Integrand size = 27, antiderivative size = 304 \[ \int \frac {x}{a c+b c x^3+d \sqrt {a+b x^3}} \, dx=-\frac {d x^2 \sqrt {1+\frac {b x^3}{a}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},1,\frac {5}{3},-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )}{2 \left (a c^2-d^2\right ) \sqrt {a+b x^3}}-\frac {\arctan \left (\frac {1-\frac {2 \sqrt [3]{b} c^{2/3} x}{\sqrt [3]{a c^2-d^2}}}{\sqrt {3}}\right )}{\sqrt {3} b^{2/3} \sqrt [3]{c} \sqrt [3]{a c^2-d^2}}-\frac {\log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 b^{2/3} \sqrt [3]{c} \sqrt [3]{a c^2-d^2}}+\frac {\log \left (\left (a c^2-d^2\right )^{2/3}-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+b^{2/3} c^{4/3} x^2\right )}{6 b^{2/3} \sqrt [3]{c} \sqrt [3]{a c^2-d^2}} \] Output:
-1/2*d*x^2*(1+b*x^3/a)^(1/2)*AppellF1(2/3,1/2,1,5/3,-b*x^3/a,-b*c^2*x^3/(a *c^2-d^2))/(a*c^2-d^2)/(b*x^3+a)^(1/2)-1/3*arctan(1/3*(1-2*b^(1/3)*c^(2/3) *x/(a*c^2-d^2)^(1/3))*3^(1/2))*3^(1/2)/b^(2/3)/c^(1/3)/(a*c^2-d^2)^(1/3)-1 /3*ln((a*c^2-d^2)^(1/3)+b^(1/3)*c^(2/3)*x)/b^(2/3)/c^(1/3)/(a*c^2-d^2)^(1/ 3)+1/6*ln((a*c^2-d^2)^(2/3)-b^(1/3)*c^(2/3)*(a*c^2-d^2)^(1/3)*x+b^(2/3)*c^ (4/3)*x^2)/b^(2/3)/c^(1/3)/(a*c^2-d^2)^(1/3)
Time = 10.34 (sec) , antiderivative size = 252, normalized size of antiderivative = 0.83 \[ \int \frac {x}{a c+b c x^3+d \sqrt {a+b x^3}} \, dx=-\frac {d x^2 \sqrt {1+\frac {b x^3}{a}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},1,\frac {5}{3},-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )}{\left (2 a c^2-2 d^2\right ) \sqrt {a+b x^3}}+\frac {2 \sqrt {3} \arctan \left (\frac {-1+\frac {2 \sqrt [3]{b} c^{2/3} x}{\sqrt [3]{a c^2-d^2}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )+\log \left (\left (a c^2-d^2\right )^{2/3}-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+b^{2/3} c^{4/3} x^2\right )}{6 b^{2/3} \sqrt [3]{c} \sqrt [3]{a c^2-d^2}} \] Input:
Integrate[x/(a*c + b*c*x^3 + d*Sqrt[a + b*x^3]),x]
Output:
-((d*x^2*Sqrt[1 + (b*x^3)/a]*AppellF1[2/3, 1/2, 1, 5/3, -((b*x^3)/a), -((b *c^2*x^3)/(a*c^2 - d^2))])/((2*a*c^2 - 2*d^2)*Sqrt[a + b*x^3])) + (2*Sqrt[ 3]*ArcTan[(-1 + (2*b^(1/3)*c^(2/3)*x)/(a*c^2 - d^2)^(1/3))/Sqrt[3]] - 2*Lo g[(a*c^2 - d^2)^(1/3) + b^(1/3)*c^(2/3)*x] + Log[(a*c^2 - d^2)^(2/3) - b^( 1/3)*c^(2/3)*(a*c^2 - d^2)^(1/3)*x + b^(2/3)*c^(4/3)*x^2])/(6*b^(2/3)*c^(1 /3)*(a*c^2 - d^2)^(1/3))
Time = 1.04 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.01, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {2587, 27, 821, 16, 1013, 1012, 1142, 25, 27, 1082, 217, 1103}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x}{d \sqrt {a+b x^3}+a c+b c x^3} \, dx\) |
| \(\Big \downarrow \) 2587 |
| \(\displaystyle a c \int \frac {x}{a \left (b c^2 x^3+a c^2-d^2\right )}dx-a d \int \frac {x}{a \sqrt {b x^3+a} \left (b c^2 x^3+a c^2-d^2\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c \int \frac {x}{b c^2 x^3+a c^2-d^2}dx-d \int \frac {x}{\sqrt {b x^3+a} \left (b c^2 x^3+a c^2-d^2\right )}dx\) |
| \(\Big \downarrow \) 821 |
| \(\displaystyle c \left (\frac {\int \frac {\sqrt [3]{b} c^{2/3} x+\sqrt [3]{a c^2-d^2}}{b^{2/3} c^{4/3} x^2-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+\left (a c^2-d^2\right )^{2/3}}dx}{3 \sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2}}-\frac {\int \frac {1}{\sqrt [3]{b} c^{2/3} x+\sqrt [3]{a c^2-d^2}}dx}{3 \sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2}}\right )-d \int \frac {x}{\sqrt {b x^3+a} \left (b c^2 x^3+a c^2-d^2\right )}dx\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle c \left (\frac {\int \frac {\sqrt [3]{b} c^{2/3} x+\sqrt [3]{a c^2-d^2}}{b^{2/3} c^{4/3} x^2-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+\left (a c^2-d^2\right )^{2/3}}dx}{3 \sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2}}-\frac {\log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 b^{2/3} c^{4/3} \sqrt [3]{a c^2-d^2}}\right )-d \int \frac {x}{\sqrt {b x^3+a} \left (b c^2 x^3+a c^2-d^2\right )}dx\) |
| \(\Big \downarrow \) 1013 |
| \(\displaystyle c \left (\frac {\int \frac {\sqrt [3]{b} c^{2/3} x+\sqrt [3]{a c^2-d^2}}{b^{2/3} c^{4/3} x^2-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+\left (a c^2-d^2\right )^{2/3}}dx}{3 \sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2}}-\frac {\log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 b^{2/3} c^{4/3} \sqrt [3]{a c^2-d^2}}\right )-\frac {d \sqrt {\frac {b x^3}{a}+1} \int \frac {x}{\sqrt {\frac {b x^3}{a}+1} \left (b c^2 x^3+a c^2-d^2\right )}dx}{\sqrt {a+b x^3}}\) |
| \(\Big \downarrow \) 1012 |
| \(\displaystyle c \left (\frac {\int \frac {\sqrt [3]{b} c^{2/3} x+\sqrt [3]{a c^2-d^2}}{b^{2/3} c^{4/3} x^2-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+\left (a c^2-d^2\right )^{2/3}}dx}{3 \sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2}}-\frac {\log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 b^{2/3} c^{4/3} \sqrt [3]{a c^2-d^2}}\right )-\frac {d x^2 \sqrt {\frac {b x^3}{a}+1} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},1,\frac {5}{3},-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )}{2 \sqrt {a+b x^3} \left (a c^2-d^2\right )}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle c \left (\frac {\frac {3}{2} \sqrt [3]{a c^2-d^2} \int \frac {1}{b^{2/3} c^{4/3} x^2-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+\left (a c^2-d^2\right )^{2/3}}dx+\frac {\int -\frac {\sqrt [3]{b} c^{2/3} \left (\sqrt [3]{a c^2-d^2}-2 \sqrt [3]{b} c^{2/3} x\right )}{b^{2/3} c^{4/3} x^2-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+\left (a c^2-d^2\right )^{2/3}}dx}{2 \sqrt [3]{b} c^{2/3}}}{3 \sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2}}-\frac {\log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 b^{2/3} c^{4/3} \sqrt [3]{a c^2-d^2}}\right )-\frac {d x^2 \sqrt {\frac {b x^3}{a}+1} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},1,\frac {5}{3},-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )}{2 \sqrt {a+b x^3} \left (a c^2-d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle c \left (\frac {\frac {3}{2} \sqrt [3]{a c^2-d^2} \int \frac {1}{b^{2/3} c^{4/3} x^2-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+\left (a c^2-d^2\right )^{2/3}}dx-\frac {\int \frac {\sqrt [3]{b} c^{2/3} \left (\sqrt [3]{a c^2-d^2}-2 \sqrt [3]{b} c^{2/3} x\right )}{b^{2/3} c^{4/3} x^2-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+\left (a c^2-d^2\right )^{2/3}}dx}{2 \sqrt [3]{b} c^{2/3}}}{3 \sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2}}-\frac {\log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 b^{2/3} c^{4/3} \sqrt [3]{a c^2-d^2}}\right )-\frac {d x^2 \sqrt {\frac {b x^3}{a}+1} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},1,\frac {5}{3},-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )}{2 \sqrt {a+b x^3} \left (a c^2-d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c \left (\frac {\frac {3}{2} \sqrt [3]{a c^2-d^2} \int \frac {1}{b^{2/3} c^{4/3} x^2-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+\left (a c^2-d^2\right )^{2/3}}dx-\frac {1}{2} \int \frac {\sqrt [3]{a c^2-d^2}-2 \sqrt [3]{b} c^{2/3} x}{b^{2/3} c^{4/3} x^2-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+\left (a c^2-d^2\right )^{2/3}}dx}{3 \sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2}}-\frac {\log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 b^{2/3} c^{4/3} \sqrt [3]{a c^2-d^2}}\right )-\frac {d x^2 \sqrt {\frac {b x^3}{a}+1} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},1,\frac {5}{3},-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )}{2 \sqrt {a+b x^3} \left (a c^2-d^2\right )}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle c \left (\frac {\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{b} c^{2/3} x}{\sqrt [3]{a c^2-d^2}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{b} c^{2/3} x}{\sqrt [3]{a c^2-d^2}}\right )}{\sqrt [3]{b} c^{2/3}}-\frac {1}{2} \int \frac {\sqrt [3]{a c^2-d^2}-2 \sqrt [3]{b} c^{2/3} x}{b^{2/3} c^{4/3} x^2-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+\left (a c^2-d^2\right )^{2/3}}dx}{3 \sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2}}-\frac {\log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 b^{2/3} c^{4/3} \sqrt [3]{a c^2-d^2}}\right )-\frac {d x^2 \sqrt {\frac {b x^3}{a}+1} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},1,\frac {5}{3},-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )}{2 \sqrt {a+b x^3} \left (a c^2-d^2\right )}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle c \left (\frac {-\frac {1}{2} \int \frac {\sqrt [3]{a c^2-d^2}-2 \sqrt [3]{b} c^{2/3} x}{b^{2/3} c^{4/3} x^2-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+\left (a c^2-d^2\right )^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} c^{2/3} x}{\sqrt [3]{a c^2-d^2}}}{\sqrt {3}}\right )}{\sqrt [3]{b} c^{2/3}}}{3 \sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2}}-\frac {\log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 b^{2/3} c^{4/3} \sqrt [3]{a c^2-d^2}}\right )-\frac {d x^2 \sqrt {\frac {b x^3}{a}+1} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},1,\frac {5}{3},-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )}{2 \sqrt {a+b x^3} \left (a c^2-d^2\right )}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle c \left (\frac {\frac {\log \left (-\sqrt [3]{b} c^{2/3} x \sqrt [3]{a c^2-d^2}+\left (a c^2-d^2\right )^{2/3}+b^{2/3} c^{4/3} x^2\right )}{2 \sqrt [3]{b} c^{2/3}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} c^{2/3} x}{\sqrt [3]{a c^2-d^2}}}{\sqrt {3}}\right )}{\sqrt [3]{b} c^{2/3}}}{3 \sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2}}-\frac {\log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 b^{2/3} c^{4/3} \sqrt [3]{a c^2-d^2}}\right )-\frac {d x^2 \sqrt {\frac {b x^3}{a}+1} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},1,\frac {5}{3},-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )}{2 \sqrt {a+b x^3} \left (a c^2-d^2\right )}\) |
Input:
Int[x/(a*c + b*c*x^3 + d*Sqrt[a + b*x^3]),x]
Output:
-1/2*(d*x^2*Sqrt[1 + (b*x^3)/a]*AppellF1[2/3, 1/2, 1, 5/3, -((b*x^3)/a), - ((b*c^2*x^3)/(a*c^2 - d^2))])/((a*c^2 - d^2)*Sqrt[a + b*x^3]) + c*(-1/3*Lo g[(a*c^2 - d^2)^(1/3) + b^(1/3)*c^(2/3)*x]/(b^(2/3)*c^(4/3)*(a*c^2 - d^2)^ (1/3)) + (-((Sqrt[3]*ArcTan[(1 - (2*b^(1/3)*c^(2/3)*x)/(a*c^2 - d^2)^(1/3) )/Sqrt[3]])/(b^(1/3)*c^(2/3))) + Log[(a*c^2 - d^2)^(2/3) - b^(1/3)*c^(2/3) *(a*c^2 - d^2)^(1/3)*x + b^(2/3)*c^(4/3)*x^2]/(2*b^(1/3)*c^(2/3)))/(3*b^(1 /3)*c^(2/3)*(a*c^2 - d^2)^(1/3)))
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> Simp[-(3*Rt[a, 3]*Rt[b, 3])^(- 1) Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]*Rt[b, 3]) Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2 *x^2), x], x] /; FreeQ[{a, b}, x]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[a^p*c^q*((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[a^IntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^ n/a))^FracPart[p]) Int[(e*x)^m*(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] & & NeQ[m, n - 1] && !(IntegerQ[p] || GtQ[a, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[(u_.)/((c_) + (d_.)*(x_)^(n_) + (e_.)*Sqrt[(a_) + (b_.)*(x_)^(n_)]), x_ Symbol] :> Simp[c Int[u/(c^2 - a*e^2 + c*d*x^n), x], x] - Simp[a*e Int[ u/((c^2 - a*e^2 + c*d*x^n)*Sqrt[a + b*x^n]), x], x] /; FreeQ[{a, b, c, d, e , n}, x] && EqQ[b*c - a*d, 0]
Result contains higher order function than in optimal. Order 9 vs. order 6.
Time = 0.52 (sec) , antiderivative size = 665, normalized size of antiderivative = 2.19
| method | result | size |
| elliptic | \(\frac {\sqrt {b \,x^{3}+a}\, \left (d +c \sqrt {b \,x^{3}+a}\right ) \left (-\frac {\ln \left (x +\left (\frac {a \,c^{2}-d^{2}}{b \,c^{2}}\right )^{\frac {1}{3}}\right )}{3 b c \left (\frac {a \,c^{2}-d^{2}}{b \,c^{2}}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a \,c^{2}-d^{2}}{b \,c^{2}}\right )^{\frac {1}{3}} x +\left (\frac {a \,c^{2}-d^{2}}{b \,c^{2}}\right )^{\frac {2}{3}}\right )}{6 b c \left (\frac {a \,c^{2}-d^{2}}{b \,c^{2}}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a \,c^{2}-d^{2}}{b \,c^{2}}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b c \left (\frac {a \,c^{2}-d^{2}}{b \,c^{2}}\right )^{\frac {1}{3}}}-\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (b \,c^{2} \textit {\_Z}^{3}+a \,c^{2}-d^{2}\right )}{\sum }\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {2}\, \sqrt {\frac {i b \left (2 x +\frac {-i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}+\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {b \left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}{-3 \left (-a \,b^{2}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {-\frac {i b \left (2 x +\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}+\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}{2 \left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \left (i \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha b -i \left (-a \,b^{2}\right )^{\frac {2}{3}} \sqrt {3}+2 \underline {\hspace {1.25 ex}}\alpha ^{2} b^{2}-\left (-a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha b -\left (-a \,b^{2}\right )^{\frac {2}{3}}\right ) \operatorname {EllipticPi}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, -\frac {c^{2} \left (2 i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} b -i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha +i \sqrt {3}\, a b -3 \left (-a \,b^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha -3 a b \right )}{2 b \,d^{2}}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{2 \underline {\hspace {1.25 ex}}\alpha \sqrt {b \,x^{3}+a}}\right )}{3 d \,b^{3}}\right )}{a c +b c \,x^{3}+d \sqrt {b \,x^{3}+a}}\) | \(665\) |
| default | \(\text {Expression too large to display}\) | \(1925\) |
Input:
int(x/(a*c+b*c*x^3+d*(b*x^3+a)^(1/2)),x,method=_RETURNVERBOSE)
Output:
(b*x^3+a)^(1/2)*(d+c*(b*x^3+a)^(1/2))/(a*c+b*c*x^3+d*(b*x^3+a)^(1/2))*(-1/ 3/b/c/((a*c^2-d^2)/b/c^2)^(1/3)*ln(x+((a*c^2-d^2)/b/c^2)^(1/3))+1/6/b/c/(( a*c^2-d^2)/b/c^2)^(1/3)*ln(x^2-((a*c^2-d^2)/b/c^2)^(1/3)*x+((a*c^2-d^2)/b/ c^2)^(2/3))+1/3/b/c*3^(1/2)/((a*c^2-d^2)/b/c^2)^(1/3)*arctan(1/3*3^(1/2)*( 2/((a*c^2-d^2)/b/c^2)^(1/3)*x-1))-1/3*I/d/b^3*2^(1/2)*sum(1/_alpha*(-a*b^2 )^(1/3)*(1/2*I*b*(2*x+1/b*(-I*3^(1/2)*(-a*b^2)^(1/3)+(-a*b^2)^(1/3)))/(-a* b^2)^(1/3))^(1/2)*(b*(x-1/b*(-a*b^2)^(1/3))/(-3*(-a*b^2)^(1/3)+I*3^(1/2)*( -a*b^2)^(1/3)))^(1/2)*(-1/2*I*b*(2*x+1/b*(I*3^(1/2)*(-a*b^2)^(1/3)+(-a*b^2 )^(1/3)))/(-a*b^2)^(1/3))^(1/2)/(b*x^3+a)^(1/2)*(I*(-a*b^2)^(1/3)*3^(1/2)* _alpha*b-I*(-a*b^2)^(2/3)*3^(1/2)+2*_alpha^2*b^2-(-a*b^2)^(1/3)*_alpha*b-( -a*b^2)^(2/3))*EllipticPi(1/3*3^(1/2)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^( 1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2),-1/2/b*c^2*(2*I*3^( 1/2)*(-a*b^2)^(1/3)*_alpha^2*b-I*3^(1/2)*(-a*b^2)^(2/3)*_alpha+I*3^(1/2)*a *b-3*(-a*b^2)^(2/3)*_alpha-3*a*b)/d^2,(I*3^(1/2)/b*(-a*b^2)^(1/3)/(-3/2/b* (-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2)),_alpha=RootOf(_Z^3* b*c^2+a*c^2-d^2)))
Timed out. \[ \int \frac {x}{a c+b c x^3+d \sqrt {a+b x^3}} \, dx=\text {Timed out} \] Input:
integrate(x/(a*c+b*c*x^3+d*(b*x^3+a)^(1/2)),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {x}{a c+b c x^3+d \sqrt {a+b x^3}} \, dx=\int \frac {x}{a c + b c x^{3} + d \sqrt {a + b x^{3}}}\, dx \] Input:
integrate(x/(a*c+b*c*x**3+d*(b*x**3+a)**(1/2)),x)
Output:
Integral(x/(a*c + b*c*x**3 + d*sqrt(a + b*x**3)), x)
\[ \int \frac {x}{a c+b c x^3+d \sqrt {a+b x^3}} \, dx=\int { \frac {x}{b c x^{3} + a c + \sqrt {b x^{3} + a} d} \,d x } \] Input:
integrate(x/(a*c+b*c*x^3+d*(b*x^3+a)^(1/2)),x, algorithm="maxima")
Output:
integrate(x/(b*c*x^3 + a*c + sqrt(b*x^3 + a)*d), x)
\[ \int \frac {x}{a c+b c x^3+d \sqrt {a+b x^3}} \, dx=\int { \frac {x}{b c x^{3} + a c + \sqrt {b x^{3} + a} d} \,d x } \] Input:
integrate(x/(a*c+b*c*x^3+d*(b*x^3+a)^(1/2)),x, algorithm="giac")
Output:
integrate(x/(b*c*x^3 + a*c + sqrt(b*x^3 + a)*d), x)
Timed out. \[ \int \frac {x}{a c+b c x^3+d \sqrt {a+b x^3}} \, dx=\int \frac {x}{a\,c+d\,\sqrt {b\,x^3+a}+b\,c\,x^3} \,d x \] Input:
int(x/(a*c + d*(a + b*x^3)^(1/2) + b*c*x^3),x)
Output:
int(x/(a*c + d*(a + b*x^3)^(1/2) + b*c*x^3), x)
\[ \int \frac {x}{a c+b c x^3+d \sqrt {a+b x^3}} \, dx=\frac {-2 \sqrt {3}\, \mathit {atan} \left (\frac {\left (a \,c^{2}-d^{2}\right )^{\frac {1}{3}}-2 c^{\frac {2}{3}} b^{\frac {1}{3}} x}{\left (a \,c^{2}-d^{2}\right )^{\frac {1}{3}} \sqrt {3}}\right )-6 c^{\frac {1}{3}} b^{\frac {2}{3}} \left (a \,c^{2}-d^{2}\right )^{\frac {1}{3}} \left (\int \frac {\sqrt {b \,x^{3}+a}\, x}{b^{2} c^{2} x^{6}+2 a b \,c^{2} x^{3}-b \,d^{2} x^{3}+a^{2} c^{2}-a \,d^{2}}d x \right ) d +\mathrm {log}\left (\left (a \,c^{2}-d^{2}\right )^{\frac {2}{3}}-c^{\frac {2}{3}} b^{\frac {1}{3}} \left (a \,c^{2}-d^{2}\right )^{\frac {1}{3}} x +c^{\frac {4}{3}} b^{\frac {2}{3}} x^{2}\right )-2 \,\mathrm {log}\left (\left (a \,c^{2}-d^{2}\right )^{\frac {1}{3}}+c^{\frac {2}{3}} b^{\frac {1}{3}} x \right )}{6 c^{\frac {1}{3}} b^{\frac {2}{3}} \left (a \,c^{2}-d^{2}\right )^{\frac {1}{3}}} \] Input:
int(x/(a*c+b*c*x^3+d*(b*x^3+a)^(1/2)),x)
Output:
( - 2*sqrt(3)*atan(((a*c**2 - d**2)**(1/3) - 2*c**(2/3)*b**(1/3)*x)/((a*c* *2 - d**2)**(1/3)*sqrt(3))) - 6*c**(1/3)*b**(2/3)*(a*c**2 - d**2)**(1/3)*i nt((sqrt(a + b*x**3)*x)/(a**2*c**2 + 2*a*b*c**2*x**3 - a*d**2 + b**2*c**2* x**6 - b*d**2*x**3),x)*d + log((a*c**2 - d**2)**(2/3) - c**(2/3)*b**(1/3)* (a*c**2 - d**2)**(1/3)*x + c**(1/3)*b**(2/3)*c*x**2) - 2*log((a*c**2 - d** 2)**(1/3) + c**(2/3)*b**(1/3)*x))/(6*c**(1/3)*b**(2/3)*(a*c**2 - d**2)**(1 /3))