Integrand size = 24, antiderivative size = 99 \[ \int \frac {x^5}{\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx=\frac {c^2}{d^2 (b c-a d) \sqrt {c+d x^2}}+\frac {\sqrt {c+d x^2}}{b d^2}-\frac {a^2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{b^{3/2} (b c-a d)^{3/2}} \] Output:
c^2/d^2/(-a*d+b*c)/(d*x^2+c)^(1/2)+(d*x^2+c)^(1/2)/b/d^2-a^2*arctanh(b^(1/ 2)*(d*x^2+c)^(1/2)/(-a*d+b*c)^(1/2))/b^(3/2)/(-a*d+b*c)^(3/2)
Time = 0.23 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.06 \[ \int \frac {x^5}{\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx=\frac {a d \left (c+d x^2\right )-b c \left (2 c+d x^2\right )}{b d^2 (-b c+a d) \sqrt {c+d x^2}}-\frac {a^2 \arctan \left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {-b c+a d}}\right )}{b^{3/2} (-b c+a d)^{3/2}} \] Input:
Integrate[x^5/((a + b*x^2)*(c + d*x^2)^(3/2)),x]
Output:
(a*d*(c + d*x^2) - b*c*(2*c + d*x^2))/(b*d^2*(-(b*c) + a*d)*Sqrt[c + d*x^2 ]) - (a^2*ArcTan[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[-(b*c) + a*d]])/(b^(3/2)*( -(b*c) + a*d)^(3/2))
Time = 0.27 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.06, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {354, 98, 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.
| \(\displaystyle \int \frac {x^5}{\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 354 |
| \(\displaystyle \frac {1}{2} \int \frac {x^4}{\left (b x^2+a\right ) \left (d x^2+c\right )^{3/2}}dx^2\) |
| \(\Big \downarrow \) 98 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {a^2}{b (b c-a d) \left (b x^2+a\right ) \sqrt {d x^2+c}}+\frac {1}{b d \sqrt {d x^2+c}}+\frac {c^2}{d (a d-b c) \left (d x^2+c\right )^{3/2}}\right )dx^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (-\frac {2 a^2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{b^{3/2} (b c-a d)^{3/2}}+\frac {2 c^2}{d^2 \sqrt {c+d x^2} (b c-a d)}+\frac {2 \sqrt {c+d x^2}}{b d^2}\right )\) |
Input:
Int[x^5/((a + b*x^2)*(c + d*x^2)^(3/2)),x]
Output:
((2*c^2)/(d^2*(b*c - a*d)*Sqrt[c + d*x^2]) + (2*Sqrt[c + d*x^2])/(b*d^2) - (2*a^2*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]])/(b^(3/2)*(b*c - a*d)^(3/2)))/2
Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_))/((a_.) + (b_.)*(x _)), x_] :> Int[ExpandIntegrand[(e + f*x)^FractionalPart[p], (c + d*x)^n*(( e + f*x)^IntegerPart[p]/(a + b*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[n, 0] && LtQ[p, -1] && FractionQ[p]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x , x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ [(m - 1)/2]
Time = 0.75 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00
| method | result | size |
| pseudoelliptic | \(\frac {\sqrt {x^{2} d +c}}{b \,d^{2}}-\frac {c^{2}}{d^{2} \left (a d -b c \right ) \sqrt {x^{2} d +c}}-\frac {a^{2} \arctan \left (\frac {\sqrt {x^{2} d +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )}{\left (a d -b c \right ) b \sqrt {\left (a d -b c \right ) b}}\) | \(99\) |
| risch | \(\frac {\sqrt {x^{2} d +c}}{b \,d^{2}}-\frac {d \,a^{2} \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{2 b \left (b \sqrt {-c d}+d \sqrt {-a b}\right ) \left (d \sqrt {-a b}-b \sqrt {-c d}\right ) \sqrt {-\frac {a d -b c}{b}}}-\frac {d \,a^{2} \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{2 b \left (b \sqrt {-c d}+d \sqrt {-a b}\right ) \left (d \sqrt {-a b}-b \sqrt {-c d}\right ) \sqrt {-\frac {a d -b c}{b}}}+\frac {b \,c^{2} \sqrt {\left (x -\frac {\sqrt {-c d}}{d}\right )^{2} d +2 \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}}{2 d \left (b \sqrt {-c d}+d \sqrt {-a b}\right ) \left (d \sqrt {-a b}-b \sqrt {-c d}\right ) \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}-\frac {b \,c^{2} \sqrt {\left (x +\frac {\sqrt {-c d}}{d}\right )^{2} d -2 \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}}{2 d \left (b \sqrt {-c d}+d \sqrt {-a b}\right ) \left (d \sqrt {-a b}-b \sqrt {-c d}\right ) \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}\) | \(619\) |
| default | \(\frac {\frac {x^{2}}{d \sqrt {x^{2} d +c}}+\frac {2 c}{d^{2} \sqrt {x^{2} d +c}}}{b}+\frac {a}{b^{2} d \sqrt {x^{2} d +c}}+\frac {a^{2} \left (-\frac {b}{\left (a d -b c \right ) \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}+\frac {2 d \sqrt {-a b}\, \left (2 d \left (x -\frac {\sqrt {-a b}}{b}\right )+\frac {2 d \sqrt {-a b}}{b}\right )}{\left (a d -b c \right ) \left (-\frac {4 d \left (a d -b c \right )}{b}+\frac {4 d^{2} a}{b}\right ) \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}+\frac {b \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{\left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}}\right )}{2 b^{3}}+\frac {a^{2} \left (-\frac {b}{\left (a d -b c \right ) \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}-\frac {2 d \sqrt {-a b}\, \left (2 d \left (x +\frac {\sqrt {-a b}}{b}\right )-\frac {2 d \sqrt {-a b}}{b}\right )}{\left (a d -b c \right ) \left (-\frac {4 d \left (a d -b c \right )}{b}+\frac {4 d^{2} a}{b}\right ) \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}+\frac {b \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{\left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}}\right )}{2 b^{3}}\) | \(786\) |
Input:
int(x^5/(b*x^2+a)/(d*x^2+c)^(3/2),x,method=_RETURNVERBOSE)
Output:
(d*x^2+c)^(1/2)/b/d^2-1/d^2*c^2/(a*d-b*c)/(d*x^2+c)^(1/2)-1/(a*d-b*c)/b*a^ 2/((a*d-b*c)*b)^(1/2)*arctan((d*x^2+c)^(1/2)*b/((a*d-b*c)*b)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 251 vs. \(2 (85) = 170\).
Time = 0.11 (sec) , antiderivative size = 542, normalized size of antiderivative = 5.47 \[ \int \frac {x^5}{\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx=\left [-\frac {{\left (a^{2} d^{3} x^{2} + a^{2} c d^{2}\right )} \sqrt {b^{2} c - a b d} \log \left (\frac {b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \, {\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2} + 4 \, {\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {b^{2} c - a b d} \sqrt {d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - 4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d + a^{2} b c d^{2} + {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{4 \, {\left (b^{4} c^{3} d^{2} - 2 \, a b^{3} c^{2} d^{3} + a^{2} b^{2} c d^{4} + {\left (b^{4} c^{2} d^{3} - 2 \, a b^{3} c d^{4} + a^{2} b^{2} d^{5}\right )} x^{2}\right )}}, -\frac {{\left (a^{2} d^{3} x^{2} + a^{2} c d^{2}\right )} \sqrt {-b^{2} c + a b d} \arctan \left (-\frac {{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {-b^{2} c + a b d} \sqrt {d x^{2} + c}}{2 \, {\left (b^{2} c^{2} - a b c d + {\left (b^{2} c d - a b d^{2}\right )} x^{2}\right )}}\right ) - 2 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d + a^{2} b c d^{2} + {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{2 \, {\left (b^{4} c^{3} d^{2} - 2 \, a b^{3} c^{2} d^{3} + a^{2} b^{2} c d^{4} + {\left (b^{4} c^{2} d^{3} - 2 \, a b^{3} c d^{4} + a^{2} b^{2} d^{5}\right )} x^{2}\right )}}\right ] \] Input:
integrate(x^5/(b*x^2+a)/(d*x^2+c)^(3/2),x, algorithm="fricas")
Output:
[-1/4*((a^2*d^3*x^2 + a^2*c*d^2)*sqrt(b^2*c - a*b*d)*log((b^2*d^2*x^4 + 8* b^2*c^2 - 8*a*b*c*d + a^2*d^2 + 2*(4*b^2*c*d - 3*a*b*d^2)*x^2 + 4*(b*d*x^2 + 2*b*c - a*d)*sqrt(b^2*c - a*b*d)*sqrt(d*x^2 + c))/(b^2*x^4 + 2*a*b*x^2 + a^2)) - 4*(2*b^3*c^3 - 3*a*b^2*c^2*d + a^2*b*c*d^2 + (b^3*c^2*d - 2*a*b^ 2*c*d^2 + a^2*b*d^3)*x^2)*sqrt(d*x^2 + c))/(b^4*c^3*d^2 - 2*a*b^3*c^2*d^3 + a^2*b^2*c*d^4 + (b^4*c^2*d^3 - 2*a*b^3*c*d^4 + a^2*b^2*d^5)*x^2), -1/2*( (a^2*d^3*x^2 + a^2*c*d^2)*sqrt(-b^2*c + a*b*d)*arctan(-1/2*(b*d*x^2 + 2*b* c - a*d)*sqrt(-b^2*c + a*b*d)*sqrt(d*x^2 + c)/(b^2*c^2 - a*b*c*d + (b^2*c* d - a*b*d^2)*x^2)) - 2*(2*b^3*c^3 - 3*a*b^2*c^2*d + a^2*b*c*d^2 + (b^3*c^2 *d - 2*a*b^2*c*d^2 + a^2*b*d^3)*x^2)*sqrt(d*x^2 + c))/(b^4*c^3*d^2 - 2*a*b ^3*c^2*d^3 + a^2*b^2*c*d^4 + (b^4*c^2*d^3 - 2*a*b^3*c*d^4 + a^2*b^2*d^5)*x ^2)]
\[ \int \frac {x^5}{\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx=\int \frac {x^{5}}{\left (a + b x^{2}\right ) \left (c + d x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(x**5/(b*x**2+a)/(d*x**2+c)**(3/2),x)
Output:
Integral(x**5/((a + b*x**2)*(c + d*x**2)**(3/2)), x)
Exception generated. \[ \int \frac {x^5}{\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(x^5/(b*x^2+a)/(d*x^2+c)^(3/2),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 = 99, normalized size of antiderivative = 1.00 \[ \int \frac {x^5}{\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx=\frac {\frac {a^{2} d^{2} \arctan \left (\frac {\sqrt {d x^{2} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{{\left (b^{2} c - a b d\right )} \sqrt {-b^{2} c + a b d}} + \frac {c^{2}}{\sqrt {d x^{2} + c} {\left (b c - a d\right )}} + \frac {\sqrt {d x^{2} + c}}{b}}{d^{2}} \] Input:
integrate(x^5/(b*x^2+a)/(d*x^2+c)^(3/2),x, algorithm="giac")
Output:
(a^2*d^2*arctan(sqrt(d*x^2 + c)*b/sqrt(-b^2*c + a*b*d))/((b^2*c - a*b*d)*s qrt(-b^2*c + a*b*d)) + c^2/(sqrt(d*x^2 + c)*(b*c - a*d)) + sqrt(d*x^2 + c) /b)/d^2
Time = 1.30 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.97 \[ \int \frac {x^5}{\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx=\frac {\sqrt {d\,x^2+c}}{b\,d^2}+\frac {a^2\,\mathrm {atan}\left (\frac {\sqrt {d\,x^2+c}\,\left (b^2\,c-a\,b\,d\right )}{\sqrt {b}\,{\left (a\,d-b\,c\right )}^{3/2}}\right )}{b^{3/2}\,{\left (a\,d-b\,c\right )}^{3/2}}-\frac {c^2}{d^2\,\sqrt {d\,x^2+c}\,\left (a\,d-b\,c\right )} \] Input:
int(x^5/((a + b*x^2)*(c + d*x^2)^(3/2)),x)
Output:
(c + d*x^2)^(1/2)/(b*d^2) + (a^2*atan(((c + d*x^2)^(1/2)*(b^2*c - a*b*d))/ (b^(1/2)*(a*d - b*c)^(3/2))))/(b^(3/2)*(a*d - b*c)^(3/2)) - c^2/(d^2*(c + d*x^2)^(1/2)*(a*d - b*c))
Time = 0.22 (sec) , antiderivative size = 344, normalized size of antiderivative = 3.47 \[ \int \frac {x^5}{\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx=\frac {-\sqrt {b}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d}\, \sqrt {d \,x^{2}+c}\, b x +b c +b d \,x^{2}}{\sqrt {b}\, \sqrt {d \,x^{2}+c}\, \sqrt {a d -b c}+\sqrt {d}\, \sqrt {b}\, \sqrt {a d -b c}\, x}\right ) a^{2} c \,d^{2}-\sqrt {b}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d}\, \sqrt {d \,x^{2}+c}\, b x +b c +b d \,x^{2}}{\sqrt {b}\, \sqrt {d \,x^{2}+c}\, \sqrt {a d -b c}+\sqrt {d}\, \sqrt {b}\, \sqrt {a d -b c}\, x}\right ) a^{2} d^{3} x^{2}+\sqrt {d \,x^{2}+c}\, a^{2} b c \,d^{2}+\sqrt {d \,x^{2}+c}\, a^{2} b \,d^{3} x^{2}-3 \sqrt {d \,x^{2}+c}\, a \,b^{2} c^{2} d -2 \sqrt {d \,x^{2}+c}\, a \,b^{2} c \,d^{2} x^{2}+2 \sqrt {d \,x^{2}+c}\, b^{3} c^{3}+\sqrt {d \,x^{2}+c}\, b^{3} c^{2} d \,x^{2}}{b^{2} d^{2} \left (a^{2} d^{3} x^{2}-2 a b c \,d^{2} x^{2}+b^{2} c^{2} d \,x^{2}+a^{2} c \,d^{2}-2 a b \,c^{2} d +b^{2} c^{3}\right )} \] Input:
int(x^5/(b*x^2+a)/(d*x^2+c)^(3/2),x)
Output:
( - sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(d)*sqrt(c + d*x**2)*b*x + b*c + b*d *x**2)/(sqrt(b)*sqrt(c + d*x**2)*sqrt(a*d - b*c) + sqrt(d)*sqrt(b)*sqrt(a* d - b*c)*x))*a**2*c*d**2 - sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(d)*sqrt(c + d*x**2)*b*x + b*c + b*d*x**2)/(sqrt(b)*sqrt(c + d*x**2)*sqrt(a*d - b*c) + sqrt(d)*sqrt(b)*sqrt(a*d - b*c)*x))*a**2*d**3*x**2 + sqrt(c + d*x**2)*a**2 *b*c*d**2 + sqrt(c + d*x**2)*a**2*b*d**3*x**2 - 3*sqrt(c + d*x**2)*a*b**2* c**2*d - 2*sqrt(c + d*x**2)*a*b**2*c*d**2*x**2 + 2*sqrt(c + d*x**2)*b**3*c **3 + sqrt(c + d*x**2)*b**3*c**2*d*x**2)/(b**2*d**2*(a**2*c*d**2 + a**2*d* *3*x**2 - 2*a*b*c**2*d - 2*a*b*c*d**2*x**2 + b**2*c**3 + b**2*c**2*d*x**2) )