\(\int x^4 (d+c^2 d x^2) (a+b \text {arcsinh}(c x)) \, dx\) [1]

Optimal result

Integrand size = 22, antiderivative size = 124 \[ \int x^4 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=-\frac {2 b d \sqrt {1+c^2 x^2}}{35 c^5}-\frac {b d \left (1+c^2 x^2\right )^{3/2}}{105 c^5}+\frac {8 b d \left (1+c^2 x^2\right )^{5/2}}{175 c^5}-\frac {b d \left (1+c^2 x^2\right )^{7/2}}{49 c^5}+\frac {1}{5} d x^5 (a+b \text {arcsinh}(c x))+\frac {1}{7} c^2 d x^7 (a+b \text {arcsinh}(c x)) \]

[Out]

-1/105*b*d*(c^2*x^2+1)^(3/2)/c^5+8/175*b*d*(c^2*x^2+1)^(5/2)/c^5-1/49*b*d*(c^2*x^2+1)^(7/2)/c^5+1/5*d*x^5*(a+b
*arcsinh(c*x))+1/7*c^2*d*x^7*(a+b*arcsinh(c*x))-2/35*b*d*(c^2*x^2+1)^(1/2)/c^5

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {14, 5803, 12, 457, 78} \[ \int x^4 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\frac {1}{7} c^2 d x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} d x^5 (a+b \text {arcsinh}(c x))-\frac {b d \left (c^2 x^2+1\right )^{7/2}}{49 c^5}+\frac {8 b d \left (c^2 x^2+1\right )^{5/2}}{175 c^5}-\frac {b d \left (c^2 x^2+1\right )^{3/2}}{105 c^5}-\frac {2 b d \sqrt {c^2 x^2+1}}{35 c^5} \]

[In]

Int[x^4*(d + c^2*d*x^2)*(a + b*ArcSinh[c*x]),x]

[Out]

(-2*b*d*Sqrt[1 + c^2*x^2])/(35*c^5) - (b*d*(1 + c^2*x^2)^(3/2))/(105*c^5) + (8*b*d*(1 + c^2*x^2)^(5/2))/(175*c
^5) - (b*d*(1 + c^2*x^2)^(7/2))/(49*c^5) + (d*x^5*(a + b*ArcSinh[c*x]))/5 + (c^2*d*x^7*(a + b*ArcSinh[c*x]))/7

Rule 12

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 457

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 5803

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

Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} d x^5 (a+b \text {arcsinh}(c x))+\frac {1}{7} c^2 d x^7 (a+b \text {arcsinh}(c x))-(b c) \int \frac {d x^5 \left (7+5 c^2 x^2\right )}{35 \sqrt {1+c^2 x^2}} \, dx \\ & = \frac {1}{5} d x^5 (a+b \text {arcsinh}(c x))+\frac {1}{7} c^2 d x^7 (a+b \text {arcsinh}(c x))-\frac {1}{35} (b c d) \int \frac {x^5 \left (7+5 c^2 x^2\right )}{\sqrt {1+c^2 x^2}} \, dx \\ & = \frac {1}{5} d x^5 (a+b \text {arcsinh}(c x))+\frac {1}{7} c^2 d x^7 (a+b \text {arcsinh}(c x))-\frac {1}{70} (b c d) \text {Subst}\left (\int \frac {x^2 \left (7+5 c^2 x\right )}{\sqrt {1+c^2 x}} \, dx,x,x^2\right ) \\ & = \frac {1}{5} d x^5 (a+b \text {arcsinh}(c x))+\frac {1}{7} c^2 d x^7 (a+b \text {arcsinh}(c x))-\frac {1}{70} (b c d) \text {Subst}\left (\int \left (\frac {2}{c^4 \sqrt {1+c^2 x}}+\frac {\sqrt {1+c^2 x}}{c^4}-\frac {8 \left (1+c^2 x\right )^{3/2}}{c^4}+\frac {5 \left (1+c^2 x\right )^{5/2}}{c^4}\right ) \, dx,x,x^2\right ) \\ & = -\frac {2 b d \sqrt {1+c^2 x^2}}{35 c^5}-\frac {b d \left (1+c^2 x^2\right )^{3/2}}{105 c^5}+\frac {8 b d \left (1+c^2 x^2\right )^{5/2}}{175 c^5}-\frac {b d \left (1+c^2 x^2\right )^{7/2}}{49 c^5}+\frac {1}{5} d x^5 (a+b \text {arcsinh}(c x))+\frac {1}{7} c^2 d x^7 (a+b \text {arcsinh}(c x)) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.70 \[ \int x^4 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\frac {d \left (105 a x^5 \left (7+5 c^2 x^2\right )-\frac {b \sqrt {1+c^2 x^2} \left (152-76 c^2 x^2+57 c^4 x^4+75 c^6 x^6\right )}{c^5}+105 b x^5 \left (7+5 c^2 x^2\right ) \text {arcsinh}(c x)\right )}{3675} \]

[In]

Integrate[x^4*(d + c^2*d*x^2)*(a + b*ArcSinh[c*x]),x]

[Out]

(d*(105*a*x^5*(7 + 5*c^2*x^2) - (b*Sqrt[1 + c^2*x^2]*(152 - 76*c^2*x^2 + 57*c^4*x^4 + 75*c^6*x^6))/c^5 + 105*b
*x^5*(7 + 5*c^2*x^2)*ArcSinh[c*x]))/3675

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.97

method	result	size
parts	\(d a \left (\frac {1}{7} c^{2} x^{7}+\frac {1}{5} x^{5}\right )+\frac {d b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{7} x^{7}}{7}+\frac {\operatorname {arcsinh}\left (c x \right ) c^{5} x^{5}}{5}-\frac {c^{6} x^{6} \sqrt {c^{2} x^{2}+1}}{49}-\frac {19 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{1225}+\frac {76 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3675}-\frac {152 \sqrt {c^{2} x^{2}+1}}{3675}\right )}{c^{5}}\)	\(120\)
derivativedivides	\(\frac {d a \left (\frac {1}{7} c^{7} x^{7}+\frac {1}{5} c^{5} x^{5}\right )+d b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{7} x^{7}}{7}+\frac {\operatorname {arcsinh}\left (c x \right ) c^{5} x^{5}}{5}-\frac {c^{6} x^{6} \sqrt {c^{2} x^{2}+1}}{49}-\frac {19 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{1225}+\frac {76 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3675}-\frac {152 \sqrt {c^{2} x^{2}+1}}{3675}\right )}{c^{5}}\)	\(124\)
default	\(\frac {d a \left (\frac {1}{7} c^{7} x^{7}+\frac {1}{5} c^{5} x^{5}\right )+d b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{7} x^{7}}{7}+\frac {\operatorname {arcsinh}\left (c x \right ) c^{5} x^{5}}{5}-\frac {c^{6} x^{6} \sqrt {c^{2} x^{2}+1}}{49}-\frac {19 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{1225}+\frac {76 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3675}-\frac {152 \sqrt {c^{2} x^{2}+1}}{3675}\right )}{c^{5}}\)	\(124\)

[In]

int(x^4*(c^2*d*x^2+d)*(a+b*arcsinh(c*x)),x,method=_RETURNVERBOSE)

[Out]

d*a*(1/7*c^2*x^7+1/5*x^5)+d*b/c^5*(1/7*arcsinh(c*x)*c^7*x^7+1/5*arcsinh(c*x)*c^5*x^5-1/49*c^6*x^6*(c^2*x^2+1)^
(1/2)-19/1225*c^4*x^4*(c^2*x^2+1)^(1/2)+76/3675*c^2*x^2*(c^2*x^2+1)^(1/2)-152/3675*(c^2*x^2+1)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.91 \[ \int x^4 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\frac {525 \, a c^{7} d x^{7} + 735 \, a c^{5} d x^{5} + 105 \, {\left (5 \, b c^{7} d x^{7} + 7 \, b c^{5} d x^{5}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - {\left (75 \, b c^{6} d x^{6} + 57 \, b c^{4} d x^{4} - 76 \, b c^{2} d x^{2} + 152 \, b d\right )} \sqrt {c^{2} x^{2} + 1}}{3675 \, c^{5}} \]

[In]

integrate(x^4*(c^2*d*x^2+d)*(a+b*arcsinh(c*x)),x, algorithm="fricas")

[Out]

1/3675*(525*a*c^7*d*x^7 + 735*a*c^5*d*x^5 + 105*(5*b*c^7*d*x^7 + 7*b*c^5*d*x^5)*log(c*x + sqrt(c^2*x^2 + 1)) -
(75*b*c^6*d*x^6 + 57*b*c^4*d*x^4 - 76*b*c^2*d*x^2 + 152*b*d)*sqrt(c^2*x^2 + 1))/c^5

Sympy [A] (verification not implemented)

Time = 0.69 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.22 \[ \int x^4 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\begin {cases} \frac {a c^{2} d x^{7}}{7} + \frac {a d x^{5}}{5} + \frac {b c^{2} d x^{7} \operatorname {asinh}{\left (c x \right )}}{7} - \frac {b c d x^{6} \sqrt {c^{2} x^{2} + 1}}{49} + \frac {b d x^{5} \operatorname {asinh}{\left (c x \right )}}{5} - \frac {19 b d x^{4} \sqrt {c^{2} x^{2} + 1}}{1225 c} + \frac {76 b d x^{2} \sqrt {c^{2} x^{2} + 1}}{3675 c^{3}} - \frac {152 b d \sqrt {c^{2} x^{2} + 1}}{3675 c^{5}} & \text {for}\: c \neq 0 \\\frac {a d x^{5}}{5} & \text {otherwise} \end {cases} \]

[In]

integrate(x**4*(c**2*d*x**2+d)*(a+b*asinh(c*x)),x)

[Out]

Piecewise((a*c**2*d*x**7/7 + a*d*x**5/5 + b*c**2*d*x**7*asinh(c*x)/7 - b*c*d*x**6*sqrt(c**2*x**2 + 1)/49 + b*d
*x**5*asinh(c*x)/5 - 19*b*d*x**4*sqrt(c**2*x**2 + 1)/(1225*c) + 76*b*d*x**2*sqrt(c**2*x**2 + 1)/(3675*c**3) -
152*b*d*sqrt(c**2*x**2 + 1)/(3675*c**5), Ne(c, 0)), (a*d*x**5/5, True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.48 \[ \int x^4 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\frac {1}{7} \, a c^{2} d x^{7} + \frac {1}{5} \, a d x^{5} + \frac {1}{245} \, {\left (35 \, x^{7} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {5 \, \sqrt {c^{2} x^{2} + 1} x^{6}}{c^{2}} - \frac {6 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{6}} - \frac {16 \, \sqrt {c^{2} x^{2} + 1}}{c^{8}}\right )} c\right )} b c^{2} d + \frac {1}{75} \, {\left (15 \, x^{5} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{2}} - \frac {4 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} b d \]

[In]

integrate(x^4*(c^2*d*x^2+d)*(a+b*arcsinh(c*x)),x, algorithm="maxima")

[Out]

1/7*a*c^2*d*x^7 + 1/5*a*d*x^5 + 1/245*(35*x^7*arcsinh(c*x) - (5*sqrt(c^2*x^2 + 1)*x^6/c^2 - 6*sqrt(c^2*x^2 + 1
)*x^4/c^4 + 8*sqrt(c^2*x^2 + 1)*x^2/c^6 - 16*sqrt(c^2*x^2 + 1)/c^8)*c)*b*c^2*d + 1/75*(15*x^5*arcsinh(c*x) - (
3*sqrt(c^2*x^2 + 1)*x^4/c^2 - 4*sqrt(c^2*x^2 + 1)*x^2/c^4 + 8*sqrt(c^2*x^2 + 1)/c^6)*c)*b*d

Giac [F(-2)]

Exception generated. \[ \int x^4 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate(x^4*(c^2*d*x^2+d)*(a+b*arcsinh(c*x)),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. \[ \int x^4 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\int x^4\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,\left (d\,c^2\,x^2+d\right ) \,d x \]

[In]

int(x^4*(a + b*asinh(c*x))*(d + c^2*d*x^2),x)

[Out]

int(x^4*(a + b*asinh(c*x))*(d + c^2*d*x^2), x)