3.24.15 \(\int \frac {(-2 q+p x^3) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6} (a q^2+2 a p q x^3+b x^4+a p^2 x^6)}{x^9} \, dx\) [2315]

3.24.15.1 Optimal result

Integrand size = 69, antiderivative size = 178 \[ \int \frac {\left (-2 q+p x^3\right ) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6} \left (a q^2+2 a p q x^3+b x^4+a p^2 x^6\right )}{x^9} \, dx=\frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6} \left (a q^3+3 a p q^2 x^3+2 b q x^4-a p q^2 x^4+3 a p^2 q x^6+2 b p x^7-a p^2 q x^7+a p^3 x^9\right )}{4 x^8}+\left (2 b p q+a p^2 q^2\right ) \log (x)+\frac {1}{2} \left (-2 b p q-a p^2 q^2\right ) \log \left (q+p x^3+\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}\right ) \]

output

1/4*(p^2*x^6-2*p*q*x^4+2*p*q*x^3+q^2)^(1/2)*(a*p^3*x^9-a*p^2*q*x^7+3*a*p^2 
*q*x^6+2*b*p*x^7-a*p*q^2*x^4+3*a*p*q^2*x^3+2*b*q*x^4+a*q^3)/x^8+(a*p^2*q^2 
+2*b*p*q)*ln(x)+1/2*(-a*p^2*q^2-2*b*p*q)*ln(q+p*x^3+(p^2*x^6-2*p*q*x^4+2*p 
*q*x^3+q^2)^(1/2))
 

3.24.15.2 Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.73 \[ \int \frac {\left (-2 q+p x^3\right ) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6} \left (a q^2+2 a p q x^3+b x^4+a p^2 x^6\right )}{x^9} \, dx=\frac {\left (q+p x^3\right ) \sqrt {q^2-2 p q (-1+x) x^3+p^2 x^6} \left (2 b x^4+a \left (q^2-p q (-2+x) x^3+p^2 x^6\right )\right )}{4 x^8}+p q (2 b+a p q) \log (x)-\frac {1}{2} p q (2 b+a p q) \log \left (q+p x^3+\sqrt {q^2-2 p q (-1+x) x^3+p^2 x^6}\right ) \]

input

Integrate[((-2*q + p*x^3)*Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6]*(a*q 
^2 + 2*a*p*q*x^3 + b*x^4 + a*p^2*x^6))/x^9,x]
 

output

((q + p*x^3)*Sqrt[q^2 - 2*p*q*(-1 + x)*x^3 + p^2*x^6]*(2*b*x^4 + a*(q^2 - 
p*q*(-2 + x)*x^3 + p^2*x^6)))/(4*x^8) + p*q*(2*b + a*p*q)*Log[x] - (p*q*(2 
*b + a*p*q)*Log[q + p*x^3 + Sqrt[q^2 - 2*p*q*(-1 + x)*x^3 + p^2*x^6]])/2
 

3.24.15.3 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.

\(\displaystyle \int \frac {\left (p x^3-2 q\right ) \sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2} \left (a p^2 x^6+2 a p q x^3+a q^2+b x^4\right )}{x^9} \, dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {3 a p q^2 \sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2}}{x^6}-\frac {2 a q^3 \sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2}}{x^9}+a p^3 \sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2}-\frac {2 b q \sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2}}{x^5}+\frac {b p \sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2}}{x^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -3 a p q^2 \int \frac {\sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2}}{x^6}dx-2 a q^3 \int \frac {\sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2}}{x^9}dx+a p^3 \int \sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2}dx-2 b q \int \frac {\sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2}}{x^5}dx+b p \int \frac {\sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2}}{x^2}dx\)

input

Int[((-2*q + p*x^3)*Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6]*(a*q^2 + 2 
*a*p*q*x^3 + b*x^4 + a*p^2*x^6))/x^9,x]
 

output

$Aborted
 

3.24.15.3.1 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] 
]
 

3.24.15.4 Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.75

method	result	size
pseudoelliptic	\(\frac {-2 p q \,x^{7} \left (a p q +2 b \right ) \ln \left (\frac {p \,x^{3}+\sqrt {\frac {p^{2} x^{6}-2 p q \,x^{3} \left (-1+x \right )+q^{2}}{x^{2}}}\, x +q}{x^{2}}\right )+\left (p \,x^{3}+q \right ) \left (a \,p^{2} x^{6}+\left (-a p q +2 b \right ) x^{4}+2 a p q \,x^{3}+a \,q^{2}\right ) \sqrt {\frac {p^{2} x^{6}-2 p q \,x^{3} \left (-1+x \right )+q^{2}}{x^{2}}}}{4 x^{7}}\)	\(133\)

input

int((p*x^3-2*q)*(p^2*x^6-2*p*q*x^4+2*p*q*x^3+q^2)^(1/2)*(a*p^2*x^6+2*a*p*q 
*x^3+b*x^4+a*q^2)/x^9,x,method=_RETURNVERBOSE)
 

output

1/4*(-2*p*q*x^7*(a*p*q+2*b)*ln((p*x^3+((p^2*x^6-2*p*q*x^3*(-1+x)+q^2)/x^2) 
^(1/2)*x+q)/x^2)+(p*x^3+q)*(a*p^2*x^6+(-a*p*q+2*b)*x^4+2*a*p*q*x^3+a*q^2)* 
((p^2*x^6-2*p*q*x^3*(-1+x)+q^2)/x^2)^(1/2))/x^7
 

3.24.15.5 Fricas [F(-1)]

Timed out. \[ \int \frac {\left (-2 q+p x^3\right ) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6} \left (a q^2+2 a p q x^3+b x^4+a p^2 x^6\right )}{x^9} \, dx=\text {Timed out} \]

input

integrate((p*x^3-2*q)*(p^2*x^6-2*p*q*x^4+2*p*q*x^3+q^2)^(1/2)*(a*p^2*x^6+2 
*a*p*q*x^3+b*x^4+a*q^2)/x^9,x, algorithm="fricas")
 

output

Timed out
 

3.24.15.6 Sympy [F]

\[ \int \frac {\left (-2 q+p x^3\right ) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6} \left (a q^2+2 a p q x^3+b x^4+a p^2 x^6\right )}{x^9} \, dx=\int \frac {\left (p x^{3} - 2 q\right ) \sqrt {p^{2} x^{6} - 2 p q x^{4} + 2 p q x^{3} + q^{2}} \left (a p^{2} x^{6} + 2 a p q x^{3} + a q^{2} + b x^{4}\right )}{x^{9}}\, dx \]

input

integrate((p*x**3-2*q)*(p**2*x**6-2*p*q*x**4+2*p*q*x**3+q**2)**(1/2)*(a*p* 
*2*x**6+2*a*p*q*x**3+b*x**4+a*q**2)/x**9,x)
 

output

Integral((p*x**3 - 2*q)*sqrt(p**2*x**6 - 2*p*q*x**4 + 2*p*q*x**3 + q**2)*( 
a*p**2*x**6 + 2*a*p*q*x**3 + a*q**2 + b*x**4)/x**9, x)
 

3.24.15.7 Maxima [F]

\[ \int \frac {\left (-2 q+p x^3\right ) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6} \left (a q^2+2 a p q x^3+b x^4+a p^2 x^6\right )}{x^9} \, dx=\int { \frac {{\left (a p^{2} x^{6} + 2 \, a p q x^{3} + b x^{4} + a q^{2}\right )} \sqrt {p^{2} x^{6} - 2 \, p q x^{4} + 2 \, p q x^{3} + q^{2}} {\left (p x^{3} - 2 \, q\right )}}{x^{9}} \,d x } \]

input

integrate((p*x^3-2*q)*(p^2*x^6-2*p*q*x^4+2*p*q*x^3+q^2)^(1/2)*(a*p^2*x^6+2 
*a*p*q*x^3+b*x^4+a*q^2)/x^9,x, algorithm="maxima")
 

output

integrate((a*p^2*x^6 + 2*a*p*q*x^3 + b*x^4 + a*q^2)*sqrt(p^2*x^6 - 2*p*q*x 
^4 + 2*p*q*x^3 + q^2)*(p*x^3 - 2*q)/x^9, x)
 

3.24.15.8 Giac [F]

\[ \int \frac {\left (-2 q+p x^3\right ) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6} \left (a q^2+2 a p q x^3+b x^4+a p^2 x^6\right )}{x^9} \, dx=\int { \frac {{\left (a p^{2} x^{6} + 2 \, a p q x^{3} + b x^{4} + a q^{2}\right )} \sqrt {p^{2} x^{6} - 2 \, p q x^{4} + 2 \, p q x^{3} + q^{2}} {\left (p x^{3} - 2 \, q\right )}}{x^{9}} \,d x } \]

input

integrate((p*x^3-2*q)*(p^2*x^6-2*p*q*x^4+2*p*q*x^3+q^2)^(1/2)*(a*p^2*x^6+2 
*a*p*q*x^3+b*x^4+a*q^2)/x^9,x, algorithm="giac")
 

output

integrate((a*p^2*x^6 + 2*a*p*q*x^3 + b*x^4 + a*q^2)*sqrt(p^2*x^6 - 2*p*q*x 
^4 + 2*p*q*x^3 + q^2)*(p*x^3 - 2*q)/x^9, x)
 

3.24.15.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\left (-2 q+p x^3\right ) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6} \left (a q^2+2 a p q x^3+b x^4+a p^2 x^6\right )}{x^9} \, dx=\int -\frac {\left (2\,q-p\,x^3\right )\,\sqrt {p^2\,x^6-2\,p\,q\,x^4+2\,p\,q\,x^3+q^2}\,\left (a\,p^2\,x^6+2\,a\,p\,q\,x^3+a\,q^2+b\,x^4\right )}{x^9} \,d x \]

input

int(-((2*q - p*x^3)*(p^2*x^6 + q^2 + 2*p*q*x^3 - 2*p*q*x^4)^(1/2)*(a*q^2 + 
 b*x^4 + a*p^2*x^6 + 2*a*p*q*x^3))/x^9,x)
 

output

int(-((2*q - p*x^3)*(p^2*x^6 + q^2 + 2*p*q*x^3 - 2*p*q*x^4)^(1/2)*(a*q^2 + 
 b*x^4 + a*p^2*x^6 + 2*a*p*q*x^3))/x^9, x)