3.3.66 \(\int \frac {\sqrt {c+d x^3}}{x^2 (4 c+d x^3)} \, dx\) [266]
Optimal. Leaf size=697 \[ -\frac {\sqrt {c+d x^3}}{4 c x}+\frac {\sqrt [3]{d} \sqrt {c+d x^3}}{4 c \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}-\frac {\sqrt [3]{d} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{2} \sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{4\ 2^{2/3} \sqrt {3} c^{5/6}}+\frac {\sqrt [3]{d} \tan ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {3} \sqrt {c}}\right )}{4\ 2^{2/3} \sqrt {3} c^{5/6}}-\frac {\sqrt [3]{d} \tanh ^{-1}\left (\frac {\sqrt [6]{c} \left (\sqrt [3]{c}-\sqrt [3]{2} \sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{4\ 2^{2/3} c^{5/6}}+\frac {\sqrt [3]{d} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{12\ 2^{2/3} c^{5/6}}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{d} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt {3}\right )}{8 c^{2/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}+\frac {\sqrt [3]{d} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt {3}\right )}{2 \sqrt {2} \sqrt [4]{3} c^{2/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}} \]
[Out]
-1/8*d^(1/3)*arctanh(c^(1/6)*(c^(1/3)-2^(1/3)*d^(1/3)*x)/(d*x^3+c)^(1/2))*2^(1/3)/c^(5/6)+1/24*d^(1/3)*arctanh
((d*x^3+c)^(1/2)/c^(1/2))*2^(1/3)/c^(5/6)-1/24*d^(1/3)*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)/c^(5/6)*3^(1/2)+1/24*d^(1/3)*arctan(1/3*(d*x^3+c)^(1/2)*3^(1/2)/c^(1/2))*2^(1/3)/c^(5/6)*
3^(1/2)-1/4*(d*x^3+c)^(1/2)/c/x+1/4*d^(1/3)*(d*x^3+c)^(1/2)/c/(d^(1/3)*x+c^(1/3)*(1+3^(1/2)))+1/12*d^(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)*((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)/c^(2/3)*2^(1/2)/(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)-1/8*3^(1/4)*d^(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)/c^(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 = 0.27, antiderivative size = 697, normalized size of antiderivative = 1.00, 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 = {486, 598, 309,
224, 1891, 497} \begin {gather*} \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 (\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 \sqrt {2} \sqrt [4]{3} c^{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]{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 (\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 )}{8 c^{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 [3]{d} \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 )}{4\ 2^{2/3} \sqrt {3} c^{5/6}}+\frac {\sqrt [3]{d} \text {ArcTan}\left (\frac {\sqrt {c+d x^3}}{\sqrt {3} \sqrt {c}}\right )}{4\ 2^{2/3} \sqrt {3} c^{5/6}}-\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 )}{4\ 2^{2/3} c^{5/6}}+\frac {\sqrt [3]{d} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{12\ 2^{2/3} c^{5/6}}-\frac {\sqrt {c+d x^3}}{4 c x}+\frac {\sqrt [3]{d} \sqrt {c+d x^3}}{4 c \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[Sqrt[c + d*x^3]/(x^2*(4*c + d*x^3)),x]
[Out]
-1/4*Sqrt[c + d*x^3]/(c*x) + (d^(1/3)*Sqrt[c + d*x^3])/(4*c*((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)) - (d^(1/3)*Ar
cTan[(Sqrt[3]*c^(1/6)*(c^(1/3) + 2^(1/3)*d^(1/3)*x))/Sqrt[c + d*x^3]])/(4*2^(2/3)*Sqrt[3]*c^(5/6)) + (d^(1/3)*
ArcTan[Sqrt[c + d*x^3]/(Sqrt[3]*Sqrt[c])])/(4*2^(2/3)*Sqrt[3]*c^(5/6)) - (d^(1/3)*ArcTanh[(c^(1/6)*(c^(1/3) -
2^(1/3)*d^(1/3)*x))/Sqrt[c + d*x^3]])/(4*2^(2/3)*c^(5/6)) + (d^(1/3)*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]])/(12*2^(
2/3)*c^(5/6)) - (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^(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]) + (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^(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 486
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/(a*e*(m + 1))), x] - Dist[1/(a*e^n*(m + 1)), Int[(e*x)^(m + n)*(a + b*
x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*b*(m + 1) + n*(b*c*(p + 1) + a*d*q) + d*(b*(m + 1) + b*n*(p + q + 1))*x^n, x
], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[0, q, 1] && LtQ[m, -1] &&
IntBinomialQ[a, b, c, d, e, m, n, p, q, 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 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]
Rubi steps
\begin {align*} \int \frac {\sqrt {c+d x^3}}{x^2 \left (4 c+d x^3\right )} \, dx &=-\frac {\sqrt {c+d x^3}}{4 c x}+\frac {\int \frac {x \left (5 c d+\frac {d^2 x^3}{2}\right )}{\sqrt {c+d x^3} \left (4 c+d x^3\right )} \, dx}{4 c}\\ &=-\frac {\sqrt {c+d x^3}}{4 c x}+\frac {\int \left (\frac {d x}{2 \sqrt {c+d x^3}}+\frac {3 c d x}{\sqrt {c+d x^3} \left (4 c+d x^3\right )}\right ) \, dx}{4 c}\\ &=-\frac {\sqrt {c+d x^3}}{4 c x}+\frac {1}{4} (3 d) \int \frac {x}{\sqrt {c+d x^3} \left (4 c+d x^3\right )} \, dx+\frac {d \int \frac {x}{\sqrt {c+d x^3}} \, dx}{8 c}\\ &=-\frac {\sqrt {c+d x^3}}{4 c 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 )}{4\ 2^{2/3} \sqrt {3} c^{5/6}}+\frac {\sqrt [3]{d} \tan ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {3} \sqrt {c}}\right )}{4\ 2^{2/3} \sqrt {3} c^{5/6}}-\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 )}{4\ 2^{2/3} c^{5/6}}+\frac {\sqrt [3]{d} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{12\ 2^{2/3} c^{5/6}}+\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}+\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^{2/3}}\\ &=-\frac {\sqrt {c+d x^3}}{4 c x}+\frac {\sqrt [3]{d} \sqrt {c+d x^3}}{4 c \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 )}{4\ 2^{2/3} \sqrt {3} c^{5/6}}+\frac {\sqrt [3]{d} \tan ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {3} \sqrt {c}}\right )}{4\ 2^{2/3} \sqrt {3} c^{5/6}}-\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 )}{4\ 2^{2/3} c^{5/6}}+\frac {\sqrt [3]{d} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{12\ 2^{2/3} c^{5/6}}-\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^{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 [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^{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 = 20.07, size = 136, normalized size = 0.20 \begin {gather*} \frac {-40 c \left (c+d x^3\right )+25 c d x^3 \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 )+d^2 x^6 \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}{4 c}\right )}{160 c^2 x \sqrt {c+d x^3}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[Sqrt[c + d*x^3]/(x^2*(4*c + d*x^3)),x]
[Out]
(-40*c*(c + d*x^3) + 25*c*d*x^3*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), -1/4*(d*x^3)/c] +
d^2*x^6*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), -1/4*(d*x^3)/c])/(160*c^2*x*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 = 1306, normalized size = 1.87
| | |
| method |
result |
size |
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| elliptic |
\(\text {Expression too large to display}\) |
\(868\) |
| risch |
\(\text {Expression too large to display}\) |
\(872\) |
| default |
\(\text {Expression too large to display}\) |
\(1306\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d*x^3+c)^(1/2)/x^2/(d*x^3+4*c),x,method=_RETURNVERBOSE)
[Out]
-1/4*d/c*(-2/3*I*3^(1/2)/d*(-c*d^2)^(1/3)*(I*(x+1/2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d
/(-c*d^2)^(1/3))^(1/2)*((x-1/d*(-c*d^2)^(1/3))/(-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3)))^(1/2)*(
-I*(x+1/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2)/(d*x^3+c)^(1/2)*((-
3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*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)))+1/4/c*(-(d*x^3+c)^(1/2)/x-I*3^(1/2)*(-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))))
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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((d*x^3+c)^(1/2)/x^2/(d*x^3+4*c),x, algorithm="maxima")
[Out]
integrate(sqrt(d*x^3 + c)/((d*x^3 + 4*c)*x^2), x)
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 2.71, size = 2319, normalized size = 3.33 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x^3+c)^(1/2)/x^2/(d*x^3+4*c),x, algorithm="fricas")
[Out]
1/48*(4*sqrt(3)*(1/432)^(1/6)*c*x*(-d^2/c^5)^(1/6)*arctan(-1/3*(3*(sqrt(3)*sqrt(1/3)*c^3*d^2*x*sqrt(-d^2/c^5)
+ 2*sqrt(3)*(1/432)^(1/6)*c*d^3*x^2*(-d^2/c^5)^(1/6) + 24*sqrt(3)*(1/432)^(5/6)*(c^4*d^2*x^3 + 4*c^5*d)*(-d^2/
c^5)^(5/6))*sqrt(d*x^3 + c) + (2*sqrt(3)*(1/2)^(1/3)*(c^2*d^2*x^3 + c^3*d)*(-d^2/c^5)^(1/3) + sqrt(3)*(d^3*x^4
+ c*d^2*x) + 3*(sqrt(3)*sqrt(1/3)*c^3*d*x*sqrt(-d^2/c^5) + 2*sqrt(3)*(1/432)^(1/6)*c*d^2*x^2*(-d^2/c^5)^(1/6)
- 24*sqrt(3)*(1/432)^(5/6)*(c^4*d*x^3 - 2*c^5)*(-d^2/c^5)^(5/6))*sqrt(d*x^3 + c))*sqrt((d^5*x^9 + 60*c*d^4*x^
6 - 32*c^3*d^2 - 24*(1/2)^(2/3)*(c^4*d^3*x^7 + 5*c^5*d^2*x^4 + 4*c^6*d*x)*(-d^2/c^5)^(2/3) + 12*(1/2)^(1/3)*(c
^2*d^4*x^8 - 7*c^3*d^3*x^5 - 8*c^4*d^2*x^2)*(-d^2/c^5)^(1/3) + 12*(648*(1/432)^(5/6)*c^5*d^2*x^5*(-d^2/c^5)^(5
/6) - sqrt(1/3)*(c^3*d^3*x^6 - 16*c^4*d^2*x^3 - 8*c^5*d)*sqrt(-d^2/c^5) - (1/432)^(1/6)*(c*d^4*x^7 + 2*c^2*d^3
*x^4 - 8*c^3*d^2*x)*(-d^2/c^5)^(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)))/(d^4
*x^4 + c*d^3*x)) + 4*sqrt(3)*(1/432)^(1/6)*c*x*(-d^2/c^5)^(1/6)*arctan(-1/3*(3*(sqrt(3)*sqrt(1/3)*c^3*d^2*x*sq
rt(-d^2/c^5) + 2*sqrt(3)*(1/432)^(1/6)*c*d^3*x^2*(-d^2/c^5)^(1/6) + 24*sqrt(3)*(1/432)^(5/6)*(c^4*d^2*x^3 + 4*
c^5*d)*(-d^2/c^5)^(5/6))*sqrt(d*x^3 + c) - (2*sqrt(3)*(1/2)^(1/3)*(c^2*d^2*x^3 + c^3*d)*(-d^2/c^5)^(1/3) + sqr
t(3)*(d^3*x^4 + c*d^2*x) - 3*(sqrt(3)*sqrt(1/3)*c^3*d*x*sqrt(-d^2/c^5) + 2*sqrt(3)*(1/432)^(1/6)*c*d^2*x^2*(-d
^2/c^5)^(1/6) - 24*sqrt(3)*(1/432)^(5/6)*(c^4*d*x^3 - 2*c^5)*(-d^2/c^5)^(5/6))*sqrt(d*x^3 + c))*sqrt((d^5*x^9
+ 60*c*d^4*x^6 - 32*c^3*d^2 - 24*(1/2)^(2/3)*(c^4*d^3*x^7 + 5*c^5*d^2*x^4 + 4*c^6*d*x)*(-d^2/c^5)^(2/3) + 12*(
1/2)^(1/3)*(c^2*d^4*x^8 - 7*c^3*d^3*x^5 - 8*c^4*d^2*x^2)*(-d^2/c^5)^(1/3) - 12*(648*(1/432)^(5/6)*c^5*d^2*x^5*
(-d^2/c^5)^(5/6) - sqrt(1/3)*(c^3*d^3*x^6 - 16*c^4*d^2*x^3 - 8*c^5*d)*sqrt(-d^2/c^5) - (1/432)^(1/6)*(c*d^4*x^
7 + 2*c^2*d^3*x^4 - 8*c^3*d^2*x)*(-d^2/c^5)^(1/6))*sqrt(d*x^3 + c))/(d^3*x^9 + 12*c*d^2*x^6 + 48*c^2*d*x^3 + 6
4*c^3)))/(d^4*x^4 + c*d^3*x)) - (1/432)^(1/6)*c*x*(-d^2/c^5)^(1/6)*log((d^5*x^9 + 60*c*d^4*x^6 - 32*c^3*d^2 -
24*(1/2)^(2/3)*(c^4*d^3*x^7 + 5*c^5*d^2*x^4 + 4*c^6*d*x)*(-d^2/c^5)^(2/3) + 12*(1/2)^(1/3)*(c^2*d^4*x^8 - 7*c^
3*d^3*x^5 - 8*c^4*d^2*x^2)*(-d^2/c^5)^(1/3) + 12*(648*(1/432)^(5/6)*c^5*d^2*x^5*(-d^2/c^5)^(5/6) - sqrt(1/3)*(
c^3*d^3*x^6 - 16*c^4*d^2*x^3 - 8*c^5*d)*sqrt(-d^2/c^5) - (1/432)^(1/6)*(c*d^4*x^7 + 2*c^2*d^3*x^4 - 8*c^3*d^2*
x)*(-d^2/c^5)^(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)*c*x*(-
d^2/c^5)^(1/6)*log((d^5*x^9 + 60*c*d^4*x^6 - 32*c^3*d^2 - 24*(1/2)^(2/3)*(c^4*d^3*x^7 + 5*c^5*d^2*x^4 + 4*c^6*
d*x)*(-d^2/c^5)^(2/3) + 12*(1/2)^(1/3)*(c^2*d^4*x^8 - 7*c^3*d^3*x^5 - 8*c^4*d^2*x^2)*(-d^2/c^5)^(1/3) - 12*(64
8*(1/432)^(5/6)*c^5*d^2*x^5*(-d^2/c^5)^(5/6) - sqrt(1/3)*(c^3*d^3*x^6 - 16*c^4*d^2*x^3 - 8*c^5*d)*sqrt(-d^2/c^
5) - (1/432)^(1/6)*(c*d^4*x^7 + 2*c^2*d^3*x^4 - 8*c^3*d^2*x)*(-d^2/c^5)^(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)) + 2*(1/432)^(1/6)*c*x*(-d^2/c^5)^(1/6)*log((d^4*x^9 - 66*c*d^3*x^6 - 72*c^
2*d^2*x^3 - 32*c^3*d + 48*(1/2)^(2/3)*(c^4*d^2*x^7 - c^5*d*x^4 - 2*c^6*x)*(-d^2/c^5)^(2/3) + 12*(1/2)^(1/3)*(c
^2*d^3*x^8 - 7*c^3*d^2*x^5 - 8*c^4*d*x^2)*(-d^2/c^5)^(1/3) + 6*(1296*(1/432)^(5/6)*c^5*d*x^5*(-d^2/c^5)^(5/6)
+ sqrt(1/3)*(5*c^3*d^2*x^6 - 20*c^4*d*x^3 - 16*c^5)*sqrt(-d^2/c^5) + 2*(1/432)^(1/6)*(c*d^3*x^7 - 16*c^2*d^2*x
^4 - 8*c^3*d*x)*(-d^2/c^5)^(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)) - 2*(1/43
2)^(1/6)*c*x*(-d^2/c^5)^(1/6)*log((d^4*x^9 - 66*c*d^3*x^6 - 72*c^2*d^2*x^3 - 32*c^3*d + 48*(1/2)^(2/3)*(c^4*d^
2*x^7 - c^5*d*x^4 - 2*c^6*x)*(-d^2/c^5)^(2/3) + 12*(1/2)^(1/3)*(c^2*d^3*x^8 - 7*c^3*d^2*x^5 - 8*c^4*d*x^2)*(-d
^2/c^5)^(1/3) - 6*(1296*(1/432)^(5/6)*c^5*d*x^5*(-d^2/c^5)^(5/6) + sqrt(1/3)*(5*c^3*d^2*x^6 - 20*c^4*d*x^3 - 1
6*c^5)*sqrt(-d^2/c^5) + 2*(1/432)^(1/6)*(c*d^3*x^7 - 16*c^2*d^2*x^4 - 8*c^3*d*x)*(-d^2/c^5)^(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)) - 12*sqrt(d)*x*weierstrassZeta(0, -4*c/d, weierstrassP
Inverse(0, -4*c/d, x)) - 12*sqrt(d*x^3 + c))/(c*x)
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {c + d x^{3}}}{x^{2} \cdot \left (4 c + d x^{3}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x**3+c)**(1/2)/x**2/(d*x**3+4*c),x)
[Out]
Integral(sqrt(c + d*x**3)/(x**2*(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((d*x^3+c)^(1/2)/x^2/(d*x^3+4*c),x, algorithm="giac")
[Out]
integrate(sqrt(d*x^3 + c)/((d*x^3 + 4*c)*x^2), x)
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {d\,x^3+c}}{x^2\,\left (d\,x^3+4\,c\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((c + d*x^3)^(1/2)/(x^2*(4*c + d*x^3)),x)
[Out]
int((c + d*x^3)^(1/2)/(x^2*(4*c + d*x^3)), x)
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