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

Optimal. Leaf size=87 \[ \frac{\sqrt{d x-1} \sqrt{d x+1} \left (2 \left (3 a d^2+2 c\right )+3 b d^2 x\right )}{6 d^4}+\frac{b \cosh ^{-1}(d x)}{2 d^3}+\frac{c x^2 \sqrt{d x-1} \sqrt{d x+1}}{3 d^2} \]

[Out]

(c*x^2*Sqrt[-1 + d*x]*Sqrt[1 + d*x])/(3*d^2) + (Sqrt[-1 + d*x]*Sqrt[1 + d*x]*(2*(2*c + 3*a*d^2) + 3*b*d^2*x))/
(6*d^4) + (b*ArcCosh[d*x])/(2*d^3)

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Rubi [A]  time = 0.146262, antiderivative size = 151, normalized size of antiderivative = 1.74, number of steps used = 5, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {1610, 1809, 780, 217, 206} \[ -\frac{\left (1-d^2 x^2\right ) \left (2 \left (3 a d^2+2 c\right )+3 b d^2 x\right )}{6 d^4 \sqrt{d x-1} \sqrt{d x+1}}+\frac{b \sqrt{d^2 x^2-1} \tanh ^{-1}\left (\frac{d x}{\sqrt{d^2 x^2-1}}\right )}{2 d^3 \sqrt{d x-1} \sqrt{d x+1}}-\frac{c x^2 \left (1-d^2 x^2\right )}{3 d^2 \sqrt{d x-1} \sqrt{d x+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(c*x^2*(1 - d^2*x^2))/(3*d^2*Sqrt[-1 + d*x]*Sqrt[1 + d*x]) - ((2*(2*c + 3*a*d^2) + 3*b*d^2*x)*(1 - d^2*x^2))/
(6*d^4*Sqrt[-1 + d*x]*Sqrt[1 + d*x]) + (b*Sqrt[-1 + d^2*x^2]*ArcTanh[(d*x)/Sqrt[-1 + d^2*x^2]])/(2*d^3*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 1809

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x,
Expon[Pq, x]]}, Simp[(f*(c*x)^(m + q - 1)*(a + b*x^2)^(p + 1))/(b*c^(q - 1)*(m + q + 2*p + 1)), x] + Dist[1/(
b*(m + q + 2*p + 1)), Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*x^q
- a*f*(m + q - 1)*x^(q - 2), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x]
&& PolyQ[Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -1])

Rule 780

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

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

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

Rubi steps

\begin{align*} \int \frac{x \left (a+b x+c x^2\right )}{\sqrt{-1+d x} \sqrt{1+d x}} \, dx &=\frac{\sqrt{-1+d^2 x^2} \int \frac{x \left (a+b x+c x^2\right )}{\sqrt{-1+d^2 x^2}} \, dx}{\sqrt{-1+d x} \sqrt{1+d x}}\\ &=-\frac{c x^2 \left (1-d^2 x^2\right )}{3 d^2 \sqrt{-1+d x} \sqrt{1+d x}}+\frac{\sqrt{-1+d^2 x^2} \int \frac{x \left (2 c+3 a d^2+3 b d^2 x\right )}{\sqrt{-1+d^2 x^2}} \, dx}{3 d^2 \sqrt{-1+d x} \sqrt{1+d x}}\\ &=-\frac{c x^2 \left (1-d^2 x^2\right )}{3 d^2 \sqrt{-1+d x} \sqrt{1+d x}}-\frac{\left (2 \left (2 c+3 a d^2\right )+3 b d^2 x\right ) \left (1-d^2 x^2\right )}{6 d^4 \sqrt{-1+d x} \sqrt{1+d x}}+\frac{\left (b \sqrt{-1+d^2 x^2}\right ) \int \frac{1}{\sqrt{-1+d^2 x^2}} \, dx}{2 d^2 \sqrt{-1+d x} \sqrt{1+d x}}\\ &=-\frac{c x^2 \left (1-d^2 x^2\right )}{3 d^2 \sqrt{-1+d x} \sqrt{1+d x}}-\frac{\left (2 \left (2 c+3 a d^2\right )+3 b d^2 x\right ) \left (1-d^2 x^2\right )}{6 d^4 \sqrt{-1+d x} \sqrt{1+d x}}+\frac{\left (b \sqrt{-1+d^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{1-d^2 x^2} \, dx,x,\frac{x}{\sqrt{-1+d^2 x^2}}\right )}{2 d^2 \sqrt{-1+d x} \sqrt{1+d x}}\\ &=-\frac{c x^2 \left (1-d^2 x^2\right )}{3 d^2 \sqrt{-1+d x} \sqrt{1+d x}}-\frac{\left (2 \left (2 c+3 a d^2\right )+3 b d^2 x\right ) \left (1-d^2 x^2\right )}{6 d^4 \sqrt{-1+d x} \sqrt{1+d x}}+\frac{b \sqrt{-1+d^2 x^2} \tanh ^{-1}\left (\frac{d x}{\sqrt{-1+d^2 x^2}}\right )}{2 d^3 \sqrt{-1+d x} \sqrt{1+d x}}\\ \end{align*}

Mathematica [A]  time = 0.331289, size = 149, normalized size = 1.71 \[ \frac{\sqrt{-(d x-1)^2} \sqrt{d x+1} \left (3 d^2 (2 a+b x)+2 c \left (d^2 x^2+2\right )\right )+6 \sqrt{d x-1} \sin ^{-1}\left (\frac{\sqrt{1-d x}}{\sqrt{2}}\right ) (d (2 a d-b)+2 c)-12 \sqrt{1-d x} \tanh ^{-1}\left (\sqrt{\frac{d x-1}{d x+1}}\right ) (d (a d-b)+c)}{6 d^4 \sqrt{1-d x}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[-(-1 + d*x)^2]*Sqrt[1 + d*x]*(3*d^2*(2*a + b*x) + 2*c*(2 + d^2*x^2)) + 6*(2*c + d*(-b + 2*a*d))*Sqrt[-1
+ d*x]*ArcSin[Sqrt[1 - d*x]/Sqrt[2]] - 12*(c + d*(-b + a*d))*Sqrt[1 - d*x]*ArcTanh[Sqrt[(-1 + d*x)/(1 + d*x)]]
)/(6*d^4*Sqrt[1 - d*x])

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Maple [C]  time = 0.023, size = 137, normalized size = 1.6 \begin{align*}{\frac{{\it csgn} \left ( d \right ) }{6\,{d}^{4}}\sqrt{dx-1}\sqrt{dx+1} \left ( 2\,{\it csgn} \left ( d \right ){x}^{2}c{d}^{2}\sqrt{{d}^{2}{x}^{2}-1}+3\,{\it csgn} \left ( d \right ) \sqrt{{d}^{2}{x}^{2}-1}xb{d}^{2}+6\,{\it csgn} \left ( d \right ) \sqrt{{d}^{2}{x}^{2}-1}a{d}^{2}+4\,{\it csgn} \left ( d \right ) \sqrt{{d}^{2}{x}^{2}-1}c+3\,\ln \left ( \left ({\it csgn} \left ( d \right ) \sqrt{{d}^{2}{x}^{2}-1}+dx \right ){\it csgn} \left ( d \right ) \right ) bd \right ){\frac{1}{\sqrt{{d}^{2}{x}^{2}-1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(d*x-1)^(1/2)*(d*x+1)^(1/2)*(2*csgn(d)*x^2*c*d^2*(d^2*x^2-1)^(1/2)+3*csgn(d)*(d^2*x^2-1)^(1/2)*x*b*d^2+6*c
sgn(d)*(d^2*x^2-1)^(1/2)*a*d^2+4*csgn(d)*(d^2*x^2-1)^(1/2)*c+3*ln((csgn(d)*(d^2*x^2-1)^(1/2)+d*x)*csgn(d))*b*d
)*csgn(d)/d^4/(d^2*x^2-1)^(1/2)

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Maxima [A]  time = 1.35168, size = 147, normalized size = 1.69 \begin{align*} \frac{\sqrt{d^{2} x^{2} - 1} c x^{2}}{3 \, d^{2}} + \frac{\sqrt{d^{2} x^{2} - 1} b x}{2 \, d^{2}} + \frac{\sqrt{d^{2} x^{2} - 1} a}{d^{2}} + \frac{b \log \left (2 \, d^{2} x + 2 \, \sqrt{d^{2} x^{2} - 1} \sqrt{d^{2}}\right )}{2 \, \sqrt{d^{2}} d^{2}} + \frac{2 \, \sqrt{d^{2} x^{2} - 1} c}{3 \, d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.05241, size = 176, normalized size = 2.02 \begin{align*} -\frac{3 \, b d \log \left (-d x + \sqrt{d x + 1} \sqrt{d x - 1}\right ) -{\left (2 \, c d^{2} x^{2} + 3 \, b d^{2} x + 6 \, a d^{2} + 4 \, c\right )} \sqrt{d x + 1} \sqrt{d x - 1}}{6 \, d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C]  time = 44.7876, size = 308, normalized size = 3.54 \begin{align*} \frac{a{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{1}{4}, \frac{1}{4} & 0, 0, \frac{1}{2}, 1 \\- \frac{1}{2}, - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, 0 & \end{matrix} \middle |{\frac{1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{2}} + \frac{i a{G_{6, 6}^{2, 6}\left (\begin{matrix} -1, - \frac{3}{4}, - \frac{1}{2}, - \frac{1}{4}, 0, 1 & \\- \frac{3}{4}, - \frac{1}{4} & -1, - \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}} d^{2}} + \frac{b{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{3}{4}, - \frac{1}{4} & - \frac{1}{2}, - \frac{1}{2}, 0, 1 \\-1, - \frac{3}{4}, - \frac{1}{2}, - \frac{1}{4}, 0, 0 & \end{matrix} \middle |{\frac{1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{3}} - \frac{i b{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{3}{2}, - \frac{5}{4}, -1, - \frac{3}{4}, - \frac{1}{2}, 1 & \\- \frac{5}{4}, - \frac{3}{4} & - \frac{3}{2}, -1, -1, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{3}} + \frac{c{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{5}{4}, - \frac{3}{4} & -1, -1, - \frac{1}{2}, 1 \\- \frac{3}{2}, - \frac{5}{4}, -1, - \frac{3}{4}, - \frac{1}{2}, 0 & \end{matrix} \middle |{\frac{1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{4}} + \frac{i c{G_{6, 6}^{2, 6}\left (\begin{matrix} -2, - \frac{7}{4}, - \frac{3}{2}, - \frac{5}{4}, -1, 1 & \\- \frac{7}{4}, - \frac{5}{4} & -2, - \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}} d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 2.12368, size = 130, normalized size = 1.49 \begin{align*} -\frac{6 \, b d^{10} \log \left ({\left | -\sqrt{d x + 1} + \sqrt{d x - 1} \right |}\right ) -{\left (6 \, a d^{11} - 3 \, b d^{10} + 6 \, c d^{9} +{\left (2 \,{\left (d x + 1\right )} c d^{9} + 3 \, b d^{10} - 4 \, c d^{9}\right )}{\left (d x + 1\right )}\right )} \sqrt{d x + 1} \sqrt{d x - 1}}{3840 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3840*(6*b*d^10*log(abs(-sqrt(d*x + 1) + sqrt(d*x - 1))) - (6*a*d^11 - 3*b*d^10 + 6*c*d^9 + (2*(d*x + 1)*c*d
^9 + 3*b*d^10 - 4*c*d^9)*(d*x + 1))*sqrt(d*x + 1)*sqrt(d*x - 1))/d