\(\int \frac {1}{(c+d x) (a+b x^3)^{2/3}} \, dx\) [159]

Optimal result

Integrand size = 19, antiderivative size = 332 \[ \int \frac {1}{(c+d x) \left (a+b x^3\right )^{2/3}} \, dx=\frac {x \left (1+\frac {b x^3}{a}\right )^{2/3} \operatorname {AppellF1}\left (\frac {1}{3},\frac {2}{3},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {d^3 x^3}{c^3}\right )}{c \left (a+b x^3\right )^{2/3}}+\frac {d \arctan \left (\frac {1+\frac {2 \sqrt [3]{b c^3-a d^3} x}{c \sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} \left (b c^3-a d^3\right )^{2/3}}-\frac {d \arctan \left (\frac {1-\frac {2 d \sqrt [3]{a+b x^3}}{\sqrt [3]{b c^3-a d^3}}}{\sqrt {3}}\right )}{\sqrt {3} \left (b c^3-a d^3\right )^{2/3}}-\frac {d \log \left (c^3+d^3 x^3\right )}{3 \left (b c^3-a d^3\right )^{2/3}}+\frac {d \log \left (\frac {\sqrt [3]{b c^3-a d^3} x}{c}-\sqrt [3]{a+b x^3}\right )}{2 \left (b c^3-a d^3\right )^{2/3}}+\frac {d \log \left (\sqrt [3]{b c^3-a d^3}+d \sqrt [3]{a+b x^3}\right )}{2 \left (b c^3-a d^3\right )^{2/3}} \] Output:

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

Mathematica [F]

\[ \int \frac {1}{(c+d x) \left (a+b x^3\right )^{2/3}} \, dx=\int \frac {1}{(c+d x) \left (a+b x^3\right )^{2/3}} \, dx \] Input:

Integrate[1/((c + d*x)*(a + b*x^3)^(2/3)),x]
 

Output:

Integrate[1/((c + d*x)*(a + b*x^3)^(2/3)), x]
 

Rubi [A] (verified)

Time = 0.90 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2581, 2009}

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.

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

\(\Big \downarrow \) 2581

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

\(\Big \downarrow \) 2009

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

Input:

Int[1/((c + d*x)*(a + b*x^3)^(2/3)),x]
 

Output:

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

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2581

Int[(Px_.)*((c_) + (d_.)*(x_))^(q_)*((a_) + (b_.)*(x_)^3)^(p_.), x_Symbol] 
:> Int[ExpandIntegrand[(c^3 + d^3*x^3)^q*(a + b*x^3)^p, Px/(c^2 - c*d*x + d 
^2*x^2)^q, x], x] /; FreeQ[{a, b, c, d, p}, x] && PolyQ[Px, x] && ILtQ[q, 0 
] && RationalQ[p] && EqQ[Denominator[p], 3]
 

Maple [F]

\[\int \frac {1}{\left (d x +c \right ) \left (b \,x^{3}+a \right )^{\frac {2}{3}}}d x\]

Input:

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

Output:

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

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{(c+d x) \left (a+b x^3\right )^{2/3}} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Sympy [F]

\[ \int \frac {1}{(c+d x) \left (a+b x^3\right )^{2/3}} \, dx=\int \frac {1}{\left (a + b x^{3}\right )^{\frac {2}{3}} \left (c + d x\right )}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {1}{(c+d x) \left (a+b x^3\right )^{2/3}} \, dx=\int { \frac {1}{{\left (b x^{3} + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {1}{(c+d x) \left (a+b x^3\right )^{2/3}} \, dx=\int { \frac {1}{{\left (b x^{3} + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(c+d x) \left (a+b x^3\right )^{2/3}} \, dx=\int \frac {1}{{\left (b\,x^3+a\right )}^{2/3}\,\left (c+d\,x\right )} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {1}{(c+d x) \left (a+b x^3\right )^{2/3}} \, dx=\int \frac {1}{\left (b \,x^{3}+a \right )^{\frac {2}{3}} c +\left (b \,x^{3}+a \right )^{\frac {2}{3}} d x}d x \] Input:

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

Output:

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