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

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

[Out]

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

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 2.34, size = 0, normalized size = 0.00 \[ \int \frac {\left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{d+e x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Integrate[(a + b*ArcCosh[c*x])^(3/2)/(d + e*x^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((a+b*arccosh(c*x))^(3/2)/(e*x^2+d),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 \[ \int \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{\frac {3}{2}}}{e x^{2} + d}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.62, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{\frac {3}{2}}}{e \,x^{2}+d}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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