3.2.64 \(\int \frac {x^4 (a+b \text {sech}^{-1}(c x))}{(d+e x^2)^{3/2}} \, dx\) [164]

Optimal. Leaf size=26 \[ \text {Int}\left (\frac {x^4 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}},x\right ) \]

[Out]

Unintegrable(x^4*(a+b*arcsech(c*x))/(e*x^2+d)^(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 = {} \begin {gather*} \int \frac {x^4 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(x^4*(a + b*ArcSech[c*x]))/(d + e*x^2)^(3/2),x]

[Out]

Defer[Int][(x^4*(a + b*ArcSech[c*x]))/(d + e*x^2)^(3/2), x]

Rubi steps

\begin {align*} \int \frac {x^4 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx &=\int \frac {x^4 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx\\ \end {align*}

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Mathematica [A]
time = 7.34, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^4 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(x^4*(a + b*ArcSech[c*x]))/(d + e*x^2)^(3/2),x]

[Out]

Integrate[(x^4*(a + b*ArcSech[c*x]))/(d + e*x^2)^(3/2), x]

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Maple [A]
time = 1.19, size = 0, normalized size = 0.00 \[\int \frac {x^{4} \left (a +b \,\mathrm {arcsech}\left (c x \right )\right )}{\left (e \,x^{2}+d \right )^{\frac {3}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(a+b*arcsech(c*x))/(e*x^2+d)^(3/2),x)

[Out]

int(x^4*(a+b*arcsech(c*x))/(e*x^2+d)^(3/2),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(x^4*(a+b*arcsech(c*x))/(e*x^2+d)^(3/2),x, algorithm="maxima")

[Out]

1/2*(x^3*e^(-1)/sqrt(x^2*e + d) - 3*d*arcsinh(x*e^(1/2)/sqrt(d))*e^(-5/2) + 3*d*x*e^(-2)/sqrt(x^2*e + d))*a +
b*integrate(x^4*log(sqrt(1/(c*x) + 1)*sqrt(1/(c*x) - 1) + 1/(c*x))/(x^2*e + d)^(3/2), 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(x^4*(a+b*arcsech(c*x))/(e*x^2+d)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} \left (a + b \operatorname {asech}{\left (c x \right )}\right )}{\left (d + e x^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(a+b*asech(c*x))/(e*x**2+d)**(3/2),x)

[Out]

Integral(x**4*(a + b*asech(c*x))/(d + e*x**2)**(3/2), 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(x^4*(a+b*arcsech(c*x))/(e*x^2+d)^(3/2),x, algorithm="giac")

[Out]

integrate((b*arcsech(c*x) + a)*x^4/(e*x^2 + d)^(3/2), x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {x^4\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4*(a + b*acosh(1/(c*x))))/(d + e*x^2)^(3/2),x)

[Out]

int((x^4*(a + b*acosh(1/(c*x))))/(d + e*x^2)^(3/2), x)

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