3.645 \(\int \frac {1}{(d+e x^2) (a+b \sinh ^{-1}(c x))^{3/2}} \, dx\)

Optimal. Leaf size=25 \[ \text {Int}\left (\frac {1}{\left (d+e x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^{3/2}},x\right ) \]

[Out]

Unintegrable(1/(e*x^2+d)/(a+b*arcsinh(c*x))^(3/2),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 = {} \[ \int \frac {1}{\left (d+e x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/((d + e*x^2)*(a + b*ArcSinh[c*x])^(3/2)),x]

[Out]

Defer[Int][1/((d + e*x^2)*(a + b*ArcSinh[c*x])^(3/2)), x]

Rubi steps

\begin {align*} \int \frac {1}{\left (d+e x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^{3/2}} \, dx &=\int \frac {1}{\left (d+e x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^{3/2}} \, dx\\ \end {align*}

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Mathematica [A]  time = 0.15, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (d+e x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/((d + e*x^2)*(a + b*ArcSinh[c*x])^(3/2)),x]

[Out]

Integrate[1/((d + e*x^2)*(a + b*ArcSinh[c*x])^(3/2)), x]

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x^2+d)/(a+b*arcsinh(c*x))^(3/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (e x^{2} + d\right )} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x^2+d)/(a+b*arcsinh(c*x))^(3/2),x, algorithm="giac")

[Out]

integrate(1/((e*x^2 + d)*(b*arcsinh(c*x) + a)^(3/2)), x)

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maple [A]  time = 0.39, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (e \,x^{2}+d \right ) \left (a +b \arcsinh \left (c x \right )\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x^2+d)/(a+b*arcsinh(c*x))^(3/2),x)

[Out]

int(1/(e*x^2+d)/(a+b*arcsinh(c*x))^(3/2),x)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (e x^{2} + d\right )} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x^2+d)/(a+b*arcsinh(c*x))^(3/2),x, algorithm="maxima")

[Out]

integrate(1/((e*x^2 + d)*(b*arcsinh(c*x) + a)^(3/2)), x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {1}{{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^{3/2}\,\left (e\,x^2+d\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + b*asinh(c*x))^(3/2)*(d + e*x^2)),x)

[Out]

int(1/((a + b*asinh(c*x))^(3/2)*(d + e*x^2)), x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{\frac {3}{2}} \left (d + e x^{2}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x**2+d)/(a+b*asinh(c*x))**(3/2),x)

[Out]

Integral(1/((a + b*asinh(c*x))**(3/2)*(d + e*x**2)), x)

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