\(\int \frac {x^m (a+b \text {arcsinh}(c x))}{(d+c^2 d x^2)^3} \, dx\) [190]

Optimal result

Integrand size = 24, antiderivative size = 24 \[ \int \frac {x^m (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^3} \, dx=\frac {x^{1+m} (a+b \text {arcsinh}(c x))}{4 d^3 \left (1+c^2 x^2\right )^2}+\frac {(3-m) x^{1+m} (a+b \text {arcsinh}(c x))}{8 d^3 \left (1+c^2 x^2\right )}-\frac {b c (3-m) x^{2+m} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {2+m}{2},\frac {4+m}{2},-c^2 x^2\right )}{8 d^3 (2+m)}-\frac {b c x^{2+m} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},\frac {2+m}{2},\frac {4+m}{2},-c^2 x^2\right )}{4 d^3 (2+m)}+\frac {(1-m) (3-m) \text {Int}\left (\frac {x^m (a+b \text {arcsinh}(c x))}{d+c^2 d x^2},x\right )}{8 d^2} \]

[Out]

1/4*x^(1+m)*(a+b*arcsinh(c*x))/d^3/(c^2*x^2+1)^2+1/8*(3-m)*x^(1+m)*(a+b*arcsinh(c*x))/d^3/(c^2*x^2+1)-1/8*b*c*
(3-m)*x^(2+m)*hypergeom([3/2, 1+1/2*m],[2+1/2*m],-c^2*x^2)/d^3/(2+m)-1/4*b*c*x^(2+m)*hypergeom([5/2, 1+1/2*m],
[2+1/2*m],-c^2*x^2)/d^3/(2+m)+1/8*(1-m)*(3-m)*Unintegrable(x^m*(a+b*arcsinh(c*x))/(c^2*d*x^2+d),x)/d^2

Rubi [N/A]

Not integrable

Time = 0.19 (sec) , antiderivative size = 24, 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 = {} \[ \int \frac {x^m (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^3} \, dx=\int \frac {x^m (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^3} \, dx \]

[In]

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

[Out]

(x^(1 + m)*(a + b*ArcSinh[c*x]))/(4*d^3*(1 + c^2*x^2)^2) + ((3 - m)*x^(1 + m)*(a + b*ArcSinh[c*x]))/(8*d^3*(1
+ c^2*x^2)) - (b*c*(3 - m)*x^(2 + m)*Hypergeometric2F1[3/2, (2 + m)/2, (4 + m)/2, -(c^2*x^2)])/(8*d^3*(2 + m))
- (b*c*x^(2 + m)*Hypergeometric2F1[5/2, (2 + m)/2, (4 + m)/2, -(c^2*x^2)])/(4*d^3*(2 + m)) + ((1 - m)*(3 - m)
*Defer[Int][(x^m*(a + b*ArcSinh[c*x]))/(d + c^2*d*x^2), x])/(8*d^2)

Rubi steps \begin{align*} \text {integral}& = \frac {x^{1+m} (a+b \text {arcsinh}(c x))}{4 d^3 \left (1+c^2 x^2\right )^2}-\frac {(b c) \int \frac {x^{1+m}}{\left (1+c^2 x^2\right )^{5/2}} \, dx}{4 d^3}+\frac {(3-m) \int \frac {x^m (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^2} \, dx}{4 d} \\ & = \frac {x^{1+m} (a+b \text {arcsinh}(c x))}{4 d^3 \left (1+c^2 x^2\right )^2}+\frac {(3-m) x^{1+m} (a+b \text {arcsinh}(c x))}{8 d^3 \left (1+c^2 x^2\right )}-\frac {b c x^{2+m} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},\frac {2+m}{2},\frac {4+m}{2},-c^2 x^2\right )}{4 d^3 (2+m)}-\frac {(b c (3-m)) \int \frac {x^{1+m}}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{8 d^3}+\frac {((1-m) (3-m)) \int \frac {x^m (a+b \text {arcsinh}(c x))}{d+c^2 d x^2} \, dx}{8 d^2} \\ & = \frac {x^{1+m} (a+b \text {arcsinh}(c x))}{4 d^3 \left (1+c^2 x^2\right )^2}+\frac {(3-m) x^{1+m} (a+b \text {arcsinh}(c x))}{8 d^3 \left (1+c^2 x^2\right )}-\frac {b c (3-m) x^{2+m} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {2+m}{2},\frac {4+m}{2},-c^2 x^2\right )}{8 d^3 (2+m)}-\frac {b c x^{2+m} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},\frac {2+m}{2},\frac {4+m}{2},-c^2 x^2\right )}{4 d^3 (2+m)}+\frac {((1-m) (3-m)) \int \frac {x^m (a+b \text {arcsinh}(c x))}{d+c^2 d x^2} \, dx}{8 d^2} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 5.24 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {x^m (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^3} \, dx=\int \frac {x^m (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^3} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.29 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00

\[\int \frac {x^{m} \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}{\left (c^{2} d \,x^{2}+d \right )^{3}}d x\]

[In]

int(x^m*(a+b*arcsinh(c*x))/(c^2*d*x^2+d)^3,x)

[Out]

int(x^m*(a+b*arcsinh(c*x))/(c^2*d*x^2+d)^3,x)

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.17 \[ \int \frac {x^m (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{m}}{{\left (c^{2} d x^{2} + d\right )}^{3}} \,d x } \]

[In]

integrate(x^m*(a+b*arcsinh(c*x))/(c^2*d*x^2+d)^3,x, algorithm="fricas")

[Out]

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

Sympy [N/A]

Not integrable

Time = 117.92 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.96 \[ \int \frac {x^m (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^3} \, dx=\frac {\int \frac {a x^{m}}{c^{6} x^{6} + 3 c^{4} x^{4} + 3 c^{2} x^{2} + 1}\, dx + \int \frac {b x^{m} \operatorname {asinh}{\left (c x \right )}}{c^{6} x^{6} + 3 c^{4} x^{4} + 3 c^{2} x^{2} + 1}\, dx}{d^{3}} \]

[In]

integrate(x**m*(a+b*asinh(c*x))/(c**2*d*x**2+d)**3,x)

[Out]

(Integral(a*x**m/(c**6*x**6 + 3*c**4*x**4 + 3*c**2*x**2 + 1), x) + Integral(b*x**m*asinh(c*x)/(c**6*x**6 + 3*c
**4*x**4 + 3*c**2*x**2 + 1), x))/d**3

Maxima [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {x^m (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{m}}{{\left (c^{2} d x^{2} + d\right )}^{3}} \,d x } \]

[In]

integrate(x^m*(a+b*arcsinh(c*x))/(c^2*d*x^2+d)^3,x, algorithm="maxima")

[Out]

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

Giac [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {x^m (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{m}}{{\left (c^{2} d x^{2} + d\right )}^{3}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.71 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {x^m (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^3} \, dx=\int \frac {x^m\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{{\left (d\,c^2\,x^2+d\right )}^3} \,d x \]

[In]

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

[Out]

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