3.6.26 \(\int \frac {x^2 \sqrt {q+p x^5} (-2 q+3 p x^5)}{b x^6+a (q+p x^5)^3} \, dx\)

Optimal. Leaf size=40 \[ -\frac {2 \tan ^{-1}\left (\frac {\sqrt {b} x^3}{\sqrt {a} \left (p x^5+q\right )^{3/2}}\right )}{3 \sqrt {a} \sqrt {b}} \]

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Rubi [A]  time = 0.67, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {6714, 205} \begin {gather*} \frac {2 \tan ^{-1}\left (\frac {\sqrt {a} \left (p x^5+q\right )^{3/2}}{\sqrt {b} x^3}\right )}{3 \sqrt {a} \sqrt {b}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*ArcTan[(Sqrt[a]*(q + p*x^5)^(3/2))/(Sqrt[b]*x^3)])/(3*Sqrt[a]*Sqrt[b])

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 6714

Int[(u_)*(v_)^(r_.)*(w_)^(s_.)*((a_.)*(v_)^(p_.) + (b_.)*(w_)^(q_.))^(m_.), x_Symbol] :> With[{c = Simplify[u/
(p*w*D[v, x] - q*v*D[w, x])]}, -Dist[(c*q)/(s + 1), Subst[Int[(a + b*x^(q/(s + 1)))^m, x], x, v^(m*p + r + 1)*
w^(s + 1)], x] /; FreeQ[c, x]] /; FreeQ[{a, b, m, p, q, r, s}, x] && EqQ[p*(s + 1) + q*(m*p + r + 1), 0] && Ne
Q[s, -1] && IntegerQ[q/(s + 1)] && IntegerQ[m]

Rubi steps

\begin {align*} \int \frac {x^2 \sqrt {q+p x^5} \left (-2 q+3 p x^5\right )}{b x^6+a \left (q+p x^5\right )^3} \, dx &=\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{b+a x^2} \, dx,x,\frac {\left (q+p x^5\right )^{3/2}}{x^3}\right )\\ &=\frac {2 \tan ^{-1}\left (\frac {\sqrt {a} \left (q+p x^5\right )^{3/2}}{\sqrt {b} x^3}\right )}{3 \sqrt {a} \sqrt {b}}\\ \end {align*}

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Mathematica [F]  time = 0.37, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^2 \sqrt {q+p x^5} \left (-2 q+3 p x^5\right )}{b x^6+a \left (q+p x^5\right )^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(x^2*Sqrt[q + p*x^5]*(-2*q + 3*p*x^5))/(b*x^6 + a*(q + p*x^5)^3),x]

[Out]

Integrate[(x^2*Sqrt[q + p*x^5]*(-2*q + 3*p*x^5))/(b*x^6 + a*(q + p*x^5)^3), x]

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IntegrateAlgebraic [A]  time = 3.16, size = 40, normalized size = 1.00 \begin {gather*} -\frac {2 \tan ^{-1}\left (\frac {\sqrt {b} x^3}{\sqrt {a} \left (q+p x^5\right )^{3/2}}\right )}{3 \sqrt {a} \sqrt {b}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(x^2*Sqrt[q + p*x^5]*(-2*q + 3*p*x^5))/(b*x^6 + a*(q + p*x^5)^3),x]

[Out]

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

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fricas [B]  time = 1.21, size = 465, normalized size = 11.62 \begin {gather*} \left [-\frac {\sqrt {-a b} \log \left (\frac {a^{2} p^{6} x^{30} + 6 \, a^{2} p^{5} q x^{25} + 15 \, a^{2} p^{4} q^{2} x^{20} - 6 \, a b p^{3} x^{21} + 20 \, a^{2} p^{3} q^{3} x^{15} - 18 \, a b p^{2} q x^{16} + 15 \, a^{2} p^{2} q^{4} x^{10} - 18 \, a b p q^{2} x^{11} + b^{2} x^{12} + 6 \, a^{2} p q^{5} x^{5} - 6 \, a b q^{3} x^{6} + a^{2} q^{6} - 4 \, {\left (a p^{4} x^{23} + 4 \, a p^{3} q x^{18} + 6 \, a p^{2} q^{2} x^{13} - b p x^{14} + 4 \, a p q^{3} x^{8} - b q x^{9} + a q^{4} x^{3}\right )} \sqrt {p x^{5} + q} \sqrt {-a b}}{a^{2} p^{6} x^{30} + 6 \, a^{2} p^{5} q x^{25} + 15 \, a^{2} p^{4} q^{2} x^{20} + 2 \, a b p^{3} x^{21} + 20 \, a^{2} p^{3} q^{3} x^{15} + 6 \, a b p^{2} q x^{16} + 15 \, a^{2} p^{2} q^{4} x^{10} + 6 \, a b p q^{2} x^{11} + b^{2} x^{12} + 6 \, a^{2} p q^{5} x^{5} + 2 \, a b q^{3} x^{6} + a^{2} q^{6}}\right )}{6 \, a b}, \frac {\sqrt {a b} \arctan \left (\frac {{\left (a p^{3} x^{15} + 3 \, a p^{2} q x^{10} + 3 \, a p q^{2} x^{5} - b x^{6} + a q^{3}\right )} \sqrt {p x^{5} + q} \sqrt {a b}}{2 \, {\left (a b p^{2} x^{13} + 2 \, a b p q x^{8} + a b q^{2} x^{3}\right )}}\right )}{3 \, a b}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(p*x^5+q)^(1/2)*(3*p*x^5-2*q)/(b*x^6+a*(p*x^5+q)^3),x, algorithm="fricas")

[Out]

[-1/6*sqrt(-a*b)*log((a^2*p^6*x^30 + 6*a^2*p^5*q*x^25 + 15*a^2*p^4*q^2*x^20 - 6*a*b*p^3*x^21 + 20*a^2*p^3*q^3*
x^15 - 18*a*b*p^2*q*x^16 + 15*a^2*p^2*q^4*x^10 - 18*a*b*p*q^2*x^11 + b^2*x^12 + 6*a^2*p*q^5*x^5 - 6*a*b*q^3*x^
6 + a^2*q^6 - 4*(a*p^4*x^23 + 4*a*p^3*q*x^18 + 6*a*p^2*q^2*x^13 - b*p*x^14 + 4*a*p*q^3*x^8 - b*q*x^9 + a*q^4*x
^3)*sqrt(p*x^5 + q)*sqrt(-a*b))/(a^2*p^6*x^30 + 6*a^2*p^5*q*x^25 + 15*a^2*p^4*q^2*x^20 + 2*a*b*p^3*x^21 + 20*a
^2*p^3*q^3*x^15 + 6*a*b*p^2*q*x^16 + 15*a^2*p^2*q^4*x^10 + 6*a*b*p*q^2*x^11 + b^2*x^12 + 6*a^2*p*q^5*x^5 + 2*a
*b*q^3*x^6 + a^2*q^6))/(a*b), 1/3*sqrt(a*b)*arctan(1/2*(a*p^3*x^15 + 3*a*p^2*q*x^10 + 3*a*p*q^2*x^5 - b*x^6 +
a*q^3)*sqrt(p*x^5 + q)*sqrt(a*b)/(a*b*p^2*x^13 + 2*a*b*p*q*x^8 + a*b*q^2*x^3))/(a*b)]

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giac [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} x^{2}}{b x^{6} + {\left (p x^{5} + q\right )}^{3} a}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(p*x^5+q)^(1/2)*(3*p*x^5-2*q)/(b*x^6+a*(p*x^5+q)^3),x, algorithm="giac")

[Out]

integrate((3*p*x^5 - 2*q)*sqrt(p*x^5 + q)*x^2/(b*x^6 + (p*x^5 + q)^3*a), x)

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {x^{2} \sqrt {p \,x^{5}+q}\, \left (3 p \,x^{5}-2 q \right )}{b \,x^{6}+a \left (p \,x^{5}+q \right )^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(p*x^5+q)^(1/2)*(3*p*x^5-2*q)/(b*x^6+a*(p*x^5+q)^3),x)

[Out]

int(x^2*(p*x^5+q)^(1/2)*(3*p*x^5-2*q)/(b*x^6+a*(p*x^5+q)^3),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} x^{2}}{b x^{6} + {\left (p x^{5} + q\right )}^{3} a}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(p*x^5+q)^(1/2)*(3*p*x^5-2*q)/(b*x^6+a*(p*x^5+q)^3),x, algorithm="maxima")

[Out]

integrate((3*p*x^5 - 2*q)*sqrt(p*x^5 + q)*x^2/(b*x^6 + (p*x^5 + q)^3*a), x)

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mupad [F(-1)]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \text {Hanged} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^2*(q + p*x^5)^(1/2)*(2*q - 3*p*x^5))/(a*(q + p*x^5)^3 + b*x^6),x)

[Out]

\text{Hanged}

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} \sqrt {p x^{5} + q} \left (3 p x^{5} - 2 q\right )}{a p^{3} x^{15} + 3 a p^{2} q x^{10} + 3 a p q^{2} x^{5} + a q^{3} + b x^{6}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(p*x**5+q)**(1/2)*(3*p*x**5-2*q)/(b*x**6+a*(p*x**5+q)**3),x)

[Out]

Integral(x**2*sqrt(p*x**5 + q)*(3*p*x**5 - 2*q)/(a*p**3*x**15 + 3*a*p**2*q*x**10 + 3*a*p*q**2*x**5 + a*q**3 +
b*x**6), x)

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