3.248 \(\int \frac{(a+b \sinh ^{-1}(c+d x))^3}{\sqrt{c e+d e x}} \, dx\)

Optimal. Leaf size=77 \[ \frac{2 \sqrt{e (c+d x)} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{d e}-\frac{6 b \text{Unintegrable}\left (\frac{\sqrt{e (c+d x)} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{\sqrt{(c+d x)^2+1}},x\right )}{e} \]

[Out]

(2*Sqrt[e*(c + d*x)]*(a + b*ArcSinh[c + d*x])^3)/(d*e) - (6*b*Unintegrable[(Sqrt[e*(c + d*x)]*(a + b*ArcSinh[c
+ d*x])^2)/Sqrt[1 + (c + d*x)^2], x])/e

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Rubi [A]  time = 0.183353, 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 = {} \[ \int \frac{\left (a+b \sinh ^{-1}(c+d x)\right )^3}{\sqrt{c e+d e x}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*ArcSinh[c + d*x])^3/Sqrt[c*e + d*e*x],x]

[Out]

(2*Sqrt[e*(c + d*x)]*(a + b*ArcSinh[c + d*x])^3)/(d*e) - (6*b*Defer[Subst][Defer[Int][(Sqrt[e*x]*(a + b*ArcSin
h[x])^2)/Sqrt[1 + x^2], x], x, c + d*x])/(d*e)

Rubi steps

\begin{align*} \int \frac{\left (a+b \sinh ^{-1}(c+d x)\right )^3}{\sqrt{c e+d e x}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b \sinh ^{-1}(x)\right )^3}{\sqrt{e x}} \, dx,x,c+d x\right )}{d}\\ &=\frac{2 \sqrt{e (c+d x)} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{d e}-\frac{(6 b) \operatorname{Subst}\left (\int \frac{\sqrt{e x} \left (a+b \sinh ^{-1}(x)\right )^2}{\sqrt{1+x^2}} \, dx,x,c+d x\right )}{d e}\\ \end{align*}

Mathematica [A]  time = 8.70346, size = 0, normalized size = 0. \[ \int \frac{\left (a+b \sinh ^{-1}(c+d x)\right )^3}{\sqrt{c e+d e x}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(a + b*ArcSinh[c + d*x])^3/Sqrt[c*e + d*e*x],x]

[Out]

Integrate[(a + b*ArcSinh[c + d*x])^3/Sqrt[c*e + d*e*x], x]

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Maple [A]  time = 0.19, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+b{\it Arcsinh} \left ( dx+c \right ) \right ) ^{3}{\frac{1}{\sqrt{dex+ce}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsinh(d*x+c))^3/(d*e*x+c*e)^(1/2),x)

[Out]

int((a+b*arcsinh(d*x+c))^3/(d*e*x+c*e)^(1/2),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((a+b*arcsinh(d*x+c))^3/(d*e*x+c*e)^(1/2),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 (\frac{b^{3} \operatorname{arsinh}\left (d x + c\right )^{3} + 3 \, a b^{2} \operatorname{arsinh}\left (d x + c\right )^{2} + 3 \, a^{2} b \operatorname{arsinh}\left (d x + c\right ) + a^{3}}{\sqrt{d e x + c e}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsinh(d*x+c))^3/(d*e*x+c*e)^(1/2),x, algorithm="fricas")

[Out]

integral((b^3*arcsinh(d*x + c)^3 + 3*a*b^2*arcsinh(d*x + c)^2 + 3*a^2*b*arcsinh(d*x + c) + a^3)/sqrt(d*e*x + c
*e), x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \operatorname{asinh}{\left (c + d x \right )}\right )^{3}}{\sqrt{e \left (c + d x\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*asinh(d*x+c))**3/(d*e*x+c*e)**(1/2),x)

[Out]

Integral((a + b*asinh(c + d*x))**3/sqrt(e*(c + d*x)), x)

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Giac [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*arcsinh(d*x+c))^3/(d*e*x+c*e)^(1/2),x, algorithm="giac")

[Out]

Timed out