3.3.65 \(\int \frac {x \sqrt {c+d x^3}}{4 c+d x^3} \, dx\) [265]
Optimal. Leaf size=659 \[ \frac {2 \sqrt {c+d x^3}}{d^{2/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac {\sqrt [6]{c} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{2} \sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{2^{2/3} \sqrt {3} d^{2/3}}-\frac {\sqrt [6]{c} \tan ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {3} \sqrt {c}}\right )}{2^{2/3} \sqrt {3} d^{2/3}}+\frac {\sqrt [6]{c} \tanh ^{-1}\left (\frac {\sqrt [6]{c} \left (\sqrt [3]{c}-\sqrt [3]{2} \sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{2^{2/3} d^{2/3}}-\frac {\sqrt [6]{c} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{3\ 2^{2/3} d^{2/3}}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt {3}\right )}{d^{2/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}+\frac {2 \sqrt {2} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt {3}\right )}{\sqrt [4]{3} d^{2/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}} \]
[Out]
1/2*c^(1/6)*arctanh(c^(1/6)*(c^(1/3)-2^(1/3)*d^(1/3)*x)/(d*x^3+c)^(1/2))*2^(1/3)/d^(2/3)-1/6*c^(1/6)*arctanh((
d*x^3+c)^(1/2)/c^(1/2))*2^(1/3)/d^(2/3)+1/6*c^(1/6)*arctan(c^(1/6)*(c^(1/3)+2^(1/3)*d^(1/3)*x)*3^(1/2)/(d*x^3+
c)^(1/2))*2^(1/3)/d^(2/3)*3^(1/2)-1/6*c^(1/6)*arctan(1/3*(d*x^3+c)^(1/2)*3^(1/2)/c^(1/2))*2^(1/3)/d^(2/3)*3^(1
/2)+2*(d*x^3+c)^(1/2)/d^(2/3)/(d^(1/3)*x+c^(1/3)*(1+3^(1/2)))+2/3*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^(2/3)/(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)-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^(2/3)/(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)
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Rubi [A]
time = 0.12, antiderivative size = 659, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {495, 309, 224,
1891, 497} \begin {gather*} \frac {2 \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 )}{\sqrt [4]{3} d^{2/3} \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]{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 )}{d^{2/3} \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 [6]{c} \text {ArcTan}\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 )}{2^{2/3} \sqrt {3} d^{2/3}}-\frac {\sqrt [6]{c} \text {ArcTan}\left (\frac {\sqrt {c+d x^3}}{\sqrt {3} \sqrt {c}}\right )}{2^{2/3} \sqrt {3} d^{2/3}}+\frac {2 \sqrt {c+d x^3}}{d^{2/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac {\sqrt [6]{c} \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 )}{2^{2/3} d^{2/3}}-\frac {\sqrt [6]{c} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{3\ 2^{2/3} d^{2/3}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(x*Sqrt[c + d*x^3])/(4*c + d*x^3),x]
[Out]
(2*Sqrt[c + d*x^3])/(d^(2/3)*((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)) + (c^(1/6)*ArcTan[(Sqrt[3]*c^(1/6)*(c^(1/3)
+ 2^(1/3)*d^(1/3)*x))/Sqrt[c + d*x^3]])/(2^(2/3)*Sqrt[3]*d^(2/3)) - (c^(1/6)*ArcTan[Sqrt[c + d*x^3]/(Sqrt[3]*S
qrt[c])])/(2^(2/3)*Sqrt[3]*d^(2/3)) + (c^(1/6)*ArcTanh[(c^(1/6)*(c^(1/3) - 2^(1/3)*d^(1/3)*x))/Sqrt[c + d*x^3]
])/(2^(2/3)*d^(2/3)) - (c^(1/6)*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]])/(3*2^(2/3)*d^(2/3)) - (3^(1/4)*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]])/(d^(2/3)*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
]) + (2*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^(1/4)*d^(2/3)*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 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 495
Int[((x_)*((a_) + (b_.)*(x_)^(n_))^(p_))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Dist[b/d, Int[x*(a + b*x^n)^(p
- 1), x], x] - Dist[(b*c - a*d)/d, Int[x*((a + b*x^n)^(p - 1)/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d}, x]
&& NeQ[b*c - a*d, 0] && IGtQ[n, 0] && GtQ[p, 0] && IntBinomialQ[a, b, c, d, 1, 1, n, p, -1, x]
Rule 497
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]
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]
Rubi steps
\begin {align*} \int \frac {x \sqrt {c+d x^3}}{4 c+d x^3} \, dx &=-\left ((3 c) \int \frac {x}{\sqrt {c+d x^3} \left (4 c+d x^3\right )} \, dx\right )+\int \frac {x}{\sqrt {c+d x^3}} \, dx\\ &=\frac {\sqrt [6]{c} \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 )}{2^{2/3} \sqrt {3} d^{2/3}}-\frac {\sqrt [6]{c} \tan ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {3} \sqrt {c}}\right )}{2^{2/3} \sqrt {3} d^{2/3}}+\frac {\sqrt [6]{c} \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 )}{2^{2/3} d^{2/3}}-\frac {\sqrt [6]{c} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{3\ 2^{2/3} d^{2/3}}+\frac {\int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\sqrt {c+d x^3}} \, dx}{\sqrt [3]{d}}+\frac {\left (\sqrt {2 \left (2-\sqrt {3}\right )} \sqrt [3]{c}\right ) \int \frac {1}{\sqrt {c+d x^3}} \, dx}{\sqrt [3]{d}}\\ &=\frac {2 \sqrt {c+d x^3}}{d^{2/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac {\sqrt [6]{c} \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 )}{2^{2/3} \sqrt {3} d^{2/3}}-\frac {\sqrt [6]{c} \tan ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {3} \sqrt {c}}\right )}{2^{2/3} \sqrt {3} d^{2/3}}+\frac {\sqrt [6]{c} \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 )}{2^{2/3} d^{2/3}}-\frac {\sqrt [6]{c} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{3\ 2^{2/3} d^{2/3}}-\frac {\sqrt [4]{3} \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 )}{d^{2/3} \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 {2 \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 )}{\sqrt [4]{3} d^{2/3} \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 = 6.23, size = 63, normalized size = 0.10 \begin {gather*} \frac {x^2 \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}{4 c}\right )}{8 \sqrt {c+d x^3}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(x*Sqrt[c + d*x^3])/(4*c + d*x^3),x]
[Out]
(x^2*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, -1/2, 1, 5/3, -((d*x^3)/c), -1/4*(d*x^3)/c])/(8*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 = 0.34, size = 848, normalized size = 1.29
| | |
| method |
result |
size |
| | |
| default |
\(\text {Expression too large to display}\) |
\(848\) |
| elliptic |
\(\text {Expression too large to display}\) |
\(848\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x*(d*x^3+c)^(1/2)/(d*x^3+4*c),x,method=_RETURNVERBOSE)
[Out]
-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),1/6/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+4*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*(d*x^3+c)^(1/2)/(d*x^3+4*c),x, algorithm="maxima")
[Out]
integrate(sqrt(d*x^3 + c)*x/(d*x^3 + 4*c), x)
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 12.06, size = 3547, normalized size = 5.38 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*(d*x^3+c)^(1/2)/(d*x^3+4*c),x, algorithm="fricas")
[Out]
1/12*(4*sqrt(3)*(1/432)^(1/6)*d*(-c/d^4)^(1/6)*arctan(-1/3*(432*sqrt(3)*(1/2)^(2/3)*(c*d^8*x^16 - 39*c^2*d^7*x
^13 - 72*c^3*d^6*x^10 - 32*c^4*d^5*x^7)*(-c/d^4)^(2/3) + 24*sqrt(3)*(1/2)^(1/3)*(c*d^7*x^17 - 271*c^2*d^6*x^14
+ 112*c^3*d^5*x^11 + 1216*c^4*d^4*x^8 + 1088*c^5*d^3*x^5 + 256*c^6*d^2*x^2)*(-c/d^4)^(1/3) + 12*sqrt(1/3)*(51
84*sqrt(3)*(1/432)^(5/6)*(d^9*x^16 + 229*c*d^8*x^13 + 492*c^2*d^7*x^10 + 328*c^3*d^6*x^7 + 64*c^4*d^5*x^4)*(-c
/d^4)^(5/6) + 6*sqrt(3)*sqrt(1/3)*(d^8*x^17 + 737*c*d^7*x^14 + 2704*c^2*d^6*x^11 + 3376*c^3*d^5*x^8 + 1664*c^4
*d^4*x^5 + 256*c^5*d^3*x^2)*sqrt(-c/d^4) + sqrt(3)*(1/432)^(1/6)*(d^7*x^18 + 1098*c*d^6*x^15 - 24720*c^2*d^5*x
^12 - 56704*c^3*d^4*x^9 - 44928*c^4*d^3*x^6 - 15360*c^5*d^2*x^3 - 2048*c^6*d)*(-c/d^4)^(1/6) - 2*sqrt(d*x^3 +
c)*(sqrt(3)*(1/2)^(2/3)*(5*d^8*x^15 - 3272*c*d^7*x^12 - 12544*c^2*d^6*x^9 - 14656*c^3*d^5*x^6 - 6656*c^4*d^4*x
^3 - 1024*c^5*d^3)*(-c/d^4)^(2/3) - 864*sqrt(3)*(1/2)^(1/3)*(c*d^6*x^13 + 2*c^2*d^5*x^10 + c^3*d^4*x^7)*(-c/d^
4)^(1/3) - 3*sqrt(3)*(17*c*d^5*x^14 - 1456*c^2*d^4*x^11 - 2544*c^3*d^3*x^8 - 1408*c^4*d^2*x^5 - 256*c^5*d*x^2)
))*sqrt((6*c^2*d^2*x^8 - 42*c^3*d*x^5 - 48*c^4*x^2 - (1/2)^(2/3)*(c*d^5*x^9 + 60*c^2*d^4*x^6 - 32*c^4*d^2)*(-c
/d^4)^(2/3) - 12*(1/2)^(1/3)*(c^2*d^3*x^7 + 5*c^3*d^2*x^4 + 4*c^4*d*x)*(-c/d^4)^(1/3) + 6*(9*sqrt(1/3)*c^2*d^3
*x^5*sqrt(-c/d^4) + 72*(1/432)^(5/6)*(c*d^5*x^7 + 2*c^2*d^4*x^4 - 8*c^3*d^3*x)*(-c/d^4)^(5/6) - 2*(1/432)^(1/6
)*(c^2*d^2*x^6 - 16*c^3*d*x^3 - 8*c^4)*(-c/d^4)^(1/6))*sqrt(d*x^3 + c))/(d^3*x^9 + 12*c*d^2*x^6 + 48*c^2*d*x^3
+ 64*c^3)) + sqrt(3)*(c*d^6*x^18 - 1416*c^2*d^5*x^15 + 14352*c^3*d^4*x^12 + 44480*c^4*d^3*x^9 + 49920*c^5*d^2
*x^6 + 24576*c^6*d*x^3 + 4096*c^7) - 24*sqrt(d*x^3 + c)*(216*sqrt(3)*(1/432)^(5/6)*(31*c*d^8*x^14 + 1744*c^2*d
^7*x^11 + 2976*c^3*d^6*x^8 + 1600*c^4*d^5*x^5 + 256*c^5*d^4*x^2)*(-c/d^4)^(5/6) + 6*sqrt(3)*sqrt(1/3)*(c*d^7*x
^15 + 157*c^2*d^6*x^12 + 348*c^3*d^5*x^9 + 256*c^4*d^4*x^6 + 64*c^5*d^3*x^3)*sqrt(-c/d^4) + sqrt(3)*(1/432)^(1
/6)*(c*d^6*x^16 + 686*c^2*d^5*x^13 + 7072*c^3*d^4*x^10 + 11008*c^4*d^3*x^7 + 5888*c^5*d^2*x^4 + 1024*c^6*d*x)*
(-c/d^4)^(1/6)))/(c*d^6*x^18 + 2184*c^2*d^5*x^15 + 57696*c^3*d^4*x^12 + 125696*c^4*d^3*x^9 + 100608*c^5*d^2*x^
6 + 33792*c^6*d*x^3 + 4096*c^7)) - 4*sqrt(3)*(1/432)^(1/6)*d*(-c/d^4)^(1/6)*arctan(-1/3*(432*sqrt(3)*(1/2)^(2/
3)*(c*d^8*x^16 - 39*c^2*d^7*x^13 - 72*c^3*d^6*x^10 - 32*c^4*d^5*x^7)*(-c/d^4)^(2/3) + 24*sqrt(3)*(1/2)^(1/3)*(
c*d^7*x^17 - 271*c^2*d^6*x^14 + 112*c^3*d^5*x^11 + 1216*c^4*d^4*x^8 + 1088*c^5*d^3*x^5 + 256*c^6*d^2*x^2)*(-c/
d^4)^(1/3) - 12*sqrt(1/3)*(5184*sqrt(3)*(1/432)^(5/6)*(d^9*x^16 + 229*c*d^8*x^13 + 492*c^2*d^7*x^10 + 328*c^3*
d^6*x^7 + 64*c^4*d^5*x^4)*(-c/d^4)^(5/6) + 6*sqrt(3)*sqrt(1/3)*(d^8*x^17 + 737*c*d^7*x^14 + 2704*c^2*d^6*x^11
+ 3376*c^3*d^5*x^8 + 1664*c^4*d^4*x^5 + 256*c^5*d^3*x^2)*sqrt(-c/d^4) + sqrt(3)*(1/432)^(1/6)*(d^7*x^18 + 1098
*c*d^6*x^15 - 24720*c^2*d^5*x^12 - 56704*c^3*d^4*x^9 - 44928*c^4*d^3*x^6 - 15360*c^5*d^2*x^3 - 2048*c^6*d)*(-c
/d^4)^(1/6) + 2*sqrt(d*x^3 + c)*(sqrt(3)*(1/2)^(2/3)*(5*d^8*x^15 - 3272*c*d^7*x^12 - 12544*c^2*d^6*x^9 - 14656
*c^3*d^5*x^6 - 6656*c^4*d^4*x^3 - 1024*c^5*d^3)*(-c/d^4)^(2/3) - 864*sqrt(3)*(1/2)^(1/3)*(c*d^6*x^13 + 2*c^2*d
^5*x^10 + c^3*d^4*x^7)*(-c/d^4)^(1/3) - 3*sqrt(3)*(17*c*d^5*x^14 - 1456*c^2*d^4*x^11 - 2544*c^3*d^3*x^8 - 1408
*c^4*d^2*x^5 - 256*c^5*d*x^2)))*sqrt((6*c^2*d^2*x^8 - 42*c^3*d*x^5 - 48*c^4*x^2 - (1/2)^(2/3)*(c*d^5*x^9 + 60*
c^2*d^4*x^6 - 32*c^4*d^2)*(-c/d^4)^(2/3) - 12*(1/2)^(1/3)*(c^2*d^3*x^7 + 5*c^3*d^2*x^4 + 4*c^4*d*x)*(-c/d^4)^(
1/3) - 6*(9*sqrt(1/3)*c^2*d^3*x^5*sqrt(-c/d^4) + 72*(1/432)^(5/6)*(c*d^5*x^7 + 2*c^2*d^4*x^4 - 8*c^3*d^3*x)*(-
c/d^4)^(5/6) - 2*(1/432)^(1/6)*(c^2*d^2*x^6 - 16*c^3*d*x^3 - 8*c^4)*(-c/d^4)^(1/6))*sqrt(d*x^3 + c))/(d^3*x^9
+ 12*c*d^2*x^6 + 48*c^2*d*x^3 + 64*c^3)) + sqrt(3)*(c*d^6*x^18 - 1416*c^2*d^5*x^15 + 14352*c^3*d^4*x^12 + 4448
0*c^4*d^3*x^9 + 49920*c^5*d^2*x^6 + 24576*c^6*d*x^3 + 4096*c^7) + 24*sqrt(d*x^3 + c)*(216*sqrt(3)*(1/432)^(5/6
)*(31*c*d^8*x^14 + 1744*c^2*d^7*x^11 + 2976*c^3*d^6*x^8 + 1600*c^4*d^5*x^5 + 256*c^5*d^4*x^2)*(-c/d^4)^(5/6) +
6*sqrt(3)*sqrt(1/3)*(c*d^7*x^15 + 157*c^2*d^6*x^12 + 348*c^3*d^5*x^9 + 256*c^4*d^4*x^6 + 64*c^5*d^3*x^3)*sqrt
(-c/d^4) + sqrt(3)*(1/432)^(1/6)*(c*d^6*x^16 + 686*c^2*d^5*x^13 + 7072*c^3*d^4*x^10 + 11008*c^4*d^3*x^7 + 5888
*c^5*d^2*x^4 + 1024*c^6*d*x)*(-c/d^4)^(1/6)))/(c*d^6*x^18 + 2184*c^2*d^5*x^15 + 57696*c^3*d^4*x^12 + 125696*c^
4*d^3*x^9 + 100608*c^5*d^2*x^6 + 33792*c^6*d*x^3 + 4096*c^7)) + (1/432)^(1/6)*d*(-c/d^4)^(1/6)*log(1/12*(6*c^2
*d^2*x^8 - 42*c^3*d*x^5 - 48*c^4*x^2 - (1/2)^(2/3)*(c*d^5*x^9 + 60*c^2*d^4*x^6 - 32*c^4*d^2)*(-c/d^4)^(2/3) -
12*(1/2)^(1/3)*(c^2*d^3*x^7 + 5*c^3*d^2*x^4 + 4*c^4*d*x)*(-c/d^4)^(1/3) + 6*(9*sqrt(1/3)*c^2*d^3*x^5*sqrt(-c/d
^4) + 72*(1/432)^(5/6)*(c*d^5*x^7 + 2*c^2*d^4*x^4 - 8*c^3*d^3*x)*(-c/d^4)^(5/6) - 2*(1/432)^(1/6)*(c^2*d^2*x^6
- 16*c^3*d*x^3 - 8*c^4)*(-c/d^4)^(1/6))*sqrt(d*x^3 + c))/(d^3*x^9 + 12*c*d^2*x^6 + 48*c^2*d*x^3 + 64*c^3)) -
(1/432)^(1/6)*d*(-c/d^4)^(1/6)*log(1/12*(6*c^2*...
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \sqrt {c + d x^{3}}}{4 c + d x^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*(d*x**3+c)**(1/2)/(d*x**3+4*c),x)
[Out]
Integral(x*sqrt(c + d*x**3)/(4*c + d*x**3), 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*(d*x^3+c)^(1/2)/(d*x^3+4*c),x, algorithm="giac")
[Out]
integrate(sqrt(d*x^3 + c)*x/(d*x^3 + 4*c), x)
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x\,\sqrt {d\,x^3+c}}{d\,x^3+4\,c} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((x*(c + d*x^3)^(1/2))/(4*c + d*x^3),x)
[Out]
int((x*(c + d*x^3)^(1/2))/(4*c + d*x^3), x)
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