\(\int \frac {(a+b \text {arcsinh}(c+d x))^3}{\sqrt {c e+d e x}} \, dx\) [248]

Optimal result

Integrand size = 25, antiderivative size = 25 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^3}{\sqrt {c e+d e x}} \, dx=\frac {2 \sqrt {e (c+d x)} (a+b \text {arcsinh}(c+d x))^3}{d e}-\frac {6 b \text {Int}\left (\frac {\sqrt {e (c+d x)} (a+b \text {arcsinh}(c+d x))^2}{\sqrt {1+(c+d x)^2}},x\right )}{e} \]

[Out]

2*(a+b*arcsinh(d*x+c))^3*(e*(d*x+c))^(1/2)/d/e-6*b*Unintegrable((a+b*arcsinh(d*x+c))^2*(e*(d*x+c))^(1/2)/(1+(d
*x+c)^2)^(1/2),x)/e

Rubi [N/A]

Not integrable

Time = 0.12 (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 = {} \[ \int \frac {(a+b \text {arcsinh}(c+d x))^3}{\sqrt {c e+d e x}} \, dx=\int \frac {(a+b \text {arcsinh}(c+d x))^3}{\sqrt {c e+d e x}} \, dx \]

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

Mathematica [N/A]

Not integrable

Time = 61.73 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^3}{\sqrt {c e+d e x}} \, dx=\int \frac {(a+b \text {arcsinh}(c+d x))^3}{\sqrt {c e+d e x}} \, dx \]

[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]

Maple [N/A] (verified)

Not integrable

Time = 0.09 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92

\[\int \frac {\left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{3}}{\sqrt {d e x +c e}}d x\]

[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)

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.20 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^3}{\sqrt {c e+d e x}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{3}}{\sqrt {d e x + c e}} \,d x } \]

[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)

Sympy [N/A]

Not integrable

Time = 3.40 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^3}{\sqrt {c e+d e x}} \, dx=\int \frac {\left (a + b \operatorname {asinh}{\left (c + d x \right )}\right )^{3}}{\sqrt {e \left (c + d x\right )}}\, dx \]

[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)

Maxima [F(-2)]

Exception generated. \[ \int \frac {(a+b \text {arcsinh}(c+d x))^3}{\sqrt {c e+d e x}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((a+b*arcsinh(d*x+c))^3/(d*e*x+c*e)^(1/2),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 = 0.60 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^3}{\sqrt {c e+d e x}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{3}}{\sqrt {d e x + c e}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.80 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^3}{\sqrt {c e+d e x}} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^3}{\sqrt {c\,e+d\,e\,x}} \,d x \]

[In]

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

[Out]

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