3.9.36 \(\int e^{3 \tanh ^{-1}(a+b x)} x^2 \, dx\) [836]

Optimal. Leaf size=168 \[ \frac {\left (11-18 a+6 a^2\right ) \sqrt {1-a-b x} \sqrt {1+a+b x}}{2 b^3}+\frac {\left (11-18 a+6 a^2\right ) \sqrt {1-a-b x} (1+a+b x)^{3/2}}{6 b^3}+\frac {(1-a)^2 (1+a+b x)^{5/2}}{b^3 \sqrt {1-a-b x}}+\frac {\sqrt {1-a-b x} (1+a+b x)^{5/2}}{3 b^3}-\frac {\left (11-18 a+6 a^2\right ) \text {ArcSin}(a+b x)}{2 b^3} \]

[Out]

-1/2*(6*a^2-18*a+11)*arcsin(b*x+a)/b^3+(1-a)^2*(b*x+a+1)^(5/2)/b^3/(-b*x-a+1)^(1/2)+1/6*(6*a^2-18*a+11)*(b*x+a
+1)^(3/2)*(-b*x-a+1)^(1/2)/b^3+1/3*(b*x+a+1)^(5/2)*(-b*x-a+1)^(1/2)/b^3+1/2*(6*a^2-18*a+11)*(-b*x-a+1)^(1/2)*(
b*x+a+1)^(1/2)/b^3

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Rubi [A]
time = 0.13, antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6298, 91, 81, 52, 55, 633, 222} \begin {gather*} -\frac {\left (6 a^2-18 a+11\right ) \text {ArcSin}(a+b x)}{2 b^3}+\frac {\left (6 a^2-18 a+11\right ) \sqrt {-a-b x+1} (a+b x+1)^{3/2}}{6 b^3}+\frac {\left (6 a^2-18 a+11\right ) \sqrt {-a-b x+1} \sqrt {a+b x+1}}{2 b^3}+\frac {\sqrt {-a-b x+1} (a+b x+1)^{5/2}}{3 b^3}+\frac {(1-a)^2 (a+b x+1)^{5/2}}{b^3 \sqrt {-a-b x+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(3*ArcTanh[a + b*x])*x^2,x]

[Out]

((11 - 18*a + 6*a^2)*Sqrt[1 - a - b*x]*Sqrt[1 + a + b*x])/(2*b^3) + ((11 - 18*a + 6*a^2)*Sqrt[1 - a - b*x]*(1
+ a + b*x)^(3/2))/(6*b^3) + ((1 - a)^2*(1 + a + b*x)^(5/2))/(b^3*Sqrt[1 - a - b*x]) + (Sqrt[1 - a - b*x]*(1 +
a + b*x)^(5/2))/(3*b^3) - ((11 - 18*a + 6*a^2)*ArcSin[a + b*x])/(2*b^3)

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 55

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 81

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

Rule 91

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*c - a*d
)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d*e - c*f)*(n + 1))), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

Rule 222

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 633

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 6298

Int[E^(ArcTanh[(c_.)*((a_) + (b_.)*(x_))]*(n_.))*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Int[(d + e*x)^m*((1
+ a*c + b*c*x)^(n/2)/(1 - a*c - b*c*x)^(n/2)), x] /; FreeQ[{a, b, c, d, e, m, n}, x]

Rubi steps

\begin {align*} \int e^{3 \tanh ^{-1}(a+b x)} x^2 \, dx &=\int \frac {x^2 (1+a+b x)^{3/2}}{(1-a-b x)^{3/2}} \, dx\\ &=\frac {(1-a)^2 (1+a+b x)^{5/2}}{b^3 \sqrt {1-a-b x}}-\frac {\int \frac {(1+a+b x)^{3/2} \left ((3-2 a) (1-a) b+b^2 x\right )}{\sqrt {1-a-b x}} \, dx}{b^3}\\ &=\frac {(1-a)^2 (1+a+b x)^{5/2}}{b^3 \sqrt {1-a-b x}}+\frac {\sqrt {1-a-b x} (1+a+b x)^{5/2}}{3 b^3}-\frac {\left (11-18 a+6 a^2\right ) \int \frac {(1+a+b x)^{3/2}}{\sqrt {1-a-b x}} \, dx}{3 b^2}\\ &=\frac {\left (11-18 a+6 a^2\right ) \sqrt {1-a-b x} (1+a+b x)^{3/2}}{6 b^3}+\frac {(1-a)^2 (1+a+b x)^{5/2}}{b^3 \sqrt {1-a-b x}}+\frac {\sqrt {1-a-b x} (1+a+b x)^{5/2}}{3 b^3}-\frac {\left (11-18 a+6 a^2\right ) \int \frac {\sqrt {1+a+b x}}{\sqrt {1-a-b x}} \, dx}{2 b^2}\\ &=\frac {\left (11-18 a+6 a^2\right ) \sqrt {1-a-b x} \sqrt {1+a+b x}}{2 b^3}+\frac {\left (11-18 a+6 a^2\right ) \sqrt {1-a-b x} (1+a+b x)^{3/2}}{6 b^3}+\frac {(1-a)^2 (1+a+b x)^{5/2}}{b^3 \sqrt {1-a-b x}}+\frac {\sqrt {1-a-b x} (1+a+b x)^{5/2}}{3 b^3}-\frac {\left (11-18 a+6 a^2\right ) \int \frac {1}{\sqrt {1-a-b x} \sqrt {1+a+b x}} \, dx}{2 b^2}\\ &=\frac {\left (11-18 a+6 a^2\right ) \sqrt {1-a-b x} \sqrt {1+a+b x}}{2 b^3}+\frac {\left (11-18 a+6 a^2\right ) \sqrt {1-a-b x} (1+a+b x)^{3/2}}{6 b^3}+\frac {(1-a)^2 (1+a+b x)^{5/2}}{b^3 \sqrt {1-a-b x}}+\frac {\sqrt {1-a-b x} (1+a+b x)^{5/2}}{3 b^3}-\frac {\left (11-18 a+6 a^2\right ) \int \frac {1}{\sqrt {(1-a) (1+a)-2 a b x-b^2 x^2}} \, dx}{2 b^2}\\ &=\frac {\left (11-18 a+6 a^2\right ) \sqrt {1-a-b x} \sqrt {1+a+b x}}{2 b^3}+\frac {\left (11-18 a+6 a^2\right ) \sqrt {1-a-b x} (1+a+b x)^{3/2}}{6 b^3}+\frac {(1-a)^2 (1+a+b x)^{5/2}}{b^3 \sqrt {1-a-b x}}+\frac {\sqrt {1-a-b x} (1+a+b x)^{5/2}}{3 b^3}+\frac {\left (11-18 a+6 a^2\right ) \text {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^4}\\ &=\frac {\left (11-18 a+6 a^2\right ) \sqrt {1-a-b x} \sqrt {1+a+b x}}{2 b^3}+\frac {\left (11-18 a+6 a^2\right ) \sqrt {1-a-b x} (1+a+b x)^{3/2}}{6 b^3}+\frac {(1-a)^2 (1+a+b x)^{5/2}}{b^3 \sqrt {1-a-b x}}+\frac {\sqrt {1-a-b x} (1+a+b x)^{5/2}}{3 b^3}-\frac {\left (11-18 a+6 a^2\right ) \sin ^{-1}(a+b x)}{2 b^3}\\ \end {align*}

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Mathematica [A]
time = 0.15, size = 170, normalized size = 1.01 \begin {gather*} \frac {-\frac {\sqrt {b} \sqrt {1+a+b x} \left (-52-53 a^2+2 a^3+19 b x+7 b^2 x^2+2 b^3 x^3+a (103-16 b x)\right )}{\sqrt {1-a-b x}}+108 a \sqrt {-b} \sinh ^{-1}\left (\frac {\sqrt {-b} \sqrt {1-a-b x}}{\sqrt {2} \sqrt {b}}\right )+6 \left (11+6 a^2\right ) \sqrt {-b} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {1-a-b x}}{\sqrt {2} \sqrt {-b}}\right )}{6 b^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-((Sqrt[b]*Sqrt[1 + a + b*x]*(-52 - 53*a^2 + 2*a^3 + 19*b*x + 7*b^2*x^2 + 2*b^3*x^3 + a*(103 - 16*b*x)))/Sqrt
[1 - a - b*x]) + 108*a*Sqrt[-b]*ArcSinh[(Sqrt[-b]*Sqrt[1 - a - b*x])/(Sqrt[2]*Sqrt[b])] + 6*(11 + 6*a^2)*Sqrt[
-b]*ArcSinh[(Sqrt[b]*Sqrt[1 - a - b*x])/(Sqrt[2]*Sqrt[-b])])/(6*b^(7/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1909\) vs. \(2(144)=288\).
time = 0.08, size = 1910, normalized size = 11.37

method	result	size
risch	\(-\frac {\left (2 b^{2} x^{2}-2 a b x +2 a^{2}+9 b x -27 a +28\right ) \left (b^{2} x^{2}+2 a b x +a^{2}-1\right )}{6 b^{3} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {3 \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right ) a^{2}}{b^{2} \sqrt {b^{2}}}+\frac {9 \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right ) a}{b^{2} \sqrt {b^{2}}}-\frac {11 \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{2 b^{2} \sqrt {b^{2}}}-\frac {4 \sqrt {-\left (x +\frac {-1+a}{b}\right )^{2} b^{2}-2 b \left (x +\frac {-1+a}{b}\right )}\, a^{2}}{b^{4} \left (x -\frac {1}{b}+\frac {a}{b}\right )}+\frac {8 \sqrt {-\left (x +\frac {-1+a}{b}\right )^{2} b^{2}-2 b \left (x +\frac {-1+a}{b}\right )}\, a}{b^{4} \left (x -\frac {1}{b}+\frac {a}{b}\right )}-\frac {4 \sqrt {-\left (x +\frac {-1+a}{b}\right )^{2} b^{2}-2 b \left (x +\frac {-1+a}{b}\right )}}{b^{4} \left (x -\frac {1}{b}+\frac {a}{b}\right )}\)	\(369\)
default	\(\text {Expression too large to display}\)	\(1910\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^3*(-1/3*x^4/b^2/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)-7/3/b*a*(-1/2*x^3/b^2/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)-5/2/b*a*
(-x^2/b^2/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)-3/b*a*(x/b^2/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)-1/b*a*(1/b^2/(-b^2*x^2-2*
a*b*x-a^2+1)^(1/2)-2/b*a*(-2*b^2*x-2*a*b)/(-4*b^2*(-a^2+1)-4*b^2*a^2)/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2))-1/b^2/(b
^2)^(1/2)*arctan((b^2)^(1/2)*(x+1/b*a)/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)))+2*(-a^2+1)/b^2*(1/b^2/(-b^2*x^2-2*a*b*
x-a^2+1)^(1/2)-2/b*a*(-2*b^2*x-2*a*b)/(-4*b^2*(-a^2+1)-4*b^2*a^2)/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)))+3/2*(-a^2+1
)/b^2*(x/b^2/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)-1/b*a*(1/b^2/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)-2/b*a*(-2*b^2*x-2*a*b)
/(-4*b^2*(-a^2+1)-4*b^2*a^2)/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2))-1/b^2/(b^2)^(1/2)*arctan((b^2)^(1/2)*(x+1/b*a)/(-
b^2*x^2-2*a*b*x-a^2+1)^(1/2))))+4/3*(-a^2+1)/b^2*(-x^2/b^2/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)-3/b*a*(x/b^2/(-b^2*x
^2-2*a*b*x-a^2+1)^(1/2)-1/b*a*(1/b^2/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)-2/b*a*(-2*b^2*x-2*a*b)/(-4*b^2*(-a^2+1)-4*
b^2*a^2)/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2))-1/b^2/(b^2)^(1/2)*arctan((b^2)^(1/2)*(x+1/b*a)/(-b^2*x^2-2*a*b*x-a^2+
1)^(1/2)))+2*(-a^2+1)/b^2*(1/b^2/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)-2/b*a*(-2*b^2*x-2*a*b)/(-4*b^2*(-a^2+1)-4*b^2*
a^2)/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2))))+3*(1+a)*b^2*(-1/2*x^3/b^2/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)-5/2/b*a*(-x^2/
b^2/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)-3/b*a*(x/b^2/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)-1/b*a*(1/b^2/(-b^2*x^2-2*a*b*x-
a^2+1)^(1/2)-2/b*a*(-2*b^2*x-2*a*b)/(-4*b^2*(-a^2+1)-4*b^2*a^2)/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2))-1/b^2/(b^2)^(1
/2)*arctan((b^2)^(1/2)*(x+1/b*a)/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)))+2*(-a^2+1)/b^2*(1/b^2/(-b^2*x^2-2*a*b*x-a^2+
1)^(1/2)-2/b*a*(-2*b^2*x-2*a*b)/(-4*b^2*(-a^2+1)-4*b^2*a^2)/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)))+3/2*(-a^2+1)/b^2*
(x/b^2/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)-1/b*a*(1/b^2/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)-2/b*a*(-2*b^2*x-2*a*b)/(-4*b
^2*(-a^2+1)-4*b^2*a^2)/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2))-1/b^2/(b^2)^(1/2)*arctan((b^2)^(1/2)*(x+1/b*a)/(-b^2*x^
2-2*a*b*x-a^2+1)^(1/2))))+3*(1+a)^2*b*(-x^2/b^2/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)-3/b*a*(x/b^2/(-b^2*x^2-2*a*b*x-
a^2+1)^(1/2)-1/b*a*(1/b^2/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)-2/b*a*(-2*b^2*x-2*a*b)/(-4*b^2*(-a^2+1)-4*b^2*a^2)/(-
b^2*x^2-2*a*b*x-a^2+1)^(1/2))-1/b^2/(b^2)^(1/2)*arctan((b^2)^(1/2)*(x+1/b*a)/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)))+
2*(-a^2+1)/b^2*(1/b^2/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)-2/b*a*(-2*b^2*x-2*a*b)/(-4*b^2*(-a^2+1)-4*b^2*a^2)/(-b^2*
x^2-2*a*b*x-a^2+1)^(1/2)))+(1+a)^3*(x/b^2/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)-1/b*a*(1/b^2/(-b^2*x^2-2*a*b*x-a^2+1)
^(1/2)-2/b*a*(-2*b^2*x-2*a*b)/(-4*b^2*(-a^2+1)-4*b^2*a^2)/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2))-1/b^2/(b^2)^(1/2)*ar
ctan((b^2)^(1/2)*(x+1/b*a)/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1645 vs. \(2 (142) = 284\).
time = 0.47, size = 1645, normalized size = 9.79 \begin {gather*} -\frac {35 \, a^{5} x}{{\left (a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}} - \frac {b x^{4}}{3 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}} + \frac {265 \, {\left (a^{2} - 1\right )} a^{3} x}{6 \, {\left (a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}} + \frac {7 \, a x^{3}}{6 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}} - \frac {35 \, {\left (a^{2} - 1\right )} a^{4}}{6 \, {\left (a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} b} - \frac {61 \, {\left (a^{2} - 1\right )}^{2} a x}{6 \, {\left (a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}} + \frac {2 \, {\left (a^{3} + 3 \, a^{2} + 3 \, a + 1\right )} a^{2} x}{{\left (a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}} + \frac {45 \, {\left (a b^{2} + b^{2}\right )} a^{4} x}{{\left (a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} b^{2}} - \frac {18 \, {\left (a^{2} b + 2 \, a b + b\right )} a^{3} x}{{\left (a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} b} - \frac {35 \, a^{2} x^{2}}{6 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} b} + \frac {29 \, {\left (a^{2} - 1\right )}^{2} a^{2}}{6 \, {\left (a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} b} - \frac {{\left (a^{3} + 3 \, a^{2} + 3 \, a + 1\right )} {\left (a^{2} - 1\right )} x}{{\left (a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}} - \frac {93 \, {\left (a b^{2} + b^{2}\right )} {\left (a^{2} - 1\right )} a^{2} x}{2 \, {\left (a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} b^{2}} + \frac {15 \, {\left (a^{2} b + 2 \, a b + b\right )} {\left (a^{2} - 1\right )} a x}{{\left (a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} b} + \frac {4 \, {\left (a^{2} - 1\right )} x^{2}}{3 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} b} - \frac {3 \, {\left (a b^{2} + b^{2}\right )} x^{3}}{2 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} b^{2}} - \frac {35 \, a^{3} \arcsin \left (-\frac {b^{2} x + a b}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}}\right )}{2 \, b^{3}} + \frac {15 \, {\left (a b^{2} + b^{2}\right )} {\left (a^{2} - 1\right )} a^{3}}{2 \, {\left (a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} b^{3}} - \frac {3 \, {\left (a^{2} b + 2 \, a b + b\right )} {\left (a^{2} - 1\right )} a^{2}}{{\left (a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} b^{2}} + \frac {{\left (a^{3} + 3 \, a^{2} + 3 \, a + 1\right )} {\left (a^{2} - 1\right )} a}{{\left (a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} b} + \frac {9 \, {\left (a b^{2} + b^{2}\right )} {\left (a^{2} - 1\right )}^{2} x}{2 \, {\left (a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} b^{2}} + \frac {15 \, {\left (a b^{2} + b^{2}\right )} a x^{2}}{2 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} b^{3}} - \frac {3 \, {\left (a^{2} b + 2 \, a b + b\right )} x^{2}}{\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} b^{2}} + \frac {15 \, {\left (a^{2} - 1\right )} a \arcsin \left (-\frac {b^{2} x + a b}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}}\right )}{2 \, b^{3}} - \frac {9 \, {\left (a b^{2} + b^{2}\right )} {\left (a^{2} - 1\right )}^{2} a}{2 \, {\left (a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} b^{3}} - \frac {35 \, {\left (a^{2} - 1\right )} a^{2}}{3 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} b^{3}} + \frac {45 \, {\left (a b^{2} + b^{2}\right )} a^{2} \arcsin \left (-\frac {b^{2} x + a b}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}}\right )}{2 \, b^{5}} - \frac {9 \, {\left (a^{2} b + 2 \, a b + b\right )} a \arcsin \left (-\frac {b^{2} x + a b}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}}\right )}{b^{4}} + \frac {{\left (a^{3} + 3 \, a^{2} + 3 \, a + 1\right )} \arcsin \left (-\frac {b^{2} x + a b}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}}\right )}{b^{3}} + \frac {8 \, {\left (a^{2} - 1\right )}^{2}}{3 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} b^{3}} - \frac {9 \, {\left (a b^{2} + b^{2}\right )} {\left (a^{2} - 1\right )} \arcsin \left (-\frac {b^{2} x + a b}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}}\right )}{2 \, b^{5}} + \frac {15 \, {\left (a b^{2} + b^{2}\right )} {\left (a^{2} - 1\right )} a}{\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} b^{5}} - \frac {6 \, {\left (a^{2} b + 2 \, a b + b\right )} {\left (a^{2} - 1\right )}}{\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-35*a^5*x/((a^2*b^2 - (a^2 - 1)*b^2)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)) - 1/3*b*x^4/sqrt(-b^2*x^2 - 2*a*b*x -
a^2 + 1) + 265/6*(a^2 - 1)*a^3*x/((a^2*b^2 - (a^2 - 1)*b^2)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)) + 7/6*a*x^3/s
qrt(-b^2*x^2 - 2*a*b*x - a^2 + 1) - 35/6*(a^2 - 1)*a^4/((a^2*b^2 - (a^2 - 1)*b^2)*sqrt(-b^2*x^2 - 2*a*b*x - a^
2 + 1)*b) - 61/6*(a^2 - 1)^2*a*x/((a^2*b^2 - (a^2 - 1)*b^2)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)) + 2*(a^3 + 3*a
^2 + 3*a + 1)*a^2*x/((a^2*b^2 - (a^2 - 1)*b^2)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)) + 45*(a*b^2 + b^2)*a^4*x/((
a^2*b^2 - (a^2 - 1)*b^2)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*b^2) - 18*(a^2*b + 2*a*b + b)*a^3*x/((a^2*b^2 - (a
^2 - 1)*b^2)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*b) - 35/6*a^2*x^2/(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*b) + 29/
6*(a^2 - 1)^2*a^2/((a^2*b^2 - (a^2 - 1)*b^2)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*b) - (a^3 + 3*a^2 + 3*a + 1)*(
a^2 - 1)*x/((a^2*b^2 - (a^2 - 1)*b^2)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)) - 93/2*(a*b^2 + b^2)*(a^2 - 1)*a^2*x
/((a^2*b^2 - (a^2 - 1)*b^2)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*b^2) + 15*(a^2*b + 2*a*b + b)*(a^2 - 1)*a*x/((a
^2*b^2 - (a^2 - 1)*b^2)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*b) + 4/3*(a^2 - 1)*x^2/(sqrt(-b^2*x^2 - 2*a*b*x - a
^2 + 1)*b) - 3/2*(a*b^2 + b^2)*x^3/(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*b^2) - 35/2*a^3*arcsin(-(b^2*x + a*b)/s
qrt(a^2*b^2 - (a^2 - 1)*b^2))/b^3 + 15/2*(a*b^2 + b^2)*(a^2 - 1)*a^3/((a^2*b^2 - (a^2 - 1)*b^2)*sqrt(-b^2*x^2
- 2*a*b*x - a^2 + 1)*b^3) - 3*(a^2*b + 2*a*b + b)*(a^2 - 1)*a^2/((a^2*b^2 - (a^2 - 1)*b^2)*sqrt(-b^2*x^2 - 2*a
*b*x - a^2 + 1)*b^2) + (a^3 + 3*a^2 + 3*a + 1)*(a^2 - 1)*a/((a^2*b^2 - (a^2 - 1)*b^2)*sqrt(-b^2*x^2 - 2*a*b*x
- a^2 + 1)*b) + 9/2*(a*b^2 + b^2)*(a^2 - 1)^2*x/((a^2*b^2 - (a^2 - 1)*b^2)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*
b^2) + 15/2*(a*b^2 + b^2)*a*x^2/(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*b^3) - 3*(a^2*b + 2*a*b + b)*x^2/(sqrt(-b^
2*x^2 - 2*a*b*x - a^2 + 1)*b^2) + 15/2*(a^2 - 1)*a*arcsin(-(b^2*x + a*b)/sqrt(a^2*b^2 - (a^2 - 1)*b^2))/b^3 -
9/2*(a*b^2 + b^2)*(a^2 - 1)^2*a/((a^2*b^2 - (a^2 - 1)*b^2)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*b^3) - 35/3*(a^2
- 1)*a^2/(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*b^3) + 45/2*(a*b^2 + b^2)*a^2*arcsin(-(b^2*x + a*b)/sqrt(a^2*b^2
- (a^2 - 1)*b^2))/b^5 - 9*(a^2*b + 2*a*b + b)*a*arcsin(-(b^2*x + a*b)/sqrt(a^2*b^2 - (a^2 - 1)*b^2))/b^4 + (a
^3 + 3*a^2 + 3*a + 1)*arcsin(-(b^2*x + a*b)/sqrt(a^2*b^2 - (a^2 - 1)*b^2))/b^3 + 8/3*(a^2 - 1)^2/(sqrt(-b^2*x^
2 - 2*a*b*x - a^2 + 1)*b^3) - 9/2*(a*b^2 + b^2)*(a^2 - 1)*arcsin(-(b^2*x + a*b)/sqrt(a^2*b^2 - (a^2 - 1)*b^2))
/b^5 + 15*(a*b^2 + b^2)*(a^2 - 1)*a/(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*b^5) - 6*(a^2*b + 2*a*b + b)*(a^2 - 1)
/(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*b^4)

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Fricas [A]
time = 0.39, size = 159, normalized size = 0.95 \begin {gather*} \frac {3 \, {\left (6 \, a^{3} + {\left (6 \, a^{2} - 18 \, a + 11\right )} b x - 24 \, a^{2} + 29 \, a - 11\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 (2 \, b^{3} x^{3} + 7 \, b^{2} x^{2} + 2 \, a^{3} - {\left (16 \, a - 19\right )} b x - 53 \, a^{2} + 103 \, a - 52\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{6 \, {\left (b^{4} x + {\left (a - 1\right )} b^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(3*(6*a^3 + (6*a^2 - 18*a + 11)*b*x - 24*a^2 + 29*a - 11)*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)) + (2*b^3*x^3 + 7*b^2*x^2 + 2*a^3 - (16*a - 19)*b*x - 53*a^2 + 103*a - 52)*s
qrt(-b^2*x^2 - 2*a*b*x - a^2 + 1))/(b^4*x + (a - 1)*b^3)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} \left (a + b x + 1\right )^{3}}{\left (- \left (a + b x - 1\right ) \left (a + b x + 1\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.44, size = 166, normalized size = 0.99 \begin {gather*} \frac {1}{6} \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (x {\left (\frac {2 \, x}{b} - \frac {2 \, a b^{6} - 9 \, b^{6}}{b^{8}}\right )} + \frac {2 \, a^{2} b^{5} - 27 \, a b^{5} + 28 \, b^{5}}{b^{8}}\right )} + \frac {{\left (6 \, a^{2} - 18 \, a + 11\right )} \arcsin \left (-b x - a\right ) \mathrm {sgn}\left (b\right )}{2 \, b^{2} {\left | b \right |}} + \frac {8 \, {\left (a^{2} - 2 \, a + 1\right )}}{b^{2} {\left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b}{b^{2} x + a b} - 1\right )} {\left | b \right |}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(x*(2*x/b - (2*a*b^6 - 9*b^6)/b^8) + (2*a^2*b^5 - 27*a*b^5 + 28*b^5)/b^
8) + 1/2*(6*a^2 - 18*a + 11)*arcsin(-b*x - a)*sgn(b)/(b^2*abs(b)) + 8*(a^2 - 2*a + 1)/(b^2*((sqrt(-b^2*x^2 - 2
*a*b*x - a^2 + 1)*abs(b) + b)/(b^2*x + a*b) - 1)*abs(b))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^2\,{\left (a+b\,x+1\right )}^3}{{\left (1-{\left (a+b\,x\right )}^2\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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