\(\int \frac {(d+c^2 d x^2)^{5/2} (a+b \text {arcsinh}(c x))}{x^4} \, dx\) [143]

Optimal result

Integrand size = 26, antiderivative size = 266 \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^4} \, dx=-\frac {b c d^2 \sqrt {d+c^2 d x^2}}{6 x^2 \sqrt {1+c^2 x^2}}-\frac {b c^5 d^2 x^2 \sqrt {d+c^2 d x^2}}{4 \sqrt {1+c^2 x^2}}+\frac {5}{2} c^4 d^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))-\frac {5 c^2 d \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 x}-\frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{3 x^3}+\frac {5 c^3 d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{4 b \sqrt {1+c^2 x^2}}+\frac {7 b c^3 d^2 \sqrt {d+c^2 d x^2} \log (x)}{3 \sqrt {1+c^2 x^2}} \]

[Out]

-5/3*c^2*d*(c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(c*x))/x-1/3*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))/x^3+5/2*c^4*d^2
*x*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)-1/6*b*c*d^2*(c^2*d*x^2+d)^(1/2)/x^2/(c^2*x^2+1)^(1/2)-1/4*b*c^5*d^2*
x^2*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)+5/4*c^3*d^2*(a+b*arcsinh(c*x))^2*(c^2*d*x^2+d)^(1/2)/b/(c^2*x^2+1)^(
1/2)+7/3*b*c^3*d^2*ln(x)*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {5807, 5785, 5783, 30, 14, 272, 45} \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^4} \, dx=-\frac {5 c^2 d \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 x}-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))}{3 x^3}+\frac {5}{2} c^4 d^2 x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))+\frac {5 c^3 d^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{4 b \sqrt {c^2 x^2+1}}-\frac {b c d^2 \sqrt {c^2 d x^2+d}}{6 x^2 \sqrt {c^2 x^2+1}}-\frac {b c^5 d^2 x^2 \sqrt {c^2 d x^2+d}}{4 \sqrt {c^2 x^2+1}}+\frac {7 b c^3 d^2 \log (x) \sqrt {c^2 d x^2+d}}{3 \sqrt {c^2 x^2+1}} \]

[In]

Int[((d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/x^4,x]

[Out]

-1/6*(b*c*d^2*Sqrt[d + c^2*d*x^2])/(x^2*Sqrt[1 + c^2*x^2]) - (b*c^5*d^2*x^2*Sqrt[d + c^2*d*x^2])/(4*Sqrt[1 + c
^2*x^2]) + (5*c^4*d^2*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/2 - (5*c^2*d*(d + c^2*d*x^2)^(3/2)*(a + b*Ar
cSinh[c*x]))/(3*x) - ((d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/(3*x^3) + (5*c^3*d^2*Sqrt[d + c^2*d*x^2]*(a
+ b*ArcSinh[c*x])^2)/(4*b*Sqrt[1 + c^2*x^2]) + (7*b*c^3*d^2*Sqrt[d + c^2*d*x^2]*Log[x])/(3*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 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 272

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

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 5807

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

Rubi steps \begin{align*} \text {integral}& = -\frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{3 x^3}+\frac {1}{3} \left (5 c^2 d\right ) \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^2} \, dx+\frac {\left (b c d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {\left (1+c^2 x^2\right )^2}{x^3} \, dx}{3 \sqrt {1+c^2 x^2}} \\ & = -\frac {5 c^2 d \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 x}-\frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{3 x^3}+\left (5 c^4 d^2\right ) \int \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx+\frac {\left (b c d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\left (1+c^2 x\right )^2}{x^2} \, dx,x,x^2\right )}{6 \sqrt {1+c^2 x^2}}+\frac {\left (5 b c^3 d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {1+c^2 x^2}{x} \, dx}{3 \sqrt {1+c^2 x^2}} \\ & = \frac {5}{2} c^4 d^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))-\frac {5 c^2 d \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 x}-\frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{3 x^3}+\frac {\left (b c d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \left (c^4+\frac {1}{x^2}+\frac {2 c^2}{x}\right ) \, dx,x,x^2\right )}{6 \sqrt {1+c^2 x^2}}+\frac {\left (5 b c^3 d^2 \sqrt {d+c^2 d x^2}\right ) \int \left (\frac {1}{x}+c^2 x\right ) \, dx}{3 \sqrt {1+c^2 x^2}}+\frac {\left (5 c^4 d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{2 \sqrt {1+c^2 x^2}}-\frac {\left (5 b c^5 d^2 \sqrt {d+c^2 d x^2}\right ) \int x \, dx}{2 \sqrt {1+c^2 x^2}} \\ & = -\frac {b c d^2 \sqrt {d+c^2 d x^2}}{6 x^2 \sqrt {1+c^2 x^2}}-\frac {b c^5 d^2 x^2 \sqrt {d+c^2 d x^2}}{4 \sqrt {1+c^2 x^2}}+\frac {5}{2} c^4 d^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))-\frac {5 c^2 d \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 x}-\frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{3 x^3}+\frac {5 c^3 d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{4 b \sqrt {1+c^2 x^2}}+\frac {7 b c^3 d^2 \sqrt {d+c^2 d x^2} \log (x)}{3 \sqrt {1+c^2 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.99 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.08 \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^4} \, dx=\frac {d^2 \left (4 a \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (-2-14 c^2 x^2+3 c^4 x^4\right )+24 b c^2 x^2 \sqrt {d+c^2 d x^2} \left (-2 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)+c x \text {arcsinh}(c x)^2+2 c x \log (c x)\right )-4 b \sqrt {d+c^2 d x^2} \left (c x+2 \left (1+c^2 x^2\right )^{3/2} \text {arcsinh}(c x)-2 c^3 x^3 \log (c x)\right )+60 a c^3 \sqrt {d} x^3 \sqrt {1+c^2 x^2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )-3 b c^3 x^3 \sqrt {d+c^2 d x^2} (\cosh (2 \text {arcsinh}(c x))-2 \text {arcsinh}(c x) (\text {arcsinh}(c x)+\sinh (2 \text {arcsinh}(c x))))\right )}{24 x^3 \sqrt {1+c^2 x^2}} \]

[In]

Integrate[((d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/x^4,x]

[Out]

(d^2*(4*a*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]*(-2 - 14*c^2*x^2 + 3*c^4*x^4) + 24*b*c^2*x^2*Sqrt[d + c^2*d*x^
2]*(-2*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + c*x*ArcSinh[c*x]^2 + 2*c*x*Log[c*x]) - 4*b*Sqrt[d + c^2*d*x^2]*(c*x +
2*(1 + c^2*x^2)^(3/2)*ArcSinh[c*x] - 2*c^3*x^3*Log[c*x]) + 60*a*c^3*Sqrt[d]*x^3*Sqrt[1 + c^2*x^2]*Log[c*d*x +
Sqrt[d]*Sqrt[d + c^2*d*x^2]] - 3*b*c^3*x^3*Sqrt[d + c^2*d*x^2]*(Cosh[2*ArcSinh[c*x]] - 2*ArcSinh[c*x]*(ArcSinh
[c*x] + Sinh[2*ArcSinh[c*x]]))))/(24*x^3*Sqrt[1 + c^2*x^2])

Maple [A] (verified)

Time = 0.22 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.21

method	result	size
default	\(-\frac {a \left (c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{3 d \,x^{3}}-\frac {4 a \,c^{2} \left (c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{3 d x}+\frac {4 a \,c^{4} x \left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{3}+\frac {5 a \,c^{4} d x \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3}+\frac {5 a \,c^{4} d^{2} x \sqrt {c^{2} d \,x^{2}+d}}{2}+\frac {5 a \,c^{4} d^{3} \ln \left (\frac {c^{2} d x}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{2 \sqrt {c^{2} d}}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (12 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{4} c^{4}-6 c^{5} x^{5}+30 \operatorname {arcsinh}\left (c x \right )^{2} x^{3} c^{3}-56 \,\operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}+56 \ln \left (\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right ) x^{3} c^{3}-56 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}-3 c^{3} x^{3}-8 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}-4 c x \right ) d^{2}}{24 \sqrt {c^{2} x^{2}+1}\, x^{3}}\)	\(322\)
parts	\(-\frac {a \left (c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{3 d \,x^{3}}-\frac {4 a \,c^{2} \left (c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{3 d x}+\frac {4 a \,c^{4} x \left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{3}+\frac {5 a \,c^{4} d x \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3}+\frac {5 a \,c^{4} d^{2} x \sqrt {c^{2} d \,x^{2}+d}}{2}+\frac {5 a \,c^{4} d^{3} \ln \left (\frac {c^{2} d x}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{2 \sqrt {c^{2} d}}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (12 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{4} c^{4}-6 c^{5} x^{5}+30 \operatorname {arcsinh}\left (c x \right )^{2} x^{3} c^{3}-56 \,\operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}+56 \ln \left (\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right ) x^{3} c^{3}-56 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}-3 c^{3} x^{3}-8 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}-4 c x \right ) d^{2}}{24 \sqrt {c^{2} x^{2}+1}\, x^{3}}\)	\(322\)

[In]

int((c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))/x^4,x,method=_RETURNVERBOSE)

[Out]

-1/3*a/d/x^3*(c^2*d*x^2+d)^(7/2)-4/3*a*c^2/d/x*(c^2*d*x^2+d)^(7/2)+4/3*a*c^4*x*(c^2*d*x^2+d)^(5/2)+5/3*a*c^4*d
*x*(c^2*d*x^2+d)^(3/2)+5/2*a*c^4*d^2*x*(c^2*d*x^2+d)^(1/2)+5/2*a*c^4*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)+1/24*b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/x^3*(12*arcsinh(c*x)*(c^2*x^2+1)^(1/2)*x
^4*c^4-6*c^5*x^5+30*arcsinh(c*x)^2*x^3*c^3-56*arcsinh(c*x)*c^3*x^3+56*ln((c*x+(c^2*x^2+1)^(1/2))^2-1)*x^3*c^3-
56*arcsinh(c*x)*(c^2*x^2+1)^(1/2)*x^2*c^2-3*c^3*x^3-8*arcsinh(c*x)*(c^2*x^2+1)^(1/2)-4*c*x)*d^2

Fricas [F]

\[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^4} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{x^{4}} \,d x } \]

[In]

integrate((c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))/x^4,x, algorithm="fricas")

[Out]

integral((a*c^4*d^2*x^4 + 2*a*c^2*d^2*x^2 + a*d^2 + (b*c^4*d^2*x^4 + 2*b*c^2*d^2*x^2 + b*d^2)*arcsinh(c*x))*sq
rt(c^2*d*x^2 + d)/x^4, x)

Sympy [F]

\[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^4} \, dx=\int \frac {\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )}{x^{4}}\, dx \]

[In]

integrate((c**2*d*x**2+d)**(5/2)*(a+b*asinh(c*x))/x**4,x)

[Out]

Integral((d*(c**2*x**2 + 1))**(5/2)*(a + b*asinh(c*x))/x**4, x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^4} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate((c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))/x^4,x, algorithm="maxima")

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^4} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))/x^4,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. \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^4} \, dx=\int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^{5/2}}{x^4} \,d x \]

[In]

int(((a + b*asinh(c*x))*(d + c^2*d*x^2)^(5/2))/x^4,x)

[Out]

int(((a + b*asinh(c*x))*(d + c^2*d*x^2)^(5/2))/x^4, x)