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

Optimal. Leaf size=20 \[ \text{Unintegrable}\left (\frac{(d+e x)^m}{\left (a+b \sinh ^{-1}(c x)\right )^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0308005, 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 = {} \[ \int \frac{(d+e x)^m}{\left (a+b \sinh ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.738448, size = 0, normalized size = 0. \[ \int \frac{(d+e x)^m}{\left (a+b \sinh ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.93, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex+d \right ) ^{m}}{ \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{{\left (c^{2} x^{2} + 1\right )}^{\frac{3}{2}}{\left (e x + d\right )}^{m} +{\left (c^{3} x^{3} + c x\right )}{\left (e x + d\right )}^{m}}{a b c^{3} x^{2} + \sqrt{c^{2} x^{2} + 1} a b c^{2} x + a b c +{\left (b^{2} c^{3} x^{2} + \sqrt{c^{2} x^{2} + 1} b^{2} c^{2} x + b^{2} c\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )} + \int \frac{{\left (c^{3} e{\left (m + 1\right )} x^{3} + c^{3} d x^{2} + c e{\left (m - 1\right )} x - c d\right )}{\left (c^{2} x^{2} + 1\right )}{\left (e x + d\right )}^{m} +{\left (2 \, c^{4} e{\left (m + 1\right )} x^{4} + 2 \, c^{4} d x^{3} + c^{2} e{\left (3 \, m + 1\right )} x^{2} + c^{2} d x + e m\right )} \sqrt{c^{2} x^{2} + 1}{\left (e x + d\right )}^{m} +{\left (c^{5} e{\left (m + 1\right )} x^{5} + c^{5} d x^{4} + 2 \, c^{3} e{\left (m + 1\right )} x^{3} + 2 \, c^{3} d x^{2} + c e{\left (m + 1\right )} x + c d\right )}{\left (e x + d\right )}^{m}}{a b c^{5} e x^{5} + a b c^{5} d x^{4} + 2 \, a b c^{3} e x^{3} + 2 \, a b c^{3} d x^{2} + a b c e x + a b c d +{\left (a b c^{3} e x^{3} + a b c^{3} d x^{2}\right )}{\left (c^{2} x^{2} + 1\right )} +{\left (b^{2} c^{5} e x^{5} + b^{2} c^{5} d x^{4} + 2 \, b^{2} c^{3} e x^{3} + 2 \, b^{2} c^{3} d x^{2} + b^{2} c e x + b^{2} c d +{\left (b^{2} c^{3} e x^{3} + b^{2} c^{3} d x^{2}\right )}{\left (c^{2} x^{2} + 1\right )} + 2 \,{\left (b^{2} c^{4} e x^{4} + b^{2} c^{4} d x^{3} + b^{2} c^{2} e x^{2} + b^{2} c^{2} d x\right )} \sqrt{c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) + 2 \,{\left (a b c^{4} e x^{4} + a b c^{4} d x^{3} + a b c^{2} e x^{2} + a b c^{2} d x\right )} \sqrt{c^{2} x^{2} + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((c^2*x^2 + 1)^(3/2)*(e*x + d)^m + (c^3*x^3 + c*x)*(e*x + d)^m)/(a*b*c^3*x^2 + sqrt(c^2*x^2 + 1)*a*b*c^2*x +
a*b*c + (b^2*c^3*x^2 + sqrt(c^2*x^2 + 1)*b^2*c^2*x + b^2*c)*log(c*x + sqrt(c^2*x^2 + 1))) + integrate(((c^3*e*
(m + 1)*x^3 + c^3*d*x^2 + c*e*(m - 1)*x - c*d)*(c^2*x^2 + 1)*(e*x + d)^m + (2*c^4*e*(m + 1)*x^4 + 2*c^4*d*x^3
+ c^2*e*(3*m + 1)*x^2 + c^2*d*x + e*m)*sqrt(c^2*x^2 + 1)*(e*x + d)^m + (c^5*e*(m + 1)*x^5 + c^5*d*x^4 + 2*c^3*
e*(m + 1)*x^3 + 2*c^3*d*x^2 + c*e*(m + 1)*x + c*d)*(e*x + d)^m)/(a*b*c^5*e*x^5 + a*b*c^5*d*x^4 + 2*a*b*c^3*e*x
^3 + 2*a*b*c^3*d*x^2 + a*b*c*e*x + a*b*c*d + (a*b*c^3*e*x^3 + a*b*c^3*d*x^2)*(c^2*x^2 + 1) + (b^2*c^5*e*x^5 +
b^2*c^5*d*x^4 + 2*b^2*c^3*e*x^3 + 2*b^2*c^3*d*x^2 + b^2*c*e*x + b^2*c*d + (b^2*c^3*e*x^3 + b^2*c^3*d*x^2)*(c^2
*x^2 + 1) + 2*(b^2*c^4*e*x^4 + b^2*c^4*d*x^3 + b^2*c^2*e*x^2 + b^2*c^2*d*x)*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(
c^2*x^2 + 1)) + 2*(a*b*c^4*e*x^4 + a*b*c^4*d*x^3 + a*b*c^2*e*x^2 + a*b*c^2*d*x)*sqrt(c^2*x^2 + 1)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((e*x + d)^m/(b^2*arcsinh(c*x)^2 + 2*a*b*arcsinh(c*x) + a^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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