\(\int \frac {c+d x^3}{(a+b x^3)^{4/3}} \, dx\) [111]

Optimal result

Integrand size = 19, antiderivative size = 99 \[ \int \frac {c+d x^3}{\left (a+b x^3\right )^{4/3}} \, dx=\frac {(b c-a d) x}{a b \sqrt [3]{a+b x^3}}+\frac {d \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} b^{4/3}}-\frac {d \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{2 b^{4/3}} \] Output:

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.52 \[ \int \frac {c+d x^3}{\left (a+b x^3\right )^{4/3}} \, dx=\frac {\frac {6 \sqrt [3]{b} (b c-a d) x}{a \sqrt [3]{a+b x^3}}+2 \sqrt {3} d \arctan \left (\frac {\sqrt {3} \sqrt [3]{b} x}{\sqrt [3]{b} x+2 \sqrt [3]{a+b x^3}}\right )-2 d \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )+d \log \left (b^{2/3} x^2+\sqrt [3]{b} x \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )}{6 b^{4/3}} \] Input:

Integrate[(c + d*x^3)/(a + b*x^3)^(4/3),x]
 

Output:

((6*b^(1/3)*(b*c - a*d)*x)/(a*(a + b*x^3)^(1/3)) + 2*Sqrt[3]*d*ArcTan[(Sqr 
t[3]*b^(1/3)*x)/(b^(1/3)*x + 2*(a + b*x^3)^(1/3))] - 2*d*Log[-(b^(1/3)*x) 
+ (a + b*x^3)^(1/3)] + d*Log[b^(2/3)*x^2 + b^(1/3)*x*(a + b*x^3)^(1/3) + ( 
a + b*x^3)^(2/3)])/(6*b^(4/3))
 

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.04, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {910, 769}

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.

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

\(\Big \downarrow \) 910

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

\(\Big \downarrow \) 769

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

Input:

Int[(c + d*x^3)/(a + b*x^3)^(4/3),x]
 

Output:

((b*c - a*d)*x)/(a*b*(a + b*x^3)^(1/3)) + (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))))/b
 

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 910

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Si 
mp[(-(b*c - a*d))*x*((a + b*x^n)^(p + 1)/(a*b*n*(p + 1))), x] - Simp[(a*d - 
 b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1))   Int[(a + b*x^n)^(p + 1), x], x] /; 
FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/ 
n + p, 0])
 

Maple [A] (verified)

Time = 0.80 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.37

method	result	size
pseudoelliptic	\(-\frac {x d}{b \left (b \,x^{3}+a \right )^{\frac {1}{3}}}+\frac {x c}{a \left (b \,x^{3}+a \right )^{\frac {1}{3}}}-\frac {d \ln \left (\frac {-b^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right )}{3 b^{\frac {4}{3}}}+\frac {d \ln \left (\frac {b^{\frac {2}{3}} x^{2}+b^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right )}{6 b^{\frac {4}{3}}}-\frac {d \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}}{b^{\frac {1}{3}}}+x \right )}{3 x}\right )}{3 b^{\frac {4}{3}}}\)	\(136\)

Input:

int((d*x^3+c)/(b*x^3+a)^(4/3),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 220 vs. \(2 (81) = 162\).

Time = 0.09 (sec) , antiderivative size = 488, normalized size of antiderivative = 4.93 \[ \int \frac {c+d x^3}{\left (a+b x^3\right )^{4/3}} \, dx=\left [\frac {3 \, \sqrt {\frac {1}{3}} {\left (a b^{2} d x^{3} + a^{2} b d\right )} \sqrt {\frac {\left (-b\right )^{\frac {1}{3}}}{b}} \log \left (3 \, b x^{3} - 3 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-b\right )^{\frac {2}{3}} x^{2} - 3 \, \sqrt {\frac {1}{3}} {\left (\left (-b\right )^{\frac {1}{3}} b x^{3} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} b x^{2} + 2 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} \left (-b\right )^{\frac {2}{3}} x\right )} \sqrt {\frac {\left (-b\right )^{\frac {1}{3}}}{b}} + 2 \, a\right ) + 6 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} {\left (b^{2} c - a b d\right )} x - 2 \, {\left (a b d x^{3} + a^{2} d\right )} \left (-b\right )^{\frac {2}{3}} \log \left (\frac {\left (-b\right )^{\frac {1}{3}} x + {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right ) + {\left (a b d x^{3} + a^{2} d\right )} \left (-b\right )^{\frac {2}{3}} \log \left (\frac {\left (-b\right )^{\frac {2}{3}} x^{2} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-b\right )^{\frac {1}{3}} x + {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right )}{6 \, {\left (a b^{3} x^{3} + a^{2} b^{2}\right )}}, -\frac {6 \, \sqrt {\frac {1}{3}} {\left (a b^{2} d x^{3} + a^{2} b d\right )} \sqrt {-\frac {\left (-b\right )^{\frac {1}{3}}}{b}} \arctan \left (-\frac {\sqrt {\frac {1}{3}} {\left (\left (-b\right )^{\frac {1}{3}} x - 2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (-b\right )^{\frac {1}{3}}}{b}}}{x}\right ) - 6 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} {\left (b^{2} c - a b d\right )} x + 2 \, {\left (a b d x^{3} + a^{2} d\right )} \left (-b\right )^{\frac {2}{3}} \log \left (\frac {\left (-b\right )^{\frac {1}{3}} x + {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right ) - {\left (a b d x^{3} + a^{2} d\right )} \left (-b\right )^{\frac {2}{3}} \log \left (\frac {\left (-b\right )^{\frac {2}{3}} x^{2} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-b\right )^{\frac {1}{3}} x + {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right )}{6 \, {\left (a b^{3} x^{3} + a^{2} b^{2}\right )}}\right ] \] Input:

integrate((d*x^3+c)/(b*x^3+a)^(4/3),x, algorithm="fricas")
 

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 3.30 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.72 \[ \int \frac {c+d x^3}{\left (a+b x^3\right )^{4/3}} \, dx=\frac {c x \Gamma \left (\frac {1}{3}\right )}{3 a^{\frac {4}{3}} \sqrt [3]{1 + \frac {b x^{3}}{a}} \Gamma \left (\frac {4}{3}\right )} + \frac {d x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {4}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {4}{3}} \Gamma \left (\frac {7}{3}\right )} \] Input:

integrate((d*x**3+c)/(b*x**3+a)**(4/3),x)
 

Output:

c*x*gamma(1/3)/(3*a**(4/3)*(1 + b*x**3/a)**(1/3)*gamma(4/3)) + d*x**4*gamm 
a(4/3)*hyper((4/3, 4/3), (7/3,), b*x**3*exp_polar(I*pi)/a)/(3*a**(4/3)*gam 
ma(7/3))
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.35 \[ \int \frac {c+d x^3}{\left (a+b x^3\right )^{4/3}} \, dx=-\frac {1}{6} \, d {\left (\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (b^{\frac {1}{3}} + \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}}{3 \, b^{\frac {1}{3}}}\right )}{b^{\frac {4}{3}}} + \frac {6 \, x}{{\left (b x^{3} + a\right )}^{\frac {1}{3}} b} - \frac {\log \left (b^{\frac {2}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{\frac {1}{3}}}{x} + \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right )}{b^{\frac {4}{3}}} + \frac {2 \, \log \left (-b^{\frac {1}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}{b^{\frac {4}{3}}}\right )} + \frac {c x}{{\left (b x^{3} + a\right )}^{\frac {1}{3}} a} \] Input:

integrate((d*x^3+c)/(b*x^3+a)^(4/3),x, algorithm="maxima")
 

Output:

-1/6*d*(2*sqrt(3)*arctan(1/3*sqrt(3)*(b^(1/3) + 2*(b*x^3 + a)^(1/3)/x)/b^( 
1/3))/b^(4/3) + 6*x/((b*x^3 + a)^(1/3)*b) - log(b^(2/3) + (b*x^3 + a)^(1/3 
)*b^(1/3)/x + (b*x^3 + a)^(2/3)/x^2)/b^(4/3) + 2*log(-b^(1/3) + (b*x^3 + a 
)^(1/3)/x)/b^(4/3)) + c*x/((b*x^3 + a)^(1/3)*a)
 

Giac [F]

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

integrate((d*x^3+c)/(b*x^3+a)^(4/3),x, algorithm="giac")
 

Output:

integrate((d*x^3 + c)/(b*x^3 + a)^(4/3), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {c+d x^3}{\left (a+b x^3\right )^{4/3}} \, dx=\int \frac {d\,x^3+c}{{\left (b\,x^3+a\right )}^{4/3}} \,d x \] Input:

int((c + d*x^3)/(a + b*x^3)^(4/3),x)
 

Output:

int((c + d*x^3)/(a + b*x^3)^(4/3), x)
 

Reduce [F]

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

int((d*x^3+c)/(b*x^3+a)^(4/3),x)
 

Output:

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