Integrand size = 41, antiderivative size = 85 \[ \int \frac {\left (-q+p x^2\right ) \left (a q+b x+a p x^2\right ) \sqrt {q^2+p^2 x^4}}{x^4} \, dx=\frac {\sqrt {q^2+p^2 x^4} \left (2 a q^2+3 b q x+3 b p x^3+2 a p^2 x^4\right )}{6 x^3}+b p q \log (x)-b p q \log \left (q+p x^2+\sqrt {q^2+p^2 x^4}\right ) \]
1/6*(p^2*x^4+q^2)^(1/2)*(2*a*p^2*x^4+3*b*p*x^3+2*a*q^2+3*b*q*x)/x^3+b*p*q* ln(x)-b*p*q*ln(q+p*x^2+(p^2*x^4+q^2)^(1/2))
Time = 0.77 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-q+p x^2\right ) \left (a q+b x+a p x^2\right ) \sqrt {q^2+p^2 x^4}}{x^4} \, dx=\frac {\sqrt {q^2+p^2 x^4} \left (2 a q^2+3 b q x+3 b p x^3+2 a p^2 x^4\right )}{6 x^3}+b p q \log (x)-b p q \log \left (q+p x^2+\sqrt {q^2+p^2 x^4}\right ) \]
(Sqrt[q^2 + p^2*x^4]*(2*a*q^2 + 3*b*q*x + 3*b*p*x^3 + 2*a*p^2*x^4))/(6*x^3 ) + b*p*q*Log[x] - b*p*q*Log[q + p*x^2 + Sqrt[q^2 + p^2*x^4]]
Time = 0.57 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.22, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.293, Rules used = {2241, 25, 27, 1579, 536, 538, 224, 219, 243, 73, 221, 2023}
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.
| \(\displaystyle \int \frac {\left (p x^2-q\right ) \sqrt {p^2 x^4+q^2} \left (a p x^2+a q+b x\right )}{x^4} \, dx\) |
| \(\Big \downarrow \) 2241 |
| \(\displaystyle \int \frac {\left (p x^2-q\right ) \left (a p x^2+a q\right ) \sqrt {p^2 x^4+q^2}}{x^4}dx+\int -\frac {b \left (q-p x^2\right ) \sqrt {p^2 x^4+q^2}}{x^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int \frac {\left (p x^2-q\right ) \left (a p x^2+a q\right ) \sqrt {p^2 x^4+q^2}}{x^4}dx-\int \frac {b \left (q-p x^2\right ) \sqrt {p^2 x^4+q^2}}{x^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {\left (p x^2-q\right ) \left (a p x^2+a q\right ) \sqrt {p^2 x^4+q^2}}{x^4}dx-b \int \frac {\left (q-p x^2\right ) \sqrt {p^2 x^4+q^2}}{x^3}dx\) |
| \(\Big \downarrow \) 1579 |
| \(\displaystyle \int \frac {\left (p x^2-q\right ) \left (a p x^2+a q\right ) \sqrt {p^2 x^4+q^2}}{x^4}dx-\frac {1}{2} b \int \frac {\left (q-p x^2\right ) \sqrt {p^2 x^4+q^2}}{x^4}dx^2\) |
| \(\Big \downarrow \) 536 |
| \(\displaystyle \int \frac {\left (p x^2-q\right ) \left (a p x^2+a q\right ) \sqrt {p^2 x^4+q^2}}{x^4}dx-\frac {1}{2} b \left (\int \frac {p^2 q x^2-p q^2}{x^2 \sqrt {p^2 x^4+q^2}}dx^2-\frac {\left (p x^2+q\right ) \sqrt {p^2 x^4+q^2}}{x^2}\right )\) |
| \(\Big \downarrow \) 538 |
| \(\displaystyle \int \frac {\left (p x^2-q\right ) \left (a p x^2+a q\right ) \sqrt {p^2 x^4+q^2}}{x^4}dx-\frac {1}{2} b \left (p^2 q \int \frac {1}{\sqrt {p^2 x^4+q^2}}dx^2-p q^2 \int \frac {1}{x^2 \sqrt {p^2 x^4+q^2}}dx^2-\frac {\left (p x^2+q\right ) \sqrt {p^2 x^4+q^2}}{x^2}\right )\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \int \frac {\left (p x^2-q\right ) \left (a p x^2+a q\right ) \sqrt {p^2 x^4+q^2}}{x^4}dx-\frac {1}{2} b \left (p^2 q \int \frac {1}{1-p^2 x^4}d\frac {x^2}{\sqrt {p^2 x^4+q^2}}-p q^2 \int \frac {1}{x^2 \sqrt {p^2 x^4+q^2}}dx^2-\frac {\left (p x^2+q\right ) \sqrt {p^2 x^4+q^2}}{x^2}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \int \frac {\left (p x^2-q\right ) \left (a p x^2+a q\right ) \sqrt {p^2 x^4+q^2}}{x^4}dx-\frac {1}{2} b \left (-p q^2 \int \frac {1}{x^2 \sqrt {p^2 x^4+q^2}}dx^2+p q \text {arctanh}\left (\frac {p x^2}{\sqrt {p^2 x^4+q^2}}\right )-\frac {\left (p x^2+q\right ) \sqrt {p^2 x^4+q^2}}{x^2}\right )\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \int \frac {\left (p x^2-q\right ) \left (a p x^2+a q\right ) \sqrt {p^2 x^4+q^2}}{x^4}dx-\frac {1}{2} b \left (-\frac {1}{2} p q^2 \int \frac {1}{x^2 \sqrt {p^2 x^4+q^2}}dx^4+p q \text {arctanh}\left (\frac {p x^2}{\sqrt {p^2 x^4+q^2}}\right )-\frac {\left (p x^2+q\right ) \sqrt {p^2 x^4+q^2}}{x^2}\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \int \frac {\left (p x^2-q\right ) \left (a p x^2+a q\right ) \sqrt {p^2 x^4+q^2}}{x^4}dx-\frac {1}{2} b \left (-\frac {q^2 \int \frac {1}{\frac {\sqrt {p^2 x^4+q^2}}{p^2}-\frac {q^2}{p^2}}d\sqrt {p^2 x^4+q^2}}{p}+p q \text {arctanh}\left (\frac {p x^2}{\sqrt {p^2 x^4+q^2}}\right )-\frac {\left (p x^2+q\right ) \sqrt {p^2 x^4+q^2}}{x^2}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \int \frac {\left (p x^2-q\right ) \left (a p x^2+a q\right ) \sqrt {p^2 x^4+q^2}}{x^4}dx-\frac {1}{2} b \left (p q \text {arctanh}\left (\frac {\sqrt {p^2 x^4+q^2}}{q}\right )+p q \text {arctanh}\left (\frac {p x^2}{\sqrt {p^2 x^4+q^2}}\right )-\frac {\sqrt {p^2 x^4+q^2} \left (p x^2+q\right )}{x^2}\right )\) |
| \(\Big \downarrow \) 2023 |
| \(\displaystyle \frac {a \left (p^2 x^4+q^2\right )^{3/2}}{3 x^3}-\frac {1}{2} b \left (p q \text {arctanh}\left (\frac {\sqrt {p^2 x^4+q^2}}{q}\right )+p q \text {arctanh}\left (\frac {p x^2}{\sqrt {p^2 x^4+q^2}}\right )-\frac {\sqrt {p^2 x^4+q^2} \left (p x^2+q\right )}{x^2}\right )\) |
(a*(q^2 + p^2*x^4)^(3/2))/(3*x^3) - (b*(-(((q + p*x^2)*Sqrt[q^2 + p^2*x^4] )/x^2) + p*q*ArcTanh[(p*x^2)/Sqrt[q^2 + p^2*x^4]] + p*q*ArcTanh[Sqrt[q^2 + p^2*x^4]/q]))/2
3.12.42.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 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[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[(((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_))/(x_)^2, x_Symbol] :> S imp[(-(2*c*p - d*x))*((a + b*x^2)^p/(2*p*x)), x] + Int[(a*d + 2*b*c*p*x)*(( a + b*x^2)^(p - 1)/x), x] /; FreeQ[{a, b, c, d}, x] && GtQ[p, 0] && Integer Q[2*p]
Int[((c_) + (d_.)*(x_))/((x_)*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Simp [c Int[1/(x*Sqrt[a + b*x^2]), x], x] + Simp[d Int[1/Sqrt[a + b*x^2], x] , x] /; FreeQ[{a, b, c, d}, x]
Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, c, d, e, p, q}, x] && IntegerQ[(m + 1)/2]
Int[(Pp_)*(Qq_)^(m_.)*(Rr_)^(n_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x], r = Expon[Rr, x]}, Simp[Coeff[Pp, x, p]*x^(p - q - r + 1)*Qq ^(m + 1)*(Rr^(n + 1)/((p + m*q + n*r + 1)*Coeff[Qq, x, q]*Coeff[Rr, x, r])) , x] /; NeQ[p + m*q + n*r + 1, 0] && EqQ[(p + m*q + n*r + 1)*Coeff[Qq, x, q ]*Coeff[Rr, x, r]*Pp, Coeff[Pp, x, p]*x^(p - q - r)*((p - q - r + 1)*Qq*Rr + (m + 1)*x*Rr*D[Qq, x] + (n + 1)*x*Qq*D[Rr, x])]] /; FreeQ[{m, n}, x] && P olyQ[Pp, x] && PolyQ[Qq, x] && PolyQ[Rr, x] && NeQ[m, -1] && NeQ[n, -1]
Int[(Pr_)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_) ^4)^(p_), x_Symbol] :> Module[{r = Expon[Pr, x], k}, Int[Sum[Coeff[Pr, x, 2 *k]*x^(2*k), {k, 0, r/2 + 1}]*(f*x)^m*(d + e*x^2)^q*(a + c*x^4)^p, x] + Sim p[1/f Int[Sum[Coeff[Pr, x, 2*k + 1]*x^(2*k), {k, 0, (r + 1)/2}]*(f*x)^(m + 1)*(d + e*x^2)^q*(a + c*x^4)^p, x], x]] /; FreeQ[{a, c, d, e, f, m, p, q} , x] && PolyQ[Pr, x] && !PolyQ[Pr, x^2]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 3.81 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.02
| method | result | size |
| pseudoelliptic | \(\frac {-6 b p q \ln \left (\frac {\sqrt {p^{2} x^{4}+q^{2}}+\left (p \,x^{2}+q \right ) \operatorname {csgn}\left (p \right )}{x}\right ) \operatorname {csgn}\left (p \right ) x^{3}+2 \sqrt {p^{2} x^{4}+q^{2}}\, \left (a \,p^{2} x^{4}+\frac {3}{2} b p \,x^{3}+a \,q^{2}+\frac {3}{2} b q x \right )}{6 x^{3}}\) | \(87\) |
| elliptic | \(\frac {b \left (p \sqrt {p^{2} x^{4}+q^{2}}-\frac {q \,p^{2} \ln \left (\frac {p^{2} x^{2}}{\sqrt {p^{2}}}+\sqrt {p^{2} x^{4}+q^{2}}\right )}{\sqrt {p^{2}}}+\frac {q \sqrt {p^{2} x^{4}+q^{2}}}{x^{2}}-\frac {p \,q^{2} \ln \left (\frac {2 q^{2}+2 \sqrt {q^{2}}\, \sqrt {p^{2} x^{4}+q^{2}}}{x^{2}}\right )}{\sqrt {q^{2}}}\right )}{2}+\frac {a \left (p^{2} x^{4}+q^{2}\right )^{\frac {3}{2}}}{3 x^{3}}\) | \(138\) |
| risch | \(\frac {\sqrt {p^{2} x^{4}+q^{2}}\, q \left (2 a q +3 b x \right )}{6 x^{3}}+\frac {a \,p^{2} x \sqrt {p^{2} x^{4}+q^{2}}}{3}+\frac {p b \sqrt {p^{2} x^{4}+q^{2}}}{2}-\frac {p b \,q^{2} \ln \left (\frac {2 q^{2}+2 \sqrt {q^{2}}\, \sqrt {p^{2} x^{4}+q^{2}}}{x^{2}}\right )}{2 \sqrt {q^{2}}}-\frac {b \,p^{2} q \ln \left (\frac {p^{2} x^{2}}{\sqrt {p^{2}}}+\sqrt {p^{2} x^{4}+q^{2}}\right )}{2 \sqrt {p^{2}}}\) | \(149\) |
| default | \(a \,p^{2} \left (\frac {x \sqrt {p^{2} x^{4}+q^{2}}}{3}+\frac {2 q^{2} \sqrt {1-\frac {i p \,x^{2}}{q}}\, \sqrt {1+\frac {i p \,x^{2}}{q}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i p}{q}}, i\right )}{3 \sqrt {\frac {i p}{q}}\, \sqrt {p^{2} x^{4}+q^{2}}}\right )+p b \left (\frac {\sqrt {p^{2} x^{4}+q^{2}}}{2}-\frac {q^{2} \ln \left (\frac {2 q^{2}+2 \sqrt {q^{2}}\, \sqrt {p^{2} x^{4}+q^{2}}}{x^{2}}\right )}{2 \sqrt {q^{2}}}\right )-a \,q^{2} \left (-\frac {\sqrt {p^{2} x^{4}+q^{2}}}{3 x^{3}}+\frac {2 p^{2} \sqrt {1-\frac {i p \,x^{2}}{q}}\, \sqrt {1+\frac {i p \,x^{2}}{q}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i p}{q}}, i\right )}{3 \sqrt {\frac {i p}{q}}\, \sqrt {p^{2} x^{4}+q^{2}}}\right )-q b \left (-\frac {\left (p^{2} x^{4}+q^{2}\right )^{\frac {3}{2}}}{2 q^{2} x^{2}}+\frac {p^{2} x^{2} \sqrt {p^{2} x^{4}+q^{2}}}{2 q^{2}}+\frac {p^{2} \ln \left (\frac {p^{2} x^{2}}{\sqrt {p^{2}}}+\sqrt {p^{2} x^{4}+q^{2}}\right )}{2 \sqrt {p^{2}}}\right )\) | \(334\) |
1/6*(-6*b*p*q*ln(((p^2*x^4+q^2)^(1/2)+(p*x^2+q)*csgn(p))/x)*csgn(p)*x^3+2* (p^2*x^4+q^2)^(1/2)*(a*p^2*x^4+3/2*b*p*x^3+a*q^2+3/2*b*q*x))/x^3
Time = 0.47 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.98 \[ \int \frac {\left (-q+p x^2\right ) \left (a q+b x+a p x^2\right ) \sqrt {q^2+p^2 x^4}}{x^4} \, dx=\frac {6 \, b p q x^{3} \log \left (\frac {p x^{2} + q - \sqrt {p^{2} x^{4} + q^{2}}}{x}\right ) + {\left (2 \, a p^{2} x^{4} + 3 \, b p x^{3} + 2 \, a q^{2} + 3 \, b q x\right )} \sqrt {p^{2} x^{4} + q^{2}}}{6 \, x^{3}} \]
1/6*(6*b*p*q*x^3*log((p*x^2 + q - sqrt(p^2*x^4 + q^2))/x) + (2*a*p^2*x^4 + 3*b*p*x^3 + 2*a*q^2 + 3*b*q*x)*sqrt(p^2*x^4 + q^2))/x^3
Result contains complex when optimal does not.
Time = 2.81 (sec) , antiderivative size = 223, normalized size of antiderivative = 2.62 \[ \int \frac {\left (-q+p x^2\right ) \left (a q+b x+a p x^2\right ) \sqrt {q^2+p^2 x^4}}{x^4} \, dx=\frac {a p^{2} q x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {p^{2} x^{4} e^{i \pi }}{q^{2}}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} - \frac {a q^{3} \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {1}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {p^{2} x^{4} e^{i \pi }}{q^{2}}} \right )}}{4 x^{3} \Gamma \left (\frac {1}{4}\right )} + \frac {b p^{2} x^{2}}{2 \sqrt {\frac {p^{2} x^{4}}{q^{2}} + 1}} + \frac {b p^{2} x^{2}}{2 \sqrt {1 + \frac {q^{2}}{p^{2} x^{4}}}} - \frac {b p q \operatorname {asinh}{\left (\frac {q}{p x^{2}} \right )}}{2} - \frac {b p q \operatorname {asinh}{\left (\frac {p x^{2}}{q} \right )}}{2} + \frac {b q^{2}}{2 x^{2} \sqrt {\frac {p^{2} x^{4}}{q^{2}} + 1}} + \frac {b q^{2}}{2 x^{2} \sqrt {1 + \frac {q^{2}}{p^{2} x^{4}}}} \]
a*p**2*q*x*gamma(1/4)*hyper((-1/2, 1/4), (5/4,), p**2*x**4*exp_polar(I*pi) /q**2)/(4*gamma(5/4)) - a*q**3*gamma(-3/4)*hyper((-3/4, -1/2), (1/4,), p** 2*x**4*exp_polar(I*pi)/q**2)/(4*x**3*gamma(1/4)) + b*p**2*x**2/(2*sqrt(p** 2*x**4/q**2 + 1)) + b*p**2*x**2/(2*sqrt(1 + q**2/(p**2*x**4))) - b*p*q*asi nh(q/(p*x**2))/2 - b*p*q*asinh(p*x**2/q)/2 + b*q**2/(2*x**2*sqrt(p**2*x**4 /q**2 + 1)) + b*q**2/(2*x**2*sqrt(1 + q**2/(p**2*x**4)))
\[ \int \frac {\left (-q+p x^2\right ) \left (a q+b x+a p x^2\right ) \sqrt {q^2+p^2 x^4}}{x^4} \, dx=\int { \frac {\sqrt {p^{2} x^{4} + q^{2}} {\left (a p x^{2} + a q + b x\right )} {\left (p x^{2} - q\right )}}{x^{4}} \,d x } \]
\[ \int \frac {\left (-q+p x^2\right ) \left (a q+b x+a p x^2\right ) \sqrt {q^2+p^2 x^4}}{x^4} \, dx=\int { \frac {\sqrt {p^{2} x^{4} + q^{2}} {\left (a p x^{2} + a q + b x\right )} {\left (p x^{2} - q\right )}}{x^{4}} \,d x } \]
Timed out. \[ \int \frac {\left (-q+p x^2\right ) \left (a q+b x+a p x^2\right ) \sqrt {q^2+p^2 x^4}}{x^4} \, dx=-\int \frac {\sqrt {p^2\,x^4+q^2}\,\left (q-p\,x^2\right )\,\left (a\,p\,x^2+b\,x+a\,q\right )}{x^4} \,d x \]