Integrand size = 43, antiderivative size = 133 \[ \int \frac {\left (-q+p x^2\right ) \sqrt {q^2+p^2 x^4}}{x^2 \left (a q+b x+a p x^2\right )} \, dx=\frac {\sqrt {q^2+p^2 x^4}}{a x}+\frac {2 \sqrt {-b^2+2 a^2 p q} \arctan \left (\frac {\sqrt {-b^2+2 a^2 p q} x}{a q+b x+a p x^2+a \sqrt {q^2+p^2 x^4}}\right )}{a^2}+\frac {b \log (x)}{a^2}-\frac {b \log \left (q+p x^2+\sqrt {q^2+p^2 x^4}\right )}{a^2} \]
(p^2*x^4+q^2)^(1/2)/a/x+2*(2*a^2*p*q-b^2)^(1/2)*arctan((2*a^2*p*q-b^2)^(1/ 2)*x/(a*q+b*x+a*p*x^2+a*(p^2*x^4+q^2)^(1/2)))/a^2+b*ln(x)/a^2-b*ln(q+p*x^2 +(p^2*x^4+q^2)^(1/2))/a^2
Time = 1.49 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.95 \[ \int \frac {\left (-q+p x^2\right ) \sqrt {q^2+p^2 x^4}}{x^2 \left (a q+b x+a p x^2\right )} \, dx=\frac {a \sqrt {q^2+p^2 x^4}+2 \sqrt {-b^2+2 a^2 p q} x \arctan \left (\frac {\sqrt {-b^2+2 a^2 p q} x}{b x+a \left (q+p x^2+\sqrt {q^2+p^2 x^4}\right )}\right )+b x \log (x)-b x \log \left (q+p x^2+\sqrt {q^2+p^2 x^4}\right )}{a^2 x} \]
(a*Sqrt[q^2 + p^2*x^4] + 2*Sqrt[-b^2 + 2*a^2*p*q]*x*ArcTan[(Sqrt[-b^2 + 2* a^2*p*q]*x)/(b*x + a*(q + p*x^2 + Sqrt[q^2 + p^2*x^4]))] + b*x*Log[x] - b* x*Log[q + p*x^2 + Sqrt[q^2 + p^2*x^4]])/(a^2*x)
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 4.67 (sec) , antiderivative size = 980, normalized size of antiderivative = 7.37, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.047, Rules used = {7279, 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.
\(\displaystyle \int \frac {\left (p x^2-q\right ) \sqrt {p^2 x^4+q^2}}{x^2 \left (a p x^2+a q+b x\right )} \, dx\) |
\(\Big \downarrow \) 7279 |
\(\displaystyle \int \left (\frac {\sqrt {p^2 x^4+q^2} \left (2 a^2 p q-a b p x-b^2\right )}{a^2 q \left (a p x^2+a q+b x\right )}+\frac {b \sqrt {p^2 x^4+q^2}}{a^2 q x}-\frac {\sqrt {p^2 x^4+q^2}}{a x^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\text {arctanh}\left (\frac {\sqrt {p^2 x^4+q^2}}{q}\right ) b}{2 a^2}+\frac {\sqrt {p^2 x^4+q^2} b}{2 a^2 q}-\frac {\sqrt {b^2-2 a^2 p q} \text {arctanh}\left (\frac {\sqrt {b^2-2 a^2 p q} x}{a \sqrt {p^2 x^4+q^2}}\right )}{a^2}-\frac {\left (b-\sqrt {b^2-4 a^2 p q}\right ) \text {arctanh}\left (\frac {p x^2}{\sqrt {p^2 x^4+q^2}}\right )}{4 a^2}-\frac {\left (b+\sqrt {b^2-4 a^2 p q}\right ) \text {arctanh}\left (\frac {p x^2}{\sqrt {p^2 x^4+q^2}}\right )}{4 a^2}+\frac {\sqrt {b^2-2 a^2 p q} \left (b+\sqrt {b^2-4 a^2 p q}\right ) \sqrt {-2 p q a^2+b^2-b \sqrt {b^2-4 a^2 p q}} \text {arctanh}\left (\frac {p \left (4 a^2 q^2+\left (b-\sqrt {b^2-4 a^2 p q}\right )^2 x^2\right )}{2 \sqrt {2} \sqrt {b^2-2 a^2 p q} \sqrt {-2 p q a^2+b^2-b \sqrt {b^2-4 a^2 p q}} \sqrt {p^2 x^4+q^2}}\right )}{4 \sqrt {2} a^4 p q}+\frac {\sqrt {b^2-2 a^2 p q} \left (b-\sqrt {b^2-4 a^2 p q}\right ) \sqrt {-2 p q a^2+b^2+b \sqrt {b^2-4 a^2 p q}} \text {arctanh}\left (\frac {p \left (4 a^2 q^2+\left (b+\sqrt {b^2-4 a^2 p q}\right )^2 x^2\right )}{2 \sqrt {2} \sqrt {b^2-2 a^2 p q} \sqrt {-2 p q a^2+b^2+b \sqrt {b^2-4 a^2 p q}} \sqrt {p^2 x^4+q^2}}\right )}{4 \sqrt {2} a^4 p q}-\frac {\sqrt {p} \sqrt {q} \left (p x^2+q\right ) \sqrt {\frac {p^2 x^4+q^2}{\left (p x^2+q\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {p} x}{\sqrt {q}}\right ),\frac {1}{2}\right )}{a \sqrt {p^2 x^4+q^2}}-\frac {\left (b-\sqrt {b^2-4 a^2 p q}\right ) \sqrt {p^2 x^4+q^2}}{4 a^2 q}-\frac {\left (b+\sqrt {b^2-4 a^2 p q}\right ) \sqrt {p^2 x^4+q^2}}{4 a^2 q}+\frac {\sqrt {p^2 x^4+q^2}}{a x}+\frac {\sqrt {p} \sqrt {q} \left (b-\sqrt {b^2-4 a^2 p q}\right ) \left (p x^2+q\right ) \sqrt {\frac {p^2 x^4+q^2}{\left (p x^2+q\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {p} x}{\sqrt {q}}\right ),\frac {1}{2}\right )}{2 a \sqrt {p^2 x^4+q^2} b}+\frac {\sqrt {p} \sqrt {q} \left (b+\sqrt {b^2-4 a^2 p q}\right ) \left (p x^2+q\right ) \sqrt {\frac {p^2 x^4+q^2}{\left (p x^2+q\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {p} x}{\sqrt {q}}\right ),\frac {1}{2}\right )}{2 a \sqrt {p^2 x^4+q^2} b}\) |
(b*Sqrt[q^2 + p^2*x^4])/(2*a^2*q) - ((b - Sqrt[b^2 - 4*a^2*p*q])*Sqrt[q^2 + p^2*x^4])/(4*a^2*q) - ((b + Sqrt[b^2 - 4*a^2*p*q])*Sqrt[q^2 + p^2*x^4])/ (4*a^2*q) + Sqrt[q^2 + p^2*x^4]/(a*x) - (Sqrt[b^2 - 2*a^2*p*q]*ArcTanh[(Sq rt[b^2 - 2*a^2*p*q]*x)/(a*Sqrt[q^2 + p^2*x^4])])/a^2 - ((b - Sqrt[b^2 - 4* a^2*p*q])*ArcTanh[(p*x^2)/Sqrt[q^2 + p^2*x^4]])/(4*a^2) - ((b + Sqrt[b^2 - 4*a^2*p*q])*ArcTanh[(p*x^2)/Sqrt[q^2 + p^2*x^4]])/(4*a^2) + (Sqrt[b^2 - 2 *a^2*p*q]*(b + Sqrt[b^2 - 4*a^2*p*q])*Sqrt[b^2 - 2*a^2*p*q - b*Sqrt[b^2 - 4*a^2*p*q]]*ArcTanh[(p*(4*a^2*q^2 + (b - Sqrt[b^2 - 4*a^2*p*q])^2*x^2))/(2 *Sqrt[2]*Sqrt[b^2 - 2*a^2*p*q]*Sqrt[b^2 - 2*a^2*p*q - b*Sqrt[b^2 - 4*a^2*p *q]]*Sqrt[q^2 + p^2*x^4])])/(4*Sqrt[2]*a^4*p*q) + (Sqrt[b^2 - 2*a^2*p*q]*( b - Sqrt[b^2 - 4*a^2*p*q])*Sqrt[b^2 - 2*a^2*p*q + b*Sqrt[b^2 - 4*a^2*p*q]] *ArcTanh[(p*(4*a^2*q^2 + (b + Sqrt[b^2 - 4*a^2*p*q])^2*x^2))/(2*Sqrt[2]*Sq rt[b^2 - 2*a^2*p*q]*Sqrt[b^2 - 2*a^2*p*q + b*Sqrt[b^2 - 4*a^2*p*q]]*Sqrt[q ^2 + p^2*x^4])])/(4*Sqrt[2]*a^4*p*q) - (b*ArcTanh[Sqrt[q^2 + p^2*x^4]/q])/ (2*a^2) - (Sqrt[p]*Sqrt[q]*(q + p*x^2)*Sqrt[(q^2 + p^2*x^4)/(q + p*x^2)^2] *EllipticF[2*ArcTan[(Sqrt[p]*x)/Sqrt[q]], 1/2])/(a*Sqrt[q^2 + p^2*x^4]) + (Sqrt[p]*Sqrt[q]*(b - Sqrt[b^2 - 4*a^2*p*q])*(q + p*x^2)*Sqrt[(q^2 + p^2*x ^4)/(q + p*x^2)^2]*EllipticF[2*ArcTan[(Sqrt[p]*x)/Sqrt[q]], 1/2])/(2*a*b*S qrt[q^2 + p^2*x^4]) + (Sqrt[p]*Sqrt[q]*(b + Sqrt[b^2 - 4*a^2*p*q])*(q + p* x^2)*Sqrt[(q^2 + p^2*x^4)/(q + p*x^2)^2]*EllipticF[2*ArcTan[(Sqrt[p]*x)...
3.20.16.3.1 Defintions of rubi rules used
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 3.10 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.53
method | result | size |
pseudoelliptic | \(\frac {-b \ln \left (\frac {\sqrt {p^{2} x^{4}+q^{2}}+\left (p \,x^{2}+q \right ) \operatorname {csgn}\left (p \right )}{x}\right ) \operatorname {csgn}\left (p \right ) x a \sqrt {\frac {-2 a^{2} p q +b^{2}}{a^{2}}}+\sqrt {\frac {-2 a^{2} p q +b^{2}}{a^{2}}}\, \sqrt {p^{2} x^{4}+q^{2}}\, a^{2}+2 \left (\ln \left (\frac {a \sqrt {p^{2} x^{4}+q^{2}}\, p \sqrt {\frac {-2 a^{2} p q +b^{2}}{a^{2}}}+\left (-2 a q x -b \,x^{2}\right ) p^{2}-b p q}{\left (p \,x^{2}+q \right ) a +b x}\right )+\ln \left (2\right )\right ) x \left (a^{2} p q -\frac {b^{2}}{2}\right )}{\sqrt {\frac {-2 a^{2} p q +b^{2}}{a^{2}}}\, x \,a^{3}}\) | \(204\) |
elliptic | \(\text {Expression too large to display}\) | \(1802\) |
risch | \(\text {Expression too large to display}\) | \(6680\) |
default | \(\text {Expression too large to display}\) | \(6985\) |
2/((-2*a^2*p*q+b^2)/a^2)^(1/2)*(-1/2*b*ln(((p^2*x^4+q^2)^(1/2)+(p*x^2+q)*c sgn(p))/x)*csgn(p)*x*a*((-2*a^2*p*q+b^2)/a^2)^(1/2)+1/2*((-2*a^2*p*q+b^2)/ a^2)^(1/2)*(p^2*x^4+q^2)^(1/2)*a^2+(ln((a*(p^2*x^4+q^2)^(1/2)*p*((-2*a^2*p *q+b^2)/a^2)^(1/2)+(-2*a*q*x-b*x^2)*p^2-b*p*q)/((p*x^2+q)*a+b*x))+ln(2))*x *(a^2*p*q-1/2*b^2))/x/a^3
Timed out. \[ \int \frac {\left (-q+p x^2\right ) \sqrt {q^2+p^2 x^4}}{x^2 \left (a q+b x+a p x^2\right )} \, dx=\text {Timed out} \]
\[ \int \frac {\left (-q+p x^2\right ) \sqrt {q^2+p^2 x^4}}{x^2 \left (a q+b x+a p x^2\right )} \, dx=\int \frac {\left (p x^{2} - q\right ) \sqrt {p^{2} x^{4} + q^{2}}}{x^{2} \left (a p x^{2} + a q + b x\right )}\, dx \]
\[ \int \frac {\left (-q+p x^2\right ) \sqrt {q^2+p^2 x^4}}{x^2 \left (a q+b x+a p x^2\right )} \, dx=\int { \frac {\sqrt {p^{2} x^{4} + q^{2}} {\left (p x^{2} - q\right )}}{{\left (a p x^{2} + a q + b x\right )} x^{2}} \,d x } \]
\[ \int \frac {\left (-q+p x^2\right ) \sqrt {q^2+p^2 x^4}}{x^2 \left (a q+b x+a p x^2\right )} \, dx=\int { \frac {\sqrt {p^{2} x^{4} + q^{2}} {\left (p x^{2} - q\right )}}{{\left (a p x^{2} + a q + b x\right )} x^{2}} \,d x } \]
Timed out. \[ \int \frac {\left (-q+p x^2\right ) \sqrt {q^2+p^2 x^4}}{x^2 \left (a q+b x+a p x^2\right )} \, dx=\int -\frac {\sqrt {p^2\,x^4+q^2}\,\left (q-p\,x^2\right )}{x^2\,\left (a\,p\,x^2+b\,x+a\,q\right )} \,d x \]