Integrand size = 25, antiderivative size = 300 \[ \int \frac {1}{a c+b c x^3+d \sqrt {a+b x^3}} \, dx=-\frac {d x \sqrt {1+\frac {b x^3}{a}} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )}{\left (a c^2-d^2\right ) \sqrt {a+b x^3}}-\frac {\sqrt [3]{c} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} c^{2/3} x}{\sqrt [3]{a c^2-d^2}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{b} \left (a c^2-d^2\right )^{2/3}}+\frac {\sqrt [3]{c} \log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 \sqrt [3]{b} \left (a c^2-d^2\right )^{2/3}}-\frac {\sqrt [3]{c} \log \left (\left (a c^2-d^2\right )^{2/3}-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+b^{2/3} c^{4/3} x^2\right )}{6 \sqrt [3]{b} \left (a c^2-d^2\right )^{2/3}} \]
1/3*c^(1/3)*ln((a*c^2-d^2)^(1/3)+b^(1/3)*c^(2/3)*x)/b^(1/3)/(a*c^2-d^2)^(2 /3)-1/6*c^(1/3)*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^(1/3)/(a*c^2-d^2)^(2/3)-1/3*c^(1/3)*arctan(1/3*(1-2* b^(1/3)*c^(2/3)*x/(a*c^2-d^2)^(1/3))*3^(1/2))/b^(1/3)/(a*c^2-d^2)^(2/3)*3^ (1/2)-d*x*AppellF1(1/3,1/2,1,4/3,-b*x^3/a,-b*c^2*x^3/(a*c^2-d^2))*(1+b*x^3 /a)^(1/2)/(a*c^2-d^2)/(b*x^3+a)^(1/2)
Time = 10.46 (sec) , antiderivative size = 425, normalized size of antiderivative = 1.42 \[ \int \frac {1}{a c+b c x^3+d \sqrt {a+b x^3}} \, dx=-\frac {8 a d \left (a c^2-d^2\right ) x \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )}{\sqrt {a+b x^3} \left (a c^2-d^2+b c^2 x^3\right ) \left (8 a \left (a c^2-d^2\right ) \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )-3 b x^3 \left (2 a c^2 \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},2,\frac {7}{3},-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )+\left (a c^2-d^2\right ) \operatorname {AppellF1}\left (\frac {4}{3},\frac {3}{2},1,\frac {7}{3},-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )\right )\right )}+\frac {\sqrt [3]{c} \left (2 \sqrt {3} \arctan \left (\frac {-1+\frac {2 \sqrt [3]{b} c^{2/3} x}{\sqrt [3]{a c^2-d^2}}}{\sqrt {3}}\right )+2 \log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )-\log \left (\left (a c^2-d^2\right )^{2/3}-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+b^{2/3} c^{4/3} x^2\right )\right )}{6 \sqrt [3]{b} \left (a c^2-d^2\right )^{2/3}} \]
(-8*a*d*(a*c^2 - d^2)*x*AppellF1[1/3, 1/2, 1, 4/3, -((b*x^3)/a), -((b*c^2* x^3)/(a*c^2 - d^2))])/(Sqrt[a + b*x^3]*(a*c^2 - d^2 + b*c^2*x^3)*(8*a*(a*c ^2 - d^2)*AppellF1[1/3, 1/2, 1, 4/3, -((b*x^3)/a), -((b*c^2*x^3)/(a*c^2 - d^2))] - 3*b*x^3*(2*a*c^2*AppellF1[4/3, 1/2, 2, 7/3, -((b*x^3)/a), -((b*c^ 2*x^3)/(a*c^2 - d^2))] + (a*c^2 - d^2)*AppellF1[4/3, 3/2, 1, 7/3, -((b*x^3 )/a), -((b*c^2*x^3)/(a*c^2 - d^2))]))) + (c^(1/3)*(2*Sqrt[3]*ArcTan[(-1 + (2*b^(1/3)*c^(2/3)*x)/(a*c^2 - d^2)^(1/3))/Sqrt[3]] + 2*Log[(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^(1/3)*(a*c^2 - d^2)^(2/3) )
Time = 0.67 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.04, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {2587, 27, 750, 16, 27, 937, 936, 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.
| \(\displaystyle \int \frac {1}{d \sqrt {a+b x^3}+a c+b c x^3} \, dx\) |
| \(\Big \downarrow \) 2587 |
| \(\displaystyle a c \int \frac {1}{a b c^2 x^3+a \left (a c^2-d^2\right )}dx-a d \int \frac {1}{a \sqrt {b x^3+a} \left (b c^2 x^3+a c^2-d^2\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle a c \int \frac {1}{a b c^2 x^3+a \left (a c^2-d^2\right )}dx-d \int \frac {1}{\sqrt {b x^3+a} \left (b c^2 x^3+a c^2-d^2\right )}dx\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle a c \left (\frac {\int \frac {\sqrt [3]{a} \left (2 \sqrt [3]{a c^2-d^2}-\sqrt [3]{b} c^{2/3} x\right )}{a^{2/3} b^{2/3} c^{4/3} x^2-a^{2/3} \sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+a^{2/3} \left (a c^2-d^2\right )^{2/3}}dx}{3 a^{2/3} \left (a c^2-d^2\right )^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x+\sqrt [3]{a} \sqrt [3]{a c^2-d^2}}dx}{3 a^{2/3} \left (a c^2-d^2\right )^{2/3}}\right )-d \int \frac {1}{\sqrt {b x^3+a} \left (b c^2 x^3+a c^2-d^2\right )}dx\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle a c \left (\frac {\int \frac {\sqrt [3]{a} \left (2 \sqrt [3]{a c^2-d^2}-\sqrt [3]{b} c^{2/3} x\right )}{a^{2/3} b^{2/3} c^{4/3} x^2-a^{2/3} \sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+a^{2/3} \left (a c^2-d^2\right )^{2/3}}dx}{3 a^{2/3} \left (a c^2-d^2\right )^{2/3}}+\frac {\log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 a \sqrt [3]{b} c^{2/3} \left (a c^2-d^2\right )^{2/3}}\right )-d \int \frac {1}{\sqrt {b x^3+a} \left (b c^2 x^3+a c^2-d^2\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle a c \left (\frac {\int \frac {2 \sqrt [3]{a c^2-d^2}-\sqrt [3]{b} c^{2/3} x}{a^{2/3} b^{2/3} c^{4/3} x^2-a^{2/3} \sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+a^{2/3} \left (a c^2-d^2\right )^{2/3}}dx}{3 \sqrt [3]{a} \left (a c^2-d^2\right )^{2/3}}+\frac {\log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 a \sqrt [3]{b} c^{2/3} \left (a c^2-d^2\right )^{2/3}}\right )-d \int \frac {1}{\sqrt {b x^3+a} \left (b c^2 x^3+a c^2-d^2\right )}dx\) |
| \(\Big \downarrow \) 937 |
| \(\displaystyle a c \left (\frac {\int \frac {2 \sqrt [3]{a c^2-d^2}-\sqrt [3]{b} c^{2/3} x}{a^{2/3} b^{2/3} c^{4/3} x^2-a^{2/3} \sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+a^{2/3} \left (a c^2-d^2\right )^{2/3}}dx}{3 \sqrt [3]{a} \left (a c^2-d^2\right )^{2/3}}+\frac {\log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 a \sqrt [3]{b} c^{2/3} \left (a c^2-d^2\right )^{2/3}}\right )-\frac {d \sqrt {\frac {b x^3}{a}+1} \int \frac {1}{\sqrt {\frac {b x^3}{a}+1} \left (b c^2 x^3+a c^2-d^2\right )}dx}{\sqrt {a+b x^3}}\) |
| \(\Big \downarrow \) 936 |
| \(\displaystyle a c \left (\frac {\int \frac {2 \sqrt [3]{a c^2-d^2}-\sqrt [3]{b} c^{2/3} x}{a^{2/3} b^{2/3} c^{4/3} x^2-a^{2/3} \sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+a^{2/3} \left (a c^2-d^2\right )^{2/3}}dx}{3 \sqrt [3]{a} \left (a c^2-d^2\right )^{2/3}}+\frac {\log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 a \sqrt [3]{b} c^{2/3} \left (a c^2-d^2\right )^{2/3}}\right )-\frac {d x \sqrt {\frac {b x^3}{a}+1} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )}{\sqrt {a+b x^3} \left (a c^2-d^2\right )}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle a c \left (\frac {\frac {3}{2} \sqrt [3]{a c^2-d^2} \int \frac {1}{a^{2/3} b^{2/3} c^{4/3} x^2-a^{2/3} \sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+a^{2/3} \left (a c^2-d^2\right )^{2/3}}dx-\frac {\int -\frac {a^{2/3} \sqrt [3]{b} c^{2/3} \left (\sqrt [3]{a c^2-d^2}-2 \sqrt [3]{b} c^{2/3} x\right )}{a^{2/3} b^{2/3} c^{4/3} x^2-a^{2/3} \sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+a^{2/3} \left (a c^2-d^2\right )^{2/3}}dx}{2 a^{2/3} \sqrt [3]{b} c^{2/3}}}{3 \sqrt [3]{a} \left (a c^2-d^2\right )^{2/3}}+\frac {\log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 a \sqrt [3]{b} c^{2/3} \left (a c^2-d^2\right )^{2/3}}\right )-\frac {d x \sqrt {\frac {b x^3}{a}+1} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )}{\sqrt {a+b x^3} \left (a c^2-d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle a c \left (\frac {\frac {\int \frac {\sqrt [3]{b} c^{2/3} \left (\sqrt [3]{a c^2-d^2}-2 \sqrt [3]{b} c^{2/3} x\right )}{b^{2/3} c^{4/3} x^2-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+\left (a c^2-d^2\right )^{2/3}}dx}{2 a^{2/3} \sqrt [3]{b} c^{2/3}}+\frac {3}{2} \sqrt [3]{a c^2-d^2} \int \frac {1}{a^{2/3} b^{2/3} c^{4/3} x^2-a^{2/3} \sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+a^{2/3} \left (a c^2-d^2\right )^{2/3}}dx}{3 \sqrt [3]{a} \left (a c^2-d^2\right )^{2/3}}+\frac {\log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 a \sqrt [3]{b} c^{2/3} \left (a c^2-d^2\right )^{2/3}}\right )-\frac {d x \sqrt {\frac {b x^3}{a}+1} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )}{\sqrt {a+b x^3} \left (a c^2-d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle a c \left (\frac {\frac {\int \frac {\sqrt [3]{a c^2-d^2}-2 \sqrt [3]{b} c^{2/3} x}{b^{2/3} c^{4/3} x^2-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+\left (a c^2-d^2\right )^{2/3}}dx}{2 a^{2/3}}+\frac {3}{2} \sqrt [3]{a c^2-d^2} \int \frac {1}{a^{2/3} b^{2/3} c^{4/3} x^2-a^{2/3} \sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+a^{2/3} \left (a c^2-d^2\right )^{2/3}}dx}{3 \sqrt [3]{a} \left (a c^2-d^2\right )^{2/3}}+\frac {\log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 a \sqrt [3]{b} c^{2/3} \left (a c^2-d^2\right )^{2/3}}\right )-\frac {d x \sqrt {\frac {b x^3}{a}+1} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )}{\sqrt {a+b x^3} \left (a c^2-d^2\right )}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle a c \left (\frac {\frac {\int \frac {\sqrt [3]{a c^2-d^2}-2 \sqrt [3]{b} c^{2/3} x}{b^{2/3} c^{4/3} x^2-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+\left (a c^2-d^2\right )^{2/3}}dx}{2 a^{2/3}}+\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{b} c^{2/3} x}{\sqrt [3]{a c^2-d^2}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{b} c^{2/3} x}{\sqrt [3]{a c^2-d^2}}\right )}{a^{2/3} \sqrt [3]{b} c^{2/3}}}{3 \sqrt [3]{a} \left (a c^2-d^2\right )^{2/3}}+\frac {\log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 a \sqrt [3]{b} c^{2/3} \left (a c^2-d^2\right )^{2/3}}\right )-\frac {d x \sqrt {\frac {b x^3}{a}+1} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )}{\sqrt {a+b x^3} \left (a c^2-d^2\right )}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle a c \left (\frac {\frac {\int \frac {\sqrt [3]{a c^2-d^2}-2 \sqrt [3]{b} c^{2/3} x}{b^{2/3} c^{4/3} x^2-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+\left (a c^2-d^2\right )^{2/3}}dx}{2 a^{2/3}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} c^{2/3} x}{\sqrt [3]{a c^2-d^2}}}{\sqrt {3}}\right )}{a^{2/3} \sqrt [3]{b} c^{2/3}}}{3 \sqrt [3]{a} \left (a c^2-d^2\right )^{2/3}}+\frac {\log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 a \sqrt [3]{b} c^{2/3} \left (a c^2-d^2\right )^{2/3}}\right )-\frac {d x \sqrt {\frac {b x^3}{a}+1} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )}{\sqrt {a+b x^3} \left (a c^2-d^2\right )}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle a c \left (\frac {-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} c^{2/3} x}{\sqrt [3]{a c^2-d^2}}}{\sqrt {3}}\right )}{a^{2/3} \sqrt [3]{b} c^{2/3}}-\frac {\log \left (-\sqrt [3]{b} c^{2/3} x \sqrt [3]{a c^2-d^2}+\left (a c^2-d^2\right )^{2/3}+b^{2/3} c^{4/3} x^2\right )}{2 a^{2/3} \sqrt [3]{b} c^{2/3}}}{3 \sqrt [3]{a} \left (a c^2-d^2\right )^{2/3}}+\frac {\log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 a \sqrt [3]{b} c^{2/3} \left (a c^2-d^2\right )^{2/3}}\right )-\frac {d x \sqrt {\frac {b x^3}{a}+1} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )}{\sqrt {a+b x^3} \left (a c^2-d^2\right )}\) |
-((d*x*Sqrt[1 + (b*x^3)/a]*AppellF1[1/3, 1/2, 1, 4/3, -((b*x^3)/a), -((b*c ^2*x^3)/(a*c^2 - d^2))])/((a*c^2 - d^2)*Sqrt[a + b*x^3])) + a*c*(Log[(a*c^ 2 - d^2)^(1/3) + b^(1/3)*c^(2/3)*x]/(3*a*b^(1/3)*c^(2/3)*(a*c^2 - d^2)^(2/ 3)) + (-((Sqrt[3]*ArcTan[(1 - (2*b^(1/3)*c^(2/3)*x)/(a*c^2 - d^2)^(1/3))/S qrt[3]])/(a^(2/3)*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*a^(2/3)*b^(1/3)*c^(2 /3)))/(3*a^(1/3)*(a*c^2 - d^2)^(2/3)))
3.6.59.3.1 Defintions of rubi rules used
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[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Simp[1/(3*Rt[a, 3]^2) Int[1/ (Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]^2) Int[(2*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[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*x*AppellF1[1/n, -p, -q, 1 + 1/n, (-b)*(x^n/a), (-d)*(x^n/c) ], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
Int[((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[(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n, p, q }, x] && NeQ[b*c - a*d, 0] && NeQ[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 = 0.86 (sec) , antiderivative size = 665, normalized size of antiderivative = 2.22
| 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 {2}{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 {2}{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 {2}{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 (-b^{2} a \right )^{\frac {1}{3}} \sqrt {2}\, \sqrt {\frac {i b \left (2 x +\frac {\left (-b^{2} a \right )^{\frac {1}{3}}-i \sqrt {3}\, \left (-b^{2} a \right )^{\frac {1}{3}}}{b}\right )}{\left (-b^{2} a \right )^{\frac {1}{3}}}}\, \sqrt {\frac {b \left (x -\frac {\left (-b^{2} a \right )^{\frac {1}{3}}}{b}\right )}{-3 \left (-b^{2} a \right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-b^{2} a \right )^{\frac {1}{3}}}}\, \sqrt {-\frac {i b \left (2 x +\frac {\left (-b^{2} a \right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-b^{2} a \right )^{\frac {1}{3}}}{b}\right )}{2 \left (-b^{2} a \right )^{\frac {1}{3}}}}\, \left (i \left (-b^{2} a \right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha b -i \left (-b^{2} a \right )^{\frac {2}{3}} \sqrt {3}+2 \underline {\hspace {1.25 ex}}\alpha ^{2} b^{2}-\left (-b^{2} a \right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha b -\left (-b^{2} a \right )^{\frac {2}{3}}\right ) \Pi \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-b^{2} a \right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-b^{2} a \right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-b^{2} a \right )^{\frac {1}{3}}}}}{3}, -\frac {c^{2} \left (2 i \left (-b^{2} a \right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} b -i \left (-b^{2} a \right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha +i \sqrt {3}\, a b -3 \left (-b^{2} a \right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha -3 a b \right )}{2 b \,d^{2}}, \sqrt {\frac {i \sqrt {3}\, \left (-b^{2} a \right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-b^{2} a \right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-b^{2} a \right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{2 \underline {\hspace {1.25 ex}}\alpha ^{2} \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 | \(\text {Expression too large to display}\) | \(1621\) |
(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)^(2/3)*ln(x+((a*c^2-d^2)/b/c^2)^(1/3))-1/6/b/c/((a *c^2-d^2)/b/c^2)^(2/3)*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/((a*c^2-d^2)/b/c^2)^(2/3)*3^(1/2)*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^2*(-b^2* a)^(1/3)*(1/2*I*b*(2*x+1/b*((-b^2*a)^(1/3)-I*3^(1/2)*(-b^2*a)^(1/3)))/(-b^ 2*a)^(1/3))^(1/2)*(b*(x-1/b*(-b^2*a)^(1/3))/(-3*(-b^2*a)^(1/3)+I*3^(1/2)*( -b^2*a)^(1/3)))^(1/2)*(-1/2*I*b*(2*x+1/b*((-b^2*a)^(1/3)+I*3^(1/2)*(-b^2*a )^(1/3)))/(-b^2*a)^(1/3))^(1/2)/(b*x^3+a)^(1/2)*(I*(-b^2*a)^(1/3)*3^(1/2)* _alpha*b-I*(-b^2*a)^(2/3)*3^(1/2)+2*_alpha^2*b^2-(-b^2*a)^(1/3)*_alpha*b-( -b^2*a)^(2/3))*EllipticPi(1/3*3^(1/2)*(I*(x+1/2/b*(-b^2*a)^(1/3)-1/2*I*3^( 1/2)/b*(-b^2*a)^(1/3))*3^(1/2)*b/(-b^2*a)^(1/3))^(1/2),-1/2/b*c^2*(2*I*(-b ^2*a)^(1/3)*3^(1/2)*_alpha^2*b-I*(-b^2*a)^(2/3)*3^(1/2)*_alpha+I*3^(1/2)*a *b-3*(-b^2*a)^(2/3)*_alpha-3*a*b)/d^2,(I*3^(1/2)/b*(-b^2*a)^(1/3)/(-3/2/b* (-b^2*a)^(1/3)+1/2*I*3^(1/2)/b*(-b^2*a)^(1/3)))^(1/2)),_alpha=RootOf(_Z^3* b*c^2+a*c^2-d^2)))
Exception generated. \[ \int \frac {1}{a c+b c x^3+d \sqrt {a+b x^3}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: Not integrable (provided residues have no relations)
\[ \int \frac {1}{a c+b c x^3+d \sqrt {a+b x^3}} \, dx=\int \frac {1}{a c + b c x^{3} + d \sqrt {a + b x^{3}}}\, dx \]
\[ \int \frac {1}{a c+b c x^3+d \sqrt {a+b x^3}} \, dx=\int { \frac {1}{b c x^{3} + a c + \sqrt {b x^{3} + a} d} \,d x } \]
\[ \int \frac {1}{a c+b c x^3+d \sqrt {a+b x^3}} \, dx=\int { \frac {1}{b c x^{3} + a c + \sqrt {b x^{3} + a} d} \,d x } \]
Timed out. \[ \int \frac {1}{a c+b c x^3+d \sqrt {a+b x^3}} \, dx=\int \frac {1}{a\,c+d\,\sqrt {b\,x^3+a}+b\,c\,x^3} \,d x \]