.06 (sec) , antiderivative size = 306, normalized size of antiderivative = 0.86

method	result	size
default	\(-\frac {i \sqrt {3}\, \left (-a \,b^{2} c \right )^{\frac {1}{3}} \sqrt {2}\, \sqrt {-\frac {i \left (i \sqrt {3}\, x \left (-a \,b^{2} c \right )^{\frac {1}{3}}-2 b \sqrt {c \,x^{3}}-x \left (-a \,b^{2} c \right )^{\frac {1}{3}}\right ) \sqrt {3}}{x \left (-a \,b^{2} c \right )^{\frac {1}{3}}}}\, \sqrt {\frac {b \sqrt {c \,x^{3}}-x \left (-a \,b^{2} c \right )^{\frac {1}{3}}}{x \left (-a \,b^{2} c \right )^{\frac {1}{3}} \left (i \sqrt {3}-3\right )}}\, \sqrt {-\frac {i \left (i \sqrt {3}\, x \left (-a \,b^{2} c \right )^{\frac {1}{3}}+2 b \sqrt {c \,x^{3}}+x \left (-a \,b^{2} c \right )^{\frac {1}{3}}\right ) \sqrt {3}}{x \left (-a \,b^{2} c \right )^{\frac {1}{3}}}}\, F\left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {-\frac {i \left (i \sqrt {3}\, x \left (-a \,b^{2} c \right )^{\frac {1}{3}}-2 b \sqrt {c \,x^{3}}-x \left (-a \,b^{2} c \right )^{\frac {1}{3}}\right ) \sqrt {3}}{x \left (-a \,b^{2} c \right )^{\frac {1}{3}}}}}{6}, \sqrt {2}\, \sqrt {\frac {i \sqrt {3}}{i \sqrt {3}-3}}\right ) x +2 a +2 b \sqrt {c \,x^{3}}}{2 x \sqrt {a +b \sqrt {c \,x^{3}}}}\)	\(306\)

[In]

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

[Out]

-1/2*(I*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))*x+2*a+2*b*(c*x^3)^(1/2))/x/(a+b*(c*x^3)^(1
/2))^(1/2)

Fricas [F]

\[ \int \frac {\sqrt {a+b \sqrt {c x^3}}}{x^2} \, dx=\int { \frac {\sqrt {\sqrt {c x^{3}} b + a}}{x^{2}} \,d x } \]

[In]

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

[Out]

integral(sqrt(sqrt(c*x^3)*b + a)/x^2, x)

Sympy [F]

\[ \int \frac {\sqrt {a+b \sqrt {c x^3}}}{x^2} \, dx=\int \frac {\sqrt {a + b \sqrt {c x^{3}}}}{x^{2}}\, dx \]

[In]

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

[Out]

Integral(sqrt(a + b*sqrt(c*x**3))/x**2, x)

Maxima [F]

\[ \int \frac {\sqrt {a+b \sqrt {c x^3}}}{x^2} \, dx=\int { \frac {\sqrt {\sqrt {c x^{3}} b + a}}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {\sqrt {a+b \sqrt {c x^3}}}{x^2} \, dx=\int { \frac {\sqrt {\sqrt {c x^{3}} b + a}}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b \sqrt {c x^3}}}{x^2} \, dx=\int \frac {\sqrt {a+b\,\sqrt {c\,x^3}}}{x^2} \,d x \]

[In]

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

[Out]

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