\(\int \frac {\sqrt {a+b (c x^2)^{3/2}}}{x^2} \, dx\) [2953]

Optimal result

Integrand size = 21, antiderivative size = 661 \[ \int \frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{x^2} \, dx=-\frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{x}+\frac {3 \sqrt [3]{b} \sqrt {c x^2} \sqrt {a+b \left (c x^2\right )^{3/2}}}{x \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )}-\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} \sqrt [3]{b} \sqrt {c x^2} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right ) \sqrt {\frac {a^{2/3}+b^{2/3} c x^2-\sqrt [3]{a} \sqrt [3]{b} \sqrt {c x^2}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}}\right )|-7-4 \sqrt {3}\right )}{2 x \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )^2}} \sqrt {a+b \left (c x^2\right )^{3/2}}}+\frac {\sqrt {2} 3^{3/4} \sqrt [3]{a} \sqrt [3]{b} \sqrt {c x^2} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right ) \sqrt {\frac {a^{2/3}+b^{2/3} c x^2-\sqrt [3]{a} \sqrt [3]{b} \sqrt {c x^2}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}}\right ),-7-4 \sqrt {3}\right )}{x \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )^2}} \sqrt {a+b \left (c x^2\right )^{3/2}}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 661, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {375, 283, 309, 224, 1891} \[ \int \frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{x^2} \, dx=\frac {\sqrt {2} 3^{3/4} \sqrt [3]{a} \sqrt [3]{b} \sqrt {c x^2} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt {c x^2}+b^{2/3} c x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} \sqrt {c x^2}+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} \sqrt {c x^2}+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right )}{x \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )^2}} \sqrt {a+b \left (c x^2\right )^{3/2}}}-\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} \sqrt [3]{b} \sqrt {c x^2} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt {c x^2}+b^{2/3} c x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{b} \sqrt {c x^2}+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} \sqrt {c x^2}+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{2 x \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )^2}} \sqrt {a+b \left (c x^2\right )^{3/2}}}-\frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{x}+\frac {3 \sqrt [3]{b} \sqrt {c x^2} \sqrt {a+b \left (c x^2\right )^{3/2}}}{x \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )} \]

[In]

Int[Sqrt[a + b*(c*x^2)^(3/2)]/x^2,x]

[Out]

-(Sqrt[a + b*(c*x^2)^(3/2)]/x) + (3*b^(1/3)*Sqrt[c*x^2]*Sqrt[a + b*(c*x^2)^(3/2)])/(x*((1 + Sqrt[3])*a^(1/3) +
b^(1/3)*Sqrt[c*x^2])) - (3*3^(1/4)*Sqrt[2 - Sqrt[3]]*a^(1/3)*b^(1/3)*Sqrt[c*x^2]*(a^(1/3) + b^(1/3)*Sqrt[c*x^
2])*Sqrt[(a^(2/3) + b^(2/3)*c*x^2 - a^(1/3)*b^(1/3)*Sqrt[c*x^2])/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*Sqrt[c*x^2])
^2]*EllipticE[ArcSin[((1 - Sqrt[3])*a^(1/3) + b^(1/3)*Sqrt[c*x^2])/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*Sqrt[c*x^2
])], -7 - 4*Sqrt[3]])/(2*x*Sqrt[(a^(1/3)*(a^(1/3) + b^(1/3)*Sqrt[c*x^2]))/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*Sqr
t[c*x^2])^2]*Sqrt[a + b*(c*x^2)^(3/2)]) + (Sqrt[2]*3^(3/4)*a^(1/3)*b^(1/3)*Sqrt[c*x^2]*(a^(1/3) + b^(1/3)*Sqrt
[c*x^2])*Sqrt[(a^(2/3) + b^(2/3)*c*x^2 - a^(1/3)*b^(1/3)*Sqrt[c*x^2])/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*Sqrt[c*
x^2])^2]*EllipticF[ArcSin[((1 - Sqrt[3])*a^(1/3) + b^(1/3)*Sqrt[c*x^2])/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*Sqrt[
c*x^2])], -7 - 4*Sqrt[3]])/(x*Sqrt[(a^(1/3)*(a^(1/3) + b^(1/3)*Sqrt[c*x^2]))/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*
Sqrt[c*x^2])^2]*Sqrt[a + b*(c*x^2)^(3/2)])

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 309

Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Dist[(-(
1 - Sqrt[3]))*(s/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 375

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*((c_.)*(x_)^(q_))^(n_))^(p_.), x_Symbol] :> Dist[(d*x)^(m + 1)/(d*((c*x^q
)^(1/q))^(m + 1)), Subst[Int[x^m*(a + b*x^(n*q))^p, x], x, (c*x^q)^(1/q)], x] /; FreeQ[{a, b, c, d, m, n, p, q
}, x] && IntegerQ[n*q] && NeQ[x, (c*x^q)^(1/q)]

Rule 1891

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]/(r^2
*Sqrt[a + b*x^3]*Sqrt[s*((s + r*x)/((1 + Sqrt[3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 - Sqrt[3])*s + r*x)/((1
+ Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]], 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*} \text {integral}& = \frac {\sqrt {c x^2} \text {Subst}\left (\int \frac {\sqrt {a+b x^3}}{x^2} \, dx,x,\sqrt {c x^2}\right )}{x} \\ & = -\frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{x}+\frac {\left (3 b \sqrt {c x^2}\right ) \text {Subst}\left (\int \frac {x}{\sqrt {a+b x^3}} \, dx,x,\sqrt {c x^2}\right )}{2 x} \\ & = -\frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{x}+\frac {\left (3 b^{2/3} \sqrt {c x^2}\right ) \text {Subst}\left (\int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt {a+b x^3}} \, dx,x,\sqrt {c x^2}\right )}{2 x}-\frac {\left (3 \left (1-\sqrt {3}\right ) \sqrt [3]{a} b^{2/3} \sqrt {c x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^3}} \, dx,x,\sqrt {c x^2}\right )}{2 x} \\ & = -\frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{x}+\frac {3 \sqrt [3]{b} \sqrt {c x^2} \sqrt {a+b \left (c x^2\right )^{3/2}}}{x \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )}-\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} \sqrt [3]{b} \sqrt {c x^2} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right ) \sqrt {\frac {a^{2/3}+b^{2/3} c x^2-\sqrt [3]{a} \sqrt [3]{b} \sqrt {c x^2}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}}\right )|-7-4 \sqrt {3}\right )}{2 x \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )^2}} \sqrt {a+b \left (c x^2\right )^{3/2}}}+\frac {\sqrt {2} 3^{3/4} \sqrt [3]{a} \sqrt [3]{b} \sqrt {c x^2} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right ) \sqrt {\frac {a^{2/3}+b^{2/3} c x^2-\sqrt [3]{a} \sqrt [3]{b} \sqrt {c x^2}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}}\right )|-7-4 \sqrt {3}\right )}{x \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )^2}} \sqrt {a+b \left (c x^2\right )^{3/2}}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 1.99 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.10 \[ \int \frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{x^2} \, dx=-\frac {\sqrt {a+b \left (c x^2\right )^{3/2}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {1}{3},\frac {2}{3},-\frac {b \left (c x^2\right )^{3/2}}{a}\right )}{x \sqrt {1+\frac {b \left (c x^2\right )^{3/2}}{a}}} \]

[In]

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

[Out]

-((Sqrt[a + b*(c*x^2)^(3/2)]*Hypergeometric2F1[-1/2, -1/3, 2/3, -((b*(c*x^2)^(3/2))/a)])/(x*Sqrt[1 + (b*(c*x^2
)^(3/2))/a]))

Maple [F]

\[\int \frac {\sqrt {a +b \left (c \,x^{2}\right )^{\frac {3}{2}}}}{x^{2}}d x\]

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 1.77 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.13 \[ \int \frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{x^2} \, dx=-\frac {3 \, \sqrt {\frac {\sqrt {c x^{2}} b c}{x}} x {\rm weierstrassZeta}\left (0, -\frac {4 \, \sqrt {c x^{2}} a}{b c^{2} x}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, \sqrt {c x^{2}} a}{b c^{2} x}, x\right )\right ) + \sqrt {\sqrt {c x^{2}} b c x^{2} + a}}{x} \]

[In]

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

[Out]

-(3*sqrt(sqrt(c*x^2)*b*c/x)*x*weierstrassZeta(0, -4*sqrt(c*x^2)*a/(b*c^2*x), weierstrassPInverse(0, -4*sqrt(c*
x^2)*a/(b*c^2*x), x)) + sqrt(sqrt(c*x^2)*b*c*x^2 + a))/x

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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