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

Optimal. Leaf size=74 \[ \frac{(d+e x)^{m+1} \left (a+b \sinh ^{-1}(c x)\right )^2}{e (m+1)}-\frac{2 b c \text{Unintegrable}\left (\frac{(d+e x)^{m+1} \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{c^2 x^2+1}},x\right )}{e (m+1)} \]

[Out]

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

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Rubi [A]  time = 0.261283, 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 = {} \[ \int (d+e x)^m \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 3.93758, size = 0, normalized size = 0. \[ \int (d+e x)^m \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx \]

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 = 2.824, size = 0, normalized size = 0. \begin{align*} \int \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 [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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]

Exception raised: ValueError

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

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

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