\(\int \frac {1}{(c+d x)^2 (a+b x^3)^{2/3}} \, dx\) [160]

Optimal result

Integrand size = 19, antiderivative size = 760 \[ \int \frac {1}{(c+d x)^2 \left (a+b x^3\right )^{2/3}} \, dx=\frac {c^2 d^2 \sqrt [3]{a+b x^3}}{\left (b c^3-a d^3\right ) \left (c^3+d^3 x^3\right )}+\frac {d^4 x^2 \sqrt [3]{a+b x^3}}{\left (b c^3-a d^3\right ) \left (c^3+d^3 x^3\right )}+\frac {x \left (1+\frac {b x^3}{a}\right )^{2/3} \operatorname {AppellF1}\left (\frac {1}{3},\frac {2}{3},2,\frac {4}{3},-\frac {b x^3}{a},-\frac {d^3 x^3}{c^3}\right )}{c^2 \left (a+b x^3\right )^{2/3}}-\frac {d^3 x^4 \left (1+\frac {b x^3}{a}\right )^{2/3} \operatorname {AppellF1}\left (\frac {4}{3},\frac {2}{3},2,\frac {7}{3},-\frac {b x^3}{a},-\frac {d^3 x^3}{c^3}\right )}{2 c^5 \left (a+b x^3\right )^{2/3}}+\frac {2 a d^4 \arctan \left (\frac {1+\frac {2 \sqrt [3]{b c^3-a d^3} x}{c \sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3} c \left (b c^3-a d^3\right )^{5/3}}+\frac {2 d \left (3 b c^3-a d^3\right ) \arctan \left (\frac {1+\frac {2 \sqrt [3]{b c^3-a d^3} x}{c \sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3} c \left (b c^3-a d^3\right )^{5/3}}-\frac {2 b c^2 d \arctan \left (\frac {1-\frac {2 d \sqrt [3]{a+b x^3}}{\sqrt [3]{b c^3-a d^3}}}{\sqrt {3}}\right )}{\sqrt {3} \left (b c^3-a d^3\right )^{5/3}}-\frac {b c^2 d \log \left (c^3+d^3 x^3\right )}{3 \left (b c^3-a d^3\right )^{5/3}}-\frac {a d^4 \log \left (c^3+d^3 x^3\right )}{9 c \left (b c^3-a d^3\right )^{5/3}}-\frac {d \left (3 b c^3-a d^3\right ) \log \left (c^3+d^3 x^3\right )}{9 c \left (b c^3-a d^3\right )^{5/3}}+\frac {a d^4 \log \left (\frac {\sqrt [3]{b c^3-a d^3} x}{c}-\sqrt [3]{a+b x^3}\right )}{3 c \left (b c^3-a d^3\right )^{5/3}}+\frac {d \left (3 b c^3-a d^3\right ) \log \left (\frac {\sqrt [3]{b c^3-a d^3} x}{c}-\sqrt [3]{a+b x^3}\right )}{3 c \left (b c^3-a d^3\right )^{5/3}}+\frac {b c^2 d \log \left (\sqrt [3]{b c^3-a d^3}+d \sqrt [3]{a+b x^3}\right )}{\left (b c^3-a d^3\right )^{5/3}} \] Output:

c^2*d^2*(b*x^3+a)^(1/3)/(-a*d^3+b*c^3)/(d^3*x^3+c^3)+d^4*x^2*(b*x^3+a)^(1/ 
3)/(-a*d^3+b*c^3)/(d^3*x^3+c^3)+x*(1+b*x^3/a)^(2/3)*AppellF1(1/3,2/3,2,4/3 
,-b*x^3/a,-d^3*x^3/c^3)/c^2/(b*x^3+a)^(2/3)-1/2*d^3*x^4*(1+b*x^3/a)^(2/3)* 
AppellF1(4/3,2/3,2,7/3,-b*x^3/a,-d^3*x^3/c^3)/c^5/(b*x^3+a)^(2/3)+2/9*a*d^ 
4*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)/c/(-a*d^3+b*c^3)^(5/3)+2/9*d*(-a*d^3+3*b*c^3)*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)/c/(-a*d^3+b*c^3)^(5/3)-2 
/3*b*c^2*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)^(5/3)-1/3*b*c^2*d*ln(d^3*x^3+c^3)/(-a*d^3+b*c^3)^ 
(5/3)-1/9*a*d^4*ln(d^3*x^3+c^3)/c/(-a*d^3+b*c^3)^(5/3)-1/9*d*(-a*d^3+3*b*c 
^3)*ln(d^3*x^3+c^3)/c/(-a*d^3+b*c^3)^(5/3)+1/3*a*d^4*ln((-a*d^3+b*c^3)^(1/ 
3)*x/c-(b*x^3+a)^(1/3))/c/(-a*d^3+b*c^3)^(5/3)+1/3*d*(-a*d^3+3*b*c^3)*ln(( 
-a*d^3+b*c^3)^(1/3)*x/c-(b*x^3+a)^(1/3))/c/(-a*d^3+b*c^3)^(5/3)+b*c^2*d*ln 
((-a*d^3+b*c^3)^(1/3)+d*(b*x^3+a)^(1/3))/(-a*d^3+b*c^3)^(5/3)
 

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 1.77 (sec) , antiderivative size = 760, 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.

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

\(\Big \downarrow \) 2581

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {d^3 x^4 \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {AppellF1}\left (\frac {4}{3},\frac {2}{3},2,\frac {7}{3},-\frac {b x^3}{a},-\frac {d^3 x^3}{c^3}\right )}{2 c^5 \left (a+b x^3\right )^{2/3}}+\frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {AppellF1}\left (\frac {1}{3},\frac {2}{3},2,\frac {4}{3},-\frac {b x^3}{a},-\frac {d^3 x^3}{c^3}\right )}{c^2 \left (a+b x^3\right )^{2/3}}+\frac {2 d \left (3 b c^3-a d^3\right ) \arctan \left (\frac {\frac {2 x \sqrt [3]{b c^3-a d^3}}{c \sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{3 \sqrt {3} c \left (b c^3-a d^3\right )^{5/3}}+\frac {2 a d^4 \arctan \left (\frac {\frac {2 x \sqrt [3]{b c^3-a d^3}}{c \sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{3 \sqrt {3} c \left (b c^3-a d^3\right )^{5/3}}-\frac {2 b c^2 d \arctan \left (\frac {1-\frac {2 d \sqrt [3]{a+b x^3}}{\sqrt [3]{b c^3-a d^3}}}{\sqrt {3}}\right )}{\sqrt {3} \left (b c^3-a d^3\right )^{5/3}}-\frac {d \left (3 b c^3-a d^3\right ) \log \left (c^3+d^3 x^3\right )}{9 c \left (b c^3-a d^3\right )^{5/3}}+\frac {d \left (3 b c^3-a d^3\right ) \log \left (\frac {x \sqrt [3]{b c^3-a d^3}}{c}-\sqrt [3]{a+b x^3}\right )}{3 c \left (b c^3-a d^3\right )^{5/3}}-\frac {a d^4 \log \left (c^3+d^3 x^3\right )}{9 c \left (b c^3-a d^3\right )^{5/3}}+\frac {a d^4 \log \left (\frac {x \sqrt [3]{b c^3-a d^3}}{c}-\sqrt [3]{a+b x^3}\right )}{3 c \left (b c^3-a d^3\right )^{5/3}}+\frac {d^4 x^2 \sqrt [3]{a+b x^3}}{\left (c^3+d^3 x^3\right ) \left (b c^3-a d^3\right )}-\frac {b c^2 d \log \left (c^3+d^3 x^3\right )}{3 \left (b c^3-a d^3\right )^{5/3}}+\frac {b c^2 d \log \left (\sqrt [3]{b c^3-a d^3}+d \sqrt [3]{a+b x^3}\right )}{\left (b c^3-a d^3\right )^{5/3}}+\frac {c^2 d^2 \sqrt [3]{a+b x^3}}{\left (c^3+d^3 x^3\right ) \left (b c^3-a d^3\right )}\)

Input:

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

Output:

(c^2*d^2*(a + b*x^3)^(1/3))/((b*c^3 - a*d^3)*(c^3 + d^3*x^3)) + (d^4*x^2*( 
a + b*x^3)^(1/3))/((b*c^3 - a*d^3)*(c^3 + d^3*x^3)) + (x*(1 + (b*x^3)/a)^( 
2/3)*AppellF1[1/3, 2/3, 2, 4/3, -((b*x^3)/a), -((d^3*x^3)/c^3)])/(c^2*(a + 
 b*x^3)^(2/3)) - (d^3*x^4*(1 + (b*x^3)/a)^(2/3)*AppellF1[4/3, 2/3, 2, 7/3, 
 -((b*x^3)/a), -((d^3*x^3)/c^3)])/(2*c^5*(a + b*x^3)^(2/3)) + (2*a*d^4*Arc 
Tan[(1 + (2*(b*c^3 - a*d^3)^(1/3)*x)/(c*(a + b*x^3)^(1/3)))/Sqrt[3]])/(3*S 
qrt[3]*c*(b*c^3 - a*d^3)^(5/3)) + (2*d*(3*b*c^3 - a*d^3)*ArcTan[(1 + (2*(b 
*c^3 - a*d^3)^(1/3)*x)/(c*(a + b*x^3)^(1/3)))/Sqrt[3]])/(3*Sqrt[3]*c*(b*c^ 
3 - a*d^3)^(5/3)) - (2*b*c^2*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)^(5/3)) - (b*c^2*d*Log[c 
^3 + d^3*x^3])/(3*(b*c^3 - a*d^3)^(5/3)) - (a*d^4*Log[c^3 + d^3*x^3])/(9*c 
*(b*c^3 - a*d^3)^(5/3)) - (d*(3*b*c^3 - a*d^3)*Log[c^3 + d^3*x^3])/(9*c*(b 
*c^3 - a*d^3)^(5/3)) + (a*d^4*Log[((b*c^3 - a*d^3)^(1/3)*x)/c - (a + b*x^3 
)^(1/3)])/(3*c*(b*c^3 - a*d^3)^(5/3)) + (d*(3*b*c^3 - a*d^3)*Log[((b*c^3 - 
 a*d^3)^(1/3)*x)/c - (a + b*x^3)^(1/3)])/(3*c*(b*c^3 - a*d^3)^(5/3)) + (b* 
c^2*d*Log[(b*c^3 - a*d^3)^(1/3) + d*(a + b*x^3)^(1/3)])/(b*c^3 - a*d^3)^(5 
/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]

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

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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