Integrand size = 21, antiderivative size = 810 \[ \int \frac {\sqrt {a+b \sqrt {c x^3}}}{x^3} \, dx=-\frac {\sqrt {a+b \sqrt {c x^3}}}{2 x^2}-\frac {3 b c x \sqrt {a+b \sqrt {c x^3}}}{4 a \sqrt {c x^3}}+\frac {3 b^{4/3} c^{2/3} \sqrt {a+b \sqrt {c x^3}}}{4 a \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )}-\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b^{4/3} c^{2/3} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right ) \sqrt {\frac {a^{2/3}+b^{2/3} \sqrt [3]{c} x-\frac {\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}}\right )|-7-4 \sqrt {3}\right )}{8 a^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )^2}} \sqrt {a+b \sqrt {c x^3}}}+\frac {3^{3/4} b^{4/3} c^{2/3} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right ) \sqrt {\frac {a^{2/3}+b^{2/3} \sqrt [3]{c} x-\frac {\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}}\right ),-7-4 \sqrt {3}\right )}{2 \sqrt {2} a^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )^2}} \sqrt {a+b \sqrt {c x^3}}} \]
1/4*3^(3/4)*b^(4/3)*c^(2/3)*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))*((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^(2/3)*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)-3/8*3^(1/4)*b ^(4/3)*c^(2/3)*EllipticE((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^(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)-1/2*(a+b *(c*x^3)^(1/2))^(1/2)/x^2+3/4*b^(4/3)*c^(2/3)*(a+b*(c*x^3)^(1/2))^(1/2)/a/ (a^(1/3)*(1+3^(1/2))+b^(1/3)*c^(2/3)*x^2/(c*x^3)^(1/2))-3/4*b*c*x*(a+b*(c* x^3)^(1/2))^(1/2)/a/(c*x^3)^(1/2)
\[ \int \frac {\sqrt {a+b \sqrt {c x^3}}}{x^3} \, dx=\int \frac {\sqrt {a+b \sqrt {c x^3}}}{x^3} \, dx \]
Time = 0.67 (sec) , antiderivative size = 886, normalized size of antiderivative = 1.09, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {893, 864, 809, 847, 832, 759, 2416}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\sqrt {a+b \sqrt {c x^3}}}{x^3} \, dx\) |
| \(\Big \downarrow \) 893 |
| \(\displaystyle \int \frac {\sqrt {a+b \sqrt {c} x^{3/2}}}{x^3}dx\) |
| \(\Big \downarrow \) 864 |
| \(\displaystyle 2 \int \frac {c^{5/2} x^5 \sqrt {\frac {b \left (c x^3\right )^{3/2}}{c x^3}+a}}{\left (c x^3\right )^{5/2}}d\frac {\sqrt {c x^3}}{\sqrt {c} x}\) |
| \(\Big \downarrow \) 809 |
| \(\displaystyle 2 \left (\frac {3}{8} b \sqrt {c} \int \frac {1}{x \sqrt {\frac {b \left (c x^3\right )^{3/2}}{c x^3}+a}}d\frac {\sqrt {c x^3}}{\sqrt {c} x}-\frac {\sqrt {a+\frac {b \left (c x^3\right )^{3/2}}{c x^3}}}{4 x^2}\right )\) |
| \(\Big \downarrow \) 847 |
| \(\displaystyle 2 \left (\frac {3}{8} b \sqrt {c} \left (\frac {b \sqrt {c} \int \frac {\sqrt {c x^3}}{\sqrt {c} x \sqrt {\frac {b \left (c x^3\right )^{3/2}}{c x^3}+a}}d\frac {\sqrt {c x^3}}{\sqrt {c} x}}{2 a}-\frac {\sqrt {c} x \sqrt {a+\frac {b \left (c x^3\right )^{3/2}}{c x^3}}}{a \sqrt {c x^3}}\right )-\frac {\sqrt {a+\frac {b \left (c x^3\right )^{3/2}}{c x^3}}}{4 x^2}\right )\) |
| \(\Big \downarrow \) 832 |
| \(\displaystyle 2 \left (\frac {3}{8} b \sqrt {c} \left (\frac {b \sqrt {c} \left (\frac {\int \frac {\frac {\sqrt [3]{b} \sqrt {c x^3}}{\sqrt [3]{c} x}+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {\frac {b \left (c x^3\right )^{3/2}}{c x^3}+a}}d\frac {\sqrt {c x^3}}{\sqrt {c} x}}{\sqrt [3]{b} \sqrt [6]{c}}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} \int \frac {1}{\sqrt {\frac {b \left (c x^3\right )^{3/2}}{c x^3}+a}}d\frac {\sqrt {c x^3}}{\sqrt {c} x}}{\sqrt [3]{b} \sqrt [6]{c}}\right )}{2 a}-\frac {\sqrt {c} x \sqrt {a+\frac {b \left (c x^3\right )^{3/2}}{c x^3}}}{a \sqrt {c x^3}}\right )-\frac {\sqrt {a+\frac {b \left (c x^3\right )^{3/2}}{c x^3}}}{4 x^2}\right )\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle 2 \left (\frac {3}{8} b \sqrt {c} \left (\frac {b \sqrt {c} \left (\frac {\int \frac {\frac {\sqrt [3]{b} \sqrt {c x^3}}{\sqrt [3]{c} x}+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {\frac {b \left (c x^3\right )^{3/2}}{c x^3}+a}}d\frac {\sqrt {c x^3}}{\sqrt {c} x}}{\sqrt [3]{b} \sqrt [6]{c}}-\frac {2 \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b} \sqrt {c x^3}}{\sqrt [3]{c} x}\right ) \sqrt {\frac {a^{2/3}-\frac {\sqrt [3]{a} \sqrt [3]{b} \sqrt {c x^3}}{\sqrt [3]{c} x}+b^{2/3} \sqrt [3]{c} x}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} \sqrt {c x^3}}{\sqrt [3]{c} x}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\frac {\sqrt [3]{b} \sqrt {c x^3}}{\sqrt [3]{c} x}+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\frac {\sqrt [3]{b} \sqrt {c x^3}}{\sqrt [3]{c} x}+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt [3]{c} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b} \sqrt {c x^3}}{\sqrt [3]{c} x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} \sqrt {c x^3}}{\sqrt [3]{c} x}\right )^2}} \sqrt {a+\frac {b \left (c x^3\right )^{3/2}}{c x^3}}}\right )}{2 a}-\frac {\sqrt {c} x \sqrt {a+\frac {b \left (c x^3\right )^{3/2}}{c x^3}}}{a \sqrt {c x^3}}\right )-\frac {\sqrt {a+\frac {b \left (c x^3\right )^{3/2}}{c x^3}}}{4 x^2}\right )\) |
| \(\Big \downarrow \) 2416 |
| \(\displaystyle 2 \left (\frac {3}{8} b \sqrt {c} \left (\frac {b \sqrt {c} \left (\frac {\frac {2 \sqrt {\frac {b \left (c x^3\right )^{3/2}}{c x^3}+a}}{\sqrt [3]{b} \sqrt [6]{c} \left (\frac {\sqrt [3]{b} \sqrt {c x^3}}{\sqrt [3]{c} x}+\left (1+\sqrt {3}\right ) \sqrt [3]{a}\right )}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} \left (\frac {\sqrt [3]{b} \sqrt {c x^3}}{\sqrt [3]{c} x}+\sqrt [3]{a}\right ) \sqrt {\frac {b^{2/3} \sqrt [3]{c} x+a^{2/3}-\frac {\sqrt [3]{a} \sqrt [3]{b} \sqrt {c x^3}}{\sqrt [3]{c} x}}{\left (\frac {\sqrt [3]{b} \sqrt {c x^3}}{\sqrt [3]{c} x}+\left (1+\sqrt {3}\right ) \sqrt [3]{a}\right )^2}} E\left (\arcsin \left (\frac {\frac {\sqrt [3]{b} \sqrt {c x^3}}{\sqrt [3]{c} x}+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\frac {\sqrt [3]{b} \sqrt {c x^3}}{\sqrt [3]{c} x}+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{\sqrt [3]{b} \sqrt [6]{c} \sqrt {\frac {\sqrt [3]{a} \left (\frac {\sqrt [3]{b} \sqrt {c x^3}}{\sqrt [3]{c} x}+\sqrt [3]{a}\right )}{\left (\frac {\sqrt [3]{b} \sqrt {c x^3}}{\sqrt [3]{c} x}+\left (1+\sqrt {3}\right ) \sqrt [3]{a}\right )^2}} \sqrt {\frac {b \left (c x^3\right )^{3/2}}{c x^3}+a}}}{\sqrt [3]{b} \sqrt [6]{c}}-\frac {2 \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\frac {\sqrt [3]{b} \sqrt {c x^3}}{\sqrt [3]{c} x}+\sqrt [3]{a}\right ) \sqrt {\frac {b^{2/3} \sqrt [3]{c} x+a^{2/3}-\frac {\sqrt [3]{a} \sqrt [3]{b} \sqrt {c x^3}}{\sqrt [3]{c} x}}{\left (\frac {\sqrt [3]{b} \sqrt {c x^3}}{\sqrt [3]{c} x}+\left (1+\sqrt {3}\right ) \sqrt [3]{a}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\frac {\sqrt [3]{b} \sqrt {c x^3}}{\sqrt [3]{c} x}+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\frac {\sqrt [3]{b} \sqrt {c x^3}}{\sqrt [3]{c} x}+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt [3]{c} \sqrt {\frac {\sqrt [3]{a} \left (\frac {\sqrt [3]{b} \sqrt {c x^3}}{\sqrt [3]{c} x}+\sqrt [3]{a}\right )}{\left (\frac {\sqrt [3]{b} \sqrt {c x^3}}{\sqrt [3]{c} x}+\left (1+\sqrt {3}\right ) \sqrt [3]{a}\right )^2}} \sqrt {\frac {b \left (c x^3\right )^{3/2}}{c x^3}+a}}\right )}{2 a}-\frac {\sqrt {c} x \sqrt {\frac {b \left (c x^3\right )^{3/2}}{c x^3}+a}}{a \sqrt {c x^3}}\right )-\frac {\sqrt {\frac {b \left (c x^3\right )^{3/2}}{c x^3}+a}}{4 x^2}\right )\) |
2*(-1/4*Sqrt[a + (b*(c*x^3)^(3/2))/(c*x^3)]/x^2 + (3*b*Sqrt[c]*(-((Sqrt[c] *x*Sqrt[a + (b*(c*x^3)^(3/2))/(c*x^3)])/(a*Sqrt[c*x^3])) + (b*Sqrt[c]*(((2 *Sqrt[a + (b*(c*x^3)^(3/2))/(c*x^3)])/(b^(1/3)*c^(1/6)*((1 + Sqrt[3])*a^(1 /3) + (b^(1/3)*Sqrt[c*x^3])/(c^(1/3)*x))) - (3^(1/4)*Sqrt[2 - Sqrt[3]]*a^( 1/3)*(a^(1/3) + (b^(1/3)*Sqrt[c*x^3])/(c^(1/3)*x))*Sqrt[(a^(2/3) + b^(2/3) *c^(1/3)*x - (a^(1/3)*b^(1/3)*Sqrt[c*x^3])/(c^(1/3)*x))/((1 + Sqrt[3])*a^( 1/3) + (b^(1/3)*Sqrt[c*x^3])/(c^(1/3)*x))^2]*EllipticE[ArcSin[((1 - Sqrt[3 ])*a^(1/3) + (b^(1/3)*Sqrt[c*x^3])/(c^(1/3)*x))/((1 + Sqrt[3])*a^(1/3) + ( b^(1/3)*Sqrt[c*x^3])/(c^(1/3)*x))], -7 - 4*Sqrt[3]])/(b^(1/3)*c^(1/6)*Sqrt [(a^(1/3)*(a^(1/3) + (b^(1/3)*Sqrt[c*x^3])/(c^(1/3)*x)))/((1 + Sqrt[3])*a^ (1/3) + (b^(1/3)*Sqrt[c*x^3])/(c^(1/3)*x))^2]*Sqrt[a + (b*(c*x^3)^(3/2))/( c*x^3)]))/(b^(1/3)*c^(1/6)) - (2*(1 - Sqrt[3])*Sqrt[2 + Sqrt[3]]*a^(1/3)*( a^(1/3) + (b^(1/3)*Sqrt[c*x^3])/(c^(1/3)*x))*Sqrt[(a^(2/3) + b^(2/3)*c^(1/ 3)*x - (a^(1/3)*b^(1/3)*Sqrt[c*x^3])/(c^(1/3)*x))/((1 + Sqrt[3])*a^(1/3) + (b^(1/3)*Sqrt[c*x^3])/(c^(1/3)*x))^2]*EllipticF[ArcSin[((1 - Sqrt[3])*a^( 1/3) + (b^(1/3)*Sqrt[c*x^3])/(c^(1/3)*x))/((1 + Sqrt[3])*a^(1/3) + (b^(1/3 )*Sqrt[c*x^3])/(c^(1/3)*x))], -7 - 4*Sqrt[3]])/(3^(1/4)*b^(2/3)*c^(1/3)*Sq rt[(a^(1/3)*(a^(1/3) + (b^(1/3)*Sqrt[c*x^3])/(c^(1/3)*x)))/((1 + Sqrt[3])* a^(1/3) + (b^(1/3)*Sqrt[c*x^3])/(c^(1/3)*x))^2]*Sqrt[a + (b*(c*x^3)^(3/2)) /(c*x^3)])))/(2*a)))/8)
3.30.69.3.1 Defintions of rubi rules used
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]*Sqrt[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]
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] - Simp[b*n*(p/(c^n*(m + 1))) I nt[(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] && IntB inomialQ[a, b, c, n, m, p, x]
Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3] ], s = Denom[Rt[b/a, 3]]}, Simp[(-(1 - Sqrt[3]))*(s/r) Int[1/Sqrt[a + b*x ^3], x], x] + Simp[1/r Int[((1 - Sqrt[3])*s + r*x)/Sqrt[a + b*x^3], x], x ]] /; FreeQ[{a, b}, x] && PosQ[a]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x )^(m + 1)*((a + b*x^n)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))) Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a , b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p , x]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denomi nator[n]}, Simp[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]
Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*((c_.)*(x_)^(q_))^(n_))^(p_.), x_Symbo l] :> With[{k = Denominator[n]}, Subst[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]
Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N umer[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] - S imp[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] && Eq Q[b*c^3 - 2*(5 - 3*Sqrt[3])*a*d^3, 0]
Time = 5.05 (sec) , antiderivative size = 869, normalized size of antiderivative = 1.07
| method | result | size |
| default | \(\frac {3 i \sqrt {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}}}}\, E\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 ) \left (-a \,b^{2} c \right )^{\frac {2}{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}}}}\, x^{2}-2 i \sqrt {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 ) \left (-a \,b^{2} c \right )^{\frac {2}{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}}}}\, x^{2}+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}}}}\, E\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 ) \left (-a \,b^{2} c \right )^{\frac {2}{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}}}}\, x^{2}-12 b^{2} c \,x^{3}-20 \sqrt {c \,x^{3}}\, a b -8 a^{2}}{16 x^{2} a \sqrt {a +b \sqrt {c \,x^{3}}}}\) | \(869\) |
1/16*(3*I*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)*EllipticE(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))*(-a*b^2*c)^(2/3)*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)*x^2-2*I* 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))*( -a*b^2*c)^(2/3)*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)*x^2+3*((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)*EllipticE(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))*(-a*b^2*c)^(2/3)*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...
\[ \int \frac {\sqrt {a+b \sqrt {c x^3}}}{x^3} \, dx=\int { \frac {\sqrt {\sqrt {c x^{3}} b + a}}{x^{3}} \,d x } \]
\[ \int \frac {\sqrt {a+b \sqrt {c x^3}}}{x^3} \, dx=\int \frac {\sqrt {a + b \sqrt {c x^{3}}}}{x^{3}}\, dx \]
\[ \int \frac {\sqrt {a+b \sqrt {c x^3}}}{x^3} \, dx=\int { \frac {\sqrt {\sqrt {c x^{3}} b + a}}{x^{3}} \,d x } \]
\[ \int \frac {\sqrt {a+b \sqrt {c x^3}}}{x^3} \, dx=\int { \frac {\sqrt {\sqrt {c x^{3}} b + a}}{x^{3}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {a+b \sqrt {c x^3}}}{x^3} \, dx=\int \frac {\sqrt {a+b\,\sqrt {c\,x^3}}}{x^3} \,d x \]