Integrand size = 24, antiderivative size = 87 \[ \int \frac {x^3}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=-\frac {\sqrt {c+d x^4}}{4 (b c-a d) \left (a+b x^4\right )}+\frac {d \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^4}}{\sqrt {b c-a d}}\right )}{4 \sqrt {b} (b c-a d)^{3/2}} \] Output:
-1/4*(d*x^4+c)^(1/2)/(-a*d+b*c)/(b*x^4+a)+1/4*d*arctanh(b^(1/2)*(d*x^4+c)^ (1/2)/(-a*d+b*c)^(1/2))/b^(1/2)/(-a*d+b*c)^(3/2)
Time = 0.21 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.99 \[ \int \frac {x^3}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=\frac {1}{4} \left (-\frac {\sqrt {c+d x^4}}{(b c-a d) \left (a+b x^4\right )}+\frac {d \arctan \left (\frac {\sqrt {b} \sqrt {c+d x^4}}{\sqrt {-b c+a d}}\right )}{\sqrt {b} (-b c+a d)^{3/2}}\right ) \] Input:
Integrate[x^3/((a + b*x^4)^2*Sqrt[c + d*x^4]),x]
Output:
(-(Sqrt[c + d*x^4]/((b*c - a*d)*(a + b*x^4))) + (d*ArcTan[(Sqrt[b]*Sqrt[c + d*x^4])/Sqrt[-(b*c) + a*d]])/(Sqrt[b]*(-(b*c) + a*d)^(3/2)))/4
Time = 0.36 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {946, 52, 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.
\(\displaystyle \int \frac {x^3}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx\) |
\(\Big \downarrow \) 946 |
\(\displaystyle \frac {1}{4} \int \frac {1}{\left (b x^4+a\right )^2 \sqrt {d x^4+c}}dx^4\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {1}{4} \left (-\frac {d \int \frac {1}{\left (b x^4+a\right ) \sqrt {d x^4+c}}dx^4}{2 (b c-a d)}-\frac {\sqrt {c+d x^4}}{\left (a+b x^4\right ) (b c-a d)}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{4} \left (-\frac {\int \frac {1}{\frac {b x^8}{d}+a-\frac {b c}{d}}d\sqrt {d x^4+c}}{b c-a d}-\frac {\sqrt {c+d x^4}}{\left (a+b x^4\right ) (b c-a d)}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{4} \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^4}}{\sqrt {b c-a d}}\right )}{\sqrt {b} (b c-a d)^{3/2}}-\frac {\sqrt {c+d x^4}}{\left (a+b x^4\right ) (b c-a d)}\right )\) |
Input:
Int[x^3/((a + b*x^4)^2*Sqrt[c + d*x^4]),x]
Output:
(-(Sqrt[c + d*x^4]/((b*c - a*d)*(a + b*x^4))) + (d*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^4])/Sqrt[b*c - a*d]])/(Sqrt[b]*(b*c - a*d)^(3/2)))/4
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
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_)^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[(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] && EqQ[m - n + 1, 0]
Time = 0.15 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.82
method | result | size |
pseudoelliptic | \(\frac {\frac {\sqrt {d \,x^{4}+c}}{b \,x^{4}+a}+\frac {d \arctan \left (\frac {\sqrt {d \,x^{4}+c}\, b}{\sqrt {\left (a d -c b \right ) b}}\right )}{\sqrt {\left (a d -c b \right ) b}}}{4 a d -4 c b}\) | \(71\) |
default | \(-\frac {\sqrt {-a b}\, \sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{8 a b \left (a d -c b \right ) \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}-\frac {d \ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}-\frac {\sqrt {-a b}}{b}}\right )}{8 b \left (a d -c b \right ) \sqrt {-\frac {a d -c b}{b}}}+\frac {\sqrt {-a b}\, \sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{8 a b \left (a d -c b \right ) \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}-\frac {d \ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}+\frac {\sqrt {-a b}}{b}}\right )}{8 b \left (a d -c b \right ) \sqrt {-\frac {a d -c b}{b}}}\) | \(541\) |
elliptic | \(-\frac {\sqrt {-a b}\, \sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{8 a b \left (a d -c b \right ) \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}-\frac {d \ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}-\frac {\sqrt {-a b}}{b}}\right )}{8 b \left (a d -c b \right ) \sqrt {-\frac {a d -c b}{b}}}+\frac {\sqrt {-a b}\, \sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{8 a b \left (a d -c b \right ) \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}-\frac {d \ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}+\frac {\sqrt {-a b}}{b}}\right )}{8 b \left (a d -c b \right ) \sqrt {-\frac {a d -c b}{b}}}\) | \(541\) |
Input:
int(x^3/(b*x^4+a)^2/(d*x^4+c)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/4/(a*d-b*c)*((d*x^4+c)^(1/2)/(b*x^4+a)+d/((a*d-b*c)*b)^(1/2)*arctan((d*x ^4+c)^(1/2)*b/((a*d-b*c)*b)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 143 vs. \(2 (71) = 142\).
Time = 0.11 (sec) , antiderivative size = 302, normalized size of antiderivative = 3.47 \[ \int \frac {x^3}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=\left [-\frac {{\left (b d x^{4} + a d\right )} \sqrt {b^{2} c - a b d} \log \left (\frac {b d x^{4} + 2 \, b c - a d - 2 \, \sqrt {d x^{4} + c} \sqrt {b^{2} c - a b d}}{b x^{4} + a}\right ) + 2 \, \sqrt {d x^{4} + c} {\left (b^{2} c - a b d\right )}}{8 \, {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2} + {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{4}\right )}}, -\frac {{\left (b d x^{4} + a d\right )} \sqrt {-b^{2} c + a b d} \arctan \left (\frac {\sqrt {d x^{4} + c} \sqrt {-b^{2} c + a b d}}{b d x^{4} + b c}\right ) + \sqrt {d x^{4} + c} {\left (b^{2} c - a b d\right )}}{4 \, {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2} + {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{4}\right )}}\right ] \] Input:
integrate(x^3/(b*x^4+a)^2/(d*x^4+c)^(1/2),x, algorithm="fricas")
Output:
[-1/8*((b*d*x^4 + a*d)*sqrt(b^2*c - a*b*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))/(a*b^3*c^2 - 2*a^2*b^2*c*d + a^3*b*d^2 + (b^4*c^2 - 2*a*b^3*c *d + a^2*b^2*d^2)*x^4), -1/4*((b*d*x^4 + a*d)*sqrt(-b^2*c + a*b*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))/(a*b^3*c^2 - 2*a^2*b^2*c*d + a^3*b*d^2 + (b^4*c^2 - 2*a*b^3 *c*d + a^2*b^2*d^2)*x^4)]
\[ \int \frac {x^3}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=\int \frac {x^{3}}{\left (a + b x^{4}\right )^{2} \sqrt {c + d x^{4}}}\, dx \] Input:
integrate(x**3/(b*x**4+a)**2/(d*x**4+c)**(1/2),x)
Output:
Integral(x**3/((a + b*x**4)**2*sqrt(c + d*x**4)), x)
Exception generated. \[ \int \frac {x^3}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(x^3/(b*x^4+a)^2/(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 = 93, normalized size of antiderivative = 1.07 \[ \int \frac {x^3}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=-\frac {d \arctan \left (\frac {\sqrt {d x^{4} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{4 \, \sqrt {-b^{2} c + a b d} {\left (b c - a d\right )}} - \frac {\sqrt {d x^{4} + c} d}{4 \, {\left ({\left (d x^{4} + c\right )} b - b c + a d\right )} {\left (b c - a d\right )}} \] Input:
integrate(x^3/(b*x^4+a)^2/(d*x^4+c)^(1/2),x, algorithm="giac")
Output:
-1/4*d*arctan(sqrt(d*x^4 + c)*b/sqrt(-b^2*c + a*b*d))/(sqrt(-b^2*c + a*b*d )*(b*c - a*d)) - 1/4*sqrt(d*x^4 + c)*d/(((d*x^4 + c)*b - b*c + a*d)*(b*c - a*d))
Time = 4.17 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.97 \[ \int \frac {x^3}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=\frac {d\,\sqrt {d\,x^4+c}}{2\,\left (a\,d-b\,c\right )\,\left (2\,b\,\left (d\,x^4+c\right )+2\,a\,d-2\,b\,c\right )}+\frac {d\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d\,x^4+c}}{\sqrt {a\,d-b\,c}}\right )}{4\,\sqrt {b}\,{\left (a\,d-b\,c\right )}^{3/2}} \] Input:
int(x^3/((a + b*x^4)^2*(c + d*x^4)^(1/2)),x)
Output:
(d*(c + d*x^4)^(1/2))/(2*(a*d - b*c)*(2*b*(c + d*x^4) + 2*a*d - 2*b*c)) + (d*atan((b^(1/2)*(c + d*x^4)^(1/2))/(a*d - b*c)^(1/2)))/(4*b^(1/2)*(a*d - b*c)^(3/2))
Time = 0.23 (sec) , antiderivative size = 872, normalized size of antiderivative = 10.02 \[ \int \frac {x^3}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx =\text {Too large to display} \] Input:
int(x^3/(b*x^4+a)^2/(d*x^4+c)^(1/2),x)
Output:
(2*sqrt(d)*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*x**2 + 2*sqrt(d)*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))*b*d*x**6 + 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*c*d + 2*sqrt(b)*sq rt(a*d - b*c)*atan((sqrt(d)*sqrt(b)*sqrt(c + d*x**4)*x**2 + sqrt(b)*c + sq rt(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**2*x**4 + 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))*b*c*d*x**4 + 2*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))*b*d* *2*x**8 + sqrt(c + d*x**4)*a*b*c*d + 2*sqrt(c + d*x**4)*a*b*d**2*x**4 - sq rt(c + d*x**4)*b**2*c**2 - 2*sqrt(c + d*x**4)*b**2*c*d*x**4 + 2*sqrt(d)*a* b*c*d*x**2 + 2*sqrt(d)*a*b*d**2*x**6 - 2*sqrt(d)*b**2*c**2*x**2 - 2*sqrt(d )*b**2*c*d*x**6)/(4*b*(2*sqrt(d)*sqrt(c + d*x**4)*a**3*d**2*x**2 - 4*sqrt( d)*sqrt(c + d*x**4)*a**2*b*c*d*x**2 + 2*sqrt(d)*sqrt(c + d*x**4)*a**2*b...