3.5.18 \(\int \frac {x^7 (c+d x^3)^{3/2}}{(8 c-d x^3)^2} \, dx\) [418]
Optimal. Leaf size=681 \[ \frac {103 c x^2 \sqrt {c+d x^3}}{13 d^2}+\frac {19 x^5 \sqrt {c+d x^3}}{39 d}+\frac {5906 c^2 \sqrt {c+d x^3}}{13 d^{8/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac {x^5 \left (c+d x^3\right )^{3/2}}{3 d \left (8 c-d x^3\right )}+\frac {108 \sqrt {3} c^{13/6} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{d^{8/3}}-\frac {108 c^{13/6} \tanh ^{-1}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x^3}}\right )}{d^{8/3}}+\frac {108 c^{13/6} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{d^{8/3}}-\frac {2953 \sqrt [4]{3} \sqrt {2-\sqrt {3}} c^{7/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt {3}\right )}{13 d^{8/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}+\frac {5906 \sqrt {2} c^{7/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt {3}\right )}{13 \sqrt [4]{3} d^{8/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}} \]
[Out]
1/3*x^5*(d*x^3+c)^(3/2)/d/(-d*x^3+8*c)-108*c^(13/6)*arctanh(1/3*(c^(1/3)+d^(1/3)*x)^2/c^(1/6)/(d*x^3+c)^(1/2))
/d^(8/3)+108*c^(13/6)*arctanh(1/3*(d*x^3+c)^(1/2)/c^(1/2))/d^(8/3)+108*c^(13/6)*arctan(c^(1/6)*(c^(1/3)+d^(1/3
)*x)*3^(1/2)/(d*x^3+c)^(1/2))*3^(1/2)/d^(8/3)+103/13*c*x^2*(d*x^3+c)^(1/2)/d^2+19/39*x^5*(d*x^3+c)^(1/2)/d+590
6/13*c^2*(d*x^3+c)^(1/2)/d^(8/3)/(d^(1/3)*x+c^(1/3)*(1+3^(1/2)))+5906/39*c^(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)*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)-2953/13*3^(1/4)*c^(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)/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 = 0.64, antiderivative size = 681, normalized size of antiderivative = 1.00, number of steps
used = 16, number of rules used = 14, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.518, Rules used = {478, 595,
596, 598, 309, 224, 1891, 499, 455, 65, 212, 2163, 2170, 211} \begin {gather*} \frac {5906 \sqrt {2} c^{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 )}{13 \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 {2953 \sqrt [4]{3} \sqrt {2-\sqrt {3}} c^{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 )}{13 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 {108 \sqrt {3} c^{13/6} \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{d^{8/3}}-\frac {108 c^{13/6} \tanh ^{-1}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x^3}}\right )}{d^{8/3}}+\frac {108 c^{13/6} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{d^{8/3}}+\frac {5906 c^2 \sqrt {c+d x^3}}{13 d^{8/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac {103 c x^2 \sqrt {c+d x^3}}{13 d^2}+\frac {x^5 \left (c+d x^3\right )^{3/2}}{3 d \left (8 c-d x^3\right )}+\frac {19 x^5 \sqrt {c+d x^3}}{39 d} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(x^7*(c + d*x^3)^(3/2))/(8*c - d*x^3)^2,x]
[Out]
(103*c*x^2*Sqrt[c + d*x^3])/(13*d^2) + (19*x^5*Sqrt[c + d*x^3])/(39*d) + (5906*c^2*Sqrt[c + d*x^3])/(13*d^(8/3
)*((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)) + (x^5*(c + d*x^3)^(3/2))/(3*d*(8*c - d*x^3)) + (108*Sqrt[3]*c^(13/6)*A
rcTan[(Sqrt[3]*c^(1/6)*(c^(1/3) + d^(1/3)*x))/Sqrt[c + d*x^3]])/d^(8/3) - (108*c^(13/6)*ArcTanh[(c^(1/3) + d^(
1/3)*x)^2/(3*c^(1/6)*Sqrt[c + d*x^3])])/d^(8/3) + (108*c^(13/6)*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/d^(8/3)
- (2953*3^(1/4)*Sqrt[2 - Sqrt[3]]*c^(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]*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]])/(13*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]) + (5906*Sqrt[2]*c^(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]])/(13*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 595
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[f*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/(b*g*(m + n*(p + q + 1) + 1))), x] + Dis
t[1/(b*(m + n*(p + q + 1) + 1)), Int[(g*x)^m*(a + b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*((b*e - a*f)*(m + 1) + b
*e*n*(p + q + 1)) + (d*(b*e - a*f)*(m + 1) + f*n*q*(b*c - a*d) + b*e*d*n*(p + q + 1))*x^n, x], x], x] /; FreeQ
[{a, b, c, d, e, f, g, m, p}, x] && IGtQ[n, 0] && GtQ[q, 0] && !(EqQ[q, 1] && SimplerQ[e + f*x^n, c + d*x^n])
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 \left (c+d x^3\right )^{3/2}}{\left (8 c-d x^3\right )^2} \, dx &=\frac {x^5 \left (c+d x^3\right )^{3/2}}{3 d \left (8 c-d x^3\right )}-\frac {\int \frac {x^4 \sqrt {c+d x^3} \left (5 c+\frac {19 d x^3}{2}\right )}{8 c-d x^3} \, dx}{3 d}\\ &=\frac {19 x^5 \sqrt {c+d x^3}}{39 d}+\frac {x^5 \left (c+d x^3\right )^{3/2}}{3 d \left (8 c-d x^3\right )}+\frac {2 \int \frac {x^4 \left (-\frac {825 c^2 d}{2}-\frac {2163}{4} c d^2 x^3\right )}{\left (8 c-d x^3\right ) \sqrt {c+d x^3}} \, dx}{39 d^2}\\ &=\frac {103 c x^2 \sqrt {c+d x^3}}{13 d^2}+\frac {19 x^5 \sqrt {c+d x^3}}{39 d}+\frac {x^5 \left (c+d x^3\right )^{3/2}}{3 d \left (8 c-d x^3\right )}+\frac {4 \int \frac {x \left (-8652 c^3 d^2-\frac {62013}{4} c^2 d^3 x^3\right )}{\left (8 c-d x^3\right ) \sqrt {c+d x^3}} \, dx}{273 d^4}\\ &=\frac {103 c x^2 \sqrt {c+d x^3}}{13 d^2}+\frac {19 x^5 \sqrt {c+d x^3}}{39 d}+\frac {x^5 \left (c+d x^3\right )^{3/2}}{3 d \left (8 c-d x^3\right )}+\frac {4 \int \left (\frac {62013 c^2 d^2 x}{4 \sqrt {c+d x^3}}-\frac {132678 c^3 d^2 x}{\left (8 c-d x^3\right ) \sqrt {c+d x^3}}\right ) \, dx}{273 d^4}\\ &=\frac {103 c x^2 \sqrt {c+d x^3}}{13 d^2}+\frac {19 x^5 \sqrt {c+d x^3}}{39 d}+\frac {x^5 \left (c+d x^3\right )^{3/2}}{3 d \left (8 c-d x^3\right )}+\frac {\left (2953 c^2\right ) \int \frac {x}{\sqrt {c+d x^3}} \, dx}{13 d^2}-\frac {\left (1944 c^3\right ) \int \frac {x}{\left (8 c-d x^3\right ) \sqrt {c+d x^3}} \, dx}{d^2}\\ &=\frac {103 c x^2 \sqrt {c+d x^3}}{13 d^2}+\frac {19 x^5 \sqrt {c+d x^3}}{39 d}+\frac {x^5 \left (c+d x^3\right )^{3/2}}{3 d \left (8 c-d x^3\right )}+\frac {\left (162 c^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}{d^3}+\frac {\left (2953 c^2\right ) \int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\sqrt {c+d x^3}} \, dx}{13 d^{7/3}}-\frac {\left (162 c^{7/3}\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}{d^{7/3}}+\frac {\left (2953 \sqrt {2 \left (2-\sqrt {3}\right )} c^{7/3}\right ) \int \frac {1}{\sqrt {c+d x^3}} \, dx}{13 d^{7/3}}+\frac {\left (486 c^{8/3}\right ) \int \frac {x^2}{\left (8 c-d x^3\right ) \sqrt {c+d x^3}} \, dx}{d^{5/3}}\\ &=\frac {103 c x^2 \sqrt {c+d x^3}}{13 d^2}+\frac {19 x^5 \sqrt {c+d x^3}}{39 d}+\frac {5906 c^2 \sqrt {c+d x^3}}{13 d^{8/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac {x^5 \left (c+d x^3\right )^{3/2}}{3 d \left (8 c-d x^3\right )}-\frac {2953 \sqrt [4]{3} \sqrt {2-\sqrt {3}} c^{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 )}{13 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 {5906 \sqrt {2} c^{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 )}{13 \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 (324 c^{8/3}\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 )}{d^{8/3}}+\frac {\left (162 c^{8/3}\right ) \text {Subst}\left (\int \frac {1}{(8 c-d x) \sqrt {c+d x}} \, dx,x,x^3\right )}{d^{5/3}}-\frac {\left (648 c^{5/3}\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 )}{d^{2/3}}\\ &=\frac {103 c x^2 \sqrt {c+d x^3}}{13 d^2}+\frac {19 x^5 \sqrt {c+d x^3}}{39 d}+\frac {5906 c^2 \sqrt {c+d x^3}}{13 d^{8/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac {x^5 \left (c+d x^3\right )^{3/2}}{3 d \left (8 c-d x^3\right )}+\frac {108 \sqrt {3} c^{13/6} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{d^{8/3}}-\frac {108 c^{13/6} \tanh ^{-1}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x^3}}\right )}{d^{8/3}}-\frac {2953 \sqrt [4]{3} \sqrt {2-\sqrt {3}} c^{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 )}{13 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 {5906 \sqrt {2} c^{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 )}{13 \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 (324 c^{8/3}\right ) \text {Subst}\left (\int \frac {1}{9 c-x^2} \, dx,x,\sqrt {c+d x^3}\right )}{d^{8/3}}\\ &=\frac {103 c x^2 \sqrt {c+d x^3}}{13 d^2}+\frac {19 x^5 \sqrt {c+d x^3}}{39 d}+\frac {5906 c^2 \sqrt {c+d x^3}}{13 d^{8/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac {x^5 \left (c+d x^3\right )^{3/2}}{3 d \left (8 c-d x^3\right )}+\frac {108 \sqrt {3} c^{13/6} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{d^{8/3}}-\frac {108 c^{13/6} \tanh ^{-1}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x^3}}\right )}{d^{8/3}}+\frac {108 c^{13/6} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{d^{8/3}}-\frac {2953 \sqrt [4]{3} \sqrt {2-\sqrt {3}} c^{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 )}{13 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 {5906 \sqrt {2} c^{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 )}{13 \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 = 6.85, size = 191, normalized size = 0.28 \begin {gather*} \frac {80 x^2 \left (-412 c^3-388 c^2 d x^3+25 c d^2 x^6+d^3 x^9\right )+4120 c^2 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 )+2953 c 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 )}{520 d^2 \left (-8 c+d x^3\right ) \sqrt {c+d x^3}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(x^7*(c + d*x^3)^(3/2))/(8*c - d*x^3)^2,x]
[Out]
(80*x^2*(-412*c^3 - 388*c^2*d*x^3 + 25*c*d^2*x^6 + d^3*x^9) + 4120*c^2*x^2*(8*c - d*x^3)*Sqrt[1 + (d*x^3)/c]*A
ppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), (d*x^3)/(8*c)] + 2953*c*d*x^5*(8*c - d*x^3)*Sqrt[1 + (d*x^3)/c]*Appell
F1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), (d*x^3)/(8*c)])/(520*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 = 0.39, size = 2224, normalized size = 3.27
| | |
| method |
result |
size |
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| elliptic |
\(\text {Expression too large to display}\) |
\(921\) |
| risch |
\(\text {Expression too large to display}\) |
\(1769\) |
| default |
\(\text {Expression too large to display}\) |
\(2224\) |
| | |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^7*(d*x^3+c)^(3/2)/(-d*x^3+8*c)^2,x,method=_RETURNVERBOSE)
[Out]
1/d^2*(2/13*d*x^5*(d*x^3+c)^(1/2)+32/91*c*x^2*(d*x^3+c)^(1/2)-18/91*I*c^2*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))))+16/d^2*c*(2
/7*x^2*(d*x^3+c)^(1/2)-44/7*I*c*3^(1/2)/d*(-c*d^2)^(1/3)*(I*(x+1/2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(
1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2)*((x-1/d*(-c*d^2)^(1/3))/(-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^
(1/3)))^(1/2)*(-I*(x+1/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2)/(d*x
^3+c)^(1/2)*((-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*EllipticE(1/3*3^(1/2)*(I*(x+1/2/d*(-c*d^2)
^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2),(I*3^(1/2)/d*(-c*d^2)^(1/3)/(-3/2/d*(-c
*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3)))^(1/2))+1/d*(-c*d^2)^(1/3)*EllipticF(1/3*3^(1/2)*(I*(x+1/2/d*(-c*d
^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2),(I*3^(1/2)/d*(-c*d^2)^(1/3)/(-3/2/d*
(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3)))^(1/2)))+3*I*c/d^3*2^(1/2)*sum(1/_alpha*(-c*d^2)^(1/3)*(1/2*I*d
*(2*x+1/d*(-I*3^(1/2)*(-c*d^2)^(1/3)+(-c*d^2)^(1/3)))/(-c*d^2)^(1/3))^(1/2)*(d*(x-1/d*(-c*d^2)^(1/3))/(-3*(-c*
d^2)^(1/3)+I*3^(1/2)*(-c*d^2)^(1/3)))^(1/2)*(-1/2*I*d*(2*x+1/d*(I*3^(1/2)*(-c*d^2)^(1/3)+(-c*d^2)^(1/3)))/(-c*
d^2)^(1/3))^(1/2)/(d*x^3+c)^(1/2)*(I*(-c*d^2)^(1/3)*_alpha*3^(1/2)*d-I*3^(1/2)*(-c*d^2)^(2/3)+2*_alpha^2*d^2-(
-c*d^2)^(1/3)*_alpha*d-(-c*d^2)^(2/3))*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)))+64*c^2/d^2*(3/8*x^2*(d*x^3+c)^(1/2)/(-d*x^
3+8*c)-19/24*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)))+3/8*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)*_al
pha*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)^(3/2)/(-d*x^3+8*c)^2,x, algorithm="maxima")
[Out]
integrate((d*x^3 + c)^(3/2)*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 = 58.72, size = 3882, normalized size = 5.70 \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)^(3/2)/(-d*x^3+8*c)^2,x, algorithm="fricas")
[Out]
-1/13*(468*sqrt(3)*(d^4*x^3 - 8*c*d^3)*(c^13/d^16)^(1/6)*arctan(-1/3*(324*sqrt(3)*(3*c^13*d^16*x^16 + 784*c^14
*d^15*x^13 + 7680*c^15*d^14*x^10 + 10752*c^16*d^13*x^7 + 4096*c^17*d^12*x^4)*(c^13/d^16)^(2/3) + 36*sqrt(3)*(c
^17*d^11*x^17 + 1772*c^18*d^10*x^14 + 42592*c^19*d^9*x^11 + 96256*c^20*d^8*x^8 + 69632*c^21*d^7*x^5 + 16384*c^
22*d^6*x^2)*(c^13/d^16)^(1/3) + sqrt(3)*(c^21*d^6*x^18 + 9456*c^22*d^5*x^15 + 749184*c^23*d^4*x^12 + 3017216*c
^24*d^3*x^9 + 3489792*c^25*d^2*x^6 + 1572864*c^26*d*x^3 + 262144*c^27) + 12*sqrt(d*x^3 + c)*(12*sqrt(3)*(35*c^
11*d^18*x^14 - 14440*c^12*d^17*x^11 - 24576*c^13*d^16*x^8 - 16384*c^14*d^15*x^5 - 4096*c^15*d^14*x^2)*(c^13/d^
16)^(5/6) + 18*sqrt(3)*(c^15*d^13*x^15 - 1112*c^16*d^12*x^12 + 7296*c^17*d^11*x^9 + 11776*c^18*d^10*x^6 + 4096
*c^19*d^9*x^3)*sqrt(c^13/d^16) + sqrt(3)*(c^19*d^8*x^16 - 4768*c^20*d^7*x^13 + 362752*c^21*d^6*x^10 + 709120*c
^22*d^5*x^7 + 413696*c^23*d^4*x^4 + 65536*c^24*d^3*x)*(c^13/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 - 1024*c^4*d^15*x^4)*(c^13/d^16)^(5/6) + 18*sqrt(3)*(c^4*d^
14*x^17 - 5290*c^5*d^13*x^14 - 21152*c^6*d^12*x^11 - 47744*c^7*d^11*x^8 - 37888*c^8*d^10*x^5 - 8192*c^9*d^9*x^
2)*sqrt(c^13/d^16) + sqrt(3)*(c^8*d^9*x^18 - 7698*c^9*d^8*x^15 - 1664688*c^10*d^7*x^12 - 5524864*c^11*d^6*x^9
- 6223872*c^12*d^5*x^6 - 2703360*c^13*d^4*x^3 - 327680*c^14*d^3)*(c^13/d^16)^(1/6) + 6*sqrt(d*x^3 + c)*(sqrt(3
)*(7*c^2*d^16*x^15 + 37352*c^3*d^15*x^12 - 230336*c^4*d^14*x^9 - 515072*c^5*d^13*x^6 - 286720*c^6*d^12*x^3 - 3
2768*c^7*d^11)*(c^13/d^16)^(2/3) + 108*sqrt(3)*(53*c^7*d^10*x^13 + 1320*c^8*d^9*x^10 + 1536*c^9*d^8*x^7 + 512*
c^10*d^7*x^4)*(c^13/d^16)^(1/3) + 6*sqrt(3)*(37*c^11*d^5*x^14 + 28912*c^12*d^4*x^11 + 43584*c^13*d^3*x^8 + 209
92*c^14*d^2*x^5 + 4096*c^15*d*x^2)))*sqrt((18*c^22*d^2*x^8 + 360*c^23*d*x^5 - 144*c^24*x^2 + (c^13*d^13*x^9 -
276*c^14*d^12*x^6 - 1608*c^15*d^11*x^3 - 1088*c^16*d^10)*(c^13/d^16)^(2/3) + 6*sqrt(d*x^3 + c)*((c^11*d^15*x^7
- 28*c^12*d^14*x^4 - 272*c^13*d^13*x)*(c^13/d^16)^(5/6) - 24*(c^16*d^9*x^5 + c^17*d^8*x^2)*sqrt(c^13/d^16) +
4*(c^20*d^4*x^6 + 41*c^21*d^3*x^3 + 40*c^22*d^2)*(c^13/d^16)^(1/6)) - 18*(c^18*d^7*x^7 - 52*c^19*d^6*x^4 - 80*
c^20*d^5*x)*(c^13/d^16)^(1/3))/(d^3*x^9 - 24*c*d^2*x^6 + 192*c^2*d*x^3 - 512*c^3)))/(c^21*d^6*x^18 - 14952*c^2
2*d^5*x^15 + 2872896*c^23*d^4*x^12 + 7330304*c^24*d^3*x^9 + 6696960*c^25*d^2*x^6 + 2457600*c^26*d*x^3 + 262144
*c^27)) - 468*sqrt(3)*(d^4*x^3 - 8*c*d^3)*(c^13/d^16)^(1/6)*arctan(-1/3*(324*sqrt(3)*(3*c^13*d^16*x^16 + 784*c
^14*d^15*x^13 + 7680*c^15*d^14*x^10 + 10752*c^16*d^13*x^7 + 4096*c^17*d^12*x^4)*(c^13/d^16)^(2/3) + 36*sqrt(3)
*(c^17*d^11*x^17 + 1772*c^18*d^10*x^14 + 42592*c^19*d^9*x^11 + 96256*c^20*d^8*x^8 + 69632*c^21*d^7*x^5 + 16384
*c^22*d^6*x^2)*(c^13/d^16)^(1/3) + sqrt(3)*(c^21*d^6*x^18 + 9456*c^22*d^5*x^15 + 749184*c^23*d^4*x^12 + 301721
6*c^24*d^3*x^9 + 3489792*c^25*d^2*x^6 + 1572864*c^26*d*x^3 + 262144*c^27) - 12*sqrt(d*x^3 + c)*(12*sqrt(3)*(35
*c^11*d^18*x^14 - 14440*c^12*d^17*x^11 - 24576*c^13*d^16*x^8 - 16384*c^14*d^15*x^5 - 4096*c^15*d^14*x^2)*(c^13
/d^16)^(5/6) + 18*sqrt(3)*(c^15*d^13*x^15 - 1112*c^16*d^12*x^12 + 7296*c^17*d^11*x^9 + 11776*c^18*d^10*x^6 + 4
096*c^19*d^9*x^3)*sqrt(c^13/d^16) + sqrt(3)*(c^19*d^8*x^16 - 4768*c^20*d^7*x^13 + 362752*c^21*d^6*x^10 + 70912
0*c^22*d^5*x^7 + 413696*c^23*d^4*x^4 + 65536*c^24*d^3*x)*(c^13/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 - 1024*c^4*d^15*x^4)*(c^13/d^16)^(5/6) + 18*sqrt(3)*(c^4
*d^14*x^17 - 5290*c^5*d^13*x^14 - 21152*c^6*d^12*x^11 - 47744*c^7*d^11*x^8 - 37888*c^8*d^10*x^5 - 8192*c^9*d^9
*x^2)*sqrt(c^13/d^16) + sqrt(3)*(c^8*d^9*x^18 - 7698*c^9*d^8*x^15 - 1664688*c^10*d^7*x^12 - 5524864*c^11*d^6*x
^9 - 6223872*c^12*d^5*x^6 - 2703360*c^13*d^4*x^3 - 327680*c^14*d^3)*(c^13/d^16)^(1/6) - 6*sqrt(d*x^3 + c)*(sqr
t(3)*(7*c^2*d^16*x^15 + 37352*c^3*d^15*x^12 - 230336*c^4*d^14*x^9 - 515072*c^5*d^13*x^6 - 286720*c^6*d^12*x^3
- 32768*c^7*d^11)*(c^13/d^16)^(2/3) + 108*sqrt(3)*(53*c^7*d^10*x^13 + 1320*c^8*d^9*x^10 + 1536*c^9*d^8*x^7 + 5
12*c^10*d^7*x^4)*(c^13/d^16)^(1/3) + 6*sqrt(3)*(37*c^11*d^5*x^14 + 28912*c^12*d^4*x^11 + 43584*c^13*d^3*x^8 +
20992*c^14*d^2*x^5 + 4096*c^15*d*x^2)))*sqrt((18*c^22*d^2*x^8 + 360*c^23*d*x^5 - 144*c^24*x^2 + (c^13*d^13*x^9
- 276*c^14*d^12*x^6 - 1608*c^15*d^11*x^3 - 1088*c^16*d^10)*(c^13/d^16)^(2/3) - 6*sqrt(d*x^3 + c)*((c^11*d^15*
x^7 - 28*c^12*d^14*x^4 - 272*c^13*d^13*x)*(c^13/d^16)^(5/6) - 24*(c^16*d^9*x^5 + c^17*d^8*x^2)*sqrt(c^13/d^16)
+ 4*(c^20*d^4*x^6 + 41*c^21*d^3*x^3 + 40*c^22*d^2)*(c^13/d^16)^(1/6)) - 18*(c^18*d^7*x^7 - 52*c^19*d^6*x^4 -
80*c^20*d^5*x)*(c^13/d^16)^(1/3))/(d^3*x^9 - 24*c*d^2*x^6 + 192*c^2*d*x^3 - 512*c^3)))/(c^21*d^6*x^18 - 14952*
c^22*d^5*x^15 + 2872896*c^23*d^4*x^12 + 7330304*c^24*d^3*x^9 + 6696960*c^25*d^2*x^6 + 2457600*c^26*d*x^3 + 262
144*c^27)) + 5906*(c^2*d*x^3 - 8*c^3)*sqrt(d)*weierstrassZeta(0, -4*c/d, weierstrassPInverse(0, -4*c/d, x)) +
117*(d^4*x^3 - 8*c*d^3)*(c^13/d^16)^(1/6)*log(8...
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{7} \left (c + d x^{3}\right )^{\frac {3}{2}}}{\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)**(3/2)/(-d*x**3+8*c)**2,x)
[Out]
Integral(x**7*(c + d*x**3)**(3/2)/(-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)^(3/2)/(-d*x^3+8*c)^2,x, algorithm="giac")
[Out]
integrate((d*x^3 + c)^(3/2)*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\,{\left (d\,x^3+c\right )}^{3/2}}{{\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)^(3/2))/(8*c - d*x^3)^2,x)
[Out]
int((x^7*(c + d*x^3)^(3/2))/(8*c - d*x^3)^2, x)
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