3.5.6 \(\int \frac {x^4 \sqrt {c+d x^3}}{(8 c-d x^3)^2} \, dx\) [406]
Optimal. Leaf size=641 \[ \frac {7 \sqrt {c+d x^3}}{3 d^{5/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac {x^2 \sqrt {c+d x^3}}{3 d \left (8 c-d x^3\right )}+\frac {5 \sqrt [6]{c} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{3 \sqrt {3} d^{5/3}}-\frac {5 \sqrt [6]{c} \tanh ^{-1}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x^3}}\right )}{9 d^{5/3}}+\frac {5 \sqrt [6]{c} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{9 d^{5/3}}-\frac {7 \sqrt {2-\sqrt {3}} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt {3}\right )}{2\ 3^{3/4} d^{5/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}+\frac {7 \sqrt {2} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} d^{5/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}} \]
[Out]
-5/9*c^(1/6)*arctanh(1/3*(c^(1/3)+d^(1/3)*x)^2/c^(1/6)/(d*x^3+c)^(1/2))/d^(5/3)+5/9*c^(1/6)*arctanh(1/3*(d*x^3
+c)^(1/2)/c^(1/2))/d^(5/3)+5/9*c^(1/6)*arctan(c^(1/6)*(c^(1/3)+d^(1/3)*x)*3^(1/2)/(d*x^3+c)^(1/2))/d^(5/3)*3^(
1/2)+1/3*x^2*(d*x^3+c)^(1/2)/d/(-d*x^3+8*c)+7/3*(d*x^3+c)^(1/2)/d^(5/3)/(d^(1/3)*x+c^(1/3)*(1+3^(1/2)))+7/9*c^
(1/3)*(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^(5/3)
/(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)-7/6*3^(1/4)*c^(1/3)*(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^(5/3
)/(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 = 0.50, antiderivative size = 641, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 12, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {478, 598,
309, 224, 1891, 499, 455, 65, 212, 2163, 2170, 211} \begin {gather*} \frac {7 \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}} 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 )}{3 \sqrt [4]{3} d^{5/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {7 \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 (\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 )}{2\ 3^{3/4} d^{5/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}+\frac {5 \sqrt [6]{c} \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^{5/3}}+\frac {7 \sqrt {c+d x^3}}{3 d^{5/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}-\frac {5 \sqrt [6]{c} \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^{5/3}}+\frac {5 \sqrt [6]{c} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{9 d^{5/3}}+\frac {x^2 \sqrt {c+d x^3}}{3 d \left (8 c-d x^3\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(x^4*Sqrt[c + d*x^3])/(8*c - d*x^3)^2,x]
[Out]
(7*Sqrt[c + d*x^3])/(3*d^(5/3)*((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)) + (x^2*Sqrt[c + d*x^3])/(3*d*(8*c - d*x^3)
) + (5*c^(1/6)*ArcTan[(Sqrt[3]*c^(1/6)*(c^(1/3) + d^(1/3)*x))/Sqrt[c + d*x^3]])/(3*Sqrt[3]*d^(5/3)) - (5*c^(1/
6)*ArcTanh[(c^(1/3) + d^(1/3)*x)^2/(3*c^(1/6)*Sqrt[c + d*x^3])])/(9*d^(5/3)) + (5*c^(1/6)*ArcTanh[Sqrt[c + d*x
^3]/(3*Sqrt[c])])/(9*d^(5/3)) - (7*Sqrt[2 - Sqrt[3]]*c^(1/3)*(c^(1/3) + d^(1/3)*x)*Sqrt[(c^(2/3) - c^(1/3)*d^(
1/3)*x + d^(2/3)*x^2)/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)^2]*EllipticE[ArcSin[((1 - Sqrt[3])*c^(1/3) + d^(1/3)
*x)/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)], -7 - 4*Sqrt[3]])/(2*3^(3/4)*d^(5/3)*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*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*3^(1/4)*d^(5/3)*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 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^4 \sqrt {c+d x^3}}{\left (8 c-d x^3\right )^2} \, dx &=\frac {x^2 \sqrt {c+d x^3}}{3 d \left (8 c-d x^3\right )}-\frac {\int \frac {x \left (2 c+\frac {7 d x^3}{2}\right )}{\left (8 c-d x^3\right ) \sqrt {c+d x^3}} \, dx}{3 d}\\ &=\frac {x^2 \sqrt {c+d x^3}}{3 d \left (8 c-d x^3\right )}-\frac {\int \left (-\frac {7 x}{2 \sqrt {c+d x^3}}+\frac {30 c x}{\left (8 c-d x^3\right ) \sqrt {c+d x^3}}\right ) \, dx}{3 d}\\ &=\frac {x^2 \sqrt {c+d x^3}}{3 d \left (8 c-d x^3\right )}+\frac {7 \int \frac {x}{\sqrt {c+d x^3}} \, dx}{6 d}-\frac {(10 c) \int \frac {x}{\left (8 c-d x^3\right ) \sqrt {c+d x^3}} \, dx}{d}\\ &=\frac {x^2 \sqrt {c+d x^3}}{3 d \left (8 c-d x^3\right )}+\frac {5 \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}{6 d^2}+\frac {7 \int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\sqrt {c+d x^3}} \, dx}{6 d^{4/3}}-\frac {\left (5 \sqrt [3]{c}\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}{6 d^{4/3}}+\frac {\left (7 \sqrt {2-\sqrt {3}} \sqrt [3]{c}\right ) \int \frac {1}{\sqrt {c+d x^3}} \, dx}{3 \sqrt {2} d^{4/3}}+\frac {\left (5 c^{2/3}\right ) \int \frac {x^2}{\left (8 c-d x^3\right ) \sqrt {c+d x^3}} \, dx}{2 d^{2/3}}\\ &=\frac {7 \sqrt {c+d x^3}}{3 d^{5/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac {x^2 \sqrt {c+d x^3}}{3 d \left (8 c-d x^3\right )}-\frac {7 \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 (\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 )}{2\ 3^{3/4} d^{5/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}+\frac {7 \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}} 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 )}{3 \sqrt [4]{3} d^{5/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {\left (5 c^{2/3}\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^{5/3}}+\frac {\left (5 c^{2/3}\right ) \text {Subst}\left (\int \frac {1}{(8 c-d x) \sqrt {c+d x}} \, dx,x,x^3\right )}{6 d^{2/3}}-\frac {\left (10 \sqrt [3]{d}\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 \sqrt [3]{c}}\\ &=\frac {7 \sqrt {c+d x^3}}{3 d^{5/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac {x^2 \sqrt {c+d x^3}}{3 d \left (8 c-d x^3\right )}+\frac {5 \sqrt [6]{c} \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^{5/3}}-\frac {5 \sqrt [6]{c} \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^{5/3}}-\frac {7 \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 (\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 )}{2\ 3^{3/4} d^{5/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}+\frac {7 \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}} 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 )}{3 \sqrt [4]{3} d^{5/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}+\frac {\left (5 c^{2/3}\right ) \text {Subst}\left (\int \frac {1}{9 c-x^2} \, dx,x,\sqrt {c+d x^3}\right )}{3 d^{5/3}}\\ &=\frac {7 \sqrt {c+d x^3}}{3 d^{5/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac {x^2 \sqrt {c+d x^3}}{3 d \left (8 c-d x^3\right )}+\frac {5 \sqrt [6]{c} \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^{5/3}}-\frac {5 \sqrt [6]{c} \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^{5/3}}+\frac {5 \sqrt [6]{c} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{9 d^{5/3}}-\frac {7 \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 (\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 )}{2\ 3^{3/4} d^{5/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}+\frac {7 \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}} 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 )}{3 \sqrt [4]{3} d^{5/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 4 in
optimal.
time = 8.29, size = 167, normalized size = 0.26 \begin {gather*} \frac {80 c x^2 \left (c+d x^3\right )+10 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 )+7 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 )}{240 c d \left (8 c-d x^3\right ) \sqrt {c+d x^3}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(x^4*Sqrt[c + d*x^3])/(8*c - d*x^3)^2,x]
[Out]
(80*c*x^2*(c + d*x^3) + 10*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)] + 7*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)])/(240*c*d*(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 = 0.36, size = 1741, normalized size = 2.72
| | |
| method |
result |
size |
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| elliptic |
\(\text {Expression too large to display}\) |
\(877\) |
| default |
\(\text {Expression too large to display}\) |
\(1741\) |
| | |
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|
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^4*(d*x^3+c)^(1/2)/(-d*x^3+8*c)^2,x,method=_RETURNVERBOSE)
[Out]
8*c/d*(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/_a
lpha*(-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)*_al
pha^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)))+1/d*(-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)))
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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^4*(d*x^3+c)^(1/2)/(-d*x^3+8*c)^2,x, algorithm="maxima")
[Out]
integrate(sqrt(d*x^3 + c)*x^4/(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 = 21.50, size = 3633, normalized size = 5.67 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^4*(d*x^3+c)^(1/2)/(-d*x^3+8*c)^2,x, algorithm="fricas")
[Out]
-1/108*(36*sqrt(d*x^3 + c)*d*x^2 + 20*sqrt(3)*(d^3*x^3 - 8*c*d^2)*(c/d^10)^(1/6)*arctan(-1/3*(324*sqrt(3)*(3*c
*d^12*x^16 + 784*c^2*d^11*x^13 + 7680*c^3*d^10*x^10 + 10752*c^4*d^9*x^7 + 4096*c^5*d^8*x^4)*(c/d^10)^(2/3) + 3
6*sqrt(3)*(c*d^9*x^17 + 1772*c^2*d^8*x^14 + 42592*c^3*d^7*x^11 + 96256*c^4*d^6*x^8 + 69632*c^5*d^5*x^5 + 16384
*c^6*d^4*x^2)*(c/d^10)^(1/3) + sqrt(3)*(c*d^6*x^18 + 9456*c^2*d^5*x^15 + 749184*c^3*d^4*x^12 + 3017216*c^4*d^3
*x^9 + 3489792*c^5*d^2*x^6 + 1572864*c^6*d*x^3 + 262144*c^7) + 12*sqrt(d*x^3 + c)*(12*sqrt(3)*(35*c*d^13*x^14
- 14440*c^2*d^12*x^11 - 24576*c^3*d^11*x^8 - 16384*c^4*d^10*x^5 - 4096*c^5*d^9*x^2)*(c/d^10)^(5/6) + 18*sqrt(3
)*(c*d^10*x^15 - 1112*c^2*d^9*x^12 + 7296*c^3*d^8*x^9 + 11776*c^4*d^7*x^6 + 4096*c^5*d^6*x^3)*sqrt(c/d^10) + s
qrt(3)*(c*d^7*x^16 - 4768*c^2*d^6*x^13 + 362752*c^3*d^5*x^10 + 709120*c^4*d^4*x^7 + 413696*c^5*d^3*x^4 + 65536
*c^6*d^2*x)*(c/d^10)^(1/6)) - 2*(324*sqrt(3)*(d^14*x^16 - 1858*c*d^13*x^13 - 4176*c^2*d^12*x^10 - 3584*c^3*d^1
1*x^7 - 1024*c^4*d^10*x^4)*(c/d^10)^(5/6) + 18*sqrt(3)*(d^11*x^17 - 5290*c*d^10*x^14 - 21152*c^2*d^9*x^11 - 47
744*c^3*d^8*x^8 - 37888*c^4*d^7*x^5 - 8192*c^5*d^6*x^2)*sqrt(c/d^10) + sqrt(3)*(d^8*x^18 - 7698*c*d^7*x^15 - 1
664688*c^2*d^6*x^12 - 5524864*c^3*d^5*x^9 - 6223872*c^4*d^4*x^6 - 2703360*c^5*d^3*x^3 - 327680*c^6*d^2)*(c/d^1
0)^(1/6) + 6*sqrt(d*x^3 + c)*(sqrt(3)*(7*d^12*x^15 + 37352*c*d^11*x^12 - 230336*c^2*d^10*x^9 - 515072*c^3*d^9*
x^6 - 286720*c^4*d^8*x^3 - 32768*c^5*d^7)*(c/d^10)^(2/3) + 108*sqrt(3)*(53*c*d^8*x^13 + 1320*c^2*d^7*x^10 + 15
36*c^3*d^6*x^7 + 512*c^4*d^5*x^4)*(c/d^10)^(1/3) + 6*sqrt(3)*(37*c*d^5*x^14 + 28912*c^2*d^4*x^11 + 43584*c^3*d
^3*x^8 + 20992*c^4*d^2*x^5 + 4096*c^5*d*x^2)))*sqrt((18*c^2*d^2*x^8 + 360*c^3*d*x^5 - 144*c^4*x^2 + (c*d^9*x^9
- 276*c^2*d^8*x^6 - 1608*c^3*d^7*x^3 - 1088*c^4*d^6)*(c/d^10)^(2/3) + 6*sqrt(d*x^3 + c)*((c*d^10*x^7 - 28*c^2
*d^9*x^4 - 272*c^3*d^8*x)*(c/d^10)^(5/6) - 24*(c^2*d^6*x^5 + c^3*d^5*x^2)*sqrt(c/d^10) + 4*(c^2*d^3*x^6 + 41*c
^3*d^2*x^3 + 40*c^4*d)*(c/d^10)^(1/6)) - 18*(c^2*d^5*x^7 - 52*c^3*d^4*x^4 - 80*c^4*d^3*x)*(c/d^10)^(1/3))/(d^3
*x^9 - 24*c*d^2*x^6 + 192*c^2*d*x^3 - 512*c^3)))/(c*d^6*x^18 - 14952*c^2*d^5*x^15 + 2872896*c^3*d^4*x^12 + 733
0304*c^4*d^3*x^9 + 6696960*c^5*d^2*x^6 + 2457600*c^6*d*x^3 + 262144*c^7)) - 20*sqrt(3)*(d^3*x^3 - 8*c*d^2)*(c/
d^10)^(1/6)*arctan(-1/3*(324*sqrt(3)*(3*c*d^12*x^16 + 784*c^2*d^11*x^13 + 7680*c^3*d^10*x^10 + 10752*c^4*d^9*x
^7 + 4096*c^5*d^8*x^4)*(c/d^10)^(2/3) + 36*sqrt(3)*(c*d^9*x^17 + 1772*c^2*d^8*x^14 + 42592*c^3*d^7*x^11 + 9625
6*c^4*d^6*x^8 + 69632*c^5*d^5*x^5 + 16384*c^6*d^4*x^2)*(c/d^10)^(1/3) + sqrt(3)*(c*d^6*x^18 + 9456*c^2*d^5*x^1
5 + 749184*c^3*d^4*x^12 + 3017216*c^4*d^3*x^9 + 3489792*c^5*d^2*x^6 + 1572864*c^6*d*x^3 + 262144*c^7) - 12*sqr
t(d*x^3 + c)*(12*sqrt(3)*(35*c*d^13*x^14 - 14440*c^2*d^12*x^11 - 24576*c^3*d^11*x^8 - 16384*c^4*d^10*x^5 - 409
6*c^5*d^9*x^2)*(c/d^10)^(5/6) + 18*sqrt(3)*(c*d^10*x^15 - 1112*c^2*d^9*x^12 + 7296*c^3*d^8*x^9 + 11776*c^4*d^7
*x^6 + 4096*c^5*d^6*x^3)*sqrt(c/d^10) + sqrt(3)*(c*d^7*x^16 - 4768*c^2*d^6*x^13 + 362752*c^3*d^5*x^10 + 709120
*c^4*d^4*x^7 + 413696*c^5*d^3*x^4 + 65536*c^6*d^2*x)*(c/d^10)^(1/6)) + 2*(324*sqrt(3)*(d^14*x^16 - 1858*c*d^13
*x^13 - 4176*c^2*d^12*x^10 - 3584*c^3*d^11*x^7 - 1024*c^4*d^10*x^4)*(c/d^10)^(5/6) + 18*sqrt(3)*(d^11*x^17 - 5
290*c*d^10*x^14 - 21152*c^2*d^9*x^11 - 47744*c^3*d^8*x^8 - 37888*c^4*d^7*x^5 - 8192*c^5*d^6*x^2)*sqrt(c/d^10)
+ sqrt(3)*(d^8*x^18 - 7698*c*d^7*x^15 - 1664688*c^2*d^6*x^12 - 5524864*c^3*d^5*x^9 - 6223872*c^4*d^4*x^6 - 270
3360*c^5*d^3*x^3 - 327680*c^6*d^2)*(c/d^10)^(1/6) - 6*sqrt(d*x^3 + c)*(sqrt(3)*(7*d^12*x^15 + 37352*c*d^11*x^1
2 - 230336*c^2*d^10*x^9 - 515072*c^3*d^9*x^6 - 286720*c^4*d^8*x^3 - 32768*c^5*d^7)*(c/d^10)^(2/3) + 108*sqrt(3
)*(53*c*d^8*x^13 + 1320*c^2*d^7*x^10 + 1536*c^3*d^6*x^7 + 512*c^4*d^5*x^4)*(c/d^10)^(1/3) + 6*sqrt(3)*(37*c*d^
5*x^14 + 28912*c^2*d^4*x^11 + 43584*c^3*d^3*x^8 + 20992*c^4*d^2*x^5 + 4096*c^5*d*x^2)))*sqrt((18*c^2*d^2*x^8 +
360*c^3*d*x^5 - 144*c^4*x^2 + (c*d^9*x^9 - 276*c^2*d^8*x^6 - 1608*c^3*d^7*x^3 - 1088*c^4*d^6)*(c/d^10)^(2/3)
- 6*sqrt(d*x^3 + c)*((c*d^10*x^7 - 28*c^2*d^9*x^4 - 272*c^3*d^8*x)*(c/d^10)^(5/6) - 24*(c^2*d^6*x^5 + c^3*d^5*
x^2)*sqrt(c/d^10) + 4*(c^2*d^3*x^6 + 41*c^3*d^2*x^3 + 40*c^4*d)*(c/d^10)^(1/6)) - 18*(c^2*d^5*x^7 - 52*c^3*d^4
*x^4 - 80*c^4*d^3*x)*(c/d^10)^(1/3))/(d^3*x^9 - 24*c*d^2*x^6 + 192*c^2*d*x^3 - 512*c^3)))/(c*d^6*x^18 - 14952*
c^2*d^5*x^15 + 2872896*c^3*d^4*x^12 + 7330304*c^4*d^3*x^9 + 6696960*c^5*d^2*x^6 + 2457600*c^6*d*x^3 + 262144*c
^7)) + 252*(d*x^3 - 8*c)*sqrt(d)*weierstrassZeta(0, -4*c/d, weierstrassPInverse(0, -4*c/d, x)) + 5*(d^3*x^3 -
8*c*d^2)*(c/d^10)^(1/6)*log(39062500/9*(18*c^2*d^2*x^8 + 360*c^3*d*x^5 - 144*c^4*x^2 + (c*d^9*x^9 - 276*c^2*d^
8*x^6 - 1608*c^3*d^7*x^3 - 1088*c^4*d^6)*(c/d^10)^(2/3) + 6*sqrt(d*x^3 + c)*((c*d^10*x^7 - 28*c^2*d^9*x^4 - 27
2*c^3*d^8*x)*(c/d^10)^(5/6) - 24*(c^2*d^6*x^5 + c^3*d^5*x^2)*sqrt(c/d^10) + 4*(c^2*d^3*x^6 + 41*c^3*d^2*x^3 +
40*c^4*d)*(c/d^10)^(1/6)) - 18*(c^2*d^5*x^7 - 5...
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} \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**4*(d*x**3+c)**(1/2)/(-d*x**3+8*c)**2,x)
[Out]
Integral(x**4*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^4*(d*x^3+c)^(1/2)/(-d*x^3+8*c)^2,x, algorithm="giac")
[Out]
integrate(sqrt(d*x^3 + c)*x^4/(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^4\,\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^4*(c + d*x^3)^(1/2))/(8*c - d*x^3)^2,x)
[Out]
int((x^4*(c + d*x^3)^(1/2))/(8*c - d*x^3)^2, x)
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