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 \[ -\frac {3 \left (2 a^2-2 a+1\right ) \sin ^{-1}(a+b x)}{2 b^4}+\frac {\sqrt {-a-b x+1} \sqrt {a+b x+1} ((3-2 a) b x+(1-2 a) (4-a))}{2 b^4}+\frac {(1-a) x^2 \sqrt {a+b x+1}}{b^2 \sqrt {-a-b x+1}} \]

[Out]

-3/2*(2*a^2-2*a+1)*arcsin(b*x+a)/b^4+(1-a)*x^2*(b*x+a+1)^(1/2)/b^2/(-b*x-a+1)^(1/2)+1/2*((1-2*a)*(4-a)+(3-2*a)
*b*x)*(-b*x-a+1)^(1/2)*(b*x+a+1)^(1/2)/b^4

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Rubi [A]  time = 0.17, antiderivative size = 109, normalized size of antiderivative = 1.00, 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 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 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 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]

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 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])

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*}

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Mathematica [A]  time = 0.37, 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]

-1/2*(-((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])/b^4

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fricas [A]  time = 0.67, size = 150, normalized size = 1.38 \[ \frac {3 \, {\left (2 \, a^{3} + {\left (2 \, a^{2} - 2 \, a + 1\right )} b x - 4 \, a^{2} + 3 \, a - 1\right )} \arctan \left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (b x + a\right )}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) + {\left (b^{2} x^{2} + 2 \, a^{3} - {\left (4 \, a - 1\right )} b x - 11 \, a^{2} + 13 \, a - 4\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{2 \, {\left (b^{5} x + {\left (a - 1\right )} b^{4}\right )}} \]

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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giac [A]  time = 0.24, size = 125, normalized size = 1.15 \[ \frac {1}{2} \, \sqrt {-{\left (b x + a\right )}^{2} + 1} {\left (\frac {x}{b^{3}} - \frac {5 \, a b^{6} - 2 \, b^{6}}{b^{10}}\right )} + \frac {3 \, {\left (2 \, a^{2} - 2 \, a + 1\right )} \arcsin \left (-b x - a\right ) \mathrm {sgn}\relax (b)}{2 \, b^{3} {\left | b \right |}} - \frac {2 \, {\left (a^{3} - 3 \, a^{2} + 3 \, a - 1\right )}}{b^{3} {\left (\frac {\sqrt {-{\left (b x + a\right )}^{2} + 1} {\left | b \right |} + b}{b^{2} x + a b} - 1\right )} {\left | b \right |}} \]

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))

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maple [B]  time = 0.04, size = 499, normalized size = 4.58 \[ -\frac {x^{3}}{2 b \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}+\frac {3 a \,x^{2}}{2 b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}+\frac {15 a^{2} x}{2 b^{3} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}+\frac {9 a^{3}}{2 b^{4} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {3 a^{2} \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{b^{3} \sqrt {b^{2}}}-\frac {9 a}{2 b^{4} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}+\frac {3 x}{2 b^{3} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {3 \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{2 b^{3} \sqrt {b^{2}}}-\frac {x^{2}}{b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {5 a x}{b^{3} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {a^{2}}{b^{4} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {a^{3} x}{b^{3} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {a^{4}}{b^{4} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}+\frac {3 a \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{b^{3} \sqrt {b^{2}}}+\frac {2}{b^{4} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}} \]

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 = 0.60, size = 576, normalized size = 5.28 \[ \frac {{\left (\frac {2 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a^{3}}{b^{6} x + a b^{5} - b^{5}} - \frac {3 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a^{2} b}{b^{7} x + a b^{6} + b^{6}} - \frac {3 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a^{2} b}{b^{7} x + a b^{6} - b^{6}} + \frac {3 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a^{2}}{b^{6} x + a b^{5} + b^{5}} - \frac {3 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a^{2}}{b^{6} x + a b^{5} - b^{5}} + \frac {6 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a}{b^{6} x + a b^{5} - b^{5}} - \frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} b}{b^{7} x + a b^{6} + b^{6}} - \frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} b}{b^{7} x + a b^{6} - b^{6}} + \frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{b^{6} x + a b^{5} + b^{5}} - \frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{b^{6} x + a b^{5} - b^{5}} - \frac {6 \, a^{2} \arcsin \left (b x + a\right )}{b^{5}} + \frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} x}{b^{4}} + \frac {6 \, a \arcsin \left (b x + a\right )}{b^{5}} - \frac {5 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a}{b^{5}} - \frac {3 \, \arcsin \left (b x + a\right )}{b^{5}} + \frac {2 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{b^{5}}\right )} b^{2}}{2 \, \sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}} \]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ -\int \frac {x^3\,\left (a+b\,x+1\right )}{\sqrt {1-{\left (a+b\,x\right )}^2}\,\left (a^2+2\,a\,b\,x+b^2\,x^2-1\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {x^{3}}{a \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} + b x \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} - \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}\, dx \]

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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