\(\int \frac {(a+b x^3)^{2/3}}{x^2 (a d-b d x^3)} \, dx\) [817]

Optimal result

Integrand size = 28, antiderivative size = 483 \[ \int \frac {\left (a+b x^3\right )^{2/3}}{x^2 \left (a d-b d x^3\right )} \, dx=-\frac {\left (a+b x^3\right )^{2/3}}{a d x}+\frac {2^{2/3} \sqrt [3]{b} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} a^{2/3} d}+\frac {\sqrt [3]{b} \arctan \left (\frac {1+\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3} a^{2/3} d}+\frac {b x^2 \sqrt [3]{1+\frac {b x^3}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},-\frac {b x^3}{a}\right )}{2 a d \sqrt [3]{a+b x^3}}+\frac {\sqrt [3]{b} \log \left (\frac {\left (\sqrt [3]{a}-\sqrt [3]{b} x\right )^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a}\right )}{6 \sqrt [3]{2} a^{2/3} d}+\frac {\sqrt [3]{b} \log \left (1+\frac {2^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{\left (a+b x^3\right )^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}\right )}{3 \sqrt [3]{2} a^{2/3} d}-\frac {2^{2/3} \sqrt [3]{b} \log \left (1+\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}\right )}{3 a^{2/3} d}-\frac {\sqrt [3]{b} \log \left (\frac {\sqrt [3]{b} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a}}-\frac {2^{2/3} \sqrt [3]{b} \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}\right )}{2 \sqrt [3]{2} a^{2/3} d} \] Output:

-(b*x^3+a)^(2/3)/a/d/x+1/3*2^(2/3)*b^(1/3)*arctan(1/3*(1-2*2^(1/3)*(a^(1/3 
)+b^(1/3)*x)/(b*x^3+a)^(1/3))*3^(1/2))*3^(1/2)/a^(2/3)/d+1/6*b^(1/3)*arcta 
n(1/3*(1+2^(1/3)*(a^(1/3)+b^(1/3)*x)/(b*x^3+a)^(1/3))*3^(1/2))*2^(2/3)*3^( 
1/2)/a^(2/3)/d+1/2*b*x^2*(1+b*x^3/a)^(1/3)*hypergeom([1/3, 2/3],[5/3],-b*x 
^3/a)/a/d/(b*x^3+a)^(1/3)+1/12*b^(1/3)*ln((a^(1/3)-b^(1/3)*x)^2*(a^(1/3)+b 
^(1/3)*x)/a)*2^(2/3)/a^(2/3)/d+1/6*b^(1/3)*ln(1+2^(2/3)*(a^(1/3)+b^(1/3)*x 
)^2/(b*x^3+a)^(2/3)-2^(1/3)*(a^(1/3)+b^(1/3)*x)/(b*x^3+a)^(1/3))*2^(2/3)/a 
^(2/3)/d-1/3*2^(2/3)*b^(1/3)*ln(1+2^(1/3)*(a^(1/3)+b^(1/3)*x)/(b*x^3+a)^(1 
/3))/a^(2/3)/d-1/4*b^(1/3)*ln(b^(1/3)*(a^(1/3)+b^(1/3)*x)/a^(1/3)-2^(2/3)* 
b^(1/3)*(b*x^3+a)^(1/3)/a^(1/3))*2^(2/3)/a^(2/3)/d
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.

Time = 11.08 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.28 \[ \int \frac {\left (a+b x^3\right )^{2/3}}{x^2 \left (a d-b d x^3\right )} \, dx=\frac {15 a b x^3 \sqrt [3]{1+\frac {b x^3}{a}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-\frac {b x^3}{a},\frac {b x^3}{a}\right )-2 \left (5 a \left (a+b x^3\right )+b^2 x^6 \sqrt [3]{1+\frac {b x^3}{a}} \operatorname {AppellF1}\left (\frac {5}{3},\frac {1}{3},1,\frac {8}{3},-\frac {b x^3}{a},\frac {b x^3}{a}\right )\right )}{10 a^2 d x \sqrt [3]{a+b x^3}} \] Input:

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

Output:

(15*a*b*x^3*(1 + (b*x^3)/a)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, -((b*x^3)/a), 
 (b*x^3)/a] - 2*(5*a*(a + b*x^3) + b^2*x^6*(1 + (b*x^3)/a)^(1/3)*AppellF1[ 
5/3, 1/3, 1, 8/3, -((b*x^3)/a), (b*x^3)/a]))/(10*a^2*d*x*(a + b*x^3)^(1/3) 
)
 

Rubi [A] (verified)

Time = 1.13 (sec) , antiderivative size = 467, normalized size of antiderivative = 0.97, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {975, 27, 1054, 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.

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

\(\Big \downarrow \) 975

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1054

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {b \left (\frac {2^{2/3} \sqrt [3]{a} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} b^{2/3}}+\frac {\sqrt [3]{a} \arctan \left (\frac {\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3} b^{2/3}}+\frac {\sqrt [3]{a} \log \left (\frac {2^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{\left (a+b x^3\right )^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1\right )}{3 \sqrt [3]{2} b^{2/3}}-\frac {2^{2/3} \sqrt [3]{a} \log \left (\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1\right )}{3 b^{2/3}}-\frac {\sqrt [3]{a} \log \left (\frac {\sqrt [3]{b} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a}}-\frac {2^{2/3} \sqrt [3]{b} \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}\right )}{2 \sqrt [3]{2} b^{2/3}}+\frac {\sqrt [3]{a} \log \left (\frac {\left (\sqrt [3]{a}-\sqrt [3]{b} x\right )^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a}\right )}{6 \sqrt [3]{2} b^{2/3}}+\frac {x^2 \sqrt [3]{\frac {b x^3}{a}+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},-\frac {b x^3}{a}\right )}{2 \sqrt [3]{a+b x^3}}\right )}{a d}-\frac {\left (a+b x^3\right )^{2/3}}{a d x}\)

Input:

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

Output:

-((a + b*x^3)^(2/3)/(a*d*x)) + (b*((2^(2/3)*a^(1/3)*ArcTan[(1 - (2*2^(1/3) 
*(a^(1/3) + b^(1/3)*x))/(a + b*x^3)^(1/3))/Sqrt[3]])/(Sqrt[3]*b^(2/3)) + ( 
a^(1/3)*ArcTan[(1 + (2^(1/3)*(a^(1/3) + b^(1/3)*x))/(a + b*x^3)^(1/3))/Sqr 
t[3]])/(2^(1/3)*Sqrt[3]*b^(2/3)) + (x^2*(1 + (b*x^3)/a)^(1/3)*Hypergeometr 
ic2F1[1/3, 2/3, 5/3, -((b*x^3)/a)])/(2*(a + b*x^3)^(1/3)) + (a^(1/3)*Log[( 
(a^(1/3) - b^(1/3)*x)^2*(a^(1/3) + b^(1/3)*x))/a])/(6*2^(1/3)*b^(2/3)) + ( 
a^(1/3)*Log[1 + (2^(2/3)*(a^(1/3) + b^(1/3)*x)^2)/(a + b*x^3)^(2/3) - (2^( 
1/3)*(a^(1/3) + b^(1/3)*x))/(a + b*x^3)^(1/3)])/(3*2^(1/3)*b^(2/3)) - (2^( 
2/3)*a^(1/3)*Log[1 + (2^(1/3)*(a^(1/3) + b^(1/3)*x))/(a + b*x^3)^(1/3)])/( 
3*b^(2/3)) - (a^(1/3)*Log[(b^(1/3)*(a^(1/3) + b^(1/3)*x))/a^(1/3) - (2^(2/ 
3)*b^(1/3)*(a + b*x^3)^(1/3))/a^(1/3)])/(2*2^(1/3)*b^(2/3))))/(a*d)
 

Defintions of rubi rules used

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 975

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_) 
)^(q_), x_Symbol] :> Simp[(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/ 
(a*e*(m + 1))), x] - Simp[1/(a*e^n*(m + 1))   Int[(e*x)^(m + n)*(a + b*x^n) 
^p*(c + d*x^n)^(q - 1)*Simp[c*b*(m + 1) + n*(b*c*(p + 1) + a*d*q) + d*(b*(m 
 + 1) + b*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && 
 NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[0, q, 1] && LtQ[m, -1] && IntBinomi 
alQ[a, b, c, d, e, m, n, p, q, x]
 

rule 1054

Int[(((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n 
_)))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*(a 
+ b*x^n)^p*((e + f*x^n)/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, g, 
 m, p}, x] && IGtQ[n, 0]
 

rule 2009

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

\[ \int \frac {\left (a+b x^3\right )^{2/3}}{x^2 \left (a d-b d x^3\right )} \, dx=- \frac {\int \frac {\left (a + b x^{3}\right )^{\frac {2}{3}}}{- a x^{2} + b x^{5}}\, dx}{d} \] Input:

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

Output:

-Integral((a + b*x**3)**(2/3)/(-a*x**2 + b*x**5), x)/d
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (a+b x^3\right )^{2/3}}{x^2 \left (a d-b d x^3\right )} \, dx=\frac {\int \frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{-b \,x^{5}+a \,x^{2}}d x}{d} \] Input:

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

Output:

int((a + b*x**3)**(2/3)/(a*x**2 - b*x**5),x)/d