3.5.10 \(\int \frac {\sqrt {c+d x^3}}{x^8 (8 c-d x^3)^2} \, dx\) [410]
Optimal. Leaf size=711 \[ -\frac {5 \sqrt {c+d x^3}}{672 c^2 x^7}-\frac {53 d \sqrt {c+d x^3}}{21504 c^3 x^4}-\frac {d^2 \sqrt {c+d x^3}}{5376 c^4 x}+\frac {d^{7/3} \sqrt {c+d x^3}}{5376 c^4 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac {\sqrt {c+d x^3}}{24 c x^7 \left (8 c-d x^3\right )}-\frac {13 d^{7/3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{12288 \sqrt {3} c^{23/6}}+\frac {13 d^{7/3} \tanh ^{-1}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x^3}}\right )}{36864 c^{23/6}}-\frac {13 d^{7/3} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{36864 c^{23/6}}-\frac {\sqrt {2-\sqrt {3}} d^{7/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt {3}\right )}{3584\ 3^{3/4} c^{11/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}+\frac {d^{7/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt {3}\right )}{2688 \sqrt {2} \sqrt [4]{3} c^{11/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}} \]
[Out]
13/36864*d^(7/3)*arctanh(1/3*(c^(1/3)+d^(1/3)*x)^2/c^(1/6)/(d*x^3+c)^(1/2))/c^(23/6)-13/36864*d^(7/3)*arctanh(
1/3*(d*x^3+c)^(1/2)/c^(1/2))/c^(23/6)-13/36864*d^(7/3)*arctan(c^(1/6)*(c^(1/3)+d^(1/3)*x)*3^(1/2)/(d*x^3+c)^(1
/2))/c^(23/6)*3^(1/2)-5/672*(d*x^3+c)^(1/2)/c^2/x^7-53/21504*d*(d*x^3+c)^(1/2)/c^3/x^4-1/5376*d^2*(d*x^3+c)^(1
/2)/c^4/x+1/24*(d*x^3+c)^(1/2)/c/x^7/(-d*x^3+8*c)+1/5376*d^(7/3)*(d*x^3+c)^(1/2)/c^4/(d^(1/3)*x+c^(1/3)*(1+3^(
1/2)))+1/16128*3^(3/4)*d^(7/3)*(c^(1/3)+d^(1/3)*x)*EllipticF((d^(1/3)*x+c^(1/3)*(1-3^(1/2)))/(d^(1/3)*x+c^(1/3
)*(1+3^(1/2))),I*3^(1/2)+2*I)*((c^(2/3)-c^(1/3)*d^(1/3)*x+d^(2/3)*x^2)/(d^(1/3)*x+c^(1/3)*(1+3^(1/2)))^2)^(1/2
)/c^(11/3)*2^(1/2)/(d*x^3+c)^(1/2)/(c^(1/3)*(c^(1/3)+d^(1/3)*x)/(d^(1/3)*x+c^(1/3)*(1+3^(1/2)))^2)^(1/2)-1/107
52*3^(1/4)*d^(7/3)*(c^(1/3)+d^(1/3)*x)*EllipticE((d^(1/3)*x+c^(1/3)*(1-3^(1/2)))/(d^(1/3)*x+c^(1/3)*(1+3^(1/2)
)),I*3^(1/2)+2*I)*(1/2*6^(1/2)-1/2*2^(1/2))*((c^(2/3)-c^(1/3)*d^(1/3)*x+d^(2/3)*x^2)/(d^(1/3)*x+c^(1/3)*(1+3^(
1/2)))^2)^(1/2)/c^(11/3)/(d*x^3+c)^(1/2)/(c^(1/3)*(c^(1/3)+d^(1/3)*x)/(d^(1/3)*x+c^(1/3)*(1+3^(1/2)))^2)^(1/2)
________________________________________________________________________________________
Rubi [A]
time = 0.74, antiderivative size = 711, normalized size of antiderivative = 1.00, number of steps
used = 17, number of rules used = 13, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.482, Rules used = {480, 597,
598, 309, 224, 1891, 499, 455, 65, 212, 2163, 2170, 211} \begin {gather*} \frac {d^{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}} F\left (\text {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 )}{2688 \sqrt {2} \sqrt [4]{3} c^{11/3} \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 {2-\sqrt {3}} d^{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 (\text {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 )}{3584\ 3^{3/4} c^{11/3} \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 {13 d^{7/3} \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{12288 \sqrt {3} c^{23/6}}+\frac {13 d^{7/3} \tanh ^{-1}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x^3}}\right )}{36864 c^{23/6}}-\frac {13 d^{7/3} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{36864 c^{23/6}}+\frac {d^{7/3} \sqrt {c+d x^3}}{5376 c^4 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}-\frac {d^2 \sqrt {c+d x^3}}{5376 c^4 x}-\frac {53 d \sqrt {c+d x^3}}{21504 c^3 x^4}-\frac {5 \sqrt {c+d x^3}}{672 c^2 x^7}+\frac {\sqrt {c+d x^3}}{24 c x^7 \left (8 c-d x^3\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[Sqrt[c + d*x^3]/(x^8*(8*c - d*x^3)^2),x]
[Out]
(-5*Sqrt[c + d*x^3])/(672*c^2*x^7) - (53*d*Sqrt[c + d*x^3])/(21504*c^3*x^4) - (d^2*Sqrt[c + d*x^3])/(5376*c^4*
x) + (d^(7/3)*Sqrt[c + d*x^3])/(5376*c^4*((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)) + Sqrt[c + d*x^3]/(24*c*x^7*(8*c
- d*x^3)) - (13*d^(7/3)*ArcTan[(Sqrt[3]*c^(1/6)*(c^(1/3) + d^(1/3)*x))/Sqrt[c + d*x^3]])/(12288*Sqrt[3]*c^(23
/6)) + (13*d^(7/3)*ArcTanh[(c^(1/3) + d^(1/3)*x)^2/(3*c^(1/6)*Sqrt[c + d*x^3])])/(36864*c^(23/6)) - (13*d^(7/3
)*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(36864*c^(23/6)) - (Sqrt[2 - Sqrt[3]]*d^(7/3)*(c^(1/3) + d^(1/3)*x)*Sq
rt[(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 - S
qrt[3])*c^(1/3) + d^(1/3)*x)/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)], -7 - 4*Sqrt[3]])/(3584*3^(3/4)*c^(11/3)*Sqr
t[(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^(7/3)*(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]])/(2688*Sqrt[
2]*3^(1/4)*c^(11/3)*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
])
Rule 65
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]
Rule 211
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]
Rule 212
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 224
Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt
[2 + Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sq
rt[s*((s + r*x)/((1 + Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3])*s + r*x)
], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x] && PosQ[a]
Rule 309
Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Dist[(-(
1 - Sqrt[3]))*(s/r), Int[1/Sqrt[a + b*x^3], x], x] + Dist[1/r, Int[((1 - Sqrt[3])*s + r*x)/Sqrt[a + b*x^3], x]
, x]] /; FreeQ[{a, b}, x] && PosQ[a]
Rule 455
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]
Rule 480
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*n*(p + 1))), x] + Dist[1/(a*n*(p + 1)), Int[(e*x)^m*(a + b*x^
n)^(p + 1)*(c + d*x^n)^(q - 1)*Simp[c*(m + n*(p + 1) + 1) + d*(m + n*(p + q + 1) + 1)*x^n, x], x], x] /; FreeQ
[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && LtQ[0, q, 1] && IntBinomialQ[a, b,
c, d, e, m, n, p, q, x]
Rule 499
Int[(x_)/(((a_) + (b_.)*(x_)^3)*Sqrt[(c_) + (d_.)*(x_)^3]), x_Symbol] :> With[{q = Rt[d/c, 3]}, Dist[d*(q/(4*b
)), Int[x^2/((8*c - d*x^3)*Sqrt[c + d*x^3]), x], x] + (-Dist[q^2/(12*b), Int[(1 + q*x)/((2 - q*x)*Sqrt[c + d*x
^3]), x], x] + Dist[1/(12*b*c), Int[(2*c*q^2 - 2*d*x - d*q*x^2)/((4 + 2*q*x + q^2*x^2)*Sqrt[c + d*x^3]), x], x
])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[8*b*c + a*d, 0]
Rule 597
Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*g*(m + 1))), x] + Dist[1/(a*c*
g^n*(m + 1)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e
*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] &&
IGtQ[n, 0] && LtQ[m, -1]
Rule 598
Int[(((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n_)))/((c_) + (d_.)*(x_)^(n_)), x_Sy
mbol] :> 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 1891
Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Simplify[(1 - Sqrt[3])*(d/c)]]
, s = Denom[Simplify[(1 - Sqrt[3])*(d/c)]]}, Simp[2*d*s^3*(Sqrt[a + b*x^3]/(a*r^2*((1 + Sqrt[3])*s + r*x))), x
] - Simp[3^(1/4)*Sqrt[2 - Sqrt[3]]*d*s*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]/(r^2
*Sqrt[a + b*x^3]*Sqrt[s*((s + r*x)/((1 + Sqrt[3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 - Sqrt[3])*s + r*x)/((1
+ Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b, c, d}, x] && PosQ[a] && EqQ[b*c^3 - 2*(5 - 3*Sqrt[3
])*a*d^3, 0]
Rule 2163
Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_Symbol] :> Dist[-2*(e/d), Subst[Int
[1/(9 - a*x^2), x], x, (1 + f*(x/e))^2/Sqrt[a + b*x^3]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f,
0] && EqQ[b*c^3 + 8*a*d^3, 0] && EqQ[2*d*e + c*f, 0]
Rule 2170
Int[((f_) + (g_.)*(x_) + (h_.)*(x_)^2)/(((c_) + (d_.)*(x_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^3]), x_Symbo
l] :> Dist[-2*g*h, Subst[Int[1/(2*e*h - (b*d*f - 2*a*e*h)*x^2), x], x, (1 + 2*h*(x/g))/Sqrt[a + b*x^3]], x] /;
FreeQ[{a, b, c, d, e, f, g, h}, x] && NeQ[b*d*f - 2*a*e*h, 0] && EqQ[b*g^3 - 8*a*h^3, 0] && EqQ[g^2 + 2*f*h,
0] && EqQ[b*d*f + b*c*g - 4*a*e*h, 0]
Rubi steps
\begin {align*} \int \frac {\sqrt {c+d x^3}}{x^8 \left (8 c-d x^3\right )^2} \, dx &=\frac {\sqrt {c+d x^3}}{24 c x^7 \left (8 c-d x^3\right )}-\frac {\int \frac {-10 c-\frac {17 d x^3}{2}}{x^8 \left (8 c-d x^3\right ) \sqrt {c+d x^3}} \, dx}{24 c}\\ &=-\frac {5 \sqrt {c+d x^3}}{672 c^2 x^7}+\frac {\sqrt {c+d x^3}}{24 c x^7 \left (8 c-d x^3\right )}+\frac {\int \frac {106 c^2 d+55 c d^2 x^3}{x^5 \left (8 c-d x^3\right ) \sqrt {c+d x^3}} \, dx}{1344 c^3}\\ &=-\frac {5 \sqrt {c+d x^3}}{672 c^2 x^7}-\frac {53 d \sqrt {c+d x^3}}{21504 c^3 x^4}+\frac {\sqrt {c+d x^3}}{24 c x^7 \left (8 c-d x^3\right )}-\frac {\int \frac {-64 c^3 d^2-265 c^2 d^3 x^3}{x^2 \left (8 c-d x^3\right ) \sqrt {c+d x^3}} \, dx}{43008 c^5}\\ &=-\frac {5 \sqrt {c+d x^3}}{672 c^2 x^7}-\frac {53 d \sqrt {c+d x^3}}{21504 c^3 x^4}-\frac {d^2 \sqrt {c+d x^3}}{5376 c^4 x}+\frac {\sqrt {c+d x^3}}{24 c x^7 \left (8 c-d x^3\right )}+\frac {\int \frac {x \left (2440 c^4 d^3-32 c^3 d^4 x^3\right )}{\left (8 c-d x^3\right ) \sqrt {c+d x^3}} \, dx}{344064 c^7}\\ &=-\frac {5 \sqrt {c+d x^3}}{672 c^2 x^7}-\frac {53 d \sqrt {c+d x^3}}{21504 c^3 x^4}-\frac {d^2 \sqrt {c+d x^3}}{5376 c^4 x}+\frac {\sqrt {c+d x^3}}{24 c x^7 \left (8 c-d x^3\right )}+\frac {\int \left (\frac {32 c^3 d^3 x}{\sqrt {c+d x^3}}+\frac {2184 c^4 d^3 x}{\left (8 c-d x^3\right ) \sqrt {c+d x^3}}\right ) \, dx}{344064 c^7}\\ &=-\frac {5 \sqrt {c+d x^3}}{672 c^2 x^7}-\frac {53 d \sqrt {c+d x^3}}{21504 c^3 x^4}-\frac {d^2 \sqrt {c+d x^3}}{5376 c^4 x}+\frac {\sqrt {c+d x^3}}{24 c x^7 \left (8 c-d x^3\right )}+\frac {d^3 \int \frac {x}{\sqrt {c+d x^3}} \, dx}{10752 c^4}+\frac {\left (13 d^3\right ) \int \frac {x}{\left (8 c-d x^3\right ) \sqrt {c+d x^3}} \, dx}{2048 c^3}\\ &=-\frac {5 \sqrt {c+d x^3}}{672 c^2 x^7}-\frac {53 d \sqrt {c+d x^3}}{21504 c^3 x^4}-\frac {d^2 \sqrt {c+d x^3}}{5376 c^4 x}+\frac {\sqrt {c+d x^3}}{24 c x^7 \left (8 c-d x^3\right )}-\frac {\left (13 d^2\right ) \int \frac {2 \sqrt [3]{c} d^{2/3}-2 d x-\frac {d^{4/3} x^2}{\sqrt [3]{c}}}{\left (4+\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}+\frac {d^{2/3} x^2}{c^{2/3}}\right ) \sqrt {c+d x^3}} \, dx}{24576 c^4}+\frac {d^{8/3} \int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\sqrt {c+d x^3}} \, dx}{10752 c^4}+\frac {\left (13 d^{8/3}\right ) \int \frac {1+\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}}{\left (2-\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}\right ) \sqrt {c+d x^3}} \, dx}{24576 c^{11/3}}+\frac {\left (\sqrt {\frac {1}{2} \left (2-\sqrt {3}\right )} d^{8/3}\right ) \int \frac {1}{\sqrt {c+d x^3}} \, dx}{5376 c^{11/3}}-\frac {\left (13 d^{10/3}\right ) \int \frac {x^2}{\left (8 c-d x^3\right ) \sqrt {c+d x^3}} \, dx}{8192 c^{10/3}}\\ &=-\frac {5 \sqrt {c+d x^3}}{672 c^2 x^7}-\frac {53 d \sqrt {c+d x^3}}{21504 c^3 x^4}-\frac {d^2 \sqrt {c+d x^3}}{5376 c^4 x}+\frac {d^{7/3} \sqrt {c+d x^3}}{5376 c^4 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac {\sqrt {c+d x^3}}{24 c x^7 \left (8 c-d x^3\right )}-\frac {\sqrt {2-\sqrt {3}} d^{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 (\sin ^{-1}\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 )}{3584\ 3^{3/4} c^{11/3} \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 {d^{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}} F\left (\sin ^{-1}\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 )}{2688 \sqrt {2} \sqrt [4]{3} c^{11/3} \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 {\left (13 d^{7/3}\right ) \text {Subst}\left (\int \frac {1}{9-c x^2} \, dx,x,\frac {\left (1+\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}\right )^2}{\sqrt {c+d x^3}}\right )}{12288 c^{10/3}}-\frac {\left (13 d^{10/3}\right ) \text {Subst}\left (\int \frac {1}{(8 c-d x) \sqrt {c+d x}} \, dx,x,x^3\right )}{24576 c^{10/3}}+\frac {\left (13 d^{13/3}\right ) \text {Subst}\left (\int \frac {1}{-\frac {2 d^2}{c}-6 d^2 x^2} \, dx,x,\frac {1+\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt {c+d x^3}}\right )}{6144 c^{13/3}}\\ &=-\frac {5 \sqrt {c+d x^3}}{672 c^2 x^7}-\frac {53 d \sqrt {c+d x^3}}{21504 c^3 x^4}-\frac {d^2 \sqrt {c+d x^3}}{5376 c^4 x}+\frac {d^{7/3} \sqrt {c+d x^3}}{5376 c^4 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac {\sqrt {c+d x^3}}{24 c x^7 \left (8 c-d x^3\right )}-\frac {13 d^{7/3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{12288 \sqrt {3} c^{23/6}}+\frac {13 d^{7/3} \tanh ^{-1}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x^3}}\right )}{36864 c^{23/6}}-\frac {\sqrt {2-\sqrt {3}} d^{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 (\sin ^{-1}\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 )}{3584\ 3^{3/4} c^{11/3} \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 {d^{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}} F\left (\sin ^{-1}\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 )}{2688 \sqrt {2} \sqrt [4]{3} c^{11/3} \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 {\left (13 d^{7/3}\right ) \text {Subst}\left (\int \frac {1}{9 c-x^2} \, dx,x,\sqrt {c+d x^3}\right )}{12288 c^{10/3}}\\ &=-\frac {5 \sqrt {c+d x^3}}{672 c^2 x^7}-\frac {53 d \sqrt {c+d x^3}}{21504 c^3 x^4}-\frac {d^2 \sqrt {c+d x^3}}{5376 c^4 x}+\frac {d^{7/3} \sqrt {c+d x^3}}{5376 c^4 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac {\sqrt {c+d x^3}}{24 c x^7 \left (8 c-d x^3\right )}-\frac {13 d^{7/3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{12288 \sqrt {3} c^{23/6}}+\frac {13 d^{7/3} \tanh ^{-1}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x^3}}\right )}{36864 c^{23/6}}-\frac {13 d^{7/3} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{36864 c^{23/6}}-\frac {\sqrt {2-\sqrt {3}} d^{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 (\sin ^{-1}\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 )}{3584\ 3^{3/4} c^{11/3} \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 {d^{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}} F\left (\sin ^{-1}\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 )}{2688 \sqrt {2} \sqrt [4]{3} c^{11/3} \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}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 4 in
optimal.
time = 10.12, size = 209, normalized size = 0.29 \begin {gather*} \frac {1525 c d^3 x^9 \left (8 c-d x^3\right ) \sqrt {1+\frac {d x^3}{c}} F_1\left (\frac {2}{3};\frac {1}{2},1;\frac {5}{3};-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )-8 \left (20 c \left (384 c^4+648 c^3 d x^3+243 c^2 d^2 x^6-25 c d^3 x^9-4 d^4 x^{12}\right )+d^4 x^{12} \left (8 c-d x^3\right ) \sqrt {1+\frac {d x^3}{c}} F_1\left (\frac {5}{3};\frac {1}{2},1;\frac {8}{3};-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )\right )}{3440640 c^5 x^7 \left (8 c-d x^3\right ) \sqrt {c+d x^3}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[Sqrt[c + d*x^3]/(x^8*(8*c - d*x^3)^2),x]
[Out]
(1525*c*d^3*x^9*(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)] - 8*
(20*c*(384*c^4 + 648*c^3*d*x^3 + 243*c^2*d^2*x^6 - 25*c*d^3*x^9 - 4*d^4*x^12) + d^4*x^12*(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)]))/(3440640*c^5*x^7*(8*c - d*x^3)*Sqrt[c +
d*x^3])
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.43, size = 3170, normalized size = 4.46
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| method |
result |
size |
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| elliptic |
\(\text {Expression too large to display}\) |
\(938\) |
| risch |
\(\text {Expression too large to display}\) |
\(1781\) |
| default |
\(\text {Expression too large to display}\) |
\(3170\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d*x^3+c)^(1/2)/x^8/(-d*x^3+8*c)^2,x,method=_RETURNVERBOSE)
[Out]
1/256/c^3*d*(-1/4*(d*x^3+c)^(1/2)/x^4-3/8*d*(d*x^3+c)^(1/2)/c/x-1/8*I/c*d*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))*El
lipticE(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))))+1/64/c^2*(-1/
7*(d*x^3+c)^(1/2)/x^7-3/56*d*(d*x^3+c)^(1/2)/x^4/c+15/112*d^2*(d*x^3+c)^(1/2)/c^2/x+5/112*I/c^2*d^2*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))))-3/4096*d^3/c^4*(-2/3*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)))+1/3*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)*3^(1/2)*_alpha^2*d-I*(-c*d^2)^(2/3)
*3^(1/2)*_alpha+I*3^(1/2)*c*d-3*(-c*d^2)^(2/3)*_alpha-3*c*d)/c,(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)),_alpha=RootOf(_Z^3*d-8*c)))+3/4096*d^2/c^4*(-(d*x^3+c)^(1/2)/x-I*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))))+1/512/c^3*d^3*(1/24*x^2*(d*x^3+c)^(1/2)/c/(-d*x^3+8*c)-1/72*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)/(...
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x^3+c)^(1/2)/x^8/(-d*x^3+8*c)^2,x, algorithm="maxima")
[Out]
integrate(sqrt(d*x^3 + c)/((d*x^3 - 8*c)^2*x^8), x)
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 9.37, size = 2738, normalized size = 3.85 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x^3+c)^(1/2)/x^8/(-d*x^3+8*c)^2,x, algorithm="fricas")
[Out]
-1/3096576*(364*sqrt(3)*(c^4*d*x^10 - 8*c^5*x^7)*(d^14/c^23)^(1/6)*arctan(1/9*((9*sqrt(3)*c^4*d^22*x^5*(d^14/c
^23)^(1/6) - sqrt(3)*(c^19*d^13*x^6 - 40*c^20*d^12*x^3 - 32*c^21*d^11)*(d^14/c^23)^(5/6) + 3*sqrt(3)*(5*c^12*d
^17*x^4 + 8*c^13*d^16*x)*sqrt(d^14/c^23))*sqrt(d*x^3 + c) + (18*sqrt(3)*(c^16*d^4*x^5 + c^17*d^3*x^2)*(d^14/c^
23)^(2/3) + 12*sqrt(3)*(c^8*d^9*x^6 - c^9*d^8*x^3 - 2*c^10*d^7)*(d^14/c^23)^(1/3) + 3*sqrt(3)*(d^14*x^7 + 5*c*
d^13*x^4 + 4*c^2*d^12*x) + sqrt(d*x^3 + c)*(sqrt(3)*(c^19*d^2*x^6 + 32*c^20*d*x^3 + 40*c^21)*(d^14/c^23)^(5/6)
+ 3*sqrt(3)*(7*c^12*d^6*x^4 + 4*c^13*d^5*x)*sqrt(d^14/c^23) + 9*sqrt(3)*(c^4*d^11*x^5 + 2*c^5*d^10*x^2)*(d^14
/c^23)^(1/6)))*sqrt((d^25*x^9 - 276*c*d^24*x^6 - 1608*c^2*d^23*x^3 - 1088*c^3*d^22 - 18*(c^16*d^15*x^7 - 52*c^
17*d^14*x^4 - 80*c^18*d^13*x)*(d^14/c^23)^(2/3) + 6*sqrt(d*x^3 + c)*(24*(c^20*d^12*x^5 + c^21*d^11*x^2)*(d^14/
c^23)^(5/6) - 4*(c^12*d^17*x^6 + 41*c^13*d^16*x^3 + 40*c^14*d^15)*sqrt(d^14/c^23) - (c^4*d^22*x^7 - 28*c^5*d^2
1*x^4 - 272*c^6*d^20*x)*(d^14/c^23)^(1/6)) + 18*(c^8*d^20*x^8 + 20*c^9*d^19*x^5 - 8*c^10*d^18*x^2)*(d^14/c^23)
^(1/3))/(d^3*x^9 - 24*c*d^2*x^6 + 192*c^2*d*x^3 - 512*c^3)))/(d^25*x^7 - 7*c*d^24*x^4 - 8*c^2*d^23*x)) + 364*s
qrt(3)*(c^4*d*x^10 - 8*c^5*x^7)*(d^14/c^23)^(1/6)*arctan(1/9*((9*sqrt(3)*c^4*d^22*x^5*(d^14/c^23)^(1/6) - sqrt
(3)*(c^19*d^13*x^6 - 40*c^20*d^12*x^3 - 32*c^21*d^11)*(d^14/c^23)^(5/6) + 3*sqrt(3)*(5*c^12*d^17*x^4 + 8*c^13*
d^16*x)*sqrt(d^14/c^23))*sqrt(d*x^3 + c) - (18*sqrt(3)*(c^16*d^4*x^5 + c^17*d^3*x^2)*(d^14/c^23)^(2/3) + 12*sq
rt(3)*(c^8*d^9*x^6 - c^9*d^8*x^3 - 2*c^10*d^7)*(d^14/c^23)^(1/3) + 3*sqrt(3)*(d^14*x^7 + 5*c*d^13*x^4 + 4*c^2*
d^12*x) - sqrt(d*x^3 + c)*(sqrt(3)*(c^19*d^2*x^6 + 32*c^20*d*x^3 + 40*c^21)*(d^14/c^23)^(5/6) + 3*sqrt(3)*(7*c
^12*d^6*x^4 + 4*c^13*d^5*x)*sqrt(d^14/c^23) + 9*sqrt(3)*(c^4*d^11*x^5 + 2*c^5*d^10*x^2)*(d^14/c^23)^(1/6)))*sq
rt((d^25*x^9 - 276*c*d^24*x^6 - 1608*c^2*d^23*x^3 - 1088*c^3*d^22 - 18*(c^16*d^15*x^7 - 52*c^17*d^14*x^4 - 80*
c^18*d^13*x)*(d^14/c^23)^(2/3) - 6*sqrt(d*x^3 + c)*(24*(c^20*d^12*x^5 + c^21*d^11*x^2)*(d^14/c^23)^(5/6) - 4*(
c^12*d^17*x^6 + 41*c^13*d^16*x^3 + 40*c^14*d^15)*sqrt(d^14/c^23) - (c^4*d^22*x^7 - 28*c^5*d^21*x^4 - 272*c^6*d
^20*x)*(d^14/c^23)^(1/6)) + 18*(c^8*d^20*x^8 + 20*c^9*d^19*x^5 - 8*c^10*d^18*x^2)*(d^14/c^23)^(1/3))/(d^3*x^9
- 24*c*d^2*x^6 + 192*c^2*d*x^3 - 512*c^3)))/(d^25*x^7 - 7*c*d^24*x^4 - 8*c^2*d^23*x)) + 576*(d^3*x^10 - 8*c*d^
2*x^7)*sqrt(d)*weierstrassZeta(0, -4*c/d, weierstrassPInverse(0, -4*c/d, x)) + 91*(c^4*d*x^10 - 8*c^5*x^7)*(d^
14/c^23)^(1/6)*log(137858491849*(d^25*x^9 - 276*c*d^24*x^6 - 1608*c^2*d^23*x^3 - 1088*c^3*d^22 - 18*(c^16*d^15
*x^7 - 52*c^17*d^14*x^4 - 80*c^18*d^13*x)*(d^14/c^23)^(2/3) + 6*sqrt(d*x^3 + c)*(24*(c^20*d^12*x^5 + c^21*d^11
*x^2)*(d^14/c^23)^(5/6) - 4*(c^12*d^17*x^6 + 41*c^13*d^16*x^3 + 40*c^14*d^15)*sqrt(d^14/c^23) - (c^4*d^22*x^7
- 28*c^5*d^21*x^4 - 272*c^6*d^20*x)*(d^14/c^23)^(1/6)) + 18*(c^8*d^20*x^8 + 20*c^9*d^19*x^5 - 8*c^10*d^18*x^2)
*(d^14/c^23)^(1/3))/(d^3*x^9 - 24*c*d^2*x^6 + 192*c^2*d*x^3 - 512*c^3)) - 91*(c^4*d*x^10 - 8*c^5*x^7)*(d^14/c^
23)^(1/6)*log(137858491849*(d^25*x^9 - 276*c*d^24*x^6 - 1608*c^2*d^23*x^3 - 1088*c^3*d^22 - 18*(c^16*d^15*x^7
- 52*c^17*d^14*x^4 - 80*c^18*d^13*x)*(d^14/c^23)^(2/3) - 6*sqrt(d*x^3 + c)*(24*(c^20*d^12*x^5 + c^21*d^11*x^2)
*(d^14/c^23)^(5/6) - 4*(c^12*d^17*x^6 + 41*c^13*d^16*x^3 + 40*c^14*d^15)*sqrt(d^14/c^23) - (c^4*d^22*x^7 - 28*
c^5*d^21*x^4 - 272*c^6*d^20*x)*(d^14/c^23)^(1/6)) + 18*(c^8*d^20*x^8 + 20*c^9*d^19*x^5 - 8*c^10*d^18*x^2)*(d^1
4/c^23)^(1/3))/(d^3*x^9 - 24*c*d^2*x^6 + 192*c^2*d*x^3 - 512*c^3)) - 182*(c^4*d*x^10 - 8*c^5*x^7)*(d^14/c^23)^
(1/6)*log(371293*(d^14*x^9 + 318*c*d^13*x^6 + 1200*c^2*d^12*x^3 + 640*c^3*d^11 + 18*(5*c^16*d^4*x^7 + 64*c^17*
d^3*x^4 + 32*c^18*d^2*x)*(d^14/c^23)^(2/3) + 6*sqrt(d*x^3 + c)*(6*(5*c^20*d*x^5 + 32*c^21*x^2)*(d^14/c^23)^(5/
6) + (7*c^12*d^6*x^6 + 152*c^13*d^5*x^3 + 64*c^14*d^4)*sqrt(d^14/c^23) + (c^4*d^11*x^7 + 80*c^5*d^10*x^4 + 160
*c^6*d^9*x)*(d^14/c^23)^(1/6)) + 18*(c^8*d^9*x^8 + 38*c^9*d^8*x^5 + 64*c^10*d^7*x^2)*(d^14/c^23)^(1/3))/(d^3*x
^9 - 24*c*d^2*x^6 + 192*c^2*d*x^3 - 512*c^3)) + 182*(c^4*d*x^10 - 8*c^5*x^7)*(d^14/c^23)^(1/6)*log(371293*(d^1
4*x^9 + 318*c*d^13*x^6 + 1200*c^2*d^12*x^3 + 640*c^3*d^11 + 18*(5*c^16*d^4*x^7 + 64*c^17*d^3*x^4 + 32*c^18*d^2
*x)*(d^14/c^23)^(2/3) - 6*sqrt(d*x^3 + c)*(6*(5*c^20*d*x^5 + 32*c^21*x^2)*(d^14/c^23)^(5/6) + (7*c^12*d^6*x^6
+ 152*c^13*d^5*x^3 + 64*c^14*d^4)*sqrt(d^14/c^23) + (c^4*d^11*x^7 + 80*c^5*d^10*x^4 + 160*c^6*d^9*x)*(d^14/c^2
3)^(1/6)) + 18*(c^8*d^9*x^8 + 38*c^9*d^8*x^5 + 64*c^10*d^7*x^2)*(d^14/c^23)^(1/3))/(d^3*x^9 - 24*c*d^2*x^6 + 1
92*c^2*d*x^3 - 512*c^3)) + 144*(4*d^3*x^9 + 21*c*d^2*x^6 - 264*c^2*d*x^3 - 384*c^3)*sqrt(d*x^3 + c))/(c^4*d*x^
10 - 8*c^5*x^7)
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {c + d x^{3}}}{x^{8} \left (- 8 c + d x^{3}\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x**3+c)**(1/2)/x**8/(-d*x**3+8*c)**2,x)
[Out]
Integral(sqrt(c + d*x**3)/(x**8*(-8*c + d*x**3)**2), x)
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x^3+c)^(1/2)/x^8/(-d*x^3+8*c)^2,x, algorithm="giac")
[Out]
integrate(sqrt(d*x^3 + c)/((d*x^3 - 8*c)^2*x^8), x)
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {d\,x^3+c}}{x^8\,{\left (8\,c-d\,x^3\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((c + d*x^3)^(1/2)/(x^8*(8*c - d*x^3)^2),x)
[Out]
int((c + d*x^3)^(1/2)/(x^8*(8*c - d*x^3)^2), x)
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