3.5.5 \(\int \frac {x^7 \sqrt {c+d x^3}}{(8 c-d x^3)^2} \, dx\) [405]

Optimal. Leaf size=663 \[ \frac {13 x^2 \sqrt {c+d x^3}}{21 d^2}+\frac {746 c \sqrt {c+d x^3}}{21 d^{8/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac {x^5 \sqrt {c+d x^3}}{3 d \left (8 c-d x^3\right )}+\frac {76 c^{7/6} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{3 \sqrt {3} d^{8/3}}-\frac {76 c^{7/6} \tanh ^{-1}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x^3}}\right )}{9 d^{8/3}}+\frac {76 c^{7/6} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{9 d^{8/3}}-\frac {373 \sqrt {2-\sqrt {3}} c^{4/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt {3}\right )}{7\ 3^{3/4} d^{8/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}+\frac {746 \sqrt {2} c^{4/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt {3}\right )}{21 \sqrt [4]{3} d^{8/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}} \]

[Out]

-76/9*c^(7/6)*arctanh(1/3*(c^(1/3)+d^(1/3)*x)^2/c^(1/6)/(d*x^3+c)^(1/2))/d^(8/3)+76/9*c^(7/6)*arctanh(1/3*(d*x
^3+c)^(1/2)/c^(1/2))/d^(8/3)+76/9*c^(7/6)*arctan(c^(1/6)*(c^(1/3)+d^(1/3)*x)*3^(1/2)/(d*x^3+c)^(1/2))/d^(8/3)*
3^(1/2)+13/21*x^2*(d*x^3+c)^(1/2)/d^2+1/3*x^5*(d*x^3+c)^(1/2)/d/(-d*x^3+8*c)+746/21*c*(d*x^3+c)^(1/2)/d^(8/3)/
(d^(1/3)*x+c^(1/3)*(1+3^(1/2)))+746/63*c^(4/3)*(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)*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)*3^(3/4)/d^(8/3)/(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)-373/21*3^(1/4)*c^(4/3)*(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)/d^(8/3)/(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 = 0.81, antiderivative size = 663, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 13, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.482, Rules used = {478, 596, 598, 309, 224, 1891, 499, 455, 65, 212, 2163, 2170, 211} \begin {gather*} \frac {746 \sqrt {2} 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}} 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 )}{21 \sqrt [4]{3} 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 {373 \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 (\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 )}{7\ 3^{3/4} 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 {76 c^{7/6} \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{3 \sqrt {3} d^{8/3}}-\frac {76 c^{7/6} \tanh ^{-1}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x^3}}\right )}{9 d^{8/3}}+\frac {76 c^{7/6} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{9 d^{8/3}}+\frac {746 c \sqrt {c+d x^3}}{21 d^{8/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac {13 x^2 \sqrt {c+d x^3}}{21 d^2}+\frac {x^5 \sqrt {c+d x^3}}{3 d \left (8 c-d x^3\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^7*Sqrt[c + d*x^3])/(8*c - d*x^3)^2,x]

[Out]

(13*x^2*Sqrt[c + d*x^3])/(21*d^2) + (746*c*Sqrt[c + d*x^3])/(21*d^(8/3)*((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)) +
(x^5*Sqrt[c + d*x^3])/(3*d*(8*c - d*x^3)) + (76*c^(7/6)*ArcTan[(Sqrt[3]*c^(1/6)*(c^(1/3) + d^(1/3)*x))/Sqrt[c
+ d*x^3]])/(3*Sqrt[3]*d^(8/3)) - (76*c^(7/6)*ArcTanh[(c^(1/3) + d^(1/3)*x)^2/(3*c^(1/6)*Sqrt[c + d*x^3])])/(9
*d^(8/3)) + (76*c^(7/6)*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(9*d^(8/3)) - (373*Sqrt[2 - Sqrt[3]]*c^(4/3)*(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
ipticE[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]])/(7*3^
(3/4)*d^(8/3)*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]) + (
746*Sqrt[2]*c^(4/3)*(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]])/(21*3^(1/4)*d^(8/3)*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 478

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[e^(n - 1
)*(e*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/(b*n*(p + 1))), x] - Dist[e^n/(b*n*(p + 1)), Int[(e*x)^
(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1)*Simp[c*(m - n + 1) + d*(m + n*(q - 1) + 1)*x^n, x], x], x] /;
FreeQ[{a, b, c, d, e}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[q, 0] && GtQ[m - n + 1, 0] &
& 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 596

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] - Dist[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]

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 {x^7 \sqrt {c+d x^3}}{\left (8 c-d x^3\right )^2} \, dx &=\frac {x^5 \sqrt {c+d x^3}}{3 d \left (8 c-d x^3\right )}-\frac {\int \frac {x^4 \left (5 c+\frac {13 d x^3}{2}\right )}{\left (8 c-d x^3\right ) \sqrt {c+d x^3}} \, dx}{3 d}\\ &=\frac {13 x^2 \sqrt {c+d x^3}}{21 d^2}+\frac {x^5 \sqrt {c+d x^3}}{3 d \left (8 c-d x^3\right )}-\frac {2 \int \frac {x \left (104 c^2 d+\frac {373}{2} c d^2 x^3\right )}{\left (8 c-d x^3\right ) \sqrt {c+d x^3}} \, dx}{21 d^3}\\ &=\frac {13 x^2 \sqrt {c+d x^3}}{21 d^2}+\frac {x^5 \sqrt {c+d x^3}}{3 d \left (8 c-d x^3\right )}-\frac {2 \int \left (-\frac {373 c d x}{2 \sqrt {c+d x^3}}+\frac {1596 c^2 d x}{\left (8 c-d x^3\right ) \sqrt {c+d x^3}}\right ) \, dx}{21 d^3}\\ &=\frac {13 x^2 \sqrt {c+d x^3}}{21 d^2}+\frac {x^5 \sqrt {c+d x^3}}{3 d \left (8 c-d x^3\right )}+\frac {(373 c) \int \frac {x}{\sqrt {c+d x^3}} \, dx}{21 d^2}-\frac {\left (152 c^2\right ) \int \frac {x}{\left (8 c-d x^3\right ) \sqrt {c+d x^3}} \, dx}{d^2}\\ &=\frac {13 x^2 \sqrt {c+d x^3}}{21 d^2}+\frac {x^5 \sqrt {c+d x^3}}{3 d \left (8 c-d x^3\right )}+\frac {(38 c) \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}{3 d^3}+\frac {(373 c) \int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\sqrt {c+d x^3}} \, dx}{21 d^{7/3}}-\frac {\left (38 c^{4/3}\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}{3 d^{7/3}}+\frac {\left (373 \sqrt {2 \left (2-\sqrt {3}\right )} c^{4/3}\right ) \int \frac {1}{\sqrt {c+d x^3}} \, dx}{21 d^{7/3}}+\frac {\left (38 c^{5/3}\right ) \int \frac {x^2}{\left (8 c-d x^3\right ) \sqrt {c+d x^3}} \, dx}{d^{5/3}}\\ &=\frac {13 x^2 \sqrt {c+d x^3}}{21 d^2}+\frac {746 c \sqrt {c+d x^3}}{21 d^{8/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac {x^5 \sqrt {c+d x^3}}{3 d \left (8 c-d x^3\right )}-\frac {373 \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 (\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 )}{7\ 3^{3/4} 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 {746 \sqrt {2} 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}} 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 )}{21 \sqrt [4]{3} 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 {\left (76 c^{5/3}\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 )}{3 d^{8/3}}+\frac {\left (38 c^{5/3}\right ) \text {Subst}\left (\int \frac {1}{(8 c-d x) \sqrt {c+d x}} \, dx,x,x^3\right )}{3 d^{5/3}}-\frac {\left (152 c^{2/3}\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 )}{3 d^{2/3}}\\ &=\frac {13 x^2 \sqrt {c+d x^3}}{21 d^2}+\frac {746 c \sqrt {c+d x^3}}{21 d^{8/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac {x^5 \sqrt {c+d x^3}}{3 d \left (8 c-d x^3\right )}+\frac {76 c^{7/6} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{3 \sqrt {3} d^{8/3}}-\frac {76 c^{7/6} \tanh ^{-1}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x^3}}\right )}{9 d^{8/3}}-\frac {373 \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 (\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 )}{7\ 3^{3/4} 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 {746 \sqrt {2} 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}} 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 )}{21 \sqrt [4]{3} 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 {\left (76 c^{5/3}\right ) \text {Subst}\left (\int \frac {1}{9 c-x^2} \, dx,x,\sqrt {c+d x^3}\right )}{3 d^{8/3}}\\ &=\frac {13 x^2 \sqrt {c+d x^3}}{21 d^2}+\frac {746 c \sqrt {c+d x^3}}{21 d^{8/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac {x^5 \sqrt {c+d x^3}}{3 d \left (8 c-d x^3\right )}+\frac {76 c^{7/6} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{3 \sqrt {3} d^{8/3}}-\frac {76 c^{7/6} \tanh ^{-1}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x^3}}\right )}{9 d^{8/3}}+\frac {76 c^{7/6} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{9 d^{8/3}}-\frac {373 \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 (\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 )}{7\ 3^{3/4} 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 {746 \sqrt {2} 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}} 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 )}{21 \sqrt [4]{3} 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}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
time = 5.37, size = 176, normalized size = 0.27 \begin {gather*} -\frac {80 \left (52 c^2 x^2+49 c d x^5-3 d^2 x^8\right )+520 c x^2 \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 )+373 d x^5 \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 )}{840 d^2 \left (-8 c+d x^3\right ) \sqrt {c+d x^3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^7*Sqrt[c + d*x^3])/(8*c - d*x^3)^2,x]

[Out]

-1/840*(80*(52*c^2*x^2 + 49*c*d*x^5 - 3*d^2*x^8) + 520*c*x^2*(-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)] + 373*d*x^5*(-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)])/(d^2*(-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 = 0.41, size = 2199, normalized size = 3.32

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(897\)
risch	\(\text {Expression too large to display}\)	\(1758\)
default	\(\text {Expression too large to display}\)	\(2199\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7*(d*x^3+c)^(1/2)/(-d*x^3+8*c)^2,x,method=_RETURNVERBOSE)

[Out]

1/d^2*(2/7*x^2*(d*x^3+c)^(1/2)-2/7*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))))+64*c^2/d^2*(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)/(-3/2/d*(-c*d^2)^(1/3)+1/
2*I*3^(1/2)/d*(-c*d^2)^(1/3)))^(1/2)))+1/216*I/d^3/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))*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)))+16/d^2*c*(-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))*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)))+1/3*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)))

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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(x^7*(d*x^3+c)^(1/2)/(-d*x^3+8*c)^2,x, algorithm="maxima")

[Out]

integrate(sqrt(d*x^3 + c)*x^7/(d*x^3 - 8*c)^2, x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 34.31, size = 3866, normalized size = 5.83 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(d*x^3+c)^(1/2)/(-d*x^3+8*c)^2,x, algorithm="fricas")

[Out]

-1/189*(532*sqrt(3)*(d^4*x^3 - 8*c*d^3)*(c^7/d^16)^(1/6)*arctan(-1/3*(324*sqrt(3)*(3*c^7*d^16*x^16 + 784*c^8*d
^15*x^13 + 7680*c^9*d^14*x^10 + 10752*c^10*d^13*x^7 + 4096*c^11*d^12*x^4)*(c^7/d^16)^(2/3) + 36*sqrt(3)*(c^9*d
^11*x^17 + 1772*c^10*d^10*x^14 + 42592*c^11*d^9*x^11 + 96256*c^12*d^8*x^8 + 69632*c^13*d^7*x^5 + 16384*c^14*d^
6*x^2)*(c^7/d^16)^(1/3) + sqrt(3)*(c^11*d^6*x^18 + 9456*c^12*d^5*x^15 + 749184*c^13*d^4*x^12 + 3017216*c^14*d^
3*x^9 + 3489792*c^15*d^2*x^6 + 1572864*c^16*d*x^3 + 262144*c^17) + 12*sqrt(d*x^3 + c)*(12*sqrt(3)*(35*c^6*d^18
*x^14 - 14440*c^7*d^17*x^11 - 24576*c^8*d^16*x^8 - 16384*c^9*d^15*x^5 - 4096*c^10*d^14*x^2)*(c^7/d^16)^(5/6) +
18*sqrt(3)*(c^8*d^13*x^15 - 1112*c^9*d^12*x^12 + 7296*c^10*d^11*x^9 + 11776*c^11*d^10*x^6 + 4096*c^12*d^9*x^3
)*sqrt(c^7/d^16) + sqrt(3)*(c^10*d^8*x^16 - 4768*c^11*d^7*x^13 + 362752*c^12*d^6*x^10 + 709120*c^13*d^5*x^7 +
413696*c^14*d^4*x^4 + 65536*c^15*d^3*x)*(c^7/d^16)^(1/6)) - 2*(324*sqrt(3)*(d^19*x^16 - 1858*c*d^18*x^13 - 417
6*c^2*d^17*x^10 - 3584*c^3*d^16*x^7 - 1024*c^4*d^15*x^4)*(c^7/d^16)^(5/6) + 18*sqrt(3)*(c^2*d^14*x^17 - 5290*c
^3*d^13*x^14 - 21152*c^4*d^12*x^11 - 47744*c^5*d^11*x^8 - 37888*c^6*d^10*x^5 - 8192*c^7*d^9*x^2)*sqrt(c^7/d^16
) + sqrt(3)*(c^4*d^9*x^18 - 7698*c^5*d^8*x^15 - 1664688*c^6*d^7*x^12 - 5524864*c^7*d^6*x^9 - 6223872*c^8*d^5*x
^6 - 2703360*c^9*d^4*x^3 - 327680*c^10*d^3)*(c^7/d^16)^(1/6) + 6*sqrt(d*x^3 + c)*(sqrt(3)*(7*c*d^16*x^15 + 373
52*c^2*d^15*x^12 - 230336*c^3*d^14*x^9 - 515072*c^4*d^13*x^6 - 286720*c^5*d^12*x^3 - 32768*c^6*d^11)*(c^7/d^16
)^(2/3) + 108*sqrt(3)*(53*c^4*d^10*x^13 + 1320*c^5*d^9*x^10 + 1536*c^6*d^8*x^7 + 512*c^7*d^7*x^4)*(c^7/d^16)^(
1/3) + 6*sqrt(3)*(37*c^6*d^5*x^14 + 28912*c^7*d^4*x^11 + 43584*c^8*d^3*x^8 + 20992*c^9*d^2*x^5 + 4096*c^10*d*x
^2)))*sqrt((18*c^12*d^2*x^8 + 360*c^13*d*x^5 - 144*c^14*x^2 + (c^7*d^13*x^9 - 276*c^8*d^12*x^6 - 1608*c^9*d^11
*x^3 - 1088*c^10*d^10)*(c^7/d^16)^(2/3) + 6*sqrt(d*x^3 + c)*((c^6*d^15*x^7 - 28*c^7*d^14*x^4 - 272*c^8*d^13*x)
*(c^7/d^16)^(5/6) - 24*(c^9*d^9*x^5 + c^10*d^8*x^2)*sqrt(c^7/d^16) + 4*(c^11*d^4*x^6 + 41*c^12*d^3*x^3 + 40*c^
13*d^2)*(c^7/d^16)^(1/6)) - 18*(c^10*d^7*x^7 - 52*c^11*d^6*x^4 - 80*c^12*d^5*x)*(c^7/d^16)^(1/3))/(d^3*x^9 - 2
4*c*d^2*x^6 + 192*c^2*d*x^3 - 512*c^3)))/(c^11*d^6*x^18 - 14952*c^12*d^5*x^15 + 2872896*c^13*d^4*x^12 + 733030
4*c^14*d^3*x^9 + 6696960*c^15*d^2*x^6 + 2457600*c^16*d*x^3 + 262144*c^17)) - 532*sqrt(3)*(d^4*x^3 - 8*c*d^3)*(
c^7/d^16)^(1/6)*arctan(-1/3*(324*sqrt(3)*(3*c^7*d^16*x^16 + 784*c^8*d^15*x^13 + 7680*c^9*d^14*x^10 + 10752*c^1
0*d^13*x^7 + 4096*c^11*d^12*x^4)*(c^7/d^16)^(2/3) + 36*sqrt(3)*(c^9*d^11*x^17 + 1772*c^10*d^10*x^14 + 42592*c^
11*d^9*x^11 + 96256*c^12*d^8*x^8 + 69632*c^13*d^7*x^5 + 16384*c^14*d^6*x^2)*(c^7/d^16)^(1/3) + sqrt(3)*(c^11*d
^6*x^18 + 9456*c^12*d^5*x^15 + 749184*c^13*d^4*x^12 + 3017216*c^14*d^3*x^9 + 3489792*c^15*d^2*x^6 + 1572864*c^
16*d*x^3 + 262144*c^17) - 12*sqrt(d*x^3 + c)*(12*sqrt(3)*(35*c^6*d^18*x^14 - 14440*c^7*d^17*x^11 - 24576*c^8*d
^16*x^8 - 16384*c^9*d^15*x^5 - 4096*c^10*d^14*x^2)*(c^7/d^16)^(5/6) + 18*sqrt(3)*(c^8*d^13*x^15 - 1112*c^9*d^1
2*x^12 + 7296*c^10*d^11*x^9 + 11776*c^11*d^10*x^6 + 4096*c^12*d^9*x^3)*sqrt(c^7/d^16) + sqrt(3)*(c^10*d^8*x^16
- 4768*c^11*d^7*x^13 + 362752*c^12*d^6*x^10 + 709120*c^13*d^5*x^7 + 413696*c^14*d^4*x^4 + 65536*c^15*d^3*x)*(
c^7/d^16)^(1/6)) + 2*(324*sqrt(3)*(d^19*x^16 - 1858*c*d^18*x^13 - 4176*c^2*d^17*x^10 - 3584*c^3*d^16*x^7 - 102
4*c^4*d^15*x^4)*(c^7/d^16)^(5/6) + 18*sqrt(3)*(c^2*d^14*x^17 - 5290*c^3*d^13*x^14 - 21152*c^4*d^12*x^11 - 4774
4*c^5*d^11*x^8 - 37888*c^6*d^10*x^5 - 8192*c^7*d^9*x^2)*sqrt(c^7/d^16) + sqrt(3)*(c^4*d^9*x^18 - 7698*c^5*d^8*
x^15 - 1664688*c^6*d^7*x^12 - 5524864*c^7*d^6*x^9 - 6223872*c^8*d^5*x^6 - 2703360*c^9*d^4*x^3 - 327680*c^10*d^
3)*(c^7/d^16)^(1/6) - 6*sqrt(d*x^3 + c)*(sqrt(3)*(7*c*d^16*x^15 + 37352*c^2*d^15*x^12 - 230336*c^3*d^14*x^9 -
515072*c^4*d^13*x^6 - 286720*c^5*d^12*x^3 - 32768*c^6*d^11)*(c^7/d^16)^(2/3) + 108*sqrt(3)*(53*c^4*d^10*x^13 +
1320*c^5*d^9*x^10 + 1536*c^6*d^8*x^7 + 512*c^7*d^7*x^4)*(c^7/d^16)^(1/3) + 6*sqrt(3)*(37*c^6*d^5*x^14 + 28912
*c^7*d^4*x^11 + 43584*c^8*d^3*x^8 + 20992*c^9*d^2*x^5 + 4096*c^10*d*x^2)))*sqrt((18*c^12*d^2*x^8 + 360*c^13*d*
x^5 - 144*c^14*x^2 + (c^7*d^13*x^9 - 276*c^8*d^12*x^6 - 1608*c^9*d^11*x^3 - 1088*c^10*d^10)*(c^7/d^16)^(2/3) -
6*sqrt(d*x^3 + c)*((c^6*d^15*x^7 - 28*c^7*d^14*x^4 - 272*c^8*d^13*x)*(c^7/d^16)^(5/6) - 24*(c^9*d^9*x^5 + c^1
0*d^8*x^2)*sqrt(c^7/d^16) + 4*(c^11*d^4*x^6 + 41*c^12*d^3*x^3 + 40*c^13*d^2)*(c^7/d^16)^(1/6)) - 18*(c^10*d^7*
x^7 - 52*c^11*d^6*x^4 - 80*c^12*d^5*x)*(c^7/d^16)^(1/3))/(d^3*x^9 - 24*c*d^2*x^6 + 192*c^2*d*x^3 - 512*c^3)))/
(c^11*d^6*x^18 - 14952*c^12*d^5*x^15 + 2872896*c^13*d^4*x^12 + 7330304*c^14*d^3*x^9 + 6696960*c^15*d^2*x^6 + 2
457600*c^16*d*x^3 + 262144*c^17)) + 6714*(c*d*x^3 - 8*c^2)*sqrt(d)*weierstrassZeta(0, -4*c/d, weierstrassPInve
rse(0, -4*c/d, x)) + 133*(d^4*x^3 - 8*c*d^3)*(c^7/d^16)^(1/6)*log(25715555729359765504/9*(18*c^12*d^2*x^8 + 36
0*c^13*d*x^5 - 144*c^14*x^2 + (c^7*d^13*x^9 - 2...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{7} \sqrt {c + d x^{3}}}{\left (- 8 c + d x^{3}\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7*(d*x**3+c)**(1/2)/(-d*x**3+8*c)**2,x)

[Out]

Integral(x**7*sqrt(c + d*x**3)/(-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(x^7*(d*x^3+c)^(1/2)/(-d*x^3+8*c)^2,x, algorithm="giac")

[Out]

integrate(sqrt(d*x^3 + c)*x^7/(d*x^3 - 8*c)^2, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^7\,\sqrt {d\,x^3+c}}{{\left (8\,c-d\,x^3\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^7*(c + d*x^3)^(1/2))/(8*c - d*x^3)^2,x)

[Out]

int((x^7*(c + d*x^3)^(1/2))/(8*c - d*x^3)^2, x)

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