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

Optimal. Leaf size=189 \[ -\frac {\sqrt {2} \left (\sqrt {b^2-4 a c}-b\right ) \tan ^{-1}\left (\frac {x \sqrt {b-\sqrt {b^2-4 a c}}}{\sqrt {2} \sqrt {a} \sqrt {p x^5+q}}\right )}{\sqrt {a} \sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} \sqrt {\sqrt {b^2-4 a c}+b} \tan ^{-1}\left (\frac {x \sqrt {\sqrt {b^2-4 a c}+b}}{\sqrt {2} \sqrt {a} \sqrt {p x^5+q}}\right )}{\sqrt {a} \sqrt {b^2-4 a c}} \]

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Rubi [F]  time = 1.37, 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 {\sqrt {q+p x^5} \left (-2 q+3 p x^5\right )}{c x^4+b x^2 \left (q+p x^5\right )+a \left (q+p x^5\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

-2*q*Defer[Int][Sqrt[q + p*x^5]/(a*q^2 + b*q*x^2 + c*x^4 + 2*a*p*q*x^5 + b*p*x^7 + a*p^2*x^10), x] + 3*p*Defer
[Int][(x^5*Sqrt[q + p*x^5])/(a*q^2 + b*q*x^2 + c*x^4 + 2*a*p*q*x^5 + b*p*x^7 + a*p^2*x^10), x]

Rubi steps

\begin {align*} \int \frac {\sqrt {q+p x^5} \left (-2 q+3 p x^5\right )}{c x^4+b x^2 \left (q+p x^5\right )+a \left (q+p x^5\right )^2} \, dx &=\int \left (-\frac {2 q \sqrt {q+p x^5}}{a q^2+b q x^2+c x^4+2 a p q x^5+b p x^7+a p^2 x^{10}}+\frac {3 p x^5 \sqrt {q+p x^5}}{a q^2+b q x^2+c x^4+2 a p q x^5+b p x^7+a p^2 x^{10}}\right ) \, dx\\ &=(3 p) \int \frac {x^5 \sqrt {q+p x^5}}{a q^2+b q x^2+c x^4+2 a p q x^5+b p x^7+a p^2 x^{10}} \, dx-(2 q) \int \frac {\sqrt {q+p x^5}}{a q^2+b q x^2+c x^4+2 a p q x^5+b p x^7+a p^2 x^{10}} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.41, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {q+p x^5} \left (-2 q+3 p x^5\right )}{c x^4+b x^2 \left (q+p x^5\right )+a \left (q+p x^5\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(Sqrt[q + p*x^5]*(-2*q + 3*p*x^5))/(c*x^4 + b*x^2*(q + p*x^5) + a*(q + p*x^5)^2),x]

[Out]

Integrate[(Sqrt[q + p*x^5]*(-2*q + 3*p*x^5))/(c*x^4 + b*x^2*(q + p*x^5) + a*(q + p*x^5)^2), x]

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IntegrateAlgebraic [A]  time = 13.36, size = 189, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {2} \left (-b+\sqrt {b^2-4 a c}\right ) \tan ^{-1}\left (\frac {\sqrt {b-\sqrt {b^2-4 a c}} x}{\sqrt {2} \sqrt {a} \sqrt {q+p x^5}}\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}} \tan ^{-1}\left (\frac {\sqrt {b+\sqrt {b^2-4 a c}} x}{\sqrt {2} \sqrt {a} \sqrt {q+p x^5}}\right )}{\sqrt {a} \sqrt {b^2-4 a c}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(Sqrt[q + p*x^5]*(-2*q + 3*p*x^5))/(c*x^4 + b*x^2*(q + p*x^5) + a*(q + p*x^5)^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^5])])
/(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^5])])/(Sqrt[a]*Sqrt[b^2 - 4*a*c])

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fricas [B]  time = 91.41, 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^5+q)^(1/2)*(3*p*x^5-2*q)/(c*x^4+b*x^2*(p*x^5+q)+a*(p*x^5+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^10 + 4*a*
p*q*x^5 - 2*c*x^4 + 2*a*q^2 + sqrt(2)*sqrt(p*x^5 + q)*((b^2 - 4*a*c)*x^3 - (2*(a^2*b^2 - 4*a^3*c)*p*x^6 + (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(-(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^7 + (a*b^2 - 4*a^2*c)*q*x^2)/sqrt(a^2*b^2 - 4*
a^3*c))/(a*p^2*x^10 + b*p*x^7 + 2*a*p*q*x^5 + 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^10 + 4*a*p*q*x^5 - 2*c*x^4 + 2*a*q^2 - sqrt(2)
*sqrt(p*x^5 + q)*((b^2 - 4*a*c)*x^3 - (2*(a^2*b^2 - 4*a^3*c)*p*x^6 + (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(-(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^7 + (a*b^2 - 4*a^2*c)*q*x^2)/sqrt(a^2*b^2 - 4*a^3*c))/(a*p^2*x^10 + b*p*x^7 + 2*a*p
*q*x^5 + 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^10 + 4*a*p*q*x^5 - 2*c*x^4 + 2*a*q^2 + sqrt(2)*sqrt(p*x^5 + q)*((b^2 - 4*a*c)*x^3 +
(2*(a^2*b^2 - 4*a^3*c)*p*x^6 + (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(-(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^7 + (a*b^2
- 4*a^2*c)*q*x^2)/sqrt(a^2*b^2 - 4*a^3*c))/(a*p^2*x^10 + b*p*x^7 + 2*a*p*q*x^5 + 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^10 + 4*a*p
*q*x^5 - 2*c*x^4 + 2*a*q^2 - sqrt(2)*sqrt(p*x^5 + q)*((b^2 - 4*a*c)*x^3 + (2*(a^2*b^2 - 4*a^3*c)*p*x^6 + (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(-(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^7 + (a*b^2 - 4*a^2*c)*q*x^2)/sqrt(a^2*b^2 - 4*a
^3*c))/(a*p^2*x^10 + b*p*x^7 + 2*a*p*q*x^5 + c*x^4 + b*q*x^2 + a*q^2))

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((p*x^5+q)^(1/2)*(3*p*x^5-2*q)/(c*x^4+b*x^2*(p*x^5+q)+a*(p*x^5+q)^2),x, algorithm="giac")

[Out]

Timed out

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maple [F]  time = 10.17, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {p \,x^{5}+q}\, \left (3 p \,x^{5}-2 q \right )}{c \,x^{4}+b \,x^{2} \left (p \,x^{5}+q \right )+a \left (p \,x^{5}+q \right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((p*x^5+q)^(1/2)*(3*p*x^5-2*q)/(c*x^4+b*x^2*(p*x^5+q)+a*(p*x^5+q)^2),x)

[Out]

int((p*x^5+q)^(1/2)*(3*p*x^5-2*q)/(c*x^4+b*x^2*(p*x^5+q)+a*(p*x^5+q)^2),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (3 \, p x^{5} - 2 \, q\right )} \sqrt {p x^{5} + q}}{c x^{4} + {\left (p x^{5} + q\right )} b x^{2} + {\left (p x^{5} + q\right )}^{2} a}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((p*x^5+q)^(1/2)*(3*p*x^5-2*q)/(c*x^4+b*x^2*(p*x^5+q)+a*(p*x^5+q)^2),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((q + p*x^5)^(1/2)*(2*q - 3*p*x^5))/(a*(q + p*x^5)^2 + c*x^4 + b*x^2*(q + p*x^5)),x)

[Out]

\text{Hanged}

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {p x^{5} + q} \left (3 p x^{5} - 2 q\right )}{a p^{2} x^{10} + 2 a p q x^{5} + a q^{2} + b p x^{7} + b q x^{2} + c x^{4}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((p*x**5+q)**(1/2)*(3*p*x**5-2*q)/(c*x**4+b*x**2*(p*x**5+q)+a*(p*x**5+q)**2),x)

[Out]

Integral(sqrt(p*x**5 + q)*(3*p*x**5 - 2*q)/(a*p**2*x**10 + 2*a*p*q*x**5 + a*q**2 + b*p*x**7 + b*q*x**2 + c*x**
4), x)

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