Integrand size = 52, antiderivative size = 189 \[ \int \frac {\left (-2 q+p x^3\right ) \sqrt {q+p x^3}}{c x^4+b x^2 \left (q+p x^3\right )+a \left (q+p x^3\right )^2} \, dx=-\frac {\sqrt {2} \left (-b+\sqrt {b^2-4 a c}\right ) \arctan \left (\frac {\sqrt {b-\sqrt {b^2-4 a c}} x}{\sqrt {2} \sqrt {a} \sqrt {q+p x^3}}\right )}{\sqrt {a} \sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} \sqrt {b+\sqrt {b^2-4 a c}} \arctan \left (\frac {\sqrt {b+\sqrt {b^2-4 a c}} x}{\sqrt {2} \sqrt {a} \sqrt {q+p x^3}}\right )}{\sqrt {a} \sqrt {b^2-4 a c}} \]
-2^(1/2)*(-b+(-4*a*c+b^2)^(1/2))*arctan(1/2*(b-(-4*a*c+b^2)^(1/2))^(1/2)*x *2^(1/2)/a^(1/2)/(p*x^3+q)^(1/2))/a^(1/2)/(-4*a*c+b^2)^(1/2)/(b-(-4*a*c+b^ 2)^(1/2))^(1/2)-2^(1/2)*(b+(-4*a*c+b^2)^(1/2))^(1/2)*arctan(1/2*(b+(-4*a*c +b^2)^(1/2))^(1/2)*x*2^(1/2)/a^(1/2)/(p*x^3+q)^(1/2))/a^(1/2)/(-4*a*c+b^2) ^(1/2)
Time = 1.47 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.80 \[ \int \frac {\left (-2 q+p x^3\right ) \sqrt {q+p x^3}}{c x^4+b x^2 \left (q+p x^3\right )+a \left (q+p x^3\right )^2} \, dx=\frac {\sqrt {2} \left (\sqrt {b-\sqrt {b^2-4 a c}} \arctan \left (\frac {\sqrt {b-\sqrt {b^2-4 a c}} x}{\sqrt {2} \sqrt {a} \sqrt {q+p x^3}}\right )-\sqrt {b+\sqrt {b^2-4 a c}} \arctan \left (\frac {\sqrt {b+\sqrt {b^2-4 a c}} x}{\sqrt {2} \sqrt {a} \sqrt {q+p x^3}}\right )\right )}{\sqrt {a} \sqrt {b^2-4 a c}} \]
(Sqrt[2]*(Sqrt[b - Sqrt[b^2 - 4*a*c]]*ArcTan[(Sqrt[b - Sqrt[b^2 - 4*a*c]]* x)/(Sqrt[2]*Sqrt[a]*Sqrt[q + p*x^3])] - Sqrt[b + Sqrt[b^2 - 4*a*c]]*ArcTan [(Sqrt[b + Sqrt[b^2 - 4*a*c]]*x)/(Sqrt[2]*Sqrt[a]*Sqrt[q + p*x^3])]))/(Sqr t[a]*Sqrt[b^2 - 4*a*c])
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.
| \(\displaystyle \int \frac {\left (p x^3-2 q\right ) \sqrt {p x^3+q}}{a \left (p x^3+q\right )^2+b x^2 \left (p x^3+q\right )+c x^4} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {p x^3 \sqrt {p x^3+q}}{a p^2 x^6+2 a p q x^3+a q^2+b p x^5+b q x^2+c x^4}-\frac {2 q \sqrt {p x^3+q}}{a p^2 x^6+2 a p q x^3+a q^2+b p x^5+b q x^2+c x^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle p \int \frac {x^3 \sqrt {p x^3+q}}{a p^2 x^6+b p x^5+c x^4+2 a p q x^3+b q x^2+a q^2}dx-2 q \int \frac {\sqrt {p x^3+q}}{a p^2 x^6+b p x^5+c x^4+2 a p q x^3+b q x^2+a q^2}dx\) |
3.24.73.3.1 Defintions of rubi rules used
Time = 1.08 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.79
| method | result | size |
| default | \(\frac {\sqrt {2}\, \left (-\frac {\left (-b +\sqrt {-4 a c +b^{2}}\right ) \operatorname {arctanh}\left (\frac {a \sqrt {p \,x^{3}+q}\, \sqrt {2}}{x \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) a}}\right )}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) a}}+\frac {\left (b +\sqrt {-4 a c +b^{2}}\right ) \arctan \left (\frac {a \sqrt {p \,x^{3}+q}\, \sqrt {2}}{x \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) a}}\right )}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) a}}\right )}{\sqrt {-4 a c +b^{2}}}\) | \(149\) |
| pseudoelliptic | \(\frac {\sqrt {2}\, \left (-\frac {\left (-b +\sqrt {-4 a c +b^{2}}\right ) \operatorname {arctanh}\left (\frac {a \sqrt {p \,x^{3}+q}\, \sqrt {2}}{x \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) a}}\right )}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) a}}+\frac {\left (b +\sqrt {-4 a c +b^{2}}\right ) \arctan \left (\frac {a \sqrt {p \,x^{3}+q}\, \sqrt {2}}{x \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) a}}\right )}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) a}}\right )}{\sqrt {-4 a c +b^{2}}}\) | \(149\) |
| elliptic | \(\text {Expression too large to display}\) | \(1362\) |
int((p*x^3-2*q)*(p*x^3+q)^(1/2)/(c*x^4+b*x^2*(p*x^3+q)+a*(p*x^3+q)^2),x,me thod=_RETURNVERBOSE)
2^(1/2)/(-4*a*c+b^2)^(1/2)*(-(-b+(-4*a*c+b^2)^(1/2))/((-b+(-4*a*c+b^2)^(1/ 2))*a)^(1/2)*arctanh(a*(p*x^3+q)^(1/2)/x*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))* a)^(1/2))+(b+(-4*a*c+b^2)^(1/2))/((b+(-4*a*c+b^2)^(1/2))*a)^(1/2)*arctan(a *(p*x^3+q)^(1/2)/x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*a)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 1321 vs. \(2 (134) = 268\).
Time = 2.16 (sec) , antiderivative size = 1321, normalized size of antiderivative = 6.99 \[ \int \frac {\left (-2 q+p x^3\right ) \sqrt {q+p x^3}}{c x^4+b x^2 \left (q+p x^3\right )+a \left (q+p x^3\right )^2} \, dx=\text {Too large to display} \]
integrate((p*x^3-2*q)*(p*x^3+q)^(1/2)/(c*x^4+b*x^2*(p*x^3+q)+a*(p*x^3+q)^2 ),x, algorithm="fricas")
-1/4*sqrt(2)*sqrt(-(b + (a*b^2 - 4*a^2*c)/sqrt(a^2*b^2 - 4*a^3*c))/(a*b^2 - 4*a^2*c))*log((2*a*p^2*x^6 + 4*a*p*q*x^3 - 2*c*x^4 + 2*a*q^2 + sqrt(2)*( (b^2 - 4*a*c)*x^3 - (2*(a^2*b^2 - 4*a^3*c)*p*x^4 + (a*b^3 - 4*a^2*b*c)*x^3 + 2*(a^2*b^2 - 4*a^3*c)*q*x)/sqrt(a^2*b^2 - 4*a^3*c))*sqrt(p*x^3 + q)*sqr t(-(b + (a*b^2 - 4*a^2*c)/sqrt(a^2*b^2 - 4*a^3*c))/(a*b^2 - 4*a^2*c)) - 2* ((a*b^2 - 4*a^2*c)*p*x^5 + (a*b^2 - 4*a^2*c)*q*x^2)/sqrt(a^2*b^2 - 4*a^3*c ))/(a*p^2*x^6 + b*p*x^5 + 2*a*p*q*x^3 + c*x^4 + b*q*x^2 + a*q^2)) + 1/4*sq rt(2)*sqrt(-(b + (a*b^2 - 4*a^2*c)/sqrt(a^2*b^2 - 4*a^3*c))/(a*b^2 - 4*a^2 *c))*log((2*a*p^2*x^6 + 4*a*p*q*x^3 - 2*c*x^4 + 2*a*q^2 - sqrt(2)*((b^2 - 4*a*c)*x^3 - (2*(a^2*b^2 - 4*a^3*c)*p*x^4 + (a*b^3 - 4*a^2*b*c)*x^3 + 2*(a ^2*b^2 - 4*a^3*c)*q*x)/sqrt(a^2*b^2 - 4*a^3*c))*sqrt(p*x^3 + q)*sqrt(-(b + (a*b^2 - 4*a^2*c)/sqrt(a^2*b^2 - 4*a^3*c))/(a*b^2 - 4*a^2*c)) - 2*((a*b^2 - 4*a^2*c)*p*x^5 + (a*b^2 - 4*a^2*c)*q*x^2)/sqrt(a^2*b^2 - 4*a^3*c))/(a*p ^2*x^6 + b*p*x^5 + 2*a*p*q*x^3 + c*x^4 + b*q*x^2 + a*q^2)) - 1/4*sqrt(2)*s qrt(-(b - (a*b^2 - 4*a^2*c)/sqrt(a^2*b^2 - 4*a^3*c))/(a*b^2 - 4*a^2*c))*lo g((2*a*p^2*x^6 + 4*a*p*q*x^3 - 2*c*x^4 + 2*a*q^2 + sqrt(2)*((b^2 - 4*a*c)* x^3 + (2*(a^2*b^2 - 4*a^3*c)*p*x^4 + (a*b^3 - 4*a^2*b*c)*x^3 + 2*(a^2*b^2 - 4*a^3*c)*q*x)/sqrt(a^2*b^2 - 4*a^3*c))*sqrt(p*x^3 + q)*sqrt(-(b - (a*b^2 - 4*a^2*c)/sqrt(a^2*b^2 - 4*a^3*c))/(a*b^2 - 4*a^2*c)) + 2*((a*b^2 - 4*a^ 2*c)*p*x^5 + (a*b^2 - 4*a^2*c)*q*x^2)/sqrt(a^2*b^2 - 4*a^3*c))/(a*p^2*x...
\[ \int \frac {\left (-2 q+p x^3\right ) \sqrt {q+p x^3}}{c x^4+b x^2 \left (q+p x^3\right )+a \left (q+p x^3\right )^2} \, dx=\int \frac {\left (p x^{3} - 2 q\right ) \sqrt {p x^{3} + q}}{a p^{2} x^{6} + 2 a p q x^{3} + a q^{2} + b p x^{5} + b q x^{2} + c x^{4}}\, dx \]
Integral((p*x**3 - 2*q)*sqrt(p*x**3 + q)/(a*p**2*x**6 + 2*a*p*q*x**3 + a*q **2 + b*p*x**5 + b*q*x**2 + c*x**4), x)
\[ \int \frac {\left (-2 q+p x^3\right ) \sqrt {q+p x^3}}{c x^4+b x^2 \left (q+p x^3\right )+a \left (q+p x^3\right )^2} \, dx=\int { \frac {\sqrt {p x^{3} + q} {\left (p x^{3} - 2 \, q\right )}}{c x^{4} + {\left (p x^{3} + q\right )} b x^{2} + {\left (p x^{3} + q\right )}^{2} a} \,d x } \]
integrate((p*x^3-2*q)*(p*x^3+q)^(1/2)/(c*x^4+b*x^2*(p*x^3+q)+a*(p*x^3+q)^2 ),x, algorithm="maxima")
\[ \int \frac {\left (-2 q+p x^3\right ) \sqrt {q+p x^3}}{c x^4+b x^2 \left (q+p x^3\right )+a \left (q+p x^3\right )^2} \, dx=\int { \frac {\sqrt {p x^{3} + q} {\left (p x^{3} - 2 \, q\right )}}{c x^{4} + {\left (p x^{3} + q\right )} b x^{2} + {\left (p x^{3} + q\right )}^{2} a} \,d x } \]
integrate((p*x^3-2*q)*(p*x^3+q)^(1/2)/(c*x^4+b*x^2*(p*x^3+q)+a*(p*x^3+q)^2 ),x, algorithm="giac")
Timed out. \[ \int \frac {\left (-2 q+p x^3\right ) \sqrt {q+p x^3}}{c x^4+b x^2 \left (q+p x^3\right )+a \left (q+p x^3\right )^2} \, dx=\int -\frac {\sqrt {p\,x^3+q}\,\left (2\,q-p\,x^3\right )}{a\,{\left (p\,x^3+q\right )}^2+c\,x^4+b\,x^2\,\left (p\,x^3+q\right )} \,d x \]