∫ x4√ ____ c + dx3 _ 4c+dx3 dx [264]

3.3.64 \(\int \frac {x^4 \sqrt {c+d x^3}}{4 c+d x^3} \, dx\) [264]

Optimal. Leaf size=689 \[ \frac {2 x^2 \sqrt {c+d x^3}}{7 d}-\frac {50 c \sqrt {c+d x^3}}{7 d^{5/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}-\frac {2 \sqrt [3]{2} c^{7/6} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{2} \sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{\sqrt {3} d^{5/3}}+\frac {2 \sqrt [3]{2} c^{7/6} \tan ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {3} \sqrt {c}}\right )}{\sqrt {3} d^{5/3}}-\frac {2 \sqrt [3]{2} c^{7/6} \tanh ^{-1}\left (\frac {\sqrt [6]{c} \left (\sqrt [3]{c}-\sqrt [3]{2} \sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{d^{5/3}}+\frac {2 \sqrt [3]{2} c^{7/6} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{3 d^{5/3}}+\frac {25 \sqrt [4]{3} \sqrt {2-\sqrt {3}} c^{4/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt {3}\right )}{7 d^{5/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {50 \sqrt {2} c^{4/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt {3}\right )}{7 \sqrt [4]{3} d^{5/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}} \]

[Out]

-2*2^(1/3)*c^(7/6)*arctanh(c^(1/6)*(c^(1/3)-2^(1/3)*d^(1/3)*x)/(d*x^3+c)^(1/2))/d^(5/3)+2/3*2^(1/3)*c^(7/6)*ar

ctanh((d*x^3+c)^(1/2)/c^(1/2))/d^(5/3)-2/3*2^(1/3)*c^(7/6)*arctan(c^(1/6)*(c^(1/3)+2^(1/3)*d^(1/3)*x)*3^(1/2)/

(d*x^3+c)^(1/2))/d^(5/3)*3^(1/2)+2/3*2^(1/3)*c^(7/6)*arctan(1/3*(d*x^3+c)^(1/2)*3^(1/2)/c^(1/2))/d^(5/3)*3^(1/

2)+2/7*x^2*(d*x^3+c)^(1/2)/d-50/7*c*(d*x^3+c)^(1/2)/d^(5/3)/(d^(1/3)*x+c^(1/3)*(1+3^(1/2)))-50/21*c^(4/3)*(c^(

1/3)+d^(1/3)*x)*EllipticF((d^(1/3)*x+c^(1/3)*(1-3^(1/2)))/(d^(1/3)*x+c^(1/3)*(1+3^(1/2))),I*3^(1/2)+2*I)*2^(1/

2)*((c^(2/3)-c^(1/3)*d^(1/3)*x+d^(2/3)*x^2)/(d^(1/3)*x+c^(1/3)*(1+3^(1/2)))^2)^(1/2)*3^(3/4)/d^(5/3)/(d*x^3+c)

^(1/2)/(c^(1/3)*(c^(1/3)+d^(1/3)*x)/(d^(1/3)*x+c^(1/3)*(1+3^(1/2)))^2)^(1/2)+25/7*3^(1/4)*c^(4/3)*(c^(1/3)+d^(

1/3)*x)*EllipticE((d^(1/3)*x+c^(1/3)*(1-3^(1/2)))/(d^(1/3)*x+c^(1/3)*(1+3^(1/2))),I*3^(1/2)+2*I)*(1/2*6^(1/2)-

1/2*2^(1/2))*((c^(2/3)-c^(1/3)*d^(1/3)*x+d^(2/3)*x^2)/(d^(1/3)*x+c^(1/3)*(1+3^(1/2)))^2)^(1/2)/d^(5/3)/(d*x^3+

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

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Rubi [A]
time = 0.33, antiderivative size = 689, 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 = {489, 598, 309, 224, 1891, 497} \begin {gather*} -\frac {50 \sqrt {2} c^{4/3} \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 )}{7 \sqrt [4]{3} d^{5/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}+\frac {25 \sqrt [4]{3} \sqrt {2-\sqrt {3}} c^{4/3} \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 )}{7 d^{5/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {2 \sqrt [3]{2} c^{7/6} \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 )}{\sqrt {3} d^{5/3}}+\frac {2 \sqrt [3]{2} c^{7/6} \text {ArcTan}\left (\frac {\sqrt {c+d x^3}}{\sqrt {3} \sqrt {c}}\right )}{\sqrt {3} d^{5/3}}-\frac {2 \sqrt [3]{2} c^{7/6} \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 )}{d^{5/3}}+\frac {2 \sqrt [3]{2} c^{7/6} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{3 d^{5/3}}-\frac {50 c \sqrt {c+d x^3}}{7 d^{5/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac {2 x^2 \sqrt {c+d x^3}}{7 d} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*x^2*Sqrt[c + d*x^3])/(7*d) - (50*c*Sqrt[c + d*x^3])/(7*d^(5/3)*((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)) - (2*2^

(1/3)*c^(7/6)*ArcTan[(Sqrt[3]*c^(1/6)*(c^(1/3) + 2^(1/3)*d^(1/3)*x))/Sqrt[c + d*x^3]])/(Sqrt[3]*d^(5/3)) + (2*

2^(1/3)*c^(7/6)*ArcTan[Sqrt[c + d*x^3]/(Sqrt[3]*Sqrt[c])])/(Sqrt[3]*d^(5/3)) - (2*2^(1/3)*c^(7/6)*ArcTanh[(c^(

1/6)*(c^(1/3) - 2^(1/3)*d^(1/3)*x))/Sqrt[c + d*x^3]])/d^(5/3) + (2*2^(1/3)*c^(7/6)*ArcTanh[Sqrt[c + d*x^3]/Sqr

t[c]])/(3*d^(5/3)) + (25*3^(1/4)*Sqrt[2 - Sqrt[3]]*c^(4/3)*(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]])/(7*d^(5/3)*Sqrt[(c^(1/3)*(c^(1/3) + d^(1/3)*x))/((1 +

Sqrt[3])*c^(1/3) + d^(1/3)*x)^2]*Sqrt[c + d*x^3]) - (50*Sqrt[2]*c^(4/3)*(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]])/(7*3^(1/4)*d^(5/3)*Sqrt[(c^(1/3)*(c^(1/

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

Rule 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 489

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*(m + n*(p + q) + 1))), x] - Dist[e^n/(b*(m + n*(p +

q) + 1)), Int[(e*x)^(m - n)*(a + b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[a*c*(m - n + 1) + (a*d*(m - n + 1) - n*q*(b

*c - a*d))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && GtQ[q, 0] &&

GtQ[m - n + 1, 0] && 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 {x^4 \sqrt {c+d x^3}}{4 c+d x^3} \, dx &=\frac {2 x^2 \sqrt {c+d x^3}}{7 d}-\frac {2 \int \frac {x \left (8 c^2+\frac {25}{2} c d x^3\right )}{\sqrt {c+d x^3} \left (4 c+d x^3\right )} \, dx}{7 d}\\ &=\frac {2 x^2 \sqrt {c+d x^3}}{7 d}-\frac {2 \int \left (\frac {25 c x}{2 \sqrt {c+d x^3}}-\frac {42 c^2 x}{\sqrt {c+d x^3} \left (4 c+d x^3\right )}\right ) \, dx}{7 d}\\ &=\frac {2 x^2 \sqrt {c+d x^3}}{7 d}-\frac {(25 c) \int \frac {x}{\sqrt {c+d x^3}} \, dx}{7 d}+\frac {\left (12 c^2\right ) \int \frac {x}{\sqrt {c+d x^3} \left (4 c+d x^3\right )} \, dx}{d}\\ &=\frac {2 x^2 \sqrt {c+d x^3}}{7 d}-\frac {2 \sqrt [3]{2} c^{7/6} \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 )}{\sqrt {3} d^{5/3}}+\frac {2 \sqrt [3]{2} c^{7/6} \tan ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {3} \sqrt {c}}\right )}{\sqrt {3} d^{5/3}}-\frac {2 \sqrt [3]{2} c^{7/6} \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 )}{d^{5/3}}+\frac {2 \sqrt [3]{2} c^{7/6} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{3 d^{5/3}}-\frac {(25 c) \int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\sqrt {c+d x^3}} \, dx}{7 d^{4/3}}-\frac {\left (25 \sqrt {2 \left (2-\sqrt {3}\right )} c^{4/3}\right ) \int \frac {1}{\sqrt {c+d x^3}} \, dx}{7 d^{4/3}}\\ &=\frac {2 x^2 \sqrt {c+d x^3}}{7 d}-\frac {50 c \sqrt {c+d x^3}}{7 d^{5/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}-\frac {2 \sqrt [3]{2} c^{7/6} \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 )}{\sqrt {3} d^{5/3}}+\frac {2 \sqrt [3]{2} c^{7/6} \tan ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {3} \sqrt {c}}\right )}{\sqrt {3} d^{5/3}}-\frac {2 \sqrt [3]{2} c^{7/6} \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 )}{d^{5/3}}+\frac {2 \sqrt [3]{2} c^{7/6} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{3 d^{5/3}}+\frac {25 \sqrt [4]{3} \sqrt {2-\sqrt {3}} c^{4/3} \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 )}{7 d^{5/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {50 \sqrt {2} c^{4/3} \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 )}{7 \sqrt [4]{3} d^{5/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
time = 4.10, size = 133, normalized size = 0.19 \begin {gather*} \frac {8 x^2 \left (c+d x^3\right )-8 c 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 )-5 d x^5 \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 )}{28 d \sqrt {c+d x^3}} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(8*x^2*(c + d*x^3) - 8*c*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] - 5*

d*x^5*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), -1/4*(d*x^3)/c])/(28*d*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 = 0.41, size = 1309, normalized size = 1.90


method	result	size

elliptic	\(\text {Expression too large to display}\)	\(867\)
risch	\(\text {Expression too large to display}\)	\(872\)
default	\(\text {Expression too large to display}\)	\(1309\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(2/7*x^2*(d*x^3+c)^(1/2)-2/7*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))))-4*c/d*(-2/3*I*3^(1/2)/d*(-c*d^2)^(1/3)*(I*(x+1/2

/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2)*((x-1/d*(-c*d^2)^(1/3))/(-3/

2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3)))^(1/2)*(-I*(x+1/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)

^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2)/(d*x^3+c)^(1/2)*((-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3)

)*EllipticE(1/3*3^(1/2)*(I*(x+1/2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(

1/2),(I*3^(1/2)/d*(-c*d^2)^(1/3)/(-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3)))^(1/2))+1/d*(-c*d^2)^(

1/3)*EllipticF(1/3*3^(1/2)*(I*(x+1/2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3)

)^(1/2),(I*3^(1/2)/d*(-c*d^2)^(1/3)/(-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3)))^(1/2)))+1/3*I/d^3*

2^(1/2)*sum(1/_alpha*(-c*d^2)^(1/3)*(1/2*I*d*(2*x+1/d*(-I*3^(1/2)*(-c*d^2)^(1/3)+(-c*d^2)^(1/3)))/(-c*d^2)^(1/

3))^(1/2)*(d*(x-1/d*(-c*d^2)^(1/3))/(-3*(-c*d^2)^(1/3)+I*3^(1/2)*(-c*d^2)^(1/3)))^(1/2)*(-1/2*I*d*(2*x+1/d*(I*

3^(1/2)*(-c*d^2)^(1/3)+(-c*d^2)^(1/3)))/(-c*d^2)^(1/3))^(1/2)/(d*x^3+c)^(1/2)*(I*(-c*d^2)^(1/3)*_alpha*3^(1/2)

*d-I*3^(1/2)*(-c*d^2)^(2/3)+2*_alpha^2*d^2-(-c*d^2)^(1/3)*_alpha*d-(-c*d^2)^(2/3))*EllipticPi(1/3*3^(1/2)*(I*(

x+1/2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2),1/6/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^4*(d*x^3+c)^(1/2)/(d*x^3+4*c),x, algorithm="maxima")

[Out]

integrate(sqrt(d*x^3 + c)*x^4/(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 = 19.98, size = 3827, normalized size = 5.55 \begin {gather*} \text {Too large to display} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/42*(28*sqrt(3)*(4/27)^(1/6)*d^2*(-c^7/d^10)^(1/6)*arctan(-1/3*(108*4^(2/3)*sqrt(3)*(c^7*d^12*x^16 - 39*c^8*

d^11*x^13 - 72*c^9*d^10*x^10 - 32*c^10*d^9*x^7)*(-c^7/d^10)^(2/3) + 12*4^(1/3)*sqrt(3)*(c^9*d^9*x^17 - 271*c^1

0*d^8*x^14 + 112*c^11*d^7*x^11 + 1216*c^12*d^6*x^8 + 1088*c^13*d^5*x^5 + 256*c^14*d^4*x^2)*(-c^7/d^10)^(1/3) +

3*sqrt(1/3)*(324*sqrt(3)*(4/27)^(5/6)*(d^14*x^16 + 229*c*d^13*x^13 + 492*c^2*d^12*x^10 + 328*c^3*d^11*x^7 + 6

4*c^4*d^10*x^4)*(-c^7/d^10)^(5/6) + 12*sqrt(3)*sqrt(1/3)*(c^2*d^11*x^17 + 737*c^3*d^10*x^14 + 2704*c^4*d^9*x^1

1 + 3376*c^5*d^8*x^8 + 1664*c^6*d^7*x^5 + 256*c^7*d^6*x^2)*sqrt(-c^7/d^10) + sqrt(3)*(4/27)^(1/6)*(c^4*d^8*x^1

8 + 1098*c^5*d^7*x^15 - 24720*c^6*d^6*x^12 - 56704*c^7*d^5*x^9 - 44928*c^8*d^4*x^6 - 15360*c^9*d^3*x^3 - 2048*

c^10*d^2)*(-c^7/d^10)^(1/6) - sqrt(d*x^3 + c)*(4^(2/3)*sqrt(3)*(5*c*d^12*x^15 - 3272*c^2*d^11*x^12 - 12544*c^3

*d^10*x^9 - 14656*c^4*d^9*x^6 - 6656*c^5*d^8*x^3 - 1024*c^6*d^7)*(-c^7/d^10)^(2/3) - 1728*4^(1/3)*sqrt(3)*(c^4

*d^8*x^13 + 2*c^5*d^7*x^10 + c^6*d^6*x^7)*(-c^7/d^10)^(1/3) - 12*sqrt(3)*(17*c^6*d^5*x^14 - 1456*c^7*d^4*x^11

- 2544*c^8*d^3*x^8 - 1408*c^9*d^2*x^5 - 256*c^10*d*x^2)))*sqrt((24*c^12*d^2*x^8 - 168*c^13*d*x^5 - 192*c^14*x^

2 - 4^(2/3)*(c^7*d^9*x^9 + 60*c^8*d^8*x^6 - 32*c^10*d^6)*(-c^7/d^10)^(2/3) - 24*4^(1/3)*(c^10*d^5*x^7 + 5*c^11

*d^4*x^4 + 4*c^12*d^3*x)*(-c^7/d^10)^(1/3) + 6*(36*sqrt(1/3)*c^9*d^6*x^5*sqrt(-c^7/d^10) + 9*(4/27)^(5/6)*(c^6

*d^10*x^7 + 2*c^7*d^9*x^4 - 8*c^8*d^8*x)*(-c^7/d^10)^(5/6) - 4*(4/27)^(1/6)*(c^11*d^3*x^6 - 16*c^12*d^2*x^3 -

8*c^13*d)*(-c^7/d^10)^(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^1

1*d^6*x^18 - 1416*c^12*d^5*x^15 + 14352*c^13*d^4*x^12 + 44480*c^14*d^3*x^9 + 49920*c^15*d^2*x^6 + 24576*c^16*d

*x^3 + 4096*c^17) - 6*sqrt(d*x^3 + c)*(27*sqrt(3)*(4/27)^(5/6)*(31*c^6*d^13*x^14 + 1744*c^7*d^12*x^11 + 2976*c

^8*d^11*x^8 + 1600*c^9*d^10*x^5 + 256*c^10*d^9*x^2)*(-c^7/d^10)^(5/6) + 24*sqrt(3)*sqrt(1/3)*(c^8*d^10*x^15 +

157*c^9*d^9*x^12 + 348*c^10*d^8*x^9 + 256*c^11*d^7*x^6 + 64*c^12*d^6*x^3)*sqrt(-c^7/d^10) + 2*sqrt(3)*(4/27)^(

1/6)*(c^10*d^7*x^16 + 686*c^11*d^6*x^13 + 7072*c^12*d^5*x^10 + 11008*c^13*d^4*x^7 + 5888*c^14*d^3*x^4 + 1024*c

^15*d^2*x)*(-c^7/d^10)^(1/6)))/(c^11*d^6*x^18 + 2184*c^12*d^5*x^15 + 57696*c^13*d^4*x^12 + 125696*c^14*d^3*x^9

+ 100608*c^15*d^2*x^6 + 33792*c^16*d*x^3 + 4096*c^17)) - 28*sqrt(3)*(4/27)^(1/6)*d^2*(-c^7/d^10)^(1/6)*arctan

(-1/3*(108*4^(2/3)*sqrt(3)*(c^7*d^12*x^16 - 39*c^8*d^11*x^13 - 72*c^9*d^10*x^10 - 32*c^10*d^9*x^7)*(-c^7/d^10)

^(2/3) + 12*4^(1/3)*sqrt(3)*(c^9*d^9*x^17 - 271*c^10*d^8*x^14 + 112*c^11*d^7*x^11 + 1216*c^12*d^6*x^8 + 1088*c

^13*d^5*x^5 + 256*c^14*d^4*x^2)*(-c^7/d^10)^(1/3) - 3*sqrt(1/3)*(324*sqrt(3)*(4/27)^(5/6)*(d^14*x^16 + 229*c*d

^13*x^13 + 492*c^2*d^12*x^10 + 328*c^3*d^11*x^7 + 64*c^4*d^10*x^4)*(-c^7/d^10)^(5/6) + 12*sqrt(3)*sqrt(1/3)*(c

^2*d^11*x^17 + 737*c^3*d^10*x^14 + 2704*c^4*d^9*x^11 + 3376*c^5*d^8*x^8 + 1664*c^6*d^7*x^5 + 256*c^7*d^6*x^2)*

sqrt(-c^7/d^10) + sqrt(3)*(4/27)^(1/6)*(c^4*d^8*x^18 + 1098*c^5*d^7*x^15 - 24720*c^6*d^6*x^12 - 56704*c^7*d^5*

x^9 - 44928*c^8*d^4*x^6 - 15360*c^9*d^3*x^3 - 2048*c^10*d^2)*(-c^7/d^10)^(1/6) + sqrt(d*x^3 + c)*(4^(2/3)*sqrt

(3)*(5*c*d^12*x^15 - 3272*c^2*d^11*x^12 - 12544*c^3*d^10*x^9 - 14656*c^4*d^9*x^6 - 6656*c^5*d^8*x^3 - 1024*c^6

*d^7)*(-c^7/d^10)^(2/3) - 1728*4^(1/3)*sqrt(3)*(c^4*d^8*x^13 + 2*c^5*d^7*x^10 + c^6*d^6*x^7)*(-c^7/d^10)^(1/3)

- 12*sqrt(3)*(17*c^6*d^5*x^14 - 1456*c^7*d^4*x^11 - 2544*c^8*d^3*x^8 - 1408*c^9*d^2*x^5 - 256*c^10*d*x^2)))*s

qrt((24*c^12*d^2*x^8 - 168*c^13*d*x^5 - 192*c^14*x^2 - 4^(2/3)*(c^7*d^9*x^9 + 60*c^8*d^8*x^6 - 32*c^10*d^6)*(-

c^7/d^10)^(2/3) - 24*4^(1/3)*(c^10*d^5*x^7 + 5*c^11*d^4*x^4 + 4*c^12*d^3*x)*(-c^7/d^10)^(1/3) - 6*(36*sqrt(1/3

)*c^9*d^6*x^5*sqrt(-c^7/d^10) + 9*(4/27)^(5/6)*(c^6*d^10*x^7 + 2*c^7*d^9*x^4 - 8*c^8*d^8*x)*(-c^7/d^10)^(5/6)

- 4*(4/27)^(1/6)*(c^11*d^3*x^6 - 16*c^12*d^2*x^3 - 8*c^13*d)*(-c^7/d^10)^(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^11*d^6*x^18 - 1416*c^12*d^5*x^15 + 14352*c^13*d^4*x^12 + 444

80*c^14*d^3*x^9 + 49920*c^15*d^2*x^6 + 24576*c^16*d*x^3 + 4096*c^17) + 6*sqrt(d*x^3 + c)*(27*sqrt(3)*(4/27)^(5

/6)*(31*c^6*d^13*x^14 + 1744*c^7*d^12*x^11 + 2976*c^8*d^11*x^8 + 1600*c^9*d^10*x^5 + 256*c^10*d^9*x^2)*(-c^7/d

^10)^(5/6) + 24*sqrt(3)*sqrt(1/3)*(c^8*d^10*x^15 + 157*c^9*d^9*x^12 + 348*c^10*d^8*x^9 + 256*c^11*d^7*x^6 + 64

*c^12*d^6*x^3)*sqrt(-c^7/d^10) + 2*sqrt(3)*(4/27)^(1/6)*(c^10*d^7*x^16 + 686*c^11*d^6*x^13 + 7072*c^12*d^5*x^1

0 + 11008*c^13*d^4*x^7 + 5888*c^14*d^3*x^4 + 1024*c^15*d^2*x)*(-c^7/d^10)^(1/6)))/(c^11*d^6*x^18 + 2184*c^12*d

^5*x^15 + 57696*c^13*d^4*x^12 + 125696*c^14*d^3*x^9 + 100608*c^15*d^2*x^6 + 33792*c^16*d*x^3 + 4096*c^17)) - 1

2*sqrt(d*x^3 + c)*d*x^2 + 7*(4/27)^(1/6)*d^2*(-c^7/d^10)^(1/6)*log(16384/3*(24*c^12*d^2*x^8 - 168*c^13*d*x^5 -

192*c^14*x^2 - 4^(2/3)*(c^7*d^9*x^9 + 60*c^8*d^8*x^6 - 32*c^10*d^6)*(-c^7/d^10)^(2/3) - 24*4^(1/3)*(c^10*d^5*

x^7 + 5*c^11*d^4*x^4 + 4*c^12*d^3*x)*(-c^7/d^10...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} \sqrt {c + d x^{3}}}{4 c + d x^{3}}\, dx \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

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

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