3.6.33 \(\int \frac {(a+b \cosh ^{-1}(c x))^2}{(d+e x^2)^{5/2}} \, dx\) [533]

Optimal. Leaf size=25 \[ \text {Int}\left (\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{\left (d+e x^2\right )^{5/2}},x\right ) \]

[Out]

Unintegrable((a+b*arccosh(c*x))^2/(e*x^2+d)^(5/2),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 {\left (a+b \cosh ^{-1}(c x)\right )^2}{\left (d+e x^2\right )^{5/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(a + b*ArcCosh[c*x])^2/(d + e*x^2)^(5/2),x]

[Out]

Defer[Int][(a + b*ArcCosh[c*x])^2/(d + e*x^2)^(5/2), x]

Rubi steps

\begin {align*} \int \frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{\left (d+e x^2\right )^{5/2}} \, dx &=\int \frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{\left (d+e x^2\right )^{5/2}} \, dx\\ \end {align*}

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Mathematica [A]
time = 60.60, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{\left (d+e x^2\right )^{5/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(a + b*ArcCosh[c*x])^2/(d + e*x^2)^(5/2),x]

[Out]

Integrate[(a + b*ArcCosh[c*x])^2/(d + e*x^2)^(5/2), x]

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Maple [A]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{2}}{\left (e \,x^{2}+d \right )^{\frac {5}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccosh(c*x))^2/(e*x^2+d)^(5/2),x)

[Out]

int((a+b*arccosh(c*x))^2/(e*x^2+d)^(5/2),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((a+b*arccosh(c*x))^2/(e*x^2+d)^(5/2),x, algorithm="maxima")

[Out]

1/3*a^2*(2*x/(sqrt(x^2*e + d)*d^2) + x/((x^2*e + d)^(3/2)*d)) + integrate(b^2*log(c*x + sqrt(c*x + 1)*sqrt(c*x
- 1))^2/(x^2*e + d)^(5/2) + 2*a*b*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/(x^2*e + d)^(5/2), 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((a+b*arccosh(c*x))^2/(e*x^2+d)^(5/2),x, algorithm="fricas")

[Out]

integral((b^2*arccosh(c*x)^2 + 2*a*b*arccosh(c*x) + a^2)*sqrt(x^2*e + d)/(x^6*e^3 + 3*d*x^4*e^2 + 3*d^2*x^2*e
+ d^3), x)

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}{\left (d + e x^{2}\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acosh(c*x))**2/(e*x**2+d)**(5/2),x)

[Out]

Integral((a + b*acosh(c*x))**2/(d + e*x**2)**(5/2), 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*arccosh(c*x))^2/(e*x^2+d)^(5/2),x, algorithm="giac")

[Out]

integrate((b*arccosh(c*x) + a)^2/(e*x^2 + d)^(5/2), x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2}{{\left (e\,x^2+d\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*acosh(c*x))^2/(d + e*x^2)^(5/2),x)

[Out]

int((a + b*acosh(c*x))^2/(d + e*x^2)^(5/2), x)

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