3.23.81 \(\int \frac {x (-2 q+p x^6) \sqrt {q+p x^6}}{b x^8+a (q+p x^6)^2} \, dx\)

Optimal. Leaf size=173 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^2 \sqrt {p x^6+q}}{\sqrt {a} p x^6+\sqrt {a} q-\sqrt {b} x^4}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b}}-\frac {\tanh ^{-1}\left (\frac {\frac {\sqrt [4]{a} p x^6}{\sqrt {2} \sqrt [4]{b}}+\frac {\sqrt [4]{a} q}{\sqrt {2} \sqrt [4]{b}}+\frac {\sqrt [4]{b} x^4}{\sqrt {2} \sqrt [4]{a}}}{x^2 \sqrt {p x^6+q}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.59, antiderivative size = 255, normalized size of antiderivative = 1.47, number of steps used = 10, number of rules used = 7, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {6714, 211, 1165, 628, 1162, 617, 204} \begin {gather*} \frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x^2}{\sqrt [4]{a} \sqrt {p x^6+q}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} x^2}{\sqrt [4]{a} \sqrt {p x^6+q}}+1\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b}}+\frac {\log \left (-\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^2}{\sqrt {p x^6+q}}+\sqrt {a}+\frac {\sqrt {b} x^4}{p x^6+q}\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{b}}-\frac {\log \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^2}{\sqrt {p x^6+q}}+\sqrt {a}+\frac {\sqrt {b} x^4}{p x^6+q}\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x*(-2*q + p*x^6)*Sqrt[q + p*x^6])/(b*x^8 + a*(q + p*x^6)^2),x]

[Out]

ArcTan[1 - (Sqrt[2]*b^(1/4)*x^2)/(a^(1/4)*Sqrt[q + p*x^6])]/(2*Sqrt[2]*a^(3/4)*b^(1/4)) - ArcTan[1 + (Sqrt[2]*
b^(1/4)*x^2)/(a^(1/4)*Sqrt[q + p*x^6])]/(2*Sqrt[2]*a^(3/4)*b^(1/4)) + Log[Sqrt[a] + (Sqrt[b]*x^4)/(q + p*x^6)
- (Sqrt[2]*a^(1/4)*b^(1/4)*x^2)/Sqrt[q + p*x^6]]/(4*Sqrt[2]*a^(3/4)*b^(1/4)) - Log[Sqrt[a] + (Sqrt[b]*x^4)/(q
+ p*x^6) + (Sqrt[2]*a^(1/4)*b^(1/4)*x^2)/Sqrt[q + p*x^6]]/(4*Sqrt[2]*a^(3/4)*b^(1/4))

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 211

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[
{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b
]]))

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

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 \left (-2 q+p x^6\right ) \sqrt {q+p x^6}}{b x^8+a \left (q+p x^6\right )^2} \, dx &=-\operatorname {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,\frac {x^2}{\sqrt {q+p x^6}}\right )\\ &=-\frac {\operatorname {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\frac {x^2}{\sqrt {q+p x^6}}\right )}{2 \sqrt {a}}-\frac {\operatorname {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\frac {x^2}{\sqrt {q+p x^6}}\right )}{2 \sqrt {a}}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\frac {x^2}{\sqrt {q+p x^6}}\right )}{4 \sqrt {a} \sqrt {b}}-\frac {\operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\frac {x^2}{\sqrt {q+p x^6}}\right )}{4 \sqrt {a} \sqrt {b}}+\frac {\operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\frac {x^2}{\sqrt {q+p x^6}}\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{b}}+\frac {\operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\frac {x^2}{\sqrt {q+p x^6}}\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{b}}\\ &=\frac {\log \left (\sqrt {a}+\frac {\sqrt {b} x^4}{q+p x^6}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^2}{\sqrt {q+p x^6}}\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{b}}-\frac {\log \left (\sqrt {a}+\frac {\sqrt {b} x^4}{q+p x^6}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^2}{\sqrt {q+p x^6}}\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{b}}-\frac {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} x^2}{\sqrt [4]{a} \sqrt {q+p x^6}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b}}+\frac {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} x^2}{\sqrt [4]{a} \sqrt {q+p x^6}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b}}\\ &=\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x^2}{\sqrt [4]{a} \sqrt {q+p x^6}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b}}-\frac {\tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} x^2}{\sqrt [4]{a} \sqrt {q+p x^6}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b}}+\frac {\log \left (\sqrt {a}+\frac {\sqrt {b} x^4}{q+p x^6}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^2}{\sqrt {q+p x^6}}\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{b}}-\frac {\log \left (\sqrt {a}+\frac {\sqrt {b} x^4}{q+p x^6}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^2}{\sqrt {q+p x^6}}\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{b}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [F]  time = 0.36, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \left (-2 q+p x^6\right ) \sqrt {q+p x^6}}{b x^8+a \left (q+p x^6\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(x*(-2*q + p*x^6)*Sqrt[q + p*x^6])/(b*x^8 + a*(q + p*x^6)^2),x]

[Out]

Integrate[(x*(-2*q + p*x^6)*Sqrt[q + p*x^6])/(b*x^8 + a*(q + p*x^6)^2), x]

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 27.69, size = 173, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^2 \sqrt {q+p x^6}}{\sqrt {a} q-\sqrt {b} x^4+\sqrt {a} p x^6}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b}}-\frac {\tanh ^{-1}\left (\frac {\frac {\sqrt [4]{a} q}{\sqrt {2} \sqrt [4]{b}}+\frac {\sqrt [4]{b} x^4}{\sqrt {2} \sqrt [4]{a}}+\frac {\sqrt [4]{a} p x^6}{\sqrt {2} \sqrt [4]{b}}}{x^2 \sqrt {q+p x^6}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(x*(-2*q + p*x^6)*Sqrt[q + p*x^6])/(b*x^8 + a*(q + p*x^6)^2),x]

[Out]

-1/2*ArcTan[(Sqrt[2]*a^(1/4)*b^(1/4)*x^2*Sqrt[q + p*x^6])/(Sqrt[a]*q - Sqrt[b]*x^4 + Sqrt[a]*p*x^6)]/(Sqrt[2]*
a^(3/4)*b^(1/4)) - ArcTanh[((a^(1/4)*q)/(Sqrt[2]*b^(1/4)) + (b^(1/4)*x^4)/(Sqrt[2]*a^(1/4)) + (a^(1/4)*p*x^6)/
(Sqrt[2]*b^(1/4)))/(x^2*Sqrt[q + p*x^6])]/(2*Sqrt[2]*a^(3/4)*b^(1/4))

________________________________________________________________________________________

fricas [B]  time = 1.27, size = 361, normalized size = 2.09 \begin {gather*} \frac {1}{2} \, \left (-\frac {1}{a^{3} b}\right )^{\frac {1}{4}} \arctan \left (\frac {a^{2} b x^{2} \left (-\frac {1}{a^{3} b}\right )^{\frac {3}{4}}}{\sqrt {p x^{6} + q}}\right ) + \frac {1}{8} \, \left (-\frac {1}{a^{3} b}\right )^{\frac {1}{4}} \log \left (\frac {a p^{2} x^{12} + 2 \, a p q x^{6} - b x^{8} + a q^{2} + 2 \, {\left (a b x^{6} \left (-\frac {1}{a^{3} b}\right )^{\frac {1}{4}} + {\left (a^{3} b p x^{8} + a^{3} b q x^{2}\right )} \left (-\frac {1}{a^{3} b}\right )^{\frac {3}{4}}\right )} \sqrt {p x^{6} + q} - 2 \, {\left (a^{2} b p x^{10} + a^{2} b q x^{4}\right )} \sqrt {-\frac {1}{a^{3} b}}}{a p^{2} x^{12} + 2 \, a p q x^{6} + b x^{8} + a q^{2}}\right ) - \frac {1}{8} \, \left (-\frac {1}{a^{3} b}\right )^{\frac {1}{4}} \log \left (\frac {a p^{2} x^{12} + 2 \, a p q x^{6} - b x^{8} + a q^{2} - 2 \, {\left (a b x^{6} \left (-\frac {1}{a^{3} b}\right )^{\frac {1}{4}} + {\left (a^{3} b p x^{8} + a^{3} b q x^{2}\right )} \left (-\frac {1}{a^{3} b}\right )^{\frac {3}{4}}\right )} \sqrt {p x^{6} + q} - 2 \, {\left (a^{2} b p x^{10} + a^{2} b q x^{4}\right )} \sqrt {-\frac {1}{a^{3} b}}}{a p^{2} x^{12} + 2 \, a p q x^{6} + b x^{8} + a q^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(p*x^6-2*q)*(p*x^6+q)^(1/2)/(b*x^8+a*(p*x^6+q)^2),x, algorithm="fricas")

[Out]

1/2*(-1/(a^3*b))^(1/4)*arctan(a^2*b*x^2*(-1/(a^3*b))^(3/4)/sqrt(p*x^6 + q)) + 1/8*(-1/(a^3*b))^(1/4)*log((a*p^
2*x^12 + 2*a*p*q*x^6 - b*x^8 + a*q^2 + 2*(a*b*x^6*(-1/(a^3*b))^(1/4) + (a^3*b*p*x^8 + a^3*b*q*x^2)*(-1/(a^3*b)
)^(3/4))*sqrt(p*x^6 + q) - 2*(a^2*b*p*x^10 + a^2*b*q*x^4)*sqrt(-1/(a^3*b)))/(a*p^2*x^12 + 2*a*p*q*x^6 + b*x^8
+ a*q^2)) - 1/8*(-1/(a^3*b))^(1/4)*log((a*p^2*x^12 + 2*a*p*q*x^6 - b*x^8 + a*q^2 - 2*(a*b*x^6*(-1/(a^3*b))^(1/
4) + (a^3*b*p*x^8 + a^3*b*q*x^2)*(-1/(a^3*b))^(3/4))*sqrt(p*x^6 + q) - 2*(a^2*b*p*x^10 + a^2*b*q*x^4)*sqrt(-1/
(a^3*b)))/(a*p^2*x^12 + 2*a*p*q*x^6 + b*x^8 + a*q^2))

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {p x^{6} + q} {\left (p x^{6} - 2 \, q\right )} x}{b x^{8} + {\left (p x^{6} + q\right )}^{2} a}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(p*x^6-2*q)*(p*x^6+q)^(1/2)/(b*x^8+a*(p*x^6+q)^2),x, algorithm="giac")

[Out]

integrate(sqrt(p*x^6 + q)*(p*x^6 - 2*q)*x/(b*x^8 + (p*x^6 + q)^2*a), x)

________________________________________________________________________________________

maple [F]  time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {x \left (p \,x^{6}-2 q \right ) \sqrt {p \,x^{6}+q}}{b \,x^{8}+a \left (p \,x^{6}+q \right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(p*x^6-2*q)*(p*x^6+q)^(1/2)/(b*x^8+a*(p*x^6+q)^2),x)

[Out]

int(x*(p*x^6-2*q)*(p*x^6+q)^(1/2)/(b*x^8+a*(p*x^6+q)^2),x)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {p x^{6} + q} {\left (p x^{6} - 2 \, q\right )} x}{b x^{8} + {\left (p x^{6} + q\right )}^{2} a}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(p*x^6-2*q)*(p*x^6+q)^(1/2)/(b*x^8+a*(p*x^6+q)^2),x, algorithm="maxima")

[Out]

integrate(sqrt(p*x^6 + q)*(p*x^6 - 2*q)*x/(b*x^8 + (p*x^6 + q)^2*a), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {x\,\sqrt {p\,x^6+q}\,\left (2\,q-p\,x^6\right )}{a\,{\left (p\,x^6+q\right )}^2+b\,x^8} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x*(q + p*x^6)^(1/2)*(2*q - p*x^6))/(a*(q + p*x^6)^2 + b*x^8),x)

[Out]

int(-(x*(q + p*x^6)^(1/2)*(2*q - p*x^6))/(a*(q + p*x^6)^2 + b*x^8), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \left (p x^{6} - 2 q\right ) \sqrt {p x^{6} + q}}{a p^{2} x^{12} + 2 a p q x^{6} + a q^{2} + b x^{8}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(p*x**6-2*q)*(p*x**6+q)**(1/2)/(b*x**8+a*(p*x**6+q)**2),x)

[Out]

Integral(x*(p*x**6 - 2*q)*sqrt(p*x**6 + q)/(a*p**2*x**12 + 2*a*p*q*x**6 + a*q**2 + b*x**8), x)

________________________________________________________________________________________