3.189 \(\int \frac {x^m (a+b \sinh ^{-1}(c x))}{(d+c^2 d x^2)^2} \, dx\)

Optimal. Leaf size=116 \[ \frac {(1-m) \text {Int}\left (\frac {x^m \left (a+b \sinh ^{-1}(c x)\right )}{c^2 d x^2+d},x\right )}{2 d}+\frac {x^{m+1} \left (a+b \sinh ^{-1}(c x)\right )}{2 d^2 \left (c^2 x^2+1\right )}-\frac {b c x^{m+2} \, _2F_1\left (\frac {3}{2},\frac {m+2}{2};\frac {m+4}{2};-c^2 x^2\right )}{2 d^2 (m+2)} \]

[Out]

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

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Rubi [A]  time = 0.16, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {x^m \left (a+b \sinh ^{-1}(c x)\right )}{\left (d+c^2 d x^2\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int \frac {x^m \left (a+b \sinh ^{-1}(c x)\right )}{\left (d+c^2 d x^2\right )^2} \, dx &=\frac {x^{1+m} \left (a+b \sinh ^{-1}(c x)\right )}{2 d^2 \left (1+c^2 x^2\right )}-\frac {(b c) \int \frac {x^{1+m}}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{2 d^2}+\frac {(1-m) \int \frac {x^m \left (a+b \sinh ^{-1}(c x)\right )}{d+c^2 d x^2} \, dx}{2 d}\\ &=\frac {x^{1+m} \left (a+b \sinh ^{-1}(c x)\right )}{2 d^2 \left (1+c^2 x^2\right )}-\frac {b c x^{2+m} \, _2F_1\left (\frac {3}{2},\frac {2+m}{2};\frac {4+m}{2};-c^2 x^2\right )}{2 d^2 (2+m)}+\frac {(1-m) \int \frac {x^m \left (a+b \sinh ^{-1}(c x)\right )}{d+c^2 d x^2} \, dx}{2 d}\\ \end {align*}

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Mathematica [A]  time = 5.63, size = 0, normalized size = 0.00 \[ \int \frac {x^m \left (a+b \sinh ^{-1}(c x)\right )}{\left (d+c^2 d x^2\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{m}}{c^{4} d^{2} x^{4} + 2 \, c^{2} d^{2} x^{2} + d^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{m}}{{\left (c^{2} d x^{2} + d\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.55, size = 0, normalized size = 0.00 \[ \int \frac {x^{m} \left (a +b \arcsinh \left (c x \right )\right )}{\left (c^{2} d \,x^{2}+d \right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{m}}{{\left (c^{2} d x^{2} + d\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^m\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{{\left (d\,c^2\,x^2+d\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {a x^{m}}{c^{4} x^{4} + 2 c^{2} x^{2} + 1}\, dx + \int \frac {b x^{m} \operatorname {asinh}{\left (c x \right )}}{c^{4} x^{4} + 2 c^{2} x^{2} + 1}\, dx}{d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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