Integrand size = 27, antiderivative size = 626 \[ \int \frac {\left (c+d x^3\right )^{3/2}}{x^2 \left (8 c-d x^3\right )} \, dx=-\frac {\sqrt {c+d x^3}}{8 x}-\frac {15 \sqrt [3]{d} \sqrt {c+d x^3}}{8 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}-\frac {9}{16} \sqrt {3} \sqrt [6]{c} \sqrt [3]{d} \arctan \left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )+\frac {9}{16} \sqrt [6]{c} \sqrt [3]{d} \text {arctanh}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x^3}}\right )-\frac {9}{16} \sqrt [6]{c} \sqrt [3]{d} \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )+\frac {15 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{c} \sqrt [3]{d} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt {3}\right )}{16 \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {5\ 3^{3/4} \sqrt [3]{c} \sqrt [3]{d} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right ),-7-4 \sqrt {3}\right )}{4 \sqrt {2} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}} \] Output:
-1/8*(d*x^3+c)^(1/2)/x-15*d^(1/3)*(d*x^3+c)^(1/2)/(8*(1+3^(1/2))*c^(1/3)+8 *d^(1/3)*x)-9/16*3^(1/2)*c^(1/6)*d^(1/3)*arctan(3^(1/2)*c^(1/6)*(c^(1/3)+d ^(1/3)*x)/(d*x^3+c)^(1/2))+9/16*c^(1/6)*d^(1/3)*arctanh(1/3*(c^(1/3)+d^(1/ 3)*x)^2/c^(1/6)/(d*x^3+c)^(1/2))-9/16*c^(1/6)*d^(1/3)*arctanh(1/3*(d*x^3+c )^(1/2)/c^(1/2))+15/16*3^(1/4)*(1/2*6^(1/2)-1/2*2^(1/2))*c^(1/3)*d^(1/3)*( c^(1/3)+d^(1/3)*x)*((c^(2/3)-c^(1/3)*d^(1/3)*x+d^(2/3)*x^2)/((1+3^(1/2))*c ^(1/3)+d^(1/3)*x)^2)^(1/2)*EllipticE(((1-3^(1/2))*c^(1/3)+d^(1/3)*x)/((1+3 ^(1/2))*c^(1/3)+d^(1/3)*x),I*3^(1/2)+2*I)/(c^(1/3)*(c^(1/3)+d^(1/3)*x)/((1 +3^(1/2))*c^(1/3)+d^(1/3)*x)^2)^(1/2)/(d*x^3+c)^(1/2)-5/8*3^(3/4)*c^(1/3)* d^(1/3)*(c^(1/3)+d^(1/3)*x)*((c^(2/3)-c^(1/3)*d^(1/3)*x+d^(2/3)*x^2)/((1+3 ^(1/2))*c^(1/3)+d^(1/3)*x)^2)^(1/2)*EllipticF(((1-3^(1/2))*c^(1/3)+d^(1/3) *x)/((1+3^(1/2))*c^(1/3)+d^(1/3)*x),I*3^(1/2)+2*I)*2^(1/2)/(c^(1/3)*(c^(1/ 3)+d^(1/3)*x)/((1+3^(1/2))*c^(1/3)+d^(1/3)*x)^2)^(1/2)/(d*x^3+c)^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 10.09 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.22 \[ \int \frac {\left (c+d x^3\right )^{3/2}}{x^2 \left (8 c-d x^3\right )} \, dx=\frac {-16 c \left (c+d x^3\right )+21 c d x^3 \sqrt {1+\frac {d x^3}{c}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},1,\frac {5}{3},-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )+3 d^2 x^6 \sqrt {1+\frac {d x^3}{c}} \operatorname {AppellF1}\left (\frac {5}{3},\frac {1}{2},1,\frac {8}{3},-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )}{128 c x \sqrt {c+d x^3}} \] Input:
Integrate[(c + d*x^3)^(3/2)/(x^2*(8*c - d*x^3)),x]
Output:
(-16*c*(c + d*x^3) + 21*c*d*x^3*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), (d*x^3)/(8*c)] + 3*d^2*x^6*Sqrt[1 + (d*x^3)/c]*AppellF1 [5/3, 1/2, 1, 8/3, -((d*x^3)/c), (d*x^3)/(8*c)])/(128*c*x*Sqrt[c + d*x^3])
Time = 1.62 (sec) , antiderivative size = 620, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {974, 27, 1054, 2009}
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.
| \(\displaystyle \int \frac {\left (c+d x^3\right )^{3/2}}{x^2 \left (8 c-d x^3\right )} \, dx\) |
| \(\Big \downarrow \) 974 |
| \(\displaystyle \frac {\int \frac {3 c d x \left (5 d x^3+14 c\right )}{2 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{8 c}-\frac {\sqrt {c+d x^3}}{8 x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{16} d \int \frac {x \left (5 d x^3+14 c\right )}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx-\frac {\sqrt {c+d x^3}}{8 x}\) |
| \(\Big \downarrow \) 1054 |
| \(\displaystyle \frac {3}{16} d \int \left (\frac {54 c x}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}-\frac {5 x}{\sqrt {d x^3+c}}\right )dx-\frac {\sqrt {c+d x^3}}{8 x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3}{16} d \left (-\frac {10 \sqrt {2} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} d^{2/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}+\frac {5 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt {3}\right )}{d^{2/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {3 \sqrt {3} \sqrt [6]{c} \arctan \left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{d^{2/3}}+\frac {3 \sqrt [6]{c} \text {arctanh}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x^3}}\right )}{d^{2/3}}-\frac {3 \sqrt [6]{c} \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{d^{2/3}}-\frac {10 \sqrt {c+d x^3}}{d^{2/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}\right )-\frac {\sqrt {c+d x^3}}{8 x}\) |
Input:
Int[(c + d*x^3)^(3/2)/(x^2*(8*c - d*x^3)),x]
Output:
-1/8*Sqrt[c + d*x^3]/x + (3*d*((-10*Sqrt[c + d*x^3])/(d^(2/3)*((1 + Sqrt[3 ])*c^(1/3) + d^(1/3)*x)) - (3*Sqrt[3]*c^(1/6)*ArcTan[(Sqrt[3]*c^(1/6)*(c^( 1/3) + d^(1/3)*x))/Sqrt[c + d*x^3]])/d^(2/3) + (3*c^(1/6)*ArcTanh[(c^(1/3) + d^(1/3)*x)^2/(3*c^(1/6)*Sqrt[c + d*x^3])])/d^(2/3) - (3*c^(1/6)*ArcTanh [Sqrt[c + d*x^3]/(3*Sqrt[c])])/d^(2/3) + (5*3^(1/4)*Sqrt[2 - Sqrt[3]]*c^(1 /3)*(c^(1/3) + d^(1/3)*x)*Sqrt[(c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2) /((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)^2]*EllipticE[ArcSin[((1 - Sqrt[3])*c^ (1/3) + d^(1/3)*x)/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)], -7 - 4*Sqrt[3]])/ (d^(2/3)*Sqrt[(c^(1/3)*(c^(1/3) + d^(1/3)*x))/((1 + Sqrt[3])*c^(1/3) + d^( 1/3)*x)^2]*Sqrt[c + d*x^3]) - (10*Sqrt[2]*c^(1/3)*(c^(1/3) + d^(1/3)*x)*Sq rt[(c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2)/((1 + Sqrt[3])*c^(1/3) + d^ (1/3)*x)^2]*EllipticF[ArcSin[((1 - Sqrt[3])*c^(1/3) + d^(1/3)*x)/((1 + Sqr t[3])*c^(1/3) + d^(1/3)*x)], -7 - 4*Sqrt[3]])/(3^(1/4)*d^(2/3)*Sqrt[(c^(1/ 3)*(c^(1/3) + d^(1/3)*x))/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)^2]*Sqrt[c + d*x^3])))/16
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_) )^(q_), x_Symbol] :> Simp[c*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^ (q - 1)/(a*e*(m + 1))), x] - Simp[1/(a*e^n*(m + 1)) Int[(e*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^(q - 2)*Simp[c*(c*b - a*d)*(m + 1) + c*n*(b*c*(p + 1 ) + a*d*(q - 1)) + d*((c*b - a*d)*(m + 1) + c*b*n*(p + q))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && GtQ[q , 1] && LtQ[m, -1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
Int[(((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n _)))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*(a + b*x^n)^p*((e + f*x^n)/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && IGtQ[n, 0]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 1.86 (sec) , antiderivative size = 859, normalized size of antiderivative = 1.37
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(859\) |
| risch | \(\text {Expression too large to display}\) | \(866\) |
| default | \(\text {Expression too large to display}\) | \(1339\) |
Input:
int((d*x^3+c)^(3/2)/x^2/(-d*x^3+8*c),x,method=_RETURNVERBOSE)
Output:
-1/8*(d*x^3+c)^(1/2)/x+5/8*I*3^(1/2)*(-c*d^2)^(1/3)*(I*(x+1/2/d*(-c*d^2)^( 1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2)*((x-1 /d*(-c*d^2)^(1/3))/(-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))) ^(1/2)*(-I*(x+1/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2) *d/(-c*d^2)^(1/3))^(1/2)/(d*x^3+c)^(1/2)*((-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^( 1/2)/d*(-c*d^2)^(1/3))*EllipticE(1/3*3^(1/2)*(I*(x+1/2/d*(-c*d^2)^(1/3)-1/ 2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2),(I*3^(1/2)/d *(-c*d^2)^(1/3)/(-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3)))^(1 /2))+1/d*(-c*d^2)^(1/3)*EllipticF(1/3*3^(1/2)*(I*(x+1/2/d*(-c*d^2)^(1/3)-1 /2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2),(I*3^(1/2)/ d*(-c*d^2)^(1/3)/(-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3)))^( 1/2)))-3/8*I/d^2*2^(1/2)*sum(1/_alpha*(-c*d^2)^(1/3)*(1/2*I*d*(2*x+1/d*(-I *3^(1/2)*(-c*d^2)^(1/3)+(-c*d^2)^(1/3)))/(-c*d^2)^(1/3))^(1/2)*(d*(x-1/d*( -c*d^2)^(1/3))/(-3*(-c*d^2)^(1/3)+I*3^(1/2)*(-c*d^2)^(1/3)))^(1/2)*(-1/2*I *d*(2*x+1/d*(I*3^(1/2)*(-c*d^2)^(1/3)+(-c*d^2)^(1/3)))/(-c*d^2)^(1/3))^(1/ 2)/(d*x^3+c)^(1/2)*(I*(-c*d^2)^(1/3)*_alpha*3^(1/2)*d-I*3^(1/2)*(-c*d^2)^( 2/3)+2*_alpha^2*d^2-(-c*d^2)^(1/3)*_alpha*d-(-c*d^2)^(2/3))*EllipticPi(1/3 *3^(1/2)*(I*(x+1/2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2 )*d/(-c*d^2)^(1/3))^(1/2),-1/18/d*(2*I*(-c*d^2)^(1/3)*_alpha^2*3^(1/2)*d-I *(-c*d^2)^(2/3)*_alpha*3^(1/2)+I*3^(1/2)*c*d-3*(-c*d^2)^(2/3)*_alpha-3*...
Timed out. \[ \int \frac {\left (c+d x^3\right )^{3/2}}{x^2 \left (8 c-d x^3\right )} \, dx=\text {Timed out} \] Input:
integrate((d*x^3+c)^(3/2)/x^2/(-d*x^3+8*c),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {\left (c+d x^3\right )^{3/2}}{x^2 \left (8 c-d x^3\right )} \, dx=- \int \frac {c \sqrt {c + d x^{3}}}{- 8 c x^{2} + d x^{5}}\, dx - \int \frac {d x^{3} \sqrt {c + d x^{3}}}{- 8 c x^{2} + d x^{5}}\, dx \] Input:
integrate((d*x**3+c)**(3/2)/x**2/(-d*x**3+8*c),x)
Output:
-Integral(c*sqrt(c + d*x**3)/(-8*c*x**2 + d*x**5), x) - Integral(d*x**3*sq rt(c + d*x**3)/(-8*c*x**2 + d*x**5), x)
\[ \int \frac {\left (c+d x^3\right )^{3/2}}{x^2 \left (8 c-d x^3\right )} \, dx=\int { -\frac {{\left (d x^{3} + c\right )}^{\frac {3}{2}}}{{\left (d x^{3} - 8 \, c\right )} x^{2}} \,d x } \] Input:
integrate((d*x^3+c)^(3/2)/x^2/(-d*x^3+8*c),x, algorithm="maxima")
Output:
-integrate((d*x^3 + c)^(3/2)/((d*x^3 - 8*c)*x^2), x)
\[ \int \frac {\left (c+d x^3\right )^{3/2}}{x^2 \left (8 c-d x^3\right )} \, dx=\int { -\frac {{\left (d x^{3} + c\right )}^{\frac {3}{2}}}{{\left (d x^{3} - 8 \, c\right )} x^{2}} \,d x } \] Input:
integrate((d*x^3+c)^(3/2)/x^2/(-d*x^3+8*c),x, algorithm="giac")
Output:
integrate(-(d*x^3 + c)^(3/2)/((d*x^3 - 8*c)*x^2), x)
Timed out. \[ \int \frac {\left (c+d x^3\right )^{3/2}}{x^2 \left (8 c-d x^3\right )} \, dx=\int \frac {{\left (d\,x^3+c\right )}^{3/2}}{x^2\,\left (8\,c-d\,x^3\right )} \,d x \] Input:
int((c + d*x^3)^(3/2)/(x^2*(8*c - d*x^3)),x)
Output:
int((c + d*x^3)^(3/2)/(x^2*(8*c - d*x^3)), x)
\[ \int \frac {\left (c+d x^3\right )^{3/2}}{x^2 \left (8 c-d x^3\right )} \, dx=\frac {2 \sqrt {d \,x^{3}+c}+21 \left (\int \frac {\sqrt {d \,x^{3}+c}}{-d^{2} x^{8}+7 c d \,x^{5}+8 c^{2} x^{2}}d x \right ) c^{2} x +6 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x^{4}}{-d^{2} x^{6}+7 c d \,x^{3}+8 c^{2}}d x \right ) d^{2} x}{5 x} \] Input:
int((d*x^3+c)^(3/2)/x^2/(-d*x^3+8*c),x)
Output:
(2*sqrt(c + d*x**3) + 21*int(sqrt(c + d*x**3)/(8*c**2*x**2 + 7*c*d*x**5 - d**2*x**8),x)*c**2*x + 6*int((sqrt(c + d*x**3)*x**4)/(8*c**2 + 7*c*d*x**3 - d**2*x**6),x)*d**2*x)/(5*x)