Integrand size = 27, antiderivative size = 681 \[ \int \frac {x^7 \left (c+d x^3\right )^{3/2}}{\left (8 c-d x^3\right )^2} \, dx=\frac {103 c x^2 \sqrt {c+d x^3}}{13 d^2}+\frac {19 x^5 \sqrt {c+d x^3}}{39 d}+\frac {5906 c^2 \sqrt {c+d x^3}}{13 d^{8/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac {x^5 \left (c+d x^3\right )^{3/2}}{3 d \left (8 c-d x^3\right )}+\frac {108 \sqrt {3} c^{13/6} \arctan \left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{d^{8/3}}-\frac {108 c^{13/6} \text {arctanh}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x^3}}\right )}{d^{8/3}}+\frac {108 c^{13/6} \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{d^{8/3}}-\frac {2953 \sqrt [4]{3} \sqrt {2-\sqrt {3}} c^{7/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt {3}\right )}{13 d^{8/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}+\frac {5906 \sqrt {2} c^{7/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right ),-7-4 \sqrt {3}\right )}{13 \sqrt [4]{3} d^{8/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}} \] Output:
103/13*c*x^2*(d*x^3+c)^(1/2)/d^2+19/39*x^5*(d*x^3+c)^(1/2)/d+5906/13*c^2*( d*x^3+c)^(1/2)/d^(8/3)/((1+3^(1/2))*c^(1/3)+d^(1/3)*x)+1/3*x^5*(d*x^3+c)^( 3/2)/d/(-d*x^3+8*c)+108*3^(1/2)*c^(13/6)*arctan(3^(1/2)*c^(1/6)*(c^(1/3)+d ^(1/3)*x)/(d*x^3+c)^(1/2))/d^(8/3)-108*c^(13/6)*arctanh(1/3*(c^(1/3)+d^(1/ 3)*x)^2/c^(1/6)/(d*x^3+c)^(1/2))/d^(8/3)+108*c^(13/6)*arctanh(1/3*(d*x^3+c )^(1/2)/c^(1/2))/d^(8/3)-2953/13*3^(1/4)*(1/2*6^(1/2)-1/2*2^(1/2))*c^(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)/d^(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)+5906/39*2^ (1/2)*c^(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)*3^(3/4)/d^(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 = 6.96 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.28 \[ \int \frac {x^7 \left (c+d x^3\right )^{3/2}}{\left (8 c-d x^3\right )^2} \, dx=\frac {80 x^2 \left (-412 c^3-388 c^2 d x^3+25 c d^2 x^6+d^3 x^9\right )+4120 c^2 x^2 \left (8 c-d x^3\right ) \sqrt {1+\frac {d x^3}{c}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},1,\frac {5}{3},-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )+2953 c d x^5 \left (8 c-d x^3\right ) \sqrt {1+\frac {d x^3}{c}} \operatorname {AppellF1}\left (\frac {5}{3},\frac {1}{2},1,\frac {8}{3},-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )}{520 d^2 \left (-8 c+d x^3\right ) \sqrt {c+d x^3}} \] Input:
Integrate[(x^7*(c + d*x^3)^(3/2))/(8*c - d*x^3)^2,x]
Output:
(80*x^2*(-412*c^3 - 388*c^2*d*x^3 + 25*c*d^2*x^6 + d^3*x^9) + 4120*c^2*x^2 *(8*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)] + 2953*c*d*x^5*(8*c - d*x^3)*Sqrt[1 + (d*x^3)/c]*AppellF1 [5/3, 1/2, 1, 8/3, -((d*x^3)/c), (d*x^3)/(8*c)])/(520*d^2*(-8*c + d*x^3)*S qrt[c + d*x^3])
Time = 1.88 (sec) , antiderivative size = 687, normalized size of antiderivative = 1.01, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {967, 27, 1051, 27, 1052, 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.
| \(\displaystyle \int \frac {x^7 \left (c+d x^3\right )^{3/2}}{\left (8 c-d x^3\right )^2} \, dx\) |
| \(\Big \downarrow \) 967 |
| \(\displaystyle \frac {x^5 \left (c+d x^3\right )^{3/2}}{3 d \left (8 c-d x^3\right )}-\frac {\int \frac {x^4 \sqrt {d x^3+c} \left (19 d x^3+10 c\right )}{2 \left (8 c-d x^3\right )}dx}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^5 \left (c+d x^3\right )^{3/2}}{3 d \left (8 c-d x^3\right )}-\frac {\int \frac {x^4 \sqrt {d x^3+c} \left (19 d x^3+10 c\right )}{8 c-d x^3}dx}{6 d}\) |
| \(\Big \downarrow \) 1051 |
| \(\displaystyle \frac {x^5 \left (c+d x^3\right )^{3/2}}{3 d \left (8 c-d x^3\right )}-\frac {-\frac {2 \int -\frac {3 c d x^4 \left (721 d x^3+550 c\right )}{2 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{13 d}-\frac {38}{13} x^5 \sqrt {c+d x^3}}{6 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^5 \left (c+d x^3\right )^{3/2}}{3 d \left (8 c-d x^3\right )}-\frac {\frac {3}{13} c \int \frac {x^4 \left (721 d x^3+550 c\right )}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx-\frac {38}{13} x^5 \sqrt {c+d x^3}}{6 d}\) |
| \(\Big \downarrow \) 1052 |
| \(\displaystyle \frac {x^5 \left (c+d x^3\right )^{3/2}}{3 d \left (8 c-d x^3\right )}-\frac {\frac {3}{13} c \left (\frac {2 \int \frac {7 c d x \left (2953 d x^3+1648 c\right )}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{7 d^2}-\frac {206 x^2 \sqrt {c+d x^3}}{d}\right )-\frac {38}{13} x^5 \sqrt {c+d x^3}}{6 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^5 \left (c+d x^3\right )^{3/2}}{3 d \left (8 c-d x^3\right )}-\frac {\frac {3}{13} c \left (\frac {2 c \int \frac {x \left (2953 d x^3+1648 c\right )}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{d}-\frac {206 x^2 \sqrt {c+d x^3}}{d}\right )-\frac {38}{13} x^5 \sqrt {c+d x^3}}{6 d}\) |
| \(\Big \downarrow \) 1054 |
| \(\displaystyle \frac {x^5 \left (c+d x^3\right )^{3/2}}{3 d \left (8 c-d x^3\right )}-\frac {\frac {3}{13} c \left (\frac {2 c \int \left (\frac {25272 c x}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}-\frac {2953 x}{\sqrt {d x^3+c}}\right )dx}{d}-\frac {206 x^2 \sqrt {c+d x^3}}{d}\right )-\frac {38}{13} x^5 \sqrt {c+d x^3}}{6 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {x^5 \left (c+d x^3\right )^{3/2}}{3 d \left (8 c-d x^3\right )}-\frac {\frac {3}{13} c \left (\frac {2 c \left (-\frac {5906 \sqrt {2} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} d^{2/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}+\frac {2953 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt {3}\right )}{d^{2/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {1404 \sqrt {3} \sqrt [6]{c} \arctan \left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{d^{2/3}}+\frac {1404 \sqrt [6]{c} \text {arctanh}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x^3}}\right )}{d^{2/3}}-\frac {1404 \sqrt [6]{c} \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{d^{2/3}}-\frac {5906 \sqrt {c+d x^3}}{d^{2/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}\right )}{d}-\frac {206 x^2 \sqrt {c+d x^3}}{d}\right )-\frac {38}{13} x^5 \sqrt {c+d x^3}}{6 d}\) |
Input:
Int[(x^7*(c + d*x^3)^(3/2))/(8*c - d*x^3)^2,x]
Output:
(x^5*(c + d*x^3)^(3/2))/(3*d*(8*c - d*x^3)) - ((-38*x^5*Sqrt[c + d*x^3])/1 3 + (3*c*((-206*x^2*Sqrt[c + d*x^3])/d + (2*c*((-5906*Sqrt[c + d*x^3])/(d^ (2/3)*((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)) - (1404*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) + (1404* c^(1/6)*ArcTanh[(c^(1/3) + d^(1/3)*x)^2/(3*c^(1/6)*Sqrt[c + d*x^3])])/d^(2 /3) - (1404*c^(1/6)*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/d^(2/3) + (2953* 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]*Elli pticE[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]) - (5906*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])))/d))/13)/(6*d)
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^(n - 1)*(e*x)^(m - n + 1)*(a + b*x^n)^(p + 1)* ((c + d*x^n)^q/(b*n*(p + 1))), x] - Simp[e^n/(b*n*(p + 1)) Int[(e*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1)*Simp[c*(m - n + 1) + d*(m + n*( q - 1) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[q, 0] && GtQ[m - n + 1, 0] && IntBino mialQ[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[f*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/(b*g*(m + n*(p + q + 1) + 1))), x] + Simp[1/( b*(m + n*(p + q + 1) + 1)) Int[(g*x)^m*(a + b*x^n)^p*(c + d*x^n)^(q - 1)* Simp[c*((b*e - a*f)*(m + 1) + b*e*n*(p + q + 1)) + (d*(b*e - a*f)*(m + 1) + f*n*q*(b*c - a*d) + b*e*d*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c , d, e, f, g, m, p}, x] && IGtQ[n, 0] && GtQ[q, 0] && !(EqQ[q, 1] && Simpl erQ[e + f*x^n, c + d*x^n])
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[f*g^(n - 1)*(g*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(b*d*(m + n*(p + q + 1) + 1))), x] - Simp[g^n/(b*d*(m + n*(p + q + 1) + 1)) Int[(g*x)^(m - n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m - n + 1) + (a*f*d*(m + n*q + 1) + b*( f*c*(m + n*p + 1) - e*d*(m + n*(p + q + 1) + 1)))*x^n, x], x], x] /; FreeQ[ {a, b, c, d, e, f, g, p, q}, x] && IGtQ[n, 0] && GtQ[m, n - 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 = 3.23 (sec) , antiderivative size = 921, normalized size of antiderivative = 1.35
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(921\) |
| risch | \(\text {Expression too large to display}\) | \(1769\) |
| default | \(\text {Expression too large to display}\) | \(2224\) |
Input:
int(x^7*(d*x^3+c)^(3/2)/(-d*x^3+8*c)^2,x,method=_RETURNVERBOSE)
Output:
24*c^2/d^2*x^2*(d*x^3+c)^(1/2)/(-d*x^3+8*c)+2/13*x^5*(d*x^3+c)^(1/2)/d+64/ 13*c*x^2*(d*x^3+c)^(1/2)/d^2-5906/39*I*c^2/d^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)))+72*I*c^2/d^5*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...
Leaf count of result is larger than twice the leaf count of optimal. 2580 vs. \(2 (490) = 980\).
Time = 16.00 (sec) , antiderivative size = 2580, normalized size of antiderivative = 3.79 \[ \int \frac {x^7 \left (c+d x^3\right )^{3/2}}{\left (8 c-d x^3\right )^2} \, dx=\text {Too large to display} \] Input:
integrate(x^7*(d*x^3+c)^(3/2)/(-d*x^3+8*c)^2,x, algorithm="fricas")
Output:
-1/13*(5906*(c^2*d*x^3 - 8*c^3)*sqrt(d)*weierstrassZeta(0, -4*c/d, weierst rassPInverse(0, -4*c/d, x)) + 117*(d^4*x^3 - 8*c*d^3 - sqrt(-3)*(d^4*x^3 - 8*c*d^3))*(c^13/d^16)^(1/6)*log(14693280768*((d^16*x^9 + 318*c*d^15*x^6 + 1200*c^2*d^14*x^3 + 640*c^3*d^13 + sqrt(-3)*(d^16*x^9 + 318*c*d^15*x^6 + 1200*c^2*d^14*x^3 + 640*c^3*d^13))*(c^13/d^16)^(5/6) + 6*(2*c^11*d^2*x^7 + 160*c^12*d*x^4 + 320*c^13*x - 6*(5*c^3*d^12*x^5 + 32*c^4*d^11*x^2 - sqrt( -3)*(5*c^3*d^12*x^5 + 32*c^4*d^11*x^2))*(c^13/d^16)^(2/3) - (7*c^7*d^7*x^6 + 152*c^8*d^6*x^3 + 64*c^9*d^5 + sqrt(-3)*(7*c^7*d^7*x^6 + 152*c^8*d^6*x^ 3 + 64*c^9*d^5))*(c^13/d^16)^(1/3))*sqrt(d*x^3 + c) - 36*(5*c^5*d^10*x^7 + 64*c^6*d^9*x^4 + 32*c^7*d^8*x)*sqrt(c^13/d^16) + 18*(c^9*d^5*x^8 + 38*c^1 0*d^4*x^5 + 64*c^11*d^3*x^2 - sqrt(-3)*(c^9*d^5*x^8 + 38*c^10*d^4*x^5 + 64 *c^11*d^3*x^2))*(c^13/d^16)^(1/6))/(d^3*x^9 - 24*c*d^2*x^6 + 192*c^2*d*x^3 - 512*c^3)) - 117*(d^4*x^3 - 8*c*d^3 - sqrt(-3)*(d^4*x^3 - 8*c*d^3))*(c^1 3/d^16)^(1/6)*log(-14693280768*((d^16*x^9 + 318*c*d^15*x^6 + 1200*c^2*d^14 *x^3 + 640*c^3*d^13 + sqrt(-3)*(d^16*x^9 + 318*c*d^15*x^6 + 1200*c^2*d^14* x^3 + 640*c^3*d^13))*(c^13/d^16)^(5/6) - 6*(2*c^11*d^2*x^7 + 160*c^12*d*x^ 4 + 320*c^13*x - 6*(5*c^3*d^12*x^5 + 32*c^4*d^11*x^2 - sqrt(-3)*(5*c^3*d^1 2*x^5 + 32*c^4*d^11*x^2))*(c^13/d^16)^(2/3) - (7*c^7*d^7*x^6 + 152*c^8*d^6 *x^3 + 64*c^9*d^5 + sqrt(-3)*(7*c^7*d^7*x^6 + 152*c^8*d^6*x^3 + 64*c^9*d^5 ))*(c^13/d^16)^(1/3))*sqrt(d*x^3 + c) - 36*(5*c^5*d^10*x^7 + 64*c^6*d^9...
Timed out. \[ \int \frac {x^7 \left (c+d x^3\right )^{3/2}}{\left (8 c-d x^3\right )^2} \, dx=\text {Timed out} \] Input:
integrate(x**7*(d*x**3+c)**(3/2)/(-d*x**3+8*c)**2,x)
Output:
Timed out
\[ \int \frac {x^7 \left (c+d x^3\right )^{3/2}}{\left (8 c-d x^3\right )^2} \, dx=\int { \frac {{\left (d x^{3} + c\right )}^{\frac {3}{2}} x^{7}}{{\left (d x^{3} - 8 \, c\right )}^{2}} \,d x } \] Input:
integrate(x^7*(d*x^3+c)^(3/2)/(-d*x^3+8*c)^2,x, algorithm="maxima")
Output:
integrate((d*x^3 + c)^(3/2)*x^7/(d*x^3 - 8*c)^2, x)
\[ \int \frac {x^7 \left (c+d x^3\right )^{3/2}}{\left (8 c-d x^3\right )^2} \, dx=\int { \frac {{\left (d x^{3} + c\right )}^{\frac {3}{2}} x^{7}}{{\left (d x^{3} - 8 \, c\right )}^{2}} \,d x } \] Input:
integrate(x^7*(d*x^3+c)^(3/2)/(-d*x^3+8*c)^2,x, algorithm="giac")
Output:
integrate((d*x^3 + c)^(3/2)*x^7/(d*x^3 - 8*c)^2, x)
Timed out. \[ \int \frac {x^7 \left (c+d x^3\right )^{3/2}}{\left (8 c-d x^3\right )^2} \, dx=\int \frac {x^7\,{\left (d\,x^3+c\right )}^{3/2}}{{\left (8\,c-d\,x^3\right )}^2} \,d x \] Input:
int((x^7*(c + d*x^3)^(3/2))/(8*c - d*x^3)^2,x)
Output:
int((x^7*(c + d*x^3)^(3/2))/(8*c - d*x^3)^2, x)
\[ \int \frac {x^7 \left (c+d x^3\right )^{3/2}}{\left (8 c-d x^3\right )^2} \, dx=\frac {1920 \sqrt {d \,x^{3}+c}\, c^{2} x^{2}-1392 \sqrt {d \,x^{3}+c}\, c d \,x^{5}-58 \sqrt {d \,x^{3}+c}\, d^{2} x^{8}+597192 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x^{7}}{d^{3} x^{9}-15 c \,d^{2} x^{6}+48 c^{2} d \,x^{3}+64 c^{3}}d x \right ) c^{3} d^{2}-74649 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x^{7}}{d^{3} x^{9}-15 c \,d^{2} x^{6}+48 c^{2} d \,x^{3}+64 c^{3}}d x \right ) c^{2} d^{3} x^{3}-245760 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x}{d^{3} x^{9}-15 c \,d^{2} x^{6}+48 c^{2} d \,x^{3}+64 c^{3}}d x \right ) c^{5}+30720 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x}{d^{3} x^{9}-15 c \,d^{2} x^{6}+48 c^{2} d \,x^{3}+64 c^{3}}d x \right ) c^{4} d \,x^{3}}{377 d^{2} \left (-d \,x^{3}+8 c \right )} \] Input:
int(x^7*(d*x^3+c)^(3/2)/(-d*x^3+8*c)^2,x)
Output:
(1920*sqrt(c + d*x**3)*c**2*x**2 - 1392*sqrt(c + d*x**3)*c*d*x**5 - 58*sqr t(c + d*x**3)*d**2*x**8 + 597192*int((sqrt(c + d*x**3)*x**7)/(64*c**3 + 48 *c**2*d*x**3 - 15*c*d**2*x**6 + d**3*x**9),x)*c**3*d**2 - 74649*int((sqrt( c + d*x**3)*x**7)/(64*c**3 + 48*c**2*d*x**3 - 15*c*d**2*x**6 + d**3*x**9), x)*c**2*d**3*x**3 - 245760*int((sqrt(c + d*x**3)*x)/(64*c**3 + 48*c**2*d*x **3 - 15*c*d**2*x**6 + d**3*x**9),x)*c**5 + 30720*int((sqrt(c + d*x**3)*x) /(64*c**3 + 48*c**2*d*x**3 - 15*c*d**2*x**6 + d**3*x**9),x)*c**4*d*x**3)/( 377*d**2*(8*c - d*x**3))