\(\int (c e+d e x) (a+b \text {arcsinh}(c+d x))^2 \, dx\) [130]

Optimal result

Integrand size = 21, antiderivative size = 103 \[ \int (c e+d e x) (a+b \text {arcsinh}(c+d x))^2 \, dx=\frac {b^2 e (c+d x)^2}{4 d}-\frac {b e (c+d x) \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))}{2 d}+\frac {e (a+b \text {arcsinh}(c+d x))^2}{4 d}+\frac {e (c+d x)^2 (a+b \text {arcsinh}(c+d x))^2}{2 d} \]

[Out]

1/4*b^2*e*(d*x+c)^2/d+1/4*e*(a+b*arcsinh(d*x+c))^2/d+1/2*e*(d*x+c)^2*(a+b*arcsinh(d*x+c))^2/d-1/2*b*e*(d*x+c)*
(a+b*arcsinh(d*x+c))*(1+(d*x+c)^2)^(1/2)/d

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5859, 12, 5776, 5812, 5783, 30} \[ \int (c e+d e x) (a+b \text {arcsinh}(c+d x))^2 \, dx=\frac {e (c+d x)^2 (a+b \text {arcsinh}(c+d x))^2}{2 d}-\frac {b e \sqrt {(c+d x)^2+1} (c+d x) (a+b \text {arcsinh}(c+d x))}{2 d}+\frac {e (a+b \text {arcsinh}(c+d x))^2}{4 d}+\frac {b^2 e (c+d x)^2}{4 d} \]

[In]

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

[Out]

(b^2*e*(c + d*x)^2)/(4*d) - (b*e*(c + d*x)*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x]))/(2*d) + (e*(a + b*A
rcSinh[c + d*x])^2)/(4*d) + (e*(c + d*x)^2*(a + b*ArcSinh[c + d*x])^2)/(2*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 30

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

Rule 5776

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcS
inh[c*x])^n/(d*(m + 1))), x] - Dist[b*c*(n/(d*(m + 1))), Int[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[
1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && 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 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 5859

Int[((a_.) + ArcSinh[(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*ArcSinh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x
]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int e x (a+b \text {arcsinh}(x))^2 \, dx,x,c+d x\right )}{d} \\ & = \frac {e \text {Subst}\left (\int x (a+b \text {arcsinh}(x))^2 \, dx,x,c+d x\right )}{d} \\ & = \frac {e (c+d x)^2 (a+b \text {arcsinh}(c+d x))^2}{2 d}-\frac {(b e) \text {Subst}\left (\int \frac {x^2 (a+b \text {arcsinh}(x))}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{d} \\ & = -\frac {b e (c+d x) \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))}{2 d}+\frac {e (c+d x)^2 (a+b \text {arcsinh}(c+d x))^2}{2 d}+\frac {(b e) \text {Subst}\left (\int \frac {a+b \text {arcsinh}(x)}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{2 d}+\frac {\left (b^2 e\right ) \text {Subst}(\int x \, dx,x,c+d x)}{2 d} \\ & = \frac {b^2 e (c+d x)^2}{4 d}-\frac {b e (c+d x) \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))}{2 d}+\frac {e (a+b \text {arcsinh}(c+d x))^2}{4 d}+\frac {e (c+d x)^2 (a+b \text {arcsinh}(c+d x))^2}{2 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.17 \[ \int (c e+d e x) (a+b \text {arcsinh}(c+d x))^2 \, dx=\frac {e \left (\left (2 a^2+b^2\right ) (c+d x)^2-2 a b (c+d x) \sqrt {1+(c+d x)^2}+2 a b \text {arcsinh}(c+d x)+2 b (c+d x) \left (2 a (c+d x)-b \sqrt {1+(c+d x)^2}\right ) \text {arcsinh}(c+d x)+b^2 \left (1+2 (c+d x)^2\right ) \text {arcsinh}(c+d x)^2\right )}{4 d} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.31

method	result	size
derivativedivides	\(\frac {\frac {e \,a^{2} \left (d x +c \right )^{2}}{2}+e \,b^{2} \left (\frac {\operatorname {arcsinh}\left (d x +c \right )^{2} \left (1+\left (d x +c \right )^{2}\right )}{2}-\frac {\sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )}{2}-\frac {\operatorname {arcsinh}\left (d x +c \right )^{2}}{4}+\frac {\left (d x +c \right )^{2}}{4}+\frac {1}{4}\right )+2 e a b \left (\frac {\left (d x +c \right )^{2} \operatorname {arcsinh}\left (d x +c \right )}{2}-\frac {\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{4}+\frac {\operatorname {arcsinh}\left (d x +c \right )}{4}\right )}{d}\)	\(135\)
default	\(\frac {\frac {e \,a^{2} \left (d x +c \right )^{2}}{2}+e \,b^{2} \left (\frac {\operatorname {arcsinh}\left (d x +c \right )^{2} \left (1+\left (d x +c \right )^{2}\right )}{2}-\frac {\sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )}{2}-\frac {\operatorname {arcsinh}\left (d x +c \right )^{2}}{4}+\frac {\left (d x +c \right )^{2}}{4}+\frac {1}{4}\right )+2 e a b \left (\frac {\left (d x +c \right )^{2} \operatorname {arcsinh}\left (d x +c \right )}{2}-\frac {\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{4}+\frac {\operatorname {arcsinh}\left (d x +c \right )}{4}\right )}{d}\)	\(135\)
parts	\(e \,a^{2} \left (\frac {1}{2} d \,x^{2}+c x \right )+\frac {e \,b^{2} \left (\frac {\operatorname {arcsinh}\left (d x +c \right )^{2} \left (1+\left (d x +c \right )^{2}\right )}{2}-\frac {\sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )}{2}-\frac {\operatorname {arcsinh}\left (d x +c \right )^{2}}{4}+\frac {\left (d x +c \right )^{2}}{4}+\frac {1}{4}\right )}{d}+\frac {2 e a b \left (\frac {\left (d x +c \right )^{2} \operatorname {arcsinh}\left (d x +c \right )}{2}-\frac {\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{4}+\frac {\operatorname {arcsinh}\left (d x +c \right )}{4}\right )}{d}\)	\(139\)

[In]

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

[Out]

1/d*(1/2*e*a^2*(d*x+c)^2+e*b^2*(1/2*arcsinh(d*x+c)^2*(1+(d*x+c)^2)-1/2*(1+(d*x+c)^2)^(1/2)*(d*x+c)*arcsinh(d*x
+c)-1/4*arcsinh(d*x+c)^2+1/4*(d*x+c)^2+1/4)+2*e*a*b*(1/2*(d*x+c)^2*arcsinh(d*x+c)-1/4*(d*x+c)*(1+(d*x+c)^2)^(1
/2)+1/4*arcsinh(d*x+c)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 230 vs. \(2 (93) = 186\).

Time = 0.26 (sec) , antiderivative size = 230, normalized size of antiderivative = 2.23 \[ \int (c e+d e x) (a+b \text {arcsinh}(c+d x))^2 \, dx=\frac {{\left (2 \, a^{2} + b^{2}\right )} d^{2} e x^{2} + 2 \, {\left (2 \, a^{2} + b^{2}\right )} c d e x + {\left (2 \, b^{2} d^{2} e x^{2} + 4 \, b^{2} c d e x + {\left (2 \, b^{2} c^{2} + b^{2}\right )} e\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{2} + 2 \, {\left (2 \, a b d^{2} e x^{2} + 4 \, a b c d e x + {\left (2 \, a b c^{2} + a b\right )} e - {\left (b^{2} d e x + b^{2} c e\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) - 2 \, {\left (a b d e x + a b c e\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}}{4 \, d} \]

[In]

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

[Out]

1/4*((2*a^2 + b^2)*d^2*e*x^2 + 2*(2*a^2 + b^2)*c*d*e*x + (2*b^2*d^2*e*x^2 + 4*b^2*c*d*e*x + (2*b^2*c^2 + b^2)*
e)*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))^2 + 2*(2*a*b*d^2*e*x^2 + 4*a*b*c*d*e*x + (2*a*b*c^2 + a*b)
*e - (b^2*d*e*x + b^2*c*e)*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*(a*b*d*e*x + a*b*c*e)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))/d

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 335 vs. \(2 (88) = 176\).

Time = 0.19 (sec) , antiderivative size = 335, normalized size of antiderivative = 3.25 \[ \int (c e+d e x) (a+b \text {arcsinh}(c+d x))^2 \, dx=\begin {cases} a^{2} c e x + \frac {a^{2} d e x^{2}}{2} + \frac {a b c^{2} e \operatorname {asinh}{\left (c + d x \right )}}{d} + 2 a b c e x \operatorname {asinh}{\left (c + d x \right )} - \frac {a b c e \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{2 d} + a b d e x^{2} \operatorname {asinh}{\left (c + d x \right )} - \frac {a b e x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{2} + \frac {a b e \operatorname {asinh}{\left (c + d x \right )}}{2 d} + \frac {b^{2} c^{2} e \operatorname {asinh}^{2}{\left (c + d x \right )}}{2 d} + b^{2} c e x \operatorname {asinh}^{2}{\left (c + d x \right )} + \frac {b^{2} c e x}{2} - \frac {b^{2} c e \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}{\left (c + d x \right )}}{2 d} + \frac {b^{2} d e x^{2} \operatorname {asinh}^{2}{\left (c + d x \right )}}{2} + \frac {b^{2} d e x^{2}}{4} - \frac {b^{2} e x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}{\left (c + d x \right )}}{2} + \frac {b^{2} e \operatorname {asinh}^{2}{\left (c + d x \right )}}{4 d} & \text {for}\: d \neq 0 \\c e x \left (a + b \operatorname {asinh}{\left (c \right )}\right )^{2} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((a**2*c*e*x + a**2*d*e*x**2/2 + a*b*c**2*e*asinh(c + d*x)/d + 2*a*b*c*e*x*asinh(c + d*x) - a*b*c*e*s
qrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/(2*d) + a*b*d*e*x**2*asinh(c + d*x) - a*b*e*x*sqrt(c**2 + 2*c*d*x + d**2*x
**2 + 1)/2 + a*b*e*asinh(c + d*x)/(2*d) + b**2*c**2*e*asinh(c + d*x)**2/(2*d) + b**2*c*e*x*asinh(c + d*x)**2 +
b**2*c*e*x/2 - b**2*c*e*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)/(2*d) + b**2*d*e*x**2*asinh(c + d
*x)**2/2 + b**2*d*e*x**2/4 - b**2*e*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)/2 + b**2*e*asinh(c +
d*x)**2/(4*d), Ne(d, 0)), (c*e*x*(a + b*asinh(c))**2, True))

Maxima [F]

\[ \int (c e+d e x) (a+b \text {arcsinh}(c+d x))^2 \, dx=\int { {\left (d e x + c e\right )} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{2} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int (c e+d e x) (a+b \text {arcsinh}(c+d x))^2 \, dx=\int { {\left (d e x + c e\right )} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{2} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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