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

Optimal. Leaf size=156 \[ -\frac {(-a-b x+1)^{3/2} \sqrt {a+b x+1} \left (18 a^2-2 (6 a+1) b x+10 a+7\right )}{24 b^4}-\frac {\left (8 a^3+12 a^2+12 a+3\right ) \sqrt {-a-b x+1} \sqrt {a+b x+1}}{8 b^4}-\frac {\left (8 a^3+12 a^2+12 a+3\right ) \sin ^{-1}(a+b x)}{8 b^4}-\frac {x^2 (-a-b x+1)^{3/2} \sqrt {a+b x+1}}{4 b^2} \]

[Out]

-1/8*(8*a^3+12*a^2+12*a+3)*arcsin(b*x+a)/b^4-1/4*x^2*(-b*x-a+1)^(3/2)*(b*x+a+1)^(1/2)/b^2-1/24*(-b*x-a+1)^(3/2
)*(7+10*a+18*a^2-2*(1+6*a)*b*x)*(b*x+a+1)^(1/2)/b^4-1/8*(8*a^3+12*a^2+12*a+3)*(-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 = 156, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6163, 100, 147, 50, 53, 619, 216} \[ -\frac {(-a-b x+1)^{3/2} \sqrt {a+b x+1} \left (18 a^2-2 (6 a+1) b x+10 a+7\right )}{24 b^4}-\frac {\left (8 a^3+12 a^2+12 a+3\right ) \sqrt {-a-b x+1} \sqrt {a+b x+1}}{8 b^4}-\frac {\left (8 a^3+12 a^2+12 a+3\right ) \sin ^{-1}(a+b x)}{8 b^4}-\frac {x^2 (-a-b x+1)^{3/2} \sqrt {a+b x+1}}{4 b^2} \]

Antiderivative was successfully verified.

[In]

Int[x^3/E^ArcTanh[a + b*x],x]

[Out]

-((3 + 12*a + 12*a^2 + 8*a^3)*Sqrt[1 - a - b*x]*Sqrt[1 + a + b*x])/(8*b^4) - (x^2*(1 - a - b*x)^(3/2)*Sqrt[1 +
a + b*x])/(4*b^2) - ((1 - a - b*x)^(3/2)*Sqrt[1 + a + b*x]*(7 + 10*a + 18*a^2 - 2*(1 + 6*a)*b*x))/(24*b^4) -
((3 + 12*a + 12*a^2 + 8*a^3)*ArcSin[a + b*x])/(8*b^4)

Rule 50

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 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 100

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

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 6163

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^{-\tanh ^{-1}(a+b x)} x^3 \, dx &=\int \frac {x^3 \sqrt {1-a-b x}}{\sqrt {1+a+b x}} \, dx\\ &=-\frac {x^2 (1-a-b x)^{3/2} \sqrt {1+a+b x}}{4 b^2}-\frac {\int \frac {x \sqrt {1-a-b x} \left (-2 \left (1-a^2\right )+(1+6 a) b x\right )}{\sqrt {1+a+b x}} \, dx}{4 b^2}\\ &=-\frac {x^2 (1-a-b x)^{3/2} \sqrt {1+a+b x}}{4 b^2}-\frac {(1-a-b x)^{3/2} \sqrt {1+a+b x} \left (7+10 a+18 a^2-2 (1+6 a) b x\right )}{24 b^4}-\frac {\left (3+12 a+12 a^2+8 a^3\right ) \int \frac {\sqrt {1-a-b x}}{\sqrt {1+a+b x}} \, dx}{8 b^3}\\ &=-\frac {\left (3+12 a+12 a^2+8 a^3\right ) \sqrt {1-a-b x} \sqrt {1+a+b x}}{8 b^4}-\frac {x^2 (1-a-b x)^{3/2} \sqrt {1+a+b x}}{4 b^2}-\frac {(1-a-b x)^{3/2} \sqrt {1+a+b x} \left (7+10 a+18 a^2-2 (1+6 a) b x\right )}{24 b^4}-\frac {\left (3+12 a+12 a^2+8 a^3\right ) \int \frac {1}{\sqrt {1-a-b x} \sqrt {1+a+b x}} \, dx}{8 b^3}\\ &=-\frac {\left (3+12 a+12 a^2+8 a^3\right ) \sqrt {1-a-b x} \sqrt {1+a+b x}}{8 b^4}-\frac {x^2 (1-a-b x)^{3/2} \sqrt {1+a+b x}}{4 b^2}-\frac {(1-a-b x)^{3/2} \sqrt {1+a+b x} \left (7+10 a+18 a^2-2 (1+6 a) b x\right )}{24 b^4}-\frac {\left (3+12 a+12 a^2+8 a^3\right ) \int \frac {1}{\sqrt {(1-a) (1+a)-2 a b x-b^2 x^2}} \, dx}{8 b^3}\\ &=-\frac {\left (3+12 a+12 a^2+8 a^3\right ) \sqrt {1-a-b x} \sqrt {1+a+b x}}{8 b^4}-\frac {x^2 (1-a-b x)^{3/2} \sqrt {1+a+b x}}{4 b^2}-\frac {(1-a-b x)^{3/2} \sqrt {1+a+b x} \left (7+10 a+18 a^2-2 (1+6 a) b x\right )}{24 b^4}+\frac {\left (3+12 a+12 a^2+8 a^3\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 )}{16 b^5}\\ &=-\frac {\left (3+12 a+12 a^2+8 a^3\right ) \sqrt {1-a-b x} \sqrt {1+a+b x}}{8 b^4}-\frac {x^2 (1-a-b x)^{3/2} \sqrt {1+a+b x}}{4 b^2}-\frac {(1-a-b x)^{3/2} \sqrt {1+a+b x} \left (7+10 a+18 a^2-2 (1+6 a) b x\right )}{24 b^4}-\frac {\left (3+12 a+12 a^2+8 a^3\right ) \sin ^{-1}(a+b x)}{8 b^4}\\ \end {align*}

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Mathematica [A]  time = 0.44, size = 160, normalized size = 1.03 \[ \frac {\frac {6 \left (8 a^3+12 a^2+12 a+3\right ) \sqrt {b} \sinh ^{-1}\left (\frac {\sqrt {-b} \sqrt {-a-b x+1}}{\sqrt {2} \sqrt {b}}\right )}{\sqrt {-b}}+\frac {\sqrt {a+b x+1} \left (6 a^4+38 a^3+5 a^2 (6 b x-1)+a \left (-18 b^2 x^2+50 b x-23\right )-6 b^4 x^4+14 b^3 x^3-17 b^2 x^2+25 b x-16\right )}{\sqrt {-a-b x+1}}}{24 b^4} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^3/E^ArcTanh[a + b*x],x]

[Out]

((Sqrt[1 + a + b*x]*(-16 + 38*a^3 + 6*a^4 + 25*b*x - 17*b^2*x^2 + 14*b^3*x^3 - 6*b^4*x^4 + 5*a^2*(-1 + 6*b*x)
+ a*(-23 + 50*b*x - 18*b^2*x^2)))/Sqrt[1 - a - b*x] + (6*(3 + 12*a + 12*a^2 + 8*a^3)*Sqrt[b]*ArcSinh[(Sqrt[-b]
*Sqrt[1 - a - b*x])/(Sqrt[2]*Sqrt[b])])/Sqrt[-b])/(24*b^4)

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fricas [A]  time = 0.68, size = 143, normalized size = 0.92 \[ \frac {3 \, {\left (8 \, a^{3} + 12 \, a^{2} + 12 \, a + 3\right )} \arctan \left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (b x + a\right )}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) + {\left (6 \, b^{3} x^{3} - 2 \, {\left (3 \, a + 4\right )} b^{2} x^{2} - 6 \, a^{3} + {\left (6 \, a^{2} + 20 \, a + 9\right )} b x - 44 \, a^{2} - 39 \, a - 16\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{24 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(3*(8*a^3 + 12*a^2 + 12*a + 3)*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)) + (6*b^3*x^3 - 2*(3*a + 4)*b^2*x^2 - 6*a^3 + (6*a^2 + 20*a + 9)*b*x - 44*a^2 - 39*a - 16)*sqrt(-b^2*x
^2 - 2*a*b*x - a^2 + 1))/b^4

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giac [A]  time = 0.22, size = 148, normalized size = 0.95 \[ \frac {1}{24} \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left ({\left (2 \, x {\left (\frac {3 \, x}{b} - \frac {3 \, a b^{5} + 4 \, b^{5}}{b^{7}}\right )} + \frac {6 \, a^{2} b^{4} + 20 \, a b^{4} + 9 \, b^{4}}{b^{7}}\right )} x - \frac {6 \, a^{3} b^{3} + 44 \, a^{2} b^{3} + 39 \, a b^{3} + 16 \, b^{3}}{b^{7}}\right )} + \frac {{\left (8 \, a^{3} + 12 \, a^{2} + 12 \, a + 3\right )} \arcsin \left (-b x - a\right ) \mathrm {sgn}\relax (b)}{8 \, b^{3} {\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*((2*x*(3*x/b - (3*a*b^5 + 4*b^5)/b^7) + (6*a^2*b^4 + 20*a*b^4 + 9*b^4)
/b^7)*x - (6*a^3*b^3 + 44*a^2*b^3 + 39*a*b^3 + 16*b^3)/b^7) + 1/8*(8*a^3 + 12*a^2 + 12*a + 3)*arcsin(-b*x - a)
*sgn(b)/(b^3*abs(b))

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maple [B]  time = 0.05, size = 809, normalized size = 5.19 \[ \frac {3 a^{2} x \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{2 b^{3}}+\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 )}{2 b^{3} \sqrt {b^{2}}}+\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 )}{2 b^{3} \sqrt {b^{2}}}-\frac {\arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {1+a}{b}-\frac {1}{b}\right )}{\sqrt {-\left (x +\frac {1+a}{b}\right )^{2} b^{2}+2 b \left (x +\frac {1+a}{b}\right )}}\right ) a^{3}}{b^{3} \sqrt {b^{2}}}-\frac {3 \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {1+a}{b}-\frac {1}{b}\right )}{\sqrt {-\left (x +\frac {1+a}{b}\right )^{2} b^{2}+2 b \left (x +\frac {1+a}{b}\right )}}\right ) a^{2}}{b^{3} \sqrt {b^{2}}}-\frac {3 \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {1+a}{b}-\frac {1}{b}\right )}{\sqrt {-\left (x +\frac {1+a}{b}\right )^{2} b^{2}+2 b \left (x +\frac {1+a}{b}\right )}}\right ) a}{b^{3} \sqrt {b^{2}}}+\frac {3 a x \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{2 b^{3}}-\frac {\sqrt {-\left (x +\frac {1+a}{b}\right )^{2} b^{2}+2 b \left (x +\frac {1+a}{b}\right )}}{b^{4}}+\frac {\left (-b^{2} x^{2}-2 a b x -a^{2}+1\right )^{\frac {3}{2}}}{3 b^{4}}+\frac {3 a^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{2 b^{4}}-\frac {\sqrt {-\left (x +\frac {1+a}{b}\right )^{2} b^{2}+2 b \left (x +\frac {1+a}{b}\right )}\, a^{3}}{b^{4}}-\frac {3 \sqrt {-\left (x +\frac {1+a}{b}\right )^{2} b^{2}+2 b \left (x +\frac {1+a}{b}\right )}\, a^{2}}{b^{4}}-\frac {3 \sqrt {-\left (x +\frac {1+a}{b}\right )^{2} b^{2}+2 b \left (x +\frac {1+a}{b}\right )}\, a}{b^{4}}-\frac {\arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {1+a}{b}-\frac {1}{b}\right )}{\sqrt {-\left (x +\frac {1+a}{b}\right )^{2} b^{2}+2 b \left (x +\frac {1+a}{b}\right )}}\right )}{b^{3} \sqrt {b^{2}}}+\frac {5 a \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{8 b^{4}}-\frac {x \left (-b^{2} x^{2}-2 a b x -a^{2}+1\right )^{\frac {3}{2}}}{4 b^{3}}+\frac {3 a \left (-b^{2} x^{2}-2 a b x -a^{2}+1\right )^{\frac {3}{2}}}{4 b^{4}}+\frac {3 a^{3} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{2 b^{4}}+\frac {5 x \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{8 b^{3}}+\frac {5 \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}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*a^2/b^3*x*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)+3/2*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))+3/2*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))-1/b^3/(b^2)
^(1/2)*arctan((b^2)^(1/2)*(x+(1+a)/b-1/b)/(-(x+(1+a)/b)^2*b^2+2*b*(x+(1+a)/b))^(1/2))*a^3-3/b^3/(b^2)^(1/2)*ar
ctan((b^2)^(1/2)*(x+(1+a)/b-1/b)/(-(x+(1+a)/b)^2*b^2+2*b*(x+(1+a)/b))^(1/2))*a^2-3/b^3/(b^2)^(1/2)*arctan((b^2
)^(1/2)*(x+(1+a)/b-1/b)/(-(x+(1+a)/b)^2*b^2+2*b*(x+(1+a)/b))^(1/2))*a+3/2*a/b^3*x*(-b^2*x^2-2*a*b*x-a^2+1)^(1/
2)-1/b^4*(-(x+(1+a)/b)^2*b^2+2*b*(x+(1+a)/b))^(1/2)+1/3/b^4*(-b^2*x^2-2*a*b*x-a^2+1)^(3/2)+3/2*a^2/b^4*(-b^2*x
^2-2*a*b*x-a^2+1)^(1/2)-1/b^4*(-(x+(1+a)/b)^2*b^2+2*b*(x+(1+a)/b))^(1/2)*a^3-3/b^4*(-(x+(1+a)/b)^2*b^2+2*b*(x+
(1+a)/b))^(1/2)*a^2-3/b^4*(-(x+(1+a)/b)^2*b^2+2*b*(x+(1+a)/b))^(1/2)*a-1/b^3/(b^2)^(1/2)*arctan((b^2)^(1/2)*(x
+(1+a)/b-1/b)/(-(x+(1+a)/b)^2*b^2+2*b*(x+(1+a)/b))^(1/2))+5/8*a/b^4*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)-1/4/b^3*x*(
-b^2*x^2-2*a*b*x-a^2+1)^(3/2)+3/4/b^4*a*(-b^2*x^2-2*a*b*x-a^2+1)^(3/2)+3/2*a^3/b^4*(-b^2*x^2-2*a*b*x-a^2+1)^(1
/2)+5/8/b^3*x*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)+5/8/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))

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maxima [B]  time = 0.42, size = 338, normalized size = 2.17 \[ \frac {3 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a^{2} x}{2 \, b^{3}} - \frac {a^{3} \arcsin \left (b x + a\right )}{b^{4}} + \frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a^{3}}{2 \, b^{4}} - \frac {{\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac {3}{2}} x}{4 \, b^{3}} + \frac {3 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a x}{2 \, b^{3}} - \frac {3 \, a^{2} \arcsin \left (b x + a\right )}{2 \, b^{4}} + \frac {3 \, {\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac {3}{2}} a}{4 \, b^{4}} - \frac {3 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a^{2}}{2 \, b^{4}} + \frac {5 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} x}{8 \, b^{3}} - \frac {3 \, a \arcsin \left (b x + a\right )}{2 \, b^{4}} + \frac {{\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac {3}{2}}}{3 \, b^{4}} - \frac {19 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a}{8 \, b^{4}} - \frac {3 \, \arcsin \left (b x + a\right )}{8 \, b^{4}} - \frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*a^2*x/b^3 - a^3*arcsin(b*x + a)/b^4 + 1/2*sqrt(-b^2*x^2 - 2*a*b*x - a^2
+ 1)*a^3/b^4 - 1/4*(-b^2*x^2 - 2*a*b*x - a^2 + 1)^(3/2)*x/b^3 + 3/2*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*a*x/b^
3 - 3/2*a^2*arcsin(b*x + a)/b^4 + 3/4*(-b^2*x^2 - 2*a*b*x - a^2 + 1)^(3/2)*a/b^4 - 3/2*sqrt(-b^2*x^2 - 2*a*b*x
- a^2 + 1)*a^2/b^4 + 5/8*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*x/b^3 - 3/2*a*arcsin(b*x + a)/b^4 + 1/3*(-b^2*x^2
- 2*a*b*x - a^2 + 1)^(3/2)/b^4 - 19/8*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*a/b^4 - 3/8*arcsin(b*x + a)/b^4 - sq
rt(-b^2*x^2 - 2*a*b*x - a^2 + 1)/b^4

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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