3.7.60 \(\int \frac {\sqrt {d+e x^2}}{(a+b \sinh ^{-1}(c x))^2} \, dx\) [660]

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

[Out]

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

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Mathematica [A]
time = 2.94, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\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 \[\int \frac {\sqrt {e \,x^{2}+d}}{\left (a +b \arcsinh \left (c x \right )\right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

-((c^2*x^2 + 1)^(3/2)*sqrt(x^2*e + d) + (c^3*x^3 + c*x)*sqrt(x^2*e + d))/(a*b*c^3*x^2 + sqrt(c^2*x^2 + 1)*a*b*
c^2*x + a*b*c + (b^2*c^3*x^2 + sqrt(c^2*x^2 + 1)*b^2*c^2*x + b^2*c)*log(c*x + sqrt(c^2*x^2 + 1))) + integrate(
((2*c^3*x^4*e + c^3*d*x^2 - c*d)*(c^2*x^2 + 1)*sqrt(x^2*e + d) + (4*c^4*x^5*e + 2*(c^4*d + 2*c^2*e)*x^3 + (c^2
*d + e)*x)*sqrt(c^2*x^2 + 1)*sqrt(x^2*e + d) + (2*c^5*x^6*e + (c^5*d + 4*c^3*e)*x^4 + 2*(c^3*d + c*e)*x^2 + c*
d)*sqrt(x^2*e + d))/(a*b*c^5*x^6*e + (a*b*c^5*d + 2*a*b*c^3*e)*x^4 + a*b*c*d + (2*a*b*c^3*d + a*b*c*e)*x^2 + (
a*b*c^3*x^4*e + a*b*c^3*d*x^2)*(c^2*x^2 + 1) + (b^2*c^5*x^6*e + (b^2*c^5*d + 2*b^2*c^3*e)*x^4 + b^2*c*d + (2*b
^2*c^3*d + b^2*c*e)*x^2 + (b^2*c^3*x^4*e + b^2*c^3*d*x^2)*(c^2*x^2 + 1) + 2*(b^2*c^4*x^5*e + b^2*c^2*d*x + (b^
2*c^4*d + b^2*c^2*e)*x^3)*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1)) + 2*(a*b*c^4*x^5*e + a*b*c^2*d*x + (
a*b*c^4*d + a*b*c^2*e)*x^3)*sqrt(c^2*x^2 + 1)), x)

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(d + e*x**2)/(a + b*asinh(c*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((e*x^2+d)^(1/2)/(a+b*arcsinh(c*x))^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 \frac {\sqrt {e\,x^2+d}}{{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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