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

Optimal. Leaf size=23 \[ \text {Int}\left (\left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right ),x\right ) \]

[Out]

Unintegrable((e*x^2+d)^(3/2)*(a+b*arcsec(c*x)),x)

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Rubi [A]  time = 0.04, 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 = {} \[ \int \left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right ) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 5.53, size = 0, normalized size = 0.00 \[ \int \left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right ) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.78, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a e x^{2} + a d + {\left (b e x^{2} + b d\right )} \operatorname {arcsec}\left (c x\right )\right )} \sqrt {e x^{2} + d}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 2.26, size = 0, normalized size = 0.00 \[ \int \left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\mathrm {arcsec}\left (c x \right )\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{8} \, {\left (2 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} x + 3 \, \sqrt {e x^{2} + d} d x + \frac {3 \, d^{2} \operatorname {arsinh}\left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {e}}\right )} a + b \int {\left (e x^{2} + d\right )}^{\frac {3}{2}} \arctan \left (\sqrt {c x + 1} \sqrt {c x - 1}\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(2*(e*x^2 + d)^(3/2)*x + 3*sqrt(e*x^2 + d)*d*x + 3*d^2*arcsinh(e*x/sqrt(d*e))/sqrt(e))*a + b*integrate((e*
x^2 + d)^(3/2)*arctan(sqrt(c*x + 1)*sqrt(c*x - 1)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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