\(\int \frac {\sqrt {c+d x^3}}{x^2 (8 c-d x^3)} \, dx\) [467]

Optimal result

Integrand size = 27, antiderivative size = 632 \[ \int \frac {\sqrt {c+d x^3}}{x^2 \left (8 c-d x^3\right )} \, dx=-\frac {\sqrt {c+d x^3}}{8 c x}+\frac {\sqrt [3]{d} \sqrt {c+d x^3}}{8 c \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}-\frac {\sqrt {3} \sqrt [3]{d} \arctan \left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{16 c^{5/6}}+\frac {\sqrt [3]{d} \text {arctanh}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x^3}}\right )}{16 c^{5/6}}-\frac {\sqrt [3]{d} \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{16 c^{5/6}}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{d} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt {3}\right )}{16 c^{2/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}+\frac {\sqrt [3]{d} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right ),-7-4 \sqrt {3}\right )}{4 \sqrt {2} \sqrt [4]{3} c^{2/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}} \] Output:

-1/8*(d*x^3+c)^(1/2)/c/x+1/8*d^(1/3)*(d*x^3+c)^(1/2)/c/((1+3^(1/2))*c^(1/3 
)+d^(1/3)*x)-1/16*3^(1/2)*d^(1/3)*arctan(3^(1/2)*c^(1/6)*(c^(1/3)+d^(1/3)* 
x)/(d*x^3+c)^(1/2))/c^(5/6)+1/16*d^(1/3)*arctanh(1/3*(c^(1/3)+d^(1/3)*x)^2 
/c^(1/6)/(d*x^3+c)^(1/2))/c^(5/6)-1/16*d^(1/3)*arctanh(1/3*(d*x^3+c)^(1/2) 
/c^(1/2))/c^(5/6)-1/16*3^(1/4)*(1/2*6^(1/2)-1/2*2^(1/2))*d^(1/3)*(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^(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)+1/24*d^(1/3)*(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)*3^(3/4)/c^(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)
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 11.10 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.22 \[ \int \frac {\sqrt {c+d x^3}}{x^2 \left (8 c-d x^3\right )} \, dx=\frac {-80 c \left (c+d x^3\right )+65 c d x^3 \sqrt {1+\frac {d x^3}{c}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},1,\frac {5}{3},-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )-d^2 x^6 \sqrt {1+\frac {d x^3}{c}} \operatorname {AppellF1}\left (\frac {5}{3},\frac {1}{2},1,\frac {8}{3},-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )}{640 c^2 x \sqrt {c+d x^3}} \] Input:

Integrate[Sqrt[c + d*x^3]/(x^2*(8*c - d*x^3)),x]
 

Output:

(-80*c*(c + d*x^3) + 65*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)] - d^2*x^6*Sqrt[1 + (d*x^3)/c]*AppellF1[5 
/3, 1/2, 1, 8/3, -((d*x^3)/c), (d*x^3)/(8*c)])/(640*c^2*x*Sqrt[c + d*x^3])
 

Rubi [A] (verified)

Time = 1.73 (sec) , antiderivative size = 625, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {975, 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.

\(\displaystyle \int \frac {\sqrt {c+d x^3}}{x^2 \left (8 c-d x^3\right )} \, dx\)

\(\Big \downarrow \) 975

\(\displaystyle \frac {\int \frac {d x \left (26 c-d x^3\right )}{2 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{8 c}-\frac {\sqrt {c+d x^3}}{8 c x}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {d \int \frac {x \left (26 c-d x^3\right )}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{16 c}-\frac {\sqrt {c+d x^3}}{8 c x}\)

\(\Big \downarrow \) 1054

\(\displaystyle \frac {d \int \left (\frac {18 c x}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}+\frac {x}{\sqrt {d x^3+c}}\right )dx}{16 c}-\frac {\sqrt {c+d x^3}}{8 c x}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {d \left (\frac {2 \sqrt {2} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} d^{2/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt {3}\right )}{d^{2/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {\sqrt {3} \sqrt [6]{c} \arctan \left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{d^{2/3}}+\frac {\sqrt [6]{c} \text {arctanh}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x^3}}\right )}{d^{2/3}}-\frac {\sqrt [6]{c} \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{d^{2/3}}+\frac {2 \sqrt {c+d x^3}}{d^{2/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}\right )}{16 c}-\frac {\sqrt {c+d x^3}}{8 c x}\)

Input:

Int[Sqrt[c + d*x^3]/(x^2*(8*c - d*x^3)),x]
 

Output:

-1/8*Sqrt[c + d*x^3]/(c*x) + (d*((2*Sqrt[c + d*x^3])/(d^(2/3)*((1 + Sqrt[3 
])*c^(1/3) + d^(1/3)*x)) - (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) + (c^(1/6)*ArcTanh[(c^(1/3) + d 
^(1/3)*x)^2/(3*c^(1/6)*Sqrt[c + d*x^3])])/d^(2/3) - (c^(1/6)*ArcTanh[Sqrt[ 
c + d*x^3]/(3*Sqrt[c])])/d^(2/3) - (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 + S 
qrt[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]) + (2*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]))) 
/(16*c)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 975

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_) 
)^(q_), x_Symbol] :> Simp[(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/ 
(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 - 1)*Simp[c*b*(m + 1) + n*(b*c*(p + 1) + a*d*q) + d*(b*(m 
 + 1) + b*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && 
 NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[0, q, 1] && LtQ[m, -1] && IntBinomi 
alQ[a, b, c, d, e, m, n, p, q, x]
 

rule 1054

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]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 1.80 (sec) , antiderivative size = 868, normalized size of antiderivative = 1.37

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(868\)
risch	\(\text {Expression too large to display}\)	\(872\)
default	\(\text {Expression too large to display}\)	\(1306\)

Input:

int((d*x^3+c)^(1/2)/x^2/(-d*x^3+8*c),x,method=_RETURNVERBOSE)
 

Output:

-1/8*(d*x^3+c)^(1/2)/c/x-1/24*I/c*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)))-1/24*I/d^2/c*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))*Ellipt 
icPi(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)*_...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2219 vs. \(2 (444) = 888\).

Time = 0.46 (sec) , antiderivative size = 2219, normalized size of antiderivative = 3.51 \[ \int \frac {\sqrt {c+d x^3}}{x^2 \left (8 c-d x^3\right )} \, dx=\text {Too large to display} \] Input:

integrate((d*x^3+c)^(1/2)/x^2/(-d*x^3+8*c),x, algorithm="fricas")
 

Output:

1/192*(2*c*x*(d^2/c^5)^(1/6)*log((d^4*x^9 + 318*c*d^3*x^6 + 1200*c^2*d^2*x 
^3 + 640*c^3*d + 18*(5*c^4*d^2*x^7 + 64*c^5*d*x^4 + 32*c^6*x)*(d^2/c^5)^(2 
/3) + 6*sqrt(d*x^3 + c)*(6*(5*c^5*d*x^5 + 32*c^6*x^2)*(d^2/c^5)^(5/6) + (7 
*c^3*d^2*x^6 + 152*c^4*d*x^3 + 64*c^5)*sqrt(d^2/c^5) + (c*d^3*x^7 + 80*c^2 
*d^2*x^4 + 160*c^3*d*x)*(d^2/c^5)^(1/6)) + 18*(c^2*d^3*x^8 + 38*c^3*d^2*x^ 
5 + 64*c^4*d*x^2)*(d^2/c^5)^(1/3))/(d^3*x^9 - 24*c*d^2*x^6 + 192*c^2*d*x^3 
 - 512*c^3)) - 2*c*x*(d^2/c^5)^(1/6)*log((d^4*x^9 + 318*c*d^3*x^6 + 1200*c 
^2*d^2*x^3 + 640*c^3*d + 18*(5*c^4*d^2*x^7 + 64*c^5*d*x^4 + 32*c^6*x)*(d^2 
/c^5)^(2/3) - 6*sqrt(d*x^3 + c)*(6*(5*c^5*d*x^5 + 32*c^6*x^2)*(d^2/c^5)^(5 
/6) + (7*c^3*d^2*x^6 + 152*c^4*d*x^3 + 64*c^5)*sqrt(d^2/c^5) + (c*d^3*x^7 
+ 80*c^2*d^2*x^4 + 160*c^3*d*x)*(d^2/c^5)^(1/6)) + 18*(c^2*d^3*x^8 + 38*c^ 
3*d^2*x^5 + 64*c^4*d*x^2)*(d^2/c^5)^(1/3))/(d^3*x^9 - 24*c*d^2*x^6 + 192*c 
^2*d*x^3 - 512*c^3)) - 24*sqrt(d)*x*weierstrassZeta(0, -4*c/d, weierstrass 
PInverse(0, -4*c/d, x)) + (sqrt(-3)*c*x + c*x)*(d^2/c^5)^(1/6)*log((d^4*x^ 
9 + 318*c*d^3*x^6 + 1200*c^2*d^2*x^3 + 640*c^3*d - 9*(5*c^4*d^2*x^7 + 64*c 
^5*d*x^4 + 32*c^6*x + sqrt(-3)*(5*c^4*d^2*x^7 + 64*c^5*d*x^4 + 32*c^6*x))* 
(d^2/c^5)^(2/3) + 3*sqrt(d*x^3 + c)*(6*(5*c^5*d*x^5 + 32*c^6*x^2 - sqrt(-3 
)*(5*c^5*d*x^5 + 32*c^6*x^2))*(d^2/c^5)^(5/6) - 2*(7*c^3*d^2*x^6 + 152*c^4 
*d*x^3 + 64*c^5)*sqrt(d^2/c^5) + (c*d^3*x^7 + 80*c^2*d^2*x^4 + 160*c^3*d*x 
 + sqrt(-3)*(c*d^3*x^7 + 80*c^2*d^2*x^4 + 160*c^3*d*x))*(d^2/c^5)^(1/6)...
 

Sympy [F]

\[ \int \frac {\sqrt {c+d x^3}}{x^2 \left (8 c-d x^3\right )} \, dx=- \int \frac {\sqrt {c + d x^{3}}}{- 8 c x^{2} + d x^{5}}\, dx \] Input:

integrate((d*x**3+c)**(1/2)/x**2/(-d*x**3+8*c),x)
 

Output:

-Integral(sqrt(c + d*x**3)/(-8*c*x**2 + d*x**5), x)
 

Maxima [F]

\[ \int \frac {\sqrt {c+d x^3}}{x^2 \left (8 c-d x^3\right )} \, dx=\int { -\frac {\sqrt {d x^{3} + c}}{{\left (d x^{3} - 8 \, c\right )} x^{2}} \,d x } \] Input:

integrate((d*x^3+c)^(1/2)/x^2/(-d*x^3+8*c),x, algorithm="maxima")
 

Output:

-integrate(sqrt(d*x^3 + c)/((d*x^3 - 8*c)*x^2), x)
 

Giac [F]

\[ \int \frac {\sqrt {c+d x^3}}{x^2 \left (8 c-d x^3\right )} \, dx=\int { -\frac {\sqrt {d x^{3} + c}}{{\left (d x^{3} - 8 \, c\right )} x^{2}} \,d x } \] Input:

integrate((d*x^3+c)^(1/2)/x^2/(-d*x^3+8*c),x, algorithm="giac")
 

Output:

integrate(-sqrt(d*x^3 + c)/((d*x^3 - 8*c)*x^2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {c+d x^3}}{x^2 \left (8 c-d x^3\right )} \, dx=\int \frac {\sqrt {d\,x^3+c}}{x^2\,\left (8\,c-d\,x^3\right )} \,d x \] Input:

int((c + d*x^3)^(1/2)/(x^2*(8*c - d*x^3)),x)
 

Output:

int((c + d*x^3)^(1/2)/(x^2*(8*c - d*x^3)), x)
 

Reduce [F]

\[ \int \frac {\sqrt {c+d x^3}}{x^2 \left (8 c-d x^3\right )} \, dx=\int \frac {\sqrt {d \,x^{3}+c}}{-d \,x^{5}+8 c \,x^{2}}d x \] Input:

int((d*x^3+c)^(1/2)/x^2/(-d*x^3+8*c),x)
 

Output:

int(sqrt(c + d*x**3)/(8*c*x**2 - d*x**5),x)