Integrand size = 20, antiderivative size = 92 \[ \int \frac {x \left (c+d x^4\right )}{\left (a+b x^4\right )^{3/4}} \, dx=\frac {d x^2 \sqrt [4]{a+b x^4}}{3 b}+\frac {\sqrt {a} (3 b c-2 a d) \left (1+\frac {b x^4}{a}\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right ),2\right )}{3 b^{3/2} \left (a+b x^4\right )^{3/4}} \] Output:
1/3*d*x^2*(b*x^4+a)^(1/4)/b+1/3*a^(1/2)*(-2*a*d+3*b*c)*(1+b*x^4/a)^(3/4)*I nverseJacobiAM(1/2*arctan(b^(1/2)*x^2/a^(1/2)),2^(1/2))/b^(3/2)/(b*x^4+a)^ (3/4)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.05 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.82 \[ \int \frac {x \left (c+d x^4\right )}{\left (a+b x^4\right )^{3/4}} \, dx=\frac {x^2 \left (2 d \left (a+b x^4\right )+(3 b c-2 a d) \left (1+\frac {b x^4}{a}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {3}{2},-\frac {b x^4}{a}\right )\right )}{6 b \left (a+b x^4\right )^{3/4}} \] Input:
Integrate[(x*(c + d*x^4))/(a + b*x^4)^(3/4),x]
Output:
(x^2*(2*d*(a + b*x^4) + (3*b*c - 2*a*d)*(1 + (b*x^4)/a)^(3/4)*Hypergeometr ic2F1[1/2, 3/4, 3/2, -((b*x^4)/a)]))/(6*b*(a + b*x^4)^(3/4))
Time = 0.34 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {959, 807, 231, 229}
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.
\(\displaystyle \int \frac {x \left (c+d x^4\right )}{\left (a+b x^4\right )^{3/4}} \, dx\) |
\(\Big \downarrow \) 959 |
\(\displaystyle \frac {(3 b c-2 a d) \int \frac {x}{\left (b x^4+a\right )^{3/4}}dx}{3 b}+\frac {d x^2 \sqrt [4]{a+b x^4}}{3 b}\) |
\(\Big \downarrow \) 807 |
\(\displaystyle \frac {(3 b c-2 a d) \int \frac {1}{\left (b x^4+a\right )^{3/4}}dx^2}{6 b}+\frac {d x^2 \sqrt [4]{a+b x^4}}{3 b}\) |
\(\Big \downarrow \) 231 |
\(\displaystyle \frac {\left (\frac {b x^4}{a}+1\right )^{3/4} (3 b c-2 a d) \int \frac {1}{\left (\frac {b x^4}{a}+1\right )^{3/4}}dx^2}{6 b \left (a+b x^4\right )^{3/4}}+\frac {d x^2 \sqrt [4]{a+b x^4}}{3 b}\) |
\(\Big \downarrow \) 229 |
\(\displaystyle \frac {\sqrt {a} \left (\frac {b x^4}{a}+1\right )^{3/4} (3 b c-2 a d) \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right ),2\right )}{3 b^{3/2} \left (a+b x^4\right )^{3/4}}+\frac {d x^2 \sqrt [4]{a+b x^4}}{3 b}\) |
Input:
Int[(x*(c + d*x^4))/(a + b*x^4)^(3/4),x]
Output:
(d*x^2*(a + b*x^4)^(1/4))/(3*b) + (Sqrt[a]*(3*b*c - 2*a*d)*(1 + (b*x^4)/a) ^(3/4)*EllipticF[ArcTan[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(3*b^(3/2)*(a + b*x^ 4)^(3/4))
Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Simp[(2/(a^(3/4)*Rt[b/a, 2]) )*EllipticF[(1/2)*ArcTan[Rt[b/a, 2]*x], 2], x] /; FreeQ[{a, b}, x] && GtQ[a , 0] && PosQ[b/a]
Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Simp[(1 + b*(x^2/a))^(3/4)/( a + b*x^2)^(3/4) Int[1/(1 + b*(x^2/a))^(3/4), x], x] /; FreeQ[{a, b}, x] && PosQ[a]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _)), x_Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(b*e*(m + n*(p + 1) + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m + n*(p + 1) + 1)) Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0] && NeQ[m + n*(p + 1) + 1, 0]
\[\int \frac {x \left (d \,x^{4}+c \right )}{\left (b \,x^{4}+a \right )^{\frac {3}{4}}}d x\]
Input:
int(x*(d*x^4+c)/(b*x^4+a)^(3/4),x)
Output:
int(x*(d*x^4+c)/(b*x^4+a)^(3/4),x)
\[ \int \frac {x \left (c+d x^4\right )}{\left (a+b x^4\right )^{3/4}} \, dx=\int { \frac {{\left (d x^{4} + c\right )} x}{{\left (b x^{4} + a\right )}^{\frac {3}{4}}} \,d x } \] Input:
integrate(x*(d*x^4+c)/(b*x^4+a)^(3/4),x, algorithm="fricas")
Output:
integral((d*x^5 + c*x)/(b*x^4 + a)^(3/4), x)
Result contains complex when optimal does not.
Time = 3.79 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.65 \[ \int \frac {x \left (c+d x^4\right )}{\left (a+b x^4\right )^{3/4}} \, dx=\frac {c x^{2} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {3}{2} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{4}}} + \frac {d x^{6} {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {3}{2} \\ \frac {5}{2} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{6 a^{\frac {3}{4}}} \] Input:
integrate(x*(d*x**4+c)/(b*x**4+a)**(3/4),x)
Output:
c*x**2*hyper((1/2, 3/4), (3/2,), b*x**4*exp_polar(I*pi)/a)/(2*a**(3/4)) + d*x**6*hyper((3/4, 3/2), (5/2,), b*x**4*exp_polar(I*pi)/a)/(6*a**(3/4))
\[ \int \frac {x \left (c+d x^4\right )}{\left (a+b x^4\right )^{3/4}} \, dx=\int { \frac {{\left (d x^{4} + c\right )} x}{{\left (b x^{4} + a\right )}^{\frac {3}{4}}} \,d x } \] Input:
integrate(x*(d*x^4+c)/(b*x^4+a)^(3/4),x, algorithm="maxima")
Output:
integrate((d*x^4 + c)*x/(b*x^4 + a)^(3/4), x)
\[ \int \frac {x \left (c+d x^4\right )}{\left (a+b x^4\right )^{3/4}} \, dx=\int { \frac {{\left (d x^{4} + c\right )} x}{{\left (b x^{4} + a\right )}^{\frac {3}{4}}} \,d x } \] Input:
integrate(x*(d*x^4+c)/(b*x^4+a)^(3/4),x, algorithm="giac")
Output:
integrate((d*x^4 + c)*x/(b*x^4 + a)^(3/4), x)
Timed out. \[ \int \frac {x \left (c+d x^4\right )}{\left (a+b x^4\right )^{3/4}} \, dx=\int \frac {x\,\left (d\,x^4+c\right )}{{\left (b\,x^4+a\right )}^{3/4}} \,d x \] Input:
int((x*(c + d*x^4))/(a + b*x^4)^(3/4),x)
Output:
int((x*(c + d*x^4))/(a + b*x^4)^(3/4), x)
\[ \int \frac {x \left (c+d x^4\right )}{\left (a+b x^4\right )^{3/4}} \, dx=\left (\int \frac {x^{5}}{\left (b \,x^{4}+a \right )^{\frac {3}{4}}}d x \right ) d +\left (\int \frac {x}{\left (b \,x^{4}+a \right )^{\frac {3}{4}}}d x \right ) c \] Input:
int(x*(d*x^4+c)/(b*x^4+a)^(3/4),x)
Output:
int(x**5/(a + b*x**4)**(3/4),x)*d + int(x/(a + b*x**4)**(3/4),x)*c