3.251 \(\int \frac{(a+b \sinh ^{-1}(c+d x))^3}{(c e+d e x)^{7/2}} \, dx\)

Optimal. Leaf size=81 \[ \frac{6 b \text{Unintegrable}\left (\frac{\left (a+b \sinh ^{-1}(c+d x)\right )^2}{\sqrt{(c+d x)^2+1} (e (c+d x))^{5/2}},x\right )}{5 e}-\frac{2 \left (a+b \sinh ^{-1}(c+d x)\right )^3}{5 d e (e (c+d x))^{5/2}} \]

[Out]

(-2*(a + b*ArcSinh[c + d*x])^3)/(5*d*e*(e*(c + d*x))^(5/2)) + (6*b*Unintegrable[(a + b*ArcSinh[c + d*x])^2/((e
*(c + d*x))^(5/2)*Sqrt[1 + (c + d*x)^2]), x])/(5*e)

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Rubi [A]  time = 0.216786, 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 = {} \[ \int \frac{\left (a+b \sinh ^{-1}(c+d x)\right )^3}{(c e+d e x)^{7/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*ArcSinh[c + d*x])^3/(c*e + d*e*x)^(7/2),x]

[Out]

(-2*(a + b*ArcSinh[c + d*x])^3)/(5*d*e*(e*(c + d*x))^(5/2)) + (6*b*Defer[Subst][Defer[Int][(a + b*ArcSinh[x])^
2/((e*x)^(5/2)*Sqrt[1 + x^2]), x], x, c + d*x])/(5*d*e)

Rubi steps

\begin{align*} \int \frac{\left (a+b \sinh ^{-1}(c+d x)\right )^3}{(c e+d e x)^{7/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b \sinh ^{-1}(x)\right )^3}{(e x)^{7/2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac{2 \left (a+b \sinh ^{-1}(c+d x)\right )^3}{5 d e (e (c+d x))^{5/2}}+\frac{(6 b) \operatorname{Subst}\left (\int \frac{\left (a+b \sinh ^{-1}(x)\right )^2}{(e x)^{5/2} \sqrt{1+x^2}} \, dx,x,c+d x\right )}{5 d e}\\ \end{align*}

Mathematica [A]  time = 69.217, size = 0, normalized size = 0. \[ \int \frac{\left (a+b \sinh ^{-1}(c+d x)\right )^3}{(c e+d e x)^{7/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(a + b*ArcSinh[c + d*x])^3/(c*e + d*e*x)^(7/2),x]

[Out]

Integrate[(a + b*ArcSinh[c + d*x])^3/(c*e + d*e*x)^(7/2), x]

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Maple [A]  time = 0.25, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+b{\it Arcsinh} \left ( dx+c \right ) \right ) ^{3} \left ( dex+ce \right ) ^{-{\frac{7}{2}}}}\, 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)^(7/2),x)

[Out]

int((a+b*arcsinh(d*x+c))^3/(d*e*x+c*e)^(7/2),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)^(7/2),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{{\left (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}\right )} \sqrt{d e x + c e}}{d^{4} e^{4} x^{4} + 4 \, c d^{3} e^{4} x^{3} + 6 \, c^{2} d^{2} e^{4} x^{2} + 4 \, c^{3} d e^{4} x + c^{4} e^{4}}, 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)^(7/2),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)/(d^4*e^4*x^4 + 4*c*d^3*e^4*x^3 + 6*c^2*d^2*e^4*x^2 + 4*c^3*d*e^4*x + c^4*e^4), x)

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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((a+b*asinh(d*x+c))**3/(d*e*x+c*e)**(7/2),x)

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arsinh}\left (d x + c\right ) + a\right )}^{3}}{{\left (d e x + c e\right )}^{\frac{7}{2}}}\,{d x} \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)^(7/2),x, algorithm="giac")

[Out]

integrate((b*arcsinh(d*x + c) + a)^3/(d*e*x + c*e)^(7/2), x)