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

Optimal. Leaf size=165 \[ -\frac {\left (4 a^2+3\right ) a e^{2 \cosh ^{-1}(a+b x)}}{16 b^4}+\frac {\left (4 a^2+3\right ) a \cosh ^{-1}(a+b x)}{8 b^4}+\frac {\left (6 a^2+1\right ) e^{-\cosh ^{-1}(a+b x)}}{8 b^4}+\frac {\left (6 a^2+1\right ) e^{3 \cosh ^{-1}(a+b x)}}{24 b^4}-\frac {3 a e^{-2 \cosh ^{-1}(a+b x)}}{16 b^4}-\frac {3 a e^{4 \cosh ^{-1}(a+b x)}}{32 b^4}+\frac {e^{-3 \cosh ^{-1}(a+b x)}}{48 b^4}+\frac {e^{5 \cosh ^{-1}(a+b x)}}{80 b^4} \]

[Out]

1/48/b^4/(b*x+a+(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2))^3-3/16*a/b^4/(b*x+a+(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2))^2+1/8*(6
*a^2+1)/b^4/(b*x+a+(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2))-1/16*a*(4*a^2+3)*(b*x+a+(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2))^2
/b^4+1/24*(6*a^2+1)*(b*x+a+(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2))^3/b^4-3/32*a*(b*x+a+(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2
))^4/b^4+1/80*(b*x+a+(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2))^5/b^4+1/8*a*(4*a^2+3)*arccosh(b*x+a)/b^4

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Rubi [A]  time = 0.16, antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5899, 2282, 12, 1628} \[ -\frac {\left (4 a^2+3\right ) a e^{2 \cosh ^{-1}(a+b x)}}{16 b^4}+\frac {\left (4 a^2+3\right ) a \cosh ^{-1}(a+b x)}{8 b^4}+\frac {\left (6 a^2+1\right ) e^{-\cosh ^{-1}(a+b x)}}{8 b^4}+\frac {\left (6 a^2+1\right ) e^{3 \cosh ^{-1}(a+b x)}}{24 b^4}-\frac {3 a e^{-2 \cosh ^{-1}(a+b x)}}{16 b^4}-\frac {3 a e^{4 \cosh ^{-1}(a+b x)}}{32 b^4}+\frac {e^{-3 \cosh ^{-1}(a+b x)}}{48 b^4}+\frac {e^{5 \cosh ^{-1}(a+b x)}}{80 b^4} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcCosh[a + b*x]*x^3,x]

[Out]

1/(48*b^4*E^(3*ArcCosh[a + b*x])) - (3*a)/(16*b^4*E^(2*ArcCosh[a + b*x])) + (1 + 6*a^2)/(8*b^4*E^ArcCosh[a + b
*x]) - (a*(3 + 4*a^2)*E^(2*ArcCosh[a + b*x]))/(16*b^4) + ((1 + 6*a^2)*E^(3*ArcCosh[a + b*x]))/(24*b^4) - (3*a*
E^(4*ArcCosh[a + b*x]))/(32*b^4) + E^(5*ArcCosh[a + b*x])/(80*b^4) + (a*(3 + 4*a^2)*ArcCosh[a + b*x])/(8*b^4)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 1628

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 5899

Int[(f_)^(ArcCosh[(a_.) + (b_.)*(x_)]^(n_.)*(c_.))*(x_)^(m_.), x_Symbol] :> Dist[1/b, Subst[Int[(-(a/b) + Cosh
[x]/b)^m*f^(c*x^n)*Sinh[x], x], x, ArcCosh[a + b*x]], x] /; FreeQ[{a, b, c, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int e^{\cosh ^{-1}(a+b x)} x^3 \, dx &=\frac {\operatorname {Subst}\left (\int e^x \left (-\frac {a}{b}+\frac {\cosh (x)}{b}\right )^3 \sinh (x) \, dx,x,\cosh ^{-1}(a+b x)\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\left (-1+x^2\right ) \left (1-2 a x+x^2\right )^3}{16 b^3 x^4} \, dx,x,e^{\cosh ^{-1}(a+b x)}\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\left (-1+x^2\right ) \left (1-2 a x+x^2\right )^3}{x^4} \, dx,x,e^{\cosh ^{-1}(a+b x)}\right )}{16 b^4}\\ &=\frac {\operatorname {Subst}\left (\int \left (-\frac {1}{x^4}+\frac {6 a}{x^3}-\frac {2 \left (1+6 a^2\right )}{x^2}+\frac {2 a \left (3+4 a^2\right )}{x}-2 a \left (3+4 a^2\right ) x+2 \left (1+6 a^2\right ) x^2-6 a x^3+x^4\right ) \, dx,x,e^{\cosh ^{-1}(a+b x)}\right )}{16 b^4}\\ &=\frac {e^{-3 \cosh ^{-1}(a+b x)}}{48 b^4}-\frac {3 a e^{-2 \cosh ^{-1}(a+b x)}}{16 b^4}+\frac {\left (1+6 a^2\right ) e^{-\cosh ^{-1}(a+b x)}}{8 b^4}-\frac {a \left (3+4 a^2\right ) e^{2 \cosh ^{-1}(a+b x)}}{16 b^4}+\frac {\left (1+6 a^2\right ) e^{3 \cosh ^{-1}(a+b x)}}{24 b^4}-\frac {3 a e^{4 \cosh ^{-1}(a+b x)}}{32 b^4}+\frac {e^{5 \cosh ^{-1}(a+b x)}}{80 b^4}+\frac {a \left (3+4 a^2\right ) \cosh ^{-1}(a+b x)}{8 b^4}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 138, normalized size = 0.84 \[ \frac {15 a \left (4 a^2+3\right ) \log \left (\sqrt {a+b x-1} \sqrt {a+b x+1}+a+b x\right )+\sqrt {a+b x-1} \sqrt {a+b x+1} \left (-6 a^4-2 \left (3 a^2+4\right ) b^2 x^2+\left (6 a^2+29\right ) a b x-83 a^2+6 a b^3 x^3+24 b^4 x^4-16\right )+30 a b^4 x^4+24 b^5 x^5}{120 b^4} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^ArcCosh[a + b*x]*x^3,x]

[Out]

(30*a*b^4*x^4 + 24*b^5*x^5 + Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x]*(-16 - 83*a^2 - 6*a^4 + a*(29 + 6*a^2)*b*x -
2*(4 + 3*a^2)*b^2*x^2 + 6*a*b^3*x^3 + 24*b^4*x^4) + 15*a*(3 + 4*a^2)*Log[a + b*x + Sqrt[-1 + a + b*x]*Sqrt[1
+ a + b*x]])/(120*b^4)

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fricas [A]  time = 0.78, size = 133, normalized size = 0.81 \[ \frac {24 \, b^{5} x^{5} + 30 \, a b^{4} x^{4} + {\left (24 \, b^{4} x^{4} + 6 \, a b^{3} x^{3} - 2 \, {\left (3 \, a^{2} + 4\right )} b^{2} x^{2} - 6 \, a^{4} + {\left (6 \, a^{3} + 29 \, a\right )} b x - 83 \, a^{2} - 16\right )} \sqrt {b x + a + 1} \sqrt {b x + a - 1} - 15 \, {\left (4 \, a^{3} + 3 \, a\right )} \log \left (-b x + \sqrt {b x + a + 1} \sqrt {b x + a - 1} - a\right )}{120 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/120*(24*b^5*x^5 + 30*a*b^4*x^4 + (24*b^4*x^4 + 6*a*b^3*x^3 - 2*(3*a^2 + 4)*b^2*x^2 - 6*a^4 + (6*a^3 + 29*a)*
b*x - 83*a^2 - 16)*sqrt(b*x + a + 1)*sqrt(b*x + a - 1) - 15*(4*a^3 + 3*a)*log(-b*x + sqrt(b*x + a + 1)*sqrt(b*
x + a - 1) - a))/b^4

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giac [B]  time = 0.25, size = 564, normalized size = 3.42 \[ \frac {24 \, b^{2} x^{5} + 30 \, a b x^{4} + 5 \, {\left ({\left (b x + a + 1\right )} {\left (2 \, {\left (b x + a + 1\right )} {\left (\frac {3 \, {\left (b x + a + 1\right )}}{b^{3}} - \frac {12 \, a b^{12} + 13 \, b^{12}}{b^{15}}\right )} + \frac {36 \, a^{2} b^{12} + 84 \, a b^{12} + 43 \, b^{12}}{b^{15}}\right )} - \frac {3 \, {\left (8 \, a^{3} b^{12} + 36 \, a^{2} b^{12} + 36 \, a b^{12} + 13 \, b^{12}\right )}}{b^{15}}\right )} \sqrt {b x + a + 1} \sqrt {b x + a - 1} + 5 \, {\left ({\left ({\left (b x + a + 1\right )} {\left (2 \, {\left (b x + a + 1\right )} {\left (\frac {3 \, {\left (b x + a + 1\right )}}{b^{3}} - \frac {12 \, a b^{12} + 13 \, b^{12}}{b^{15}}\right )} + \frac {36 \, a^{2} b^{12} + 84 \, a b^{12} + 43 \, b^{12}}{b^{15}}\right )} - \frac {3 \, {\left (8 \, a^{3} b^{12} + 36 \, a^{2} b^{12} + 36 \, a b^{12} + 13 \, b^{12}\right )}}{b^{15}}\right )} \sqrt {b x + a + 1} \sqrt {b x + a - 1} - \frac {6 \, {\left (8 \, a^{3} + 12 \, a^{2} + 12 \, a + 3\right )} \log \left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}{b^{3}}\right )} a + {\left ({\left ({\left (2 \, {\left (b x + a + 1\right )} {\left (3 \, {\left (b x + a + 1\right )} {\left (\frac {4 \, {\left (b x + a + 1\right )}}{b^{4}} - \frac {20 \, a b^{20} + 21 \, b^{20}}{b^{24}}\right )} + \frac {120 \, a^{2} b^{20} + 260 \, a b^{20} + 133 \, b^{20}}{b^{24}}\right )} - \frac {5 \, {\left (48 \, a^{3} b^{20} + 168 \, a^{2} b^{20} + 172 \, a b^{20} + 59 \, b^{20}\right )}}{b^{24}}\right )} {\left (b x + a + 1\right )} + \frac {15 \, {\left (8 \, a^{4} b^{20} + 48 \, a^{3} b^{20} + 72 \, a^{2} b^{20} + 52 \, a b^{20} + 13 \, b^{20}\right )}}{b^{24}}\right )} \sqrt {b x + a + 1} \sqrt {b x + a - 1} + \frac {30 \, {\left (8 \, a^{4} + 16 \, a^{3} + 24 \, a^{2} + 12 \, a + 3\right )} \log \left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}{b^{4}}\right )} b - \frac {30 \, {\left (8 \, a^{3} + 12 \, a^{2} + 12 \, a + 3\right )} \log \left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}{b^{3}}}{120 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/120*(24*b^2*x^5 + 30*a*b*x^4 + 5*((b*x + a + 1)*(2*(b*x + a + 1)*(3*(b*x + a + 1)/b^3 - (12*a*b^12 + 13*b^12
)/b^15) + (36*a^2*b^12 + 84*a*b^12 + 43*b^12)/b^15) - 3*(8*a^3*b^12 + 36*a^2*b^12 + 36*a*b^12 + 13*b^12)/b^15)
*sqrt(b*x + a + 1)*sqrt(b*x + a - 1) + 5*(((b*x + a + 1)*(2*(b*x + a + 1)*(3*(b*x + a + 1)/b^3 - (12*a*b^12 +
13*b^12)/b^15) + (36*a^2*b^12 + 84*a*b^12 + 43*b^12)/b^15) - 3*(8*a^3*b^12 + 36*a^2*b^12 + 36*a*b^12 + 13*b^12
)/b^15)*sqrt(b*x + a + 1)*sqrt(b*x + a - 1) - 6*(8*a^3 + 12*a^2 + 12*a + 3)*log(sqrt(b*x + a + 1) - sqrt(b*x +
a - 1))/b^3)*a + (((2*(b*x + a + 1)*(3*(b*x + a + 1)*(4*(b*x + a + 1)/b^4 - (20*a*b^20 + 21*b^20)/b^24) + (12
0*a^2*b^20 + 260*a*b^20 + 133*b^20)/b^24) - 5*(48*a^3*b^20 + 168*a^2*b^20 + 172*a*b^20 + 59*b^20)/b^24)*(b*x +
a + 1) + 15*(8*a^4*b^20 + 48*a^3*b^20 + 72*a^2*b^20 + 52*a*b^20 + 13*b^20)/b^24)*sqrt(b*x + a + 1)*sqrt(b*x +
a - 1) + 30*(8*a^4 + 16*a^3 + 24*a^2 + 12*a + 3)*log(sqrt(b*x + a + 1) - sqrt(b*x + a - 1))/b^4)*b - 30*(8*a^
3 + 12*a^2 + 12*a + 3)*log(sqrt(b*x + a + 1) - sqrt(b*x + a - 1))/b^3)/b

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maple [C]  time = 0.04, size = 376, normalized size = 2.28 \[ \frac {\sqrt {b x +a -1}\, \sqrt {b x +a +1}\, \left (24 \,\mathrm {csgn}\relax (b ) x^{4} b^{4} \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}+6 \,\mathrm {csgn}\relax (b ) x^{3} a \,b^{3} \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}-6 \,\mathrm {csgn}\relax (b ) x^{2} a^{2} b^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}+6 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \mathrm {csgn}\relax (b ) x \,a^{3} b -8 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \mathrm {csgn}\relax (b ) x^{2} b^{2}-6 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \mathrm {csgn}\relax (b ) a^{4}+29 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \mathrm {csgn}\relax (b ) x a b -83 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \mathrm {csgn}\relax (b ) a^{2}+60 \ln \left (\left (\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \mathrm {csgn}\relax (b )+b x +a \right ) \mathrm {csgn}\relax (b )\right ) a^{3}-16 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \mathrm {csgn}\relax (b )+45 \ln \left (\left (\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \mathrm {csgn}\relax (b )+b x +a \right ) \mathrm {csgn}\relax (b )\right ) a \right ) \mathrm {csgn}\relax (b )}{120 b^{4} \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}}+\frac {b \,x^{5}}{5}+\frac {x^{4} a}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/120*(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2)*(24*csgn(b)*x^4*b^4*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)+6*csgn(b)*x^3*a*b^3*(b
^2*x^2+2*a*b*x+a^2-1)^(1/2)-6*csgn(b)*x^2*a^2*b^2*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)+6*(b^2*x^2+2*a*b*x+a^2-1)^(1/2
)*csgn(b)*x*a^3*b-8*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)*csgn(b)*x^2*b^2-6*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)*csgn(b)*a^4+
29*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)*csgn(b)*x*a*b-83*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)*csgn(b)*a^2+60*ln(((b^2*x^2+2*
a*b*x+a^2-1)^(1/2)*csgn(b)+b*x+a)*csgn(b))*a^3-16*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)*csgn(b)+45*ln(((b^2*x^2+2*a*b*
x+a^2-1)^(1/2)*csgn(b)+b*x+a)*csgn(b))*a)*csgn(b)/b^4/(b^2*x^2+2*a*b*x+a^2-1)^(1/2)+1/5*b*x^5+1/4*x^4*a

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maxima [B]  time = 0.33, size = 495, normalized size = 3.00 \[ \frac {1}{5} \, b x^{5} + \frac {1}{4} \, a x^{4} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )}^{\frac {3}{2}} x^{2}}{5 \, b^{2}} - \frac {{\left (a^{2} - 1\right )} a^{3} \log \left (2 \, b^{2} x + 2 \, a b + 2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} b\right )}{5 \, b^{4}} - \frac {7 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )}^{\frac {3}{2}} a x}{20 \, b^{3}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} {\left (a^{2} - 1\right )} a x}{5 \, b^{3}} + \frac {{\left (a^{2} - 1\right )}^{2} a \log \left (2 \, b^{2} x + 2 \, a b + 2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} b\right )}{5 \, b^{4}} + \frac {7 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )}^{\frac {3}{2}} a^{2}}{12 \, b^{4}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} {\left (a^{2} - 1\right )} a^{2}}{5 \, b^{4}} + \frac {7 \, {\left (5 \, a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}\right )} a^{3} \log \left (2 \, b^{2} x + 2 \, a b + 2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} b\right )}{40 \, b^{6}} - \frac {2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )}^{\frac {3}{2}} {\left (a^{2} - 1\right )}}{15 \, b^{4}} - \frac {7 \, {\left (5 \, a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} a x}{40 \, b^{5}} - \frac {7 \, {\left (5 \, a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}\right )} {\left (a^{2} - 1\right )} a \log \left (2 \, b^{2} x + 2 \, a b + 2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} b\right )}{40 \, b^{6}} - \frac {7 \, {\left (5 \, a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} a^{2}}{40 \, b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*b*x^5 + 1/4*a*x^4 + 1/5*(b^2*x^2 + 2*a*b*x + a^2 - 1)^(3/2)*x^2/b^2 - 1/5*(a^2 - 1)*a^3*log(2*b^2*x + 2*a*
b + 2*sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1)*b)/b^4 - 7/20*(b^2*x^2 + 2*a*b*x + a^2 - 1)^(3/2)*a*x/b^3 + 1/5*sqrt(b
^2*x^2 + 2*a*b*x + a^2 - 1)*(a^2 - 1)*a*x/b^3 + 1/5*(a^2 - 1)^2*a*log(2*b^2*x + 2*a*b + 2*sqrt(b^2*x^2 + 2*a*b
*x + a^2 - 1)*b)/b^4 + 7/12*(b^2*x^2 + 2*a*b*x + a^2 - 1)^(3/2)*a^2/b^4 + 1/5*sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1
)*(a^2 - 1)*a^2/b^4 + 7/40*(5*a^2*b^2 - (a^2 - 1)*b^2)*a^3*log(2*b^2*x + 2*a*b + 2*sqrt(b^2*x^2 + 2*a*b*x + a^
2 - 1)*b)/b^6 - 2/15*(b^2*x^2 + 2*a*b*x + a^2 - 1)^(3/2)*(a^2 - 1)/b^4 - 7/40*(5*a^2*b^2 - (a^2 - 1)*b^2)*sqrt
(b^2*x^2 + 2*a*b*x + a^2 - 1)*a*x/b^5 - 7/40*(5*a^2*b^2 - (a^2 - 1)*b^2)*(a^2 - 1)*a*log(2*b^2*x + 2*a*b + 2*s
qrt(b^2*x^2 + 2*a*b*x + a^2 - 1)*b)/b^6 - 7/40*(5*a^2*b^2 - (a^2 - 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1)*a
^2/b^6

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mupad [B]  time = 69.81, size = 1408, normalized size = 8.53 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*x^4)/4 - ((((a - 1)^(1/2) - (a + b*x - 1)^(1/2))*((3*a)/2 + 2*a^3))/(b^4*((a + 1)^(1/2) - (a + b*x + 1)^(1/
2))) + (((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^19*((3*a)/2 + 2*a^3))/(b^4*((a + 1)^(1/2) - (a + b*x + 1)^(1/2))
^19) - (((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^3*((29*a)/2 + (58*a^3)/3))/(b^4*((a + 1)^(1/2) - (a + b*x + 1)^(
1/2))^3) - (((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^17*((29*a)/2 + (58*a^3)/3))/(b^4*((a + 1)^(1/2) - (a + b*x +
1)^(1/2))^17) - (((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^5*(654*a - (4552*a^3)/3 + (3584*a^5)/5))/(b^4*((a + 1)
^(1/2) - (a + b*x + 1)^(1/2))^5) - (((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^15*(654*a - (4552*a^3)/3 + (3584*a^5
)/5))/(b^4*((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^15) - (((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^7*(4622*a - 1602
4*a^3 + 11776*a^5))/(b^4*((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^7) - (((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^13*
(4622*a - 16024*a^3 + 11776*a^5))/(b^4*((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^13) - (((a - 1)^(1/2) - (a + b*x
- 1)^(1/2))^9*(11095*a - 48012*a^3 + 39936*a^5))/(b^4*((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^9) - (((a - 1)^(1/
2) - (a + b*x - 1)^(1/2))^11*(11095*a - 48012*a^3 + 39936*a^5))/(b^4*((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^11)
+ (((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^4*(a - 1)^(1/2)*(a + 1)^(1/2)*(64*a^4 - 128*a^2 + 64))/(b^4*((a + 1)
^(1/2) - (a + b*x + 1)^(1/2))^4) + (((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^16*(a - 1)^(1/2)*(a + 1)^(1/2)*(64*a
^4 - 128*a^2 + 64))/(b^4*((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^16) + (((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^6*
(a - 1)^(1/2)*(a + 1)^(1/2)*(3712*a^4 - (12544*a^2)/3 + 1408/3))/(b^4*((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^6)
+ (((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^14*(a - 1)^(1/2)*(a + 1)^(1/2)*(3712*a^4 - (12544*a^2)/3 + 1408/3))/
(b^4*((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^14) + (((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^8*(a - 1)^(1/2)*(a + 1
)^(1/2)*(25536*a^4 - (56960*a^2)/3 + 4928/3))/(b^4*((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^8) + (((a - 1)^(1/2)
- (a + b*x - 1)^(1/2))^12*(a - 1)^(1/2)*(a + 1)^(1/2)*(25536*a^4 - (56960*a^2)/3 + 4928/3))/(b^4*((a + 1)^(1/2
) - (a + b*x + 1)^(1/2))^12) + (((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^10*(a - 1)^(1/2)*(a + 1)^(1/2)*((231168*
a^4)/5 - (160256*a^2)/5 + 11008/5))/(b^4*((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^10))/((45*((a - 1)^(1/2) - (a +
b*x - 1)^(1/2))^4)/((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^4 - (10*((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^2)/((a
+ 1)^(1/2) - (a + b*x + 1)^(1/2))^2 - (120*((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^6)/((a + 1)^(1/2) - (a + b*x
+ 1)^(1/2))^6 + (210*((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^8)/((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^8 - (252*
((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^10)/((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^10 + (210*((a - 1)^(1/2) - (a
+ b*x - 1)^(1/2))^12)/((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^12 - (120*((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^14
)/((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^14 + (45*((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^16)/((a + 1)^(1/2) - (a
+ b*x + 1)^(1/2))^16 - (10*((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^18)/((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^18
+ ((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^20/((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^20 + 1) + (b*x^5)/5 + (a*ata
nh(((a - 1)^(1/2) - (a + b*x - 1)^(1/2))/((a + 1)^(1/2) - (a + b*x + 1)^(1/2)))*(4*a^2 + 3))/(2*b^4)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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