3.26.57 \(\int \frac {(-q+2 p x^3) \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6} (b q x+c x^2+b p x^4+a (q+p x^3)^2)}{x^5} \, dx\) [2557]

3.26.57.1 Optimal result

Integrand size = 70, antiderivative size = 215 \[ \int \frac {\left (-q+2 p x^3\right ) \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6} \left (b q x+c x^2+b p x^4+a \left (q+p x^3\right )^2\right )}{x^5} \, dx=\frac {\sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6} \left (3 a q^3+4 b q^2 x+6 c q x^2-3 a p q^2 x^2-8 b p q x^3+9 a p q^2 x^3+8 b p q x^4+6 c p x^5-3 a p^2 q x^5+9 a p^2 q x^6+4 b p^2 x^7+3 a p^3 x^9\right )}{12 x^4}+\frac {1}{2} \left (2 c p q+a p^2 q^2\right ) \log (x)+\frac {1}{2} \left (-2 c p q-a p^2 q^2\right ) \log \left (q+p x^3+\sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}\right ) \]

output

1/12*(p^2*x^6+2*p*q*x^3-2*p*q*x^2+q^2)^(1/2)*(3*a*p^3*x^9+9*a*p^2*q*x^6+4* 
b*p^2*x^7-3*a*p^2*q*x^5+9*a*p*q^2*x^3+8*b*p*q*x^4+6*c*p*x^5-3*a*p*q^2*x^2- 
8*b*p*q*x^3+3*a*q^3+4*b*q^2*x+6*c*q*x^2)/x^4+1/2*(a*p^2*q^2+2*c*p*q)*ln(x) 
+1/2*(-a*p^2*q^2-2*c*p*q)*ln(q+p*x^3+(p^2*x^6+2*p*q*x^3-2*p*q*x^2+q^2)^(1/ 
2))
 

3.26.57.2 Mathematica [A] (verified)

Time = 0.54 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.82 \[ \int \frac {\left (-q+2 p x^3\right ) \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6} \left (b q x+c x^2+b p x^4+a \left (q+p x^3\right )^2\right )}{x^5} \, dx=\frac {\sqrt {q^2+2 p q (-1+x) x^2+p^2 x^6} \left (3 a \left (q^3+p^3 x^9+p q^2 x^2 (-1+3 x)+p^2 q x^5 (-1+3 x)\right )+2 x \left (3 c x \left (q+p x^3\right )+2 b \left (q^2+2 p q (-1+x) x^2+p^2 x^6\right )\right )\right )}{12 x^4}+\frac {1}{2} p q (2 c+a p q) \log (x)-\frac {1}{2} p q (2 c+a p q) \log \left (q+p x^3+\sqrt {q^2+2 p q (-1+x) x^2+p^2 x^6}\right ) \]

input

Integrate[((-q + 2*p*x^3)*Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^6]*(b*q 
*x + c*x^2 + b*p*x^4 + a*(q + p*x^3)^2))/x^5,x]
 

output

(Sqrt[q^2 + 2*p*q*(-1 + x)*x^2 + p^2*x^6]*(3*a*(q^3 + p^3*x^9 + p*q^2*x^2* 
(-1 + 3*x) + p^2*q*x^5*(-1 + 3*x)) + 2*x*(3*c*x*(q + p*x^3) + 2*b*(q^2 + 2 
*p*q*(-1 + x)*x^2 + p^2*x^6))))/(12*x^4) + (p*q*(2*c + a*p*q)*Log[x])/2 - 
(p*q*(2*c + a*p*q)*Log[q + p*x^3 + Sqrt[q^2 + 2*p*q*(-1 + x)*x^2 + p^2*x^6 
]])/2
 

3.26.57.3 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.

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

input

Int[((-q + 2*p*x^3)*Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^6]*(b*q*x + c 
*x^2 + b*p*x^4 + a*(q + p*x^3)^2))/x^5,x]
 

output

$Aborted
 

3.26.57.3.1 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] 
]
 

3.26.57.4 Maple [A] (verified)

Time = 0.28 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.82

method	result	size
pseudoelliptic	\(\frac {\left (3 a \,q^{3}+\left (9 a p \,x^{3}-3 a p \,x^{2}+4 b x \right ) q^{2}+\left (9 a \,p^{2} x^{6}-3 a \,p^{2} x^{5}+8 b p \,x^{4}-8 b p \,x^{3}+6 c \,x^{2}\right ) q +3 a \,p^{3} x^{9}+4 b \,p^{2} x^{7}+6 c p \,x^{5}\right ) \sqrt {p^{2} x^{6}+2 p q \,x^{2} \left (-1+x \right )+q^{2}}-6 p q \,x^{4} \left (a p q +2 c \right ) \ln \left (\frac {q +p \,x^{3}+\sqrt {p^{2} x^{6}+2 p q \,x^{2} \left (-1+x \right )+q^{2}}}{x}\right )}{12 x^{4}}\)	\(177\)

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

output

1/12*((3*a*q^3+(9*a*p*x^3-3*a*p*x^2+4*b*x)*q^2+(9*a*p^2*x^6-3*a*p^2*x^5+8* 
b*p*x^4-8*b*p*x^3+6*c*x^2)*q+3*a*p^3*x^9+4*b*p^2*x^7+6*c*p*x^5)*(p^2*x^6+2 
*p*q*x^2*(-1+x)+q^2)^(1/2)-6*p*q*x^4*(a*p*q+2*c)*ln((q+p*x^3+(p^2*x^6+2*p* 
q*x^2*(-1+x)+q^2)^(1/2))/x))/x^4
 

3.26.57.5 Fricas [F(-1)]

Timed out. \[ \int \frac {\left (-q+2 p x^3\right ) \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6} \left (b q x+c x^2+b p x^4+a \left (q+p x^3\right )^2\right )}{x^5} \, dx=\text {Timed out} \]

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

output

Timed out
 

3.26.57.6 Sympy [F]

\[ \int \frac {\left (-q+2 p x^3\right ) \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6} \left (b q x+c x^2+b p x^4+a \left (q+p x^3\right )^2\right )}{x^5} \, dx=\int \frac {\left (2 p x^{3} - q\right ) \sqrt {p^{2} x^{6} + 2 p q x^{3} - 2 p q x^{2} + q^{2}} \left (a p^{2} x^{6} + 2 a p q x^{3} + a q^{2} + b p x^{4} + b q x + c x^{2}\right )}{x^{5}}\, dx \]

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)*(b*q* 
x+c*x**2+b*p*x**4+a*(p*x**3+q)**2)/x**5,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)*( 
a*p**2*x**6 + 2*a*p*q*x**3 + a*q**2 + b*p*x**4 + b*q*x + c*x**2)/x**5, x)
 

3.26.57.7 Maxima [F]

\[ \int \frac {\left (-q+2 p x^3\right ) \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6} \left (b q x+c x^2+b p x^4+a \left (q+p x^3\right )^2\right )}{x^5} \, dx=\int { \frac {\sqrt {p^{2} x^{6} + 2 \, p q x^{3} - 2 \, p q x^{2} + q^{2}} {\left (b p x^{4} + {\left (p x^{3} + q\right )}^{2} a + b q x + c x^{2}\right )} {\left (2 \, p x^{3} - q\right )}}{x^{5}} \,d x } \]

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

output

integrate(sqrt(p^2*x^6 + 2*p*q*x^3 - 2*p*q*x^2 + q^2)*(b*p*x^4 + (p*x^3 + 
q)^2*a + b*q*x + c*x^2)*(2*p*x^3 - q)/x^5, x)
 

3.26.57.8 Giac [F]

\[ \int \frac {\left (-q+2 p x^3\right ) \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6} \left (b q x+c x^2+b p x^4+a \left (q+p x^3\right )^2\right )}{x^5} \, dx=\int { \frac {\sqrt {p^{2} x^{6} + 2 \, p q x^{3} - 2 \, p q x^{2} + q^{2}} {\left (b p x^{4} + {\left (p x^{3} + q\right )}^{2} a + b q x + c x^{2}\right )} {\left (2 \, p x^{3} - q\right )}}{x^{5}} \,d x } \]

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

output

integrate(sqrt(p^2*x^6 + 2*p*q*x^3 - 2*p*q*x^2 + q^2)*(b*p*x^4 + (p*x^3 + 
q)^2*a + b*q*x + c*x^2)*(2*p*x^3 - q)/x^5, x)
 

3.26.57.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\left (-q+2 p x^3\right ) \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6} \left (b q x+c x^2+b p x^4+a \left (q+p x^3\right )^2\right )}{x^5} \, dx=-\int \frac {\left (q-2\,p\,x^3\right )\,\left (a\,{\left (p\,x^3+q\right )}^2+c\,x^2+b\,p\,x^4+b\,q\,x\right )\,\sqrt {p^2\,x^6+2\,p\,q\,x^3-2\,p\,q\,x^2+q^2}}{x^5} \,d x \]

input

int(-((q - 2*p*x^3)*(a*(q + p*x^3)^2 + c*x^2 + b*p*x^4 + b*q*x)*(p^2*x^6 + 
 q^2 - 2*p*q*x^2 + 2*p*q*x^3)^(1/2))/x^5,x)
 

output

-int(((q - 2*p*x^3)*(a*(q + p*x^3)^2 + c*x^2 + b*p*x^4 + b*q*x)*(p^2*x^6 + 
 q^2 - 2*p*q*x^2 + 2*p*q*x^3)^(1/2))/x^5, x)