Integrand size = 24, antiderivative size = 134 \[ \int \sqrt [3]{a+b x^4} \left (A+B x^2+C x^4\right ) \, dx=\frac {3 C x \left (a+b x^4\right )^{4/3}}{19 b}+\frac {(19 A b-3 a C) x \sqrt [3]{a+b x^4} \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},\frac {1}{4},\frac {5}{4},-\frac {b x^4}{a}\right )}{19 b \sqrt [3]{1+\frac {b x^4}{a}}}+\frac {B x^3 \sqrt [3]{a+b x^4} \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},\frac {3}{4},\frac {7}{4},-\frac {b x^4}{a}\right )}{3 \sqrt [3]{1+\frac {b x^4}{a}}} \] Output:
3/19*C*x*(b*x^4+a)^(4/3)/b+1/19*(19*A*b-3*C*a)*x*(b*x^4+a)^(1/3)*hypergeom ([-1/3, 1/4],[5/4],-b*x^4/a)/b/(1+b*x^4/a)^(1/3)+1/3*B*x^3*(b*x^4+a)^(1/3) *hypergeom([-1/3, 3/4],[7/4],-b*x^4/a)/(1+b*x^4/a)^(1/3)
Time = 10.04 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.77 \[ \int \sqrt [3]{a+b x^4} \left (A+B x^2+C x^4\right ) \, dx=\frac {\sqrt [3]{a+b x^4} \left (15 A x \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},\frac {1}{4},\frac {5}{4},-\frac {b x^4}{a}\right )+5 B x^3 \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},\frac {3}{4},\frac {7}{4},-\frac {b x^4}{a}\right )+3 C x^5 \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},\frac {5}{4},\frac {9}{4},-\frac {b x^4}{a}\right )\right )}{15 \sqrt [3]{1+\frac {b x^4}{a}}} \] Input:
Integrate[(a + b*x^4)^(1/3)*(A + B*x^2 + C*x^4),x]
Output:
((a + b*x^4)^(1/3)*(15*A*x*Hypergeometric2F1[-1/3, 1/4, 5/4, -((b*x^4)/a)] + 5*B*x^3*Hypergeometric2F1[-1/3, 3/4, 7/4, -((b*x^4)/a)] + 3*C*x^5*Hyper geometric2F1[-1/3, 5/4, 9/4, -((b*x^4)/a)]))/(15*(1 + (b*x^4)/a)^(1/3))
Time = 0.48 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.13, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2432, 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.
\(\displaystyle \int \sqrt [3]{a+b x^4} \left (A+B x^2+C x^4\right ) \, dx\) |
\(\Big \downarrow \) 2432 |
\(\displaystyle \int \left (A \sqrt [3]{a+b x^4}+B x^2 \sqrt [3]{a+b x^4}+C x^4 \sqrt [3]{a+b x^4}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {A x \sqrt [3]{a+b x^4} \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},\frac {1}{4},\frac {5}{4},-\frac {b x^4}{a}\right )}{\sqrt [3]{\frac {b x^4}{a}+1}}+\frac {B x^3 \sqrt [3]{a+b x^4} \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},\frac {3}{4},\frac {7}{4},-\frac {b x^4}{a}\right )}{3 \sqrt [3]{\frac {b x^4}{a}+1}}+\frac {C x^5 \sqrt [3]{a+b x^4} \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},\frac {5}{4},\frac {9}{4},-\frac {b x^4}{a}\right )}{5 \sqrt [3]{\frac {b x^4}{a}+1}}\) |
Input:
Int[(a + b*x^4)^(1/3)*(A + B*x^2 + C*x^4),x]
Output:
(A*x*(a + b*x^4)^(1/3)*Hypergeometric2F1[-1/3, 1/4, 5/4, -((b*x^4)/a)])/(1 + (b*x^4)/a)^(1/3) + (B*x^3*(a + b*x^4)^(1/3)*Hypergeometric2F1[-1/3, 3/4 , 7/4, -((b*x^4)/a)])/(3*(1 + (b*x^4)/a)^(1/3)) + (C*x^5*(a + b*x^4)^(1/3) *Hypergeometric2F1[-1/3, 5/4, 9/4, -((b*x^4)/a)])/(5*(1 + (b*x^4)/a)^(1/3) )
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[ Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, n, p}, x] && (PolyQ[Pq, x] || Poly Q[Pq, x^n])
\[\int \left (b \,x^{4}+a \right )^{\frac {1}{3}} \left (C \,x^{4}+B \,x^{2}+A \right )d x\]
Input:
int((b*x^4+a)^(1/3)*(C*x^4+B*x^2+A),x)
Output:
int((b*x^4+a)^(1/3)*(C*x^4+B*x^2+A),x)
\[ \int \sqrt [3]{a+b x^4} \left (A+B x^2+C x^4\right ) \, dx=\int { {\left (C x^{4} + B x^{2} + A\right )} {\left (b x^{4} + a\right )}^{\frac {1}{3}} \,d x } \] Input:
integrate((b*x^4+a)^(1/3)*(C*x^4+B*x^2+A),x, algorithm="fricas")
Output:
integral((C*x^4 + B*x^2 + A)*(b*x^4 + a)^(1/3), x)
Result contains complex when optimal does not.
Time = 1.38 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.93 \[ \int \sqrt [3]{a+b x^4} \left (A+B x^2+C x^4\right ) \, dx=\frac {A \sqrt [3]{a} x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} + \frac {B \sqrt [3]{a} x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} + \frac {C \sqrt [3]{a} x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {9}{4}\right )} \] Input:
integrate((b*x**4+a)**(1/3)*(C*x**4+B*x**2+A),x)
Output:
A*a**(1/3)*x*gamma(1/4)*hyper((-1/3, 1/4), (5/4,), b*x**4*exp_polar(I*pi)/ a)/(4*gamma(5/4)) + B*a**(1/3)*x**3*gamma(3/4)*hyper((-1/3, 3/4), (7/4,), b*x**4*exp_polar(I*pi)/a)/(4*gamma(7/4)) + C*a**(1/3)*x**5*gamma(5/4)*hype r((-1/3, 5/4), (9/4,), b*x**4*exp_polar(I*pi)/a)/(4*gamma(9/4))
\[ \int \sqrt [3]{a+b x^4} \left (A+B x^2+C x^4\right ) \, dx=\int { {\left (C x^{4} + B x^{2} + A\right )} {\left (b x^{4} + a\right )}^{\frac {1}{3}} \,d x } \] Input:
integrate((b*x^4+a)^(1/3)*(C*x^4+B*x^2+A),x, algorithm="maxima")
Output:
integrate((C*x^4 + B*x^2 + A)*(b*x^4 + a)^(1/3), x)
\[ \int \sqrt [3]{a+b x^4} \left (A+B x^2+C x^4\right ) \, dx=\int { {\left (C x^{4} + B x^{2} + A\right )} {\left (b x^{4} + a\right )}^{\frac {1}{3}} \,d x } \] Input:
integrate((b*x^4+a)^(1/3)*(C*x^4+B*x^2+A),x, algorithm="giac")
Output:
integrate((C*x^4 + B*x^2 + A)*(b*x^4 + a)^(1/3), x)
Timed out. \[ \int \sqrt [3]{a+b x^4} \left (A+B x^2+C x^4\right ) \, dx=\int {\left (b\,x^4+a\right )}^{1/3}\,\left (C\,x^4+B\,x^2+A\right ) \,d x \] Input:
int((a + b*x^4)^(1/3)*(A + B*x^2 + C*x^4),x)
Output:
int((a + b*x^4)^(1/3)*(A + B*x^2 + C*x^4), x)
\[ \int \sqrt [3]{a+b x^4} \left (A+B x^2+C x^4\right ) \, dx=\frac {741 \left (b \,x^{4}+a \right )^{\frac {1}{3}} a b x +156 \left (b \,x^{4}+a \right )^{\frac {1}{3}} a c x +399 \left (b \,x^{4}+a \right )^{\frac {1}{3}} b^{2} x^{3}+273 \left (b \,x^{4}+a \right )^{\frac {1}{3}} b c \,x^{5}+988 \left (\int \frac {1}{\left (b \,x^{4}+a \right )^{\frac {2}{3}}}d x \right ) a^{2} b -156 \left (\int \frac {1}{\left (b \,x^{4}+a \right )^{\frac {2}{3}}}d x \right ) a^{2} c +532 \left (\int \frac {x^{2}}{\left (b \,x^{4}+a \right )^{\frac {2}{3}}}d x \right ) a \,b^{2}}{1729 b} \] Input:
int((b*x^4+a)^(1/3)*(C*x^4+B*x^2+A),x)
Output:
(741*(a + b*x**4)**(1/3)*a*b*x + 156*(a + b*x**4)**(1/3)*a*c*x + 399*(a + b*x**4)**(1/3)*b**2*x**3 + 273*(a + b*x**4)**(1/3)*b*c*x**5 + 988*int((a + b*x**4)**(1/3)/(a + b*x**4),x)*a**2*b - 156*int((a + b*x**4)**(1/3)/(a + b*x**4),x)*a**2*c + 532*int(((a + b*x**4)**(1/3)*x**2)/(a + b*x**4),x)*a*b **2)/(1729*b)