3.7.52 \(\int \sqrt {d+e x^2} (a+b \sinh ^{-1}(c x))^2 \, dx\) [652]

Optimal. Leaf size=25 \[ \text {Int}\left (\sqrt {d+e x^2} \left (a+b \sinh ^{-1}(c x)\right )^2,x\right ) \]

[Out]

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

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Rubi [A]
time = 0.03, 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 \sqrt {d+e x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 8.31, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {d+e x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(c^2*d-%e>0)', see `assume?` fo
r more detai

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [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((a+b*arcsinh(c*x))^2*(e*x^2+d)^(1/2),x, algorithm="giac")

[Out]

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

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,\sqrt {e\,x^2+d} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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