\(\int (c e+d e x)^m (a+b \text {arcsinh}(c+d x))^3 \, dx\) [136]

Optimal result

Integrand size = 23, antiderivative size = 23 \[ \int (c e+d e x)^m (a+b \text {arcsinh}(c+d x))^3 \, dx=\frac {(e (c+d x))^{1+m} (a+b \text {arcsinh}(c+d x))^3}{d e (1+m)}-\frac {3 b \text {Int}\left (\frac {(e (c+d x))^{1+m} (a+b \text {arcsinh}(c+d x))^2}{\sqrt {1+(c+d x)^2}},x\right )}{e (1+m)} \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.13 (sec) , antiderivative size = 23, 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 = {} \[ \int (c e+d e x)^m (a+b \text {arcsinh}(c+d x))^3 \, dx=\int (c e+d e x)^m (a+b \text {arcsinh}(c+d x))^3 \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 1.50 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int (c e+d e x)^m (a+b \text {arcsinh}(c+d x))^3 \, dx=\int (c e+d e x)^m (a+b \text {arcsinh}(c+d x))^3 \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 1.06 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00

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

[In]

int((d*e*x+c*e)^m*(a+b*arcsinh(d*x+c))^3,x)

[Out]

int((d*e*x+c*e)^m*(a+b*arcsinh(d*x+c))^3,x)

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.39 \[ \int (c e+d e x)^m (a+b \text {arcsinh}(c+d x))^3 \, dx=\int { {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{3} {\left (d e x + c e\right )}^{m} \,d x } \]

[In]

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

Sympy [N/A]

Not integrable

Time = 17.30 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int (c e+d e x)^m (a+b \text {arcsinh}(c+d x))^3 \, dx=\int \left (e \left (c + d x\right )\right )^{m} \left (a + b \operatorname {asinh}{\left (c + d x \right )}\right )^{3}\, dx \]

[In]

integrate((d*e*x+c*e)**m*(a+b*asinh(d*x+c))**3,x)

[Out]

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

Maxima [N/A]

Not integrable

Time = 3.98 (sec) , antiderivative size = 716, normalized size of antiderivative = 31.13 \[ \int (c e+d e x)^m (a+b \text {arcsinh}(c+d x))^3 \, dx=\int { {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{3} {\left (d e x + c e\right )}^{m} \,d x } \]

[In]

integrate((d*e*x+c*e)^m*(a+b*arcsinh(d*x+c))^3,x, algorithm="maxima")

[Out]

(b^3*d*e^m*x + b^3*c*e^m)*(d*x + c)^m*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))^3/(d*(m + 1)) + (d*e*x
+ c*e)^(m + 1)*a^3/(d*e*(m + 1)) + integrate(-3*(((b^3*c^2*e^m - (c^2*e^m*(m + 1) + e^m*(m + 1))*a*b^2 - (a*b^
2*d^2*e^m*(m + 1) - b^3*d^2*e^m)*x^2 - 2*(a*b^2*c*d*e^m*(m + 1) - b^3*c*d*e^m)*x)*sqrt(d^2*x^2 + 2*c*d*x + c^2
+ 1)*(d*x + c)^m - ((c^3*e^m*(m + 1) + c*e^m*(m + 1))*a*b^2 - (c^3*e^m + c*e^m)*b^3 + (a*b^2*d^3*e^m*(m + 1)
- b^3*d^3*e^m)*x^3 + 3*(a*b^2*c*d^2*e^m*(m + 1) - b^3*c*d^2*e^m)*x^2 + ((3*c^2*d*e^m*(m + 1) + d*e^m*(m + 1))*
a*b^2 - (3*c^2*d*e^m + d*e^m)*b^3)*x)*(d*x + c)^m)*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))^2 - ((a^2*
b*d^2*e^m*(m + 1)*x^2 + 2*a^2*b*c*d*e^m*(m + 1)*x + (c^2*e^m*(m + 1) + e^m*(m + 1))*a^2*b)*sqrt(d^2*x^2 + 2*c*
d*x + c^2 + 1)*(d*x + c)^m + (a^2*b*d^3*e^m*(m + 1)*x^3 + 3*a^2*b*c*d^2*e^m*(m + 1)*x^2 + (3*c^2*d*e^m*(m + 1)
+ d*e^m*(m + 1))*a^2*b*x + (c^3*e^m*(m + 1) + c*e^m*(m + 1))*a^2*b)*(d*x + c)^m)*log(d*x + c + sqrt(d^2*x^2 +
2*c*d*x + c^2 + 1)))/(d^3*(m + 1)*x^3 + 3*c*d^2*(m + 1)*x^2 + c^3*(m + 1) + c*(m + 1) + (3*c^2*d*(m + 1) + d*
(m + 1))*x + (d^2*(m + 1)*x^2 + 2*c*d*(m + 1)*x + c^2*(m + 1) + m + 1)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)), x)

Giac [N/A]

Not integrable

Time = 0.82 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int (c e+d e x)^m (a+b \text {arcsinh}(c+d x))^3 \, dx=\int { {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{3} {\left (d e x + c e\right )}^{m} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.77 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int (c e+d e x)^m (a+b \text {arcsinh}(c+d x))^3 \, dx=\int {\left (c\,e+d\,e\,x\right )}^m\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^3 \,d x \]

[In]

int((c*e + d*e*x)^m*(a + b*asinh(c + d*x))^3,x)

[Out]

int((c*e + d*e*x)^m*(a + b*asinh(c + d*x))^3, x)