3.7.36 \(\int \frac {(a+b \sinh ^{-1}(c x))^{3/2}}{d+e x^2} \, dx\) [636]

Optimal. Leaf size=25 \[ \text {Int}\left (\frac {\left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{d+e x^2},x\right ) \]

[Out]

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

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 1.91, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{d+e x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \arcsinh \left (c x \right )\right )^{\frac {3}{2}}}{e \,x^{2}+d}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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