3.24.76 \(\int \frac {\sqrt {q+p x^5} (-2 q+3 p x^5)}{c x^4+b x^2 (q+p x^5)+a (q+p x^5)^2} \, dx\) [2376]

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

[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^5+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^5+q)^(1/2))/a^(1/2)/(-4*a*c+b^2)^(1/2)

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Rubi [F]
time = 0.94, 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 [A]
time = 6.91, size = 151, normalized size = 0.80 \begin {gather*} \frac {\sqrt {2} \left (\sqrt {b-\sqrt {b^2-4 a c}} \text {ArcTan}\left (\frac {\sqrt {b-\sqrt {b^2-4 a c}} x}{\sqrt {2} \sqrt {a} \sqrt {q+p x^5}}\right )-\sqrt {b+\sqrt {b^2-4 a c}} \text {ArcTan}\left (\frac {\sqrt {b+\sqrt {b^2-4 a c}} x}{\sqrt {2} \sqrt {a} \sqrt {q+p x^5}}\right )\right )}{\sqrt {a} \sqrt {b^2-4 a c}} \end {gather*}

Antiderivative was successfully verified.

[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]

(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[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])]))/(S
qrt[a]*Sqrt[b^2 - 4*a*c])

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Maple [F]
time = 8.34, 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*} \text {Failed to integrate} \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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1321 vs. \(2 (134) = 268\).
time = 50.72, size = 1321, normalized size = 6.99 \begin {gather*} -\frac {1}{4} \, \sqrt {2} \sqrt {-\frac {b + \frac {a b^{2} - 4 \, a^{2} c}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}}{a b^{2} - 4 \, a^{2} c}} \log \left (\frac {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} {\left ({\left (b^{2} - 4 \, a c\right )} x^{3} - \frac {2 \, {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} p x^{6} + {\left (a b^{3} - 4 \, a^{2} b c\right )} x^{3} + 2 \, {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} q x}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}\right )} \sqrt {-\frac {b + \frac {a b^{2} - 4 \, a^{2} c}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}}{a b^{2} - 4 \, a^{2} c}} - \frac {2 \, {\left ({\left (a b^{2} - 4 \, a^{2} c\right )} p x^{7} + {\left (a b^{2} - 4 \, a^{2} c\right )} q x^{2}\right )}}{\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}}\right ) + \frac {1}{4} \, \sqrt {2} \sqrt {-\frac {b + \frac {a b^{2} - 4 \, a^{2} c}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}}{a b^{2} - 4 \, a^{2} c}} \log \left (\frac {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} {\left ({\left (b^{2} - 4 \, a c\right )} x^{3} - \frac {2 \, {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} p x^{6} + {\left (a b^{3} - 4 \, a^{2} b c\right )} x^{3} + 2 \, {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} q x}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}\right )} \sqrt {-\frac {b + \frac {a b^{2} - 4 \, a^{2} c}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}}{a b^{2} - 4 \, a^{2} c}} - \frac {2 \, {\left ({\left (a b^{2} - 4 \, a^{2} c\right )} p x^{7} + {\left (a b^{2} - 4 \, a^{2} c\right )} q x^{2}\right )}}{\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}}\right ) - \frac {1}{4} \, \sqrt {2} \sqrt {-\frac {b - \frac {a b^{2} - 4 \, a^{2} c}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}}{a b^{2} - 4 \, a^{2} c}} \log \left (\frac {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} {\left ({\left (b^{2} - 4 \, a c\right )} x^{3} + \frac {2 \, {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} p x^{6} + {\left (a b^{3} - 4 \, a^{2} b c\right )} x^{3} + 2 \, {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} q x}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}\right )} \sqrt {-\frac {b - \frac {a b^{2} - 4 \, a^{2} c}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}}{a b^{2} - 4 \, a^{2} c}} + \frac {2 \, {\left ({\left (a b^{2} - 4 \, a^{2} c\right )} p x^{7} + {\left (a b^{2} - 4 \, a^{2} c\right )} q x^{2}\right )}}{\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}}\right ) + \frac {1}{4} \, \sqrt {2} \sqrt {-\frac {b - \frac {a b^{2} - 4 \, a^{2} c}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}}{a b^{2} - 4 \, a^{2} c}} \log \left (\frac {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} {\left ({\left (b^{2} - 4 \, a c\right )} x^{3} + \frac {2 \, {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} p x^{6} + {\left (a b^{3} - 4 \, a^{2} b c\right )} x^{3} + 2 \, {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} q x}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}\right )} \sqrt {-\frac {b - \frac {a b^{2} - 4 \, a^{2} c}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}}{a b^{2} - 4 \, a^{2} c}} + \frac {2 \, {\left ({\left (a b^{2} - 4 \, a^{2} c\right )} p x^{7} + {\left (a b^{2} - 4 \, a^{2} c\right )} q x^{2}\right )}}{\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}}\right ) \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="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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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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Giac [F(-1)] Timed out
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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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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