\(\int (f+g x) (d+c^2 d x^2)^{5/2} (a+b \text {arcsinh}(c x)) \, dx\) [45]

Optimal result

Integrand size = 28, antiderivative size = 494 \[ \int (f+g x) \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=-\frac {b d^2 g x \sqrt {d+c^2 d x^2}}{7 c \sqrt {1+c^2 x^2}}-\frac {25 b c d^2 f x^2 \sqrt {d+c^2 d x^2}}{96 \sqrt {1+c^2 x^2}}-\frac {b c d^2 g x^3 \sqrt {d+c^2 d x^2}}{7 \sqrt {1+c^2 x^2}}-\frac {5 b c^3 d^2 f x^4 \sqrt {d+c^2 d x^2}}{96 \sqrt {1+c^2 x^2}}-\frac {3 b c^3 d^2 g x^5 \sqrt {d+c^2 d x^2}}{35 \sqrt {1+c^2 x^2}}-\frac {b c^5 d^2 g x^7 \sqrt {d+c^2 d x^2}}{49 \sqrt {1+c^2 x^2}}-\frac {b d^2 f \left (1+c^2 x^2\right )^{5/2} \sqrt {d+c^2 d x^2}}{36 c}+\frac {5}{16} d^2 f x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {5}{24} d^2 f x \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {1}{6} d^2 f x \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {d^2 g \left (1+c^2 x^2\right )^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{7 c^2}+\frac {5 d^2 f \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{32 b c \sqrt {1+c^2 x^2}} \]

[Out]

-1/36*b*d^2*f*(c^2*x^2+1)^(5/2)*(c^2*d*x^2+d)^(1/2)/c+5/16*d^2*f*x*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)+5/24
*d^2*f*x*(c^2*x^2+1)*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)+1/6*d^2*f*x*(c^2*x^2+1)^2*(a+b*arcsinh(c*x))*(c^2*
d*x^2+d)^(1/2)+1/7*d^2*g*(c^2*x^2+1)^3*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/c^2-1/7*b*d^2*g*x*(c^2*d*x^2+d)^
(1/2)/c/(c^2*x^2+1)^(1/2)-25/96*b*c*d^2*f*x^2*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-1/7*b*c*d^2*g*x^3*(c^2*d*x
^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-5/96*b*c^3*d^2*f*x^4*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-3/35*b*c^3*d^2*g*x^5*
(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-1/49*b*c^5*d^2*g*x^7*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)+5/32*d^2*f*(a
+b*arcsinh(c*x))^2*(c^2*d*x^2+d)^(1/2)/b/c/(c^2*x^2+1)^(1/2)

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 494, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {5845, 5838, 5786, 5785, 5783, 30, 14, 267, 5798, 200} \[ \int (f+g x) \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {1}{6} d^2 f x \left (c^2 x^2+1\right )^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))+\frac {5}{16} d^2 f x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))+\frac {5}{24} d^2 f x \left (c^2 x^2+1\right ) \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))+\frac {5 d^2 f \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{32 b c \sqrt {c^2 x^2+1}}+\frac {d^2 g \left (c^2 x^2+1\right )^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{7 c^2}-\frac {25 b c d^2 f x^2 \sqrt {c^2 d x^2+d}}{96 \sqrt {c^2 x^2+1}}-\frac {b d^2 f \left (c^2 x^2+1\right )^{5/2} \sqrt {c^2 d x^2+d}}{36 c}-\frac {b d^2 g x \sqrt {c^2 d x^2+d}}{7 c \sqrt {c^2 x^2+1}}-\frac {b c d^2 g x^3 \sqrt {c^2 d x^2+d}}{7 \sqrt {c^2 x^2+1}}-\frac {b c^5 d^2 g x^7 \sqrt {c^2 d x^2+d}}{49 \sqrt {c^2 x^2+1}}-\frac {5 b c^3 d^2 f x^4 \sqrt {c^2 d x^2+d}}{96 \sqrt {c^2 x^2+1}}-\frac {3 b c^3 d^2 g x^5 \sqrt {c^2 d x^2+d}}{35 \sqrt {c^2 x^2+1}} \]

[In]

Int[(f + g*x)*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]),x]

[Out]

-1/7*(b*d^2*g*x*Sqrt[d + c^2*d*x^2])/(c*Sqrt[1 + c^2*x^2]) - (25*b*c*d^2*f*x^2*Sqrt[d + c^2*d*x^2])/(96*Sqrt[1
+ c^2*x^2]) - (b*c*d^2*g*x^3*Sqrt[d + c^2*d*x^2])/(7*Sqrt[1 + c^2*x^2]) - (5*b*c^3*d^2*f*x^4*Sqrt[d + c^2*d*x
^2])/(96*Sqrt[1 + c^2*x^2]) - (3*b*c^3*d^2*g*x^5*Sqrt[d + c^2*d*x^2])/(35*Sqrt[1 + c^2*x^2]) - (b*c^5*d^2*g*x^
7*Sqrt[d + c^2*d*x^2])/(49*Sqrt[1 + c^2*x^2]) - (b*d^2*f*(1 + c^2*x^2)^(5/2)*Sqrt[d + c^2*d*x^2])/(36*c) + (5*
d^2*f*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/16 + (5*d^2*f*x*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2]*(a + b*Arc
Sinh[c*x]))/24 + (d^2*f*x*(1 + c^2*x^2)^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/6 + (d^2*g*(1 + c^2*x^2)^3
*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(7*c^2) + (5*d^2*f*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(32*
b*c*Sqrt[1 + c^2*x^2])

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 30

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

Rule 200

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rule 267

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -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 5785

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

Rule 5786

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

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 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]))

Rule 5845

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

Rubi steps \begin{align*} \text {integral}& = \frac {\left (d^2 \sqrt {d+c^2 d x^2}\right ) \int (f+g x) \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx}{\sqrt {1+c^2 x^2}} \\ & = \frac {\left (d^2 \sqrt {d+c^2 d x^2}\right ) \int \left (f \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))+g x \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))\right ) \, dx}{\sqrt {1+c^2 x^2}} \\ & = \frac {\left (d^2 f \sqrt {d+c^2 d x^2}\right ) \int \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx}{\sqrt {1+c^2 x^2}}+\frac {\left (d^2 g \sqrt {d+c^2 d x^2}\right ) \int x \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx}{\sqrt {1+c^2 x^2}} \\ & = \frac {1}{6} d^2 f x \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {d^2 g \left (1+c^2 x^2\right )^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{7 c^2}+\frac {\left (5 d^2 f \sqrt {d+c^2 d x^2}\right ) \int \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx}{6 \sqrt {1+c^2 x^2}}-\frac {\left (b c d^2 f \sqrt {d+c^2 d x^2}\right ) \int x \left (1+c^2 x^2\right )^2 \, dx}{6 \sqrt {1+c^2 x^2}}-\frac {\left (b d^2 g \sqrt {d+c^2 d x^2}\right ) \int \left (1+c^2 x^2\right )^3 \, dx}{7 c \sqrt {1+c^2 x^2}} \\ & = -\frac {b d^2 f \left (1+c^2 x^2\right )^{5/2} \sqrt {d+c^2 d x^2}}{36 c}+\frac {5}{24} d^2 f x \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {1}{6} d^2 f x \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {d^2 g \left (1+c^2 x^2\right )^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{7 c^2}+\frac {\left (5 d^2 f \sqrt {d+c^2 d x^2}\right ) \int \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \, dx}{8 \sqrt {1+c^2 x^2}}-\frac {\left (5 b c d^2 f \sqrt {d+c^2 d x^2}\right ) \int x \left (1+c^2 x^2\right ) \, dx}{24 \sqrt {1+c^2 x^2}}-\frac {\left (b d^2 g \sqrt {d+c^2 d x^2}\right ) \int \left (1+3 c^2 x^2+3 c^4 x^4+c^6 x^6\right ) \, dx}{7 c \sqrt {1+c^2 x^2}} \\ & = -\frac {b d^2 g x \sqrt {d+c^2 d x^2}}{7 c \sqrt {1+c^2 x^2}}-\frac {b c d^2 g x^3 \sqrt {d+c^2 d x^2}}{7 \sqrt {1+c^2 x^2}}-\frac {3 b c^3 d^2 g x^5 \sqrt {d+c^2 d x^2}}{35 \sqrt {1+c^2 x^2}}-\frac {b c^5 d^2 g x^7 \sqrt {d+c^2 d x^2}}{49 \sqrt {1+c^2 x^2}}-\frac {b d^2 f \left (1+c^2 x^2\right )^{5/2} \sqrt {d+c^2 d x^2}}{36 c}+\frac {5}{16} d^2 f x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {5}{24} d^2 f x \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {1}{6} d^2 f x \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {d^2 g \left (1+c^2 x^2\right )^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{7 c^2}+\frac {\left (5 d^2 f \sqrt {d+c^2 d x^2}\right ) \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{16 \sqrt {1+c^2 x^2}}-\frac {\left (5 b c d^2 f \sqrt {d+c^2 d x^2}\right ) \int \left (x+c^2 x^3\right ) \, dx}{24 \sqrt {1+c^2 x^2}}-\frac {\left (5 b c d^2 f \sqrt {d+c^2 d x^2}\right ) \int x \, dx}{16 \sqrt {1+c^2 x^2}} \\ & = -\frac {b d^2 g x \sqrt {d+c^2 d x^2}}{7 c \sqrt {1+c^2 x^2}}-\frac {25 b c d^2 f x^2 \sqrt {d+c^2 d x^2}}{96 \sqrt {1+c^2 x^2}}-\frac {b c d^2 g x^3 \sqrt {d+c^2 d x^2}}{7 \sqrt {1+c^2 x^2}}-\frac {5 b c^3 d^2 f x^4 \sqrt {d+c^2 d x^2}}{96 \sqrt {1+c^2 x^2}}-\frac {3 b c^3 d^2 g x^5 \sqrt {d+c^2 d x^2}}{35 \sqrt {1+c^2 x^2}}-\frac {b c^5 d^2 g x^7 \sqrt {d+c^2 d x^2}}{49 \sqrt {1+c^2 x^2}}-\frac {b d^2 f \left (1+c^2 x^2\right )^{5/2} \sqrt {d+c^2 d x^2}}{36 c}+\frac {5}{16} d^2 f x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {5}{24} d^2 f x \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {1}{6} d^2 f x \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {d^2 g \left (1+c^2 x^2\right )^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{7 c^2}+\frac {5 d^2 f \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{32 b c \sqrt {1+c^2 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.89 (sec) , antiderivative size = 390, normalized size of antiderivative = 0.79 \[ \int (f+g x) \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {-d^3 \left (1+c^2 x^2\right ) \left (-1680 a \sqrt {1+c^2 x^2} \left (48 g \left (1+c^2 x^2\right )^3+7 c^2 f x \left (33+26 c^2 x^2+8 c^4 x^4\right )\right )+b c \left (2304 g x \left (35+35 c^2 x^2+21 c^4 x^4+5 c^6 x^6\right )+245 f \left (299+792 c^2 x^2+312 c^4 x^4+64 c^6 x^6\right )\right )\right )+88200 b c d^3 f \left (1+c^2 x^2\right ) \text {arcsinh}(c x)^2+176400 a c d^{5/2} f \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )+420 b d^3 \left (1+c^2 x^2\right ) \text {arcsinh}(c x) \left (192 g \sqrt {1+c^2 x^2}+576 c^2 g x^2 \sqrt {1+c^2 x^2}+576 c^4 g x^4 \sqrt {1+c^2 x^2}+192 c^6 g x^6 \sqrt {1+c^2 x^2}+315 c f \sinh (2 \text {arcsinh}(c x))+63 c f \sinh (4 \text {arcsinh}(c x))+7 c f \sinh (6 \text {arcsinh}(c x))\right )}{564480 c^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}} \]

[In]

Integrate[(f + g*x)*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]),x]

[Out]

(-(d^3*(1 + c^2*x^2)*(-1680*a*Sqrt[1 + c^2*x^2]*(48*g*(1 + c^2*x^2)^3 + 7*c^2*f*x*(33 + 26*c^2*x^2 + 8*c^4*x^4
)) + b*c*(2304*g*x*(35 + 35*c^2*x^2 + 21*c^4*x^4 + 5*c^6*x^6) + 245*f*(299 + 792*c^2*x^2 + 312*c^4*x^4 + 64*c^
6*x^6)))) + 88200*b*c*d^3*f*(1 + c^2*x^2)*ArcSinh[c*x]^2 + 176400*a*c*d^(5/2)*f*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2
*d*x^2]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]] + 420*b*d^3*(1 + c^2*x^2)*ArcSinh[c*x]*(192*g*Sqrt[1 + c^2*x^
2] + 576*c^2*g*x^2*Sqrt[1 + c^2*x^2] + 576*c^4*g*x^4*Sqrt[1 + c^2*x^2] + 192*c^6*g*x^6*Sqrt[1 + c^2*x^2] + 315
*c*f*Sinh[2*ArcSinh[c*x]] + 63*c*f*Sinh[4*ArcSinh[c*x]] + 7*c*f*Sinh[6*ArcSinh[c*x]]))/(564480*c^2*Sqrt[1 + c^
2*x^2]*Sqrt[d + c^2*d*x^2])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1678\) vs. \(2(430)=860\).

Time = 1.00 (sec) , antiderivative size = 1679, normalized size of antiderivative = 3.40

method	result	size
default	\(\text {Expression too large to display}\)	\(1679\)
parts	\(\text {Expression too large to display}\)	\(1679\)

[In]

int((g*x+f)*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x)),x,method=_RETURNVERBOSE)

[Out]

1/6*a*f*x*(c^2*d*x^2+d)^(5/2)+5/24*a*f*d*x*(c^2*d*x^2+d)^(3/2)+5/16*a*f*d^2*x*(c^2*d*x^2+d)^(1/2)+5/16*a*f*d^3
*ln(c^2*d*x/(c^2*d)^(1/2)+(c^2*d*x^2+d)^(1/2))/(c^2*d)^(1/2)+1/7*a*g*(c^2*d*x^2+d)^(7/2)/c^2/d+b*(5/32*(d*(c^2
*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/c*f*arcsinh(c*x)^2*d^2+1/6272*(d*(c^2*x^2+1))^(1/2)*(64*c^8*x^8+64*c^7*x^7*(c
^2*x^2+1)^(1/2)+144*c^6*x^6+112*c^5*x^5*(c^2*x^2+1)^(1/2)+104*c^4*x^4+56*c^3*x^3*(c^2*x^2+1)^(1/2)+25*c^2*x^2+
7*c*x*(c^2*x^2+1)^(1/2)+1)*g*(-1+7*arcsinh(c*x))*d^2/c^2/(c^2*x^2+1)+1/2304*(d*(c^2*x^2+1))^(1/2)*(32*c^7*x^7+
32*c^6*x^6*(c^2*x^2+1)^(1/2)+64*c^5*x^5+48*c^4*x^4*(c^2*x^2+1)^(1/2)+38*c^3*x^3+18*c^2*x^2*(c^2*x^2+1)^(1/2)+6
*c*x+(c^2*x^2+1)^(1/2))*f*(-1+6*arcsinh(c*x))*d^2/c/(c^2*x^2+1)+1/640*(d*(c^2*x^2+1))^(1/2)*(16*c^6*x^6+16*c^5
*x^5*(c^2*x^2+1)^(1/2)+28*c^4*x^4+20*c^3*x^3*(c^2*x^2+1)^(1/2)+13*c^2*x^2+5*c*x*(c^2*x^2+1)^(1/2)+1)*g*(-1+5*a
rcsinh(c*x))*d^2/c^2/(c^2*x^2+1)+3/512*(d*(c^2*x^2+1))^(1/2)*(8*c^5*x^5+8*c^4*x^4*(c^2*x^2+1)^(1/2)+12*c^3*x^3
+8*c^2*x^2*(c^2*x^2+1)^(1/2)+4*c*x+(c^2*x^2+1)^(1/2))*f*(-1+4*arcsinh(c*x))*d^2/c/(c^2*x^2+1)+1/128*(d*(c^2*x^
2+1))^(1/2)*(4*c^4*x^4+4*c^3*x^3*(c^2*x^2+1)^(1/2)+5*c^2*x^2+3*c*x*(c^2*x^2+1)^(1/2)+1)*g*(-1+3*arcsinh(c*x))*
d^2/c^2/(c^2*x^2+1)+15/256*(d*(c^2*x^2+1))^(1/2)*(2*c^3*x^3+2*c^2*x^2*(c^2*x^2+1)^(1/2)+2*c*x+(c^2*x^2+1)^(1/2
))*f*(-1+2*arcsinh(c*x))*d^2/c/(c^2*x^2+1)+5/128*(d*(c^2*x^2+1))^(1/2)*(c^2*x^2+c*x*(c^2*x^2+1)^(1/2)+1)*g*(-1
+arcsinh(c*x))*d^2/c^2/(c^2*x^2+1)+5/128*(d*(c^2*x^2+1))^(1/2)*(c^2*x^2-c*x*(c^2*x^2+1)^(1/2)+1)*g*(arcsinh(c*
x)+1)*d^2/c^2/(c^2*x^2+1)+15/256*(d*(c^2*x^2+1))^(1/2)*(2*c^3*x^3-2*c^2*x^2*(c^2*x^2+1)^(1/2)+2*c*x-(c^2*x^2+1
)^(1/2))*f*(1+2*arcsinh(c*x))*d^2/c/(c^2*x^2+1)+1/128*(d*(c^2*x^2+1))^(1/2)*(4*c^4*x^4-4*c^3*x^3*(c^2*x^2+1)^(
1/2)+5*c^2*x^2-3*c*x*(c^2*x^2+1)^(1/2)+1)*g*(3*arcsinh(c*x)+1)*d^2/c^2/(c^2*x^2+1)+3/512*(d*(c^2*x^2+1))^(1/2)
*(8*c^5*x^5-8*c^4*x^4*(c^2*x^2+1)^(1/2)+12*c^3*x^3-8*c^2*x^2*(c^2*x^2+1)^(1/2)+4*c*x-(c^2*x^2+1)^(1/2))*f*(1+4
*arcsinh(c*x))*d^2/c/(c^2*x^2+1)+1/640*(d*(c^2*x^2+1))^(1/2)*(16*c^6*x^6-16*c^5*x^5*(c^2*x^2+1)^(1/2)+28*c^4*x
^4-20*c^3*x^3*(c^2*x^2+1)^(1/2)+13*c^2*x^2-5*c*x*(c^2*x^2+1)^(1/2)+1)*g*(1+5*arcsinh(c*x))*d^2/c^2/(c^2*x^2+1)
+1/2304*(d*(c^2*x^2+1))^(1/2)*(32*c^7*x^7-32*c^6*x^6*(c^2*x^2+1)^(1/2)+64*c^5*x^5-48*c^4*x^4*(c^2*x^2+1)^(1/2)
+38*c^3*x^3-18*c^2*x^2*(c^2*x^2+1)^(1/2)+6*c*x-(c^2*x^2+1)^(1/2))*f*(1+6*arcsinh(c*x))*d^2/c/(c^2*x^2+1)+1/627
2*(d*(c^2*x^2+1))^(1/2)*(64*c^8*x^8-64*c^7*x^7*(c^2*x^2+1)^(1/2)+144*c^6*x^6-112*c^5*x^5*(c^2*x^2+1)^(1/2)+104
*c^4*x^4-56*c^3*x^3*(c^2*x^2+1)^(1/2)+25*c^2*x^2-7*c*x*(c^2*x^2+1)^(1/2)+1)*g*(1+7*arcsinh(c*x))*d^2/c^2/(c^2*
x^2+1))

Fricas [F]

\[ \int (f+g x) \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\int { {\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (g x + f\right )} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} \,d x } \]

[In]

integrate((g*x+f)*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x)),x, algorithm="fricas")

[Out]

integral((a*c^4*d^2*g*x^5 + a*c^4*d^2*f*x^4 + 2*a*c^2*d^2*g*x^3 + 2*a*c^2*d^2*f*x^2 + a*d^2*g*x + a*d^2*f + (b
*c^4*d^2*g*x^5 + b*c^4*d^2*f*x^4 + 2*b*c^2*d^2*g*x^3 + 2*b*c^2*d^2*f*x^2 + b*d^2*g*x + b*d^2*f)*arcsinh(c*x))*
sqrt(c^2*d*x^2 + d), x)

Sympy [F]

\[ \int (f+g x) \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\int \left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \left (a + b \operatorname {asinh}{\left (c x \right )}\right ) \left (f + g x\right )\, dx \]

[In]

integrate((g*x+f)*(c**2*d*x**2+d)**(5/2)*(a+b*asinh(c*x)),x)

[Out]

Integral((d*(c**2*x**2 + 1))**(5/2)*(a + b*asinh(c*x))*(f + g*x), x)

Maxima [F(-2)]

Exception generated. \[ \int (f+g x) \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate((g*x+f)*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x)),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negative exponent.

Giac [F(-2)]

Exception generated. \[ \int (f+g x) \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate((g*x+f)*(c^2*d*x^2+d)^(5/2)*(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. \[ \int (f+g x) \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\int \left (f+g\,x\right )\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^{5/2} \,d x \]

[In]

int((f + g*x)*(a + b*asinh(c*x))*(d + c^2*d*x^2)^(5/2),x)

[Out]

int((f + g*x)*(a + b*asinh(c*x))*(d + c^2*d*x^2)^(5/2), x)