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

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

[Out]

Unintegrable(1/(e*x^2+d)^(5/2)/(a+b*arcsinh(c*x)),x)

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Rubi [A]  time = 0.05, 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 = {} \[ \int \frac {1}{\left (d+e x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/((d + e*x^2)^(5/2)*(a + b*ArcSinh[c*x])),x]

[Out]

Defer[Int][1/((d + e*x^2)^(5/2)*(a + b*ArcSinh[c*x])), x]

Rubi steps

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

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Mathematica [A]  time = 3.72, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (d+e x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/((d + e*x^2)^(5/2)*(a + b*ArcSinh[c*x])),x]

[Out]

Integrate[1/((d + e*x^2)^(5/2)*(a + b*ArcSinh[c*x])), x]

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fricas [A]  time = 0.80, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e x^{2} + d}}{a e^{3} x^{6} + 3 \, a d e^{2} x^{4} + 3 \, a d^{2} e x^{2} + a d^{3} + {\left (b e^{3} x^{6} + 3 \, b d e^{2} x^{4} + 3 \, b d^{2} e x^{2} + b d^{3}\right )} \operatorname {arsinh}\left (c x\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x^2+d)^(5/2)/(a+b*arcsinh(c*x)),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (e x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x^2+d)^(5/2)/(a+b*arcsinh(c*x)),x, algorithm="giac")

[Out]

integrate(1/((e*x^2 + d)^(5/2)*(b*arcsinh(c*x) + a)), x)

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maple [A]  time = 0.25, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (e \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \arcsinh \left (c x \right )\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x^2+d)^(5/2)/(a+b*arcsinh(c*x)),x, algorithm="maxima")

[Out]

integrate(1/((e*x^2 + d)^(5/2)*(b*arcsinh(c*x) + a)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a + b \operatorname {asinh}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x**2+d)**(5/2)/(a+b*asinh(c*x)),x)

[Out]

Integral(1/((a + b*asinh(c*x))*(d + e*x**2)**(5/2)), x)

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