\(\int \frac {(-2 q+p x^3) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^3 (a q+b x^2+a p x^3)} \, dx\) [2323]

Optimal result

Integrand size = 59, antiderivative size = 180 \[ \int \frac {\left (-2 q+p x^3\right ) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^3 \left (a q+b x^2+a p x^3\right )} \, dx=\frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{a x^2}+\frac {2 \sqrt {-b^2+2 a^2 p q} \arctan \left (\frac {\sqrt {-b^2+2 a^2 p q} x^2}{a q+b x^2+a p x^3+a \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}\right )}{a^2}+\frac {2 b \log (x)}{a^2}-\frac {b \log \left (q+p x^3+\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}\right )}{a^2} \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.77 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.93 \[ \int \frac {\left (-2 q+p x^3\right ) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^3 \left (a q+b x^2+a p x^3\right )} \, dx=\frac {a \sqrt {q^2-2 p q (-1+x) x^3+p^2 x^6}+2 \sqrt {-b^2+2 a^2 p q} x^2 \arctan \left (\frac {\sqrt {-b^2+2 a^2 p q} x^2}{b x^2+a \left (q+p x^3+\sqrt {q^2-2 p q (-1+x) x^3+p^2 x^6}\right )}\right )+2 b x^2 \log (x)-b x^2 \log \left (q+p x^3+\sqrt {q^2-2 p q (-1+x) x^3+p^2 x^6}\right )}{a^2 x^2} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.28 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.35

method	result	size
pseudoelliptic	\(\frac {\sqrt {\frac {p^{2} x^{6}-2 p q \,x^{3} \left (-1+x \right )+q^{2}}{x^{2}}}\, a^{2} \sqrt {\frac {-2 a^{2} p q +b^{2}}{a^{2}}}+2 x \left (-\frac {b \ln \left (\frac {p \,x^{3}+\sqrt {\frac {p^{2} x^{6}-2 p q \,x^{3} \left (-1+x \right )+q^{2}}{x^{2}}}\, x +q}{x^{2}}\right ) a \sqrt {\frac {-2 a^{2} p q +b^{2}}{a^{2}}}}{2}+\left (\ln \left (\frac {-2 a p q \,x^{2}-b p \,x^{3}+\sqrt {\frac {p^{2} x^{6}-2 p q \,x^{3} \left (-1+x \right )+q^{2}}{x^{2}}}\, \sqrt {\frac {-2 a^{2} p q +b^{2}}{a^{2}}}\, a x -q b}{a p \,x^{3}+b \,x^{2}+a q}\right )+\ln \left (2\right )\right ) \left (a^{2} p q -\frac {b^{2}}{2}\right )\right )}{\sqrt {\frac {-2 a^{2} p q +b^{2}}{a^{2}}}\, x \,a^{3}}\)	\(243\)

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

[Out]

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

Fricas [F(-1)]

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

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

[Out]

Timed out

Sympy [F(-1)]

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

[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)/x**3/(a*p*x**3+b*x**2+a*q),x)

[Out]

Timed out

Maxima [F]

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

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

[Out]

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

Giac [F]

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

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

[Out]

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

Mupad [F(-1)]

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

[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))/(x^3*(a*q + b*x^2 + a*p*x^3)),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))/(x^3*(a*q + b*x^2 + a*p*x^3)), x)