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

Optimal. Leaf size=398 \[ -\frac {3 e^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{32 c^4}-\frac {4 b d e^2 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^3}-\frac {3 b e^3 x \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{16 c^3}-\frac {3 d^2 e \left (a+b \cosh ^{-1}(c x)\right )^2}{4 c^2}-\frac {d^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{4 e}-\frac {2 b d^3 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{c}-\frac {3 b d^2 e x \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{2 c}-\frac {2 b d e^2 x^2 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{3 c}+\frac {(d+e x)^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{4 e}-\frac {b e^3 x^3 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{8 c}+\frac {4 b^2 d e^2 x}{3 c^2}+\frac {3 b^2 e^3 x^2}{32 c^2}+2 b^2 d^3 x+\frac {3}{4} b^2 d^2 e x^2+\frac {2}{9} b^2 d e^2 x^3+\frac {1}{32} b^2 e^3 x^4 \]

[Out]

2*b^2*d^3*x+4/3*b^2*d*e^2*x/c^2+3/4*b^2*d^2*e*x^2+3/32*b^2*e^3*x^2/c^2+2/9*b^2*d*e^2*x^3+1/32*b^2*e^3*x^4-1/4*
d^4*(a+b*arccosh(c*x))^2/e-3/4*d^2*e*(a+b*arccosh(c*x))^2/c^2-3/32*e^3*(a+b*arccosh(c*x))^2/c^4+1/4*(e*x+d)^4*
(a+b*arccosh(c*x))^2/e-2*b*d^3*(a+b*arccosh(c*x))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c-4/3*b*d*e^2*(a+b*arccosh(c*x))
*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^3-3/2*b*d^2*e*x*(a+b*arccosh(c*x))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c-3/16*b*e^3*x*(
a+b*arccosh(c*x))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^3-2/3*b*d*e^2*x^2*(a+b*arccosh(c*x))*(c*x-1)^(1/2)*(c*x+1)^(1/
2)/c-1/8*b*e^3*x^3*(a+b*arccosh(c*x))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c

________________________________________________________________________________________

Rubi [A]  time = 1.69, antiderivative size = 398, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {5802, 5822, 5676, 5718, 8, 5759, 30} \[ -\frac {3 d^2 e \left (a+b \cosh ^{-1}(c x)\right )^2}{4 c^2}-\frac {4 b d e^2 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^3}-\frac {3 e^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{32 c^4}-\frac {3 b e^3 x \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{16 c^3}-\frac {d^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{4 e}-\frac {3 b d^2 e x \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{2 c}-\frac {2 b d^3 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{c}-\frac {2 b d e^2 x^2 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{3 c}+\frac {(d+e x)^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{4 e}-\frac {b e^3 x^3 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{8 c}+\frac {4 b^2 d e^2 x}{3 c^2}+\frac {3 b^2 e^3 x^2}{32 c^2}+\frac {3}{4} b^2 d^2 e x^2+2 b^2 d^3 x+\frac {2}{9} b^2 d e^2 x^3+\frac {1}{32} b^2 e^3 x^4 \]

Antiderivative was successfully verified.

[In]

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

[Out]

2*b^2*d^3*x + (4*b^2*d*e^2*x)/(3*c^2) + (3*b^2*d^2*e*x^2)/4 + (3*b^2*e^3*x^2)/(32*c^2) + (2*b^2*d*e^2*x^3)/9 +
(b^2*e^3*x^4)/32 - (2*b*d^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/c - (4*b*d*e^2*Sqrt[-1 + c*x]*
Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(3*c^3) - (3*b*d^2*e*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/
(2*c) - (3*b*e^3*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(16*c^3) - (2*b*d*e^2*x^2*Sqrt[-1 + c*x]
*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(3*c) - (b*e^3*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(8*
c) - (d^4*(a + b*ArcCosh[c*x])^2)/(4*e) - (3*d^2*e*(a + b*ArcCosh[c*x])^2)/(4*c^2) - (3*e^3*(a + b*ArcCosh[c*x
])^2)/(32*c^4) + ((d + e*x)^4*(a + b*ArcCosh[c*x])^2)/(4*e)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 5676

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol]
:> Simp[(a + b*ArcCosh[c*x])^(n + 1)/(b*c*Sqrt[-(d1*d2)]*(n + 1)), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n},
x] && EqQ[e1, c*d1] && EqQ[e2, -(c*d2)] && GtQ[d1, 0] && LtQ[d2, 0] && NeQ[n, -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 5802

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

Rule 5822

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

Rubi steps

\begin {align*} \int (d+e x)^3 \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx &=\frac {(d+e x)^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{4 e}-\frac {(b c) \int \frac {(d+e x)^4 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{2 e}\\ &=\frac {(d+e x)^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{4 e}-\frac {(b c) \int \left (\frac {d^4 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {4 d^3 e x \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {6 d^2 e^2 x^2 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {4 d e^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {e^4 x^4 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}\right ) \, dx}{2 e}\\ &=\frac {(d+e x)^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{4 e}-\left (2 b c d^3\right ) \int \frac {x \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx-\frac {\left (b c d^4\right ) \int \frac {a+b \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{2 e}-\left (3 b c d^2 e\right ) \int \frac {x^2 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx-\left (2 b c d e^2\right ) \int \frac {x^3 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx-\frac {1}{2} \left (b c e^3\right ) \int \frac {x^4 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=-\frac {2 b d^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{c}-\frac {3 b d^2 e x \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{2 c}-\frac {2 b d e^2 x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c}-\frac {b e^3 x^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{8 c}-\frac {d^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{4 e}+\frac {(d+e x)^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{4 e}+\left (2 b^2 d^3\right ) \int 1 \, dx+\frac {1}{2} \left (3 b^2 d^2 e\right ) \int x \, dx-\frac {\left (3 b d^2 e\right ) \int \frac {a+b \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{2 c}+\frac {1}{3} \left (2 b^2 d e^2\right ) \int x^2 \, dx-\frac {\left (4 b d e^2\right ) \int \frac {x \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{3 c}+\frac {1}{8} \left (b^2 e^3\right ) \int x^3 \, dx-\frac {\left (3 b e^3\right ) \int \frac {x^2 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{8 c}\\ &=2 b^2 d^3 x+\frac {3}{4} b^2 d^2 e x^2+\frac {2}{9} b^2 d e^2 x^3+\frac {1}{32} b^2 e^3 x^4-\frac {2 b d^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{c}-\frac {4 b d e^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^3}-\frac {3 b d^2 e x \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{2 c}-\frac {3 b e^3 x \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{16 c^3}-\frac {2 b d e^2 x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c}-\frac {b e^3 x^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{8 c}-\frac {d^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{4 e}-\frac {3 d^2 e \left (a+b \cosh ^{-1}(c x)\right )^2}{4 c^2}+\frac {(d+e x)^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{4 e}+\frac {\left (4 b^2 d e^2\right ) \int 1 \, dx}{3 c^2}-\frac {\left (3 b e^3\right ) \int \frac {a+b \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{16 c^3}+\frac {\left (3 b^2 e^3\right ) \int x \, dx}{16 c^2}\\ &=2 b^2 d^3 x+\frac {4 b^2 d e^2 x}{3 c^2}+\frac {3}{4} b^2 d^2 e x^2+\frac {3 b^2 e^3 x^2}{32 c^2}+\frac {2}{9} b^2 d e^2 x^3+\frac {1}{32} b^2 e^3 x^4-\frac {2 b d^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{c}-\frac {4 b d e^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^3}-\frac {3 b d^2 e x \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{2 c}-\frac {3 b e^3 x \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{16 c^3}-\frac {2 b d e^2 x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c}-\frac {b e^3 x^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{8 c}-\frac {d^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{4 e}-\frac {3 d^2 e \left (a+b \cosh ^{-1}(c x)\right )^2}{4 c^2}-\frac {3 e^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{32 c^4}+\frac {(d+e x)^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{4 e}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.85, size = 386, normalized size = 0.97 \[ \frac {c \left (72 a^2 c^3 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )-6 a b \sqrt {c x-1} \sqrt {c x+1} \left (c^2 \left (96 d^3+72 d^2 e x+32 d e^2 x^2+6 e^3 x^3\right )+e^2 (64 d+9 e x)\right )+b^2 c x \left (c^2 \left (576 d^3+216 d^2 e x+64 d e^2 x^2+9 e^3 x^3\right )+3 e^2 (128 d+9 e x)\right )\right )-54 a b e \left (8 c^2 d^2+e^2\right ) \log \left (c x+\sqrt {c x-1} \sqrt {c x+1}\right )-6 b c \cosh ^{-1}(c x) \left (b \sqrt {c x-1} \sqrt {c x+1} \left (c^2 \left (96 d^3+72 d^2 e x+32 d e^2 x^2+6 e^3 x^3\right )+e^2 (64 d+9 e x)\right )-24 a c^3 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )\right )+9 b^2 \cosh ^{-1}(c x)^2 \left (8 c^4 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )-24 c^2 d^2 e-3 e^3\right )}{288 c^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c*(72*a^2*c^3*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) - 6*a*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(e^2*(64*d +
9*e*x) + c^2*(96*d^3 + 72*d^2*e*x + 32*d*e^2*x^2 + 6*e^3*x^3)) + b^2*c*x*(3*e^2*(128*d + 9*e*x) + c^2*(576*d^
3 + 216*d^2*e*x + 64*d*e^2*x^2 + 9*e^3*x^3))) - 6*b*c*(-24*a*c^3*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3)
+ b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(e^2*(64*d + 9*e*x) + c^2*(96*d^3 + 72*d^2*e*x + 32*d*e^2*x^2 + 6*e^3*x^3)))
*ArcCosh[c*x] + 9*b^2*(-24*c^2*d^2*e - 3*e^3 + 8*c^4*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3))*ArcCosh[c*
x]^2 - 54*a*b*e*(8*c^2*d^2 + e^2)*Log[c*x + Sqrt[-1 + c*x]*Sqrt[1 + c*x]])/(288*c^4)

________________________________________________________________________________________

fricas [A]  time = 0.61, size = 472, normalized size = 1.19 \[ \frac {9 \, {\left (8 \, a^{2} + b^{2}\right )} c^{4} e^{3} x^{4} + 32 \, {\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{4} d e^{2} x^{3} + 27 \, {\left (8 \, {\left (2 \, a^{2} + b^{2}\right )} c^{4} d^{2} e + b^{2} c^{2} e^{3}\right )} x^{2} + 9 \, {\left (8 \, b^{2} c^{4} e^{3} x^{4} + 32 \, b^{2} c^{4} d e^{2} x^{3} + 48 \, b^{2} c^{4} d^{2} e x^{2} + 32 \, b^{2} c^{4} d^{3} x - 24 \, b^{2} c^{2} d^{2} e - 3 \, b^{2} e^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )^{2} + 96 \, {\left (3 \, {\left (a^{2} + 2 \, b^{2}\right )} c^{4} d^{3} + 4 \, b^{2} c^{2} d e^{2}\right )} x + 6 \, {\left (24 \, a b c^{4} e^{3} x^{4} + 96 \, a b c^{4} d e^{2} x^{3} + 144 \, a b c^{4} d^{2} e x^{2} + 96 \, a b c^{4} d^{3} x - 72 \, a b c^{2} d^{2} e - 9 \, a b e^{3} - {\left (6 \, b^{2} c^{3} e^{3} x^{3} + 32 \, b^{2} c^{3} d e^{2} x^{2} + 96 \, b^{2} c^{3} d^{3} + 64 \, b^{2} c d e^{2} + 9 \, {\left (8 \, b^{2} c^{3} d^{2} e + b^{2} c e^{3}\right )} x\right )} \sqrt {c^{2} x^{2} - 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - 6 \, {\left (6 \, a b c^{3} e^{3} x^{3} + 32 \, a b c^{3} d e^{2} x^{2} + 96 \, a b c^{3} d^{3} + 64 \, a b c d e^{2} + 9 \, {\left (8 \, a b c^{3} d^{2} e + a b c e^{3}\right )} x\right )} \sqrt {c^{2} x^{2} - 1}}{288 \, c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/288*(9*(8*a^2 + b^2)*c^4*e^3*x^4 + 32*(9*a^2 + 2*b^2)*c^4*d*e^2*x^3 + 27*(8*(2*a^2 + b^2)*c^4*d^2*e + b^2*c^
2*e^3)*x^2 + 9*(8*b^2*c^4*e^3*x^4 + 32*b^2*c^4*d*e^2*x^3 + 48*b^2*c^4*d^2*e*x^2 + 32*b^2*c^4*d^3*x - 24*b^2*c^
2*d^2*e - 3*b^2*e^3)*log(c*x + sqrt(c^2*x^2 - 1))^2 + 96*(3*(a^2 + 2*b^2)*c^4*d^3 + 4*b^2*c^2*d*e^2)*x + 6*(24
*a*b*c^4*e^3*x^4 + 96*a*b*c^4*d*e^2*x^3 + 144*a*b*c^4*d^2*e*x^2 + 96*a*b*c^4*d^3*x - 72*a*b*c^2*d^2*e - 9*a*b*
e^3 - (6*b^2*c^3*e^3*x^3 + 32*b^2*c^3*d*e^2*x^2 + 96*b^2*c^3*d^3 + 64*b^2*c*d*e^2 + 9*(8*b^2*c^3*d^2*e + b^2*c
*e^3)*x)*sqrt(c^2*x^2 - 1))*log(c*x + sqrt(c^2*x^2 - 1)) - 6*(6*a*b*c^3*e^3*x^3 + 32*a*b*c^3*d*e^2*x^2 + 96*a*
b*c^3*d^3 + 64*a*b*c*d*e^2 + 9*(8*a*b*c^3*d^2*e + a*b*c*e^3)*x)*sqrt(c^2*x^2 - 1))/c^4

________________________________________________________________________________________

giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> An error occurred running a Giac command:INPUT:sage2OUTPUT:sym2poly/r2sym(co
nst gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

________________________________________________________________________________________

maple [B]  time = 0.08, size = 791, normalized size = 1.99 \[ 2 b^{2} d^{3} x +\frac {a^{2} d^{4}}{4 e}-\frac {3 b^{2} \mathrm {arccosh}\left (c x \right )^{2} e^{3}}{32 c^{4}}+b^{2} \mathrm {arccosh}\left (c x \right )^{2} x \,d^{3}+\frac {b^{2} \mathrm {arccosh}\left (c x \right )^{2} x^{4} e^{3}}{4}+a^{2} e^{2} x^{3} d +\frac {3 a^{2} e \,x^{2} d^{2}}{2}-\frac {3 a b e \sqrt {c x -1}\, \sqrt {c x +1}\, d^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{2 c^{2} \sqrt {c^{2} x^{2}-1}}-\frac {2 b^{2} \mathrm {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, d^{3}}{c}-\frac {2 a b \sqrt {c x -1}\, \sqrt {c x +1}\, d^{3}}{c}+2 a b \,e^{2} \mathrm {arccosh}\left (c x \right ) x^{3} d +3 a b e \,\mathrm {arccosh}\left (c x \right ) x^{2} d^{2}-\frac {2 a b \,e^{2} \sqrt {c x -1}\, \sqrt {c x +1}\, x^{2} d}{3 c}-\frac {a b \sqrt {c x -1}\, \sqrt {c x +1}\, d^{4} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{2 e \sqrt {c^{2} x^{2}-1}}-\frac {2 b^{2} \mathrm {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, x^{2} d \,e^{2}}{3 c}-\frac {3 a b e \sqrt {c x -1}\, \sqrt {c x +1}\, d^{2} x}{2 c}-\frac {3 b^{2} \mathrm {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, x \,d^{2} e}{2 c}-\frac {3 a b \,e^{3} \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{16 c^{4} \sqrt {c^{2} x^{2}-1}}+\frac {a b \,\mathrm {arccosh}\left (c x \right ) d^{4}}{2 e}+b^{2} \mathrm {arccosh}\left (c x \right )^{2} x^{3} d \,e^{2}+\frac {3 b^{2} \mathrm {arccosh}\left (c x \right )^{2} x^{2} d^{2} e}{2}+\frac {a b \,e^{3} \mathrm {arccosh}\left (c x \right ) x^{4}}{2}+2 a b \,\mathrm {arccosh}\left (c x \right ) x \,d^{3}+\frac {3 b^{2} d^{2} e \,x^{2}}{4}+\frac {3 b^{2} e^{3} x^{2}}{32 c^{2}}+\frac {2 b^{2} d \,e^{2} x^{3}}{9}-\frac {4 b^{2} \mathrm {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, d \,e^{2}}{3 c^{3}}-\frac {a b \,e^{3} \sqrt {c x -1}\, \sqrt {c x +1}\, x^{3}}{8 c}-\frac {4 a b \,e^{2} \sqrt {c x -1}\, \sqrt {c x +1}\, d}{3 c^{3}}-\frac {3 a b \,e^{3} \sqrt {c x -1}\, \sqrt {c x +1}\, x}{16 c^{3}}-\frac {b^{2} \mathrm {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, x^{3} e^{3}}{8 c}-\frac {3 b^{2} \mathrm {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, x \,e^{3}}{16 c^{3}}+\frac {b^{2} e^{3} x^{4}}{32}-\frac {3 b^{2} \mathrm {arccosh}\left (c x \right )^{2} d^{2} e}{4 c^{2}}+a^{2} x \,d^{3}+\frac {a^{2} e^{3} x^{4}}{4}+\frac {4 b^{2} d \,e^{2} x}{3 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*b^2*d^3*x+1/4*a^2/e*d^4-3/32/c^4*b^2*arccosh(c*x)^2*e^3+b^2*arccosh(c*x)^2*x*d^3+1/4*b^2*arccosh(c*x)^2*x^4*
e^3+a^2*e^2*x^3*d+3/2*a^2*e*x^2*d^2-3/2/c^2*a*b*e*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^2*x^2-1)^(1/2)*d^2*ln(c*x+(c^
2*x^2-1)^(1/2))-2/c*b^2*arccosh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)*d^3-2/c*a*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)*d^3+2
*a*b*e^2*arccosh(c*x)*x^3*d+3*a*b*e*arccosh(c*x)*x^2*d^2-2/3/c*a*b*e^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*x^2*d-1/2*a
*b/e*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^2*x^2-1)^(1/2)*d^4*ln(c*x+(c^2*x^2-1)^(1/2))-2/3/c*b^2*arccosh(c*x)*(c*x-1
)^(1/2)*(c*x+1)^(1/2)*x^2*d*e^2-3/2/c*a*b*e*(c*x-1)^(1/2)*(c*x+1)^(1/2)*d^2*x-3/2/c*b^2*arccosh(c*x)*(c*x-1)^(
1/2)*(c*x+1)^(1/2)*x*d^2*e-3/16/c^4*a*b*e^3*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^2*x^2-1)^(1/2)*ln(c*x+(c^2*x^2-1)^(
1/2))+1/2*a*b/e*arccosh(c*x)*d^4+b^2*arccosh(c*x)^2*x^3*d*e^2+3/2*b^2*arccosh(c*x)^2*x^2*d^2*e+1/2*a*b*e^3*arc
cosh(c*x)*x^4+2*a*b*arccosh(c*x)*x*d^3+3/4*b^2*d^2*e*x^2+3/32*b^2*e^3*x^2/c^2+2/9*b^2*d*e^2*x^3-4/3/c^3*b^2*ar
ccosh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)*d*e^2-1/8/c*a*b*e^3*(c*x-1)^(1/2)*(c*x+1)^(1/2)*x^3-4/3/c^3*a*b*e^2*(c*
x-1)^(1/2)*(c*x+1)^(1/2)*d-3/16/c^3*a*b*e^3*(c*x-1)^(1/2)*(c*x+1)^(1/2)*x-1/8/c*b^2*arccosh(c*x)*(c*x-1)^(1/2)
*(c*x+1)^(1/2)*x^3*e^3-3/16/c^3*b^2*arccosh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)*x*e^3+1/32*b^2*e^3*x^4-3/4/c^2*b^
2*arccosh(c*x)^2*d^2*e+a^2*x*d^3+1/4*a^2*e^3*x^4+4/3*b^2*d*e^2*x/c^2

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{4} \, a^{2} e^{3} x^{4} + a^{2} d e^{2} x^{3} + b^{2} d^{3} x \operatorname {arcosh}\left (c x\right )^{2} + \frac {3}{2} \, a^{2} d^{2} e x^{2} + \frac {3}{2} \, {\left (2 \, x^{2} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x}{c^{2}} + \frac {\log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{3}}\right )}\right )} a b d^{2} e + \frac {2}{3} \, {\left (3 \, x^{3} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} a b d e^{2} + \frac {1}{16} \, {\left (8 \, x^{4} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {2 \, \sqrt {c^{2} x^{2} - 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {c^{2} x^{2} - 1} x}{c^{4}} + \frac {3 \, \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{5}}\right )} c\right )} a b e^{3} + 2 \, b^{2} d^{3} {\left (x - \frac {\sqrt {c^{2} x^{2} - 1} \operatorname {arcosh}\left (c x\right )}{c}\right )} + a^{2} d^{3} x + \frac {2 \, {\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )} a b d^{3}}{c} + \frac {1}{4} \, {\left (b^{2} e^{3} x^{4} + 4 \, b^{2} d e^{2} x^{3} + 6 \, b^{2} d^{2} e x^{2}\right )} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )^{2} - \int \frac {{\left (b^{2} c^{3} e^{3} x^{6} + 4 \, b^{2} c^{3} d e^{2} x^{5} - 4 \, b^{2} c d e^{2} x^{3} - 6 \, b^{2} c d^{2} e x^{2} + {\left (6 \, c^{3} d^{2} e - c e^{3}\right )} b^{2} x^{4} + {\left (b^{2} c^{2} e^{3} x^{5} + 4 \, b^{2} c^{2} d e^{2} x^{4} + 6 \, b^{2} c^{2} d^{2} e x^{3}\right )} \sqrt {c x + 1} \sqrt {c x - 1}\right )} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{2 \, {\left (c^{3} x^{3} + {\left (c^{2} x^{2} - 1\right )} \sqrt {c x + 1} \sqrt {c x - 1} - c x\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*a^2*e^3*x^4 + a^2*d*e^2*x^3 + b^2*d^3*x*arccosh(c*x)^2 + 3/2*a^2*d^2*e*x^2 + 3/2*(2*x^2*arccosh(c*x) - c*(
sqrt(c^2*x^2 - 1)*x/c^2 + log(2*c^2*x + 2*sqrt(c^2*x^2 - 1)*c)/c^3))*a*b*d^2*e + 2/3*(3*x^3*arccosh(c*x) - c*(
sqrt(c^2*x^2 - 1)*x^2/c^2 + 2*sqrt(c^2*x^2 - 1)/c^4))*a*b*d*e^2 + 1/16*(8*x^4*arccosh(c*x) - (2*sqrt(c^2*x^2 -
1)*x^3/c^2 + 3*sqrt(c^2*x^2 - 1)*x/c^4 + 3*log(2*c^2*x + 2*sqrt(c^2*x^2 - 1)*c)/c^5)*c)*a*b*e^3 + 2*b^2*d^3*(
x - sqrt(c^2*x^2 - 1)*arccosh(c*x)/c) + a^2*d^3*x + 2*(c*x*arccosh(c*x) - sqrt(c^2*x^2 - 1))*a*b*d^3/c + 1/4*(
b^2*e^3*x^4 + 4*b^2*d*e^2*x^3 + 6*b^2*d^2*e*x^2)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))^2 - integrate(1/2*(b^2
*c^3*e^3*x^6 + 4*b^2*c^3*d*e^2*x^5 - 4*b^2*c*d*e^2*x^3 - 6*b^2*c*d^2*e*x^2 + (6*c^3*d^2*e - c*e^3)*b^2*x^4 + (
b^2*c^2*e^3*x^5 + 4*b^2*c^2*d*e^2*x^4 + 6*b^2*c^2*d^2*e*x^3)*sqrt(c*x + 1)*sqrt(c*x - 1))*log(c*x + sqrt(c*x +
1)*sqrt(c*x - 1))/(c^3*x^3 + (c^2*x^2 - 1)*sqrt(c*x + 1)*sqrt(c*x - 1) - c*x), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 3.70, size = 750, normalized size = 1.88 \[ \begin {cases} a^{2} d^{3} x + \frac {3 a^{2} d^{2} e x^{2}}{2} + a^{2} d e^{2} x^{3} + \frac {a^{2} e^{3} x^{4}}{4} + 2 a b d^{3} x \operatorname {acosh}{\left (c x \right )} + 3 a b d^{2} e x^{2} \operatorname {acosh}{\left (c x \right )} + 2 a b d e^{2} x^{3} \operatorname {acosh}{\left (c x \right )} + \frac {a b e^{3} x^{4} \operatorname {acosh}{\left (c x \right )}}{2} - \frac {2 a b d^{3} \sqrt {c^{2} x^{2} - 1}}{c} - \frac {3 a b d^{2} e x \sqrt {c^{2} x^{2} - 1}}{2 c} - \frac {2 a b d e^{2} x^{2} \sqrt {c^{2} x^{2} - 1}}{3 c} - \frac {a b e^{3} x^{3} \sqrt {c^{2} x^{2} - 1}}{8 c} - \frac {3 a b d^{2} e \operatorname {acosh}{\left (c x \right )}}{2 c^{2}} - \frac {4 a b d e^{2} \sqrt {c^{2} x^{2} - 1}}{3 c^{3}} - \frac {3 a b e^{3} x \sqrt {c^{2} x^{2} - 1}}{16 c^{3}} - \frac {3 a b e^{3} \operatorname {acosh}{\left (c x \right )}}{16 c^{4}} + b^{2} d^{3} x \operatorname {acosh}^{2}{\left (c x \right )} + 2 b^{2} d^{3} x + \frac {3 b^{2} d^{2} e x^{2} \operatorname {acosh}^{2}{\left (c x \right )}}{2} + \frac {3 b^{2} d^{2} e x^{2}}{4} + b^{2} d e^{2} x^{3} \operatorname {acosh}^{2}{\left (c x \right )} + \frac {2 b^{2} d e^{2} x^{3}}{9} + \frac {b^{2} e^{3} x^{4} \operatorname {acosh}^{2}{\left (c x \right )}}{4} + \frac {b^{2} e^{3} x^{4}}{32} - \frac {2 b^{2} d^{3} \sqrt {c^{2} x^{2} - 1} \operatorname {acosh}{\left (c x \right )}}{c} - \frac {3 b^{2} d^{2} e x \sqrt {c^{2} x^{2} - 1} \operatorname {acosh}{\left (c x \right )}}{2 c} - \frac {2 b^{2} d e^{2} x^{2} \sqrt {c^{2} x^{2} - 1} \operatorname {acosh}{\left (c x \right )}}{3 c} - \frac {b^{2} e^{3} x^{3} \sqrt {c^{2} x^{2} - 1} \operatorname {acosh}{\left (c x \right )}}{8 c} - \frac {3 b^{2} d^{2} e \operatorname {acosh}^{2}{\left (c x \right )}}{4 c^{2}} + \frac {4 b^{2} d e^{2} x}{3 c^{2}} + \frac {3 b^{2} e^{3} x^{2}}{32 c^{2}} - \frac {4 b^{2} d e^{2} \sqrt {c^{2} x^{2} - 1} \operatorname {acosh}{\left (c x \right )}}{3 c^{3}} - \frac {3 b^{2} e^{3} x \sqrt {c^{2} x^{2} - 1} \operatorname {acosh}{\left (c x \right )}}{16 c^{3}} - \frac {3 b^{2} e^{3} \operatorname {acosh}^{2}{\left (c x \right )}}{32 c^{4}} & \text {for}\: c \neq 0 \\\left (a + \frac {i \pi b}{2}\right )^{2} \left (d^{3} x + \frac {3 d^{2} e x^{2}}{2} + d e^{2} x^{3} + \frac {e^{3} x^{4}}{4}\right ) & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**2*d**3*x + 3*a**2*d**2*e*x**2/2 + a**2*d*e**2*x**3 + a**2*e**3*x**4/4 + 2*a*b*d**3*x*acosh(c*x)
+ 3*a*b*d**2*e*x**2*acosh(c*x) + 2*a*b*d*e**2*x**3*acosh(c*x) + a*b*e**3*x**4*acosh(c*x)/2 - 2*a*b*d**3*sqrt(c
**2*x**2 - 1)/c - 3*a*b*d**2*e*x*sqrt(c**2*x**2 - 1)/(2*c) - 2*a*b*d*e**2*x**2*sqrt(c**2*x**2 - 1)/(3*c) - a*b
*e**3*x**3*sqrt(c**2*x**2 - 1)/(8*c) - 3*a*b*d**2*e*acosh(c*x)/(2*c**2) - 4*a*b*d*e**2*sqrt(c**2*x**2 - 1)/(3*
c**3) - 3*a*b*e**3*x*sqrt(c**2*x**2 - 1)/(16*c**3) - 3*a*b*e**3*acosh(c*x)/(16*c**4) + b**2*d**3*x*acosh(c*x)*
*2 + 2*b**2*d**3*x + 3*b**2*d**2*e*x**2*acosh(c*x)**2/2 + 3*b**2*d**2*e*x**2/4 + b**2*d*e**2*x**3*acosh(c*x)**
2 + 2*b**2*d*e**2*x**3/9 + b**2*e**3*x**4*acosh(c*x)**2/4 + b**2*e**3*x**4/32 - 2*b**2*d**3*sqrt(c**2*x**2 - 1
)*acosh(c*x)/c - 3*b**2*d**2*e*x*sqrt(c**2*x**2 - 1)*acosh(c*x)/(2*c) - 2*b**2*d*e**2*x**2*sqrt(c**2*x**2 - 1)
*acosh(c*x)/(3*c) - b**2*e**3*x**3*sqrt(c**2*x**2 - 1)*acosh(c*x)/(8*c) - 3*b**2*d**2*e*acosh(c*x)**2/(4*c**2)
+ 4*b**2*d*e**2*x/(3*c**2) + 3*b**2*e**3*x**2/(32*c**2) - 4*b**2*d*e**2*sqrt(c**2*x**2 - 1)*acosh(c*x)/(3*c**
3) - 3*b**2*e**3*x*sqrt(c**2*x**2 - 1)*acosh(c*x)/(16*c**3) - 3*b**2*e**3*acosh(c*x)**2/(32*c**4), Ne(c, 0)),
((a + I*pi*b/2)**2*(d**3*x + 3*d**2*e*x**2/2 + d*e**2*x**3 + e**3*x**4/4), True))

________________________________________________________________________________________