\(\int \frac {\sqrt {q+p x^5} (-2 q+3 p x^5)}{c x^4+b x^2 (q+p x^5)+a (q+p x^5)^2} \, dx\) [2376]

Optimal result

Integrand size = 53, antiderivative size = 189 \[ \int \frac {\sqrt {q+p x^5} \left (-2 q+3 p x^5\right )}{c x^4+b x^2 \left (q+p x^5\right )+a \left (q+p x^5\right )^2} \, dx=-\frac {\sqrt {2} \left (-b+\sqrt {b^2-4 a c}\right ) \arctan \left (\frac {\sqrt {b-\sqrt {b^2-4 a c}} x}{\sqrt {2} \sqrt {a} \sqrt {q+p x^5}}\right )}{\sqrt {a} \sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} \sqrt {b+\sqrt {b^2-4 a c}} \arctan \left (\frac {\sqrt {b+\sqrt {b^2-4 a c}} x}{\sqrt {2} \sqrt {a} \sqrt {q+p x^5}}\right )}{\sqrt {a} \sqrt {b^2-4 a c}} \] Output:

-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^5+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^5+q)^(1/2))/a^(1/2)/(-4*a*c+b^2) 
^(1/2)
 

Mathematica [A] (verified)

Time = 7.22 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.80 \[ \int \frac {\sqrt {q+p x^5} \left (-2 q+3 p x^5\right )}{c x^4+b x^2 \left (q+p x^5\right )+a \left (q+p x^5\right )^2} \, dx=\frac {\sqrt {2} \left (\sqrt {b-\sqrt {b^2-4 a c}} \arctan \left (\frac {\sqrt {b-\sqrt {b^2-4 a c}} x}{\sqrt {2} \sqrt {a} \sqrt {q+p x^5}}\right )-\sqrt {b+\sqrt {b^2-4 a c}} \arctan \left (\frac {\sqrt {b+\sqrt {b^2-4 a c}} x}{\sqrt {2} \sqrt {a} \sqrt {q+p x^5}}\right )\right )}{\sqrt {a} \sqrt {b^2-4 a c}} \] Input:

Integrate[(Sqrt[q + p*x^5]*(-2*q + 3*p*x^5))/(c*x^4 + b*x^2*(q + p*x^5) + 
a*(q + p*x^5)^2),x]
 

Output:

(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^5])] - 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^5])]))/(Sqr 
t[a]*Sqrt[b^2 - 4*a*c])
 

Rubi [F]

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.

\(\displaystyle \int \frac {\sqrt {p x^5+q} \left (3 p x^5-2 q\right )}{a \left (p x^5+q\right )^2+b x^2 \left (p x^5+q\right )+c x^4} \, dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {3 p x^5 \sqrt {p x^5+q}}{a p^2 x^{10}+2 a p q x^5+a q^2+b p x^7+b q x^2+c x^4}-\frac {2 q \sqrt {p x^5+q}}{a p^2 x^{10}+2 a p q x^5+a q^2+b p x^7+b q x^2+c x^4}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 3 p \int \frac {x^5 \sqrt {p x^5+q}}{a p^2 x^{10}+b p x^7+2 a p q x^5+c x^4+b q x^2+a q^2}dx-2 q \int \frac {\sqrt {p x^5+q}}{a p^2 x^{10}+b p x^7+2 a p q x^5+c x^4+b q x^2+a q^2}dx\)

Input:

Int[(Sqrt[q + p*x^5]*(-2*q + 3*p*x^5))/(c*x^4 + b*x^2*(q + p*x^5) + a*(q + 
 p*x^5)^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 6.04 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.79

method	result	size
pseudoelliptic	\(\frac {\sqrt {2}\, \left (-\frac {\left (-b +\sqrt {-4 a c +b^{2}}\right ) \operatorname {arctanh}\left (\frac {a \sqrt {p \,x^{5}+q}\, \sqrt {2}}{x \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) a}}\right )}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) a}}+\frac {\left (b +\sqrt {-4 a c +b^{2}}\right ) \arctan \left (\frac {a \sqrt {p \,x^{5}+q}\, \sqrt {2}}{x \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) a}}\right )}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) a}}\right )}{\sqrt {-4 a c +b^{2}}}\)	\(149\)

Input:

int((p*x^5+q)^(1/2)*(3*p*x^5-2*q)/(c*x^4+b*x^2*(p*x^5+q)+a*(p*x^5+q)^2),x, 
method=_RETURNVERBOSE)
 

Output:

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^5+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^5+q)^(1/2)/x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*a)^(1/2)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1321 vs. \(2 (134) = 268\).

Time = 45.79 (sec) , antiderivative size = 1321, normalized size of antiderivative = 6.99 \[ \int \frac {\sqrt {q+p x^5} \left (-2 q+3 p x^5\right )}{c x^4+b x^2 \left (q+p x^5\right )+a \left (q+p x^5\right )^2} \, dx=\text {Too large to display} \] Input:

integrate((p*x^5+q)^(1/2)*(3*p*x^5-2*q)/(c*x^4+b*x^2*(p*x^5+q)+a*(p*x^5+q) 
^2),x, algorithm="fricas")
 

Output:

-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^10 + 4*a*p*q*x^5 - 2*c*x^4 + 2*a*q^2 + sqrt(2)* 
sqrt(p*x^5 + q)*((b^2 - 4*a*c)*x^3 - (2*(a^2*b^2 - 4*a^3*c)*p*x^6 + (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))*sq 
rt(-(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^7 + (a*b^2 - 4*a^2*c)*q*x^2)/sqrt(a^2*b^2 - 4*a^3* 
c))/(a*p^2*x^10 + b*p*x^7 + 2*a*p*q*x^5 + c*x^4 + b*q*x^2 + a*q^2)) + 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^10 + 4*a*p*q*x^5 - 2*c*x^4 + 2*a*q^2 - sqrt(2)*sqrt( 
p*x^5 + q)*((b^2 - 4*a*c)*x^3 - (2*(a^2*b^2 - 4*a^3*c)*p*x^6 + (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(-( 
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^7 + (a*b^2 - 4*a^2*c)*q*x^2)/sqrt(a^2*b^2 - 4*a^3*c))/( 
a*p^2*x^10 + b*p*x^7 + 2*a*p*q*x^5 + c*x^4 + b*q*x^2 + a*q^2)) - 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^10 + 4*a*p*q*x^5 - 2*c*x^4 + 2*a*q^2 + sqrt(2)*sqrt(p*x^5 
 + q)*((b^2 - 4*a*c)*x^3 + (2*(a^2*b^2 - 4*a^3*c)*p*x^6 + (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(-(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^7 + (a*b^2 - 4*a^2*c)*q*x^2)/sqrt(a^2*b^2 - 4*a^3*c))/(a*...
 

Sympy [F]

\[ \int \frac {\sqrt {q+p x^5} \left (-2 q+3 p x^5\right )}{c x^4+b x^2 \left (q+p x^5\right )+a \left (q+p x^5\right )^2} \, dx=\int \frac {\sqrt {p x^{5} + q} \left (3 p x^{5} - 2 q\right )}{a p^{2} x^{10} + 2 a p q x^{5} + a q^{2} + b p x^{7} + b q x^{2} + c x^{4}}\, dx \] Input:

integrate((p*x**5+q)**(1/2)*(3*p*x**5-2*q)/(c*x**4+b*x**2*(p*x**5+q)+a*(p* 
x**5+q)**2),x)
 

Output:

Integral(sqrt(p*x**5 + q)*(3*p*x**5 - 2*q)/(a*p**2*x**10 + 2*a*p*q*x**5 + 
a*q**2 + b*p*x**7 + b*q*x**2 + c*x**4), x)
 

Maxima [F]

\[ \int \frac {\sqrt {q+p x^5} \left (-2 q+3 p x^5\right )}{c x^4+b x^2 \left (q+p x^5\right )+a \left (q+p x^5\right )^2} \, dx=\int { \frac {{\left (3 \, p x^{5} - 2 \, q\right )} \sqrt {p x^{5} + q}}{c x^{4} + {\left (p x^{5} + q\right )} b x^{2} + {\left (p x^{5} + q\right )}^{2} a} \,d x } \] Input:

integrate((p*x^5+q)^(1/2)*(3*p*x^5-2*q)/(c*x^4+b*x^2*(p*x^5+q)+a*(p*x^5+q) 
^2),x, algorithm="maxima")
 

Output:

integrate((3*p*x^5 - 2*q)*sqrt(p*x^5 + q)/(c*x^4 + (p*x^5 + q)*b*x^2 + (p* 
x^5 + q)^2*a), x)
 

Giac [F(-1)]

Timed out. \[ \int \frac {\sqrt {q+p x^5} \left (-2 q+3 p x^5\right )}{c x^4+b x^2 \left (q+p x^5\right )+a \left (q+p x^5\right )^2} \, dx=\text {Timed out} \] Input:

integrate((p*x^5+q)^(1/2)*(3*p*x^5-2*q)/(c*x^4+b*x^2*(p*x^5+q)+a*(p*x^5+q) 
^2),x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {q+p x^5} \left (-2 q+3 p x^5\right )}{c x^4+b x^2 \left (q+p x^5\right )+a \left (q+p x^5\right )^2} \, dx=\text {Hanged} \] Input:

int(-((q + p*x^5)^(1/2)*(2*q - 3*p*x^5))/(a*(q + p*x^5)^2 + c*x^4 + b*x^2* 
(q + p*x^5)),x)
 

Output:

\text{Hanged}
 

Reduce [F]

\[ \int \frac {\sqrt {q+p x^5} \left (-2 q+3 p x^5\right )}{c x^4+b x^2 \left (q+p x^5\right )+a \left (q+p x^5\right )^2} \, dx=-2 \left (\int \frac {\sqrt {p \,x^{5}+q}}{a \,p^{2} x^{10}+b p \,x^{7}+2 a p q \,x^{5}+c \,x^{4}+b q \,x^{2}+a \,q^{2}}d x \right ) q +3 \left (\int \frac {\sqrt {p \,x^{5}+q}\, x^{5}}{a \,p^{2} x^{10}+b p \,x^{7}+2 a p q \,x^{5}+c \,x^{4}+b q \,x^{2}+a \,q^{2}}d x \right ) p \] Input:

int((p*x^5+q)^(1/2)*(3*p*x^5-2*q)/(c*x^4+b*x^2*(p*x^5+q)+a*(p*x^5+q)^2),x)
 

Output:

 - 2*int(sqrt(p*x**5 + q)/(a*p**2*x**10 + 2*a*p*q*x**5 + a*q**2 + b*p*x**7 
 + b*q*x**2 + c*x**4),x)*q + 3*int((sqrt(p*x**5 + q)*x**5)/(a*p**2*x**10 + 
 2*a*p*q*x**5 + a*q**2 + b*p*x**7 + b*q*x**2 + c*x**4),x)*p