Integrand size = 24, antiderivative size = 74 \[ \int \frac {x^7}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=\frac {\sqrt {c+d x^4}}{2 b d}+\frac {a \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^4}}{\sqrt {b c-a d}}\right )}{2 b^{3/2} \sqrt {b c-a d}} \] Output:
1/2*(d*x^4+c)^(1/2)/b/d+1/2*a*arctanh(b^(1/2)*(d*x^4+c)^(1/2)/(-a*d+b*c)^( 1/2))/b^(3/2)/(-a*d+b*c)^(1/2)
Time = 0.11 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.99 \[ \int \frac {x^7}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=\frac {1}{2} \left (\frac {\sqrt {c+d x^4}}{b d}-\frac {a \arctan \left (\frac {\sqrt {b} \sqrt {c+d x^4}}{\sqrt {-b c+a d}}\right )}{b^{3/2} \sqrt {-b c+a d}}\right ) \] Input:
Integrate[x^7/((a + b*x^4)*Sqrt[c + d*x^4]),x]
Output:
(Sqrt[c + d*x^4]/(b*d) - (a*ArcTan[(Sqrt[b]*Sqrt[c + d*x^4])/Sqrt[-(b*c) + a*d]])/(b^(3/2)*Sqrt[-(b*c) + a*d]))/2
Time = 0.34 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {948, 90, 73, 221}
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.
\(\displaystyle \int \frac {x^7}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx\) |
\(\Big \downarrow \) 948 |
\(\displaystyle \frac {1}{4} \int \frac {x^4}{\left (b x^4+a\right ) \sqrt {d x^4+c}}dx^4\) |
\(\Big \downarrow \) 90 |
\(\displaystyle \frac {1}{4} \left (\frac {2 \sqrt {c+d x^4}}{b d}-\frac {a \int \frac {1}{\left (b x^4+a\right ) \sqrt {d x^4+c}}dx^4}{b}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{4} \left (\frac {2 \sqrt {c+d x^4}}{b d}-\frac {2 a \int \frac {1}{\frac {b x^8}{d}+a-\frac {b c}{d}}d\sqrt {d x^4+c}}{b d}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{4} \left (\frac {2 a \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^4}}{\sqrt {b c-a d}}\right )}{b^{3/2} \sqrt {b c-a d}}+\frac {2 \sqrt {c+d x^4}}{b d}\right )\) |
Input:
Int[x^7/((a + b*x^4)*Sqrt[c + d*x^4]),x]
Output:
((2*Sqrt[c + d*x^4])/(b*d) + (2*a*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^4])/Sqrt[b *c - a*d]])/(b^(3/2)*Sqrt[b*c - a*d]))/4
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
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 = 0.15 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.80
method | result | size |
pseudoelliptic | \(\frac {\frac {\sqrt {d \,x^{4}+c}}{d}-\frac {a \arctan \left (\frac {\sqrt {d \,x^{4}+c}\, b}{\sqrt {\left (a d -c b \right ) b}}\right )}{\sqrt {\left (a d -c b \right ) b}}}{2 b}\) | \(59\) |
risch | \(\frac {\sqrt {d \,x^{4}+c}}{2 b d}+\frac {a \ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}-\frac {\sqrt {-a b}}{b}}\right )}{4 b^{2} \sqrt {-\frac {a d -c b}{b}}}+\frac {a \ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}+\frac {\sqrt {-a b}}{b}}\right )}{4 b^{2} \sqrt {-\frac {a d -c b}{b}}}\) | \(335\) |
elliptic | \(\frac {\sqrt {d \,x^{4}+c}}{2 b d}+\frac {a \ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}-\frac {\sqrt {-a b}}{b}}\right )}{4 b^{2} \sqrt {-\frac {a d -c b}{b}}}+\frac {a \ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}+\frac {\sqrt {-a b}}{b}}\right )}{4 b^{2} \sqrt {-\frac {a d -c b}{b}}}\) | \(335\) |
default | \(\frac {\sqrt {d \,x^{4}+c}}{2 b d}-\frac {a \left (-\frac {\ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}-\frac {\sqrt {-a b}}{b}}\right )}{4 b \sqrt {-\frac {a d -c b}{b}}}-\frac {\ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}+\frac {\sqrt {-a b}}{b}}\right )}{4 b \sqrt {-\frac {a d -c b}{b}}}\right )}{b}\) | \(340\) |
Input:
int(x^7/(b*x^4+a)/(d*x^4+c)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/2/b*(1/d*(d*x^4+c)^(1/2)-a/((a*d-b*c)*b)^(1/2)*arctan((d*x^4+c)^(1/2)*b/ ((a*d-b*c)*b)^(1/2)))
Time = 0.10 (sec) , antiderivative size = 205, normalized size of antiderivative = 2.77 \[ \int \frac {x^7}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=\left [\frac {\sqrt {b^{2} c - a b d} a d \log \left (\frac {b d x^{4} + 2 \, b c - a d + 2 \, \sqrt {d x^{4} + c} \sqrt {b^{2} c - a b d}}{b x^{4} + a}\right ) + 2 \, \sqrt {d x^{4} + c} {\left (b^{2} c - a b d\right )}}{4 \, {\left (b^{3} c d - a b^{2} d^{2}\right )}}, -\frac {\sqrt {-b^{2} c + a b d} a d \arctan \left (\frac {\sqrt {d x^{4} + c} \sqrt {-b^{2} c + a b d}}{b d x^{4} + b c}\right ) - \sqrt {d x^{4} + c} {\left (b^{2} c - a b d\right )}}{2 \, {\left (b^{3} c d - a b^{2} d^{2}\right )}}\right ] \] Input:
integrate(x^7/(b*x^4+a)/(d*x^4+c)^(1/2),x, algorithm="fricas")
Output:
[1/4*(sqrt(b^2*c - a*b*d)*a*d*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*sqrt(d*x^4 + c)*(b^2*c - a*b*d))/ (b^3*c*d - a*b^2*d^2), -1/2*(sqrt(-b^2*c + a*b*d)*a*d*arctan(sqrt(d*x^4 + c)*sqrt(-b^2*c + a*b*d)/(b*d*x^4 + b*c)) - sqrt(d*x^4 + c)*(b^2*c - a*b*d) )/(b^3*c*d - a*b^2*d^2)]
\[ \int \frac {x^7}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=\int \frac {x^{7}}{\left (a + b x^{4}\right ) \sqrt {c + d x^{4}}}\, dx \] Input:
integrate(x**7/(b*x**4+a)/(d*x**4+c)**(1/2),x)
Output:
Integral(x**7/((a + b*x**4)*sqrt(c + d*x**4)), x)
Exception generated. \[ \int \frac {x^7}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(x^7/(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 = 64, normalized size of antiderivative = 0.86 \[ \int \frac {x^7}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=-\frac {\frac {a d \arctan \left (\frac {\sqrt {d x^{4} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{\sqrt {-b^{2} c + a b d} b} - \frac {\sqrt {d x^{4} + c}}{b}}{2 \, d} \] Input:
integrate(x^7/(b*x^4+a)/(d*x^4+c)^(1/2),x, algorithm="giac")
Output:
-1/2*(a*d*arctan(sqrt(d*x^4 + c)*b/sqrt(-b^2*c + a*b*d))/(sqrt(-b^2*c + a* b*d)*b) - sqrt(d*x^4 + c)/b)/d
Time = 3.92 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.78 \[ \int \frac {x^7}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=\frac {\sqrt {d\,x^4+c}}{2\,b\,d}-\frac {a\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d\,x^4+c}}{\sqrt {a\,d-b\,c}}\right )}{2\,b^{3/2}\,\sqrt {a\,d-b\,c}} \] Input:
int(x^7/((a + b*x^4)*(c + d*x^4)^(1/2)),x)
Output:
(c + d*x^4)^(1/2)/(2*b*d) - (a*atan((b^(1/2)*(c + d*x^4)^(1/2))/(a*d - b*c )^(1/2)))/(2*b^(3/2)*(a*d - b*c)^(1/2))
Time = 0.21 (sec) , antiderivative size = 295, normalized size of antiderivative = 3.99 \[ \int \frac {x^7}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=\frac {-\sqrt {b}\, \sqrt {d \,x^{4}+c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d}\, \sqrt {b}\, \sqrt {d \,x^{4}+c}\, x^{2}+\sqrt {b}\, c +\sqrt {b}\, d \,x^{4}}{\sqrt {d \,x^{4}+c}\, \sqrt {a d -b c}+\sqrt {d}\, \sqrt {a d -b c}\, x^{2}}\right ) a d -\sqrt {d}\, \sqrt {b}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d}\, \sqrt {b}\, \sqrt {d \,x^{4}+c}\, x^{2}+\sqrt {b}\, c +\sqrt {b}\, d \,x^{4}}{\sqrt {d \,x^{4}+c}\, \sqrt {a d -b c}+\sqrt {d}\, \sqrt {a d -b c}\, x^{2}}\right ) a d \,x^{2}+\sqrt {d}\, \sqrt {d \,x^{4}+c}\, a b d \,x^{2}-\sqrt {d}\, \sqrt {d \,x^{4}+c}\, b^{2} c \,x^{2}+a b c d +a b \,d^{2} x^{4}-b^{2} c^{2}-b^{2} c d \,x^{4}}{2 b^{2} d \left (\sqrt {d \,x^{4}+c}\, a d -\sqrt {d \,x^{4}+c}\, b c +\sqrt {d}\, a d \,x^{2}-\sqrt {d}\, b c \,x^{2}\right )} \] Input:
int(x^7/(b*x^4+a)/(d*x^4+c)^(1/2),x)
Output:
( - 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*d - 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*d*x **2 + sqrt(d)*sqrt(c + d*x**4)*a*b*d*x**2 - sqrt(d)*sqrt(c + d*x**4)*b**2* c*x**2 + a*b*c*d + a*b*d**2*x**4 - b**2*c**2 - b**2*c*d*x**4)/(2*b**2*d*(s qrt(c + d*x**4)*a*d - sqrt(c + d*x**4)*b*c + sqrt(d)*a*d*x**2 - sqrt(d)*b* c*x**2))