Integrand size = 22, antiderivative size = 91 \[ \int \frac {x^2 \left (c+d x^4\right )}{\left (a+b x^4\right )^{7/4}} \, dx=\frac {(b c-a d) x^3}{3 a b \left (a+b x^4\right )^{3/4}}-\frac {d \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 b^{7/4}}+\frac {d \text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 b^{7/4}} \] Output:
1/3*(-a*d+b*c)*x^3/a/b/(b*x^4+a)^(3/4)-1/2*d*arctan(b^(1/4)*x/(b*x^4+a)^(1 /4))/b^(7/4)+1/2*d*arctanh(b^(1/4)*x/(b*x^4+a)^(1/4))/b^(7/4)
Time = 0.75 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.95 \[ \int \frac {x^2 \left (c+d x^4\right )}{\left (a+b x^4\right )^{7/4}} \, dx=\frac {\frac {2 b^{3/4} (b c-a d) x^3}{a \left (a+b x^4\right )^{3/4}}-3 d \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )+3 d \text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{6 b^{7/4}} \] Input:
Integrate[(x^2*(c + d*x^4))/(a + b*x^4)^(7/4),x]
Output:
((2*b^(3/4)*(b*c - a*d)*x^3)/(a*(a + b*x^4)^(3/4)) - 3*d*ArcTan[(b^(1/4)*x )/(a + b*x^4)^(1/4)] + 3*d*ArcTanh[(b^(1/4)*x)/(a + b*x^4)^(1/4)])/(6*b^(7 /4))
Time = 0.37 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {954, 854, 827, 216, 219}
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.
| \(\displaystyle \int \frac {x^2 \left (c+d x^4\right )}{\left (a+b x^4\right )^{7/4}} \, dx\) |
| \(\Big \downarrow \) 954 |
| \(\displaystyle \frac {d \int \frac {x^2}{\left (b x^4+a\right )^{3/4}}dx}{b}+\frac {x^3 (b c-a d)}{3 a b \left (a+b x^4\right )^{3/4}}\) |
| \(\Big \downarrow \) 854 |
| \(\displaystyle \frac {d \int \frac {x^2}{\sqrt {b x^4+a} \left (1-\frac {b x^4}{b x^4+a}\right )}d\frac {x}{\sqrt [4]{b x^4+a}}}{b}+\frac {x^3 (b c-a d)}{3 a b \left (a+b x^4\right )^{3/4}}\) |
| \(\Big \downarrow \) 827 |
| \(\displaystyle \frac {d \left (\frac {\int \frac {1}{1-\frac {\sqrt {b} x^2}{\sqrt {b x^4+a}}}d\frac {x}{\sqrt [4]{b x^4+a}}}{2 \sqrt {b}}-\frac {\int \frac {1}{\frac {\sqrt {b} x^2}{\sqrt {b x^4+a}}+1}d\frac {x}{\sqrt [4]{b x^4+a}}}{2 \sqrt {b}}\right )}{b}+\frac {x^3 (b c-a d)}{3 a b \left (a+b x^4\right )^{3/4}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {d \left (\frac {\int \frac {1}{1-\frac {\sqrt {b} x^2}{\sqrt {b x^4+a}}}d\frac {x}{\sqrt [4]{b x^4+a}}}{2 \sqrt {b}}-\frac {\arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 b^{3/4}}\right )}{b}+\frac {x^3 (b c-a d)}{3 a b \left (a+b x^4\right )^{3/4}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 b^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 b^{3/4}}\right )}{b}+\frac {x^3 (b c-a d)}{3 a b \left (a+b x^4\right )^{3/4}}\) |
Input:
Int[(x^2*(c + d*x^4))/(a + b*x^4)^(7/4),x]
Output:
((b*c - a*d)*x^3)/(3*a*b*(a + b*x^4)^(3/4)) + (d*(-1/2*ArcTan[(b^(1/4)*x)/ (a + b*x^4)^(1/4)]/b^(3/4) + ArcTanh[(b^(1/4)*x)/(a + b*x^4)^(1/4)]/(2*b^( 3/4))))/b
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b) Int[1/(r + s*x^2), x], x] - Simp[s/(2*b) Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ [a/b, 0]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + (m + 1)/n) Subst[Int[x^m/(1 - b*x^n)^(p + (m + 1)/n + 1), x], x, x/(a + b*x^n )^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, - 2^(-1)] && IntegersQ[m, p + (m + 1)/n]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _)), x_Symbol] :> Simp[(b*c - a*d)*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*b* e*(m + 1))), x] + Simp[d/b Int[(e*x)^m*(a + b*x^n)^(p + 1), x], x] /; Fre eQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n*(p + 1) + 1, 0] && NeQ[m, -1]
Time = 0.18 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.07
| method | result | size |
| pseudoelliptic | \(\frac {\frac {\ln \left (\frac {x \,b^{\frac {1}{4}}+\left (b \,x^{4}+a \right )^{\frac {1}{4}}}{-x \,b^{\frac {1}{4}}+\left (b \,x^{4}+a \right )^{\frac {1}{4}}}\right ) a d}{4 b^{\frac {7}{4}}}+\frac {\arctan \left (\frac {\left (b \,x^{4}+a \right )^{\frac {1}{4}}}{x \,b^{\frac {1}{4}}}\right ) a d}{2 b^{\frac {7}{4}}}-\frac {\left (a d -c b \right ) x^{3}}{3 b \left (b \,x^{4}+a \right )^{\frac {3}{4}}}}{a}\) | \(97\) |
Input:
int(x^2*(d*x^4+c)/(b*x^4+a)^(7/4),x,method=_RETURNVERBOSE)
Output:
(1/4*ln((x*b^(1/4)+(b*x^4+a)^(1/4))/(-x*b^(1/4)+(b*x^4+a)^(1/4)))*a/b^(7/4 )*d+1/2*arctan((b*x^4+a)^(1/4)/x/b^(1/4))*a/b^(7/4)*d-1/3*(a*d-b*c)*x^3/b/ (b*x^4+a)^(3/4))/a
Result contains complex when optimal does not.
Time = 0.09 (sec) , antiderivative size = 273, normalized size of antiderivative = 3.00 \[ \int \frac {x^2 \left (c+d x^4\right )}{\left (a+b x^4\right )^{7/4}} \, dx=\frac {4 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}} {\left (b c - a d\right )} x^{3} + 3 \, {\left (a b^{2} x^{4} + a^{2} b\right )} \left (\frac {d^{4}}{b^{7}}\right )^{\frac {1}{4}} \log \left (\frac {b^{2} x \left (\frac {d^{4}}{b^{7}}\right )^{\frac {1}{4}} + {\left (b x^{4} + a\right )}^{\frac {1}{4}} d}{x}\right ) - 3 \, {\left (a b^{2} x^{4} + a^{2} b\right )} \left (\frac {d^{4}}{b^{7}}\right )^{\frac {1}{4}} \log \left (-\frac {b^{2} x \left (\frac {d^{4}}{b^{7}}\right )^{\frac {1}{4}} - {\left (b x^{4} + a\right )}^{\frac {1}{4}} d}{x}\right ) - 3 \, {\left (-i \, a b^{2} x^{4} - i \, a^{2} b\right )} \left (\frac {d^{4}}{b^{7}}\right )^{\frac {1}{4}} \log \left (\frac {i \, b^{2} x \left (\frac {d^{4}}{b^{7}}\right )^{\frac {1}{4}} + {\left (b x^{4} + a\right )}^{\frac {1}{4}} d}{x}\right ) - 3 \, {\left (i \, a b^{2} x^{4} + i \, a^{2} b\right )} \left (\frac {d^{4}}{b^{7}}\right )^{\frac {1}{4}} \log \left (\frac {-i \, b^{2} x \left (\frac {d^{4}}{b^{7}}\right )^{\frac {1}{4}} + {\left (b x^{4} + a\right )}^{\frac {1}{4}} d}{x}\right )}{12 \, {\left (a b^{2} x^{4} + a^{2} b\right )}} \] Input:
integrate(x^2*(d*x^4+c)/(b*x^4+a)^(7/4),x, algorithm="fricas")
Output:
1/12*(4*(b*x^4 + a)^(1/4)*(b*c - a*d)*x^3 + 3*(a*b^2*x^4 + a^2*b)*(d^4/b^7 )^(1/4)*log((b^2*x*(d^4/b^7)^(1/4) + (b*x^4 + a)^(1/4)*d)/x) - 3*(a*b^2*x^ 4 + a^2*b)*(d^4/b^7)^(1/4)*log(-(b^2*x*(d^4/b^7)^(1/4) - (b*x^4 + a)^(1/4) *d)/x) - 3*(-I*a*b^2*x^4 - I*a^2*b)*(d^4/b^7)^(1/4)*log((I*b^2*x*(d^4/b^7) ^(1/4) + (b*x^4 + a)^(1/4)*d)/x) - 3*(I*a*b^2*x^4 + I*a^2*b)*(d^4/b^7)^(1/ 4)*log((-I*b^2*x*(d^4/b^7)^(1/4) + (b*x^4 + a)^(1/4)*d)/x))/(a*b^2*x^4 + a ^2*b)
Result contains complex when optimal does not.
Time = 8.04 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.80 \[ \int \frac {x^2 \left (c+d x^4\right )}{\left (a+b x^4\right )^{7/4}} \, dx=\frac {c x^{3} \Gamma \left (\frac {3}{4}\right )}{4 a^{\frac {7}{4}} \left (1 + \frac {b x^{4}}{a}\right )^{\frac {3}{4}} \Gamma \left (\frac {7}{4}\right )} + \frac {d x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {7}{4}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {7}{4}} \Gamma \left (\frac {11}{4}\right )} \] Input:
integrate(x**2*(d*x**4+c)/(b*x**4+a)**(7/4),x)
Output:
c*x**3*gamma(3/4)/(4*a**(7/4)*(1 + b*x**4/a)**(3/4)*gamma(7/4)) + d*x**7*g amma(7/4)*hyper((7/4, 7/4), (11/4,), b*x**4*exp_polar(I*pi)/a)/(4*a**(7/4) *gamma(11/4))
Time = 0.11 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.24 \[ \int \frac {x^2 \left (c+d x^4\right )}{\left (a+b x^4\right )^{7/4}} \, dx=\frac {c x^{3}}{3 \, {\left (b x^{4} + a\right )}^{\frac {3}{4}} a} - \frac {1}{12} \, {\left (\frac {4 \, x^{3}}{{\left (b x^{4} + a\right )}^{\frac {3}{4}} b} - \frac {3 \, {\left (\frac {2 \, \arctan \left (\frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{b^{\frac {1}{4}} x}\right )}{b^{\frac {3}{4}}} - \frac {\log \left (-\frac {b^{\frac {1}{4}} - \frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{x}}{b^{\frac {1}{4}} + \frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{x}}\right )}{b^{\frac {3}{4}}}\right )}}{b}\right )} d \] Input:
integrate(x^2*(d*x^4+c)/(b*x^4+a)^(7/4),x, algorithm="maxima")
Output:
1/3*c*x^3/((b*x^4 + a)^(3/4)*a) - 1/12*(4*x^3/((b*x^4 + a)^(3/4)*b) - 3*(2 *arctan((b*x^4 + a)^(1/4)/(b^(1/4)*x))/b^(3/4) - log(-(b^(1/4) - (b*x^4 + a)^(1/4)/x)/(b^(1/4) + (b*x^4 + a)^(1/4)/x))/b^(3/4))/b)*d
\[ \int \frac {x^2 \left (c+d x^4\right )}{\left (a+b x^4\right )^{7/4}} \, dx=\int { \frac {{\left (d x^{4} + c\right )} x^{2}}{{\left (b x^{4} + a\right )}^{\frac {7}{4}}} \,d x } \] Input:
integrate(x^2*(d*x^4+c)/(b*x^4+a)^(7/4),x, algorithm="giac")
Output:
integrate((d*x^4 + c)*x^2/(b*x^4 + a)^(7/4), x)
Timed out. \[ \int \frac {x^2 \left (c+d x^4\right )}{\left (a+b x^4\right )^{7/4}} \, dx=\int \frac {x^2\,\left (d\,x^4+c\right )}{{\left (b\,x^4+a\right )}^{7/4}} \,d x \] Input:
int((x^2*(c + d*x^4))/(a + b*x^4)^(7/4),x)
Output:
int((x^2*(c + d*x^4))/(a + b*x^4)^(7/4), x)
\[ \int \frac {x^2 \left (c+d x^4\right )}{\left (a+b x^4\right )^{7/4}} \, dx=\left (\int \frac {x^{6}}{\left (b \,x^{4}+a \right )^{\frac {3}{4}} a +\left (b \,x^{4}+a \right )^{\frac {3}{4}} b \,x^{4}}d x \right ) d +\left (\int \frac {x^{2}}{\left (b \,x^{4}+a \right )^{\frac {3}{4}} a +\left (b \,x^{4}+a \right )^{\frac {3}{4}} b \,x^{4}}d x \right ) c \] Input:
int(x^2*(d*x^4+c)/(b*x^4+a)^(7/4),x)
Output:
int(x**6/((a + b*x**4)**(3/4)*a + (a + b*x**4)**(3/4)*b*x**4),x)*d + int(x **2/((a + b*x**4)**(3/4)*a + (a + b*x**4)**(3/4)*b*x**4),x)*c