\(\int x^m \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \, dx\) [323]

Optimal result

Integrand size = 28, antiderivative size = 28 \[ \int x^m \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \, dx=-\frac {2 b c x^{2+m} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{(2+m)^2 \sqrt {1+c^2 x^2}}+\frac {x^{1+m} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2+m}+\frac {2 b^2 c^2 x^{3+m} \sqrt {d+c^2 d x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3+m}{2},\frac {5+m}{2},-c^2 x^2\right )}{(2+m)^2 (3+m) \sqrt {1+c^2 x^2}}+\frac {d \text {Int}\left (\frac {x^m (a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}},x\right )}{2+m} \]

[Out]

x^(1+m)*(a+b*arcsinh(c*x))^2*(c^2*d*x^2+d)^(1/2)/(2+m)-2*b*c*x^(2+m)*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/(2
+m)^2/(c^2*x^2+1)^(1/2)+2*b^2*c^2*x^(3+m)*hypergeom([1/2, 3/2+1/2*m],[5/2+1/2*m],-c^2*x^2)*(c^2*d*x^2+d)^(1/2)
/(2+m)^2/(3+m)/(c^2*x^2+1)^(1/2)+d*Unintegrable(x^m*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^(1/2),x)/(2+m)

Rubi [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int x^m \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \, dx=\int x^m \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \, dx \]

[In]

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

[Out]

(-2*b*c*x^(2 + m)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/((2 + m)^2*Sqrt[1 + c^2*x^2]) + (x^(1 + m)*Sqrt[d
+ c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(2 + m) + (2*b^2*c^2*x^(3 + m)*Sqrt[d + c^2*d*x^2]*Hypergeometric2F1[1/2,
(3 + m)/2, (5 + m)/2, -(c^2*x^2)])/((2 + m)^2*(3 + m)*Sqrt[1 + c^2*x^2]) + (d*Defer[Int][(x^m*(a + b*ArcSinh[
c*x])^2)/Sqrt[d + c^2*d*x^2], x])/(2 + m)

Rubi steps \begin{align*} \text {integral}& = \frac {x^{1+m} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2+m}+\frac {d \int \frac {x^m (a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx}{2+m}-\frac {\left (2 b c \sqrt {d+c^2 d x^2}\right ) \int x^{1+m} (a+b \text {arcsinh}(c x)) \, dx}{(2+m) \sqrt {1+c^2 x^2}} \\ & = -\frac {2 b c x^{2+m} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{(2+m)^2 \sqrt {1+c^2 x^2}}+\frac {x^{1+m} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2+m}+\frac {d \int \frac {x^m (a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx}{2+m}+\frac {\left (2 b^2 c^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {x^{2+m}}{\sqrt {1+c^2 x^2}} \, dx}{(2+m)^2 \sqrt {1+c^2 x^2}} \\ & = -\frac {2 b c x^{2+m} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{(2+m)^2 \sqrt {1+c^2 x^2}}+\frac {x^{1+m} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2+m}+\frac {2 b^2 c^2 x^{3+m} \sqrt {d+c^2 d x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3+m}{2},\frac {5+m}{2},-c^2 x^2\right )}{(2+m)^2 (3+m) \sqrt {1+c^2 x^2}}+\frac {d \int \frac {x^m (a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx}{2+m} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.66 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int x^m \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \, dx=\int x^m \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.71 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93

\[\int x^{m} \sqrt {c^{2} d \,x^{2}+d}\, \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{2}d x\]

[In]

int(x^m*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^2,x)

[Out]

int(x^m*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^2,x)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.43 \[ \int x^m \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \, dx=\int { \sqrt {c^{2} d x^{2} + d} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{m} \,d x } \]

[In]

integrate(x^m*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^2,x, algorithm="fricas")

[Out]

integral(sqrt(c^2*d*x^2 + d)*(b^2*arcsinh(c*x)^2 + 2*a*b*arcsinh(c*x) + a^2)*x^m, x)

Sympy [N/A]

Not integrable

Time = 21.29 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int x^m \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \, dx=\int x^{m} \sqrt {d \left (c^{2} x^{2} + 1\right )} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}\, dx \]

[In]

integrate(x**m*(c**2*d*x**2+d)**(1/2)*(a+b*asinh(c*x))**2,x)

[Out]

Integral(x**m*sqrt(d*(c**2*x**2 + 1))*(a + b*asinh(c*x))**2, x)

Maxima [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int x^m \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \, dx=\int { \sqrt {c^{2} d x^{2} + d} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{m} \,d x } \]

[In]

integrate(x^m*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^2,x, algorithm="maxima")

[Out]

integrate(sqrt(c^2*d*x^2 + d)*(b*arcsinh(c*x) + a)^2*x^m, x)

Giac [F(-2)]

Exception generated. \[ \int x^m \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x^m*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^2,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 [N/A]

Not integrable

Time = 3.18 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int x^m \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \, dx=\int x^m\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,\sqrt {d\,c^2\,x^2+d} \,d x \]

[In]

int(x^m*(a + b*asinh(c*x))^2*(d + c^2*d*x^2)^(1/2),x)

[Out]

int(x^m*(a + b*asinh(c*x))^2*(d + c^2*d*x^2)^(1/2), x)