3.277 \(\int \frac{1}{x^2 \sqrt{c+d x^3} (4 c+d x^3)} \, dx\)

Optimal. Leaf size=697 \[ \frac{\sqrt [3]{d} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right ),-7-4 \sqrt{3}\right )}{2 \sqrt{2} \sqrt [4]{3} c^{5/3} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}-\frac{\sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt [3]{d} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{d} x+\left (1-\sqrt{3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt{3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt{3}\right )}{8 c^{5/3} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}-\frac{\sqrt{c+d x^3}}{4 c^2 x}+\frac{\sqrt [3]{d} \sqrt{c+d x^3}}{4 c^2 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac{\sqrt [3]{d} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{2} \sqrt [3]{d} x\right )}{\sqrt{c+d x^3}}\right )}{12\ 2^{2/3} \sqrt{3} c^{11/6}}-\frac{\sqrt [3]{d} \tan ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{3} \sqrt{c}}\right )}{12\ 2^{2/3} \sqrt{3} c^{11/6}}+\frac{\sqrt [3]{d} \tanh ^{-1}\left (\frac{\sqrt [6]{c} \left (\sqrt [3]{c}-\sqrt [3]{2} \sqrt [3]{d} x\right )}{\sqrt{c+d x^3}}\right )}{12\ 2^{2/3} c^{11/6}}-\frac{\sqrt [3]{d} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{36\ 2^{2/3} c^{11/6}} \]

[Out]

-Sqrt[c + d*x^3]/(4*c^2*x) + (d^(1/3)*Sqrt[c + d*x^3])/(4*c^2*((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)) + (d^(1/3)*
ArcTan[(Sqrt[3]*c^(1/6)*(c^(1/3) + 2^(1/3)*d^(1/3)*x))/Sqrt[c + d*x^3]])/(12*2^(2/3)*Sqrt[3]*c^(11/6)) - (d^(1
/3)*ArcTan[Sqrt[c + d*x^3]/(Sqrt[3]*Sqrt[c])])/(12*2^(2/3)*Sqrt[3]*c^(11/6)) + (d^(1/3)*ArcTanh[(c^(1/6)*(c^(1
/3) - 2^(1/3)*d^(1/3)*x))/Sqrt[c + d*x^3]])/(12*2^(2/3)*c^(11/6)) - (d^(1/3)*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]])
/(36*2^(2/3)*c^(11/6)) - (3^(1/4)*Sqrt[2 - Sqrt[3]]*d^(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]])/(8*c^(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]) + (d^(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]])/(2*Sqrt[2]*3^(1/4)*c^(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])

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Rubi [A]  time = 0.388117, antiderivative size = 697, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {480, 584, 303, 218, 1877, 484} \[ \frac{\sqrt [3]{d} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{d} x+\left (1-\sqrt{3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt{3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt{3}\right )}{2 \sqrt{2} \sqrt [4]{3} c^{5/3} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}-\frac{\sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt [3]{d} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{d} x+\left (1-\sqrt{3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt{3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt{3}\right )}{8 c^{5/3} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}-\frac{\sqrt{c+d x^3}}{4 c^2 x}+\frac{\sqrt [3]{d} \sqrt{c+d x^3}}{4 c^2 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac{\sqrt [3]{d} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{2} \sqrt [3]{d} x\right )}{\sqrt{c+d x^3}}\right )}{12\ 2^{2/3} \sqrt{3} c^{11/6}}-\frac{\sqrt [3]{d} \tan ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{3} \sqrt{c}}\right )}{12\ 2^{2/3} \sqrt{3} c^{11/6}}+\frac{\sqrt [3]{d} \tanh ^{-1}\left (\frac{\sqrt [6]{c} \left (\sqrt [3]{c}-\sqrt [3]{2} \sqrt [3]{d} x\right )}{\sqrt{c+d x^3}}\right )}{12\ 2^{2/3} c^{11/6}}-\frac{\sqrt [3]{d} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{36\ 2^{2/3} c^{11/6}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*Sqrt[c + d*x^3]*(4*c + d*x^3)),x]

[Out]

-Sqrt[c + d*x^3]/(4*c^2*x) + (d^(1/3)*Sqrt[c + d*x^3])/(4*c^2*((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)) + (d^(1/3)*
ArcTan[(Sqrt[3]*c^(1/6)*(c^(1/3) + 2^(1/3)*d^(1/3)*x))/Sqrt[c + d*x^3]])/(12*2^(2/3)*Sqrt[3]*c^(11/6)) - (d^(1
/3)*ArcTan[Sqrt[c + d*x^3]/(Sqrt[3]*Sqrt[c])])/(12*2^(2/3)*Sqrt[3]*c^(11/6)) + (d^(1/3)*ArcTanh[(c^(1/6)*(c^(1
/3) - 2^(1/3)*d^(1/3)*x))/Sqrt[c + d*x^3]])/(12*2^(2/3)*c^(11/6)) - (d^(1/3)*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]])
/(36*2^(2/3)*c^(11/6)) - (3^(1/4)*Sqrt[2 - Sqrt[3]]*d^(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]])/(8*c^(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]) + (d^(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]])/(2*Sqrt[2]*3^(1/4)*c^(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 480

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[((e*x)^(m
+ 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*c*e*(m + 1)), x] - Dist[1/(a*c*e^n*(m + 1)), Int[(e*x)^(m +
n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[(b*c + a*d)*(m + n + 1) + n*(b*c*p + a*d*q) + b*d*(m + n*(p + q + 2) + 1)*
x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[m, -1] && IntBino
mialQ[a, b, c, d, e, m, n, p, q, x]

Rule 584

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 303

Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Dist[(Sq
rt[2]*s)/(Sqrt[2 + Sqrt[3]]*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 218

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(2*Sqr
t[2 + Sqrt[3]]*(s + r*x)*Sqrt[(s^2 - r*s*x + r^2*x^2)/((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]])/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[(s*(s + r*x))/((1 + Sqr
t[3])*s + r*x)^2]), x]] /; FreeQ[{a, b}, x] && PosQ[a]

Rule 1877

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]*Elli
pticE[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]])/(r^2*Sqrt[a + b*x^3]*Sqrt[(s*(
s + r*x))/((1 + Sqrt[3])*s + r*x)^2]), x]] /; FreeQ[{a, b, c, d}, x] && PosQ[a] && EqQ[b*c^3 - 2*(5 - 3*Sqrt[3
])*a*d^3, 0]

Rule 484

Int[(x_)/(((a_) + (b_.)*(x_)^3)*Sqrt[(c_) + (d_.)*(x_)^3]), x_Symbol] :> With[{q = Rt[d/c, 3]}, Simp[(q*ArcTan
h[Sqrt[c + d*x^3]/Rt[c, 2]])/(9*2^(2/3)*b*Rt[c, 2]), x] + (-Simp[(q*ArcTanh[(Rt[c, 2]*(1 - 2^(1/3)*q*x))/Sqrt[
c + d*x^3]])/(3*2^(2/3)*b*Rt[c, 2]), x] + Simp[(q*ArcTan[Sqrt[c + d*x^3]/(Sqrt[3]*Rt[c, 2])])/(3*2^(2/3)*Sqrt[
3]*b*Rt[c, 2]), x] - Simp[(q*ArcTan[(Sqrt[3]*Rt[c, 2]*(1 + 2^(1/3)*q*x))/Sqrt[c + d*x^3]])/(3*2^(2/3)*Sqrt[3]*
b*Rt[c, 2]), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[4*b*c - a*d, 0] && PosQ[c]

Rubi steps

\begin{align*} \int \frac{1}{x^2 \sqrt{c+d x^3} \left (4 c+d x^3\right )} \, dx &=-\frac{\sqrt{c+d x^3}}{4 c^2 x}+\frac{\int \frac{x \left (c d+\frac{d^2 x^3}{2}\right )}{\sqrt{c+d x^3} \left (4 c+d x^3\right )} \, dx}{4 c^2}\\ &=-\frac{\sqrt{c+d x^3}}{4 c^2 x}+\frac{\int \left (\frac{d x}{2 \sqrt{c+d x^3}}-\frac{c d x}{\sqrt{c+d x^3} \left (4 c+d x^3\right )}\right ) \, dx}{4 c^2}\\ &=-\frac{\sqrt{c+d x^3}}{4 c^2 x}+\frac{d \int \frac{x}{\sqrt{c+d x^3}} \, dx}{8 c^2}-\frac{d \int \frac{x}{\sqrt{c+d x^3} \left (4 c+d x^3\right )} \, dx}{4 c}\\ &=-\frac{\sqrt{c+d x^3}}{4 c^2 x}+\frac{\sqrt [3]{d} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{2} \sqrt [3]{d} x\right )}{\sqrt{c+d x^3}}\right )}{12\ 2^{2/3} \sqrt{3} c^{11/6}}-\frac{\sqrt [3]{d} \tan ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{3} \sqrt{c}}\right )}{12\ 2^{2/3} \sqrt{3} c^{11/6}}+\frac{\sqrt [3]{d} \tanh ^{-1}\left (\frac{\sqrt [6]{c} \left (\sqrt [3]{c}-\sqrt [3]{2} \sqrt [3]{d} x\right )}{\sqrt{c+d x^3}}\right )}{12\ 2^{2/3} c^{11/6}}-\frac{\sqrt [3]{d} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{36\ 2^{2/3} c^{11/6}}+\frac{d^{2/3} \int \frac{\left (1-\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\sqrt{c+d x^3}} \, dx}{8 c^2}+\frac{\left (\sqrt{\frac{1}{2} \left (2-\sqrt{3}\right )} d^{2/3}\right ) \int \frac{1}{\sqrt{c+d x^3}} \, dx}{4 c^{5/3}}\\ &=-\frac{\sqrt{c+d x^3}}{4 c^2 x}+\frac{\sqrt [3]{d} \sqrt{c+d x^3}}{4 c^2 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac{\sqrt [3]{d} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{2} \sqrt [3]{d} x\right )}{\sqrt{c+d x^3}}\right )}{12\ 2^{2/3} \sqrt{3} c^{11/6}}-\frac{\sqrt [3]{d} \tan ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{3} \sqrt{c}}\right )}{12\ 2^{2/3} \sqrt{3} c^{11/6}}+\frac{\sqrt [3]{d} \tanh ^{-1}\left (\frac{\sqrt [6]{c} \left (\sqrt [3]{c}-\sqrt [3]{2} \sqrt [3]{d} x\right )}{\sqrt{c+d x^3}}\right )}{12\ 2^{2/3} c^{11/6}}-\frac{\sqrt [3]{d} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{36\ 2^{2/3} c^{11/6}}-\frac{\sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt [3]{d} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\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 )}{8 c^{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{\sqrt [3]{d} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} 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 )}{2 \sqrt{2} \sqrt [4]{3} c^{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*}

Mathematica [C]  time = 0.074898, size = 136, normalized size = 0.2 \[ \frac{d^2 x^6 \sqrt{\frac{d x^3}{c}+1} F_1\left (\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right )+5 c d x^3 \sqrt{\frac{d x^3}{c}+1} F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right )-40 c \left (c+d x^3\right )}{160 c^3 x \sqrt{c+d x^3}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(x^2*Sqrt[c + d*x^3]*(4*c + d*x^3)),x]

[Out]

(-40*c*(c + d*x^3) + 5*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)/(4*c)] +
d^2*x^6*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), -(d*x^3)/(4*c)])/(160*c^3*x*Sqrt[c + d*x^
3])

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Maple [C]  time = 0.019, size = 874, normalized size = 1.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/(d*x^3+4*c)/(d*x^3+c)^(1/2),x)

[Out]

1/4/c*(-(d*x^3+c)^(1/2)/c/x-1/3*I/c*3^(1/2)*(-d^2*c)^(1/3)*(I*(x+1/2/d*(-d^2*c)^(1/3)-1/2*I*3^(1/2)/d*(-d^2*c)
^(1/3))*3^(1/2)*d/(-d^2*c)^(1/3))^(1/2)*((x-1/d*(-d^2*c)^(1/3))/(-3/2/d*(-d^2*c)^(1/3)+1/2*I*3^(1/2)/d*(-d^2*c
)^(1/3)))^(1/2)*(-I*(x+1/2/d*(-d^2*c)^(1/3)+1/2*I*3^(1/2)/d*(-d^2*c)^(1/3))*3^(1/2)*d/(-d^2*c)^(1/3))^(1/2)/(d
*x^3+c)^(1/2)*((-3/2/d*(-d^2*c)^(1/3)+1/2*I*3^(1/2)/d*(-d^2*c)^(1/3))*EllipticE(1/3*3^(1/2)*(I*(x+1/2/d*(-d^2*
c)^(1/3)-1/2*I*3^(1/2)/d*(-d^2*c)^(1/3))*3^(1/2)*d/(-d^2*c)^(1/3))^(1/2),(I*3^(1/2)/d*(-d^2*c)^(1/3)/(-3/2/d*(
-d^2*c)^(1/3)+1/2*I*3^(1/2)/d*(-d^2*c)^(1/3)))^(1/2))+1/d*(-d^2*c)^(1/3)*EllipticF(1/3*3^(1/2)*(I*(x+1/2/d*(-d
^2*c)^(1/3)-1/2*I*3^(1/2)/d*(-d^2*c)^(1/3))*3^(1/2)*d/(-d^2*c)^(1/3))^(1/2),(I*3^(1/2)/d*(-d^2*c)^(1/3)/(-3/2/
d*(-d^2*c)^(1/3)+1/2*I*3^(1/2)/d*(-d^2*c)^(1/3)))^(1/2))))+1/36*I/d^2/c^2*2^(1/2)*sum(1/_alpha*(-d^2*c)^(1/3)*
(1/2*I*d*(2*x+1/d*(-I*3^(1/2)*(-d^2*c)^(1/3)+(-d^2*c)^(1/3)))/(-d^2*c)^(1/3))^(1/2)*(d*(x-1/d*(-d^2*c)^(1/3))/
(-3*(-d^2*c)^(1/3)+I*3^(1/2)*(-d^2*c)^(1/3)))^(1/2)*(-1/2*I*d*(2*x+1/d*(I*3^(1/2)*(-d^2*c)^(1/3)+(-d^2*c)^(1/3
)))/(-d^2*c)^(1/3))^(1/2)/(d*x^3+c)^(1/2)*(I*(-d^2*c)^(1/3)*_alpha*3^(1/2)*d-I*3^(1/2)*(-d^2*c)^(2/3)+2*_alpha
^2*d^2-(-d^2*c)^(1/3)*_alpha*d-(-d^2*c)^(2/3))*EllipticPi(1/3*3^(1/2)*(I*(x+1/2/d*(-d^2*c)^(1/3)-1/2*I*3^(1/2)
/d*(-d^2*c)^(1/3))*3^(1/2)*d/(-d^2*c)^(1/3))^(1/2),1/6/d*(2*I*(-d^2*c)^(1/3)*3^(1/2)*_alpha^2*d-I*(-d^2*c)^(2/
3)*3^(1/2)*_alpha+I*3^(1/2)*c*d-3*(-d^2*c)^(2/3)*_alpha-3*c*d)/c,(I*3^(1/2)/d*(-d^2*c)^(1/3)/(-3/2/d*(-d^2*c)^
(1/3)+1/2*I*3^(1/2)/d*(-d^2*c)^(1/3)))^(1/2)),_alpha=RootOf(_Z^3*d+4*c))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (d x^{3} + 4 \, c\right )} \sqrt{d x^{3} + c} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(d*x^3+4*c)/(d*x^3+c)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/((d*x^3 + 4*c)*sqrt(d*x^3 + c)*x^2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{d x^{3} + c}}{d^{2} x^{8} + 5 \, c d x^{5} + 4 \, c^{2} x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(d*x^3+4*c)/(d*x^3+c)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(d*x^3 + c)/(d^2*x^8 + 5*c*d*x^5 + 4*c^2*x^2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{2} \sqrt{c + d x^{3}} \left (4 c + d x^{3}\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**2/(d*x**3+4*c)/(d*x**3+c)**(1/2),x)

[Out]

Integral(1/(x**2*sqrt(c + d*x**3)*(4*c + d*x**3)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (d x^{3} + 4 \, c\right )} \sqrt{d x^{3} + c} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(d*x^3+4*c)/(d*x^3+c)^(1/2),x, algorithm="giac")

[Out]

integrate(1/((d*x^3 + 4*c)*sqrt(d*x^3 + c)*x^2), x)