Integrand size = 62, antiderivative size = 159 \[ \int \frac {\left (-q+2 p x^3\right ) \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{x \left (b x^2+a \left (q+p x^3\right )^2\right )} \, dx=\frac {\sqrt {b+2 a p q} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {b+2 a p q} x^2}{a q^2+b x^2+2 a p q x^3+a p^2 x^6+\left (a q+a p x^3\right ) \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}}\right )}{a \sqrt {b}}-\frac {\log (x)}{a}+\frac {\log \left (q+p x^3+\sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}\right )}{a} \]
(2*a*p*q+b)^(1/2)*arctanh(b^(1/2)*(2*a*p*q+b)^(1/2)*x^2/(a*q^2+b*x^2+2*a*p *q*x^3+a*p^2*x^6+(a*p*x^3+a*q)*(p^2*x^6+2*p*q*x^3-2*p*q*x^2+q^2)^(1/2)))/a /b^(1/2)-ln(x)/a+ln(q+p*x^3+(p^2*x^6+2*p*q*x^3-2*p*q*x^2+q^2)^(1/2))/a
Time = 0.63 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.82 \[ \int \frac {\left (-q+2 p x^3\right ) \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{x \left (b x^2+a \left (q+p x^3\right )^2\right )} \, dx=-\frac {-\frac {\sqrt {b+2 a p q} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {b+2 a p q} x^2}{b x^2+a \left (q+p x^3\right ) \left (q+p x^3+\sqrt {q^2+2 p q (-1+x) x^2+p^2 x^6}\right )}\right )}{\sqrt {b}}+\log (x)-\log \left (q+p x^3+\sqrt {q^2+2 p q (-1+x) x^2+p^2 x^6}\right )}{a} \]
Integrate[((-q + 2*p*x^3)*Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^6])/(x* (b*x^2 + a*(q + p*x^3)^2)),x]
-((-((Sqrt[b + 2*a*p*q]*ArcTanh[(Sqrt[b]*Sqrt[b + 2*a*p*q]*x^2)/(b*x^2 + a *(q + p*x^3)*(q + p*x^3 + Sqrt[q^2 + 2*p*q*(-1 + x)*x^2 + p^2*x^6]))])/Sqr t[b]) + Log[x] - Log[q + p*x^3 + Sqrt[q^2 + 2*p*q*(-1 + x)*x^2 + p^2*x^6]] )/a)
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.
\(\displaystyle \int \frac {\left (2 p x^3-q\right ) \sqrt {p^2 x^6+2 p q x^3-2 p q x^2+q^2}}{x \left (a \left (p x^3+q\right )^2+b x^2\right )} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {x \sqrt {p^2 x^6+2 p q x^3-2 p q x^2+q^2} \left (a p^2 x^4+4 a p q x+b\right )}{a q \left (a p^2 x^6+2 a p q x^3+a q^2+b x^2\right )}-\frac {\sqrt {p^2 x^6+2 p q x^3-2 p q x^2+q^2}}{a q x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 4 p \int \frac {x^2 \sqrt {p^2 x^6+2 p q x^3-2 p q x^2+q^2}}{a p^2 x^6+2 a p q x^3+b x^2+a q^2}dx+\frac {b \int \frac {x \sqrt {p^2 x^6+2 p q x^3-2 p q x^2+q^2}}{a p^2 x^6+2 a p q x^3+b x^2+a q^2}dx}{a q}+\frac {p^2 \int \frac {x^5 \sqrt {p^2 x^6+2 p q x^3-2 p q x^2+q^2}}{a p^2 x^6+2 a p q x^3+b x^2+a q^2}dx}{q}-\frac {\int \frac {\sqrt {p^2 x^6+2 p q x^3-2 p q x^2+q^2}}{x}dx}{a q}\) |
Int[((-q + 2*p*x^3)*Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^6])/(x*(b*x^2 + a*(q + p*x^3)^2)),x]
3.22.67.3.1 Defintions of rubi rules used
Time = 0.28 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.69
method | result | size |
pseudoelliptic | \(-\frac {-\sqrt {-a b}\, \ln \left (\frac {q +p \,x^{3}+\sqrt {p^{2} x^{6}+2 p q \,x^{2} \left (-1+x \right )+q^{2}}}{x}\right ) \sqrt {\frac {-2 a p q -b}{a}}+\left (\ln \left (\frac {\sqrt {p^{2} x^{6}+2 p q \,x^{2} \left (-1+x \right )+q^{2}}\, \sqrt {\frac {-2 a p q -b}{a}}\, a +\left (-p \,x^{3}-q \right ) \sqrt {-a b}-2 q a p x}{\left (p \,x^{3}+q \right ) a +\sqrt {-a b}\, x}\right )-\ln \left (\frac {\sqrt {p^{2} x^{6}+2 p q \,x^{2} \left (-1+x \right )+q^{2}}\, \sqrt {\frac {-2 a p q -b}{a}}\, a +\left (p \,x^{3}+q \right ) \sqrt {-a b}-2 q a p x}{\left (p \,x^{3}+q \right ) a -\sqrt {-a b}\, x}\right )\right ) \left (a p q +\frac {b}{2}\right )}{\sqrt {\frac {-2 a p q -b}{a}}\, \sqrt {-a b}\, a}\) | \(268\) |
int((2*p*x^3-q)*(p^2*x^6+2*p*q*x^3-2*p*q*x^2+q^2)^(1/2)/x/(b*x^2+a*(p*x^3+ q)^2),x,method=_RETURNVERBOSE)
-1/((-2*a*p*q-b)/a)^(1/2)/(-a*b)^(1/2)*(-(-a*b)^(1/2)*ln((q+p*x^3+(p^2*x^6 +2*p*q*x^2*(-1+x)+q^2)^(1/2))/x)*((-2*a*p*q-b)/a)^(1/2)+(ln(((p^2*x^6+2*p* q*x^2*(-1+x)+q^2)^(1/2)*((-2*a*p*q-b)/a)^(1/2)*a+(-p*x^3-q)*(-a*b)^(1/2)-2 *q*a*p*x)/((p*x^3+q)*a+(-a*b)^(1/2)*x))-ln(((p^2*x^6+2*p*q*x^2*(-1+x)+q^2) ^(1/2)*((-2*a*p*q-b)/a)^(1/2)*a+(p*x^3+q)*(-a*b)^(1/2)-2*q*a*p*x)/((p*x^3+ q)*a-(-a*b)^(1/2)*x)))*(a*p*q+1/2*b))/a
Timed out. \[ \int \frac {\left (-q+2 p x^3\right ) \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{x \left (b x^2+a \left (q+p x^3\right )^2\right )} \, dx=\text {Timed out} \]
integrate((2*p*x^3-q)*(p^2*x^6+2*p*q*x^3-2*p*q*x^2+q^2)^(1/2)/x/(b*x^2+a*( p*x^3+q)^2),x, algorithm="fricas")
\[ \int \frac {\left (-q+2 p x^3\right ) \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{x \left (b x^2+a \left (q+p x^3\right )^2\right )} \, dx=\int \frac {\left (2 p x^{3} - q\right ) \sqrt {p^{2} x^{6} + 2 p q x^{3} - 2 p q x^{2} + q^{2}}}{x \left (a p^{2} x^{6} + 2 a p q x^{3} + a q^{2} + b x^{2}\right )}\, dx \]
integrate((2*p*x**3-q)*(p**2*x**6+2*p*q*x**3-2*p*q*x**2+q**2)**(1/2)/x/(b* x**2+a*(p*x**3+q)**2),x)
Integral((2*p*x**3 - q)*sqrt(p**2*x**6 + 2*p*q*x**3 - 2*p*q*x**2 + q**2)/( x*(a*p**2*x**6 + 2*a*p*q*x**3 + a*q**2 + b*x**2)), x)
\[ \int \frac {\left (-q+2 p x^3\right ) \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{x \left (b x^2+a \left (q+p x^3\right )^2\right )} \, dx=\int { \frac {\sqrt {p^{2} x^{6} + 2 \, p q x^{3} - 2 \, p q x^{2} + q^{2}} {\left (2 \, p x^{3} - q\right )}}{{\left ({\left (p x^{3} + q\right )}^{2} a + b x^{2}\right )} x} \,d x } \]
integrate((2*p*x^3-q)*(p^2*x^6+2*p*q*x^3-2*p*q*x^2+q^2)^(1/2)/x/(b*x^2+a*( p*x^3+q)^2),x, algorithm="maxima")
integrate(sqrt(p^2*x^6 + 2*p*q*x^3 - 2*p*q*x^2 + q^2)*(2*p*x^3 - q)/(((p*x ^3 + q)^2*a + b*x^2)*x), x)
\[ \int \frac {\left (-q+2 p x^3\right ) \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{x \left (b x^2+a \left (q+p x^3\right )^2\right )} \, dx=\int { \frac {\sqrt {p^{2} x^{6} + 2 \, p q x^{3} - 2 \, p q x^{2} + q^{2}} {\left (2 \, p x^{3} - q\right )}}{{\left ({\left (p x^{3} + q\right )}^{2} a + b x^{2}\right )} x} \,d x } \]
integrate((2*p*x^3-q)*(p^2*x^6+2*p*q*x^3-2*p*q*x^2+q^2)^(1/2)/x/(b*x^2+a*( p*x^3+q)^2),x, algorithm="giac")
integrate(sqrt(p^2*x^6 + 2*p*q*x^3 - 2*p*q*x^2 + q^2)*(2*p*x^3 - q)/(((p*x ^3 + q)^2*a + b*x^2)*x), x)
Timed out. \[ \int \frac {\left (-q+2 p x^3\right ) \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{x \left (b x^2+a \left (q+p x^3\right )^2\right )} \, dx=\int -\frac {\left (q-2\,p\,x^3\right )\,\sqrt {p^2\,x^6+2\,p\,q\,x^3-2\,p\,q\,x^2+q^2}}{x\,\left (a\,{\left (p\,x^3+q\right )}^2+b\,x^2\right )} \,d x \]
int(-((q - 2*p*x^3)*(p^2*x^6 + q^2 - 2*p*q*x^2 + 2*p*q*x^3)^(1/2))/(x*(a*( q + p*x^3)^2 + b*x^2)),x)