3.188 \(\int \frac {x^m (a+b \sinh ^{-1}(c x))}{d+c^2 d x^2} \, dx\)

Optimal. Leaf size=27 \[ \text {Int}\left (\frac {x^m \left (a+b \sinh ^{-1}(c x)\right )}{c^2 d x^2+d},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.07, 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 = {} \[ \int \frac {x^m \left (a+b \sinh ^{-1}(c x)\right )}{d+c^2 d x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 3.81, size = 0, normalized size = 0.00 \[ \int \frac {x^m \left (a+b \sinh ^{-1}(c x)\right )}{d+c^2 d x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{m}}{c^{2} d x^{2} + d}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{m}}{c^{2} d x^{2} + d}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.48, size = 0, normalized size = 0.00 \[ \int \frac {x^{m} \left (a +b \arcsinh \left (c x \right )\right )}{c^{2} d \,x^{2}+d}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{m}}{c^{2} d x^{2} + d}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {x^m\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{d\,c^2\,x^2+d} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {a x^{m}}{c^{2} x^{2} + 1}\, dx + \int \frac {b x^{m} \operatorname {asinh}{\left (c x \right )}}{c^{2} x^{2} + 1}\, dx}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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