Integrand size = 25, antiderivative size = 627 \[ \int \frac {x \left (c+d x^3\right )^{3/2}}{8 c-d x^3} \, dx=-\frac {2}{7} x^2 \sqrt {c+d x^3}-\frac {132 c \sqrt {c+d x^3}}{7 d^{2/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}-\frac {9 \sqrt {3} c^{7/6} \arctan \left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{2 d^{2/3}}+\frac {9 c^{7/6} \text {arctanh}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x^3}}\right )}{2 d^{2/3}}-\frac {9 c^{7/6} \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{2 d^{2/3}}+\frac {66 \sqrt [4]{3} \sqrt {2-\sqrt {3}} c^{4/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt {3}\right )}{7 d^{2/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {44 \sqrt {2} 3^{3/4} c^{4/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right ),-7-4 \sqrt {3}\right )}{7 d^{2/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}} \] Output:
-2/7*x^2*(d*x^3+c)^(1/2)-132/7*c*(d*x^3+c)^(1/2)/d^(2/3)/((1+3^(1/2))*c^(1 /3)+d^(1/3)*x)-9/2*3^(1/2)*c^(7/6)*arctan(3^(1/2)*c^(1/6)*(c^(1/3)+d^(1/3) *x)/(d*x^3+c)^(1/2))/d^(2/3)+9/2*c^(7/6)*arctanh(1/3*(c^(1/3)+d^(1/3)*x)^2 /c^(1/6)/(d*x^3+c)^(1/2))/d^(2/3)-9/2*c^(7/6)*arctanh(1/3*(d*x^3+c)^(1/2)/ c^(1/2))/d^(2/3)+66/7*3^(1/4)*(1/2*6^(1/2)-1/2*2^(1/2))*c^(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)/d^(2/3)/(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)-44/7*2^(1/2)*3^(3/4)* c^(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)/d^(2/3)/(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 = 10.09 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.20 \[ \int \frac {x \left (c+d x^3\right )^{3/2}}{8 c-d x^3} \, dx=\frac {x^2 \left (-160 \left (c+d x^3\right )+195 c \sqrt {1+\frac {d x^3}{c}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},1,\frac {5}{3},-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )+132 d x^3 \sqrt {1+\frac {d x^3}{c}} \operatorname {AppellF1}\left (\frac {5}{3},\frac {1}{2},1,\frac {8}{3},-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )\right )}{560 \sqrt {c+d x^3}} \] Input:
Integrate[(x*(c + d*x^3)^(3/2))/(8*c - d*x^3),x]
Output:
(x^2*(-160*(c + d*x^3) + 195*c*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5 /3, -((d*x^3)/c), (d*x^3)/(8*c)] + 132*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)]))/(560*Sqrt[c + d*x^3])
Time = 1.61 (sec) , antiderivative size = 626, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {977, 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.
| \(\displaystyle \int \frac {x \left (c+d x^3\right )^{3/2}}{8 c-d x^3} \, dx\) |
| \(\Big \downarrow \) 977 |
| \(\displaystyle -\frac {2 \int -\frac {3 c d x \left (22 d x^3+13 c\right )}{2 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{7 d}-\frac {2}{7} x^2 \sqrt {c+d x^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{7} c \int \frac {x \left (22 d x^3+13 c\right )}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx-\frac {2}{7} x^2 \sqrt {c+d x^3}\) |
| \(\Big \downarrow \) 1054 |
| \(\displaystyle \frac {3}{7} c \int \left (\frac {189 c x}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}-\frac {22 x}{\sqrt {d x^3+c}}\right )dx-\frac {2}{7} x^2 \sqrt {c+d x^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3}{7} c \left (-\frac {44 \sqrt {2} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} d^{2/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}+\frac {22 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt {3}\right )}{d^{2/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {21 \sqrt {3} \sqrt [6]{c} \arctan \left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{2 d^{2/3}}+\frac {21 \sqrt [6]{c} \text {arctanh}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x^3}}\right )}{2 d^{2/3}}-\frac {21 \sqrt [6]{c} \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{2 d^{2/3}}-\frac {44 \sqrt {c+d x^3}}{d^{2/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}\right )-\frac {2}{7} x^2 \sqrt {c+d x^3}\) |
Input:
Int[(x*(c + d*x^3)^(3/2))/(8*c - d*x^3),x]
Output:
(-2*x^2*Sqrt[c + d*x^3])/7 + (3*c*((-44*Sqrt[c + d*x^3])/(d^(2/3)*((1 + Sq rt[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)*ArcTan h[(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)) + (22*3^(1/4)*Sq rt[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[ArcS in[((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 + Sq rt[3])*c^(1/3) + d^(1/3)*x)^2]*Sqrt[c + d*x^3]) - (44*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 + S qrt[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])))/7
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_)*((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 = 1.68 (sec) , antiderivative size = 864, normalized size of antiderivative = 1.38
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(864\) |
| elliptic | \(\text {Expression too large to display}\) | \(864\) |
| risch | \(\text {Expression too large to display}\) | \(866\) |
Input:
int(x*(d*x^3+c)^(3/2)/(-d*x^3+8*c),x,method=_RETURNVERBOSE)
Output:
-2/7*x^2*(d*x^3+c)^(1/2)+44/7*I*c*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)))-3*I*c/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))*Ellipti cPi(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*3^(1/2)*c*d-3*(-c*d^2)^(2/3)*_a...
Leaf count of result is larger than twice the leaf count of optimal. 2368 vs. \(2 (439) = 878\).
Time = 2.54 (sec) , antiderivative size = 2368, normalized size of antiderivative = 3.78 \[ \int \frac {x \left (c+d x^3\right )^{3/2}}{8 c-d x^3} \, dx=\text {Too large to display} \] Input:
integrate(x*(d*x^3+c)^(3/2)/(-d*x^3+8*c),x, algorithm="fricas")
Output:
-1/56*(16*sqrt(d*x^3 + c)*d*x^2 - 1056*c*sqrt(d)*weierstrassZeta(0, -4*c/d , weierstrassPInverse(0, -4*c/d, x)) + 21*(c^7/d^4)^(1/6)*(sqrt(-3)*d - d) *log(59049/4*((d^6*x^9 + 318*c*d^5*x^6 + 1200*c^2*d^4*x^3 + 640*c^3*d^3 + sqrt(-3)*(d^6*x^9 + 318*c*d^5*x^6 + 1200*c^2*d^4*x^3 + 640*c^3*d^3))*(c^7/ d^4)^(5/6) + 6*(2*c^6*d^2*x^7 + 160*c^7*d*x^4 + 320*c^8*x - 6*(5*c^2*d^4*x ^5 + 32*c^3*d^3*x^2 - sqrt(-3)*(5*c^2*d^4*x^5 + 32*c^3*d^3*x^2))*(c^7/d^4) ^(2/3) - (7*c^4*d^3*x^6 + 152*c^5*d^2*x^3 + 64*c^6*d + sqrt(-3)*(7*c^4*d^3 *x^6 + 152*c^5*d^2*x^3 + 64*c^6*d))*(c^7/d^4)^(1/3))*sqrt(d*x^3 + c) - 36* (5*c^3*d^4*x^7 + 64*c^4*d^3*x^4 + 32*c^5*d^2*x)*sqrt(c^7/d^4) + 18*(c^5*d^ 3*x^8 + 38*c^6*d^2*x^5 + 64*c^7*d*x^2 - sqrt(-3)*(c^5*d^3*x^8 + 38*c^6*d^2 *x^5 + 64*c^7*d*x^2))*(c^7/d^4)^(1/6))/(d^3*x^9 - 24*c*d^2*x^6 + 192*c^2*d *x^3 - 512*c^3)) - 21*(c^7/d^4)^(1/6)*(sqrt(-3)*d - d)*log(-59049/4*((d^6* x^9 + 318*c*d^5*x^6 + 1200*c^2*d^4*x^3 + 640*c^3*d^3 + sqrt(-3)*(d^6*x^9 + 318*c*d^5*x^6 + 1200*c^2*d^4*x^3 + 640*c^3*d^3))*(c^7/d^4)^(5/6) - 6*(2*c ^6*d^2*x^7 + 160*c^7*d*x^4 + 320*c^8*x - 6*(5*c^2*d^4*x^5 + 32*c^3*d^3*x^2 - sqrt(-3)*(5*c^2*d^4*x^5 + 32*c^3*d^3*x^2))*(c^7/d^4)^(2/3) - (7*c^4*d^3 *x^6 + 152*c^5*d^2*x^3 + 64*c^6*d + sqrt(-3)*(7*c^4*d^3*x^6 + 152*c^5*d^2* x^3 + 64*c^6*d))*(c^7/d^4)^(1/3))*sqrt(d*x^3 + c) - 36*(5*c^3*d^4*x^7 + 64 *c^4*d^3*x^4 + 32*c^5*d^2*x)*sqrt(c^7/d^4) + 18*(c^5*d^3*x^8 + 38*c^6*d^2* x^5 + 64*c^7*d*x^2 - sqrt(-3)*(c^5*d^3*x^8 + 38*c^6*d^2*x^5 + 64*c^7*d*...
\[ \int \frac {x \left (c+d x^3\right )^{3/2}}{8 c-d x^3} \, dx=- \int \frac {c x \sqrt {c + d x^{3}}}{- 8 c + d x^{3}}\, dx - \int \frac {d x^{4} \sqrt {c + d x^{3}}}{- 8 c + d x^{3}}\, dx \] Input:
integrate(x*(d*x**3+c)**(3/2)/(-d*x**3+8*c),x)
Output:
-Integral(c*x*sqrt(c + d*x**3)/(-8*c + d*x**3), x) - Integral(d*x**4*sqrt( c + d*x**3)/(-8*c + d*x**3), x)
\[ \int \frac {x \left (c+d x^3\right )^{3/2}}{8 c-d x^3} \, dx=\int { -\frac {{\left (d x^{3} + c\right )}^{\frac {3}{2}} x}{d x^{3} - 8 \, c} \,d x } \] Input:
integrate(x*(d*x^3+c)^(3/2)/(-d*x^3+8*c),x, algorithm="maxima")
Output:
-integrate((d*x^3 + c)^(3/2)*x/(d*x^3 - 8*c), x)
\[ \int \frac {x \left (c+d x^3\right )^{3/2}}{8 c-d x^3} \, dx=\int { -\frac {{\left (d x^{3} + c\right )}^{\frac {3}{2}} x}{d x^{3} - 8 \, c} \,d x } \] Input:
integrate(x*(d*x^3+c)^(3/2)/(-d*x^3+8*c),x, algorithm="giac")
Output:
integrate(-(d*x^3 + c)^(3/2)*x/(d*x^3 - 8*c), x)
Timed out. \[ \int \frac {x \left (c+d x^3\right )^{3/2}}{8 c-d x^3} \, dx=\int \frac {x\,{\left (d\,x^3+c\right )}^{3/2}}{8\,c-d\,x^3} \,d x \] Input:
int((x*(c + d*x^3)^(3/2))/(8*c - d*x^3),x)
Output:
int((x*(c + d*x^3)^(3/2))/(8*c - d*x^3), x)
\[ \int \frac {x \left (c+d x^3\right )^{3/2}}{8 c-d x^3} \, dx=-\frac {2 \sqrt {d \,x^{3}+c}\, x^{2}}{7}+\frac {66 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x^{4}}{-d^{2} x^{6}+7 c d \,x^{3}+8 c^{2}}d x \right ) c d}{7}+\frac {39 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x}{-d^{2} x^{6}+7 c d \,x^{3}+8 c^{2}}d x \right ) c^{2}}{7} \] Input:
int(x*(d*x^3+c)^(3/2)/(-d*x^3+8*c),x)
Output:
( - 2*sqrt(c + d*x**3)*x**2 + 66*int((sqrt(c + d*x**3)*x**4)/(8*c**2 + 7*c *d*x**3 - d**2*x**6),x)*c*d + 39*int((sqrt(c + d*x**3)*x)/(8*c**2 + 7*c*d* x**3 - d**2*x**6),x)*c**2)/7