Integrand size = 27, antiderivative size = 645 \[ \int \frac {x^4 \left (c+d x^3\right )^{3/2}}{8 c-d x^3} \, dx=-\frac {240 c x^2 \sqrt {c+d x^3}}{91 d}-\frac {2}{13} x^5 \sqrt {c+d x^3}-\frac {13782 c^2 \sqrt {c+d x^3}}{91 d^{5/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}-\frac {36 \sqrt {3} c^{13/6} \arctan \left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{d^{5/3}}+\frac {36 c^{13/6} \text {arctanh}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x^3}}\right )}{d^{5/3}}-\frac {36 c^{13/6} \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{d^{5/3}}+\frac {6891 \sqrt [4]{3} \sqrt {2-\sqrt {3}} c^{7/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt {3}\right )}{91 d^{5/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {4594 \sqrt {2} 3^{3/4} c^{7/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right ),-7-4 \sqrt {3}\right )}{91 d^{5/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}} \] Output:
-240/91*c*x^2*(d*x^3+c)^(1/2)/d-2/13*x^5*(d*x^3+c)^(1/2)-13782/91*c^2*(d*x ^3+c)^(1/2)/d^(5/3)/((1+3^(1/2))*c^(1/3)+d^(1/3)*x)-36*3^(1/2)*c^(13/6)*ar ctan(3^(1/2)*c^(1/6)*(c^(1/3)+d^(1/3)*x)/(d*x^3+c)^(1/2))/d^(5/3)+36*c^(13 /6)*arctanh(1/3*(c^(1/3)+d^(1/3)*x)^2/c^(1/6)/(d*x^3+c)^(1/2))/d^(5/3)-36* c^(13/6)*arctanh(1/3*(d*x^3+c)^(1/2)/c^(1/2))/d^(5/3)+6891/91*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 ^(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)-4594/91*2^(1/2)*3^(3/4)*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)/d^(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 = 7.97 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.23 \[ \int \frac {x^4 \left (c+d x^3\right )^{3/2}}{8 c-d x^3} \, dx=\frac {-80 \left (120 c^2 x^2+127 c d x^5+7 d^2 x^8\right )+9600 c^2 x^2 \sqrt {1+\frac {d x^3}{c}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},1,\frac {5}{3},-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )+6891 c d x^5 \sqrt {1+\frac {d x^3}{c}} \operatorname {AppellF1}\left (\frac {5}{3},\frac {1}{2},1,\frac {8}{3},-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )}{3640 d \sqrt {c+d x^3}} \] Input:
Integrate[(x^4*(c + d*x^3)^(3/2))/(8*c - d*x^3),x]
Output:
(-80*(120*c^2*x^2 + 127*c*d*x^5 + 7*d^2*x^8) + 9600*c^2*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)] + 6891*c*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)])/(3640*d*Sqrt[c + d*x^3])
Time = 1.74 (sec) , antiderivative size = 650, normalized size of antiderivative = 1.01, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {977, 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.
| \(\displaystyle \int \frac {x^4 \left (c+d x^3\right )^{3/2}}{8 c-d x^3} \, dx\) |
| \(\Big \downarrow \) 977 |
| \(\displaystyle -\frac {2 \int -\frac {3 c d x^4 \left (40 d x^3+31 c\right )}{2 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{13 d}-\frac {2}{13} x^5 \sqrt {c+d x^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{13} c \int \frac {x^4 \left (40 d x^3+31 c\right )}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx-\frac {2}{13} x^5 \sqrt {c+d x^3}\) |
| \(\Big \downarrow \) 1052 |
| \(\displaystyle \frac {3}{13} c \left (\frac {2 \int \frac {c d x \left (2297 d x^3+1280 c\right )}{2 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{7 d^2}-\frac {80 x^2 \sqrt {c+d x^3}}{7 d}\right )-\frac {2}{13} x^5 \sqrt {c+d x^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{13} c \left (\frac {c \int \frac {x \left (2297 d x^3+1280 c\right )}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{7 d}-\frac {80 x^2 \sqrt {c+d x^3}}{7 d}\right )-\frac {2}{13} x^5 \sqrt {c+d x^3}\) |
| \(\Big \downarrow \) 1054 |
| \(\displaystyle \frac {3}{13} c \left (\frac {c \int \left (\frac {19656 c x}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}-\frac {2297 x}{\sqrt {d x^3+c}}\right )dx}{7 d}-\frac {80 x^2 \sqrt {c+d x^3}}{7 d}\right )-\frac {2}{13} x^5 \sqrt {c+d x^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3}{13} c \left (\frac {c \left (-\frac {4594 \sqrt {2} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} d^{2/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}+\frac {2297 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt {3}\right )}{d^{2/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {1092 \sqrt {3} \sqrt [6]{c} \arctan \left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{d^{2/3}}+\frac {1092 \sqrt [6]{c} \text {arctanh}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x^3}}\right )}{d^{2/3}}-\frac {1092 \sqrt [6]{c} \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{d^{2/3}}-\frac {4594 \sqrt {c+d x^3}}{d^{2/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}\right )}{7 d}-\frac {80 x^2 \sqrt {c+d x^3}}{7 d}\right )-\frac {2}{13} x^5 \sqrt {c+d x^3}\) |
Input:
Int[(x^4*(c + d*x^3)^(3/2))/(8*c - d*x^3),x]
Output:
(-2*x^5*Sqrt[c + d*x^3])/13 + (3*c*((-80*x^2*Sqrt[c + d*x^3])/(7*d) + (c*( (-4594*Sqrt[c + d*x^3])/(d^(2/3)*((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)) - (1 092*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) + (1092*c^(1/6)*ArcTanh[(c^(1/3) + d^(1/3)*x)^2/(3*c^(1 /6)*Sqrt[c + d*x^3])])/d^(2/3) - (1092*c^(1/6)*ArcTanh[Sqrt[c + d*x^3]/(3* Sqrt[c])])/d^(2/3) + (2297*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]) - (4594*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]*Ell ipticF[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])))/(7*d) ))/13
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 = 1.78 (sec) , antiderivative size = 884, normalized size of antiderivative = 1.37
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(884\) |
| elliptic | \(\text {Expression too large to display}\) | \(886\) |
| default | \(\text {Expression too large to display}\) | \(1344\) |
Input:
int(x^4*(d*x^3+c)^(3/2)/(-d*x^3+8*c),x,method=_RETURNVERBOSE)
Output:
-2/91*x^2*(7*d*x^3+120*c)/d*(d*x^3+c)^(1/2)-3/91*c^2/d*(-4594/3*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)))+728*I/d^3*2^(1/2)*sum(1/_alp ha*(-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...
Leaf count of result is larger than twice the leaf count of optimal. 2442 vs. \(2 (459) = 918\).
Time = 10.16 (sec) , antiderivative size = 2442, normalized size of antiderivative = 3.79 \[ \int \frac {x^4 \left (c+d x^3\right )^{3/2}}{8 c-d x^3} \, dx=\text {Too large to display} \] Input:
integrate(x^4*(d*x^3+c)^(3/2)/(-d*x^3+8*c),x, algorithm="fricas")
Output:
1/91*(13782*c^2*sqrt(d)*weierstrassZeta(0, -4*c/d, weierstrassPInverse(0, -4*c/d, x)) + 546*(c^13/d^10)^(1/6)*d^2*log(60466176*((d^11*x^9 + 318*c*d^ 10*x^6 + 1200*c^2*d^9*x^3 + 640*c^3*d^8)*(c^13/d^10)^(5/6) + 6*(c^11*d^2*x ^7 + 80*c^12*d*x^4 + 160*c^13*x + 6*(5*c^3*d^8*x^5 + 32*c^4*d^7*x^2)*(c^13 /d^10)^(2/3) + (7*c^7*d^5*x^6 + 152*c^8*d^4*x^3 + 64*c^9*d^3)*(c^13/d^10)^ (1/3))*sqrt(d*x^3 + c) + 18*(5*c^5*d^7*x^7 + 64*c^6*d^6*x^4 + 32*c^7*d^5*x )*sqrt(c^13/d^10) + 18*(c^9*d^4*x^8 + 38*c^10*d^3*x^5 + 64*c^11*d^2*x^2)*( c^13/d^10)^(1/6))/(d^3*x^9 - 24*c*d^2*x^6 + 192*c^2*d*x^3 - 512*c^3)) - 54 6*(c^13/d^10)^(1/6)*d^2*log(-60466176*((d^11*x^9 + 318*c*d^10*x^6 + 1200*c ^2*d^9*x^3 + 640*c^3*d^8)*(c^13/d^10)^(5/6) - 6*(c^11*d^2*x^7 + 80*c^12*d* x^4 + 160*c^13*x + 6*(5*c^3*d^8*x^5 + 32*c^4*d^7*x^2)*(c^13/d^10)^(2/3) + (7*c^7*d^5*x^6 + 152*c^8*d^4*x^3 + 64*c^9*d^3)*(c^13/d^10)^(1/3))*sqrt(d*x ^3 + c) + 18*(5*c^5*d^7*x^7 + 64*c^6*d^6*x^4 + 32*c^7*d^5*x)*sqrt(c^13/d^1 0) + 18*(c^9*d^4*x^8 + 38*c^10*d^3*x^5 + 64*c^11*d^2*x^2)*(c^13/d^10)^(1/6 ))/(d^3*x^9 - 24*c*d^2*x^6 + 192*c^2*d*x^3 - 512*c^3)) - 273*(c^13/d^10)^( 1/6)*(sqrt(-3)*d^2 - d^2)*log(60466176*((d^11*x^9 + 318*c*d^10*x^6 + 1200* c^2*d^9*x^3 + 640*c^3*d^8 + sqrt(-3)*(d^11*x^9 + 318*c*d^10*x^6 + 1200*c^2 *d^9*x^3 + 640*c^3*d^8))*(c^13/d^10)^(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^8*x^5 + 32*c^4*d^7*x^2 - sqrt(-3)*(5*c^3*d ^8*x^5 + 32*c^4*d^7*x^2))*(c^13/d^10)^(2/3) - (7*c^7*d^5*x^6 + 152*c^8*...
\[ \int \frac {x^4 \left (c+d x^3\right )^{3/2}}{8 c-d x^3} \, dx=- \int \frac {c x^{4} \sqrt {c + d x^{3}}}{- 8 c + d x^{3}}\, dx - \int \frac {d x^{7} \sqrt {c + d x^{3}}}{- 8 c + d x^{3}}\, dx \] Input:
integrate(x**4*(d*x**3+c)**(3/2)/(-d*x**3+8*c),x)
Output:
-Integral(c*x**4*sqrt(c + d*x**3)/(-8*c + d*x**3), x) - Integral(d*x**7*sq rt(c + d*x**3)/(-8*c + d*x**3), x)
\[ \int \frac {x^4 \left (c+d x^3\right )^{3/2}}{8 c-d x^3} \, dx=\int { -\frac {{\left (d x^{3} + c\right )}^{\frac {3}{2}} x^{4}}{d x^{3} - 8 \, c} \,d x } \] Input:
integrate(x^4*(d*x^3+c)^(3/2)/(-d*x^3+8*c),x, algorithm="maxima")
Output:
-integrate((d*x^3 + c)^(3/2)*x^4/(d*x^3 - 8*c), x)
\[ \int \frac {x^4 \left (c+d x^3\right )^{3/2}}{8 c-d x^3} \, dx=\int { -\frac {{\left (d x^{3} + c\right )}^{\frac {3}{2}} x^{4}}{d x^{3} - 8 \, c} \,d x } \] Input:
integrate(x^4*(d*x^3+c)^(3/2)/(-d*x^3+8*c),x, algorithm="giac")
Output:
integrate(-(d*x^3 + c)^(3/2)*x^4/(d*x^3 - 8*c), x)
Timed out. \[ \int \frac {x^4 \left (c+d x^3\right )^{3/2}}{8 c-d x^3} \, dx=\int \frac {x^4\,{\left (d\,x^3+c\right )}^{3/2}}{8\,c-d\,x^3} \,d x \] Input:
int((x^4*(c + d*x^3)^(3/2))/(8*c - d*x^3),x)
Output:
int((x^4*(c + d*x^3)^(3/2))/(8*c - d*x^3), x)
\[ \int \frac {x^4 \left (c+d x^3\right )^{3/2}}{8 c-d x^3} \, dx=\frac {-240 \sqrt {d \,x^{3}+c}\, c \,x^{2}-14 \sqrt {d \,x^{3}+c}\, d \,x^{5}+6891 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x^{4}}{-d^{2} x^{6}+7 c d \,x^{3}+8 c^{2}}d x \right ) c^{2} d +3840 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x}{-d^{2} x^{6}+7 c d \,x^{3}+8 c^{2}}d x \right ) c^{3}}{91 d} \] Input:
int(x^4*(d*x^3+c)^(3/2)/(-d*x^3+8*c),x)
Output:
( - 240*sqrt(c + d*x**3)*c*x**2 - 14*sqrt(c + d*x**3)*d*x**5 + 6891*int((s qrt(c + d*x**3)*x**4)/(8*c**2 + 7*c*d*x**3 - d**2*x**6),x)*c**2*d + 3840*i nt((sqrt(c + d*x**3)*x)/(8*c**2 + 7*c*d*x**3 - d**2*x**6),x)*c**3)/(91*d)