\(\int \frac {x^3 (a+b x^3)^{2/3}}{c+d x^3} \, dx\) [700]

Optimal result

Integrand size = 24, antiderivative size = 272 \[ \int \frac {x^3 \left (a+b x^3\right )^{2/3}}{c+d x^3} \, dx=\frac {x \left (a+b x^3\right )^{2/3}}{3 d}-\frac {(3 b c-2 a d) \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3} \sqrt [3]{b} d^2}+\frac {\sqrt [3]{c} (b c-a d)^{2/3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} d^2}+\frac {\sqrt [3]{c} (b c-a d)^{2/3} \log \left (c+d x^3\right )}{6 d^2}-\frac {\sqrt [3]{c} (b c-a d)^{2/3} \log \left (\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 d^2}+\frac {(3 b c-2 a d) \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{6 \sqrt [3]{b} d^2} \] Output:

1/3*x*(b*x^3+a)^(2/3)/d-1/9*(-2*a*d+3*b*c)*arctan(1/3*(1+2*b^(1/3)*x/(b*x^ 
3+a)^(1/3))*3^(1/2))*3^(1/2)/b^(1/3)/d^2+1/3*c^(1/3)*(-a*d+b*c)^(2/3)*arct 
an(1/3*(1+2*(-a*d+b*c)^(1/3)*x/c^(1/3)/(b*x^3+a)^(1/3))*3^(1/2))*3^(1/2)/d 
^2+1/6*c^(1/3)*(-a*d+b*c)^(2/3)*ln(d*x^3+c)/d^2-1/2*c^(1/3)*(-a*d+b*c)^(2/ 
3)*ln((-a*d+b*c)^(1/3)*x/c^(1/3)-(b*x^3+a)^(1/3))/d^2+1/6*(-2*a*d+3*b*c)*l 
n(-b^(1/3)*x+(b*x^3+a)^(1/3))/b^(1/3)/d^2
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 4.03 (sec) , antiderivative size = 466, normalized size of antiderivative = 1.71 \[ \int \frac {x^3 \left (a+b x^3\right )^{2/3}}{c+d x^3} \, dx=\frac {12 d x \left (a+b x^3\right )^{2/3}-\frac {4 \sqrt {3} (3 b c-2 a d) \arctan \left (\frac {\sqrt {3} \sqrt [3]{b} x}{\sqrt [3]{b} x+2 \sqrt [3]{a+b x^3}}\right )}{\sqrt [3]{b}}-6 \sqrt {-6+6 i \sqrt {3}} \sqrt [3]{c} (b c-a d)^{2/3} \arctan \left (\frac {3 \sqrt [3]{b c-a d} x}{\sqrt {3} \sqrt [3]{b c-a d} x-\left (3 i+\sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{a+b x^3}}\right )+\frac {4 (3 b c-2 a d) \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{b}}+6 \left (1+i \sqrt {3}\right ) \sqrt [3]{c} (b c-a d)^{2/3} \log \left (2 \sqrt [3]{b c-a d} x+\left (1+i \sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{a+b x^3}\right )+\frac {2 (-3 b c+2 a d) \log \left (b^{2/3} x^2+\sqrt [3]{b} x \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )}{\sqrt [3]{b}}-3 i \left (-i+\sqrt {3}\right ) \sqrt [3]{c} (b c-a d)^{2/3} \log \left (2 (b c-a d)^{2/3} x^2+\left (-1-i \sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{b c-a d} x \sqrt [3]{a+b x^3}+i \left (i+\sqrt {3}\right ) c^{2/3} \left (a+b x^3\right )^{2/3}\right )}{36 d^2} \] Input:

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

Output:

(12*d*x*(a + b*x^3)^(2/3) - (4*Sqrt[3]*(3*b*c - 2*a*d)*ArcTan[(Sqrt[3]*b^( 
1/3)*x)/(b^(1/3)*x + 2*(a + b*x^3)^(1/3))])/b^(1/3) - 6*Sqrt[-6 + (6*I)*Sq 
rt[3]]*c^(1/3)*(b*c - a*d)^(2/3)*ArcTan[(3*(b*c - a*d)^(1/3)*x)/(Sqrt[3]*( 
b*c - a*d)^(1/3)*x - (3*I + Sqrt[3])*c^(1/3)*(a + b*x^3)^(1/3))] + (4*(3*b 
*c - 2*a*d)*Log[-(b^(1/3)*x) + (a + b*x^3)^(1/3)])/b^(1/3) + 6*(1 + I*Sqrt 
[3])*c^(1/3)*(b*c - a*d)^(2/3)*Log[2*(b*c - a*d)^(1/3)*x + (1 + I*Sqrt[3]) 
*c^(1/3)*(a + b*x^3)^(1/3)] + (2*(-3*b*c + 2*a*d)*Log[b^(2/3)*x^2 + b^(1/3 
)*x*(a + b*x^3)^(1/3) + (a + b*x^3)^(2/3)])/b^(1/3) - (3*I)*(-I + Sqrt[3]) 
*c^(1/3)*(b*c - a*d)^(2/3)*Log[2*(b*c - a*d)^(2/3)*x^2 + (-1 - I*Sqrt[3])* 
c^(1/3)*(b*c - a*d)^(1/3)*x*(a + b*x^3)^(1/3) + I*(I + Sqrt[3])*c^(2/3)*(a 
 + b*x^3)^(2/3)])/(36*d^2)
 

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {978, 1026, 769, 901}

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.

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

\(\Big \downarrow \) 978

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

\(\Big \downarrow \) 1026

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

\(\Big \downarrow \) 769

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

\(\Big \downarrow \) 901

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 769

Int[((a_) + (b_.)*(x_)^3)^(-1/3), x_Symbol] :> Simp[ArcTan[(1 + 2*Rt[b, 3]* 
(x/(a + b*x^3)^(1/3)))/Sqrt[3]]/(Sqrt[3]*Rt[b, 3]), x] - Simp[Log[(a + b*x^ 
3)^(1/3) - Rt[b, 3]*x]/(2*Rt[b, 3]), x] /; FreeQ[{a, b}, x]
 

rule 901

Int[1/(((a_) + (b_.)*(x_)^3)^(1/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> Wit 
h[{q = Rt[(b*c - a*d)/c, 3]}, Simp[ArcTan[(1 + (2*q*x)/(a + b*x^3)^(1/3))/S 
qrt[3]]/(Sqrt[3]*c*q), x] + (-Simp[Log[q*x - (a + b*x^3)^(1/3)]/(2*c*q), x] 
 + Simp[Log[c + d*x^3]/(6*c*q), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0]
 

rule 978

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

rule 1026

Int[(((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n_)))/((c_) + (d_.)* 
(x_)^(n_)), x_Symbol] :> Simp[f/d   Int[(a + b*x^n)^p, x], x] + Simp[(d*e - 
 c*f)/d   Int[(a + b*x^n)^p/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e, f, 
 p, n}, x]
 

Maple [A] (verified)

Time = 2.01 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.18

method	result	size
pseudoelliptic	\(-\frac {\left (-d x \left (b \,x^{3}+a \right )^{\frac {2}{3}} b^{\frac {1}{3}}+\frac {2 \left (\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (b^{\frac {1}{3}} x +2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right )}{3 b^{\frac {1}{3}} x}\right )+\ln \left (\frac {-b^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right )-\frac {\ln \left (\frac {b^{\frac {2}{3}} x^{2}+b^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right )}{2}\right ) \left (a d -\frac {3 b c}{2}\right )}{3}\right ) \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}}+\left (\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x -2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right )}{3 \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x}\right )+\ln \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right )-\frac {\ln \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {2}{3}} x^{2}-\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right )}{2}\right ) \left (b^{\frac {1}{3}} a d -b^{\frac {4}{3}} c \right )}{3 \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} b^{\frac {1}{3}} d^{2}}\)	\(321\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 521 vs. \(2 (219) = 438\).

Time = 0.29 (sec) , antiderivative size = 1091, normalized size of antiderivative = 4.01 \[ \int \frac {x^3 \left (a+b x^3\right )^{2/3}}{c+d x^3} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/18*(6*(b*x^3 + a)^(2/3)*b*d*x - 3*sqrt(1/3)*(3*b^2*c - 2*a*b*d)*sqrt((- 
b)^(1/3)/b)*log(3*b*x^3 - 3*(b*x^3 + a)^(1/3)*(-b)^(2/3)*x^2 - 3*sqrt(1/3) 
*((-b)^(1/3)*b*x^3 - (b*x^3 + a)^(1/3)*b*x^2 + 2*(b*x^3 + a)^(2/3)*(-b)^(2 
/3)*x)*sqrt((-b)^(1/3)/b) + 2*a) + 6*sqrt(3)*(-b^2*c^3 + 2*a*b*c^2*d - a^2 
*c*d^2)^(1/3)*b*arctan(1/3*(sqrt(3)*(b*c - a*d)*x - 2*sqrt(3)*(-b^2*c^3 + 
2*a*b*c^2*d - a^2*c*d^2)^(1/3)*(b*x^3 + a)^(1/3))/((b*c - a*d)*x)) + 2*(3* 
b*c - 2*a*d)*(-b)^(2/3)*log(((-b)^(1/3)*x + (b*x^3 + a)^(1/3))/x) - (3*b*c 
 - 2*a*d)*(-b)^(2/3)*log(((-b)^(2/3)*x^2 - (b*x^3 + a)^(1/3)*(-b)^(1/3)*x 
+ (b*x^3 + a)^(2/3))/x^2) + 6*(-b^2*c^3 + 2*a*b*c^2*d - a^2*c*d^2)^(1/3)*b 
*log(((-b^2*c^3 + 2*a*b*c^2*d - a^2*c*d^2)^(2/3)*x - (b*x^3 + a)^(1/3)*(b* 
c^2 - a*c*d))/x) - 3*(-b^2*c^3 + 2*a*b*c^2*d - a^2*c*d^2)^(1/3)*b*log(((-b 
^2*c^3 + 2*a*b*c^2*d - a^2*c*d^2)^(1/3)*(b*c - a*d)*x^2 - (-b^2*c^3 + 2*a* 
b*c^2*d - a^2*c*d^2)^(2/3)*(b*x^3 + a)^(1/3)*x - (b*x^3 + a)^(2/3)*(b*c^2 
- a*c*d))/x^2))/(b*d^2), 1/18*(6*(b*x^3 + a)^(2/3)*b*d*x + 6*sqrt(1/3)*(3* 
b^2*c - 2*a*b*d)*sqrt(-(-b)^(1/3)/b)*arctan(-sqrt(1/3)*((-b)^(1/3)*x - 2*( 
b*x^3 + a)^(1/3))*sqrt(-(-b)^(1/3)/b)/x) + 6*sqrt(3)*(-b^2*c^3 + 2*a*b*c^2 
*d - a^2*c*d^2)^(1/3)*b*arctan(1/3*(sqrt(3)*(b*c - a*d)*x - 2*sqrt(3)*(-b^ 
2*c^3 + 2*a*b*c^2*d - a^2*c*d^2)^(1/3)*(b*x^3 + a)^(1/3))/((b*c - a*d)*x)) 
 + 2*(3*b*c - 2*a*d)*(-b)^(2/3)*log(((-b)^(1/3)*x + (b*x^3 + a)^(1/3))/x) 
- (3*b*c - 2*a*d)*(-b)^(2/3)*log(((-b)^(2/3)*x^2 - (b*x^3 + a)^(1/3)*(-...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^3 \left (a+b x^3\right )^{2/3}}{c+d x^3} \, dx=\frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}} x -\left (\int \frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{b d \,x^{6}+a d \,x^{3}+b c \,x^{3}+a c}d x \right ) a c +2 \left (\int \frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}} x^{3}}{b d \,x^{6}+a d \,x^{3}+b c \,x^{3}+a c}d x \right ) a d -3 \left (\int \frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}} x^{3}}{b d \,x^{6}+a d \,x^{3}+b c \,x^{3}+a c}d x \right ) b c}{3 d} \] Input:

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

Output:

((a + b*x**3)**(2/3)*x - int((a + b*x**3)**(2/3)/(a*c + a*d*x**3 + b*c*x** 
3 + b*d*x**6),x)*a*c + 2*int(((a + b*x**3)**(2/3)*x**3)/(a*c + a*d*x**3 + 
b*c*x**3 + b*d*x**6),x)*a*d - 3*int(((a + b*x**3)**(2/3)*x**3)/(a*c + a*d* 
x**3 + b*c*x**3 + b*d*x**6),x)*b*c)/(3*d)