3.5.3 \(\int \frac {x^m \sqrt {1+c^2 x^2}}{a+b \sinh ^{-1}(c x)} \, dx\) [403]

Optimal. Leaf size=30 \[ \text {Int}\left (\frac {x^m \sqrt {1+c^2 x^2}}{a+b \sinh ^{-1}(c x)},x\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.08, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {x^m \sqrt {1+c^2 x^2}}{a+b \sinh ^{-1}(c x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(x^m*Sqrt[1 + c^2*x^2])/(a + b*ArcSinh[c*x]),x]

[Out]

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

Rubi steps

\begin {align*} \int \frac {x^m \sqrt {1+c^2 x^2}}{a+b \sinh ^{-1}(c x)} \, dx &=\int \frac {x^m \sqrt {1+c^2 x^2}}{a+b \sinh ^{-1}(c x)} \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.09, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^m \sqrt {1+c^2 x^2}}{a+b \sinh ^{-1}(c x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(x^m*Sqrt[1 + c^2*x^2])/(a + b*ArcSinh[c*x]),x]

[Out]

Integrate[(x^m*Sqrt[1 + c^2*x^2])/(a + b*ArcSinh[c*x]), x]

________________________________________________________________________________________

Maple [A]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {x^{m} \sqrt {c^{2} x^{2}+1}}{a +b \arcsinh \left (c x \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(c^2*x^2 + 1)*x^m/(b*arcsinh(c*x) + a), x)

________________________________________________________________________________________

Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(c^2*x^2 + 1)*x^m/(b*arcsinh(c*x) + a), x)

________________________________________________________________________________________

Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{m} \sqrt {c^{2} x^{2} + 1}}{a + b \operatorname {asinh}{\left (c x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(c^2*x^2+1)^(1/2)/(a+b*arcsinh(c*x)),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 [A]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {x^m\,\sqrt {c^2\,x^2+1}}{a+b\,\mathrm {asinh}\left (c\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________