\(\int (d+e x)^3 (a+b \text {arcsinh}(c x))^2 \, dx\) [12]

Optimal result

Integrand size = 18, antiderivative size = 368 \[ \int (d+e x)^3 (a+b \text {arcsinh}(c x))^2 \, dx=2 b^2 d^3 x-\frac {4 b^2 d e^2 x}{3 c^2}+\frac {3}{4} b^2 d^2 e x^2-\frac {3 b^2 e^3 x^2}{32 c^2}+\frac {2}{9} b^2 d e^2 x^3+\frac {1}{32} b^2 e^3 x^4-\frac {2 b d^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{c}+\frac {4 b d e^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{3 c^3}-\frac {3 b d^2 e x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{2 c}+\frac {3 b e^3 x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{16 c^3}-\frac {2 b d e^2 x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{3 c}-\frac {b e^3 x^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{8 c}-\frac {d^4 (a+b \text {arcsinh}(c x))^2}{4 e}+\frac {3 d^2 e (a+b \text {arcsinh}(c x))^2}{4 c^2}-\frac {3 e^3 (a+b \text {arcsinh}(c x))^2}{32 c^4}+\frac {(d+e x)^4 (a+b \text {arcsinh}(c x))^2}{4 e} \]

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

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {5828, 5838, 5783, 5798, 8, 5812, 30} \[ \int (d+e x)^3 (a+b \text {arcsinh}(c x))^2 \, dx=-\frac {3 e^3 (a+b \text {arcsinh}(c x))^2}{32 c^4}-\frac {2 b d^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c}-\frac {3 b d^2 e x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{2 c}+\frac {3 d^2 e (a+b \text {arcsinh}(c x))^2}{4 c^2}-\frac {2 b d e^2 x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{3 c}-\frac {b e^3 x^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{8 c}+\frac {4 b d e^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{3 c^3}+\frac {3 b e^3 x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{16 c^3}-\frac {d^4 (a+b \text {arcsinh}(c x))^2}{4 e}+\frac {(d+e x)^4 (a+b \text {arcsinh}(c x))^2}{4 e}-\frac {4 b^2 d e^2 x}{3 c^2}-\frac {3 b^2 e^3 x^2}{32 c^2}+2 b^2 d^3 x+\frac {3}{4} b^2 d^2 e x^2+\frac {2}{9} b^2 d e^2 x^3+\frac {1}{32} b^2 e^3 x^4 \]

[In]

Int[(d + e*x)^3*(a + b*ArcSinh[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^2*x^2]*(a + b*ArcSinh[c*x]))/c + (4*b*d*e^2*Sqrt[1 + c^2*x^2]*(a + b*A
rcSinh[c*x]))/(3*c^3) - (3*b*d^2*e*x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(2*c) + (3*b*e^3*x*Sqrt[1 + c^2*x
^2]*(a + b*ArcSinh[c*x]))/(16*c^3) - (2*b*d*e^2*x^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(3*c) - (b*e^3*x^3
*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(8*c) - (d^4*(a + b*ArcSinh[c*x])^2)/(4*e) + (3*d^2*e*(a + b*ArcSinh[
c*x])^2)/(4*c^2) - (3*e^3*(a + b*ArcSinh[c*x])^2)/(32*c^4) + ((d + e*x)^4*(a + b*ArcSinh[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 5783

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(1/(b*c*(n + 1)))*S
imp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSinh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ
[e, c^2*d] && NeQ[n, -1]

Rule 5798

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

Rule 5812

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

Rule 5828

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*
((a + b*ArcSinh[c*x])^n/(e*(m + 1))), x] - Dist[b*c*(n/(e*(m + 1))), Int[(d + e*x)^(m + 1)*((a + b*ArcSinh[c*x
])^(n - 1)/Sqrt[1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 5838

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol]
:> Int[ExpandIntegrand[(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, (f + g*x)^m, x], x] /; FreeQ[{a, b, c, d, e, f, g
}, x] && EqQ[e, c^2*d] && IGtQ[m, 0] && IntegerQ[p + 1/2] && GtQ[d, 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*} \text {integral}& = \frac {(d+e x)^4 (a+b \text {arcsinh}(c x))^2}{4 e}-\frac {(b c) \int \frac {(d+e x)^4 (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}} \, dx}{2 e} \\ & = \frac {(d+e x)^4 (a+b \text {arcsinh}(c x))^2}{4 e}-\frac {(b c) \int \left (\frac {d^4 (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}}+\frac {4 d^3 e x (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}}+\frac {6 d^2 e^2 x^2 (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}}+\frac {4 d e^3 x^3 (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}}+\frac {e^4 x^4 (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}}\right ) \, dx}{2 e} \\ & = \frac {(d+e x)^4 (a+b \text {arcsinh}(c x))^2}{4 e}-\left (2 b c d^3\right ) \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}} \, dx-\frac {\left (b c d^4\right ) \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{2 e}-\left (3 b c d^2 e\right ) \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}} \, dx-\left (2 b c d e^2\right ) \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}} \, dx-\frac {1}{2} \left (b c e^3\right ) \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}} \, dx \\ & = -\frac {2 b d^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{c}-\frac {3 b d^2 e x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{2 c}-\frac {2 b d e^2 x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{3 c}-\frac {b e^3 x^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{8 c}-\frac {d^4 (a+b \text {arcsinh}(c x))^2}{4 e}+\frac {(d+e x)^4 (a+b \text {arcsinh}(c x))^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 \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}} \, 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 (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}} \, 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 (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}} \, 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^2 x^2} (a+b \text {arcsinh}(c x))}{c}+\frac {4 b d e^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{3 c^3}-\frac {3 b d^2 e x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{2 c}+\frac {3 b e^3 x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{16 c^3}-\frac {2 b d e^2 x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{3 c}-\frac {b e^3 x^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{8 c}-\frac {d^4 (a+b \text {arcsinh}(c x))^2}{4 e}+\frac {3 d^2 e (a+b \text {arcsinh}(c x))^2}{4 c^2}+\frac {(d+e x)^4 (a+b \text {arcsinh}(c x))^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 \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}} \, 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^2 x^2} (a+b \text {arcsinh}(c x))}{c}+\frac {4 b d e^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{3 c^3}-\frac {3 b d^2 e x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{2 c}+\frac {3 b e^3 x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{16 c^3}-\frac {2 b d e^2 x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{3 c}-\frac {b e^3 x^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{8 c}-\frac {d^4 (a+b \text {arcsinh}(c x))^2}{4 e}+\frac {3 d^2 e (a+b \text {arcsinh}(c x))^2}{4 c^2}-\frac {3 e^3 (a+b \text {arcsinh}(c x))^2}{32 c^4}+\frac {(d+e x)^4 (a+b \text {arcsinh}(c x))^2}{4 e} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.36 (sec) , antiderivative size = 354, normalized size of antiderivative = 0.96 \[ \int (d+e x)^3 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {c \left (72 a^2 c^3 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )-6 a b \sqrt {1+c^2 x^2} \left (-e^2 (64 d+9 e x)+c^2 \left (96 d^3+72 d^2 e x+32 d e^2 x^2+6 e^3 x^3\right )\right )+b^2 c x \left (-3 e^2 (128 d+9 e x)+c^2 \left (576 d^3+216 d^2 e x+64 d e^2 x^2+9 e^3 x^3\right )\right )\right )-6 b \left (-3 a \left (24 c^2 d^2 e-3 e^3+8 c^4 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )\right )+b c \sqrt {1+c^2 x^2} \left (-e^2 (64 d+9 e x)+c^2 \left (96 d^3+72 d^2 e x+32 d e^2 x^2+6 e^3 x^3\right )\right )\right ) \text {arcsinh}(c x)+9 b^2 \left (24 c^2 d^2 e-3 e^3+8 c^4 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )\right ) \text {arcsinh}(c x)^2}{288 c^4} \]

[In]

Integrate[(d + e*x)^3*(a + b*ArcSinh[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^2*x^2]*(-(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*(-3*a*(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)) + b*c*Sqrt[1 + c^2*x^2]*(-(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)))*ArcSinh[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))*
ArcSinh[c*x]^2)/(288*c^4)

Maple [A] (verified)

Time = 0.44 (sec) , antiderivative size = 526, normalized size of antiderivative = 1.43

method	result	size
derivativedivides	\(\frac {\frac {a^{2} \left (e c x +c d \right )^{4}}{4 c^{3} e}+\frac {b^{2} \left (\frac {e^{3} \left (8 \operatorname {arcsinh}\left (c x \right )^{2} x^{4} c^{4}-4 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+c^{4} x^{4}+6 \,\operatorname {arcsinh}\left (c x \right ) c x \sqrt {c^{2} x^{2}+1}-3 \operatorname {arcsinh}\left (c x \right )^{2}-3 c^{2} x^{2}-3\right )}{32}+\frac {d c \,e^{2} \left (9 \operatorname {arcsinh}\left (c x \right )^{2} x^{3} c^{3}-6 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}+2 c^{3} x^{3}+12 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}-12 c x \right )}{9}+\frac {3 d^{2} c^{2} e \left (2 \operatorname {arcsinh}\left (c x \right )^{2} x^{2} c^{2}-2 \,\operatorname {arcsinh}\left (c x \right ) c x \sqrt {c^{2} x^{2}+1}+\operatorname {arcsinh}\left (c x \right )^{2}+c^{2} x^{2}+1\right )}{4}+d^{3} c^{3} \left (\operatorname {arcsinh}\left (c x \right )^{2} x c -2 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}+2 c x \right )\right )}{c^{3}}+\frac {2 a b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{4} d^{4}}{4 e}+\operatorname {arcsinh}\left (c x \right ) c^{4} d^{3} x +\frac {3 e \,\operatorname {arcsinh}\left (c x \right ) c^{4} d^{2} x^{2}}{2}+e^{2} \operatorname {arcsinh}\left (c x \right ) c^{4} d \,x^{3}+\frac {e^{3} \operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}}{4}-\frac {c^{4} d^{4} \operatorname {arcsinh}\left (c x \right )+e^{4} \left (\frac {c^{3} x^{3} \sqrt {c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {c^{2} x^{2}+1}}{8}+\frac {3 \,\operatorname {arcsinh}\left (c x \right )}{8}\right )+4 d^{3} c^{3} e \sqrt {c^{2} x^{2}+1}+6 d^{2} c^{2} e^{2} \left (\frac {c x \sqrt {c^{2} x^{2}+1}}{2}-\frac {\operatorname {arcsinh}\left (c x \right )}{2}\right )+4 d c \,e^{3} \left (\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {c^{2} x^{2}+1}}{3}\right )}{4 e}\right )}{c^{3}}}{c}\)	\(526\)
default	\(\frac {\frac {a^{2} \left (e c x +c d \right )^{4}}{4 c^{3} e}+\frac {b^{2} \left (\frac {e^{3} \left (8 \operatorname {arcsinh}\left (c x \right )^{2} x^{4} c^{4}-4 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+c^{4} x^{4}+6 \,\operatorname {arcsinh}\left (c x \right ) c x \sqrt {c^{2} x^{2}+1}-3 \operatorname {arcsinh}\left (c x \right )^{2}-3 c^{2} x^{2}-3\right )}{32}+\frac {d c \,e^{2} \left (9 \operatorname {arcsinh}\left (c x \right )^{2} x^{3} c^{3}-6 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}+2 c^{3} x^{3}+12 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}-12 c x \right )}{9}+\frac {3 d^{2} c^{2} e \left (2 \operatorname {arcsinh}\left (c x \right )^{2} x^{2} c^{2}-2 \,\operatorname {arcsinh}\left (c x \right ) c x \sqrt {c^{2} x^{2}+1}+\operatorname {arcsinh}\left (c x \right )^{2}+c^{2} x^{2}+1\right )}{4}+d^{3} c^{3} \left (\operatorname {arcsinh}\left (c x \right )^{2} x c -2 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}+2 c x \right )\right )}{c^{3}}+\frac {2 a b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{4} d^{4}}{4 e}+\operatorname {arcsinh}\left (c x \right ) c^{4} d^{3} x +\frac {3 e \,\operatorname {arcsinh}\left (c x \right ) c^{4} d^{2} x^{2}}{2}+e^{2} \operatorname {arcsinh}\left (c x \right ) c^{4} d \,x^{3}+\frac {e^{3} \operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}}{4}-\frac {c^{4} d^{4} \operatorname {arcsinh}\left (c x \right )+e^{4} \left (\frac {c^{3} x^{3} \sqrt {c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {c^{2} x^{2}+1}}{8}+\frac {3 \,\operatorname {arcsinh}\left (c x \right )}{8}\right )+4 d^{3} c^{3} e \sqrt {c^{2} x^{2}+1}+6 d^{2} c^{2} e^{2} \left (\frac {c x \sqrt {c^{2} x^{2}+1}}{2}-\frac {\operatorname {arcsinh}\left (c x \right )}{2}\right )+4 d c \,e^{3} \left (\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {c^{2} x^{2}+1}}{3}\right )}{4 e}\right )}{c^{3}}}{c}\)	\(526\)
parts	\(\frac {a^{2} \left (e x +d \right )^{4}}{4 e}+\frac {b^{2} \left (72 \operatorname {arcsinh}\left (c x \right )^{2} c^{4} x^{4} e^{3}+288 \operatorname {arcsinh}\left (c x \right )^{2} c^{4} x^{3} d \,e^{2}+432 \operatorname {arcsinh}\left (c x \right )^{2} c^{4} x^{2} d^{2} e +288 \operatorname {arcsinh}\left (c x \right )^{2} c^{4} x \,d^{3}-36 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3} e^{3}-192 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{3} x^{2} d \,e^{2}-432 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{3} x \,d^{2} e -576 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{3} d^{3}+216 \operatorname {arcsinh}\left (c x \right )^{2} c^{2} d^{2} e +9 c^{4} x^{4} e^{3}+64 c^{4} d \,e^{2} x^{3}+216 c^{4} x^{2} d^{2} e +576 x \,c^{4} d^{3}+54 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c x \,e^{3}+384 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c d \,e^{2}-27 \operatorname {arcsinh}\left (c x \right )^{2} e^{3}-27 c^{2} x^{2} e^{3}-384 c^{2} x d \,e^{2}+216 c^{2} d^{2} e -27 e^{3}\right )}{288 c^{4}}+\frac {2 a b \left (\frac {c \,e^{3} \operatorname {arcsinh}\left (c x \right ) x^{4}}{4}+c \,e^{2} \operatorname {arcsinh}\left (c x \right ) x^{3} d +\frac {3 c \,\operatorname {arcsinh}\left (c x \right ) d^{2} e \,x^{2}}{2}+\operatorname {arcsinh}\left (c x \right ) c x \,d^{3}+\frac {c \,\operatorname {arcsinh}\left (c x \right ) d^{4}}{4 e}-\frac {c^{4} d^{4} \operatorname {arcsinh}\left (c x \right )+e^{4} \left (\frac {c^{3} x^{3} \sqrt {c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {c^{2} x^{2}+1}}{8}+\frac {3 \,\operatorname {arcsinh}\left (c x \right )}{8}\right )+4 d^{3} c^{3} e \sqrt {c^{2} x^{2}+1}+6 d^{2} c^{2} e^{2} \left (\frac {c x \sqrt {c^{2} x^{2}+1}}{2}-\frac {\operatorname {arcsinh}\left (c x \right )}{2}\right )+4 d c \,e^{3} \left (\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {c^{2} x^{2}+1}}{3}\right )}{4 c^{3} e}\right )}{c}\)	\(573\)

[In]

int((e*x+d)^3*(a+b*arcsinh(c*x))^2,x,method=_RETURNVERBOSE)

[Out]

1/c*(1/4*a^2/c^3*(c*e*x+c*d)^4/e+b^2/c^3*(1/32*e^3*(8*arcsinh(c*x)^2*x^4*c^4-4*arcsinh(c*x)*(c^2*x^2+1)^(1/2)*
x^3*c^3+c^4*x^4+6*arcsinh(c*x)*c*x*(c^2*x^2+1)^(1/2)-3*arcsinh(c*x)^2-3*c^2*x^2-3)+1/9*d*c*e^2*(9*arcsinh(c*x)
^2*x^3*c^3-6*arcsinh(c*x)*(c^2*x^2+1)^(1/2)*x^2*c^2+2*c^3*x^3+12*arcsinh(c*x)*(c^2*x^2+1)^(1/2)-12*c*x)+3/4*d^
2*c^2*e*(2*arcsinh(c*x)^2*x^2*c^2-2*arcsinh(c*x)*c*x*(c^2*x^2+1)^(1/2)+arcsinh(c*x)^2+c^2*x^2+1)+d^3*c^3*(arcs
inh(c*x)^2*x*c-2*arcsinh(c*x)*(c^2*x^2+1)^(1/2)+2*c*x))+2*a*b/c^3*(1/4/e*arcsinh(c*x)*c^4*d^4+arcsinh(c*x)*c^4
*d^3*x+3/2*e*arcsinh(c*x)*c^4*d^2*x^2+e^2*arcsinh(c*x)*c^4*d*x^3+1/4*e^3*arcsinh(c*x)*c^4*x^4-1/4/e*(c^4*d^4*a
rcsinh(c*x)+e^4*(1/4*c^3*x^3*(c^2*x^2+1)^(1/2)-3/8*c*x*(c^2*x^2+1)^(1/2)+3/8*arcsinh(c*x))+4*d^3*c^3*e*(c^2*x^
2+1)^(1/2)+6*d^2*c^2*e^2*(1/2*c*x*(c^2*x^2+1)^(1/2)-1/2*arcsinh(c*x))+4*d*c*e^3*(1/3*c^2*x^2*(c^2*x^2+1)^(1/2)
-2/3*(c^2*x^2+1)^(1/2)))))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 475, normalized size of antiderivative = 1.29 \[ \int (d+e x)^3 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {9 \, {\left (8 \, a^{2} + b^{2}\right )} c^{4} e^{3} x^{4} + 32 \, {\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{4} d e^{2} x^{3} + 27 \, {\left (8 \, {\left (2 \, a^{2} + b^{2}\right )} c^{4} d^{2} e - b^{2} c^{2} e^{3}\right )} x^{2} + 9 \, {\left (8 \, b^{2} c^{4} e^{3} x^{4} + 32 \, b^{2} c^{4} d e^{2} x^{3} + 48 \, b^{2} c^{4} d^{2} e x^{2} + 32 \, b^{2} c^{4} d^{3} x + 24 \, b^{2} c^{2} d^{2} e - 3 \, b^{2} e^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 96 \, {\left (3 \, {\left (a^{2} + 2 \, b^{2}\right )} c^{4} d^{3} - 4 \, b^{2} c^{2} d e^{2}\right )} x + 6 \, {\left (24 \, a b c^{4} e^{3} x^{4} + 96 \, a b c^{4} d e^{2} x^{3} + 144 \, a b c^{4} d^{2} e x^{2} + 96 \, a b c^{4} d^{3} x + 72 \, a b c^{2} d^{2} e - 9 \, a b e^{3} - {\left (6 \, b^{2} c^{3} e^{3} x^{3} + 32 \, b^{2} c^{3} d e^{2} x^{2} + 96 \, b^{2} c^{3} d^{3} - 64 \, b^{2} c d e^{2} + 9 \, {\left (8 \, b^{2} c^{3} d^{2} e - b^{2} c e^{3}\right )} x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - 6 \, {\left (6 \, a b c^{3} e^{3} x^{3} + 32 \, a b c^{3} d e^{2} x^{2} + 96 \, a b c^{3} d^{3} - 64 \, a b c d e^{2} + 9 \, {\left (8 \, a b c^{3} d^{2} e - a b c e^{3}\right )} x\right )} \sqrt {c^{2} x^{2} + 1}}{288 \, c^{4}} \]

[In]

integrate((e*x+d)^3*(a+b*arcsinh(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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 743 vs. \(2 (364) = 728\).

Time = 0.45 (sec) , antiderivative size = 743, normalized size of antiderivative = 2.02 \[ \int (d+e x)^3 (a+b \text {arcsinh}(c x))^2 \, dx=\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 {asinh}{\left (c x \right )} + 3 a b d^{2} e x^{2} \operatorname {asinh}{\left (c x \right )} + 2 a b d e^{2} x^{3} \operatorname {asinh}{\left (c x \right )} + \frac {a b e^{3} x^{4} \operatorname {asinh}{\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 {asinh}{\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 {asinh}{\left (c x \right )}}{16 c^{4}} + b^{2} d^{3} x \operatorname {asinh}^{2}{\left (c x \right )} + 2 b^{2} d^{3} x + \frac {3 b^{2} d^{2} e x^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{2} + \frac {3 b^{2} d^{2} e x^{2}}{4} + b^{2} d e^{2} x^{3} \operatorname {asinh}^{2}{\left (c x \right )} + \frac {2 b^{2} d e^{2} x^{3}}{9} + \frac {b^{2} e^{3} x^{4} \operatorname {asinh}^{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 {asinh}{\left (c x \right )}}{c} - \frac {3 b^{2} d^{2} e x \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{2 c} - \frac {2 b^{2} d e^{2} x^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{3 c} - \frac {b^{2} e^{3} x^{3} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{8 c} + \frac {3 b^{2} d^{2} e \operatorname {asinh}^{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 {asinh}{\left (c x \right )}}{3 c^{3}} + \frac {3 b^{2} e^{3} x \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{16 c^{3}} - \frac {3 b^{2} e^{3} \operatorname {asinh}^{2}{\left (c x \right )}}{32 c^{4}} & \text {for}\: c \neq 0 \\a^{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} \]

[In]

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

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 590, normalized size of antiderivative = 1.60 \[ \int (d+e x)^3 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {1}{4} \, b^{2} e^{3} x^{4} \operatorname {arsinh}\left (c x\right )^{2} + b^{2} d e^{2} x^{3} \operatorname {arsinh}\left (c x\right )^{2} + \frac {1}{4} \, a^{2} e^{3} x^{4} + \frac {3}{2} \, b^{2} d^{2} e x^{2} \operatorname {arsinh}\left (c x\right )^{2} + a^{2} d e^{2} x^{3} + b^{2} d^{3} x \operatorname {arsinh}\left (c x\right )^{2} + \frac {3}{2} \, a^{2} d^{2} e x^{2} + \frac {3}{2} \, {\left (2 \, x^{2} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x}{c^{2}} - \frac {\operatorname {arsinh}\left (c x\right )}{c^{3}}\right )}\right )} a b d^{2} e + \frac {3}{4} \, {\left (c^{2} {\left (\frac {x^{2}}{c^{2}} - \frac {\log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2}}{c^{4}}\right )} - 2 \, c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x}{c^{2}} - \frac {\operatorname {arsinh}\left (c x\right )}{c^{3}}\right )} \operatorname {arsinh}\left (c x\right )\right )} b^{2} d^{2} e + \frac {2}{3} \, {\left (3 \, x^{3} \operatorname {arsinh}\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 {2}{9} \, {\left (3 \, c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )} \operatorname {arsinh}\left (c x\right ) - \frac {c^{2} x^{3} - 6 \, x}{c^{2}}\right )} b^{2} d e^{2} + \frac {1}{16} \, {\left (8 \, x^{4} \operatorname {arsinh}\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 \, \operatorname {arsinh}\left (c x\right )}{c^{5}}\right )} c\right )} a b e^{3} + \frac {1}{32} \, {\left ({\left (\frac {x^{4}}{c^{2}} - \frac {3 \, x^{2}}{c^{4}} + \frac {3 \, \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2}}{c^{6}}\right )} c^{2} - 2 \, {\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 \, \operatorname {arsinh}\left (c x\right )}{c^{5}}\right )} c \operatorname {arsinh}\left (c x\right )\right )} b^{2} e^{3} + 2 \, b^{2} d^{3} {\left (x - \frac {\sqrt {c^{2} x^{2} + 1} \operatorname {arsinh}\left (c x\right )}{c}\right )} + a^{2} d^{3} x + \frac {2 \, {\left (c x \operatorname {arsinh}\left (c x\right ) - \sqrt {c^{2} x^{2} + 1}\right )} a b d^{3}}{c} \]

[In]

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

[Out]

1/4*b^2*e^3*x^4*arcsinh(c*x)^2 + b^2*d*e^2*x^3*arcsinh(c*x)^2 + 1/4*a^2*e^3*x^4 + 3/2*b^2*d^2*e*x^2*arcsinh(c*
x)^2 + a^2*d*e^2*x^3 + b^2*d^3*x*arcsinh(c*x)^2 + 3/2*a^2*d^2*e*x^2 + 3/2*(2*x^2*arcsinh(c*x) - c*(sqrt(c^2*x^
2 + 1)*x/c^2 - arcsinh(c*x)/c^3))*a*b*d^2*e + 3/4*(c^2*(x^2/c^2 - log(c*x + sqrt(c^2*x^2 + 1))^2/c^4) - 2*c*(s
qrt(c^2*x^2 + 1)*x/c^2 - arcsinh(c*x)/c^3)*arcsinh(c*x))*b^2*d^2*e + 2/3*(3*x^3*arcsinh(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 - 2/9*(3*c*(sqrt(c^2*x^2 + 1)*x^2/c^2 - 2*sqrt(c^2*x^2 + 1
)/c^4)*arcsinh(c*x) - (c^2*x^3 - 6*x)/c^2)*b^2*d*e^2 + 1/16*(8*x^4*arcsinh(c*x) - (2*sqrt(c^2*x^2 + 1)*x^3/c^2
- 3*sqrt(c^2*x^2 + 1)*x/c^4 + 3*arcsinh(c*x)/c^5)*c)*a*b*e^3 + 1/32*((x^4/c^2 - 3*x^2/c^4 + 3*log(c*x + sqrt(
c^2*x^2 + 1))^2/c^6)*c^2 - 2*(2*sqrt(c^2*x^2 + 1)*x^3/c^2 - 3*sqrt(c^2*x^2 + 1)*x/c^4 + 3*arcsinh(c*x)/c^5)*c*
arcsinh(c*x))*b^2*e^3 + 2*b^2*d^3*(x - sqrt(c^2*x^2 + 1)*arcsinh(c*x)/c) + a^2*d^3*x + 2*(c*x*arcsinh(c*x) - s
qrt(c^2*x^2 + 1))*a*b*d^3/c

Giac [F(-2)]

Exception generated. \[ \int (d+e x)^3 (a+b \text {arcsinh}(c x))^2 \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate((e*x+d)^3*(a+b*arcsinh(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

Mupad [F(-1)]

Timed out. \[ \int (d+e x)^3 (a+b \text {arcsinh}(c x))^2 \, dx=\int {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d+e\,x\right )}^3 \,d x \]

[In]

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

[Out]

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