3.38 \(\int \frac{a+b x+c x^2}{x^3 \sqrt{-1+d x} \sqrt{1+d x}} \, dx\)

Optimal. Leaf size=83 \[ \frac{1}{2} \left (a d^2+2 c\right ) \tan ^{-1}\left (\sqrt{d x-1} \sqrt{d x+1}\right )+\frac{a \sqrt{d x-1} \sqrt{d x+1}}{2 x^2}+\frac{b \sqrt{d x-1} \sqrt{d x+1}}{x} \]

[Out]

(a*Sqrt[-1 + d*x]*Sqrt[1 + d*x])/(2*x^2) + (b*Sqrt[-1 + d*x]*Sqrt[1 + d*x])/x + ((2*c + a*d^2)*ArcTan[Sqrt[-1
+ d*x]*Sqrt[1 + d*x]])/2

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Rubi [A]  time = 0.191248, antiderivative size = 129, normalized size of antiderivative = 1.55, number of steps used = 6, number of rules used = 6, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {1610, 1807, 807, 266, 63, 205} \[ \frac{\sqrt{d^2 x^2-1} \left (a d^2+2 c\right ) \tan ^{-1}\left (\sqrt{d^2 x^2-1}\right )}{2 \sqrt{d x-1} \sqrt{d x+1}}-\frac{a \left (1-d^2 x^2\right )}{2 x^2 \sqrt{d x-1} \sqrt{d x+1}}-\frac{b \left (1-d^2 x^2\right )}{x \sqrt{d x-1} \sqrt{d x+1}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x + c*x^2)/(x^3*Sqrt[-1 + d*x]*Sqrt[1 + d*x]),x]

[Out]

-(a*(1 - d^2*x^2))/(2*x^2*Sqrt[-1 + d*x]*Sqrt[1 + d*x]) - (b*(1 - d^2*x^2))/(x*Sqrt[-1 + d*x]*Sqrt[1 + d*x]) +
((2*c + a*d^2)*Sqrt[-1 + d^2*x^2]*ArcTan[Sqrt[-1 + d^2*x^2]])/(2*Sqrt[-1 + d*x]*Sqrt[1 + d*x])

Rule 1610

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Dist[((
a + b*x)^FracPart[m]*(c + d*x)^FracPart[m])/(a*c + b*d*x^2)^FracPart[m], Int[Px*(a*c + b*d*x^2)^m*(e + f*x)^p,
x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x] && EqQ[b*c + a*d, 0] && EqQ[m, n] &&  !Intege
rQ[m]

Rule 1807

Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, c*x, x],
R = PolynomialRemainder[Pq, c*x, x]}, Simp[(R*(c*x)^(m + 1)*(a + b*x^2)^(p + 1))/(a*c*(m + 1)), x] + Dist[1/(
a*c*(m + 1)), Int[(c*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*c*(m + 1)*Q - b*R*(m + 2*p + 3)*x, x], x], x]] /;
FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && LtQ[m, -1] && (IntegerQ[2*p] || NeQ[Expon[Pq, x], 1])

Rule 807

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Simp[((e*f - d*g
)*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 + a*e^2)), x] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e^2
), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0]
&& EqQ[Simplify[m + 2*p + 3], 0]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

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]

Rubi steps

\begin{align*} \int \frac{a+b x+c x^2}{x^3 \sqrt{-1+d x} \sqrt{1+d x}} \, dx &=\frac{\sqrt{-1+d^2 x^2} \int \frac{a+b x+c x^2}{x^3 \sqrt{-1+d^2 x^2}} \, dx}{\sqrt{-1+d x} \sqrt{1+d x}}\\ &=-\frac{a \left (1-d^2 x^2\right )}{2 x^2 \sqrt{-1+d x} \sqrt{1+d x}}+\frac{\sqrt{-1+d^2 x^2} \int \frac{2 b+\left (2 c+a d^2\right ) x}{x^2 \sqrt{-1+d^2 x^2}} \, dx}{2 \sqrt{-1+d x} \sqrt{1+d x}}\\ &=-\frac{a \left (1-d^2 x^2\right )}{2 x^2 \sqrt{-1+d x} \sqrt{1+d x}}-\frac{b \left (1-d^2 x^2\right )}{x \sqrt{-1+d x} \sqrt{1+d x}}+\frac{\left (\left (2 c+a d^2\right ) \sqrt{-1+d^2 x^2}\right ) \int \frac{1}{x \sqrt{-1+d^2 x^2}} \, dx}{2 \sqrt{-1+d x} \sqrt{1+d x}}\\ &=-\frac{a \left (1-d^2 x^2\right )}{2 x^2 \sqrt{-1+d x} \sqrt{1+d x}}-\frac{b \left (1-d^2 x^2\right )}{x \sqrt{-1+d x} \sqrt{1+d x}}+\frac{\left (\left (2 c+a d^2\right ) \sqrt{-1+d^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{-1+d^2 x}} \, dx,x,x^2\right )}{4 \sqrt{-1+d x} \sqrt{1+d x}}\\ &=-\frac{a \left (1-d^2 x^2\right )}{2 x^2 \sqrt{-1+d x} \sqrt{1+d x}}-\frac{b \left (1-d^2 x^2\right )}{x \sqrt{-1+d x} \sqrt{1+d x}}+\frac{\left (\left (2 c+a d^2\right ) \sqrt{-1+d^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{d^2}+\frac{x^2}{d^2}} \, dx,x,\sqrt{-1+d^2 x^2}\right )}{2 d^2 \sqrt{-1+d x} \sqrt{1+d x}}\\ &=-\frac{a \left (1-d^2 x^2\right )}{2 x^2 \sqrt{-1+d x} \sqrt{1+d x}}-\frac{b \left (1-d^2 x^2\right )}{x \sqrt{-1+d x} \sqrt{1+d x}}+\frac{\left (2 c+a d^2\right ) \sqrt{-1+d^2 x^2} \tan ^{-1}\left (\sqrt{-1+d^2 x^2}\right )}{2 \sqrt{-1+d x} \sqrt{1+d x}}\\ \end{align*}

Mathematica [A]  time = 0.116759, size = 82, normalized size = 0.99 \[ \frac{\left (d^2 x^2-1\right ) (a+2 b x)+x^2 \sqrt{d^2 x^2-1} \left (a d^2+2 c\right ) \tan ^{-1}\left (\sqrt{d^2 x^2-1}\right )}{2 x^2 \sqrt{d x-1} \sqrt{d x+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x + c*x^2)/(x^3*Sqrt[-1 + d*x]*Sqrt[1 + d*x]),x]

[Out]

((a + 2*b*x)*(-1 + d^2*x^2) + (2*c + a*d^2)*x^2*Sqrt[-1 + d^2*x^2]*ArcTan[Sqrt[-1 + d^2*x^2]])/(2*x^2*Sqrt[-1
+ d*x]*Sqrt[1 + d*x])

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Maple [C]  time = 0., size = 103, normalized size = 1.2 \begin{align*} -{\frac{ \left ({\it csgn} \left ( d \right ) \right ) ^{2}}{2\,{x}^{2}}\sqrt{dx-1}\sqrt{dx+1} \left ( \arctan \left ({\frac{1}{\sqrt{{d}^{2}{x}^{2}-1}}} \right ){x}^{2}a{d}^{2}+2\,\arctan \left ({\frac{1}{\sqrt{{d}^{2}{x}^{2}-1}}} \right ){x}^{2}c-2\,\sqrt{{d}^{2}{x}^{2}-1}xb-\sqrt{{d}^{2}{x}^{2}-1}a \right ){\frac{1}{\sqrt{{d}^{2}{x}^{2}-1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x+a)/x^3/(d*x-1)^(1/2)/(d*x+1)^(1/2),x)

[Out]

-1/2*(d*x-1)^(1/2)*(d*x+1)^(1/2)*csgn(d)^2*(arctan(1/(d^2*x^2-1)^(1/2))*x^2*a*d^2+2*arctan(1/(d^2*x^2-1)^(1/2)
)*x^2*c-2*(d^2*x^2-1)^(1/2)*x*b-(d^2*x^2-1)^(1/2)*a)/(d^2*x^2-1)^(1/2)/x^2

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Maxima [A]  time = 4.01953, size = 88, normalized size = 1.06 \begin{align*} -\frac{1}{2} \, a d^{2} \arcsin \left (\frac{1}{\sqrt{d^{2}}{\left | x \right |}}\right ) - c \arcsin \left (\frac{1}{\sqrt{d^{2}}{\left | x \right |}}\right ) + \frac{\sqrt{d^{2} x^{2} - 1} b}{x} + \frac{\sqrt{d^{2} x^{2} - 1} a}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*a*d^2*arcsin(1/(sqrt(d^2)*abs(x))) - c*arcsin(1/(sqrt(d^2)*abs(x))) + sqrt(d^2*x^2 - 1)*b/x + 1/2*sqrt(d^
2*x^2 - 1)*a/x^2

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Fricas [A]  time = 1.17296, size = 173, normalized size = 2.08 \begin{align*} \frac{2 \, b d x^{2} + 2 \,{\left (a d^{2} + 2 \, c\right )} x^{2} \arctan \left (-d x + \sqrt{d x + 1} \sqrt{d x - 1}\right ) +{\left (2 \, b x + a\right )} \sqrt{d x + 1} \sqrt{d x - 1}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)/x^3/(d*x-1)^(1/2)/(d*x+1)^(1/2),x, algorithm="fricas")

[Out]

1/2*(2*b*d*x^2 + 2*(a*d^2 + 2*c)*x^2*arctan(-d*x + sqrt(d*x + 1)*sqrt(d*x - 1)) + (2*b*x + a)*sqrt(d*x + 1)*sq
rt(d*x - 1))/x^2

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Sympy [C]  time = 33.6278, size = 212, normalized size = 2.55 \begin{align*} - \frac{a d^{2}{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{7}{4}, \frac{9}{4}, 1 & 2, 2, \frac{5}{2} \\\frac{3}{2}, \frac{7}{4}, 2, \frac{9}{4}, \frac{5}{2} & 0 \end{matrix} \middle |{\frac{1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} + \frac{i a d^{2}{G_{6, 6}^{2, 6}\left (\begin{matrix} 1, \frac{5}{4}, \frac{3}{2}, \frac{7}{4}, 2, 1 & \\\frac{5}{4}, \frac{7}{4} & 1, \frac{3}{2}, \frac{3}{2}, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} - \frac{b d{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{5}{4}, \frac{7}{4}, 1 & \frac{3}{2}, \frac{3}{2}, 2 \\1, \frac{5}{4}, \frac{3}{2}, \frac{7}{4}, 2 & 0 \end{matrix} \middle |{\frac{1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} - \frac{i b d{G_{6, 6}^{2, 6}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4}, 1, \frac{5}{4}, \frac{3}{2}, 1 & \\\frac{3}{4}, \frac{5}{4} & \frac{1}{2}, 1, 1, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} - \frac{c{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{3}{4}, \frac{5}{4}, 1 & 1, 1, \frac{3}{2} \\\frac{1}{2}, \frac{3}{4}, 1, \frac{5}{4}, \frac{3}{2} & 0 \end{matrix} \middle |{\frac{1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} + \frac{i c{G_{6, 6}^{2, 6}\left (\begin{matrix} 0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, 1 & \\\frac{1}{4}, \frac{3}{4} & 0, \frac{1}{2}, \frac{1}{2}, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)/x**3/(d*x-1)**(1/2)/(d*x+1)**(1/2),x)

[Out]

-a*d**2*meijerg(((7/4, 9/4, 1), (2, 2, 5/2)), ((3/2, 7/4, 2, 9/4, 5/2), (0,)), 1/(d**2*x**2))/(4*pi**(3/2)) +
I*a*d**2*meijerg(((1, 5/4, 3/2, 7/4, 2, 1), ()), ((5/4, 7/4), (1, 3/2, 3/2, 0)), exp_polar(2*I*pi)/(d**2*x**2)
)/(4*pi**(3/2)) - b*d*meijerg(((5/4, 7/4, 1), (3/2, 3/2, 2)), ((1, 5/4, 3/2, 7/4, 2), (0,)), 1/(d**2*x**2))/(4
*pi**(3/2)) - I*b*d*meijerg(((1/2, 3/4, 1, 5/4, 3/2, 1), ()), ((3/4, 5/4), (1/2, 1, 1, 0)), exp_polar(2*I*pi)/
(d**2*x**2))/(4*pi**(3/2)) - c*meijerg(((3/4, 5/4, 1), (1, 1, 3/2)), ((1/2, 3/4, 1, 5/4, 3/2), (0,)), 1/(d**2*
x**2))/(4*pi**(3/2)) + I*c*meijerg(((0, 1/4, 1/2, 3/4, 1, 1), ()), ((1/4, 3/4), (0, 1/2, 1/2, 0)), exp_polar(2
*I*pi)/(d**2*x**2))/(4*pi**(3/2))

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Giac [B]  time = 1.98699, size = 196, normalized size = 2.36 \begin{align*} -\frac{{\left (a d^{3} + 2 \, c d\right )} \arctan \left (\frac{1}{2} \,{\left (\sqrt{d x + 1} - \sqrt{d x - 1}\right )}^{2}\right ) + \frac{2 \,{\left (a d^{3}{\left (\sqrt{d x + 1} - \sqrt{d x - 1}\right )}^{6} - 4 \, b d^{2}{\left (\sqrt{d x + 1} - \sqrt{d x - 1}\right )}^{4} - 4 \, a d^{3}{\left (\sqrt{d x + 1} - \sqrt{d x - 1}\right )}^{2} - 16 \, b d^{2}\right )}}{{\left ({\left (\sqrt{d x + 1} - \sqrt{d x - 1}\right )}^{4} + 4\right )}^{2}}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((a*d^3 + 2*c*d)*arctan(1/2*(sqrt(d*x + 1) - sqrt(d*x - 1))^2) + 2*(a*d^3*(sqrt(d*x + 1) - sqrt(d*x - 1))^6 -
4*b*d^2*(sqrt(d*x + 1) - sqrt(d*x - 1))^4 - 4*a*d^3*(sqrt(d*x + 1) - sqrt(d*x - 1))^2 - 16*b*d^2)/((sqrt(d*x
+ 1) - sqrt(d*x - 1))^4 + 4)^2)/d