\(\int \frac {x^6}{(a+b x^3)^{4/3} (c+d x^3)} \, dx\) [770]

Optimal result

Integrand size = 24, antiderivative size = 260 \[ \int \frac {x^6}{\left (a+b x^3\right )^{4/3} \left (c+d x^3\right )} \, dx=\frac {a x}{b (b c-a d) \sqrt [3]{a+b x^3}}+\frac {\arctan \left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} b^{4/3} d}-\frac {c^{4/3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} d (b c-a d)^{4/3}}-\frac {c^{4/3} \log \left (c+d x^3\right )}{6 d (b c-a d)^{4/3}}+\frac {c^{4/3} \log \left (\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 d (b c-a d)^{4/3}}-\frac {\log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{2 b^{4/3} d} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 4.56 (sec) , antiderivative size = 466, normalized size of antiderivative = 1.79 \[ \int \frac {x^6}{\left (a+b x^3\right )^{4/3} \left (c+d x^3\right )} \, dx=\frac {1}{12} \left (\frac {12 a x}{\left (b^2 c-a b d\right ) \sqrt [3]{a+b x^3}}+\frac {4 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{b} x}{\sqrt [3]{b} x+2 \sqrt [3]{a+b x^3}}\right )}{b^{4/3} d}+\frac {2 \sqrt {-6+6 i \sqrt {3}} c^{4/3} \arctan \left (\frac {3 \sqrt [3]{b c-a d} x}{\sqrt {3} \sqrt [3]{b c-a d} x-\left (3 i+\sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{a+b x^3}}\right )}{d (b c-a d)^{4/3}}-\frac {4 \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{b^{4/3} d}-\frac {2 i \left (-i+\sqrt {3}\right ) c^{4/3} \log \left (2 \sqrt [3]{b c-a d} x+\left (1+i \sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{a+b x^3}\right )}{d (b c-a d)^{4/3}}+\frac {2 \log \left (b^{2/3} x^2+\sqrt [3]{b} x \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )}{b^{4/3} d}+\frac {\left (1+i \sqrt {3}\right ) c^{4/3} \log \left (2 (b c-a d)^{2/3} x^2+\left (-1-i \sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{b c-a d} x \sqrt [3]{a+b x^3}+i \left (i+\sqrt {3}\right ) c^{2/3} \left (a+b x^3\right )^{2/3}\right )}{d (b c-a d)^{4/3}}\right ) \] Input:

Integrate[x^6/((a + b*x^3)^(4/3)*(c + d*x^3)),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.09, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {970, 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.

\(\displaystyle \int \frac {x^6}{\left (a+b x^3\right )^{4/3} \left (c+d x^3\right )} \, dx\)

\(\Big \downarrow \) 970

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

\(\Big \downarrow \) 1026

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

\(\Big \downarrow \) 769

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

\(\Big \downarrow \) 901

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

Input:

Int[x^6/((a + b*x^3)^(4/3)*(c + d*x^3)),x]
 

Output:

(a*x)/(b*(b*c - a*d)*(a + b*x^3)^(1/3)) - ((b*c^2*(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 - ((b*c - 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)/ 
(b*(b*c - a*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 970

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ 
))^(q_), x_Symbol] :> Simp[(-a)*e^(2*n - 1)*(e*x)^(m - 2*n + 1)*(a + b*x^n) 
^(p + 1)*((c + d*x^n)^(q + 1)/(b*n*(b*c - a*d)*(p + 1))), x] + Simp[e^(2*n) 
/(b*n*(b*c - a*d)*(p + 1))   Int[(e*x)^(m - 2*n)*(a + b*x^n)^(p + 1)*(c + d 
*x^n)^q*Simp[a*c*(m - 2*n + 1) + (a*d*(m - n + n*q + 1) + b*c*n*(p + 1))*x^ 
n, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[ 
n, 0] && LtQ[p, -1] && GtQ[m - n + 1, n] && 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 = 1.41 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.31

method	result	size
pseudoelliptic	\(\frac {\left (-\left (a d -b c \right ) \left (\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (b^{\frac {1}{3}} x +2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right )}{3 b^{\frac {1}{3}} x}\right )+\ln \left (\frac {-b^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right )-\frac {\ln \left (\frac {b^{\frac {2}{3}} x^{2}+b^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right )}{2}\right ) \left (b \,x^{3}+a \right )^{\frac {1}{3}}-3 b^{\frac {1}{3}} a d x \right ) \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}}+c \left (b \,x^{3}+a \right )^{\frac {1}{3}} \left (\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x -2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right )}{3 \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x}\right )+\ln \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right )-\frac {\ln \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {2}{3}} x^{2}-\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right )}{2}\right ) b^{\frac {4}{3}}}{3 \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} b^{\frac {4}{3}} \left (a d -b c \right ) d}\)	\(341\)

Input:

int(x^6/(b*x^3+a)^(4/3)/(d*x^3+c),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 539 vs. \(2 (211) = 422\).

Time = 0.12 (sec) , antiderivative size = 1127, normalized size of antiderivative = 4.33 \[ \int \frac {x^6}{\left (a+b x^3\right )^{4/3} \left (c+d x^3\right )} \, dx=\text {Too large to display} \] Input:

integrate(x^6/(b*x^3+a)^(4/3)/(d*x^3+c),x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {x^6}{\left (a+b x^3\right )^{4/3} \left (c+d x^3\right )} \, dx=\int \frac {x^{6}}{\left (a + b x^{3}\right )^{\frac {4}{3}} \left (c + d x^{3}\right )}\, dx \] Input:

integrate(x**6/(b*x**3+a)**(4/3)/(d*x**3+c),x)
 

Output:

Integral(x**6/((a + b*x**3)**(4/3)*(c + d*x**3)), x)
 

Maxima [F]

\[ \int \frac {x^6}{\left (a+b x^3\right )^{4/3} \left (c+d x^3\right )} \, dx=\int { \frac {x^{6}}{{\left (b x^{3} + a\right )}^{\frac {4}{3}} {\left (d x^{3} + c\right )}} \,d x } \] Input:

integrate(x^6/(b*x^3+a)^(4/3)/(d*x^3+c),x, algorithm="maxima")
 

Output:

integrate(x^6/((b*x^3 + a)^(4/3)*(d*x^3 + c)), x)
 

Giac [F]

\[ \int \frac {x^6}{\left (a+b x^3\right )^{4/3} \left (c+d x^3\right )} \, dx=\int { \frac {x^{6}}{{\left (b x^{3} + a\right )}^{\frac {4}{3}} {\left (d x^{3} + c\right )}} \,d x } \] Input:

integrate(x^6/(b*x^3+a)^(4/3)/(d*x^3+c),x, algorithm="giac")
 

Output:

integrate(x^6/((b*x^3 + a)^(4/3)*(d*x^3 + c)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^6}{\left (a+b x^3\right )^{4/3} \left (c+d x^3\right )} \, dx=\int \frac {x^6}{{\left (b\,x^3+a\right )}^{4/3}\,\left (d\,x^3+c\right )} \,d x \] Input:

int(x^6/((a + b*x^3)^(4/3)*(c + d*x^3)),x)
 

Output:

int(x^6/((a + b*x^3)^(4/3)*(c + d*x^3)), x)
 

Reduce [F]

\[ \int \frac {x^6}{\left (a+b x^3\right )^{4/3} \left (c+d x^3\right )} \, dx=\int \frac {x^{6}}{\left (b \,x^{3}+a \right )^{\frac {1}{3}} a c +\left (b \,x^{3}+a \right )^{\frac {1}{3}} a d \,x^{3}+\left (b \,x^{3}+a \right )^{\frac {1}{3}} b c \,x^{3}+\left (b \,x^{3}+a \right )^{\frac {1}{3}} b d \,x^{6}}d x \] Input:

int(x^6/(b*x^3+a)^(4/3)/(d*x^3+c),x)
                                                                                    
                                                                                    
 

Output:

int(x**6/((a + b*x**3)**(1/3)*a*c + (a + b*x**3)**(1/3)*a*d*x**3 + (a + b* 
x**3)**(1/3)*b*c*x**3 + (a + b*x**3)**(1/3)*b*d*x**6),x)