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

Optimal. Leaf size=189 \[ -\frac {\sqrt {2} \left (\sqrt {b^2-4 a c}-b\right ) \tan ^{-1}\left (\frac {x \sqrt {b-\sqrt {b^2-4 a c}}}{\sqrt {2} \sqrt {a} \sqrt {p x^3+q}}\right )}{\sqrt {a} \sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} \sqrt {\sqrt {b^2-4 a c}+b} \tan ^{-1}\left (\frac {x \sqrt {\sqrt {b^2-4 a c}+b}}{\sqrt {2} \sqrt {a} \sqrt {p x^3+q}}\right )}{\sqrt {a} \sqrt {b^2-4 a c}} \]

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Rubi [F]  time = 1.34, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \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 \end {gather*}

Verification is not applicable to the result.

[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*} \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 \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*}

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Mathematica [C]  time = 6.68, size = 16759, normalized size = 88.67 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[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]

Result too large to show

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IntegrateAlgebraic [A]  time = 1.65, size = 189, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {2} \left (\sqrt {b^2-4 a c}-b\right ) \tan ^{-1}\left (\frac {x \sqrt {b-\sqrt {b^2-4 a c}}}{\sqrt {2} \sqrt {a} \sqrt {p x^3+q}}\right )}{\sqrt {a} \sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} \sqrt {\sqrt {b^2-4 a c}+b} \tan ^{-1}\left (\frac {x \sqrt {\sqrt {b^2-4 a c}+b}}{\sqrt {2} \sqrt {a} \sqrt {p x^3+q}}\right )}{\sqrt {a} \sqrt {b^2-4 a c}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-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]*(-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[a]*Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]])) - (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[a]*Sqrt[b^2 - 4*a*c])

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fricas [B]  time = 4.53, size = 1321, normalized size = 6.99

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[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))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {sage}_{0} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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maple [C]  time = 0.45, size = 1362, normalized size = 7.21

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[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)

[Out]

-2/3*I/a*3^(1/2)/p*(-q*p^2)^(1/3)*(I*(x+1/2/p*(-q*p^2)^(1/3)-1/2*I*3^(1/2)/p*(-q*p^2)^(1/3))*3^(1/2)*p/(-q*p^2
)^(1/3))^(1/2)*((x-1/p*(-q*p^2)^(1/3))/(-3/2/p*(-q*p^2)^(1/3)+1/2*I*3^(1/2)/p*(-q*p^2)^(1/3)))^(1/2)*(-I*(x+1/
2/p*(-q*p^2)^(1/3)+1/2*I*3^(1/2)/p*(-q*p^2)^(1/3))*3^(1/2)*p/(-q*p^2)^(1/3))^(1/2)/(p*x^3+q)^(1/2)*EllipticF(1
/3*3^(1/2)*(I*(x+1/2/p*(-q*p^2)^(1/3)-1/2*I*3^(1/2)/p*(-q*p^2)^(1/3))*3^(1/2)*p/(-q*p^2)^(1/3))^(1/2),(I*3^(1/
2)/p*(-q*p^2)^(1/3)/(-3/2/p*(-q*p^2)^(1/3)+1/2*I*3^(1/2)/p*(-q*p^2)^(1/3)))^(1/2))-I/a/p^2/q^2/c*2^(1/2)*sum((
_alpha^5*b*p+3*_alpha^3*a*p*q+_alpha^4*c+_alpha^2*b*q+3*a*q^2)/_alpha/(6*_alpha^4*a*p^2+5*_alpha^3*b*p+6*_alph
a*a*p*q+4*_alpha^2*c+2*b*q)*(-q*p^2)^(1/3)*(1/2*I*p*(2*x+1/p*(-I*3^(1/2)*(-q*p^2)^(1/3)+(-q*p^2)^(1/3)))/(-q*p
^2)^(1/3))^(1/2)*(p*(x-1/p*(-q*p^2)^(1/3))/(-3*(-q*p^2)^(1/3)+I*3^(1/2)*(-q*p^2)^(1/3)))^(1/2)*(-1/2*I*p*(2*x+
1/p*(I*3^(1/2)*(-q*p^2)^(1/3)+(-q*p^2)^(1/3)))/(-q*p^2)^(1/3))^(1/2)/(p*x^3+q)^(1/2)*(2*q*p^2*(_alpha^4*a*p^2+
_alpha^3*b*p+_alpha*a*p*q+_alpha^2*c)+I*(-q*p^2)^(1/3)*3^(1/2)*a*q^2*p^2+I*(-q*p^2)^(2/3)*3^(1/2)*_alpha^2*a*q
*p^2+(-q*p^2)^(2/3)*_alpha^5*a*p^3+I*(-q*p^2)^(1/3)*p^3*3^(1/2)*_alpha^3*a*q+I*(-q*p^2)^(2/3)*3^(1/2)*_alpha^4
*b*p^2+I*(-q*p^2)^(2/3)*3^(1/2)*_alpha^3*p*c+(-q*p^2)^(2/3)*_alpha^4*b*p^2-I*(-q*p^2)^(2/3)*3^(1/2)*q*c-(-q*p^
2)^(1/3)*_alpha^3*a*p^3*q+(-q*p^2)^(2/3)*_alpha^2*a*p^2*q+I*(-q*p^2)^(2/3)*p^3*3^(1/2)*_alpha^5*a+(-q*p^2)^(2/
3)*_alpha^3*c*p+I*(-q*p^2)^(1/3)*3^(1/2)*_alpha*q*p*c-(-q*p^2)^(1/3)*_alpha^2*b*p^2*q+I*(-q*p^2)^(1/3)*3^(1/2)
*_alpha^2*b*q*p^2-(-q*p^2)^(1/3)*a*p^2*q^2-(-q*p^2)^(1/3)*_alpha*c*p*q-(-q*p^2)^(2/3)*c*q)*EllipticPi(1/3*3^(1
/2)*(I*(x+1/2/p*(-q*p^2)^(1/3)-1/2*I*3^(1/2)/p*(-q*p^2)^(1/3))*3^(1/2)*p/(-q*p^2)^(1/3))^(1/2),1/2/p/q*(I*3^(1
/2)*_alpha^3*c*p^2+I*(-q*p^2)^(2/3)*3^(1/2)*_alpha*c-2*I*(-q*p^2)^(1/3)*3^(1/2)*_alpha^3*b*p^2+I*(-q*p^2)^(2/3
)*3^(1/2)*a*p*q-3*p^4*_alpha^5*a-I*3^(1/2)*c*p*q+I*3^(1/2)*_alpha^5*a*p^4+I*(-q*p^2)^(2/3)*3^(1/2)*_alpha^2*b*
p+3*(-q*p^2)^(2/3)*_alpha^3*a*p^2-2*I*(-q*p^2)^(1/3)*3^(1/2)*_alpha^4*a*p^3+I*(-q*p^2)^(2/3)*3^(1/2)*_alpha^3*
a*p^2-3*b*p^3*_alpha^4+I*3^(1/2)*_alpha^4*b*p^3-2*I*(-q*p^2)^(1/3)*3^(1/2)*_alpha^2*c*p-3*p^3*_alpha^2*a*q-2*I
*(-q*p^2)^(1/3)*3^(1/2)*_alpha*a*p^2*q+3*(-q*p^2)^(2/3)*_alpha^2*b*p-3*_alpha^3*p^2*c+3*(-q*p^2)^(2/3)*a*q*p+I
*3^(1/2)*_alpha^2*a*p^3*q+3*(-q*p^2)^(2/3)*_alpha*c+3*q*c*p)/c,(I*3^(1/2)/p*(-q*p^2)^(1/3)/(-3/2/p*(-q*p^2)^(1
/3)+1/2*I*3^(1/2)/p*(-q*p^2)^(1/3)))^(1/2)),_alpha=RootOf(_Z^6*a*p^2+_Z^5*b*p+2*_Z^3*a*p*q+_Z^4*c+_Z^2*b*q+a*q
^2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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