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

Optimal. Leaf size=28 \[ \frac{\text{Unintegrable}\left (\frac{\left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{c+d x},x\right )}{e} \]

[Out]

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

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Rubi [A]  time = 0.105253, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{c e+d e x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 1.07201, size = 0, normalized size = 0. \[ \int \frac{\left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{c e+d e x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.154, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{dex+ce} \left ( a+b{\it Arcsinh} \left ( dx+c \right ) \right ) ^{{\frac{5}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arsinh}\left (d x + c\right ) + a\right )}^{\frac{5}{2}}}{d e x + c e}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arsinh}\left (d x + c\right ) + a\right )}^{\frac{5}{2}}}{d e x + c e}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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