3.869 \(\int \frac{e^{\tanh ^{-1}(a+b x)} x^3}{1-a^2-2 a b x-b^2 x^2} \, dx\)

Optimal. Leaf size=109 \[ -\frac{3 \left (2 a^2-2 a+1\right ) \sin ^{-1}(a+b x)}{2 b^4}+\frac{(1-a) x^2 \sqrt{a+b x+1}}{b^2 \sqrt{-a-b x+1}}+\frac{\sqrt{-a-b x+1} \sqrt{a+b x+1} ((3-2 a) b x+(1-2 a) (4-a))}{2 b^4} \]

[Out]

((1 - a)*x^2*Sqrt[1 + a + b*x])/(b^2*Sqrt[1 - a - b*x]) + (Sqrt[1 - a - b*x]*Sqrt[1 + a + b*x]*((1 - 2*a)*(4 -
a) + (3 - 2*a)*b*x))/(2*b^4) - (3*(1 - 2*a + 2*a^2)*ArcSin[a + b*x])/(2*b^4)

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Rubi [A]  time = 0.168588, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {6164, 98, 147, 53, 619, 216} \[ -\frac{3 \left (2 a^2-2 a+1\right ) \sin ^{-1}(a+b x)}{2 b^4}+\frac{(1-a) x^2 \sqrt{a+b x+1}}{b^2 \sqrt{-a-b x+1}}+\frac{\sqrt{-a-b x+1} \sqrt{a+b x+1} ((3-2 a) b x+(1-2 a) (4-a))}{2 b^4} \]

Antiderivative was successfully verified.

[In]

Int[(E^ArcTanh[a + b*x]*x^3)/(1 - a^2 - 2*a*b*x - b^2*x^2),x]

[Out]

((1 - a)*x^2*Sqrt[1 + a + b*x])/(b^2*Sqrt[1 - a - b*x]) + (Sqrt[1 - a - b*x]*Sqrt[1 + a + b*x]*((1 - 2*a)*(4 -
a) + (3 - 2*a)*b*x))/(2*b^4) - (3*(1 - 2*a + 2*a^2)*ArcSin[a + b*x])/(2*b^4)

Rule 6164

Int[E^(ArcTanh[(a_) + (b_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[(c/
(1 - a^2))^p, Int[u*(1 - a - b*x)^(p - n/2)*(1 + a + b*x)^(p + n/2), x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]
&& EqQ[b*d - 2*a*e, 0] && EqQ[b^2*c + e*(1 - a^2), 0] && (IntegerQ[p] || GtQ[c/(1 - a^2), 0])

Rule 98

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c -
a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^(p + 1))/(b*(b*e - a*f)*(m + 1)), x] + Dist[1/(b*(b*e - a*
f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d
*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p
] || IntegersQ[p, m + n])

Rule 147

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

Rule 53

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

Rule 619

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*((-4*c)/(b^2 - 4*a*c))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

Rule 216

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

Rubi steps

\begin{align*} \int \frac{e^{\tanh ^{-1}(a+b x)} x^3}{1-a^2-2 a b x-b^2 x^2} \, dx &=\int \frac{x^3}{(1-a-b x)^{3/2} \sqrt{1+a+b x}} \, dx\\ &=\frac{(1-a) x^2 \sqrt{1+a+b x}}{b^2 \sqrt{1-a-b x}}-\frac{\int \frac{x \left (2 \left (1-a^2\right )+(3-2 a) b x\right )}{\sqrt{1-a-b x} \sqrt{1+a+b x}} \, dx}{b^2}\\ &=\frac{(1-a) x^2 \sqrt{1+a+b x}}{b^2 \sqrt{1-a-b x}}+\frac{\sqrt{1-a-b x} \sqrt{1+a+b x} ((1-2 a) (4-a)+(3-2 a) b x)}{2 b^4}-\frac{\left (3 \left (1-2 a+2 a^2\right )\right ) \int \frac{1}{\sqrt{1-a-b x} \sqrt{1+a+b x}} \, dx}{2 b^3}\\ &=\frac{(1-a) x^2 \sqrt{1+a+b x}}{b^2 \sqrt{1-a-b x}}+\frac{\sqrt{1-a-b x} \sqrt{1+a+b x} ((1-2 a) (4-a)+(3-2 a) b x)}{2 b^4}-\frac{\left (3 \left (1-2 a+2 a^2\right )\right ) \int \frac{1}{\sqrt{(1-a) (1+a)-2 a b x-b^2 x^2}} \, dx}{2 b^3}\\ &=\frac{(1-a) x^2 \sqrt{1+a+b x}}{b^2 \sqrt{1-a-b x}}+\frac{\sqrt{1-a-b x} \sqrt{1+a+b x} ((1-2 a) (4-a)+(3-2 a) b x)}{2 b^4}+\frac{\left (3 \left (1-2 a+2 a^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{4 b^2}}} \, dx,x,-2 a b-2 b^2 x\right )}{4 b^5}\\ &=\frac{(1-a) x^2 \sqrt{1+a+b x}}{b^2 \sqrt{1-a-b x}}+\frac{\sqrt{1-a-b x} \sqrt{1+a+b x} ((1-2 a) (4-a)+(3-2 a) b x)}{2 b^4}-\frac{3 \left (1-2 a+2 a^2\right ) \sin ^{-1}(a+b x)}{2 b^4}\\ \end{align*}

Mathematica [A]  time = 0.338711, size = 90, normalized size = 0.83 \[ -\frac{3 \left (2 a^2-2 a+1\right ) \sin ^{-1}(a+b x)-\frac{\sqrt{-a^2-2 a b x-b^2 x^2+1} \left (2 a^3-11 a^2+a (13-4 b x)+b^2 x^2+b x-4\right )}{a+b x-1}}{2 b^4} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(E^ArcTanh[a + b*x]*x^3)/(1 - a^2 - 2*a*b*x - b^2*x^2),x]

[Out]

-(-((Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]*(-4 - 11*a^2 + 2*a^3 + b*x + b^2*x^2 + a*(13 - 4*b*x)))/(-1 + a + b*x))
+ 3*(1 - 2*a + 2*a^2)*ArcSin[a + b*x])/(2*b^4)

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Maple [B]  time = 0.043, size = 499, normalized size = 4.6 \begin{align*} -{\frac{{x}^{3}}{2\,b}{\frac{1}{\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}}}+{\frac{3\,a{x}^{2}}{2\,{b}^{2}}{\frac{1}{\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}}}+{\frac{15\,{a}^{2}x}{2\,{b}^{3}}{\frac{1}{\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}}}+{\frac{9\,{a}^{3}}{2\,{b}^{4}}{\frac{1}{\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}}}-3\,{\frac{{a}^{2}}{{b}^{3}\sqrt{{b}^{2}}}\arctan \left ({\frac{\sqrt{{b}^{2}}}{\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}} \left ( x+{\frac{a}{b}} \right ) } \right ) }-{\frac{9\,a}{2\,{b}^{4}}{\frac{1}{\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}}}+{\frac{3\,x}{2\,{b}^{3}}{\frac{1}{\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}}}-{\frac{3}{2\,{b}^{3}}\arctan \left ({\sqrt{{b}^{2}} \left ( x+{\frac{a}{b}} \right ){\frac{1}{\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}}} \right ){\frac{1}{\sqrt{{b}^{2}}}}}-{\frac{{x}^{2}}{{b}^{2}}{\frac{1}{\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}}}-5\,{\frac{ax}{{b}^{3}\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}}-{\frac{{a}^{2}}{{b}^{4}}{\frac{1}{\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}}}-{\frac{x{a}^{3}}{{b}^{3}}{\frac{1}{\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}}}-{\frac{{a}^{4}}{{b}^{4}}{\frac{1}{\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}}}+3\,{\frac{a}{{b}^{3}\sqrt{{b}^{2}}}\arctan \left ({\frac{\sqrt{{b}^{2}}}{\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}} \left ( x+{\frac{a}{b}} \right ) } \right ) }+2\,{\frac{1}{{b}^{4}\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/b*x^3/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)+3/2/b^2*a*x^2/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)+15/2/b^3*a^2*x/(-b^2*x^
2-2*a*b*x-a^2+1)^(1/2)+9/2/b^4*a^3/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)-3*a^2/b^3/(b^2)^(1/2)*arctan((b^2)^(1/2)*(x+
a/b)/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2))-9/2/b^4*a/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)+3/2/b^3*x/(-b^2*x^2-2*a*b*x-a^2+
1)^(1/2)-3/2/b^3/(b^2)^(1/2)*arctan((b^2)^(1/2)*(x+a/b)/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2))-x^2/b^2/(-b^2*x^2-2*a*
b*x-a^2+1)^(1/2)-5*a/b^3*x/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)-a^2/b^4/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)-a^3/b^3/(-b^2
*x^2-2*a*b*x-a^2+1)^(1/2)*x-a^4/b^4/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)+3*a/b^3/(b^2)^(1/2)*arctan((b^2)^(1/2)*(x+a
/b)/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2))+2/b^4/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)

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Maxima [B]  time = 2.1036, size = 1358, normalized size = 12.46 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*a^3/(b^6*x + a*b^5 + sqrt(b^2)*b^4) - sqrt(-b^2*x^2 - 2*a*b*x - a^2 +
1)*a^3/(b^6*x + a*b^5 - sqrt(b^2)*b^4) - sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*a^3/(sqrt(b^2)*b^5*x + a*sqrt(b^2
)*b^4 + b^5) - sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*a^3/(sqrt(b^2)*b^5*x + a*sqrt(b^2)*b^4 - b^5) + 3*sqrt(-b^2*
x^2 - 2*a*b*x - a^2 + 1)*a^2*sqrt(b^2)/(b^7*x + a*b^6 + sqrt(b^2)*b^5) + 3*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*
a^2*sqrt(b^2)/(b^7*x + a*b^6 - sqrt(b^2)*b^5) - 3*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*a^2/(b^6*x + a*b^5 + sqrt
(b^2)*b^4) + 3*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*a^2/(b^6*x + a*b^5 - sqrt(b^2)*b^4) + 3*sqrt(-b^2*x^2 - 2*a*
b*x - a^2 + 1)*a/(b^6*x + a*b^5 + sqrt(b^2)*b^4) - 3*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*a/(b^6*x + a*b^5 - sqr
t(b^2)*b^4) - 3*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*a/(sqrt(b^2)*b^5*x + a*sqrt(b^2)*b^4 + b^5) - 3*sqrt(-b^2*x
^2 - 2*a*b*x - a^2 + 1)*a/(sqrt(b^2)*b^5*x + a*sqrt(b^2)*b^4 - b^5) + sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*sqrt(
b^2)/(b^7*x + a*b^6 + sqrt(b^2)*b^5) + sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*sqrt(b^2)/(b^7*x + a*b^6 - sqrt(b^2)
*b^5) - sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)/(b^6*x + a*b^5 + sqrt(b^2)*b^4) + sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1
)/(b^6*x + a*b^5 - sqrt(b^2)*b^4) + 6*a^2*arcsin(sqrt(b^2)*x + a*sqrt(b^2)/b)/b^5 - sqrt(-b^2*x^2 - 2*a*b*x -
a^2 + 1)*sqrt(b^2)*x/b^5 - 6*a*arcsin(sqrt(b^2)*x + a*sqrt(b^2)/b)/b^5 + 5*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*
a*sqrt(b^2)/b^6 + 3*arcsin(sqrt(b^2)*x + a*sqrt(b^2)/b)/b^5 - 2*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*sqrt(b^2)/b
^6)*b^2/sqrt(a^2*b^2 - (a^2 - 1)*b^2)

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Fricas [A]  time = 1.84489, size = 344, normalized size = 3.16 \begin{align*} \frac{3 \,{\left (2 \, a^{3} +{\left (2 \, a^{2} - 2 \, a + 1\right )} b x - 4 \, a^{2} + 3 \, a - 1\right )} \arctan \left (\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}{\left (b x + a\right )}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) +{\left (b^{2} x^{2} + 2 \, a^{3} -{\left (4 \, a - 1\right )} b x - 11 \, a^{2} + 13 \, a - 4\right )} \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{2 \,{\left (b^{5} x +{\left (a - 1\right )} b^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(3*(2*a^3 + (2*a^2 - 2*a + 1)*b*x - 4*a^2 + 3*a - 1)*arctan(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(b*x + a)/(
b^2*x^2 + 2*a*b*x + a^2 - 1)) + (b^2*x^2 + 2*a^3 - (4*a - 1)*b*x - 11*a^2 + 13*a - 4)*sqrt(-b^2*x^2 - 2*a*b*x
- a^2 + 1))/(b^5*x + (a - 1)*b^4)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{x^{3}}{a \sqrt{- a^{2} - 2 a b x - b^{2} x^{2} + 1} + b x \sqrt{- a^{2} - 2 a b x - b^{2} x^{2} + 1} - \sqrt{- a^{2} - 2 a b x - b^{2} x^{2} + 1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(x**3/(a*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1) + b*x*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1) - sqrt(-a*
*2 - 2*a*b*x - b**2*x**2 + 1)), x)

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Giac [A]  time = 1.26801, size = 169, normalized size = 1.55 \begin{align*} \frac{1}{2} \, \sqrt{-{\left (b x + a\right )}^{2} + 1}{\left (\frac{x}{b^{3}} - \frac{5 \, a b^{6} - 2 \, b^{6}}{b^{10}}\right )} + \frac{3 \,{\left (2 \, a^{2} - 2 \, a + 1\right )} \arcsin \left (-b x - a\right ) \mathrm{sgn}\left (b\right )}{2 \, b^{3}{\left | b \right |}} - \frac{2 \,{\left (a^{3} - 3 \, a^{2} + 3 \, a - 1\right )}}{b^{3}{\left (\frac{\sqrt{-{\left (b x + a\right )}^{2} + 1}{\left | b \right |} + b}{b^{2} x + a b} - 1\right )}{\left | b \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(-(b*x + a)^2 + 1)*(x/b^3 - (5*a*b^6 - 2*b^6)/b^10) + 3/2*(2*a^2 - 2*a + 1)*arcsin(-b*x - a)*sgn(b)/(b
^3*abs(b)) - 2*(a^3 - 3*a^2 + 3*a - 1)/(b^3*((sqrt(-(b*x + a)^2 + 1)*abs(b) + b)/(b^2*x + a*b) - 1)*abs(b))