Integrand size = 27, antiderivative size = 669 \[ \int \frac {x^7 \left (c+d x^3\right )^{3/2}}{8 c-d x^3} \, dx=-\frac {36534 c^2 x^2 \sqrt {c+d x^3}}{1729 d^2}-\frac {348 c x^5 \sqrt {c+d x^3}}{247 d}-\frac {2}{19} x^8 \sqrt {c+d x^3}-\frac {2094648 c^3 \sqrt {c+d x^3}}{1729 d^{8/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}-\frac {288 \sqrt {3} c^{19/6} \arctan \left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{d^{8/3}}+\frac {288 c^{19/6} \text {arctanh}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x^3}}\right )}{d^{8/3}}-\frac {288 c^{19/6} \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{d^{8/3}}+\frac {1047324 \sqrt [4]{3} \sqrt {2-\sqrt {3}} c^{10/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt {3}\right )}{1729 d^{8/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {698216 \sqrt {2} 3^{3/4} c^{10/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right ),-7-4 \sqrt {3}\right )}{1729 d^{8/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}} \] Output:
-36534/1729*c^2*x^2*(d*x^3+c)^(1/2)/d^2-348/247*c*x^5*(d*x^3+c)^(1/2)/d-2/ 19*x^8*(d*x^3+c)^(1/2)-2094648/1729*c^3*(d*x^3+c)^(1/2)/d^(8/3)/((1+3^(1/2 ))*c^(1/3)+d^(1/3)*x)-288*3^(1/2)*c^(19/6)*arctan(3^(1/2)*c^(1/6)*(c^(1/3) +d^(1/3)*x)/(d*x^3+c)^(1/2))/d^(8/3)+288*c^(19/6)*arctanh(1/3*(c^(1/3)+d^( 1/3)*x)^2/c^(1/6)/(d*x^3+c)^(1/2))/d^(8/3)-288*c^(19/6)*arctanh(1/3*(d*x^3 +c)^(1/2)/c^(1/2))/d^(8/3)+1047324/1729*3^(1/4)*(1/2*6^(1/2)-1/2*2^(1/2))* c^(10/3)*(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)-69 8216/1729*2^(1/2)*3^(3/4)*c^(10/3)*(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)/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 = 8.28 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.24 \[ \int \frac {x^7 \left (c+d x^3\right )^{3/2}}{8 c-d x^3} \, dx=\frac {-20 x^2 \left (18267 c^3+19485 c^2 d x^3+1309 c d^2 x^6+91 d^3 x^9\right )+365340 c^3 x^2 \sqrt {1+\frac {d x^3}{c}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},1,\frac {5}{3},-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )+261831 c^2 d x^5 \sqrt {1+\frac {d x^3}{c}} \operatorname {AppellF1}\left (\frac {5}{3},\frac {1}{2},1,\frac {8}{3},-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )}{17290 d^2 \sqrt {c+d x^3}} \] Input:
Integrate[(x^7*(c + d*x^3)^(3/2))/(8*c - d*x^3),x]
Output:
(-20*x^2*(18267*c^3 + 19485*c^2*d*x^3 + 1309*c*d^2*x^6 + 91*d^3*x^9) + 365 340*c^3*x^2*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), ( d*x^3)/(8*c)] + 261831*c^2*d*x^5*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), (d*x^3)/(8*c)])/(17290*d^2*Sqrt[c + d*x^3])
Time = 1.85 (sec) , antiderivative size = 680, normalized size of antiderivative = 1.02, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {977, 27, 1052, 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.
| \(\displaystyle \int \frac {x^7 \left (c+d x^3\right )^{3/2}}{8 c-d x^3} \, dx\) |
| \(\Big \downarrow \) 977 |
| \(\displaystyle -\frac {2 \int -\frac {3 c d x^7 \left (58 d x^3+49 c\right )}{2 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{19 d}-\frac {2}{19} x^8 \sqrt {c+d x^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{19} c \int \frac {x^7 \left (58 d x^3+49 c\right )}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx-\frac {2}{19} x^8 \sqrt {c+d x^3}\) |
| \(\Big \downarrow \) 1052 |
| \(\displaystyle \frac {3}{19} c \left (\frac {2 \int \frac {c d x^4 \left (6089 d x^3+4640 c\right )}{2 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{13 d^2}-\frac {116 x^5 \sqrt {c+d x^3}}{13 d}\right )-\frac {2}{19} x^8 \sqrt {c+d x^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{19} c \left (\frac {c \int \frac {x^4 \left (6089 d x^3+4640 c\right )}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{13 d}-\frac {116 x^5 \sqrt {c+d x^3}}{13 d}\right )-\frac {2}{19} x^8 \sqrt {c+d x^3}\) |
| \(\Big \downarrow \) 1052 |
| \(\displaystyle \frac {3}{19} c \left (\frac {c \left (\frac {2 \int \frac {2 c d x \left (87277 d x^3+48712 c\right )}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{7 d^2}-\frac {12178 x^2 \sqrt {c+d x^3}}{7 d}\right )}{13 d}-\frac {116 x^5 \sqrt {c+d x^3}}{13 d}\right )-\frac {2}{19} x^8 \sqrt {c+d x^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{19} c \left (\frac {c \left (\frac {4 c \int \frac {x \left (87277 d x^3+48712 c\right )}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{7 d}-\frac {12178 x^2 \sqrt {c+d x^3}}{7 d}\right )}{13 d}-\frac {116 x^5 \sqrt {c+d x^3}}{13 d}\right )-\frac {2}{19} x^8 \sqrt {c+d x^3}\) |
| \(\Big \downarrow \) 1054 |
| \(\displaystyle \frac {3}{19} c \left (\frac {c \left (\frac {4 c \int \left (\frac {746928 c x}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}-\frac {87277 x}{\sqrt {d x^3+c}}\right )dx}{7 d}-\frac {12178 x^2 \sqrt {c+d x^3}}{7 d}\right )}{13 d}-\frac {116 x^5 \sqrt {c+d x^3}}{13 d}\right )-\frac {2}{19} x^8 \sqrt {c+d x^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3}{19} c \left (\frac {c \left (\frac {4 c \left (-\frac {174554 \sqrt {2} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} d^{2/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}+\frac {87277 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt {3}\right )}{d^{2/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {41496 \sqrt {3} \sqrt [6]{c} \arctan \left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{d^{2/3}}+\frac {41496 \sqrt [6]{c} \text {arctanh}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x^3}}\right )}{d^{2/3}}-\frac {41496 \sqrt [6]{c} \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{d^{2/3}}-\frac {174554 \sqrt {c+d x^3}}{d^{2/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}\right )}{7 d}-\frac {12178 x^2 \sqrt {c+d x^3}}{7 d}\right )}{13 d}-\frac {116 x^5 \sqrt {c+d x^3}}{13 d}\right )-\frac {2}{19} x^8 \sqrt {c+d x^3}\) |
Input:
Int[(x^7*(c + d*x^3)^(3/2))/(8*c - d*x^3),x]
Output:
(-2*x^8*Sqrt[c + d*x^3])/19 + (3*c*((-116*x^5*Sqrt[c + d*x^3])/(13*d) + (c *((-12178*x^2*Sqrt[c + d*x^3])/(7*d) + (4*c*((-174554*Sqrt[c + d*x^3])/(d^ (2/3)*((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)) - (41496*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) + (4149 6*c^(1/6)*ArcTanh[(c^(1/3) + d^(1/3)*x)^2/(3*c^(1/6)*Sqrt[c + d*x^3])])/d^ (2/3) - (41496*c^(1/6)*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/d^(2/3) + (87 277*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]) - (174554*Sq rt[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 - S qrt[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])))/(7*d)))/(13*d)))/19
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[d*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n) ^(q - 1)/(b*e*(m + n*(p + q) + 1))), x] + Simp[1/(b*(m + n*(p + q) + 1)) Int[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^(q - 2)*Simp[c*((c*b - a*d)*(m + 1) + c*b*n*(p + q)) + (d*(c*b - a*d)*(m + 1) + d*n*(q - 1)*(b*c - a*d) + c*b*d* n*(p + q))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && GtQ[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[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 = 2.23 (sec) , antiderivative size = 895, normalized size of antiderivative = 1.34
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(895\) |
| elliptic | \(\text {Expression too large to display}\) | \(906\) |
| default | \(\text {Expression too large to display}\) | \(1840\) |
Input:
int(x^7*(d*x^3+c)^(3/2)/(-d*x^3+8*c),x,method=_RETURNVERBOSE)
Output:
-2/1729*x^2*(91*d^2*x^6+1218*c*d*x^3+18267*c^2)/d^2*(d*x^3+c)^(1/2)-12/172 9*c^3/d^2*(-174554/3*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) ))+27664*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-...
Leaf count of result is larger than twice the leaf count of optimal. 2453 vs. \(2 (479) = 958\).
Time = 30.40 (sec) , antiderivative size = 2453, normalized size of antiderivative = 3.67 \[ \int \frac {x^7 \left (c+d x^3\right )^{3/2}}{8 c-d x^3} \, dx=\text {Too large to display} \] Input:
integrate(x^7*(d*x^3+c)^(3/2)/(-d*x^3+8*c),x, algorithm="fricas")
Output:
2/1729*(1047324*c^3*sqrt(d)*weierstrassZeta(0, -4*c/d, weierstrassPInverse (0, -4*c/d, x)) + 41496*(c^19/d^16)^(1/6)*d^3*log(1981355655168*((d^16*x^9 + 318*c*d^15*x^6 + 1200*c^2*d^14*x^3 + 640*c^3*d^13)*(c^19/d^16)^(5/6) + 6*(c^16*d^2*x^7 + 80*c^17*d*x^4 + 160*c^18*x + 6*(5*c^4*d^12*x^5 + 32*c^5* d^11*x^2)*(c^19/d^16)^(2/3) + (7*c^10*d^7*x^6 + 152*c^11*d^6*x^3 + 64*c^12 *d^5)*(c^19/d^16)^(1/3))*sqrt(d*x^3 + c) + 18*(5*c^7*d^10*x^7 + 64*c^8*d^9 *x^4 + 32*c^9*d^8*x)*sqrt(c^19/d^16) + 18*(c^13*d^5*x^8 + 38*c^14*d^4*x^5 + 64*c^15*d^3*x^2)*(c^19/d^16)^(1/6))/(d^3*x^9 - 24*c*d^2*x^6 + 192*c^2*d* x^3 - 512*c^3)) - 41496*(c^19/d^16)^(1/6)*d^3*log(-1981355655168*((d^16*x^ 9 + 318*c*d^15*x^6 + 1200*c^2*d^14*x^3 + 640*c^3*d^13)*(c^19/d^16)^(5/6) - 6*(c^16*d^2*x^7 + 80*c^17*d*x^4 + 160*c^18*x + 6*(5*c^4*d^12*x^5 + 32*c^5 *d^11*x^2)*(c^19/d^16)^(2/3) + (7*c^10*d^7*x^6 + 152*c^11*d^6*x^3 + 64*c^1 2*d^5)*(c^19/d^16)^(1/3))*sqrt(d*x^3 + c) + 18*(5*c^7*d^10*x^7 + 64*c^8*d^ 9*x^4 + 32*c^9*d^8*x)*sqrt(c^19/d^16) + 18*(c^13*d^5*x^8 + 38*c^14*d^4*x^5 + 64*c^15*d^3*x^2)*(c^19/d^16)^(1/6))/(d^3*x^9 - 24*c*d^2*x^6 + 192*c^2*d *x^3 - 512*c^3)) - 20748*(sqrt(-3)*d^3 - d^3)*(c^19/d^16)^(1/6)*log(198135 5655168*((d^16*x^9 + 318*c*d^15*x^6 + 1200*c^2*d^14*x^3 + 640*c^3*d^13 + s qrt(-3)*(d^16*x^9 + 318*c*d^15*x^6 + 1200*c^2*d^14*x^3 + 640*c^3*d^13))*(c ^19/d^16)^(5/6) + 6*(2*c^16*d^2*x^7 + 160*c^17*d*x^4 + 320*c^18*x - 6*(5*c ^4*d^12*x^5 + 32*c^5*d^11*x^2 - sqrt(-3)*(5*c^4*d^12*x^5 + 32*c^5*d^11*...
\[ \int \frac {x^7 \left (c+d x^3\right )^{3/2}}{8 c-d x^3} \, dx=- \int \frac {c x^{7} \sqrt {c + d x^{3}}}{- 8 c + d x^{3}}\, dx - \int \frac {d x^{10} \sqrt {c + d x^{3}}}{- 8 c + d x^{3}}\, dx \] Input:
integrate(x**7*(d*x**3+c)**(3/2)/(-d*x**3+8*c),x)
Output:
-Integral(c*x**7*sqrt(c + d*x**3)/(-8*c + d*x**3), x) - Integral(d*x**10*s qrt(c + d*x**3)/(-8*c + d*x**3), x)
\[ \int \frac {x^7 \left (c+d x^3\right )^{3/2}}{8 c-d x^3} \, dx=\int { -\frac {{\left (d x^{3} + c\right )}^{\frac {3}{2}} x^{7}}{d x^{3} - 8 \, c} \,d x } \] Input:
integrate(x^7*(d*x^3+c)^(3/2)/(-d*x^3+8*c),x, algorithm="maxima")
Output:
-integrate((d*x^3 + c)^(3/2)*x^7/(d*x^3 - 8*c), x)
\[ \int \frac {x^7 \left (c+d x^3\right )^{3/2}}{8 c-d x^3} \, dx=\int { -\frac {{\left (d x^{3} + c\right )}^{\frac {3}{2}} x^{7}}{d x^{3} - 8 \, c} \,d x } \] Input:
integrate(x^7*(d*x^3+c)^(3/2)/(-d*x^3+8*c),x, algorithm="giac")
Output:
integrate(-(d*x^3 + c)^(3/2)*x^7/(d*x^3 - 8*c), x)
Timed out. \[ \int \frac {x^7 \left (c+d x^3\right )^{3/2}}{8 c-d x^3} \, dx=\int \frac {x^7\,{\left (d\,x^3+c\right )}^{3/2}}{8\,c-d\,x^3} \,d x \] Input:
int((x^7*(c + d*x^3)^(3/2))/(8*c - d*x^3),x)
Output:
int((x^7*(c + d*x^3)^(3/2))/(8*c - d*x^3), x)
\[ \int \frac {x^7 \left (c+d x^3\right )^{3/2}}{8 c-d x^3} \, dx=\frac {-\frac {36534 \sqrt {d \,x^{3}+c}\, c^{2} x^{2}}{1729}-\frac {348 \sqrt {d \,x^{3}+c}\, c d \,x^{5}}{247}-\frac {2 \sqrt {d \,x^{3}+c}\, d^{2} x^{8}}{19}+\frac {1047324 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x^{4}}{-d^{2} x^{6}+7 c d \,x^{3}+8 c^{2}}d x \right ) c^{3} d}{1729}+\frac {584544 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x}{-d^{2} x^{6}+7 c d \,x^{3}+8 c^{2}}d x \right ) c^{4}}{1729}}{d^{2}} \] Input:
int(x^7*(d*x^3+c)^(3/2)/(-d*x^3+8*c),x)
Output:
(2*( - 18267*sqrt(c + d*x**3)*c**2*x**2 - 1218*sqrt(c + d*x**3)*c*d*x**5 - 91*sqrt(c + d*x**3)*d**2*x**8 + 523662*int((sqrt(c + d*x**3)*x**4)/(8*c** 2 + 7*c*d*x**3 - d**2*x**6),x)*c**3*d + 292272*int((sqrt(c + d*x**3)*x)/(8 *c**2 + 7*c*d*x**3 - d**2*x**6),x)*c**4))/(1729*d**2)