\(\int x (d+c^2 d x^2) (a+b \text {arcsinh}(c x)) \, dx\) [4]

Optimal result

Integrand size = 20, antiderivative size = 87 \[ \int x \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=-\frac {3 b d x \sqrt {1+c^2 x^2}}{32 c}-\frac {b d x \left (1+c^2 x^2\right )^{3/2}}{16 c}-\frac {3 b d \text {arcsinh}(c x)}{32 c^2}+\frac {d \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))}{4 c^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {5798, 201, 221} \[ \int x \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\frac {d \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {3 b d \text {arcsinh}(c x)}{32 c^2}-\frac {b d x \left (c^2 x^2+1\right )^{3/2}}{16 c}-\frac {3 b d x \sqrt {c^2 x^2+1}}{32 c} \]

[In]

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

[Out]

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

Rule 201

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(n*p + 1)), x] + Dist[a*n*(p/(n*p + 1)),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 221

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt[a])]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

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]

Rubi steps \begin{align*} \text {integral}& = \frac {d \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {(b d) \int \left (1+c^2 x^2\right )^{3/2} \, dx}{4 c} \\ & = -\frac {b d x \left (1+c^2 x^2\right )^{3/2}}{16 c}+\frac {d \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {(3 b d) \int \sqrt {1+c^2 x^2} \, dx}{16 c} \\ & = -\frac {3 b d x \sqrt {1+c^2 x^2}}{32 c}-\frac {b d x \left (1+c^2 x^2\right )^{3/2}}{16 c}+\frac {d \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {(3 b d) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{32 c} \\ & = -\frac {3 b d x \sqrt {1+c^2 x^2}}{32 c}-\frac {b d x \left (1+c^2 x^2\right )^{3/2}}{16 c}-\frac {3 b d \text {arcsinh}(c x)}{32 c^2}+\frac {d \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))}{4 c^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.89 \[ \int x \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\frac {d \left (c x \left (8 a c x \left (2+c^2 x^2\right )-b \sqrt {1+c^2 x^2} \left (5+2 c^2 x^2\right )\right )+b \left (5+16 c^2 x^2+8 c^4 x^4\right ) \text {arcsinh}(c x)\right )}{32 c^2} \]

[In]

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

[Out]

(d*(c*x*(8*a*c*x*(2 + c^2*x^2) - b*Sqrt[1 + c^2*x^2]*(5 + 2*c^2*x^2)) + b*(5 + 16*c^2*x^2 + 8*c^4*x^4)*ArcSinh
[c*x]))/(32*c^2)

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.98

method	result	size
derivativedivides	\(\frac {\frac {d a \left (c^{2} x^{2}+1\right )^{2}}{4}+d b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}}{4}+\frac {\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}}{2}+\frac {5 \,\operatorname {arcsinh}\left (c x \right )}{32}-\frac {c x \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{16}-\frac {3 c x \sqrt {c^{2} x^{2}+1}}{32}\right )}{c^{2}}\)	\(85\)
default	\(\frac {\frac {d a \left (c^{2} x^{2}+1\right )^{2}}{4}+d b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}}{4}+\frac {\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}}{2}+\frac {5 \,\operatorname {arcsinh}\left (c x \right )}{32}-\frac {c x \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{16}-\frac {3 c x \sqrt {c^{2} x^{2}+1}}{32}\right )}{c^{2}}\)	\(85\)
parts	\(\frac {d a \left (c^{2} x^{2}+1\right )^{2}}{4 c^{2}}+\frac {d b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}}{4}+\frac {\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}}{2}+\frac {5 \,\operatorname {arcsinh}\left (c x \right )}{32}-\frac {c x \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{16}-\frac {3 c x \sqrt {c^{2} x^{2}+1}}{32}\right )}{c^{2}}\)	\(87\)

[In]

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

[Out]

1/c^2*(1/4*d*a*(c^2*x^2+1)^2+d*b*(1/4*arcsinh(c*x)*c^4*x^4+1/2*arcsinh(c*x)*c^2*x^2+5/32*arcsinh(c*x)-1/16*c*x
*(c^2*x^2+1)^(3/2)-3/32*c*x*(c^2*x^2+1)^(1/2)))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.13 \[ \int x \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\frac {8 \, a c^{4} d x^{4} + 16 \, a c^{2} d x^{2} + {\left (8 \, b c^{4} d x^{4} + 16 \, b c^{2} d x^{2} + 5 \, b d\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - {\left (2 \, b c^{3} d x^{3} + 5 \, b c d x\right )} \sqrt {c^{2} x^{2} + 1}}{32 \, c^{2}} \]

[In]

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

[Out]

1/32*(8*a*c^4*d*x^4 + 16*a*c^2*d*x^2 + (8*b*c^4*d*x^4 + 16*b*c^2*d*x^2 + 5*b*d)*log(c*x + sqrt(c^2*x^2 + 1)) -
(2*b*c^3*d*x^3 + 5*b*c*d*x)*sqrt(c^2*x^2 + 1))/c^2

Sympy [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.34 \[ \int x \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\begin {cases} \frac {a c^{2} d x^{4}}{4} + \frac {a d x^{2}}{2} + \frac {b c^{2} d x^{4} \operatorname {asinh}{\left (c x \right )}}{4} - \frac {b c d x^{3} \sqrt {c^{2} x^{2} + 1}}{16} + \frac {b d x^{2} \operatorname {asinh}{\left (c x \right )}}{2} - \frac {5 b d x \sqrt {c^{2} x^{2} + 1}}{32 c} + \frac {5 b d \operatorname {asinh}{\left (c x \right )}}{32 c^{2}} & \text {for}\: c \neq 0 \\\frac {a d x^{2}}{2} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((a*c**2*d*x**4/4 + a*d*x**2/2 + b*c**2*d*x**4*asinh(c*x)/4 - b*c*d*x**3*sqrt(c**2*x**2 + 1)/16 + b*d
*x**2*asinh(c*x)/2 - 5*b*d*x*sqrt(c**2*x**2 + 1)/(32*c) + 5*b*d*asinh(c*x)/(32*c**2), Ne(c, 0)), (a*d*x**2/2,
True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.46 \[ \int x \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\frac {1}{4} \, a c^{2} d x^{4} + \frac {1}{32} \, {\left (8 \, x^{4} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {2 \, \sqrt {c^{2} x^{2} + 1} x^{3}}{c^{2}} - \frac {3 \, \sqrt {c^{2} x^{2} + 1} x}{c^{4}} + \frac {3 \, \operatorname {arsinh}\left (c x\right )}{c^{5}}\right )} c\right )} b c^{2} d + \frac {1}{2} \, a d x^{2} + \frac {1}{4} \, {\left (2 \, x^{2} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x}{c^{2}} - \frac {\operatorname {arsinh}\left (c x\right )}{c^{3}}\right )}\right )} b d \]

[In]

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

[Out]

1/4*a*c^2*d*x^4 + 1/32*(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*arcs
inh(c*x)/c^5)*c)*b*c^2*d + 1/2*a*d*x^2 + 1/4*(2*x^2*arcsinh(c*x) - c*(sqrt(c^2*x^2 + 1)*x/c^2 - arcsinh(c*x)/c
^3))*b*d

Giac [F(-2)]

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

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

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

[Out]

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