Integrand size = 24, antiderivative size = 236 \[ \int \frac {x \left (d+c^2 d x^2\right )}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=-\frac {2 d x \left (1+c^2 x^2\right )^{3/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}+\frac {d e^{\frac {4 a}{b}} \sqrt {\pi } \text {erf}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^2}+\frac {d e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{2 b^{3/2} c^2}+\frac {d e^{-\frac {4 a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^2}+\frac {d e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{2 b^{3/2} c^2} \] Output:
-2*d*x*(c^2*x^2+1)^(3/2)/b/c/(a+b*arcsinh(c*x))^(1/2)+1/4*d*exp(4*a/b)*Pi^ (1/2)*erf(2*(a+b*arcsinh(c*x))^(1/2)/b^(1/2))/b^(3/2)/c^2+1/4*d*exp(2*a/b) *2^(1/2)*Pi^(1/2)*erf(2^(1/2)*(a+b*arcsinh(c*x))^(1/2)/b^(1/2))/b^(3/2)/c^ 2+1/4*d*Pi^(1/2)*erfi(2*(a+b*arcsinh(c*x))^(1/2)/b^(1/2))/b^(3/2)/c^2/exp( 4*a/b)+1/4*d*2^(1/2)*Pi^(1/2)*erfi(2^(1/2)*(a+b*arcsinh(c*x))^(1/2)/b^(1/2 ))/b^(3/2)/c^2/exp(2*a/b)
Time = 0.34 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.96 \[ \int \frac {x \left (d+c^2 d x^2\right )}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=\frac {d e^{-\frac {4 a}{b}} \left (\sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )+\sqrt {2} e^{\frac {2 a}{b}} \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )-e^{\frac {4 a}{b}} \left (\sqrt {2} e^{\frac {2 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {1}{2},\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )+e^{\frac {4 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {1}{2},\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )+2 \sinh (2 \text {arcsinh}(c x))+\sinh (4 \text {arcsinh}(c x))\right )\right )}{4 b c^2 \sqrt {a+b \text {arcsinh}(c x)}} \] Input:
Integrate[(x*(d + c^2*d*x^2))/(a + b*ArcSinh[c*x])^(3/2),x]
Output:
(d*(Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[1/2, (-4*(a + b*ArcSinh[c*x]))/b ] + Sqrt[2]*E^((2*a)/b)*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[1/2, (-2*(a + b*ArcSinh[c*x]))/b] - E^((4*a)/b)*(Sqrt[2]*E^((2*a)/b)*Sqrt[a/b + ArcSin h[c*x]]*Gamma[1/2, (2*(a + b*ArcSinh[c*x]))/b] + E^((4*a)/b)*Sqrt[a/b + Ar cSinh[c*x]]*Gamma[1/2, (4*(a + b*ArcSinh[c*x]))/b] + 2*Sinh[2*ArcSinh[c*x] ] + Sinh[4*ArcSinh[c*x]])))/(4*b*c^2*E^((4*a)/b)*Sqrt[a + b*ArcSinh[c*x]])
Time = 2.29 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.14, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6229, 6206, 3042, 3793, 2009, 6234, 5971, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {x \left (c^2 d x^2+d\right )}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx\) |
\(\Big \downarrow \) 6229 |
\(\displaystyle \frac {2 d \int \frac {\sqrt {c^2 x^2+1}}{\sqrt {a+b \text {arcsinh}(c x)}}dx}{b c}+\frac {8 c d \int \frac {x^2 \sqrt {c^2 x^2+1}}{\sqrt {a+b \text {arcsinh}(c x)}}dx}{b}-\frac {2 d x \left (c^2 x^2+1\right )^{3/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}\) |
\(\Big \downarrow \) 6206 |
\(\displaystyle \frac {2 d \int \frac {\cosh ^2\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{b^2 c^2}+\frac {8 c d \int \frac {x^2 \sqrt {c^2 x^2+1}}{\sqrt {a+b \text {arcsinh}(c x)}}dx}{b}-\frac {2 d x \left (c^2 x^2+1\right )^{3/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 d \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c x))}{b}+\frac {\pi }{2}\right )^2}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{b^2 c^2}+\frac {8 c d \int \frac {x^2 \sqrt {c^2 x^2+1}}{\sqrt {a+b \text {arcsinh}(c x)}}dx}{b}-\frac {2 d x \left (c^2 x^2+1\right )^{3/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}\) |
\(\Big \downarrow \) 3793 |
\(\displaystyle \frac {2 d \int \left (\frac {\cosh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{2 \sqrt {a+b \text {arcsinh}(c x)}}+\frac {1}{2 \sqrt {a+b \text {arcsinh}(c x)}}\right )d(a+b \text {arcsinh}(c x))}{b^2 c^2}+\frac {8 c d \int \frac {x^2 \sqrt {c^2 x^2+1}}{\sqrt {a+b \text {arcsinh}(c x)}}dx}{b}-\frac {2 d x \left (c^2 x^2+1\right )^{3/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {8 c d \int \frac {x^2 \sqrt {c^2 x^2+1}}{\sqrt {a+b \text {arcsinh}(c x)}}dx}{b}+\frac {2 d \left (\frac {1}{4} \sqrt {\frac {\pi }{2}} \sqrt {b} e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{4} \sqrt {\frac {\pi }{2}} \sqrt {b} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\sqrt {a+b \text {arcsinh}(c x)}\right )}{b^2 c^2}-\frac {2 d x \left (c^2 x^2+1\right )^{3/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}\) |
\(\Big \downarrow \) 6234 |
\(\displaystyle \frac {8 d \int \frac {\cosh ^2\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh ^2\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{b^2 c^2}+\frac {2 d \left (\frac {1}{4} \sqrt {\frac {\pi }{2}} \sqrt {b} e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{4} \sqrt {\frac {\pi }{2}} \sqrt {b} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\sqrt {a+b \text {arcsinh}(c x)}\right )}{b^2 c^2}-\frac {2 d x \left (c^2 x^2+1\right )^{3/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}\) |
\(\Big \downarrow \) 5971 |
\(\displaystyle \frac {8 d \int \left (\frac {\cosh \left (\frac {4 a}{b}-\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{8 \sqrt {a+b \text {arcsinh}(c x)}}-\frac {1}{8 \sqrt {a+b \text {arcsinh}(c x)}}\right )d(a+b \text {arcsinh}(c x))}{b^2 c^2}+\frac {2 d \left (\frac {1}{4} \sqrt {\frac {\pi }{2}} \sqrt {b} e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{4} \sqrt {\frac {\pi }{2}} \sqrt {b} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\sqrt {a+b \text {arcsinh}(c x)}\right )}{b^2 c^2}-\frac {2 d x \left (c^2 x^2+1\right )^{3/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {8 d \left (\frac {1}{32} \sqrt {\pi } \sqrt {b} e^{\frac {4 a}{b}} \text {erf}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{32} \sqrt {\pi } \sqrt {b} e^{-\frac {4 a}{b}} \text {erfi}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{4} \sqrt {a+b \text {arcsinh}(c x)}\right )}{b^2 c^2}+\frac {2 d \left (\frac {1}{4} \sqrt {\frac {\pi }{2}} \sqrt {b} e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{4} \sqrt {\frac {\pi }{2}} \sqrt {b} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\sqrt {a+b \text {arcsinh}(c x)}\right )}{b^2 c^2}-\frac {2 d x \left (c^2 x^2+1\right )^{3/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}\) |
Input:
Int[(x*(d + c^2*d*x^2))/(a + b*ArcSinh[c*x])^(3/2),x]
Output:
(-2*d*x*(1 + c^2*x^2)^(3/2))/(b*c*Sqrt[a + b*ArcSinh[c*x]]) + (8*d*(-1/4*S qrt[a + b*ArcSinh[c*x]] + (Sqrt[b]*E^((4*a)/b)*Sqrt[Pi]*Erf[(2*Sqrt[a + b* ArcSinh[c*x]])/Sqrt[b]])/32 + (Sqrt[b]*Sqrt[Pi]*Erfi[(2*Sqrt[a + b*ArcSinh [c*x]])/Sqrt[b]])/(32*E^((4*a)/b))))/(b^2*c^2) + (2*d*(Sqrt[a + b*ArcSinh[ c*x]] + (Sqrt[b]*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c* x]])/Sqrt[b]])/4 + (Sqrt[b]*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c* x]])/Sqrt[b]])/(4*E^((2*a)/b))))/(b^2*c^2)
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f , m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & & IGtQ[p, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(1/(b*c))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Subst[Int [x^n*Cosh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a , b, c, d, e, n}, x] && EqQ[e, c^2*d] && IGtQ[2*p, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.)*((d_) + (e_ .)*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^m*Sqrt[1 + c^2*x^2]*(d + e*x^2)^p *((a + b*ArcSinh[c*x])^(n + 1)/(b*c*(n + 1))), x] + (-Simp[f*(m/(b*c*(n + 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] - Simp[c*((m + 2*p + 1)/(b*f* (n + 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] && LtQ[n, -1] && IGtQ[2*p, 0] && NeQ[m + 2*p + 1 , 0] && IGtQ[m, -3]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_) ^2)^(p_.), x_Symbol] :> Simp[(1/(b*c^(m + 1)))*Simp[(d + e*x^2)^p/(1 + c^2* x^2)^p] Subst[Int[x^n*Sinh[-a/b + x/b]^m*Cosh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c^2*d] && IGtQ[2*p + 2, 0] && IGtQ[m, 0]
\[\int \frac {x \left (c^{2} d \,x^{2}+d \right )}{\left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{\frac {3}{2}}}d x\]
Input:
int(x*(c^2*d*x^2+d)/(a+b*arcsinh(x*c))^(3/2),x)
Output:
int(x*(c^2*d*x^2+d)/(a+b*arcsinh(x*c))^(3/2),x)
Exception generated. \[ \int \frac {x \left (d+c^2 d x^2\right )}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x*(c^2*d*x^2+d)/(a+b*arcsinh(c*x))^(3/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {x \left (d+c^2 d x^2\right )}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=d \left (\int \frac {x}{a \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} + b \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} \operatorname {asinh}{\left (c x \right )}}\, dx + \int \frac {c^{2} x^{3}}{a \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} + b \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} \operatorname {asinh}{\left (c x \right )}}\, dx\right ) \] Input:
integrate(x*(c**2*d*x**2+d)/(a+b*asinh(c*x))**(3/2),x)
Output:
d*(Integral(x/(a*sqrt(a + b*asinh(c*x)) + b*sqrt(a + b*asinh(c*x))*asinh(c *x)), x) + Integral(c**2*x**3/(a*sqrt(a + b*asinh(c*x)) + b*sqrt(a + b*asi nh(c*x))*asinh(c*x)), x))
\[ \int \frac {x \left (d+c^2 d x^2\right )}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )} x}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(x*(c^2*d*x^2+d)/(a+b*arcsinh(c*x))^(3/2),x, algorithm="maxima")
Output:
integrate((c^2*d*x^2 + d)*x/(b*arcsinh(c*x) + a)^(3/2), x)
Exception generated. \[ \int \frac {x \left (d+c^2 d x^2\right )}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(x*(c^2*d*x^2+d)/(a+b*arcsinh(c*x))^(3/2),x, algorithm="giac")
Output:
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve cteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {x \left (d+c^2 d x^2\right )}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=\int \frac {x\,\left (d\,c^2\,x^2+d\right )}{{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^{3/2}} \,d x \] Input:
int((x*(d + c^2*d*x^2))/(a + b*asinh(c*x))^(3/2),x)
Output:
int((x*(d + c^2*d*x^2))/(a + b*asinh(c*x))^(3/2), x)
\[ \int \frac {x \left (d+c^2 d x^2\right )}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=d \left (\left (\int \frac {\sqrt {\mathit {asinh} \left (c x \right ) b +a}\, x^{3}}{\mathit {asinh} \left (c x \right )^{2} b^{2}+2 \mathit {asinh} \left (c x \right ) a b +a^{2}}d x \right ) c^{2}+\int \frac {\sqrt {\mathit {asinh} \left (c x \right ) b +a}\, x}{\mathit {asinh} \left (c x \right )^{2} b^{2}+2 \mathit {asinh} \left (c x \right ) a b +a^{2}}d x \right ) \] Input:
int(x*(c^2*d*x^2+d)/(a+b*asinh(c*x))^(3/2),x)
Output:
d*(int((sqrt(asinh(c*x)*b + a)*x**3)/(asinh(c*x)**2*b**2 + 2*asinh(c*x)*a* b + a**2),x)*c**2 + int((sqrt(asinh(c*x)*b + a)*x)/(asinh(c*x)**2*b**2 + 2 *asinh(c*x)*a*b + a**2),x))