3.253 \(\int (c e+d e x)^{5/2} (a+b \sinh ^{-1}(c+d x))^4 \, dx\)

Optimal. Leaf size=81 \[ \frac{2 (e (c+d x))^{7/2} \left (a+b \sinh ^{-1}(c+d x)\right )^4}{7 d e}-\frac{8 b \text{Unintegrable}\left (\frac{(e (c+d x))^{7/2} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{\sqrt{(c+d x)^2+1}},x\right )}{7 e} \]

[Out]

(2*(e*(c + d*x))^(7/2)*(a + b*ArcSinh[c + d*x])^4)/(7*d*e) - (8*b*Unintegrable[((e*(c + d*x))^(7/2)*(a + b*Arc
Sinh[c + d*x])^3)/Sqrt[1 + (c + d*x)^2], x])/(7*e)

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Rubi [A]  time = 0.206969, 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 = {} \[ \int (c e+d e x)^{5/2} \left (a+b \sinh ^{-1}(c+d x)\right )^4 \, dx \]

Verification is Not applicable to the result.

[In]

Int[(c*e + d*e*x)^(5/2)*(a + b*ArcSinh[c + d*x])^4,x]

[Out]

(2*(e*(c + d*x))^(7/2)*(a + b*ArcSinh[c + d*x])^4)/(7*d*e) - (8*b*Defer[Subst][Defer[Int][((e*x)^(7/2)*(a + b*
ArcSinh[x])^3)/Sqrt[1 + x^2], x], x, c + d*x])/(7*d*e)

Rubi steps

\begin{align*} \int (c e+d e x)^{5/2} \left (a+b \sinh ^{-1}(c+d x)\right )^4 \, dx &=\frac{\operatorname{Subst}\left (\int (e x)^{5/2} \left (a+b \sinh ^{-1}(x)\right )^4 \, dx,x,c+d x\right )}{d}\\ &=\frac{2 (e (c+d x))^{7/2} \left (a+b \sinh ^{-1}(c+d x)\right )^4}{7 d e}-\frac{(8 b) \operatorname{Subst}\left (\int \frac{(e x)^{7/2} \left (a+b \sinh ^{-1}(x)\right )^3}{\sqrt{1+x^2}} \, dx,x,c+d x\right )}{7 d e}\\ \end{align*}

Mathematica [A]  time = 122.473, size = 0, normalized size = 0. \[ \int (c e+d e x)^{5/2} \left (a+b \sinh ^{-1}(c+d x)\right )^4 \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(c*e + d*e*x)^(5/2)*(a + b*ArcSinh[c + d*x])^4,x]

[Out]

Integrate[(c*e + d*e*x)^(5/2)*(a + b*ArcSinh[c + d*x])^4, x]

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Maple [A]  time = 0.261, size = 0, normalized size = 0. \begin{align*} \int \left ( dex+ce \right ) ^{{\frac{5}{2}}} \left ( a+b{\it Arcsinh} \left ( dx+c \right ) \right ) ^{4}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*e*x+c*e)^(5/2)*(a+b*arcsinh(d*x+c))^4,x)

[Out]

int((d*e*x+c*e)^(5/2)*(a+b*arcsinh(d*x+c))^4,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((d*e*x+c*e)^(5/2)*(a+b*arcsinh(d*x+c))^4,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 (a^{4} d^{2} e^{2} x^{2} + 2 \, a^{4} c d e^{2} x + a^{4} c^{2} e^{2} +{\left (b^{4} d^{2} e^{2} x^{2} + 2 \, b^{4} c d e^{2} x + b^{4} c^{2} e^{2}\right )} \operatorname{arsinh}\left (d x + c\right )^{4} + 4 \,{\left (a b^{3} d^{2} e^{2} x^{2} + 2 \, a b^{3} c d e^{2} x + a b^{3} c^{2} e^{2}\right )} \operatorname{arsinh}\left (d x + c\right )^{3} + 6 \,{\left (a^{2} b^{2} d^{2} e^{2} x^{2} + 2 \, a^{2} b^{2} c d e^{2} x + a^{2} b^{2} c^{2} e^{2}\right )} \operatorname{arsinh}\left (d x + c\right )^{2} + 4 \,{\left (a^{3} b d^{2} e^{2} x^{2} + 2 \, a^{3} b c d e^{2} x + a^{3} b c^{2} e^{2}\right )} \operatorname{arsinh}\left (d x + c\right )\right )} \sqrt{d e x + c e}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^(5/2)*(a+b*arcsinh(d*x+c))^4,x, algorithm="fricas")

[Out]

integral((a^4*d^2*e^2*x^2 + 2*a^4*c*d*e^2*x + a^4*c^2*e^2 + (b^4*d^2*e^2*x^2 + 2*b^4*c*d*e^2*x + b^4*c^2*e^2)*
arcsinh(d*x + c)^4 + 4*(a*b^3*d^2*e^2*x^2 + 2*a*b^3*c*d*e^2*x + a*b^3*c^2*e^2)*arcsinh(d*x + c)^3 + 6*(a^2*b^2
*d^2*e^2*x^2 + 2*a^2*b^2*c*d*e^2*x + a^2*b^2*c^2*e^2)*arcsinh(d*x + c)^2 + 4*(a^3*b*d^2*e^2*x^2 + 2*a^3*b*c*d*
e^2*x + a^3*b*c^2*e^2)*arcsinh(d*x + c))*sqrt(d*e*x + c*e), 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((d*e*x+c*e)**(5/2)*(a+b*asinh(d*x+c))**4,x)

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d e x + c e\right )}^{\frac{5}{2}}{\left (b \operatorname{arsinh}\left (d x + c\right ) + a\right )}^{4}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^(5/2)*(a+b*arcsinh(d*x+c))^4,x, algorithm="giac")

[Out]

integrate((d*e*x + c*e)^(5/2)*(b*arcsinh(d*x + c) + a)^4, x)