Integrand size = 26, antiderivative size = 136 \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=-\frac {b x \sqrt {1+c^2 x^2}}{c^3 d \sqrt {d+c^2 d x^2}}+\frac {a+b \text {arcsinh}(c x)}{c^4 d \sqrt {d+c^2 d x^2}}+\frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{c^4 d^2}-\frac {b \sqrt {1+c^2 x^2} \arctan (c x)}{c^4 d \sqrt {d+c^2 d x^2}} \] Output:
-b*x*(c^2*x^2+1)^(1/2)/c^3/d/(c^2*d*x^2+d)^(1/2)+(a+b*arcsinh(c*x))/c^4/d/ (c^2*d*x^2+d)^(1/2)+(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))/c^4/d^2-b*(c^2* x^2+1)^(1/2)*arctan(c*x)/c^4/d/(c^2*d*x^2+d)^(1/2)
Time = 0.32 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.96 \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\frac {\sqrt {d+c^2 d x^2} \left (-b c x-b c^3 x^3+2 a \sqrt {1+c^2 x^2}+a c^2 x^2 \sqrt {1+c^2 x^2}+b \sqrt {1+c^2 x^2} \left (2+c^2 x^2\right ) \text {arcsinh}(c x)-\left (b+b c^2 x^2\right ) \arctan (c x)\right )}{c^4 d^2 \left (1+c^2 x^2\right )^{3/2}} \] Input:
Integrate[(x^3*(a + b*ArcSinh[c*x]))/(d + c^2*d*x^2)^(3/2),x]
Output:
(Sqrt[d + c^2*d*x^2]*(-(b*c*x) - b*c^3*x^3 + 2*a*Sqrt[1 + c^2*x^2] + a*c^2 *x^2*Sqrt[1 + c^2*x^2] + b*Sqrt[1 + c^2*x^2]*(2 + c^2*x^2)*ArcSinh[c*x] - (b + b*c^2*x^2)*ArcTan[c*x]))/(c^4*d^2*(1 + c^2*x^2)^(3/2))
Time = 0.39 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.77, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {6219, 27, 299, 216}
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.
\(\displaystyle \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\left (c^2 d x^2+d\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 6219 |
\(\displaystyle -\frac {b c \sqrt {c^2 d x^2+d} \int \frac {c^2 x^2+2}{c^4 d^2 \left (c^2 x^2+1\right )}dx}{\sqrt {c^2 x^2+1}}+\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{c^4 d^2}+\frac {a+b \text {arcsinh}(c x)}{c^4 d \sqrt {c^2 d x^2+d}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {b \sqrt {c^2 d x^2+d} \int \frac {c^2 x^2+2}{c^2 x^2+1}dx}{c^3 d^2 \sqrt {c^2 x^2+1}}+\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{c^4 d^2}+\frac {a+b \text {arcsinh}(c x)}{c^4 d \sqrt {c^2 d x^2+d}}\) |
\(\Big \downarrow \) 299 |
\(\displaystyle -\frac {b \sqrt {c^2 d x^2+d} \left (\int \frac {1}{c^2 x^2+1}dx+x\right )}{c^3 d^2 \sqrt {c^2 x^2+1}}+\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{c^4 d^2}+\frac {a+b \text {arcsinh}(c x)}{c^4 d \sqrt {c^2 d x^2+d}}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{c^4 d^2}+\frac {a+b \text {arcsinh}(c x)}{c^4 d \sqrt {c^2 d x^2+d}}-\frac {b \left (\frac {\arctan (c x)}{c}+x\right ) \sqrt {c^2 d x^2+d}}{c^3 d^2 \sqrt {c^2 x^2+1}}\) |
Input:
Int[(x^3*(a + b*ArcSinh[c*x]))/(d + c^2*d*x^2)^(3/2),x]
Output:
(a + b*ArcSinh[c*x])/(c^4*d*Sqrt[d + c^2*d*x^2]) + (Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(c^4*d^2) - (b*Sqrt[d + c^2*d*x^2]*(x + ArcTan[c*x]/c) )/(c^3*d^2*Sqrt[1 + c^2*x^2])
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x *((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 *p + 3)) Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && NeQ[2*p + 3, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_ ), x_Symbol] :> With[{u = IntHide[x^m*(d + e*x^2)^p, x]}, Simp[(a + b*ArcSi nh[c*x]) u, x] - Simp[b*c*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]] Int[S implifyIntegrand[u/Sqrt[d + e*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x ] && EqQ[e, c^2*d] && IntegerQ[p - 1/2] && NeQ[p, -2^(-1)] && (IGtQ[(m + 1) /2, 0] || ILtQ[(m + 2*p + 3)/2, 0])
Result contains complex when optimal does not.
Time = 0.94 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.29
method | result | size |
default | \(a \left (\frac {x^{2}}{c^{2} d \sqrt {c^{2} d \,x^{2}+d}}+\frac {2}{d \,c^{4} \sqrt {c^{2} d \,x^{2}+d}}\right )+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}+i \sqrt {c^{2} x^{2}+1}\, \ln \left (x c +\sqrt {c^{2} x^{2}+1}-i\right )-i \sqrt {c^{2} x^{2}+1}\, \ln \left (x c +\sqrt {c^{2} x^{2}+1}+i\right )-\sqrt {c^{2} x^{2}+1}\, x c +2 \,\operatorname {arcsinh}\left (x c \right )\right )}{d^{2} c^{4} \left (c^{2} x^{2}+1\right )}\) | \(176\) |
parts | \(a \left (\frac {x^{2}}{c^{2} d \sqrt {c^{2} d \,x^{2}+d}}+\frac {2}{d \,c^{4} \sqrt {c^{2} d \,x^{2}+d}}\right )+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}+i \sqrt {c^{2} x^{2}+1}\, \ln \left (x c +\sqrt {c^{2} x^{2}+1}-i\right )-i \sqrt {c^{2} x^{2}+1}\, \ln \left (x c +\sqrt {c^{2} x^{2}+1}+i\right )-\sqrt {c^{2} x^{2}+1}\, x c +2 \,\operatorname {arcsinh}\left (x c \right )\right )}{d^{2} c^{4} \left (c^{2} x^{2}+1\right )}\) | \(176\) |
Input:
int(x^3*(a+b*arcsinh(x*c))/(c^2*d*x^2+d)^(3/2),x,method=_RETURNVERBOSE)
Output:
a*(x^2/c^2/d/(c^2*d*x^2+d)^(1/2)+2/d/c^4/(c^2*d*x^2+d)^(1/2))+b*(d*(c^2*x^ 2+1))^(1/2)*(arcsinh(x*c)*c^2*x^2+I*(c^2*x^2+1)^(1/2)*ln(x*c+(c^2*x^2+1)^( 1/2)-I)-I*(c^2*x^2+1)^(1/2)*ln(x*c+(c^2*x^2+1)^(1/2)+I)-(c^2*x^2+1)^(1/2)* x*c+2*arcsinh(x*c))/d^2/c^4/(c^2*x^2+1)
Time = 0.15 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.22 \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\frac {{\left (b c^{2} x^{2} + b\right )} \sqrt {d} \arctan \left (\frac {2 \, \sqrt {c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} + 1} c \sqrt {d} x}{c^{4} d x^{4} - d}\right ) + 2 \, {\left (b c^{2} x^{2} + 2 \, b\right )} \sqrt {c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + 2 \, {\left (a c^{2} x^{2} - \sqrt {c^{2} x^{2} + 1} b c x + 2 \, a\right )} \sqrt {c^{2} d x^{2} + d}}{2 \, {\left (c^{6} d^{2} x^{2} + c^{4} d^{2}\right )}} \] Input:
integrate(x^3*(a+b*arcsinh(c*x))/(c^2*d*x^2+d)^(3/2),x, algorithm="fricas" )
Output:
1/2*((b*c^2*x^2 + b)*sqrt(d)*arctan(2*sqrt(c^2*d*x^2 + d)*sqrt(c^2*x^2 + 1 )*c*sqrt(d)*x/(c^4*d*x^4 - d)) + 2*(b*c^2*x^2 + 2*b)*sqrt(c^2*d*x^2 + d)*l og(c*x + sqrt(c^2*x^2 + 1)) + 2*(a*c^2*x^2 - sqrt(c^2*x^2 + 1)*b*c*x + 2*a )*sqrt(c^2*d*x^2 + d))/(c^6*d^2*x^2 + c^4*d^2)
\[ \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\int \frac {x^{3} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )}{\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(x**3*(a+b*asinh(c*x))/(c**2*d*x**2+d)**(3/2),x)
Output:
Integral(x**3*(a + b*asinh(c*x))/(d*(c**2*x**2 + 1))**(3/2), x)
Time = 0.12 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.88 \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=-b c {\left (\frac {x}{c^{4} d^{\frac {3}{2}}} + \frac {\arctan \left (c x\right )}{c^{5} d^{\frac {3}{2}}}\right )} + b {\left (\frac {x^{2}}{\sqrt {c^{2} d x^{2} + d} c^{2} d} + \frac {2}{\sqrt {c^{2} d x^{2} + d} c^{4} d}\right )} \operatorname {arsinh}\left (c x\right ) + a {\left (\frac {x^{2}}{\sqrt {c^{2} d x^{2} + d} c^{2} d} + \frac {2}{\sqrt {c^{2} d x^{2} + d} c^{4} d}\right )} \] Input:
integrate(x^3*(a+b*arcsinh(c*x))/(c^2*d*x^2+d)^(3/2),x, algorithm="maxima" )
Output:
-b*c*(x/(c^4*d^(3/2)) + arctan(c*x)/(c^5*d^(3/2))) + b*(x^2/(sqrt(c^2*d*x^ 2 + d)*c^2*d) + 2/(sqrt(c^2*d*x^2 + d)*c^4*d))*arcsinh(c*x) + a*(x^2/(sqrt (c^2*d*x^2 + d)*c^2*d) + 2/(sqrt(c^2*d*x^2 + d)*c^4*d))
Exception generated. \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^3*(a+b*arcsinh(c*x))/(c^2*d*x^2+d)^(3/2),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\int \frac {x^3\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{{\left (d\,c^2\,x^2+d\right )}^{3/2}} \,d x \] Input:
int((x^3*(a + b*asinh(c*x)))/(d + c^2*d*x^2)^(3/2),x)
Output:
int((x^3*(a + b*asinh(c*x)))/(d + c^2*d*x^2)^(3/2), x)
\[ \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\frac {\sqrt {c^{2} x^{2}+1}\, a \,c^{2} x^{2}+2 \sqrt {c^{2} x^{2}+1}\, a +\left (\int \frac {\mathit {asinh} \left (c x \right ) x^{3}}{\sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) b \,c^{6} x^{2}+\left (\int \frac {\mathit {asinh} \left (c x \right ) x^{3}}{\sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) b \,c^{4}}{\sqrt {d}\, c^{4} d \left (c^{2} x^{2}+1\right )} \] Input:
int(x^3*(a+b*asinh(c*x))/(c^2*d*x^2+d)^(3/2),x)
Output:
(sqrt(c**2*x**2 + 1)*a*c**2*x**2 + 2*sqrt(c**2*x**2 + 1)*a + int((asinh(c* x)*x**3)/(sqrt(c**2*x**2 + 1)*c**2*x**2 + sqrt(c**2*x**2 + 1)),x)*b*c**6*x **2 + int((asinh(c*x)*x**3)/(sqrt(c**2*x**2 + 1)*c**2*x**2 + sqrt(c**2*x** 2 + 1)),x)*b*c**4)/(sqrt(d)*c**4*d*(c**2*x**2 + 1))