3.6.57 \(\int \frac {x^3}{a c+b c x^3+d \sqrt {a+b x^3}} \, dx\) [557]

3.6.57.1 Optimal result

Integrand size = 29, antiderivative size = 311 \[ \int \frac {x^3}{a c+b c x^3+d \sqrt {a+b x^3}} \, dx=\frac {x}{b c}-\frac {d x^4 \sqrt {1+\frac {b x^3}{a}} \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},1,\frac {7}{3},-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )}{4 \left (a c^2-d^2\right ) \sqrt {a+b x^3}}+\frac {\sqrt [3]{a c^2-d^2} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} c^{2/3} x}{\sqrt [3]{a c^2-d^2}}}{\sqrt {3}}\right )}{\sqrt {3} b^{4/3} c^{5/3}}-\frac {\sqrt [3]{a c^2-d^2} \log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 b^{4/3} c^{5/3}}+\frac {\sqrt [3]{a c^2-d^2} \log \left (\left (a c^2-d^2\right )^{2/3}-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+b^{2/3} c^{4/3} x^2\right )}{6 b^{4/3} c^{5/3}} \]

output

x/b/c-1/3*(a*c^2-d^2)^(1/3)*ln((a*c^2-d^2)^(1/3)+b^(1/3)*c^(2/3)*x)/b^(4/3 
)/c^(5/3)+1/6*(a*c^2-d^2)^(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^(4/3)/c^(5/3)+1/3*(a*c^2-d^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))/b^(4/3)/c^(5 
/3)*3^(1/2)-1/4*d*x^4*AppellF1(4/3,1/2,1,7/3,-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)
 

3.6.57.2 Mathematica [A] (verified)

Time = 10.43 (sec) , antiderivative size = 296, normalized size of antiderivative = 0.95 \[ \int \frac {x^3}{a c+b c x^3+d \sqrt {a+b x^3}} \, dx=-\frac {d x^4 \sqrt {1+\frac {b x^3}{a}} \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},1,\frac {7}{3},-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )}{\left (4 a c^2-4 d^2\right ) \sqrt {a+b x^3}}+\frac {6 \sqrt [3]{b} c^{2/3} x-2 \sqrt {3} \sqrt [3]{a c^2-d^2} \arctan \left (\frac {-1+\frac {2 \sqrt [3]{b} c^{2/3} x}{\sqrt [3]{a c^2-d^2}}}{\sqrt {3}}\right )-2 \sqrt [3]{a c^2-d^2} \log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )+\sqrt [3]{a c^2-d^2} \log \left (\left (a c^2-d^2\right )^{2/3}-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+b^{2/3} c^{4/3} x^2\right )}{6 b^{4/3} c^{5/3}} \]

input

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

output

-((d*x^4*Sqrt[1 + (b*x^3)/a]*AppellF1[4/3, 1/2, 1, 7/3, -((b*x^3)/a), -((b 
*c^2*x^3)/(a*c^2 - d^2))])/((4*a*c^2 - 4*d^2)*Sqrt[a + b*x^3])) + (6*b^(1/ 
3)*c^(2/3)*x - 2*Sqrt[3]*(a*c^2 - d^2)^(1/3)*ArcTan[(-1 + (2*b^(1/3)*c^(2/ 
3)*x)/(a*c^2 - d^2)^(1/3))/Sqrt[3]] - 2*(a*c^2 - d^2)^(1/3)*Log[(a*c^2 - d 
^2)^(1/3) + b^(1/3)*c^(2/3)*x] + (a*c^2 - d^2)^(1/3)*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])/(6*b^(4 
/3)*c^(5/3))
 

3.6.57.3 Rubi [A] (verified)

Time = 0.78 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.05, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.448, Rules used = {2587, 27, 843, 750, 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.

\(\displaystyle \int \frac {x^3}{d \sqrt {a+b x^3}+a c+b c x^3} \, dx\)

\(\Big \downarrow \) 2587

\(\displaystyle a c \int \frac {x^3}{a \left (b c^2 x^3+a c^2-d^2\right )}dx-a d \int \frac {x^3}{a \sqrt {b x^3+a} \left (b c^2 x^3+a c^2-d^2\right )}dx\)

\(\Big \downarrow \) 27

\(\displaystyle c \int \frac {x^3}{b c^2 x^3+a c^2-d^2}dx-d \int \frac {x^3}{\sqrt {b x^3+a} \left (b c^2 x^3+a c^2-d^2\right )}dx\)

\(\Big \downarrow \) 843

\(\displaystyle c \left (\frac {x}{b c^2}-\frac {\left (a c^2-d^2\right ) \int \frac {1}{b c^2 x^3+a c^2-d^2}dx}{b c^2}\right )-d \int \frac {x^3}{\sqrt {b x^3+a} \left (b c^2 x^3+a c^2-d^2\right )}dx\)

\(\Big \downarrow \) 750

\(\displaystyle c \left (\frac {x}{b c^2}-\frac {\left (a c^2-d^2\right ) \left (\frac {\int \frac {2 \sqrt [3]{a c^2-d^2}-\sqrt [3]{b} c^{2/3} x}{b^{2/3} c^{4/3} x^2-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+\left (a c^2-d^2\right )^{2/3}}dx}{3 \left (a c^2-d^2\right )^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{b} c^{2/3} x+\sqrt [3]{a c^2-d^2}}dx}{3 \left (a c^2-d^2\right )^{2/3}}\right )}{b c^2}\right )-d \int \frac {x^3}{\sqrt {b x^3+a} \left (b c^2 x^3+a c^2-d^2\right )}dx\)

\(\Big \downarrow \) 16

\(\displaystyle c \left (\frac {x}{b c^2}-\frac {\left (a c^2-d^2\right ) \left (\frac {\int \frac {2 \sqrt [3]{a c^2-d^2}-\sqrt [3]{b} c^{2/3} x}{b^{2/3} c^{4/3} x^2-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+\left (a c^2-d^2\right )^{2/3}}dx}{3 \left (a c^2-d^2\right )^{2/3}}+\frac {\log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 \sqrt [3]{b} c^{2/3} \left (a c^2-d^2\right )^{2/3}}\right )}{b c^2}\right )-d \int \frac {x^3}{\sqrt {b x^3+a} \left (b c^2 x^3+a c^2-d^2\right )}dx\)

\(\Big \downarrow \) 1013

\(\displaystyle c \left (\frac {x}{b c^2}-\frac {\left (a c^2-d^2\right ) \left (\frac {\int \frac {2 \sqrt [3]{a c^2-d^2}-\sqrt [3]{b} c^{2/3} x}{b^{2/3} c^{4/3} x^2-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+\left (a c^2-d^2\right )^{2/3}}dx}{3 \left (a c^2-d^2\right )^{2/3}}+\frac {\log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 \sqrt [3]{b} c^{2/3} \left (a c^2-d^2\right )^{2/3}}\right )}{b c^2}\right )-\frac {d \sqrt {\frac {b x^3}{a}+1} \int \frac {x^3}{\sqrt {\frac {b x^3}{a}+1} \left (b c^2 x^3+a c^2-d^2\right )}dx}{\sqrt {a+b x^3}}\)

\(\Big \downarrow \) 1012

\(\displaystyle c \left (\frac {x}{b c^2}-\frac {\left (a c^2-d^2\right ) \left (\frac {\int \frac {2 \sqrt [3]{a c^2-d^2}-\sqrt [3]{b} c^{2/3} x}{b^{2/3} c^{4/3} x^2-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+\left (a c^2-d^2\right )^{2/3}}dx}{3 \left (a c^2-d^2\right )^{2/3}}+\frac {\log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 \sqrt [3]{b} c^{2/3} \left (a c^2-d^2\right )^{2/3}}\right )}{b c^2}\right )-\frac {d x^4 \sqrt {\frac {b x^3}{a}+1} \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},1,\frac {7}{3},-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )}{4 \sqrt {a+b x^3} \left (a c^2-d^2\right )}\)

\(\Big \downarrow \) 1142

\(\displaystyle c \left (\frac {x}{b c^2}-\frac {\left (a c^2-d^2\right ) \left (\frac {\frac {3}{2} \sqrt [3]{a c^2-d^2} \int \frac {1}{b^{2/3} c^{4/3} x^2-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+\left (a c^2-d^2\right )^{2/3}}dx-\frac {\int -\frac {\sqrt [3]{b} c^{2/3} \left (\sqrt [3]{a c^2-d^2}-2 \sqrt [3]{b} c^{2/3} x\right )}{b^{2/3} c^{4/3} x^2-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+\left (a c^2-d^2\right )^{2/3}}dx}{2 \sqrt [3]{b} c^{2/3}}}{3 \left (a c^2-d^2\right )^{2/3}}+\frac {\log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 \sqrt [3]{b} c^{2/3} \left (a c^2-d^2\right )^{2/3}}\right )}{b c^2}\right )-\frac {d x^4 \sqrt {\frac {b x^3}{a}+1} \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},1,\frac {7}{3},-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )}{4 \sqrt {a+b x^3} \left (a c^2-d^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle c \left (\frac {x}{b c^2}-\frac {\left (a c^2-d^2\right ) \left (\frac {\frac {3}{2} \sqrt [3]{a c^2-d^2} \int \frac {1}{b^{2/3} c^{4/3} x^2-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+\left (a c^2-d^2\right )^{2/3}}dx+\frac {\int \frac {\sqrt [3]{b} c^{2/3} \left (\sqrt [3]{a c^2-d^2}-2 \sqrt [3]{b} c^{2/3} x\right )}{b^{2/3} c^{4/3} x^2-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+\left (a c^2-d^2\right )^{2/3}}dx}{2 \sqrt [3]{b} c^{2/3}}}{3 \left (a c^2-d^2\right )^{2/3}}+\frac {\log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 \sqrt [3]{b} c^{2/3} \left (a c^2-d^2\right )^{2/3}}\right )}{b c^2}\right )-\frac {d x^4 \sqrt {\frac {b x^3}{a}+1} \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},1,\frac {7}{3},-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )}{4 \sqrt {a+b x^3} \left (a c^2-d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle c \left (\frac {x}{b c^2}-\frac {\left (a c^2-d^2\right ) \left (\frac {\frac {3}{2} \sqrt [3]{a c^2-d^2} \int \frac {1}{b^{2/3} c^{4/3} x^2-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+\left (a c^2-d^2\right )^{2/3}}dx+\frac {1}{2} \int \frac {\sqrt [3]{a c^2-d^2}-2 \sqrt [3]{b} c^{2/3} x}{b^{2/3} c^{4/3} x^2-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+\left (a c^2-d^2\right )^{2/3}}dx}{3 \left (a c^2-d^2\right )^{2/3}}+\frac {\log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 \sqrt [3]{b} c^{2/3} \left (a c^2-d^2\right )^{2/3}}\right )}{b c^2}\right )-\frac {d x^4 \sqrt {\frac {b x^3}{a}+1} \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},1,\frac {7}{3},-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )}{4 \sqrt {a+b x^3} \left (a c^2-d^2\right )}\)

\(\Big \downarrow \) 1082

\(\displaystyle c \left (\frac {x}{b c^2}-\frac {\left (a c^2-d^2\right ) \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a c^2-d^2}-2 \sqrt [3]{b} c^{2/3} x}{b^{2/3} c^{4/3} x^2-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+\left (a c^2-d^2\right )^{2/3}}dx+\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{b} c^{2/3} x}{\sqrt [3]{a c^2-d^2}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{b} c^{2/3} x}{\sqrt [3]{a c^2-d^2}}\right )}{\sqrt [3]{b} c^{2/3}}}{3 \left (a c^2-d^2\right )^{2/3}}+\frac {\log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 \sqrt [3]{b} c^{2/3} \left (a c^2-d^2\right )^{2/3}}\right )}{b c^2}\right )-\frac {d x^4 \sqrt {\frac {b x^3}{a}+1} \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},1,\frac {7}{3},-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )}{4 \sqrt {a+b x^3} \left (a c^2-d^2\right )}\)

\(\Big \downarrow \) 217

\(\displaystyle c \left (\frac {x}{b c^2}-\frac {\left (a c^2-d^2\right ) \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a c^2-d^2}-2 \sqrt [3]{b} c^{2/3} x}{b^{2/3} c^{4/3} x^2-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+\left (a c^2-d^2\right )^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} c^{2/3} x}{\sqrt [3]{a c^2-d^2}}}{\sqrt {3}}\right )}{\sqrt [3]{b} c^{2/3}}}{3 \left (a c^2-d^2\right )^{2/3}}+\frac {\log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 \sqrt [3]{b} c^{2/3} \left (a c^2-d^2\right )^{2/3}}\right )}{b c^2}\right )-\frac {d x^4 \sqrt {\frac {b x^3}{a}+1} \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},1,\frac {7}{3},-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )}{4 \sqrt {a+b x^3} \left (a c^2-d^2\right )}\)

\(\Big \downarrow \) 1103

\(\displaystyle c \left (\frac {x}{b c^2}-\frac {\left (a c^2-d^2\right ) \left (\frac {-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} c^{2/3} x}{\sqrt [3]{a c^2-d^2}}}{\sqrt {3}}\right )}{\sqrt [3]{b} c^{2/3}}-\frac {\log \left (-\sqrt [3]{b} c^{2/3} x \sqrt [3]{a c^2-d^2}+\left (a c^2-d^2\right )^{2/3}+b^{2/3} c^{4/3} x^2\right )}{2 \sqrt [3]{b} c^{2/3}}}{3 \left (a c^2-d^2\right )^{2/3}}+\frac {\log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 \sqrt [3]{b} c^{2/3} \left (a c^2-d^2\right )^{2/3}}\right )}{b c^2}\right )-\frac {d x^4 \sqrt {\frac {b x^3}{a}+1} \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},1,\frac {7}{3},-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )}{4 \sqrt {a+b x^3} \left (a c^2-d^2\right )}\)

input

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

output

-1/4*(d*x^4*Sqrt[1 + (b*x^3)/a]*AppellF1[4/3, 1/2, 1, 7/3, -((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*(x/(b*c^ 
2) - ((a*c^2 - d^2)*(Log[(a*c^2 - d^2)^(1/3) + b^(1/3)*c^(2/3)*x]/(3*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))/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*(a*c^2 - d^2)^(2/3))))/(b*c^2))
 

3.6.57.3.1 Defintions of rubi rules used

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

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])
 

rule 750

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]
 

rule 843

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n 
 - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*(m + n*p + 1))), x] - Simp[ 
a*c^n*((m - n + 1)/(b*(m + n*p + 1)))   Int[(c*x)^(m - n)*(a + b*x^n)^p, x] 
, x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n* 
p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]
 

rule 1012

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])
 

rule 1013

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])
 

rule 1082

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]
 

rule 1103

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]
 

rule 1142

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]
 

rule 2587

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]
 

3.6.57.4 Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 6.

Time = 0.99 (sec) , antiderivative size = 994, normalized size of antiderivative = 3.20

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(994\)
default	\(\text {Expression too large to display}\)	\(1672\)

input

int(x^3/(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))*(c*( 
1/b/c^2*x+(1/3/b/c^2/((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^2/((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^2/((a*c^2-d^2)/b/c^2)^(2/3)*3^(1/2)*a 
rctan(1/3*3^(1/2)*(2/((a*c^2-d^2)/b/c^2)^(1/3)*x-1)))*(-a*c^2+d^2)/b/c^2)+ 
2/3*I*d/b^2/c^2*3^(1/2)*(-b^2*a)^(1/3)*(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)*((x-1/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)*(-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)/(b*x^3+a)^(1/2)*EllipticF(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),(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))+1/3*I/d/b^4/c^2*2^(1/2)*sum((a*c^2-d^2)/_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)^...
 

3.6.57.5 Fricas [F(-1)]

Timed out. \[ \int \frac {x^3}{a c+b c x^3+d \sqrt {a+b x^3}} \, dx=\text {Timed out} \]

input

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

output

Timed out
 

3.6.57.6 Sympy [F]

\[ \int \frac {x^3}{a c+b c x^3+d \sqrt {a+b x^3}} \, dx=\int \frac {x^{3}}{a c + b c x^{3} + d \sqrt {a + b x^{3}}}\, dx \]

input

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

output

Integral(x**3/(a*c + b*c*x**3 + d*sqrt(a + b*x**3)), x)
 

3.6.57.7 Maxima [F]

\[ \int \frac {x^3}{a c+b c x^3+d \sqrt {a+b x^3}} \, dx=\int { \frac {x^{3}}{b c x^{3} + a c + \sqrt {b x^{3} + a} d} \,d x } \]

input

integrate(x^3/(a*c+b*c*x^3+d*(b*x^3+a)^(1/2)),x, algorithm="maxima")
 

output

integrate(x^3/(b*c*x^3 + a*c + sqrt(b*x^3 + a)*d), x)
 

3.6.57.8 Giac [F]

\[ \int \frac {x^3}{a c+b c x^3+d \sqrt {a+b x^3}} \, dx=\int { \frac {x^{3}}{b c x^{3} + a c + \sqrt {b x^{3} + a} d} \,d x } \]

input

integrate(x^3/(a*c+b*c*x^3+d*(b*x^3+a)^(1/2)),x, algorithm="giac")
 

output

integrate(x^3/(b*c*x^3 + a*c + sqrt(b*x^3 + a)*d), x)
 

3.6.57.9 Mupad [F(-1)]

Timed out. \[ \int \frac {x^3}{a c+b c x^3+d \sqrt {a+b x^3}} \, dx=\int \frac {x^3}{a\,c+d\,\sqrt {b\,x^3+a}+b\,c\,x^3} \,d x \]

input

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

output

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