3.22.67 \(\int \frac {(-q+2 p x^3) \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{x (b x^2+a (q+p x^3)^2)} \, dx\) [2167]

3.22.67.1 Optimal result

Integrand size = 62, antiderivative size = 159 \[ \int \frac {\left (-q+2 p x^3\right ) \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{x \left (b x^2+a \left (q+p x^3\right )^2\right )} \, dx=\frac {\sqrt {b+2 a p q} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {b+2 a p q} x^2}{a q^2+b x^2+2 a p q x^3+a p^2 x^6+\left (a q+a p x^3\right ) \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}}\right )}{a \sqrt {b}}-\frac {\log (x)}{a}+\frac {\log \left (q+p x^3+\sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}\right )}{a} \]

output

(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
 

3.22.67.2 Mathematica [A] (verified)

Time = 0.63 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.82 \[ \int \frac {\left (-q+2 p x^3\right ) \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{x \left (b x^2+a \left (q+p x^3\right )^2\right )} \, dx=-\frac {-\frac {\sqrt {b+2 a p q} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {b+2 a p q} x^2}{b x^2+a \left (q+p x^3\right ) \left (q+p x^3+\sqrt {q^2+2 p q (-1+x) x^2+p^2 x^6}\right )}\right )}{\sqrt {b}}+\log (x)-\log \left (q+p x^3+\sqrt {q^2+2 p q (-1+x) x^2+p^2 x^6}\right )}{a} \]

input

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]
 

output

-((-((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)
 

3.22.67.3 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.

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

\(\Big \downarrow \) 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

\(\displaystyle 4 p \int \frac {x^2 \sqrt {p^2 x^6+2 p q x^3-2 p q x^2+q^2}}{a p^2 x^6+2 a p q x^3+b x^2+a q^2}dx+\frac {b \int \frac {x \sqrt {p^2 x^6+2 p q x^3-2 p q x^2+q^2}}{a p^2 x^6+2 a p q x^3+b x^2+a q^2}dx}{a q}+\frac {p^2 \int \frac {x^5 \sqrt {p^2 x^6+2 p q x^3-2 p q x^2+q^2}}{a p^2 x^6+2 a p q x^3+b x^2+a q^2}dx}{q}-\frac {\int \frac {\sqrt {p^2 x^6+2 p q x^3-2 p q x^2+q^2}}{x}dx}{a q}\)

input

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]
 

output

$Aborted
 

3.22.67.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

3.22.67.4 Maple [A] (verified)

Time = 0.28 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.69

method	result	size
pseudoelliptic	\(-\frac {-\sqrt {-a b}\, \ln \left (\frac {q +p \,x^{3}+\sqrt {p^{2} x^{6}+2 p q \,x^{2} \left (-1+x \right )+q^{2}}}{x}\right ) \sqrt {\frac {-2 a p q -b}{a}}+\left (\ln \left (\frac {\sqrt {p^{2} x^{6}+2 p q \,x^{2} \left (-1+x \right )+q^{2}}\, \sqrt {\frac {-2 a p q -b}{a}}\, a +\left (-p \,x^{3}-q \right ) \sqrt {-a b}-2 q a p x}{\left (p \,x^{3}+q \right ) a +\sqrt {-a b}\, x}\right )-\ln \left (\frac {\sqrt {p^{2} x^{6}+2 p q \,x^{2} \left (-1+x \right )+q^{2}}\, \sqrt {\frac {-2 a p q -b}{a}}\, a +\left (p \,x^{3}+q \right ) \sqrt {-a b}-2 q a p x}{\left (p \,x^{3}+q \right ) a -\sqrt {-a b}\, x}\right )\right ) \left (a p q +\frac {b}{2}\right )}{\sqrt {\frac {-2 a p q -b}{a}}\, \sqrt {-a b}\, a}\)	\(268\)

input

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)
 

output

-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
 

3.22.67.5 Fricas [F(-1)]

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

input

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")
 

output

Timed out
 

3.22.67.6 Sympy [F]

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

input

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)
 

output

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)
 

3.22.67.7 Maxima [F]

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

input

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")
 

output

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)
 

3.22.67.8 Giac [F]

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

input

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")
 

output

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)
 

3.22.67.9 Mupad [F(-1)]

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

input

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)
 

output

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)