3.9.35 \(\int e^{3 \tanh ^{-1}(a+b x)} x^3 \, dx\) [835]

Optimal. Leaf size=187 \[ \frac {3 \left (17-44 a+36 a^2-8 a^3\right ) \sqrt {1-a-b x} \sqrt {1+a+b x}}{8 b^4}+\frac {2 x^3 (1+a+b x)^{3/2}}{b \sqrt {1-a-b x}}+\frac {9 x^2 \sqrt {1-a-b x} (1+a+b x)^{3/2}}{4 b^2}+\frac {\sqrt {1-a-b x} (1+a+b x)^{3/2} \left (29-54 a+22 a^2+2 (11-10 a) b x\right )}{8 b^4}-\frac {3 \left (17-44 a+36 a^2-8 a^3\right ) \text {ArcSin}(a+b x)}{8 b^4} \]

[Out]

-3/8*(-8*a^3+36*a^2-44*a+17)*arcsin(b*x+a)/b^4+2*x^3*(b*x+a+1)^(3/2)/b/(-b*x-a+1)^(1/2)+9/4*x^2*(b*x+a+1)^(3/2
)*(-b*x-a+1)^(1/2)/b^2+1/8*(b*x+a+1)^(3/2)*(29-54*a+22*a^2+2*(11-10*a)*b*x)*(-b*x-a+1)^(1/2)/b^4+3/8*(-8*a^3+3
6*a^2-44*a+17)*(-b*x-a+1)^(1/2)*(b*x+a+1)^(1/2)/b^4

________________________________________________________________________________________

Rubi [A]
time = 0.13, antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {6298, 99, 158, 152, 52, 55, 633, 222} \begin {gather*} \frac {\sqrt {-a-b x+1} (a+b x+1)^{3/2} \left (22 a^2+2 (11-10 a) b x-54 a+29\right )}{8 b^4}-\frac {3 \left (-8 a^3+36 a^2-44 a+17\right ) \text {ArcSin}(a+b x)}{8 b^4}+\frac {3 \left (-8 a^3+36 a^2-44 a+17\right ) \sqrt {-a-b x+1} \sqrt {a+b x+1}}{8 b^4}+\frac {9 x^2 \sqrt {-a-b x+1} (a+b x+1)^{3/2}}{4 b^2}+\frac {2 x^3 (a+b x+1)^{3/2}}{b \sqrt {-a-b x+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(3*ArcTanh[a + b*x])*x^3,x]

[Out]

(3*(17 - 44*a + 36*a^2 - 8*a^3)*Sqrt[1 - a - b*x]*Sqrt[1 + a + b*x])/(8*b^4) + (2*x^3*(1 + a + b*x)^(3/2))/(b*
Sqrt[1 - a - b*x]) + (9*x^2*Sqrt[1 - a - b*x]*(1 + a + b*x)^(3/2))/(4*b^2) + (Sqrt[1 - a - b*x]*(1 + a + b*x)^
(3/2)*(29 - 54*a + 22*a^2 + 2*(11 - 10*a)*b*x))/(8*b^4) - (3*(17 - 44*a + 36*a^2 - 8*a^3)*ArcSin[a + b*x])/(8*
b^4)

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 99

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

Rule 152

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 158

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

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^3 \, dx &=\int \frac {x^3 (1+a+b x)^{3/2}}{(1-a-b x)^{3/2}} \, dx\\ &=\frac {2 x^3 (1+a+b x)^{3/2}}{b \sqrt {1-a-b x}}-\frac {2 \int \frac {x^2 \sqrt {1+a+b x} \left (3 (1+a)+\frac {9 b x}{2}\right )}{\sqrt {1-a-b x}} \, dx}{b}\\ &=\frac {2 x^3 (1+a+b x)^{3/2}}{b \sqrt {1-a-b x}}+\frac {9 x^2 \sqrt {1-a-b x} (1+a+b x)^{3/2}}{4 b^2}+\frac {\int \frac {x \sqrt {1+a+b x} \left (-9 (1-a) (1+a) b-\frac {3}{2} (11-10 a) b^2 x\right )}{\sqrt {1-a-b x}} \, dx}{2 b^3}\\ &=\frac {2 x^3 (1+a+b x)^{3/2}}{b \sqrt {1-a-b x}}+\frac {9 x^2 \sqrt {1-a-b x} (1+a+b x)^{3/2}}{4 b^2}+\frac {\sqrt {1-a-b x} (1+a+b x)^{3/2} \left (29-54 a+22 a^2+2 (11-10 a) b x\right )}{8 b^4}-\frac {\left (3 \left (17-44 a+36 a^2-8 a^3\right )\right ) \int \frac {\sqrt {1+a+b x}}{\sqrt {1-a-b x}} \, dx}{8 b^3}\\ &=\frac {3 \left (17-44 a+36 a^2-8 a^3\right ) \sqrt {1-a-b x} \sqrt {1+a+b x}}{8 b^4}+\frac {2 x^3 (1+a+b x)^{3/2}}{b \sqrt {1-a-b x}}+\frac {9 x^2 \sqrt {1-a-b x} (1+a+b x)^{3/2}}{4 b^2}+\frac {\sqrt {1-a-b x} (1+a+b x)^{3/2} \left (29-54 a+22 a^2+2 (11-10 a) b x\right )}{8 b^4}-\frac {\left (3 \left (17-44 a+36 a^2-8 a^3\right )\right ) \int \frac {1}{\sqrt {1-a-b x} \sqrt {1+a+b x}} \, dx}{8 b^3}\\ &=\frac {3 \left (17-44 a+36 a^2-8 a^3\right ) \sqrt {1-a-b x} \sqrt {1+a+b x}}{8 b^4}+\frac {2 x^3 (1+a+b x)^{3/2}}{b \sqrt {1-a-b x}}+\frac {9 x^2 \sqrt {1-a-b x} (1+a+b x)^{3/2}}{4 b^2}+\frac {\sqrt {1-a-b x} (1+a+b x)^{3/2} \left (29-54 a+22 a^2+2 (11-10 a) b x\right )}{8 b^4}-\frac {\left (3 \left (17-44 a+36 a^2-8 a^3\right )\right ) \int \frac {1}{\sqrt {(1-a) (1+a)-2 a b x-b^2 x^2}} \, dx}{8 b^3}\\ &=\frac {3 \left (17-44 a+36 a^2-8 a^3\right ) \sqrt {1-a-b x} \sqrt {1+a+b x}}{8 b^4}+\frac {2 x^3 (1+a+b x)^{3/2}}{b \sqrt {1-a-b x}}+\frac {9 x^2 \sqrt {1-a-b x} (1+a+b x)^{3/2}}{4 b^2}+\frac {\sqrt {1-a-b x} (1+a+b x)^{3/2} \left (29-54 a+22 a^2+2 (11-10 a) b x\right )}{8 b^4}+\frac {\left (3 \left (17-44 a+36 a^2-8 a^3\right )\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 )}{16 b^5}\\ &=\frac {3 \left (17-44 a+36 a^2-8 a^3\right ) \sqrt {1-a-b x} \sqrt {1+a+b x}}{8 b^4}+\frac {2 x^3 (1+a+b x)^{3/2}}{b \sqrt {1-a-b x}}+\frac {9 x^2 \sqrt {1-a-b x} (1+a+b x)^{3/2}}{4 b^2}+\frac {\sqrt {1-a-b x} (1+a+b x)^{3/2} \left (29-54 a+22 a^2+2 (11-10 a) b x\right )}{8 b^4}-\frac {3 \left (17-44 a+36 a^2-8 a^3\right ) \sin ^{-1}(a+b x)}{8 b^4}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.20, size = 203, normalized size = 1.09 \begin {gather*} \frac {-\frac {\sqrt {b} \sqrt {1+a+b x} \left (-80+78 a^3-2 a^4+29 b x+11 b^2 x^2+6 b^3 x^3+2 b^4 x^4+a^2 (-233+22 b x)+a \left (237-54 b x-10 b^2 x^2\right )\right )}{\sqrt {1-a-b x}}+24 a \left (11+2 a^2\right ) \sqrt {-b} \sinh ^{-1}\left (\frac {\sqrt {-b} \sqrt {1-a-b x}}{\sqrt {2} \sqrt {b}}\right )+6 \left (17+36 a^2\right ) \sqrt {-b} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {1-a-b x}}{\sqrt {2} \sqrt {-b}}\right )}{8 b^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-((Sqrt[b]*Sqrt[1 + a + b*x]*(-80 + 78*a^3 - 2*a^4 + 29*b*x + 11*b^2*x^2 + 6*b^3*x^3 + 2*b^4*x^4 + a^2*(-233
+ 22*b*x) + a*(237 - 54*b*x - 10*b^2*x^2)))/Sqrt[1 - a - b*x]) + 24*a*(11 + 2*a^2)*Sqrt[-b]*ArcSinh[(Sqrt[-b]*
Sqrt[1 - a - b*x])/(Sqrt[2]*Sqrt[b])] + 6*(17 + 36*a^2)*Sqrt[-b]*ArcSinh[(Sqrt[b]*Sqrt[1 - a - b*x])/(Sqrt[2]*
Sqrt[-b])])/(8*b^(9/2))

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(3195\) vs. \(2(163)=326\).
time = 0.08, size = 3196, normalized size = 17.09

method	result	size
risch	\(\frac {\left (-2 b^{3} x^{3}+2 a \,b^{2} x^{2}-2 a^{2} b x -8 b^{2} x^{2}+2 a^{3}+20 a b x -44 a^{2}-19 b x +93 a -48\right ) \left (b^{2} x^{2}+2 a b x +a^{2}-1\right )}{8 b^{4} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}+\frac {3 \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right ) a^{3}}{b^{3} \sqrt {b^{2}}}-\frac {27 \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right ) a^{2}}{2 b^{3} \sqrt {b^{2}}}+\frac {33 \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right ) a}{2 b^{3} \sqrt {b^{2}}}-\frac {51 \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{8 b^{3} \sqrt {b^{2}}}+\frac {4 \sqrt {-\left (x +\frac {-1+a}{b}\right )^{2} b^{2}-2 b \left (x +\frac {-1+a}{b}\right )}\, a^{3}}{b^{5} \left (x -\frac {1}{b}+\frac {a}{b}\right )}-\frac {12 \sqrt {-\left (x +\frac {-1+a}{b}\right )^{2} b^{2}-2 b \left (x +\frac {-1+a}{b}\right )}\, a^{2}}{b^{5} \left (x -\frac {1}{b}+\frac {a}{b}\right )}+\frac {12 \sqrt {-\left (x +\frac {-1+a}{b}\right )^{2} b^{2}-2 b \left (x +\frac {-1+a}{b}\right )}\, a}{b^{5} \left (x -\frac {1}{b}+\frac {a}{b}\right )}-\frac {4 \sqrt {-\left (x +\frac {-1+a}{b}\right )^{2} b^{2}-2 b \left (x +\frac {-1+a}{b}\right )}}{b^{5} \left (x -\frac {1}{b}+\frac {a}{b}\right )}\)	\(500\)
default	\(\text {Expression too large to display}\)	\(3196\)

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^3,x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 2349 vs. \(2 (163) = 326\).
time = 0.49, size = 2349, normalized size = 12.56 \begin {gather*} \text {Too large to display} \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^3,x, algorithm="maxima")

[Out]

-1/4*b*x^5/sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1) + 315/4*a^6*x/((a^2*b^2 - (a^2 - 1)*b^2)*sqrt(-b^2*x^2 - 2*a*b*x
- a^2 + 1)*b) + 3/4*a*x^4/sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1) - 945/8*(a^2 - 1)*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) - 21/8*a^2*x^3/(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*b) + 105/8*(a^2 -
1)*a^5/((a^2*b^2 - (a^2 - 1)*b^2)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*b^2) - 105*(a*b^2 + b^2)*a^5*x/((a^2*b^2
- (a^2 - 1)*b^2)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*b^3) + 45*(a^2*b + 2*a*b + b)*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) + 169/4*(a^2 - 1)^2*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) - 6*(a^3 + 3*a^2 + 3*a + 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)*b) + 105/8*a^3*x^2/(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*b^2) + 5/8*(a^2 - 1)*x^3/(sqrt(-b^2*x^2 - 2
*a*b*x - a^2 + 1)*b) - (a*b^2 + b^2)*x^4/(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*b^2) - 14*(a^2 - 1)^2*a^3/((a^2*b
^2 - (a^2 - 1)*b^2)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*b^2) + 265/2*(a*b^2 + b^2)*(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)*b^3) - 93/2*(a^2*b + 2*a*b + b)*(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) - 15/8*(a^2 - 1)^3*x/((a^2*b^2 - (a^2 - 1)*b^2)*sqrt(-b
^2*x^2 - 2*a*b*x - a^2 + 1)*b) + 5*(a^3 + 3*a^2 + 3*a + 1)*(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) - 49/8*(a^2 - 1)*a*x^2/(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*b^2) + 7/2*(a*b^2 + b^2
)*a*x^3/(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*b^3) - 3/2*(a^2*b + 2*a*b + b)*x^3/(sqrt(-b^2*x^2 - 2*a*b*x - a^2
+ 1)*b^2) + 315/8*a^4*arcsin(-(b^2*x + a*b)/sqrt(a^2*b^2 - (a^2 - 1)*b^2))/b^4 - 35/2*(a*b^2 + b^2)*(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^4) + 15/2*(a^2*b + 2*a*b + b)*(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) + 15/8*(a^2 - 1)^3*a/((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^2/((a^2*b^2 - (a^2 - 1)*b
^2)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*b^2) - 61/2*(a*b^2 + b^2)*(a^2 - 1)^2*a*x/((a^2*b^2 - (a^2 - 1)*b^2)*sq
rt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*b^3) + 9/2*(a^2*b + 2*a*b + b)*(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) - 35/2*(a*b^2 + b^2)*a^2*x^2/(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*b^4) + 15/
2*(a^2*b + 2*a*b + b)*a*x^2/(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*b^3) - (a^3 + 3*a^2 + 3*a + 1)*x^2/(sqrt(-b^2*
x^2 - 2*a*b*x - a^2 + 1)*b^2) - 105/4*(a^2 - 1)*a^2*arcsin(-(b^2*x + a*b)/sqrt(a^2*b^2 - (a^2 - 1)*b^2))/b^4 +
29/2*(a*b^2 + b^2)*(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^4) + 105/4
*(a^2 - 1)*a^3/(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*b^4) - 9/2*(a^2*b + 2*a*b + b)*(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) + 4*(a*b^2 + b^2)*(a^2 - 1)*x^2/(sqrt(-b^2*x^2 - 2*a*b*x
- a^2 + 1)*b^4) - 105/2*(a*b^2 + b^2)*a^3*arcsin(-(b^2*x + a*b)/sqrt(a^2*b^2 - (a^2 - 1)*b^2))/b^6 + 45/2*(a^2
*b + 2*a*b + b)*a^2*arcsin(-(b^2*x + a*b)/sqrt(a^2*b^2 - (a^2 - 1)*b^2))/b^5 + 15/8*(a^2 - 1)^2*arcsin(-(b^2*x
+ a*b)/sqrt(a^2*b^2 - (a^2 - 1)*b^2))/b^4 - 3*(a^3 + 3*a^2 + 3*a + 1)*a*arcsin(-(b^2*x + a*b)/sqrt(a^2*b^2 -
(a^2 - 1)*b^2))/b^4 - 49/4*(a^2 - 1)^2*a/(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*b^4) + 45/2*(a*b^2 + b^2)*(a^2 -
1)*a*arcsin(-(b^2*x + a*b)/sqrt(a^2*b^2 - (a^2 - 1)*b^2))/b^6 - 9/2*(a^2*b + 2*a*b + b)*(a^2 - 1)*arcsin(-(b^2
*x + a*b)/sqrt(a^2*b^2 - (a^2 - 1)*b^2))/b^5 - 35*(a*b^2 + b^2)*(a^2 - 1)*a^2/(sqrt(-b^2*x^2 - 2*a*b*x - a^2 +
1)*b^6) + 15*(a^2*b + 2*a*b + b)*(a^2 - 1)*a/(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*b^5) - 2*(a^3 + 3*a^2 + 3*a
+ 1)*(a^2 - 1)/(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*b^4) + 8*(a*b^2 + b^2)*(a^2 - 1)^2/(sqrt(-b^2*x^2 - 2*a*b*x
- a^2 + 1)*b^6)

________________________________________________________________________________________

Fricas [A]
time = 0.41, size = 192, normalized size = 1.03 \begin {gather*} -\frac {3 \, {\left (8 \, a^{4} - 44 \, a^{3} + {\left (8 \, a^{3} - 36 \, a^{2} + 44 \, a - 17\right )} b x + 80 \, a^{2} - 61 \, a + 17\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^{4} x^{4} + 6 \, b^{3} x^{3} - {\left (10 \, a - 11\right )} b^{2} x^{2} - 2 \, a^{4} + 78 \, a^{3} + {\left (22 \, a^{2} - 54 \, a + 29\right )} b x - 233 \, a^{2} + 237 \, a - 80\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{8 \, {\left (b^{5} x + {\left (a - 1\right )} b^{4}\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^3,x, algorithm="fricas")

[Out]

-1/8*(3*(8*a^4 - 44*a^3 + (8*a^3 - 36*a^2 + 44*a - 17)*b*x + 80*a^2 - 61*a + 17)*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^4*x^4 + 6*b^3*x^3 - (10*a - 11)*b^2*x^2 - 2*a^4 +
78*a^3 + (22*a^2 - 54*a + 29)*b*x - 233*a^2 + 237*a - 80)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1))/(b^5*x + (a - 1
)*b^4)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3} \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**3,x)

[Out]

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

________________________________________________________________________________________

Giac [A]
time = 0.43, size = 211, normalized size = 1.13 \begin {gather*} \frac {1}{8} \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left ({\left (2 \, x {\left (\frac {x}{b} - \frac {a b^{11} - 4 \, b^{11}}{b^{13}}\right )} + \frac {2 \, a^{2} b^{10} - 20 \, a b^{10} + 19 \, b^{10}}{b^{13}}\right )} x - \frac {2 \, a^{3} b^{9} - 44 \, a^{2} b^{9} + 93 \, a b^{9} - 48 \, b^{9}}{b^{13}}\right )} - \frac {3 \, {\left (8 \, a^{3} - 36 \, a^{2} + 44 \, a - 17\right )} \arcsin \left (-b x - a\right ) \mathrm {sgn}\left (b\right )}{8 \, b^{3} {\left | b \right |}} - \frac {8 \, {\left (a^{3} - 3 \, a^{2} + 3 \, a - 1\right )}}{b^{3} {\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^3,x, algorithm="giac")

[Out]

1/8*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*((2*x*(x/b - (a*b^11 - 4*b^11)/b^13) + (2*a^2*b^10 - 20*a*b^10 + 19*b^1
0)/b^13)*x - (2*a^3*b^9 - 44*a^2*b^9 + 93*a*b^9 - 48*b^9)/b^13) - 3/8*(8*a^3 - 36*a^2 + 44*a - 17)*arcsin(-b*x
- a)*sgn(b)/(b^3*abs(b)) - 8*(a^3 - 3*a^2 + 3*a - 1)/(b^3*((sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)/(b
^2*x + a*b) - 1)*abs(b))

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________