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

Optimal result

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

[Out]

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))

Rubi [F]

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

[In]

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]

[Out]

a*p^3*Defer[Int][Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6], x] - 2*a*q^3*Defer[Int][Sqrt[q^2 + 2*p*q*x^3 - 2
*p*q*x^4 + p^2*x^6]/x^9, x] - 3*a*p*q^2*Defer[Int][Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6]/x^6, x] - 2*b*q
*Defer[Int][Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6]/x^5, x] + b*p*Defer[Int][Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*
x^4 + p^2*x^6]/x^2, x]

Rubi steps \begin{align*} \text {integral}& = \int \left (a p^3 \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}-\frac {2 a q^3 \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^9}-\frac {3 a p q^2 \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^6}-\frac {2 b q \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^5}+\frac {b p \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^2}\right ) \, dx \\ & = (b p) \int \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^2} \, dx+\left (a p^3\right ) \int \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6} \, dx-(2 b q) \int \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^5} \, dx-\left (3 a p q^2\right ) \int \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^6} \, dx-\left (2 a q^3\right ) \int \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^9} \, dx \\ \end{align*}

Mathematica [A] (verified)

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

[In]

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]

[Out]

((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

Maple [A] (verified)

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

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

[In]

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=_RETU
RNVERBOSE)

[Out]

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

Fricas [F(-1)]

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

[In]

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, algor
ithm="fricas")

[Out]

Timed out

Sympy [F]

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

[In]

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)

[Out]

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)

Maxima [F]

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

[In]

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, algor
ithm="maxima")

[Out]

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)

Giac [F]

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

[In]

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, algor
ithm="giac")

[Out]

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)

Mupad [F(-1)]

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

[In]

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)

[Out]

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)