\(\int \frac {b+a x^2}{\sqrt [3]{b^2 x^2+a^3 x^3}} \, dx\) [2550]

Optimal result

Integrand size = 27, antiderivative size = 215 \[ \int \frac {b+a x^2}{\sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\frac {\left (-4 b^2+3 a^3 x\right ) \left (b^2 x^2+a^3 x^3\right )^{2/3}}{6 a^5 x}+\frac {\left (9 a^5 b+2 b^4\right ) \arctan \left (\frac {\sqrt {3} a x}{a x+2 \sqrt [3]{b^2 x^2+a^3 x^3}}\right )}{3 \sqrt {3} a^6}+\frac {\left (-9 a^5 b-2 b^4\right ) \log \left (-a x+\sqrt [3]{b^2 x^2+a^3 x^3}\right )}{9 a^6}+\frac {\left (9 a^5 b+2 b^4\right ) \log \left (a^2 x^2+a x \sqrt [3]{b^2 x^2+a^3 x^3}+\left (b^2 x^2+a^3 x^3\right )^{2/3}\right )}{18 a^6} \] Output:

1/6*(3*a^3*x-4*b^2)*(a^3*x^3+b^2*x^2)^(2/3)/a^5/x+1/9*(9*a^5*b+2*b^4)*arct 
an(3^(1/2)*a*x/(a*x+2*(a^3*x^3+b^2*x^2)^(1/3)))*3^(1/2)/a^6+1/9*(-9*a^5*b- 
2*b^4)*ln(-a*x+(a^3*x^3+b^2*x^2)^(1/3))/a^6+1/18*(9*a^5*b+2*b^4)*ln(a^2*x^ 
2+a*x*(a^3*x^3+b^2*x^2)^(1/3)+(a^3*x^3+b^2*x^2)^(2/3))/a^6
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.56 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.46 \[ \int \frac {b+a x^2}{\sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\frac {x^{2/3} \left (-12 a b^4 \sqrt [3]{x}-3 a^4 b^2 x^{4/3}+9 a^7 x^{7/3}+2 \sqrt {3} b \left (9 a^5+2 b^3\right ) \sqrt [3]{b^2+a^3 x} \arctan \left (\frac {\sqrt {3} a \sqrt [3]{x}}{a \sqrt [3]{x}+2 \sqrt [3]{b^2+a^3 x}}\right )-2 b \left (9 a^5+2 b^3\right ) \sqrt [3]{b^2+a^3 x} \log \left (a^6 \left (a \sqrt [3]{x}-\sqrt [3]{b^2+a^3 x}\right )\right )+9 a^5 b \sqrt [3]{b^2+a^3 x} \log \left (a^2 x^{2/3}+a \sqrt [3]{x} \sqrt [3]{b^2+a^3 x}+\left (b^2+a^3 x\right )^{2/3}\right )+2 b^4 \sqrt [3]{b^2+a^3 x} \log \left (a^2 x^{2/3}+a \sqrt [3]{x} \sqrt [3]{b^2+a^3 x}+\left (b^2+a^3 x\right )^{2/3}\right )\right )}{18 a^6 \sqrt [3]{x^2 \left (b^2+a^3 x\right )}} \] Input:

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

Output:

(x^(2/3)*(-12*a*b^4*x^(1/3) - 3*a^4*b^2*x^(4/3) + 9*a^7*x^(7/3) + 2*Sqrt[3 
]*b*(9*a^5 + 2*b^3)*(b^2 + a^3*x)^(1/3)*ArcTan[(Sqrt[3]*a*x^(1/3))/(a*x^(1 
/3) + 2*(b^2 + a^3*x)^(1/3))] - 2*b*(9*a^5 + 2*b^3)*(b^2 + a^3*x)^(1/3)*Lo 
g[a^6*(a*x^(1/3) - (b^2 + a^3*x)^(1/3))] + 9*a^5*b*(b^2 + a^3*x)^(1/3)*Log 
[a^2*x^(2/3) + a*x^(1/3)*(b^2 + a^3*x)^(1/3) + (b^2 + a^3*x)^(2/3)] + 2*b^ 
4*(b^2 + a^3*x)^(1/3)*Log[a^2*x^(2/3) + a*x^(1/3)*(b^2 + a^3*x)^(1/3) + (b 
^2 + a^3*x)^(2/3)]))/(18*a^6*(x^2*(b^2 + a^3*x))^(1/3))
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(467\) vs. \(2(215)=430\).

Time = 0.70 (sec) , antiderivative size = 467, normalized size of antiderivative = 2.17, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2450, 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.

\(\displaystyle \int \frac {a x^2+b}{\sqrt [3]{a^3 x^3+b^2 x^2}} \, dx\)

\(\Big \downarrow \) 2450

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\sqrt {3} b x^{2/3} \sqrt [3]{a^3 x+b^2} \arctan \left (\frac {2 \sqrt [3]{a^3 x+b^2}}{\sqrt {3} a \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{a \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {b x^{2/3} \log (x) \sqrt [3]{a^3 x+b^2}}{2 a \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {3 b x^{2/3} \sqrt [3]{a^3 x+b^2} \log \left (\frac {\sqrt [3]{a^3 x+b^2}}{a \sqrt [3]{x}}-1\right )}{2 a \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {2 b^4 x^{2/3} \sqrt [3]{a^3 x+b^2} \arctan \left (\frac {2 \sqrt [3]{a^3 x+b^2}}{\sqrt {3} a \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{3 \sqrt {3} a^6 \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {b^4 x^{2/3} \log (x) \sqrt [3]{a^3 x+b^2}}{9 a^6 \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {b^4 x^{2/3} \sqrt [3]{a^3 x+b^2} \log \left (\frac {\sqrt [3]{a^3 x+b^2}}{a \sqrt [3]{x}}-1\right )}{3 a^6 \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {2 b^2 \left (a^3 x^3+b^2 x^2\right )^{2/3}}{3 a^5 x}+\frac {\left (a^3 x^3+b^2 x^2\right )^{2/3}}{2 a^2}\)

Input:

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

Output:

(b^2*x^2 + a^3*x^3)^(2/3)/(2*a^2) - (2*b^2*(b^2*x^2 + a^3*x^3)^(2/3))/(3*a 
^5*x) - (Sqrt[3]*b*x^(2/3)*(b^2 + a^3*x)^(1/3)*ArcTan[1/Sqrt[3] + (2*(b^2 
+ a^3*x)^(1/3))/(Sqrt[3]*a*x^(1/3))])/(a*(b^2*x^2 + a^3*x^3)^(1/3)) - (2*b 
^4*x^(2/3)*(b^2 + a^3*x)^(1/3)*ArcTan[1/Sqrt[3] + (2*(b^2 + a^3*x)^(1/3))/ 
(Sqrt[3]*a*x^(1/3))])/(3*Sqrt[3]*a^6*(b^2*x^2 + a^3*x^3)^(1/3)) - (b*x^(2/ 
3)*(b^2 + a^3*x)^(1/3)*Log[x])/(2*a*(b^2*x^2 + a^3*x^3)^(1/3)) - (b^4*x^(2 
/3)*(b^2 + a^3*x)^(1/3)*Log[x])/(9*a^6*(b^2*x^2 + a^3*x^3)^(1/3)) - (3*b*x 
^(2/3)*(b^2 + a^3*x)^(1/3)*Log[-1 + (b^2 + a^3*x)^(1/3)/(a*x^(1/3))])/(2*a 
*(b^2*x^2 + a^3*x^3)^(1/3)) - (b^4*x^(2/3)*(b^2 + a^3*x)^(1/3)*Log[-1 + (b 
^2 + a^3*x)^(1/3)/(a*x^(1/3))])/(3*a^6*(b^2*x^2 + a^3*x^3)^(1/3))
 

Defintions of rubi rules used

rule 2009

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

rule 2450

Int[(Pq_)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[Expan 
dIntegrand[Pq*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, j, n, p}, x] && (Po 
lyQ[Pq, x] || PolyQ[Pq, x^n]) &&  !IntegerQ[p] && NeQ[n, j]
 

Maple [A] (verified)

Time = 0.40 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.16

method	result	size
pseudoelliptic	\(-\frac {\left (-\frac {x \left (a^{5}+\frac {2 b^{3}}{9}\right ) b \ln \left (\frac {a^{2} x^{2}+\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}} a x +\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {2}{3}}}{x^{2}}\right )}{2}+\sqrt {3}\, b x \left (a^{5}+\frac {2 b^{3}}{9}\right ) \arctan \left (\frac {\left (a x +2 \left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a x}\right )+x \left (a^{5}+\frac {2 b^{3}}{9}\right ) b \ln \left (\frac {-a x +\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}}}{x}\right )-\frac {\left (a^{3} x -\frac {4 b^{2}}{3}\right ) \left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {2}{3}} a}{2}\right ) x^{3} b^{4}}{{\left (a x -\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}}\right )}^{2} {\left (\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {2}{3}}+a x \left (a x +\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}}\right )\right )}^{2} a^{6}}\)	\(249\)

Input:

int((a*x^2+b)/(a^3*x^3+b^2*x^2)^(1/3),x,method=_RETURNVERBOSE)
 

Output:

-(-1/2*x*(a^5+2/9*b^3)*b*ln((a^2*x^2+(x^2*(a^3*x+b^2))^(1/3)*a*x+(x^2*(a^3 
*x+b^2))^(2/3))/x^2)+3^(1/2)*b*x*(a^5+2/9*b^3)*arctan(1/3*(a*x+2*(x^2*(a^3 
*x+b^2))^(1/3))*3^(1/2)/a/x)+x*(a^5+2/9*b^3)*b*ln((-a*x+(x^2*(a^3*x+b^2))^ 
(1/3))/x)-1/2*(a^3*x-4/3*b^2)*(x^2*(a^3*x+b^2))^(2/3)*a)*x^3*b^4/(a*x-(x^2 
*(a^3*x+b^2))^(1/3))^2/((x^2*(a^3*x+b^2))^(2/3)+a*x*(a*x+(x^2*(a^3*x+b^2)) 
^(1/3)))^2/a^6
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.96 \[ \int \frac {b+a x^2}{\sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=-\frac {2 \, \sqrt {3} {\left (9 \, a^{5} b + 2 \, b^{4}\right )} x \arctan \left (\frac {\sqrt {3} a x + 2 \, \sqrt {3} {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}}}{3 \, a x}\right ) + 2 \, {\left (9 \, a^{5} b + 2 \, b^{4}\right )} x \log \left (-\frac {a x - {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - {\left (9 \, a^{5} b + 2 \, b^{4}\right )} x \log \left (\frac {a^{2} x^{2} + {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}} a x + {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right ) - 3 \, {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {2}{3}} {\left (3 \, a^{4} x - 4 \, a b^{2}\right )}}{18 \, a^{6} x} \] Input:

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

Output:

-1/18*(2*sqrt(3)*(9*a^5*b + 2*b^4)*x*arctan(1/3*(sqrt(3)*a*x + 2*sqrt(3)*( 
a^3*x^3 + b^2*x^2)^(1/3))/(a*x)) + 2*(9*a^5*b + 2*b^4)*x*log(-(a*x - (a^3* 
x^3 + b^2*x^2)^(1/3))/x) - (9*a^5*b + 2*b^4)*x*log((a^2*x^2 + (a^3*x^3 + b 
^2*x^2)^(1/3)*a*x + (a^3*x^3 + b^2*x^2)^(2/3))/x^2) - 3*(a^3*x^3 + b^2*x^2 
)^(2/3)*(3*a^4*x - 4*a*b^2))/(a^6*x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.55 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.91 \[ \int \frac {b+a x^2}{\sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=-\frac {\frac {2 \, \sqrt {3} {\left (9 \, a^{5} b^{3} + 2 \, b^{6}\right )} \arctan \left (\frac {\sqrt {3} {\left (a + 2 \, {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {1}{3}}\right )}}{3 \, a}\right )}{a^{6}} - \frac {{\left (9 \, a^{5} b^{3} + 2 \, b^{6}\right )} \log \left (a^{2} + {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {1}{3}} a + {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {2}{3}}\right )}{a^{6}} + \frac {2 \, {\left (9 \, a^{5} b^{3} + 2 \, b^{6}\right )} \log \left ({\left | -a + {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {1}{3}} \right |}\right )}{a^{6}} - \frac {3 \, {\left (7 \, {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {2}{3}} a^{3} b^{6} - 4 \, {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {5}{3}} b^{6}\right )} x^{2}}{a^{5} b^{4}}}{18 \, b^{2}} \] Input:

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

Output:

-1/18*(2*sqrt(3)*(9*a^5*b^3 + 2*b^6)*arctan(1/3*sqrt(3)*(a + 2*(a^3 + b^2/ 
x)^(1/3))/a)/a^6 - (9*a^5*b^3 + 2*b^6)*log(a^2 + (a^3 + b^2/x)^(1/3)*a + ( 
a^3 + b^2/x)^(2/3))/a^6 + 2*(9*a^5*b^3 + 2*b^6)*log(abs(-a + (a^3 + b^2/x) 
^(1/3)))/a^6 - 3*(7*(a^3 + b^2/x)^(2/3)*a^3*b^6 - 4*(a^3 + b^2/x)^(5/3)*b^ 
6)*x^2/(a^5*b^4))/b^2
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {b+a x^2}{\sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\left (\int \frac {x^{\frac {4}{3}}}{\left (a^{3} x +b^{2}\right )^{\frac {1}{3}}}d x \right ) a +\left (\int \frac {1}{x^{\frac {2}{3}} \left (a^{3} x +b^{2}\right )^{\frac {1}{3}}}d x \right ) b \] Input:

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

Output:

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