Integrand size = 27, antiderivative size = 675 \[ \int \frac {\left (c+d x^3\right )^{3/2}}{x^8 \left (8 c-d x^3\right )} \, dx=-\frac {\sqrt {c+d x^3}}{56 x^7}-\frac {75 d \sqrt {c+d x^3}}{1792 c x^4}-\frac {3 d^2 \sqrt {c+d x^3}}{56 c^2 x}+\frac {3 d^{7/3} \sqrt {c+d x^3}}{56 c^2 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}-\frac {9 \sqrt {3} d^{7/3} \arctan \left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{1024 c^{11/6}}+\frac {9 d^{7/3} \text {arctanh}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x^3}}\right )}{1024 c^{11/6}}-\frac {9 d^{7/3} \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{1024 c^{11/6}}-\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} d^{7/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt {3}\right )}{112 c^{5/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}+\frac {3^{3/4} d^{7/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right ),-7-4 \sqrt {3}\right )}{28 \sqrt {2} c^{5/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}} \] Output:
-1/56*(d*x^3+c)^(1/2)/x^7-75/1792*d*(d*x^3+c)^(1/2)/c/x^4-3/56*d^2*(d*x^3+ c)^(1/2)/c^2/x+3/56*d^(7/3)*(d*x^3+c)^(1/2)/c^2/((1+3^(1/2))*c^(1/3)+d^(1/ 3)*x)-9/1024*3^(1/2)*d^(7/3)*arctan(3^(1/2)*c^(1/6)*(c^(1/3)+d^(1/3)*x)/(d *x^3+c)^(1/2))/c^(11/6)+9/1024*d^(7/3)*arctanh(1/3*(c^(1/3)+d^(1/3)*x)^2/c ^(1/6)/(d*x^3+c)^(1/2))/c^(11/6)-9/1024*d^(7/3)*arctanh(1/3*(d*x^3+c)^(1/2 )/c^(1/2))/c^(11/6)-3/112*3^(1/4)*(1/2*6^(1/2)-1/2*2^(1/2))*d^(7/3)*(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^(5/3)/(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)+1/56*3^(3/4)*d^(7 /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^(5/3)/(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 = 167, normalized size of antiderivative = 0.25 \[ \int \frac {\left (c+d x^3\right )^{3/2}}{x^8 \left (8 c-d x^3\right )} \, dx=\frac {6675 c d^3 x^9 \sqrt {1+\frac {d x^3}{c}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},1,\frac {5}{3},-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )-32 \left (5 c \left (32 c^3+107 c^2 d x^3+171 c d^2 x^6+96 d^3 x^9\right )+6 d^4 x^{12} \sqrt {1+\frac {d x^3}{c}} \operatorname {AppellF1}\left (\frac {5}{3},\frac {1}{2},1,\frac {8}{3},-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )\right )}{286720 c^3 x^7 \sqrt {c+d x^3}} \] Input:
Integrate[(c + d*x^3)^(3/2)/(x^8*(8*c - d*x^3)),x]
Output:
(6675*c*d^3*x^9*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c ), (d*x^3)/(8*c)] - 32*(5*c*(32*c^3 + 107*c^2*d*x^3 + 171*c*d^2*x^6 + 96*d ^3*x^9) + 6*d^4*x^12*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x ^3)/c), (d*x^3)/(8*c)]))/(286720*c^3*x^7*Sqrt[c + d*x^3])
Time = 1.86 (sec) , antiderivative size = 681, normalized size of antiderivative = 1.01, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {974, 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.
| \(\displaystyle \int \frac {\left (c+d x^3\right )^{3/2}}{x^8 \left (8 c-d x^3\right )} \, dx\) |
| \(\Big \downarrow \) 974 |
| \(\displaystyle \frac {\int \frac {3 c d \left (41 d x^3+50 c\right )}{2 x^5 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{56 c}-\frac {\sqrt {c+d x^3}}{56 x^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{112} d \int \frac {41 d x^3+50 c}{x^5 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx-\frac {\sqrt {c+d x^3}}{56 x^7}\) |
| \(\Big \downarrow \) 1053 |
| \(\displaystyle \frac {3}{112} d \left (-\frac {\int -\frac {c d \left (125 d x^3+512 c\right )}{x^2 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{32 c^2}-\frac {25 \sqrt {c+d x^3}}{16 c x^4}\right )-\frac {\sqrt {c+d x^3}}{56 x^7}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3}{112} d \left (\frac {\int \frac {c d \left (125 d x^3+512 c\right )}{x^2 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{32 c^2}-\frac {25 \sqrt {c+d x^3}}{16 c x^4}\right )-\frac {\sqrt {c+d x^3}}{56 x^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{112} d \left (\frac {d \int \frac {125 d x^3+512 c}{x^2 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{32 c}-\frac {25 \sqrt {c+d x^3}}{16 c x^4}\right )-\frac {\sqrt {c+d x^3}}{56 x^7}\) |
| \(\Big \downarrow \) 1053 |
| \(\displaystyle \frac {3}{112} d \left (\frac {d \left (-\frac {\int -\frac {8 c d x \left (445 c-32 d x^3\right )}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{8 c^2}-\frac {64 \sqrt {c+d x^3}}{c x}\right )}{32 c}-\frac {25 \sqrt {c+d x^3}}{16 c x^4}\right )-\frac {\sqrt {c+d x^3}}{56 x^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{112} d \left (\frac {d \left (\frac {d \int \frac {x \left (445 c-32 d x^3\right )}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{c}-\frac {64 \sqrt {c+d x^3}}{c x}\right )}{32 c}-\frac {25 \sqrt {c+d x^3}}{16 c x^4}\right )-\frac {\sqrt {c+d x^3}}{56 x^7}\) |
| \(\Big \downarrow \) 1054 |
| \(\displaystyle \frac {3}{112} d \left (\frac {d \left (\frac {d \int \left (\frac {189 c x}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}+\frac {32 x}{\sqrt {d x^3+c}}\right )dx}{c}-\frac {64 \sqrt {c+d x^3}}{c x}\right )}{32 c}-\frac {25 \sqrt {c+d x^3}}{16 c x^4}\right )-\frac {\sqrt {c+d x^3}}{56 x^7}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3}{112} d \left (\frac {d \left (\frac {d \left (\frac {64 \sqrt {2} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} d^{2/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {32 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt {3}\right )}{d^{2/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {21 \sqrt {3} \sqrt [6]{c} \arctan \left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{2 d^{2/3}}+\frac {21 \sqrt [6]{c} \text {arctanh}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x^3}}\right )}{2 d^{2/3}}-\frac {21 \sqrt [6]{c} \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{2 d^{2/3}}+\frac {64 \sqrt {c+d x^3}}{d^{2/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}\right )}{c}-\frac {64 \sqrt {c+d x^3}}{c x}\right )}{32 c}-\frac {25 \sqrt {c+d x^3}}{16 c x^4}\right )-\frac {\sqrt {c+d x^3}}{56 x^7}\) |
Input:
Int[(c + d*x^3)^(3/2)/(x^8*(8*c - d*x^3)),x]
Output:
-1/56*Sqrt[c + d*x^3]/x^7 + (3*d*((-25*Sqrt[c + d*x^3])/(16*c*x^4) + (d*(( -64*Sqrt[c + d*x^3])/(c*x) + (d*((64*Sqrt[c + d*x^3])/(d^(2/3)*((1 + Sqrt[ 3])*c^(1/3) + d^(1/3)*x)) - (21*Sqrt[3]*c^(1/6)*ArcTan[(Sqrt[3]*c^(1/6)*(c ^(1/3) + d^(1/3)*x))/Sqrt[c + d*x^3]])/(2*d^(2/3)) + (21*c^(1/6)*ArcTanh[( c^(1/3) + d^(1/3)*x)^2/(3*c^(1/6)*Sqrt[c + d*x^3])])/(2*d^(2/3)) - (21*c^( 1/6)*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(2*d^(2/3)) - (32*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]) + (64*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)))/112
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_.)*((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 = 3.32 (sec) , antiderivative size = 895, normalized size of antiderivative = 1.33
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(895\) |
| elliptic | \(\text {Expression too large to display}\) | \(903\) |
| default | \(\text {Expression too large to display}\) | \(2306\) |
Input:
int((d*x^3+c)^(3/2)/x^8/(-d*x^3+8*c),x,method=_RETURNVERBOSE)
Output:
-1/1792*(d*x^3+c)^(1/2)*(96*d^2*x^6+75*c*d*x^3+32*c^2)/x^7/c^2+3/3584*d^3/ c^2*(-64/3*I*3^(1/2)/d*(-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)))-7*I/d^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^2)^(1/3)*_alpha^2*3^(1/2)*d-I*(-c*d^2)^(2/...
Leaf count of result is larger than twice the leaf count of optimal. 2442 vs. \(2 (479) = 958\).
Time = 1.76 (sec) , antiderivative size = 2442, normalized size of antiderivative = 3.62 \[ \int \frac {\left (c+d x^3\right )^{3/2}}{x^8 \left (8 c-d x^3\right )} \, dx=\text {Too large to display} \] Input:
integrate((d*x^3+c)^(3/2)/x^8/(-d*x^3+8*c),x, algorithm="fricas")
Output:
-1/28672*(1536*d^(5/2)*x^7*weierstrassZeta(0, -4*c/d, weierstrassPInverse( 0, -4*c/d, x)) - 42*(d^14/c^11)^(1/6)*c^2*x^7*log(6561*(d^14*x^9 + 318*c*d ^13*x^6 + 1200*c^2*d^12*x^3 + 640*c^3*d^11 + 18*(5*c^8*d^4*x^7 + 64*c^9*d^ 3*x^4 + 32*c^10*d^2*x)*(d^14/c^11)^(2/3) + 6*sqrt(d*x^3 + c)*(6*(5*c^10*d* x^5 + 32*c^11*x^2)*(d^14/c^11)^(5/6) + (7*c^6*d^6*x^6 + 152*c^7*d^5*x^3 + 64*c^8*d^4)*sqrt(d^14/c^11) + (c^2*d^11*x^7 + 80*c^3*d^10*x^4 + 160*c^4*d^ 9*x)*(d^14/c^11)^(1/6)) + 18*(c^4*d^9*x^8 + 38*c^5*d^8*x^5 + 64*c^6*d^7*x^ 2)*(d^14/c^11)^(1/3))/(d^3*x^9 - 24*c*d^2*x^6 + 192*c^2*d*x^3 - 512*c^3)) + 42*(d^14/c^11)^(1/6)*c^2*x^7*log(6561*(d^14*x^9 + 318*c*d^13*x^6 + 1200* c^2*d^12*x^3 + 640*c^3*d^11 + 18*(5*c^8*d^4*x^7 + 64*c^9*d^3*x^4 + 32*c^10 *d^2*x)*(d^14/c^11)^(2/3) - 6*sqrt(d*x^3 + c)*(6*(5*c^10*d*x^5 + 32*c^11*x ^2)*(d^14/c^11)^(5/6) + (7*c^6*d^6*x^6 + 152*c^7*d^5*x^3 + 64*c^8*d^4)*sqr t(d^14/c^11) + (c^2*d^11*x^7 + 80*c^3*d^10*x^4 + 160*c^4*d^9*x)*(d^14/c^11 )^(1/6)) + 18*(c^4*d^9*x^8 + 38*c^5*d^8*x^5 + 64*c^6*d^7*x^2)*(d^14/c^11)^ (1/3))/(d^3*x^9 - 24*c*d^2*x^6 + 192*c^2*d*x^3 - 512*c^3)) - 21*(sqrt(-3)* c^2*x^7 + c^2*x^7)*(d^14/c^11)^(1/6)*log(6561*(d^14*x^9 + 318*c*d^13*x^6 + 1200*c^2*d^12*x^3 + 640*c^3*d^11 - 9*(5*c^8*d^4*x^7 + 64*c^9*d^3*x^4 + 32 *c^10*d^2*x + sqrt(-3)*(5*c^8*d^4*x^7 + 64*c^9*d^3*x^4 + 32*c^10*d^2*x))*( d^14/c^11)^(2/3) + 3*sqrt(d*x^3 + c)*(6*(5*c^10*d*x^5 + 32*c^11*x^2 - sqrt (-3)*(5*c^10*d*x^5 + 32*c^11*x^2))*(d^14/c^11)^(5/6) - 2*(7*c^6*d^6*x^6...
\[ \int \frac {\left (c+d x^3\right )^{3/2}}{x^8 \left (8 c-d x^3\right )} \, dx=- \int \frac {c \sqrt {c + d x^{3}}}{- 8 c x^{8} + d x^{11}}\, dx - \int \frac {d x^{3} \sqrt {c + d x^{3}}}{- 8 c x^{8} + d x^{11}}\, dx \] Input:
integrate((d*x**3+c)**(3/2)/x**8/(-d*x**3+8*c),x)
Output:
-Integral(c*sqrt(c + d*x**3)/(-8*c*x**8 + d*x**11), x) - Integral(d*x**3*s qrt(c + d*x**3)/(-8*c*x**8 + d*x**11), x)
\[ \int \frac {\left (c+d x^3\right )^{3/2}}{x^8 \left (8 c-d x^3\right )} \, dx=\int { -\frac {{\left (d x^{3} + c\right )}^{\frac {3}{2}}}{{\left (d x^{3} - 8 \, c\right )} x^{8}} \,d x } \] Input:
integrate((d*x^3+c)^(3/2)/x^8/(-d*x^3+8*c),x, algorithm="maxima")
Output:
-integrate((d*x^3 + c)^(3/2)/((d*x^3 - 8*c)*x^8), x)
\[ \int \frac {\left (c+d x^3\right )^{3/2}}{x^8 \left (8 c-d x^3\right )} \, dx=\int { -\frac {{\left (d x^{3} + c\right )}^{\frac {3}{2}}}{{\left (d x^{3} - 8 \, c\right )} x^{8}} \,d x } \] Input:
integrate((d*x^3+c)^(3/2)/x^8/(-d*x^3+8*c),x, algorithm="giac")
Output:
integrate(-(d*x^3 + c)^(3/2)/((d*x^3 - 8*c)*x^8), x)
Timed out. \[ \int \frac {\left (c+d x^3\right )^{3/2}}{x^8 \left (8 c-d x^3\right )} \, dx=\int \frac {{\left (d\,x^3+c\right )}^{3/2}}{x^8\,\left (8\,c-d\,x^3\right )} \,d x \] Input:
int((c + d*x^3)^(3/2)/(x^8*(8*c - d*x^3)),x)
Output:
int((c + d*x^3)^(3/2)/(x^8*(8*c - d*x^3)), x)
\[ \int \frac {\left (c+d x^3\right )^{3/2}}{x^8 \left (8 c-d x^3\right )} \, dx=\frac {-2 \sqrt {d \,x^{3}+c}+150 \left (\int \frac {\sqrt {d \,x^{3}+c}}{-d^{2} x^{11}+7 c d \,x^{8}+8 c^{2} x^{5}}d x \right ) c d \,x^{7}+123 \left (\int \frac {\sqrt {d \,x^{3}+c}}{-d^{2} x^{8}+7 c d \,x^{5}+8 c^{2} x^{2}}d x \right ) d^{2} x^{7}}{112 x^{7}} \] Input:
int((d*x^3+c)^(3/2)/x^8/(-d*x^3+8*c),x)
Output:
( - 2*sqrt(c + d*x**3) + 150*int(sqrt(c + d*x**3)/(8*c**2*x**5 + 7*c*d*x** 8 - d**2*x**11),x)*c*d*x**7 + 123*int(sqrt(c + d*x**3)/(8*c**2*x**2 + 7*c* d*x**5 - d**2*x**8),x)*d**2*x**7)/(112*x**7)