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

Optimal result

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

[Out]

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))

Rubi [F]

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

[In]

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]

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \left (2 c p \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}-\frac {a q^3 \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{x^5}-\frac {b q^2 \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{x^4}-\frac {c q \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{x^3}+\frac {b p q \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{x}+3 a p^2 q x \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}+2 b p^2 x^2 \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}+2 a p^3 x^4 \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}\right ) \, dx \\ & = (2 c p) \int \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6} \, dx+\left (2 b p^2\right ) \int x^2 \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6} \, dx+\left (2 a p^3\right ) \int x^4 \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6} \, dx-(c q) \int \frac {\sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{x^3} \, dx+(b p q) \int \frac {\sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{x} \, dx+\left (3 a p^2 q\right ) \int x \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6} \, dx-\left (b q^2\right ) \int \frac {\sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{x^4} \, dx-\left (a q^3\right ) \int \frac {\sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{x^5} \, dx \\ \end{align*}

Mathematica [A] (verified)

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

[In]

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]

[Out]

(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

Maple [A] (verified)

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

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

[In]

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=_RETU
RNVERBOSE)

[Out]

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

Fricas [F(-1)]

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

[In]

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, algor
ithm="fricas")

[Out]

Timed out

Sympy [F]

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

[In]

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)

[Out]

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)

Maxima [F]

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

[In]

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, algor
ithm="maxima")

[Out]

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)

Giac [F]

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

[In]

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, algor
ithm="giac")

[Out]

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)

Mupad [F(-1)]

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

[In]

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)

[Out]

-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)