3.156 \(\int \frac {x^6 (a+b \sec ^{-1}(c x))}{(d+e x^2)^{5/2}} \, dx\)

Optimal. Leaf size=26 \[ \text {Int}\left (\frac {x^6 \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}},x\right ) \]

[Out]

Unintegrable(x^6*(a+b*arcsec(c*x))/(e*x^2+d)^(5/2),x)

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Rubi [A]  time = 0.12, 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 = {} \[ \int \frac {x^6 \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(x^6*(a + b*ArcSec[c*x]))/(d + e*x^2)^(5/2),x]

[Out]

Defer[Int][(x^6*(a + b*ArcSec[c*x]))/(d + e*x^2)^(5/2), x]

Rubi steps

\begin {align*} \int \frac {x^6 \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx &=\int \frac {x^6 \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx\\ \end {align*}

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Mathematica [A]  time = 10.32, size = 0, normalized size = 0.00 \[ \int \frac {x^6 \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(x^6*(a + b*ArcSec[c*x]))/(d + e*x^2)^(5/2),x]

[Out]

Integrate[(x^6*(a + b*ArcSec[c*x]))/(d + e*x^2)^(5/2), x]

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fricas [A]  time = 1.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{6} \operatorname {arcsec}\left (c x\right ) + a x^{6}\right )} \sqrt {e x^{2} + d}}{e^{3} x^{6} + 3 \, d e^{2} x^{4} + 3 \, d^{2} e x^{2} + d^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(a+b*arcsec(c*x))/(e*x^2+d)^(5/2),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arcsec}\left (c x\right ) + a\right )} x^{6}}{{\left (e x^{2} + d\right )}^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(a+b*arcsec(c*x))/(e*x^2+d)^(5/2),x, algorithm="giac")

[Out]

integrate((b*arcsec(c*x) + a)*x^6/(e*x^2 + d)^(5/2), x)

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maple [A]  time = 3.69, size = 0, normalized size = 0.00 \[ \int \frac {x^{6} \left (a +b \,\mathrm {arcsec}\left (c x \right )\right )}{\left (e \,x^{2}+d \right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*(a+b*arcsec(c*x))/(e*x^2+d)^(5/2),x)

[Out]

int(x^6*(a+b*arcsec(c*x))/(e*x^2+d)^(5/2),x)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{6} \, {\left (\frac {3 \, x^{5}}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} e} + \frac {5 \, d x {\left (\frac {3 \, x^{2}}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} e} + \frac {2 \, d}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} e^{2}}\right )}}{e} + \frac {5 \, d x}{\sqrt {e x^{2} + d} e^{3}} - \frac {15 \, d \operatorname {arsinh}\left (\frac {e x}{\sqrt {d e}}\right )}{e^{\frac {7}{2}}}\right )} a + b \int \frac {x^{6} \arctan \left (\sqrt {c x + 1} \sqrt {c x - 1}\right )}{{\left (e^{2} x^{4} + 2 \, d e x^{2} + d^{2}\right )} \sqrt {e x^{2} + d}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(a+b*arcsec(c*x))/(e*x^2+d)^(5/2),x, algorithm="maxima")

[Out]

1/6*(3*x^5/((e*x^2 + d)^(3/2)*e) + 5*d*x*(3*x^2/((e*x^2 + d)^(3/2)*e) + 2*d/((e*x^2 + d)^(3/2)*e^2))/e + 5*d*x
/(sqrt(e*x^2 + d)*e^3) - 15*d*arcsinh(e*x/sqrt(d*e))/e^(7/2))*a + b*integrate(x^6*arctan(sqrt(c*x + 1)*sqrt(c*
x - 1))/((e^2*x^4 + 2*d*e*x^2 + d^2)*sqrt(e*x^2 + d)), x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {x^6\,\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^6*(a + b*acos(1/(c*x))))/(d + e*x^2)^(5/2),x)

[Out]

int((x^6*(a + b*acos(1/(c*x))))/(d + e*x^2)^(5/2), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6*(a+b*asec(c*x))/(e*x**2+d)**(5/2),x)

[Out]

Timed out

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