\(\int \frac {(-2 q+p x^3) \sqrt {q+p x^3}}{c x^4+b x^2 (q+p x^3)+a (q+p x^3)^2} \, dx\) [2373]

Optimal result

Integrand size = 52, antiderivative size = 189 \[ \int \frac {\left (-2 q+p x^3\right ) \sqrt {q+p x^3}}{c x^4+b x^2 \left (q+p x^3\right )+a \left (q+p x^3\right )^2} \, dx=-\frac {\sqrt {2} \left (-b+\sqrt {b^2-4 a c}\right ) \arctan \left (\frac {\sqrt {b-\sqrt {b^2-4 a c}} x}{\sqrt {2} \sqrt {a} \sqrt {q+p x^3}}\right )}{\sqrt {a} \sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} \sqrt {b+\sqrt {b^2-4 a c}} \arctan \left (\frac {\sqrt {b+\sqrt {b^2-4 a c}} x}{\sqrt {2} \sqrt {a} \sqrt {q+p x^3}}\right )}{\sqrt {a} \sqrt {b^2-4 a c}} \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 1.47 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.80 \[ \int \frac {\left (-2 q+p x^3\right ) \sqrt {q+p x^3}}{c x^4+b x^2 \left (q+p x^3\right )+a \left (q+p x^3\right )^2} \, dx=\frac {\sqrt {2} \left (\sqrt {b-\sqrt {b^2-4 a c}} \arctan \left (\frac {\sqrt {b-\sqrt {b^2-4 a c}} x}{\sqrt {2} \sqrt {a} \sqrt {q+p x^3}}\right )-\sqrt {b+\sqrt {b^2-4 a c}} \arctan \left (\frac {\sqrt {b+\sqrt {b^2-4 a c}} x}{\sqrt {2} \sqrt {a} \sqrt {q+p x^3}}\right )\right )}{\sqrt {a} \sqrt {b^2-4 a c}} \]

[In]

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

[Out]

(Sqrt[2]*(Sqrt[b - Sqrt[b^2 - 4*a*c]]*ArcTan[(Sqrt[b - Sqrt[b^2 - 4*a*c]]*x)/(Sqrt[2]*Sqrt[a]*Sqrt[q + p*x^3])
] - Sqrt[b + Sqrt[b^2 - 4*a*c]]*ArcTan[(Sqrt[b + Sqrt[b^2 - 4*a*c]]*x)/(Sqrt[2]*Sqrt[a]*Sqrt[q + p*x^3])]))/(S
qrt[a]*Sqrt[b^2 - 4*a*c])

Maple [A] (verified)

Time = 1.08 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.79

method	result	size
default	\(\frac {\sqrt {2}\, \left (-\frac {\left (-b +\sqrt {-4 a c +b^{2}}\right ) \operatorname {arctanh}\left (\frac {a \sqrt {p \,x^{3}+q}\, \sqrt {2}}{x \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) a}}\right )}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) a}}+\frac {\left (b +\sqrt {-4 a c +b^{2}}\right ) \arctan \left (\frac {a \sqrt {p \,x^{3}+q}\, \sqrt {2}}{x \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) a}}\right )}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) a}}\right )}{\sqrt {-4 a c +b^{2}}}\)	\(149\)
pseudoelliptic	\(\frac {\sqrt {2}\, \left (-\frac {\left (-b +\sqrt {-4 a c +b^{2}}\right ) \operatorname {arctanh}\left (\frac {a \sqrt {p \,x^{3}+q}\, \sqrt {2}}{x \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) a}}\right )}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) a}}+\frac {\left (b +\sqrt {-4 a c +b^{2}}\right ) \arctan \left (\frac {a \sqrt {p \,x^{3}+q}\, \sqrt {2}}{x \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) a}}\right )}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) a}}\right )}{\sqrt {-4 a c +b^{2}}}\)	\(149\)
elliptic	\(\text {Expression too large to display}\)	\(1362\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1321 vs. \(2 (134) = 268\).

Time = 2.16 (sec) , antiderivative size = 1321, normalized size of antiderivative = 6.99 \[ \int \frac {\left (-2 q+p x^3\right ) \sqrt {q+p x^3}}{c x^4+b x^2 \left (q+p x^3\right )+a \left (q+p x^3\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/4*sqrt(2)*sqrt(-(b + (a*b^2 - 4*a^2*c)/sqrt(a^2*b^2 - 4*a^3*c))/(a*b^2 - 4*a^2*c))*log((2*a*p^2*x^6 + 4*a*p
*q*x^3 - 2*c*x^4 + 2*a*q^2 + sqrt(2)*((b^2 - 4*a*c)*x^3 - (2*(a^2*b^2 - 4*a^3*c)*p*x^4 + (a*b^3 - 4*a^2*b*c)*x
^3 + 2*(a^2*b^2 - 4*a^3*c)*q*x)/sqrt(a^2*b^2 - 4*a^3*c))*sqrt(p*x^3 + q)*sqrt(-(b + (a*b^2 - 4*a^2*c)/sqrt(a^2
*b^2 - 4*a^3*c))/(a*b^2 - 4*a^2*c)) - 2*((a*b^2 - 4*a^2*c)*p*x^5 + (a*b^2 - 4*a^2*c)*q*x^2)/sqrt(a^2*b^2 - 4*a
^3*c))/(a*p^2*x^6 + b*p*x^5 + 2*a*p*q*x^3 + c*x^4 + b*q*x^2 + a*q^2)) + 1/4*sqrt(2)*sqrt(-(b + (a*b^2 - 4*a^2*
c)/sqrt(a^2*b^2 - 4*a^3*c))/(a*b^2 - 4*a^2*c))*log((2*a*p^2*x^6 + 4*a*p*q*x^3 - 2*c*x^4 + 2*a*q^2 - sqrt(2)*((
b^2 - 4*a*c)*x^3 - (2*(a^2*b^2 - 4*a^3*c)*p*x^4 + (a*b^3 - 4*a^2*b*c)*x^3 + 2*(a^2*b^2 - 4*a^3*c)*q*x)/sqrt(a^
2*b^2 - 4*a^3*c))*sqrt(p*x^3 + q)*sqrt(-(b + (a*b^2 - 4*a^2*c)/sqrt(a^2*b^2 - 4*a^3*c))/(a*b^2 - 4*a^2*c)) - 2
*((a*b^2 - 4*a^2*c)*p*x^5 + (a*b^2 - 4*a^2*c)*q*x^2)/sqrt(a^2*b^2 - 4*a^3*c))/(a*p^2*x^6 + b*p*x^5 + 2*a*p*q*x
^3 + c*x^4 + b*q*x^2 + a*q^2)) - 1/4*sqrt(2)*sqrt(-(b - (a*b^2 - 4*a^2*c)/sqrt(a^2*b^2 - 4*a^3*c))/(a*b^2 - 4*
a^2*c))*log((2*a*p^2*x^6 + 4*a*p*q*x^3 - 2*c*x^4 + 2*a*q^2 + sqrt(2)*((b^2 - 4*a*c)*x^3 + (2*(a^2*b^2 - 4*a^3*
c)*p*x^4 + (a*b^3 - 4*a^2*b*c)*x^3 + 2*(a^2*b^2 - 4*a^3*c)*q*x)/sqrt(a^2*b^2 - 4*a^3*c))*sqrt(p*x^3 + q)*sqrt(
-(b - (a*b^2 - 4*a^2*c)/sqrt(a^2*b^2 - 4*a^3*c))/(a*b^2 - 4*a^2*c)) + 2*((a*b^2 - 4*a^2*c)*p*x^5 + (a*b^2 - 4*
a^2*c)*q*x^2)/sqrt(a^2*b^2 - 4*a^3*c))/(a*p^2*x^6 + b*p*x^5 + 2*a*p*q*x^3 + c*x^4 + b*q*x^2 + a*q^2)) + 1/4*sq
rt(2)*sqrt(-(b - (a*b^2 - 4*a^2*c)/sqrt(a^2*b^2 - 4*a^3*c))/(a*b^2 - 4*a^2*c))*log((2*a*p^2*x^6 + 4*a*p*q*x^3
- 2*c*x^4 + 2*a*q^2 - sqrt(2)*((b^2 - 4*a*c)*x^3 + (2*(a^2*b^2 - 4*a^3*c)*p*x^4 + (a*b^3 - 4*a^2*b*c)*x^3 + 2*
(a^2*b^2 - 4*a^3*c)*q*x)/sqrt(a^2*b^2 - 4*a^3*c))*sqrt(p*x^3 + q)*sqrt(-(b - (a*b^2 - 4*a^2*c)/sqrt(a^2*b^2 -
4*a^3*c))/(a*b^2 - 4*a^2*c)) + 2*((a*b^2 - 4*a^2*c)*p*x^5 + (a*b^2 - 4*a^2*c)*q*x^2)/sqrt(a^2*b^2 - 4*a^3*c))/
(a*p^2*x^6 + b*p*x^5 + 2*a*p*q*x^3 + c*x^4 + b*q*x^2 + a*q^2))

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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