\(\int (c e+d e x)^{7/2} (a+b \text {arcsinh}(c+d x))^4 \, dx\) [252]

Optimal result

Integrand size = 25, antiderivative size = 25 \[ \int (c e+d e x)^{7/2} (a+b \text {arcsinh}(c+d x))^4 \, dx=\frac {2 (e (c+d x))^{9/2} (a+b \text {arcsinh}(c+d x))^4}{9 d e}-\frac {8 b \text {Int}\left (\frac {(e (c+d x))^{9/2} (a+b \text {arcsinh}(c+d x))^3}{\sqrt {1+(c+d x)^2}},x\right )}{9 e} \]

[Out]

2/9*(e*(d*x+c))^(9/2)*(a+b*arcsinh(d*x+c))^4/d/e-8/9*b*Unintegrable((e*(d*x+c))^(9/2)*(a+b*arcsinh(d*x+c))^3/(
1+(d*x+c)^2)^(1/2),x)/e

Rubi [N/A]

Not integrable

Time = 0.14 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int (c e+d e x)^{7/2} (a+b \text {arcsinh}(c+d x))^4 \, dx=\int (c e+d e x)^{7/2} (a+b \text {arcsinh}(c+d x))^4 \, dx \]

[In]

Int[(c*e + d*e*x)^(7/2)*(a + b*ArcSinh[c + d*x])^4,x]

[Out]

(2*(e*(c + d*x))^(9/2)*(a + b*ArcSinh[c + d*x])^4)/(9*d*e) - (8*b*Defer[Subst][Defer[Int][((e*x)^(9/2)*(a + b*
ArcSinh[x])^3)/Sqrt[1 + x^2], x], x, c + d*x])/(9*d*e)

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int (e x)^{7/2} (a+b \text {arcsinh}(x))^4 \, dx,x,c+d x\right )}{d} \\ & = \frac {2 (e (c+d x))^{9/2} (a+b \text {arcsinh}(c+d x))^4}{9 d e}-\frac {(8 b) \text {Subst}\left (\int \frac {(e x)^{9/2} (a+b \text {arcsinh}(x))^3}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{9 d e} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 37.75 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int (c e+d e x)^{7/2} (a+b \text {arcsinh}(c+d x))^4 \, dx=\int (c e+d e x)^{7/2} (a+b \text {arcsinh}(c+d x))^4 \, dx \]

[In]

Integrate[(c*e + d*e*x)^(7/2)*(a + b*ArcSinh[c + d*x])^4,x]

[Out]

Integrate[(c*e + d*e*x)^(7/2)*(a + b*ArcSinh[c + d*x])^4, x]

Maple [N/A] (verified)

Not integrable

Time = 0.90 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92

\[\int \left (d e x +c e \right )^{\frac {7}{2}} \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{4}d x\]

[In]

int((d*e*x+c*e)^(7/2)*(a+b*arcsinh(d*x+c))^4,x)

[Out]

int((d*e*x+c*e)^(7/2)*(a+b*arcsinh(d*x+c))^4,x)

Fricas [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 330, normalized size of antiderivative = 13.20 \[ \int (c e+d e x)^{7/2} (a+b \text {arcsinh}(c+d x))^4 \, dx=\int { {\left (d e x + c e\right )}^{\frac {7}{2}} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{4} \,d x } \]

[In]

integrate((d*e*x+c*e)^(7/2)*(a+b*arcsinh(d*x+c))^4,x, algorithm="fricas")

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int (c e+d e x)^{7/2} (a+b \text {arcsinh}(c+d x))^4 \, dx=\text {Timed out} \]

[In]

integrate((d*e*x+c*e)**(7/2)*(a+b*asinh(d*x+c))**4,x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int (c e+d e x)^{7/2} (a+b \text {arcsinh}(c+d x))^4 \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((d*e*x+c*e)^(7/2)*(a+b*arcsinh(d*x+c))^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [N/A]

Not integrable

Time = 172.17 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int (c e+d e x)^{7/2} (a+b \text {arcsinh}(c+d x))^4 \, dx=\int { {\left (d e x + c e\right )}^{\frac {7}{2}} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{4} \,d x } \]

[In]

integrate((d*e*x+c*e)^(7/2)*(a+b*arcsinh(d*x+c))^4,x, algorithm="giac")

[Out]

integrate((d*e*x + c*e)^(7/2)*(b*arcsinh(d*x + c) + a)^4, x)

Mupad [N/A]

Not integrable

Time = 2.76 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int (c e+d e x)^{7/2} (a+b \text {arcsinh}(c+d x))^4 \, dx=\int {\left (c\,e+d\,e\,x\right )}^{7/2}\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^4 \,d x \]

[In]

int((c*e + d*e*x)^(7/2)*(a + b*asinh(c + d*x))^4,x)

[Out]

int((c*e + d*e*x)^(7/2)*(a + b*asinh(c + d*x))^4, x)