3.115 \(\int (c e+d e x)^2 (a+b \cosh ^{-1}(c+d x))^3 \, dx\)

Optimal. Leaf size=262 \[ \frac {2 b^2 e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )}{9 d}+\frac {4}{3} a b^2 e^2 x-\frac {b e^2 \sqrt {c+d x-1} (c+d x)^2 \sqrt {c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{3 d}-\frac {2 b e^2 \sqrt {c+d x-1} \sqrt {c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{3 d}+\frac {e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{3 d}-\frac {2 b^3 e^2 \sqrt {c+d x-1} (c+d x)^2 \sqrt {c+d x+1}}{27 d}-\frac {40 b^3 e^2 \sqrt {c+d x-1} \sqrt {c+d x+1}}{27 d}+\frac {4 b^3 e^2 (c+d x) \cosh ^{-1}(c+d x)}{3 d} \]

[Out]

4/3*a*b^2*e^2*x+4/3*b^3*e^2*(d*x+c)*arccosh(d*x+c)/d+2/9*b^2*e^2*(d*x+c)^3*(a+b*arccosh(d*x+c))/d+1/3*e^2*(d*x
+c)^3*(a+b*arccosh(d*x+c))^3/d-40/27*b^3*e^2*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/d-2/27*b^3*e^2*(d*x+c)^2*(d*x+c-1
)^(1/2)*(d*x+c+1)^(1/2)/d-2/3*b*e^2*(a+b*arccosh(d*x+c))^2*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/d-1/3*b*e^2*(d*x+c)
^2*(a+b*arccosh(d*x+c))^2*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/d

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Rubi [A]  time = 0.47, antiderivative size = 262, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {5866, 12, 5662, 5759, 5718, 5654, 74, 100} \[ \frac {2 b^2 e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )}{9 d}+\frac {4}{3} a b^2 e^2 x-\frac {b e^2 \sqrt {c+d x-1} (c+d x)^2 \sqrt {c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{3 d}-\frac {2 b e^2 \sqrt {c+d x-1} \sqrt {c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{3 d}+\frac {e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{3 d}-\frac {2 b^3 e^2 \sqrt {c+d x-1} (c+d x)^2 \sqrt {c+d x+1}}{27 d}-\frac {40 b^3 e^2 \sqrt {c+d x-1} \sqrt {c+d x+1}}{27 d}+\frac {4 b^3 e^2 (c+d x) \cosh ^{-1}(c+d x)}{3 d} \]

Antiderivative was successfully verified.

[In]

Int[(c*e + d*e*x)^2*(a + b*ArcCosh[c + d*x])^3,x]

[Out]

(4*a*b^2*e^2*x)/3 - (40*b^3*e^2*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])/(27*d) - (2*b^3*e^2*Sqrt[-1 + c + d*x]*(
c + d*x)^2*Sqrt[1 + c + d*x])/(27*d) + (4*b^3*e^2*(c + d*x)*ArcCosh[c + d*x])/(3*d) + (2*b^2*e^2*(c + d*x)^3*(
a + b*ArcCosh[c + d*x]))/(9*d) - (2*b*e^2*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^2)/(3*
d) - (b*e^2*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^2)/(3*d) + (e^2*(c + d*x
)^3*(a + b*ArcCosh[c + d*x])^3)/(3*d)

Rule 12

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

Rule 74

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] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] &
& EqQ[a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 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 5654

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcCosh[c*x])^n, x] - Dist[b*c*n, In
t[(x*(a + b*ArcCosh[c*x])^(n - 1))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 5662

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcC
osh[c*x])^n)/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcCosh[c*x])^(n - 1))/(Sqr
t[-1 + c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 5718

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_
Symbol] :> Simp[((d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(2*e1*e2*(p + 1)), x] - Dist[
(b*n*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPart[p])/(2*c*(p + 1)*(1 + c*x)^FracPart[p]
*(-1 + c*x)^FracPart[p]), Int[(-1 + c^2*x^2)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c,
d1, e1, d2, e2, p}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && NeQ[p, -1] && IntegerQ[p + 1
/2]

Rule 5759

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_
.)*(x_)]), x_Symbol] :> Simp[(f*(f*x)^(m - 1)*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]*(a + b*ArcCosh[c*x])^n)/(e1*e2*m
), x] + (Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)^(m - 2)*(a + b*ArcCosh[c*x])^n)/(Sqrt[d1 + e1*x]*Sqrt[d2 + e2*
x]), x], x] + Dist[(b*f*n*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/(c*d1*d2*m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]), Int[(f*x)
^(m - 1)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f}, x] && EqQ[e1 - c*d1, 0]
&& EqQ[e2 + c*d2, 0] && GtQ[n, 0] && GtQ[m, 1] && IntegerQ[m]

Rule 5866

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

Rubi steps

\begin {align*} \int (c e+d e x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^3 \, dx &=\frac {\operatorname {Subst}\left (\int e^2 x^2 \left (a+b \cosh ^{-1}(x)\right )^3 \, dx,x,c+d x\right )}{d}\\ &=\frac {e^2 \operatorname {Subst}\left (\int x^2 \left (a+b \cosh ^{-1}(x)\right )^3 \, dx,x,c+d x\right )}{d}\\ &=\frac {e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{3 d}-\frac {\left (b e^2\right ) \operatorname {Subst}\left (\int \frac {x^3 \left (a+b \cosh ^{-1}(x)\right )^2}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{d}\\ &=-\frac {b e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{3 d}+\frac {e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{3 d}-\frac {\left (2 b e^2\right ) \operatorname {Subst}\left (\int \frac {x \left (a+b \cosh ^{-1}(x)\right )^2}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{3 d}+\frac {\left (2 b^2 e^2\right ) \operatorname {Subst}\left (\int x^2 \left (a+b \cosh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{3 d}\\ &=\frac {2 b^2 e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )}{9 d}-\frac {2 b e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{3 d}-\frac {b e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{3 d}+\frac {e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{3 d}+\frac {\left (4 b^2 e^2\right ) \operatorname {Subst}\left (\int \left (a+b \cosh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{3 d}-\frac {\left (2 b^3 e^2\right ) \operatorname {Subst}\left (\int \frac {x^3}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{9 d}\\ &=\frac {4}{3} a b^2 e^2 x-\frac {2 b^3 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{27 d}+\frac {2 b^2 e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )}{9 d}-\frac {2 b e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{3 d}-\frac {b e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{3 d}+\frac {e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{3 d}-\frac {\left (2 b^3 e^2\right ) \operatorname {Subst}\left (\int \frac {2 x}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{27 d}+\frac {\left (4 b^3 e^2\right ) \operatorname {Subst}\left (\int \cosh ^{-1}(x) \, dx,x,c+d x\right )}{3 d}\\ &=\frac {4}{3} a b^2 e^2 x-\frac {2 b^3 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{27 d}+\frac {4 b^3 e^2 (c+d x) \cosh ^{-1}(c+d x)}{3 d}+\frac {2 b^2 e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )}{9 d}-\frac {2 b e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{3 d}-\frac {b e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{3 d}+\frac {e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{3 d}-\frac {\left (4 b^3 e^2\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{27 d}-\frac {\left (4 b^3 e^2\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{3 d}\\ &=\frac {4}{3} a b^2 e^2 x-\frac {40 b^3 e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x}}{27 d}-\frac {2 b^3 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{27 d}+\frac {4 b^3 e^2 (c+d x) \cosh ^{-1}(c+d x)}{3 d}+\frac {2 b^2 e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )}{9 d}-\frac {2 b e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{3 d}-\frac {b e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{3 d}+\frac {e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{3 d}\\ \end {align*}

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Mathematica [A]  time = 0.43, size = 296, normalized size = 1.13 \[ \frac {e^2 \left (a \left (3 a^2+2 b^2\right ) (c+d x)^3+\frac {1}{3} b \sqrt {c+d x-1} \sqrt {c+d x+1} \left (-\left (\left (9 a^2+2 b^2\right ) (c+d x)^2\right )-2 \left (9 a^2+20 b^2\right )\right )-b \cosh ^{-1}(c+d x) \left (-9 a^2 (c+d x)^3+6 a b \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2+12 a b \sqrt {c+d x-1} \sqrt {c+d x+1}-2 b^2 (c+d x)^3-12 b^2 (c+d x)\right )+12 a b^2 (c+d x)-3 b^2 \cosh ^{-1}(c+d x)^2 \left (-3 a (c+d x)^3+b \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2+2 b \sqrt {c+d x-1} \sqrt {c+d x+1}\right )+3 b^3 (c+d x)^3 \cosh ^{-1}(c+d x)^3\right )}{9 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*e + d*e*x)^2*(a + b*ArcCosh[c + d*x])^3,x]

[Out]

(e^2*(12*a*b^2*(c + d*x) + a*(3*a^2 + 2*b^2)*(c + d*x)^3 + (b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]*(-2*(9*a^2
+ 20*b^2) - (9*a^2 + 2*b^2)*(c + d*x)^2))/3 - b*(-12*b^2*(c + d*x) - 9*a^2*(c + d*x)^3 - 2*b^2*(c + d*x)^3 + 1
2*a*b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x] + 6*a*b*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x])*ArcCosh[c
+ d*x] - 3*b^2*(-3*a*(c + d*x)^3 + 2*b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x] + b*Sqrt[-1 + c + d*x]*(c + d*x)^
2*Sqrt[1 + c + d*x])*ArcCosh[c + d*x]^2 + 3*b^3*(c + d*x)^3*ArcCosh[c + d*x]^3))/(9*d)

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fricas [B]  time = 2.08, size = 607, normalized size = 2.32 \[ \frac {3 \, {\left (3 \, a^{3} + 2 \, a b^{2}\right )} d^{3} e^{2} x^{3} + 9 \, {\left (3 \, a^{3} + 2 \, a b^{2}\right )} c d^{2} e^{2} x^{2} + 9 \, {\left (4 \, a b^{2} + {\left (3 \, a^{3} + 2 \, a b^{2}\right )} c^{2}\right )} d e^{2} x + 9 \, {\left (b^{3} d^{3} e^{2} x^{3} + 3 \, b^{3} c d^{2} e^{2} x^{2} + 3 \, b^{3} c^{2} d e^{2} x + b^{3} c^{3} e^{2}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )^{3} + 9 \, {\left (3 \, a b^{2} d^{3} e^{2} x^{3} + 9 \, a b^{2} c d^{2} e^{2} x^{2} + 9 \, a b^{2} c^{2} d e^{2} x + 3 \, a b^{2} c^{3} e^{2} - {\left (b^{3} d^{2} e^{2} x^{2} + 2 \, b^{3} c d e^{2} x + {\left (b^{3} c^{2} + 2 \, b^{3}\right )} e^{2}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )^{2} + 3 \, {\left ({\left (9 \, a^{2} b + 2 \, b^{3}\right )} d^{3} e^{2} x^{3} + 3 \, {\left (9 \, a^{2} b + 2 \, b^{3}\right )} c d^{2} e^{2} x^{2} + 3 \, {\left (4 \, b^{3} + {\left (9 \, a^{2} b + 2 \, b^{3}\right )} c^{2}\right )} d e^{2} x + {\left (12 \, b^{3} c + {\left (9 \, a^{2} b + 2 \, b^{3}\right )} c^{3}\right )} e^{2} - 6 \, {\left (a b^{2} d^{2} e^{2} x^{2} + 2 \, a b^{2} c d e^{2} x + {\left (a b^{2} c^{2} + 2 \, a b^{2}\right )} e^{2}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - {\left ({\left (9 \, a^{2} b + 2 \, b^{3}\right )} d^{2} e^{2} x^{2} + 2 \, {\left (9 \, a^{2} b + 2 \, b^{3}\right )} c d e^{2} x + {\left (18 \, a^{2} b + 40 \, b^{3} + {\left (9 \, a^{2} b + 2 \, b^{3}\right )} c^{2}\right )} e^{2}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}}{27 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^2*(a+b*arccosh(d*x+c))^3,x, algorithm="fricas")

[Out]

1/27*(3*(3*a^3 + 2*a*b^2)*d^3*e^2*x^3 + 9*(3*a^3 + 2*a*b^2)*c*d^2*e^2*x^2 + 9*(4*a*b^2 + (3*a^3 + 2*a*b^2)*c^2
)*d*e^2*x + 9*(b^3*d^3*e^2*x^3 + 3*b^3*c*d^2*e^2*x^2 + 3*b^3*c^2*d*e^2*x + b^3*c^3*e^2)*log(d*x + c + sqrt(d^2
*x^2 + 2*c*d*x + c^2 - 1))^3 + 9*(3*a*b^2*d^3*e^2*x^3 + 9*a*b^2*c*d^2*e^2*x^2 + 9*a*b^2*c^2*d*e^2*x + 3*a*b^2*
c^3*e^2 - (b^3*d^2*e^2*x^2 + 2*b^3*c*d*e^2*x + (b^3*c^2 + 2*b^3)*e^2)*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1))*log(d
*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1))^2 + 3*((9*a^2*b + 2*b^3)*d^3*e^2*x^3 + 3*(9*a^2*b + 2*b^3)*c*d^2*e
^2*x^2 + 3*(4*b^3 + (9*a^2*b + 2*b^3)*c^2)*d*e^2*x + (12*b^3*c + (9*a^2*b + 2*b^3)*c^3)*e^2 - 6*(a*b^2*d^2*e^2
*x^2 + 2*a*b^2*c*d*e^2*x + (a*b^2*c^2 + 2*a*b^2)*e^2)*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1))*log(d*x + c + sqrt(d^
2*x^2 + 2*c*d*x + c^2 - 1)) - ((9*a^2*b + 2*b^3)*d^2*e^2*x^2 + 2*(9*a^2*b + 2*b^3)*c*d*e^2*x + (18*a^2*b + 40*
b^3 + (9*a^2*b + 2*b^3)*c^2)*e^2)*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1))/d

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d e x + c e\right )}^{2} {\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^2*(a+b*arccosh(d*x+c))^3,x, algorithm="giac")

[Out]

integrate((d*e*x + c*e)^2*(b*arccosh(d*x + c) + a)^3, x)

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maple [A]  time = 0.05, size = 326, normalized size = 1.24 \[ \frac {\frac {\left (d x +c \right )^{3} e^{2} a^{3}}{3}+e^{2} b^{3} \left (\frac {\left (d x +c \right )^{3} \mathrm {arccosh}\left (d x +c \right )^{3}}{3}-\frac {2 \mathrm {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{3}-\frac {\left (d x +c \right )^{2} \mathrm {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{3}+\frac {4 \left (d x +c \right ) \mathrm {arccosh}\left (d x +c \right )}{3}-\frac {40 \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{27}+\frac {2 \,\mathrm {arccosh}\left (d x +c \right ) \left (d x +c \right )^{3}}{9}-\frac {2 \left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{27}\right )+3 e^{2} a \,b^{2} \left (\frac {\left (d x +c \right )^{3} \mathrm {arccosh}\left (d x +c \right )^{2}}{3}-\frac {4 \,\mathrm {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{9}-\frac {2 \,\mathrm {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )^{2}}{9}+\frac {4 d x}{9}+\frac {4 c}{9}+\frac {2 \left (d x +c \right )^{3}}{27}\right )+3 e^{2} a^{2} b \left (\frac {\mathrm {arccosh}\left (d x +c \right ) \left (d x +c \right )^{3}}{3}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (\left (d x +c \right )^{2}+2\right )}{9}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*e*x+c*e)^2*(a+b*arccosh(d*x+c))^3,x)

[Out]

1/d*(1/3*(d*x+c)^3*e^2*a^3+e^2*b^3*(1/3*(d*x+c)^3*arccosh(d*x+c)^3-2/3*arccosh(d*x+c)^2*(d*x+c-1)^(1/2)*(d*x+c
+1)^(1/2)-1/3*(d*x+c)^2*arccosh(d*x+c)^2*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)+4/3*(d*x+c)*arccosh(d*x+c)-40/27*(d*x
+c-1)^(1/2)*(d*x+c+1)^(1/2)+2/9*arccosh(d*x+c)*(d*x+c)^3-2/27*(d*x+c)^2*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))+3*e^2
*a*b^2*(1/3*(d*x+c)^3*arccosh(d*x+c)^2-4/9*arccosh(d*x+c)*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)-2/9*arccosh(d*x+c)*(
d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*(d*x+c)^2+4/9*d*x+4/9*c+2/27*(d*x+c)^3)+3*e^2*a^2*b*(1/3*arccosh(d*x+c)*(d*x+c)
^3-1/9*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*((d*x+c)^2+2)))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^2*(a+b*arccosh(d*x+c))^3,x, algorithm="maxima")

[Out]

1/3*a^3*d^2*e^2*x^3 + a^3*c*d*e^2*x^2 + 3/2*(2*x^2*arccosh(d*x + c) - d*(3*c^2*log(2*d^2*x + 2*c*d + 2*sqrt(d^
2*x^2 + 2*c*d*x + c^2 - 1)*d)/d^3 + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*x/d^2 - (c^2 - 1)*log(2*d^2*x + 2*c*d +
2*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*d)/d^3 - 3*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*c/d^3))*a^2*b*c*d*e^2 + 1/6*(
6*x^3*arccosh(d*x + c) - d*(2*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*x^2/d^2 - 15*c^3*log(2*d^2*x + 2*c*d + 2*sqrt(
d^2*x^2 + 2*c*d*x + c^2 - 1)*d)/d^4 - 5*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*c*x/d^3 + 9*(c^2 - 1)*c*log(2*d^2*x
+ 2*c*d + 2*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*d)/d^4 + 15*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*c^2/d^4 - 4*sqrt(d
^2*x^2 + 2*c*d*x + c^2 - 1)*(c^2 - 1)/d^4))*a^2*b*d^2*e^2 + a^3*c^2*e^2*x + 3*((d*x + c)*arccosh(d*x + c) - sq
rt((d*x + c)^2 - 1))*a^2*b*c^2*e^2/d + 1/3*(b^3*d^2*e^2*x^3 + 3*b^3*c*d*e^2*x^2 + 3*b^3*c^2*e^2*x)*log(d*x + s
qrt(d*x + c + 1)*sqrt(d*x + c - 1) + c)^3 + integrate(((3*a*b^2*d^5*e^2 - b^3*d^5*e^2)*x^5 + 5*(3*a*b^2*c*d^4*
e^2 - b^3*c*d^4*e^2)*x^4 + 3*(c^5*e^2 - c^3*e^2)*a*b^2 + (3*(10*c^2*d^3*e^2 - d^3*e^2)*a*b^2 - (10*c^2*d^3*e^2
- d^3*e^2)*b^3)*x^3 + 3*((10*c^3*d^2*e^2 - 3*c*d^2*e^2)*a*b^2 - (3*c^3*d^2*e^2 - c*d^2*e^2)*b^3)*x^2 + ((3*a*
b^2*d^4*e^2 - b^3*d^4*e^2)*x^4 + 3*(c^4*e^2 - c^2*e^2)*a*b^2 + 4*(3*a*b^2*c*d^3*e^2 - b^3*c*d^3*e^2)*x^3 - 3*(
2*b^3*c^2*d^2*e^2 - (6*c^2*d^2*e^2 - d^2*e^2)*a*b^2)*x^2 - 3*(b^3*c^3*d*e^2 - 2*(2*c^3*d*e^2 - c*d*e^2)*a*b^2)
*x)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + 3*((5*c^4*d*e^2 - 3*c^2*d*e^2)*a*b^2 - (c^4*d*e^2 - c^2*d*e^2)*b^3)*
x)*log(d*x + sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c)^2/(d^3*x^3 + 3*c*d^2*x^2 + c^3 + (d^2*x^2 + 2*c*d*x + c^
2 - 1)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + (3*c^2*d - d)*x - c), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*e + d*e*x)^2*(a + b*acosh(c + d*x))^3,x)

[Out]

int((c*e + d*e*x)^2*(a + b*acosh(c + d*x))^3, x)

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sympy [A]  time = 4.72, size = 1173, normalized size = 4.48 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)**2*(a+b*acosh(d*x+c))**3,x)

[Out]

Piecewise((a**3*c**2*e**2*x + a**3*c*d*e**2*x**2 + a**3*d**2*e**2*x**3/3 + a**2*b*c**3*e**2*acosh(c + d*x)/d +
3*a**2*b*c**2*e**2*x*acosh(c + d*x) - a**2*b*c**2*e**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)/(3*d) + 3*a**2*b*
c*d*e**2*x**2*acosh(c + d*x) - 2*a**2*b*c*e**2*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)/3 + a**2*b*d**2*e**2*x**
3*acosh(c + d*x) - a**2*b*d*e**2*x**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)/3 - 2*a**2*b*e**2*sqrt(c**2 + 2*c*d
*x + d**2*x**2 - 1)/(3*d) + a*b**2*c**3*e**2*acosh(c + d*x)**2/d + 3*a*b**2*c**2*e**2*x*acosh(c + d*x)**2 + 2*
a*b**2*c**2*e**2*x/3 - 2*a*b**2*c**2*e**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)/(3*d) + 3*a*b**2
*c*d*e**2*x**2*acosh(c + d*x)**2 + 2*a*b**2*c*d*e**2*x**2/3 - 4*a*b**2*c*e**2*x*sqrt(c**2 + 2*c*d*x + d**2*x**
2 - 1)*acosh(c + d*x)/3 + a*b**2*d**2*e**2*x**3*acosh(c + d*x)**2 + 2*a*b**2*d**2*e**2*x**3/9 - 2*a*b**2*d*e**
2*x**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)/3 + 4*a*b**2*e**2*x/3 - 4*a*b**2*e**2*sqrt(c**2 + 2
*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)/(3*d) + b**3*c**3*e**2*acosh(c + d*x)**3/(3*d) + 2*b**3*c**3*e**2*acosh
(c + d*x)/(9*d) + b**3*c**2*e**2*x*acosh(c + d*x)**3 + 2*b**3*c**2*e**2*x*acosh(c + d*x)/3 - b**3*c**2*e**2*sq
rt(c**2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)**2/(3*d) - 2*b**3*c**2*e**2*sqrt(c**2 + 2*c*d*x + d**2*x**2
- 1)/(27*d) + b**3*c*d*e**2*x**2*acosh(c + d*x)**3 + 2*b**3*c*d*e**2*x**2*acosh(c + d*x)/3 - 2*b**3*c*e**2*x*s
qrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)**2/3 - 4*b**3*c*e**2*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)
/27 + 4*b**3*c*e**2*acosh(c + d*x)/(3*d) + b**3*d**2*e**2*x**3*acosh(c + d*x)**3/3 + 2*b**3*d**2*e**2*x**3*aco
sh(c + d*x)/9 - b**3*d*e**2*x**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)**2/3 - 2*b**3*d*e**2*x**2
*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)/27 + 4*b**3*e**2*x*acosh(c + d*x)/3 - 2*b**3*e**2*sqrt(c**2 + 2*c*d*x +
d**2*x**2 - 1)*acosh(c + d*x)**2/(3*d) - 40*b**3*e**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)/(27*d), Ne(d, 0)),
(c**2*e**2*x*(a + b*acosh(c))**3, True))

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