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

Optimal. Leaf size=29 \[ \frac {\text {Int}\left (\frac {1}{(c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}},x\right )}{e} \]

[Out]

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

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Rubi [A]  time = 0.11, 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 = {} \[ \int \frac {1}{(c e+d e x) \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.08, size = 0, normalized size = 0.00 \[ \int \frac {1}{(c e+d e x) \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.22, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (d e x +c e \right ) \left (a +b \arcsinh \left (d x +c \right )\right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{\left (c\,e+d\,e\,x\right )\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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