Integrand size = 24, antiderivative size = 104 \[ \int \frac {x^{11}}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=-\frac {(b c+a d) \sqrt {c+d x^4}}{2 b^2 d^2}+\frac {\left (c+d x^4\right )^{3/2}}{6 b d^2}-\frac {a^2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^4}}{\sqrt {b c-a d}}\right )}{2 b^{5/2} \sqrt {b c-a d}} \] Output:
-1/2*(a*d+b*c)*(d*x^4+c)^(1/2)/b^2/d^2+1/6*(d*x^4+c)^(3/2)/b/d^2-1/2*a^2*a rctanh(b^(1/2)*(d*x^4+c)^(1/2)/(-a*d+b*c)^(1/2))/b^(5/2)/(-a*d+b*c)^(1/2)
Time = 0.24 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.88 \[ \int \frac {x^{11}}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=\frac {\sqrt {c+d x^4} \left (-2 b c-3 a d+b d x^4\right )}{6 b^2 d^2}+\frac {a^2 \arctan \left (\frac {\sqrt {b} \sqrt {c+d x^4}}{\sqrt {-b c+a d}}\right )}{2 b^{5/2} \sqrt {-b c+a d}} \] Input:
Integrate[x^11/((a + b*x^4)*Sqrt[c + d*x^4]),x]
Output:
(Sqrt[c + d*x^4]*(-2*b*c - 3*a*d + b*d*x^4))/(6*b^2*d^2) + (a^2*ArcTan[(Sq rt[b]*Sqrt[c + d*x^4])/Sqrt[-(b*c) + a*d]])/(2*b^(5/2)*Sqrt[-(b*c) + a*d])
Time = 0.45 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {948, 99, 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.
\(\displaystyle \int \frac {x^{11}}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx\) |
\(\Big \downarrow \) 948 |
\(\displaystyle \frac {1}{4} \int \frac {x^8}{\left (b x^4+a\right ) \sqrt {d x^4+c}}dx^4\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \frac {1}{4} \int \left (\frac {a^2}{b^2 \left (b x^4+a\right ) \sqrt {d x^4+c}}+\frac {\sqrt {d x^4+c}}{b d}+\frac {-b c-a d}{b^2 d \sqrt {d x^4+c}}\right )dx^4\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{4} \left (-\frac {2 a^2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^4}}{\sqrt {b c-a d}}\right )}{b^{5/2} \sqrt {b c-a d}}-\frac {2 \sqrt {c+d x^4} (a d+b c)}{b^2 d^2}+\frac {2 \left (c+d x^4\right )^{3/2}}{3 b d^2}\right )\) |
Input:
Int[x^11/((a + b*x^4)*Sqrt[c + d*x^4]),x]
Output:
((-2*(b*c + a*d)*Sqrt[c + d*x^4])/(b^2*d^2) + (2*(c + d*x^4)^(3/2))/(3*b*d ^2) - (2*a^2*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^4])/Sqrt[b*c - a*d]])/(b^(5/2)* Sqrt[b*c - a*d]))/4
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. ), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^ p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ [b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
Time = 0.21 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.89
method | result | size |
pseudoelliptic | \(-\frac {-\arctan \left (\frac {\sqrt {d \,x^{4}+c}\, b}{\sqrt {\left (a d -c b \right ) b}}\right ) a^{2} d^{2}+\sqrt {\left (a d -c b \right ) b}\, \sqrt {d \,x^{4}+c}\, \left (\frac {\left (-d \,x^{4}+2 c \right ) b}{3}+a d \right )}{2 \sqrt {\left (a d -c b \right ) b}\, b^{2} d^{2}}\) | \(93\) |
risch | \(-\frac {\left (-d b \,x^{4}+3 a d +2 c b \right ) \sqrt {d \,x^{4}+c}}{6 d^{2} b^{2}}-\frac {a^{2} \ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}-\frac {\sqrt {-a b}}{b}}\right )}{4 b^{3} \sqrt {-\frac {a d -c b}{b}}}-\frac {a^{2} \ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}+\frac {\sqrt {-a b}}{b}}\right )}{4 b^{3} \sqrt {-\frac {a d -c b}{b}}}\) | \(355\) |
default | \(-\frac {\sqrt {d \,x^{4}+c}\, \left (-d \,x^{4}+2 c \right )}{6 b \,d^{2}}+\frac {a^{2} \left (-\frac {\ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}-\frac {\sqrt {-a b}}{b}}\right )}{4 b \sqrt {-\frac {a d -c b}{b}}}-\frac {\ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}+\frac {\sqrt {-a b}}{b}}\right )}{4 b \sqrt {-\frac {a d -c b}{b}}}\right )}{b^{2}}-\frac {a \sqrt {d \,x^{4}+c}}{2 b^{2} d}\) | \(369\) |
elliptic | \(\frac {x^{4} \sqrt {d \,x^{4}+c}}{6 b d}-\frac {c \sqrt {d \,x^{4}+c}}{3 b \,d^{2}}-\frac {a \sqrt {d \,x^{4}+c}}{2 b^{2} d}-\frac {a^{2} \ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}-\frac {\sqrt {-a b}}{b}}\right )}{4 b^{3} \sqrt {-\frac {a d -c b}{b}}}-\frac {a^{2} \ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}+\frac {\sqrt {-a b}}{b}}\right )}{4 b^{3} \sqrt {-\frac {a d -c b}{b}}}\) | \(378\) |
Input:
int(x^11/(b*x^4+a)/(d*x^4+c)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/2/((a*d-b*c)*b)^(1/2)*(-arctan((d*x^4+c)^(1/2)*b/((a*d-b*c)*b)^(1/2))*a ^2*d^2+((a*d-b*c)*b)^(1/2)*(d*x^4+c)^(1/2)*(1/3*(-d*x^4+2*c)*b+a*d))/b^2/d ^2
Time = 0.11 (sec) , antiderivative size = 289, normalized size of antiderivative = 2.78 \[ \int \frac {x^{11}}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=\left [\frac {3 \, \sqrt {b^{2} c - a b d} a^{2} d^{2} \log \left (\frac {b d x^{4} + 2 \, b c - a d - 2 \, \sqrt {d x^{4} + c} \sqrt {b^{2} c - a b d}}{b x^{4} + a}\right ) - 2 \, {\left (2 \, b^{3} c^{2} + a b^{2} c d - 3 \, a^{2} b d^{2} - {\left (b^{3} c d - a b^{2} d^{2}\right )} x^{4}\right )} \sqrt {d x^{4} + c}}{12 \, {\left (b^{4} c d^{2} - a b^{3} d^{3}\right )}}, \frac {3 \, \sqrt {-b^{2} c + a b d} a^{2} d^{2} \arctan \left (\frac {\sqrt {d x^{4} + c} \sqrt {-b^{2} c + a b d}}{b d x^{4} + b c}\right ) - {\left (2 \, b^{3} c^{2} + a b^{2} c d - 3 \, a^{2} b d^{2} - {\left (b^{3} c d - a b^{2} d^{2}\right )} x^{4}\right )} \sqrt {d x^{4} + c}}{6 \, {\left (b^{4} c d^{2} - a b^{3} d^{3}\right )}}\right ] \] Input:
integrate(x^11/(b*x^4+a)/(d*x^4+c)^(1/2),x, algorithm="fricas")
Output:
[1/12*(3*sqrt(b^2*c - a*b*d)*a^2*d^2*log((b*d*x^4 + 2*b*c - a*d - 2*sqrt(d *x^4 + c)*sqrt(b^2*c - a*b*d))/(b*x^4 + a)) - 2*(2*b^3*c^2 + a*b^2*c*d - 3 *a^2*b*d^2 - (b^3*c*d - a*b^2*d^2)*x^4)*sqrt(d*x^4 + c))/(b^4*c*d^2 - a*b^ 3*d^3), 1/6*(3*sqrt(-b^2*c + a*b*d)*a^2*d^2*arctan(sqrt(d*x^4 + c)*sqrt(-b ^2*c + a*b*d)/(b*d*x^4 + b*c)) - (2*b^3*c^2 + a*b^2*c*d - 3*a^2*b*d^2 - (b ^3*c*d - a*b^2*d^2)*x^4)*sqrt(d*x^4 + c))/(b^4*c*d^2 - a*b^3*d^3)]
\[ \int \frac {x^{11}}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=\int \frac {x^{11}}{\left (a + b x^{4}\right ) \sqrt {c + d x^{4}}}\, dx \] Input:
integrate(x**11/(b*x**4+a)/(d*x**4+c)**(1/2),x)
Output:
Integral(x**11/((a + b*x**4)*sqrt(c + d*x**4)), x)
Exception generated. \[ \int \frac {x^{11}}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(x^11/(b*x^4+a)/(d*x^4+c)^(1/2),x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for m ore detail
Time = 0.12 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.98 \[ \int \frac {x^{11}}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=\frac {\frac {3 \, a^{2} d^{2} \arctan \left (\frac {\sqrt {d x^{4} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{\sqrt {-b^{2} c + a b d} b^{2}} + \frac {{\left (d x^{4} + c\right )}^{\frac {3}{2}} b^{2} - 3 \, \sqrt {d x^{4} + c} b^{2} c - 3 \, \sqrt {d x^{4} + c} a b d}{b^{3}}}{6 \, d^{2}} \] Input:
integrate(x^11/(b*x^4+a)/(d*x^4+c)^(1/2),x, algorithm="giac")
Output:
1/6*(3*a^2*d^2*arctan(sqrt(d*x^4 + c)*b/sqrt(-b^2*c + a*b*d))/(sqrt(-b^2*c + a*b*d)*b^2) + ((d*x^4 + c)^(3/2)*b^2 - 3*sqrt(d*x^4 + c)*b^2*c - 3*sqrt (d*x^4 + c)*a*b*d)/b^3)/d^2
Time = 3.98 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.98 \[ \int \frac {x^{11}}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=\frac {{\left (d\,x^4+c\right )}^{3/2}}{6\,b\,d^2}-\left (\frac {c}{b\,d^2}+\frac {2\,a\,d^3-2\,b\,c\,d^2}{4\,b^2\,d^4}\right )\,\sqrt {d\,x^4+c}+\frac {a^2\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d\,x^4+c}}{\sqrt {a\,d-b\,c}}\right )}{2\,b^{5/2}\,\sqrt {a\,d-b\,c}} \] Input:
int(x^11/((a + b*x^4)*(c + d*x^4)^(1/2)),x)
Output:
(c + d*x^4)^(3/2)/(6*b*d^2) - (c/(b*d^2) + (2*a*d^3 - 2*b*c*d^2)/(4*b^2*d^ 4))*(c + d*x^4)^(1/2) + (a^2*atan((b^(1/2)*(c + d*x^4)^(1/2))/(a*d - b*c)^ (1/2)))/(2*b^(5/2)*(a*d - b*c)^(1/2))
Time = 0.25 (sec) , antiderivative size = 794, normalized size of antiderivative = 7.63 \[ \int \frac {x^{11}}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx =\text {Too large to display} \] Input:
int(x^11/(b*x^4+a)/(d*x^4+c)^(1/2),x)
Output:
(3*sqrt(b)*sqrt(c + d*x**4)*sqrt(a*d - b*c)*atan((sqrt(d)*sqrt(b)*sqrt(c + d*x**4)*x**2 + sqrt(b)*c + sqrt(b)*d*x**4)/(sqrt(c + d*x**4)*sqrt(a*d - b *c) + sqrt(d)*sqrt(a*d - b*c)*x**2))*a**2*c*d**2 + 12*sqrt(b)*sqrt(c + d*x **4)*sqrt(a*d - b*c)*atan((sqrt(d)*sqrt(b)*sqrt(c + d*x**4)*x**2 + sqrt(b) *c + sqrt(b)*d*x**4)/(sqrt(c + d*x**4)*sqrt(a*d - b*c) + sqrt(d)*sqrt(a*d - b*c)*x**2))*a**2*d**3*x**4 + 9*sqrt(d)*sqrt(b)*sqrt(a*d - b*c)*atan((sqr t(d)*sqrt(b)*sqrt(c + d*x**4)*x**2 + sqrt(b)*c + sqrt(b)*d*x**4)/(sqrt(c + d*x**4)*sqrt(a*d - b*c) + sqrt(d)*sqrt(a*d - b*c)*x**2))*a**2*c*d**2*x**2 + 12*sqrt(d)*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(d)*sqrt(b)*sqrt(c + d*x** 4)*x**2 + sqrt(b)*c + sqrt(b)*d*x**4)/(sqrt(c + d*x**4)*sqrt(a*d - b*c) + sqrt(d)*sqrt(a*d - b*c)*x**2))*a**2*d**3*x**6 - 9*sqrt(d)*sqrt(c + d*x**4) *a**2*b*c*d**2*x**2 - 12*sqrt(d)*sqrt(c + d*x**4)*a**2*b*d**3*x**6 + 3*sqr t(d)*sqrt(c + d*x**4)*a*b**2*c**2*d*x**2 + 7*sqrt(d)*sqrt(c + d*x**4)*a*b* *2*c*d**2*x**6 + 4*sqrt(d)*sqrt(c + d*x**4)*a*b**2*d**3*x**10 + 6*sqrt(d)* sqrt(c + d*x**4)*b**3*c**3*x**2 + 5*sqrt(d)*sqrt(c + d*x**4)*b**3*c**2*d*x **6 - 4*sqrt(d)*sqrt(c + d*x**4)*b**3*c*d**2*x**10 - 3*a**2*b*c**2*d**2 - 15*a**2*b*c*d**3*x**4 - 12*a**2*b*d**4*x**8 + a*b**2*c**3*d + 6*a*b**2*c** 2*d**2*x**4 + 9*a*b**2*c*d**3*x**8 + 4*a*b**2*d**4*x**12 + 2*b**3*c**4 + 9 *b**3*c**3*d*x**4 + 3*b**3*c**2*d**2*x**8 - 4*b**3*c*d**3*x**12)/(6*b**3*d **2*(sqrt(c + d*x**4)*a*c*d + 4*sqrt(c + d*x**4)*a*d**2*x**4 - sqrt(c +...