\(\int \frac {(-2 q+p x^6) \sqrt {q+p x^6}}{x^3 (a q+b x^4+a p x^6)} \, dx\) [730]

Optimal result

Integrand size = 41, antiderivative size = 56 \[ \int \frac {\left (-2 q+p x^6\right ) \sqrt {q+p x^6}}{x^3 \left (a q+b x^4+a p x^6\right )} \, dx=\frac {\sqrt {q+p x^6}}{a x^2}+\frac {\sqrt {b} \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a} \sqrt {q+p x^6}}\right )}{a^{3/2}} \] Output:

(p*x^6+q)^(1/2)/a/x^2+b^(1/2)*arctan(b^(1/2)*x^2/a^(1/2)/(p*x^6+q)^(1/2))/ 
a^(3/2)
 

Mathematica [A] (verified)

Time = 9.02 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-2 q+p x^6\right ) \sqrt {q+p x^6}}{x^3 \left (a q+b x^4+a p x^6\right )} \, dx=\frac {\sqrt {q+p x^6}}{a x^2}+\frac {\sqrt {b} \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a} \sqrt {q+p x^6}}\right )}{a^{3/2}} \] Input:

Integrate[((-2*q + p*x^6)*Sqrt[q + p*x^6])/(x^3*(a*q + b*x^4 + a*p*x^6)),x 
]
 

Output:

Sqrt[q + p*x^6]/(a*x^2) + (Sqrt[b]*ArcTan[(Sqrt[b]*x^2)/(Sqrt[a]*Sqrt[q + 
p*x^6])])/a^(3/2)
 

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.

\(\displaystyle \int \frac {\left (p x^6-2 q\right ) \sqrt {p x^6+q}}{x^3 \left (a p x^6+a q+b x^4\right )} \, dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {x \sqrt {p x^6+q} \left (3 a p x^2+2 b\right )}{a \left (a p x^6+a q+b x^4\right )}-\frac {2 \sqrt {p x^6+q}}{a x^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {b \text {Subst}\left (\int \frac {\sqrt {p x^3+q}}{a p x^3+b x^2+a q}dx,x,x^2\right )}{a}+\frac {3}{2} p \text {Subst}\left (\int \frac {x \sqrt {p x^3+q}}{a p x^3+b x^2+a q}dx,x,x^2\right )-\frac {\sqrt {2} 3^{3/4} \sqrt [3]{p} \sqrt [3]{q} \left (\sqrt [3]{p} x^2+\sqrt [3]{q}\right ) \sqrt {\frac {p^{2/3} x^4-\sqrt [3]{p} \sqrt [3]{q} x^2+q^{2/3}}{\left (\sqrt [3]{p} x^2+\left (1+\sqrt {3}\right ) \sqrt [3]{q}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{p} x^2+\left (1-\sqrt {3}\right ) \sqrt [3]{q}}{\sqrt [3]{p} x^2+\left (1+\sqrt {3}\right ) \sqrt [3]{q}}\right ),-7-4 \sqrt {3}\right )}{a \sqrt {\frac {\sqrt [3]{q} \left (\sqrt [3]{p} x^2+\sqrt [3]{q}\right )}{\left (\sqrt [3]{p} x^2+\left (1+\sqrt {3}\right ) \sqrt [3]{q}\right )^2}} \sqrt {p x^6+q}}+\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{p} \sqrt [3]{q} \left (\sqrt [3]{p} x^2+\sqrt [3]{q}\right ) \sqrt {\frac {p^{2/3} x^4-\sqrt [3]{p} \sqrt [3]{q} x^2+q^{2/3}}{\left (\sqrt [3]{p} x^2+\left (1+\sqrt {3}\right ) \sqrt [3]{q}\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{p} x^2+\left (1-\sqrt {3}\right ) \sqrt [3]{q}}{\sqrt [3]{p} x^2+\left (1+\sqrt {3}\right ) \sqrt [3]{q}}\right )|-7-4 \sqrt {3}\right )}{2 a \sqrt {\frac {\sqrt [3]{q} \left (\sqrt [3]{p} x^2+\sqrt [3]{q}\right )}{\left (\sqrt [3]{p} x^2+\left (1+\sqrt {3}\right ) \sqrt [3]{q}\right )^2}} \sqrt {p x^6+q}}-\frac {3 \sqrt [3]{p} \sqrt {p x^6+q}}{a \left (\sqrt [3]{p} x^2+\left (1+\sqrt {3}\right ) \sqrt [3]{q}\right )}+\frac {\sqrt {p x^6+q}}{a x^2}\)

Input:

Int[((-2*q + p*x^6)*Sqrt[q + p*x^6])/(x^3*(a*q + b*x^4 + a*p*x^6)),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 1.52 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.84

method	result	size
pseudoelliptic	\(\frac {\frac {\sqrt {p \,x^{6}+q}}{x^{2}}-\frac {b \arctan \left (\frac {a \sqrt {p \,x^{6}+q}}{x^{2} \sqrt {a b}}\right )}{\sqrt {a b}}}{a}\)	\(47\)

Input:

int((p*x^6-2*q)*(p*x^6+q)^(1/2)/x^3/(a*p*x^6+b*x^4+a*q),x,method=_RETURNVE 
RBOSE)
 

Output:

1/a*((p*x^6+q)^(1/2)/x^2-b/(a*b)^(1/2)*arctan(a*(p*x^6+q)^(1/2)/x^2/(a*b)^ 
(1/2)))
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\left (-2 q+p x^6\right ) \sqrt {q+p x^6}}{x^3 \left (a q+b x^4+a p x^6\right )} \, dx=\text {Timed out} \] Input:

integrate((p*x^6-2*q)*(p*x^6+q)^(1/2)/x^3/(a*p*x^6+b*x^4+a*q),x, algorithm 
="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (-2 q+p x^6\right ) \sqrt {q+p x^6}}{x^3 \left (a q+b x^4+a p x^6\right )} \, dx=\text {Timed out} \] Input:

integrate((p*x**6-2*q)*(p*x**6+q)**(1/2)/x**3/(a*p*x**6+b*x**4+a*q),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (-2 q+p x^6\right ) \sqrt {q+p x^6}}{x^3 \left (a q+b x^4+a p x^6\right )} \, dx=\int { \frac {\sqrt {p x^{6} + q} {\left (p x^{6} - 2 \, q\right )}}{{\left (a p x^{6} + b x^{4} + a q\right )} x^{3}} \,d x } \] Input:

integrate((p*x^6-2*q)*(p*x^6+q)^(1/2)/x^3/(a*p*x^6+b*x^4+a*q),x, algorithm 
="maxima")
 

Output:

integrate(sqrt(p*x^6 + q)*(p*x^6 - 2*q)/((a*p*x^6 + b*x^4 + a*q)*x^3), x)
 

Giac [F]

\[ \int \frac {\left (-2 q+p x^6\right ) \sqrt {q+p x^6}}{x^3 \left (a q+b x^4+a p x^6\right )} \, dx=\int { \frac {\sqrt {p x^{6} + q} {\left (p x^{6} - 2 \, q\right )}}{{\left (a p x^{6} + b x^{4} + a q\right )} x^{3}} \,d x } \] Input:

integrate((p*x^6-2*q)*(p*x^6+q)^(1/2)/x^3/(a*p*x^6+b*x^4+a*q),x, algorithm 
="giac")
 

Output:

integrate(sqrt(p*x^6 + q)*(p*x^6 - 2*q)/((a*p*x^6 + b*x^4 + a*q)*x^3), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (-2 q+p x^6\right ) \sqrt {q+p x^6}}{x^3 \left (a q+b x^4+a p x^6\right )} \, dx=\int -\frac {\sqrt {p\,x^6+q}\,\left (2\,q-p\,x^6\right )}{x^3\,\left (a\,p\,x^6+b\,x^4+a\,q\right )} \,d x \] Input:

int(-((q + p*x^6)^(1/2)*(2*q - p*x^6))/(x^3*(a*q + b*x^4 + a*p*x^6)),x)
                                                                                    
                                                                                    
 

Output:

int(-((q + p*x^6)^(1/2)*(2*q - p*x^6))/(x^3*(a*q + b*x^4 + a*p*x^6)), x)
 

Reduce [F]

\[ \int \frac {\left (-2 q+p x^6\right ) \sqrt {q+p x^6}}{x^3 \left (a q+b x^4+a p x^6\right )} \, dx=\frac {\sqrt {p \,x^{6}+q}-\left (\int \frac {\sqrt {p \,x^{6}+q}\, x^{7}}{a \,p^{2} x^{12}+b p \,x^{10}+2 a p q \,x^{6}+b q \,x^{4}+a \,q^{2}}d x \right ) b p \,x^{2}+2 \left (\int \frac {\sqrt {p \,x^{6}+q}\, x}{a \,p^{2} x^{12}+b p \,x^{10}+2 a p q \,x^{6}+b q \,x^{4}+a \,q^{2}}d x \right ) b q \,x^{2}}{a \,x^{2}} \] Input:

int((p*x^6-2*q)*(p*x^6+q)^(1/2)/x^3/(a*p*x^6+b*x^4+a*q),x)
 

Output:

(sqrt(p*x**6 + q) - int((sqrt(p*x**6 + q)*x**7)/(a*p**2*x**12 + 2*a*p*q*x* 
*6 + a*q**2 + b*p*x**10 + b*q*x**4),x)*b*p*x**2 + 2*int((sqrt(p*x**6 + q)* 
x)/(a*p**2*x**12 + 2*a*p*q*x**6 + a*q**2 + b*p*x**10 + b*q*x**4),x)*b*q*x* 
*2)/(a*x**2)