3.227 \(\int \frac{1}{(c e+d e x) (a+b \sinh ^{-1}(c+d x))^{7/2}} \, dx\)

Optimal. Leaf size=28 \[ \frac{\text{Unintegrable}\left (\frac{1}{(c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}},x\right )}{e} \]

[Out]

Unintegrable[1/((c + d*x)*(a + b*ArcSinh[c + d*x])^(7/2)), x]/e

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Rubi [A]  time = 0.117297, 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 = {} \[ \int \frac{1}{(c e+d e x) \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/((c*e + d*e*x)*(a + b*ArcSinh[c + d*x])^(7/2)),x]

[Out]

Defer[Subst][Defer[Int][1/(x*(a + b*ArcSinh[x])^(7/2)), x], x, c + d*x]/(d*e)

Rubi steps

\begin{align*} \int \frac{1}{(c e+d e x) \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{e x \left (a+b \sinh ^{-1}(x)\right )^{7/2}} \, dx,x,c+d x\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x \left (a+b \sinh ^{-1}(x)\right )^{7/2}} \, dx,x,c+d x\right )}{d e}\\ \end{align*}

Mathematica [A]  time = 0.0811025, size = 0, normalized size = 0. \[ \int \frac{1}{(c e+d e x) \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/((c*e + d*e*x)*(a + b*ArcSinh[c + d*x])^(7/2)),x]

[Out]

Integrate[1/((c*e + d*e*x)*(a + b*ArcSinh[c + d*x])^(7/2)), x]

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Maple [A]  time = 0.167, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{dex+ce} \left ( a+b{\it Arcsinh} \left ( dx+c \right ) \right ) ^{-{\frac{7}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(d*e*x+c*e)/(a+b*arcsinh(d*x+c))^(7/2),x)

[Out]

int(1/(d*e*x+c*e)/(a+b*arcsinh(d*x+c))^(7/2),x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (d e x + c e\right )}{\left (b \operatorname{arsinh}\left (d x + c\right ) + a\right )}^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*e*x+c*e)/(a+b*arcsinh(d*x+c))^(7/2),x, algorithm="maxima")

[Out]

integrate(1/((d*e*x + c*e)*(b*arcsinh(d*x + c) + a)^(7/2)), 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(1/(d*e*x+c*e)/(a+b*arcsinh(d*x+c))^(7/2),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(1/(d*e*x+c*e)/(a+b*asinh(d*x+c))**(7/2),x)

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (d e x + c e\right )}{\left (b \operatorname{arsinh}\left (d x + c\right ) + a\right )}^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*e*x+c*e)/(a+b*arcsinh(d*x+c))^(7/2),x, algorithm="giac")

[Out]

integrate(1/((d*e*x + c*e)*(b*arcsinh(d*x + c) + a)^(7/2)), x)