\(\int \frac {\sqrt [3]{a}+\sqrt [3]{b} x}{(2 \sqrt [3]{a}-\sqrt [3]{b} x) \sqrt {-a-b x^3}} \, dx\) [127]

Optimal result

Integrand size = 46, antiderivative size = 53 \[ \int \frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\left (2 \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {-a-b x^3}} \, dx=\frac {2 \arctan \left (\frac {\left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{3 \sqrt [6]{a} \sqrt {-a-b x^3}}\right )}{3 \sqrt [6]{a} \sqrt [3]{b}} \] Output:

2/3*arctan(1/3*(a^(1/3)+b^(1/3)*x)^2/a^(1/6)/(-b*x^3-a)^(1/2))/a^(1/6)/b^( 
1/3)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 3.32 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\left (2 \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {-a-b x^3}} \, dx=-\frac {2 \arctan \left (\frac {3 \sqrt [6]{a} \sqrt {-a-b x^3}}{\left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}\right )}{3 \sqrt [6]{a} \sqrt [3]{b}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {2563, 216}

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.

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

\(\Big \downarrow \) 2563

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

\(\Big \downarrow \) 216

\(\displaystyle \frac {2 \arctan \left (\frac {\left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{3 \sqrt [6]{a} \sqrt {-a-b x^3}}\right )}{3 \sqrt [6]{a} \sqrt [3]{b}}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

rule 2563

Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_ 
Symbol] :> Simp[-2*(e/d)   Subst[Int[1/(9 - a*x^2), x], x, (1 + f*(x/e))^2/ 
Sqrt[a + b*x^3]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] & 
& EqQ[b*c^3 + 8*a*d^3, 0] && EqQ[2*d*e + c*f, 0]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 138 vs. \(2 (37) = 74\).

Time = 0.71 (sec) , antiderivative size = 641, normalized size of antiderivative = 12.09 \[ \int \frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\left (2 \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {-a-b x^3}} \, dx=\left [\frac {1}{6} \, a^{\frac {1}{3}} \sqrt {-\frac {1}{a b^{\frac {2}{3}}}} \log \left (\frac {b^{6} x^{18} + 7800 \, a b^{5} x^{15} + 535272 \, a^{2} b^{4} x^{12} + 5147264 \, a^{3} b^{3} x^{9} + 10516992 \, a^{4} b^{2} x^{6} + 5922816 \, a^{5} b x^{3} + 557056 \, a^{6} + 144 \, {\left (7 \, b^{5} x^{16} + 1169 \, a b^{4} x^{13} + 20266 \, a^{2} b^{3} x^{10} + 66976 \, a^{3} b^{2} x^{7} + 58112 \, a^{4} b x^{4} + 10240 \, a^{5} x\right )} a^{\frac {2}{3}} b^{\frac {1}{3}} + 72 \, {\left (b^{5} x^{17} + 581 \, a b^{4} x^{14} + 19108 \, a^{2} b^{3} x^{11} + 106336 \, a^{3} b^{2} x^{8} + 137984 \, a^{4} b x^{5} + 50176 \, a^{5} x^{2}\right )} a^{\frac {1}{3}} b^{\frac {2}{3}} + 12 \, {\left ({\left (b^{5} x^{16} + 1568 \, a b^{4} x^{13} + 72520 \, a^{2} b^{3} x^{10} + 498304 \, a^{3} b^{2} x^{7} + 625664 \, a^{4} b x^{4} + 139264 \, a^{5} x\right )} \sqrt {-b x^{3} - a} a^{\frac {2}{3}} b^{\frac {2}{3}} + 6 \, {\left (41 \, a b^{5} x^{14} + 4268 \, a^{2} b^{4} x^{11} + 52896 \, a^{3} b^{3} x^{8} + 116480 \, a^{4} b^{2} x^{5} + 48128 \, a^{5} b x^{2}\right )} \sqrt {-b x^{3} - a} a^{\frac {1}{3}} + {\left (25 \, a b^{5} x^{15} + 7202 \, a^{2} b^{4} x^{12} + 167392 \, a^{3} b^{3} x^{9} + 647296 \, a^{4} b^{2} x^{6} + 468992 \, a^{5} b x^{3} + 40960 \, a^{6}\right )} \sqrt {-b x^{3} - a} b^{\frac {1}{3}}\right )} \sqrt {-\frac {1}{a b^{\frac {2}{3}}}}}{b^{6} x^{18} - 48 \, a b^{5} x^{15} + 960 \, a^{2} b^{4} x^{12} - 10240 \, a^{3} b^{3} x^{9} + 61440 \, a^{4} b^{2} x^{6} - 196608 \, a^{5} b x^{3} + 262144 \, a^{6}}\right ), -\frac {1}{3} \, a^{\frac {1}{3}} \sqrt {\frac {1}{a b^{\frac {2}{3}}}} \arctan \left (\frac {{\left ({\left (11 \, b x^{4} - 16 \, a x\right )} \sqrt {-b x^{3} - a} a^{\frac {2}{3}} b^{\frac {2}{3}} + {\left (b^{2} x^{5} + 28 \, a b x^{2}\right )} \sqrt {-b x^{3} - a} a^{\frac {1}{3}} - {\left (17 \, a b x^{3} - 10 \, a^{2}\right )} \sqrt {-b x^{3} - a} b^{\frac {1}{3}}\right )} \sqrt {\frac {1}{a b^{\frac {2}{3}}}}}{6 \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}}\right )\right ] \] Input:

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

Output:

[1/6*a^(1/3)*sqrt(-1/(a*b^(2/3)))*log((b^6*x^18 + 7800*a*b^5*x^15 + 535272 
*a^2*b^4*x^12 + 5147264*a^3*b^3*x^9 + 10516992*a^4*b^2*x^6 + 5922816*a^5*b 
*x^3 + 557056*a^6 + 144*(7*b^5*x^16 + 1169*a*b^4*x^13 + 20266*a^2*b^3*x^10 
 + 66976*a^3*b^2*x^7 + 58112*a^4*b*x^4 + 10240*a^5*x)*a^(2/3)*b^(1/3) + 72 
*(b^5*x^17 + 581*a*b^4*x^14 + 19108*a^2*b^3*x^11 + 106336*a^3*b^2*x^8 + 13 
7984*a^4*b*x^5 + 50176*a^5*x^2)*a^(1/3)*b^(2/3) + 12*((b^5*x^16 + 1568*a*b 
^4*x^13 + 72520*a^2*b^3*x^10 + 498304*a^3*b^2*x^7 + 625664*a^4*b*x^4 + 139 
264*a^5*x)*sqrt(-b*x^3 - a)*a^(2/3)*b^(2/3) + 6*(41*a*b^5*x^14 + 4268*a^2* 
b^4*x^11 + 52896*a^3*b^3*x^8 + 116480*a^4*b^2*x^5 + 48128*a^5*b*x^2)*sqrt( 
-b*x^3 - a)*a^(1/3) + (25*a*b^5*x^15 + 7202*a^2*b^4*x^12 + 167392*a^3*b^3* 
x^9 + 647296*a^4*b^2*x^6 + 468992*a^5*b*x^3 + 40960*a^6)*sqrt(-b*x^3 - a)* 
b^(1/3))*sqrt(-1/(a*b^(2/3))))/(b^6*x^18 - 48*a*b^5*x^15 + 960*a^2*b^4*x^1 
2 - 10240*a^3*b^3*x^9 + 61440*a^4*b^2*x^6 - 196608*a^5*b*x^3 + 262144*a^6) 
), -1/3*a^(1/3)*sqrt(1/(a*b^(2/3)))*arctan(1/6*((11*b*x^4 - 16*a*x)*sqrt(- 
b*x^3 - a)*a^(2/3)*b^(2/3) + (b^2*x^5 + 28*a*b*x^2)*sqrt(-b*x^3 - a)*a^(1/ 
3) - (17*a*b*x^3 - 10*a^2)*sqrt(-b*x^3 - a)*b^(1/3))*sqrt(1/(a*b^(2/3)))/( 
b^2*x^6 + 2*a*b*x^3 + a^2))]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [B] (verification not implemented)

Time = 25.63 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.47 \[ \int \frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\left (2 \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {-a-b x^3}} \, dx=\frac {\ln \left (\frac {\left (\sqrt {-b\,x^3-a}-\sqrt {a}\,1{}\mathrm {i}\right )\,{\left (2\,a^{1/6}\,b^{1/3}\,x-\sqrt {a}+\sqrt {-b\,x^3-a}\,1{}\mathrm {i}\right )}^3}{x^3\,{\left (b^{1/3}\,x-2\,a^{1/3}\right )}^3}\right )\,1{}\mathrm {i}}{3\,a^{1/6}\,b^{1/3}} \] Input:

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

Output:

(log((((- a - b*x^3)^(1/2) - a^(1/2)*1i)*((- a - b*x^3)^(1/2)*1i - a^(1/2) 
 + 2*a^(1/6)*b^(1/3)*x)^3)/(x^3*(b^(1/3)*x - 2*a^(1/3))^3))*1i)/(3*a^(1/6) 
*b^(1/3))
 

Reduce [F]

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

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

Output:

 - i*(a**(1/3)*int(sqrt(a + b*x**3)/(2*a**(1/3)*a + 2*a**(1/3)*b*x**3 - b* 
*(1/3)*a*x - b**(1/3)*b*x**4),x) + b**(1/3)*int((sqrt(a + b*x**3)*x)/(2*a* 
*(1/3)*a + 2*a**(1/3)*b*x**3 - b**(1/3)*a*x - b**(1/3)*b*x**4),x))