Integrand size = 27, antiderivative size = 687 \[ \int \frac {\sqrt {c+d x^3}}{x^5 \left (8 c-d x^3\right )^2} \, dx=-\frac {7 \sqrt {c+d x^3}}{768 c^2 x^4}-\frac {d \sqrt {c+d x^3}}{96 c^3 x}+\frac {d^{4/3} \sqrt {c+d x^3}}{96 c^3 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac {\sqrt {c+d x^3}}{24 c x^4 \left (8 c-d x^3\right )}-\frac {17 d^{4/3} \arctan \left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{3072 \sqrt {3} c^{17/6}}+\frac {17 d^{4/3} \text {arctanh}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x^3}}\right )}{9216 c^{17/6}}-\frac {17 d^{4/3} \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{9216 c^{17/6}}-\frac {\sqrt {2-\sqrt {3}} d^{4/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt {3}\right )}{64\ 3^{3/4} c^{8/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}+\frac {d^{4/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right ),-7-4 \sqrt {3}\right )}{48 \sqrt {2} \sqrt [4]{3} c^{8/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}} \] Output:
-7/768*(d*x^3+c)^(1/2)/c^2/x^4-1/96*d*(d*x^3+c)^(1/2)/c^3/x+1/96*d^(4/3)*( d*x^3+c)^(1/2)/c^3/((1+3^(1/2))*c^(1/3)+d^(1/3)*x)+1/24*(d*x^3+c)^(1/2)/c/ x^4/(-d*x^3+8*c)-17/9216*d^(4/3)*arctan(3^(1/2)*c^(1/6)*(c^(1/3)+d^(1/3)*x )/(d*x^3+c)^(1/2))*3^(1/2)/c^(17/6)+17/9216*d^(4/3)*arctanh(1/3*(c^(1/3)+d ^(1/3)*x)^2/c^(1/6)/(d*x^3+c)^(1/2))/c^(17/6)-17/9216*d^(4/3)*arctanh(1/3* (d*x^3+c)^(1/2)/c^(1/2))/c^(17/6)-1/192*3^(1/4)*(1/2*6^(1/2)-1/2*2^(1/2))* d^(4/3)*(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^(8/3)/(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)+1/2 88*d^(4/3)*(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)*3^(3/4)/c^(8 /3)/(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 = 10.12 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.29 \[ \int \frac {\sqrt {c+d x^3}}{x^5 \left (8 c-d x^3\right )^2} \, dx=\sqrt {c+d x^3} \left (-\frac {1}{256 c^2 x^4}-\frac {5 d}{512 c^3 x}-\frac {d^2 x^2}{1536 c^3 \left (-8 c+d x^3\right )}\right )+\frac {115 d^2 x^2 \sqrt {\frac {c+d x^3}{c}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},1,\frac {5}{3},-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )}{24576 c^3 \sqrt {c+d x^3}}-\frac {d^3 x^5 \sqrt {\frac {c+d x^3}{c}} \operatorname {AppellF1}\left (\frac {5}{3},\frac {1}{2},1,\frac {8}{3},-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )}{7680 c^4 \sqrt {c+d x^3}} \] Input:
Integrate[Sqrt[c + d*x^3]/(x^5*(8*c - d*x^3)^2),x]
Output:
Sqrt[c + d*x^3]*(-1/256*1/(c^2*x^4) - (5*d)/(512*c^3*x) - (d^2*x^2)/(1536* c^3*(-8*c + d*x^3))) + (115*d^2*x^2*Sqrt[(c + d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), (d*x^3)/(8*c)])/(24576*c^3*Sqrt[c + d*x^3]) - (d^3* x^5*Sqrt[(c + d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), (d*x^3)/( 8*c)])/(7680*c^4*Sqrt[c + d*x^3])
Time = 1.88 (sec) , antiderivative size = 698, normalized size of antiderivative = 1.02, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {969, 27, 1053, 25, 27, 1053, 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.
| \(\displaystyle \int \frac {\sqrt {c+d x^3}}{x^5 \left (8 c-d x^3\right )^2} \, dx\) |
| \(\Big \downarrow \) 969 |
| \(\displaystyle \frac {\sqrt {c+d x^3}}{24 c x^4 \left (8 c-d x^3\right )}-\frac {\int -\frac {11 d x^3+14 c}{2 x^5 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{24 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {11 d x^3+14 c}{x^5 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{48 c}+\frac {\sqrt {c+d x^3}}{24 c x^4 \left (8 c-d x^3\right )}\) |
| \(\Big \downarrow \) 1053 |
| \(\displaystyle \frac {-\frac {\int -\frac {c d \left (35 d x^3+128 c\right )}{x^2 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{32 c^2}-\frac {7 \sqrt {c+d x^3}}{16 c x^4}}{48 c}+\frac {\sqrt {c+d x^3}}{24 c x^4 \left (8 c-d x^3\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {c d \left (35 d x^3+128 c\right )}{x^2 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{32 c^2}-\frac {7 \sqrt {c+d x^3}}{16 c x^4}}{48 c}+\frac {\sqrt {c+d x^3}}{24 c x^4 \left (8 c-d x^3\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {d \int \frac {35 d x^3+128 c}{x^2 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{32 c}-\frac {7 \sqrt {c+d x^3}}{16 c x^4}}{48 c}+\frac {\sqrt {c+d x^3}}{24 c x^4 \left (8 c-d x^3\right )}\) |
| \(\Big \downarrow \) 1053 |
| \(\displaystyle \frac {\frac {d \left (-\frac {\int -\frac {8 c d x \left (115 c-8 d x^3\right )}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{8 c^2}-\frac {16 \sqrt {c+d x^3}}{c x}\right )}{32 c}-\frac {7 \sqrt {c+d x^3}}{16 c x^4}}{48 c}+\frac {\sqrt {c+d x^3}}{24 c x^4 \left (8 c-d x^3\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {d \left (\frac {d \int \frac {x \left (115 c-8 d x^3\right )}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{c}-\frac {16 \sqrt {c+d x^3}}{c x}\right )}{32 c}-\frac {7 \sqrt {c+d x^3}}{16 c x^4}}{48 c}+\frac {\sqrt {c+d x^3}}{24 c x^4 \left (8 c-d x^3\right )}\) |
| \(\Big \downarrow \) 1054 |
| \(\displaystyle \frac {\frac {d \left (\frac {d \int \left (\frac {51 c x}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}+\frac {8 x}{\sqrt {d x^3+c}}\right )dx}{c}-\frac {16 \sqrt {c+d x^3}}{c x}\right )}{32 c}-\frac {7 \sqrt {c+d x^3}}{16 c x^4}}{48 c}+\frac {\sqrt {c+d x^3}}{24 c x^4 \left (8 c-d x^3\right )}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {d \left (\frac {d \left (\frac {16 \sqrt {2} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} d^{2/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {8 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt {3}\right )}{d^{2/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {17 \sqrt [6]{c} \arctan \left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{2 \sqrt {3} d^{2/3}}+\frac {17 \sqrt [6]{c} \text {arctanh}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x^3}}\right )}{6 d^{2/3}}-\frac {17 \sqrt [6]{c} \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{6 d^{2/3}}+\frac {16 \sqrt {c+d x^3}}{d^{2/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}\right )}{c}-\frac {16 \sqrt {c+d x^3}}{c x}\right )}{32 c}-\frac {7 \sqrt {c+d x^3}}{16 c x^4}}{48 c}+\frac {\sqrt {c+d x^3}}{24 c x^4 \left (8 c-d x^3\right )}\) |
Input:
Int[Sqrt[c + d*x^3]/(x^5*(8*c - d*x^3)^2),x]
Output:
Sqrt[c + d*x^3]/(24*c*x^4*(8*c - d*x^3)) + ((-7*Sqrt[c + d*x^3])/(16*c*x^4 ) + (d*((-16*Sqrt[c + d*x^3])/(c*x) + (d*((16*Sqrt[c + d*x^3])/(d^(2/3)*(( 1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)) - (17*c^(1/6)*ArcTan[(Sqrt[3]*c^(1/6)*( c^(1/3) + d^(1/3)*x))/Sqrt[c + d*x^3]])/(2*Sqrt[3]*d^(2/3)) + (17*c^(1/6)* ArcTanh[(c^(1/3) + d^(1/3)*x)^2/(3*c^(1/6)*Sqrt[c + d*x^3])])/(6*d^(2/3)) - (17*c^(1/6)*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(6*d^(2/3)) - (8*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]) + (16*Sqrt[2]*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]*EllipticF[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]])/(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])))/c))/(32*c))/(48*c)
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[(-(e*x)^(m + 1))*(a + b*x^n)^(p + 1)*((c + d*x^n )^q/(a*e*n*(p + 1))), x] + Simp[1/(a*n*(p + 1)) Int[(e*x)^m*(a + b*x^n)^( p + 1)*(c + d*x^n)^(q - 1)*Simp[c*(m + n*(p + 1) + 1) + d*(m + n*(p + q + 1 ) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && LtQ[0, q, 1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_ ))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b *x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^n*( m + 1)) Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2 ) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && IGtQ[n, 0] && LtQ[m, -1]
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 = 3.97 (sec) , antiderivative size = 919, normalized size of antiderivative = 1.34
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(919\) |
| risch | \(\text {Expression too large to display}\) | \(1770\) |
| default | \(\text {Expression too large to display}\) | \(2672\) |
Input:
int((d*x^3+c)^(1/2)/x^5/(-d*x^3+8*c)^2,x,method=_RETURNVERBOSE)
Output:
1/1536*d^2*x^2*(d*x^3+c)^(1/2)/(-d*x^3+8*c)/c^3-1/256*(d*x^3+c)^(1/2)/c^2/ x^4-5/512*d*(d*x^3+c)^(1/2)/c^3/x-1/288*I*d/c^3*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)))-17/13824*I/d/c^3*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^...
Leaf count of result is larger than twice the leaf count of optimal. 2549 vs. \(2 (490) = 980\).
Time = 1.51 (sec) , antiderivative size = 2549, normalized size of antiderivative = 3.71 \[ \int \frac {\sqrt {c+d x^3}}{x^5 \left (8 c-d x^3\right )^2} \, dx=\text {Too large to display} \] Input:
integrate((d*x^3+c)^(1/2)/x^5/(-d*x^3+8*c)^2,x, algorithm="fricas")
Output:
-1/110592*(1152*(d^2*x^7 - 8*c*d*x^4)*sqrt(d)*weierstrassZeta(0, -4*c/d, w eierstrassPInverse(0, -4*c/d, x)) - 17*(c^3*d*x^7 - 8*c^4*x^4 + sqrt(-3)*( c^3*d*x^7 - 8*c^4*x^4))*(d^8/c^17)^(1/6)*log(1419857*(d^9*x^9 + 318*c*d^8* x^6 + 1200*c^2*d^7*x^3 + 640*c^3*d^6 - 9*(5*c^12*d^3*x^7 + 64*c^13*d^2*x^4 + 32*c^14*d*x + sqrt(-3)*(5*c^12*d^3*x^7 + 64*c^13*d^2*x^4 + 32*c^14*d*x) )*(d^8/c^17)^(2/3) + 3*sqrt(d*x^3 + c)*(6*(5*c^15*d*x^5 + 32*c^16*x^2 - sq rt(-3)*(5*c^15*d*x^5 + 32*c^16*x^2))*(d^8/c^17)^(5/6) - 2*(7*c^9*d^4*x^6 + 152*c^10*d^3*x^3 + 64*c^11*d^2)*sqrt(d^8/c^17) + (c^3*d^7*x^7 + 80*c^4*d^ 6*x^4 + 160*c^5*d^5*x + sqrt(-3)*(c^3*d^7*x^7 + 80*c^4*d^6*x^4 + 160*c^5*d ^5*x))*(d^8/c^17)^(1/6)) - 9*(c^6*d^6*x^8 + 38*c^7*d^5*x^5 + 64*c^8*d^4*x^ 2 - sqrt(-3)*(c^6*d^6*x^8 + 38*c^7*d^5*x^5 + 64*c^8*d^4*x^2))*(d^8/c^17)^( 1/3))/(d^3*x^9 - 24*c*d^2*x^6 + 192*c^2*d*x^3 - 512*c^3)) + 17*(c^3*d*x^7 - 8*c^4*x^4 + sqrt(-3)*(c^3*d*x^7 - 8*c^4*x^4))*(d^8/c^17)^(1/6)*log(14198 57*(d^9*x^9 + 318*c*d^8*x^6 + 1200*c^2*d^7*x^3 + 640*c^3*d^6 - 9*(5*c^12*d ^3*x^7 + 64*c^13*d^2*x^4 + 32*c^14*d*x + sqrt(-3)*(5*c^12*d^3*x^7 + 64*c^1 3*d^2*x^4 + 32*c^14*d*x))*(d^8/c^17)^(2/3) - 3*sqrt(d*x^3 + c)*(6*(5*c^15* d*x^5 + 32*c^16*x^2 - sqrt(-3)*(5*c^15*d*x^5 + 32*c^16*x^2))*(d^8/c^17)^(5 /6) - 2*(7*c^9*d^4*x^6 + 152*c^10*d^3*x^3 + 64*c^11*d^2)*sqrt(d^8/c^17) + (c^3*d^7*x^7 + 80*c^4*d^6*x^4 + 160*c^5*d^5*x + sqrt(-3)*(c^3*d^7*x^7 + 80 *c^4*d^6*x^4 + 160*c^5*d^5*x))*(d^8/c^17)^(1/6)) - 9*(c^6*d^6*x^8 + 38*...
\[ \int \frac {\sqrt {c+d x^3}}{x^5 \left (8 c-d x^3\right )^2} \, dx=\int \frac {\sqrt {c + d x^{3}}}{x^{5} \left (- 8 c + d x^{3}\right )^{2}}\, dx \] Input:
integrate((d*x**3+c)**(1/2)/x**5/(-d*x**3+8*c)**2,x)
Output:
Integral(sqrt(c + d*x**3)/(x**5*(-8*c + d*x**3)**2), x)
\[ \int \frac {\sqrt {c+d x^3}}{x^5 \left (8 c-d x^3\right )^2} \, dx=\int { \frac {\sqrt {d x^{3} + c}}{{\left (d x^{3} - 8 \, c\right )}^{2} x^{5}} \,d x } \] Input:
integrate((d*x^3+c)^(1/2)/x^5/(-d*x^3+8*c)^2,x, algorithm="maxima")
Output:
integrate(sqrt(d*x^3 + c)/((d*x^3 - 8*c)^2*x^5), x)
\[ \int \frac {\sqrt {c+d x^3}}{x^5 \left (8 c-d x^3\right )^2} \, dx=\int { \frac {\sqrt {d x^{3} + c}}{{\left (d x^{3} - 8 \, c\right )}^{2} x^{5}} \,d x } \] Input:
integrate((d*x^3+c)^(1/2)/x^5/(-d*x^3+8*c)^2,x, algorithm="giac")
Output:
integrate(sqrt(d*x^3 + c)/((d*x^3 - 8*c)^2*x^5), x)
Timed out. \[ \int \frac {\sqrt {c+d x^3}}{x^5 \left (8 c-d x^3\right )^2} \, dx=\int \frac {\sqrt {d\,x^3+c}}{x^5\,{\left (8\,c-d\,x^3\right )}^2} \,d x \] Input:
int((c + d*x^3)^(1/2)/(x^5*(8*c - d*x^3)^2),x)
Output:
int((c + d*x^3)^(1/2)/(x^5*(8*c - d*x^3)^2), x)
\[ \int \frac {\sqrt {c+d x^3}}{x^5 \left (8 c-d x^3\right )^2} \, dx=\int \frac {\sqrt {d \,x^{3}+c}}{d^{2} x^{11}-16 c d \,x^{8}+64 c^{2} x^{5}}d x \] Input:
int((d*x^3+c)^(1/2)/x^5/(-d*x^3+8*c)^2,x)
Output:
int(sqrt(c + d*x**3)/(64*c**2*x**5 - 16*c*d*x**8 + d**2*x**11),x)