3.2.90 \(\int \frac {x^m (a+b \sinh ^{-1}(c x))}{(d+c^2 d x^2)^3} \, dx\) [190]

Optimal. Leaf size=207 \[ \frac {x^{1+m} \left (a+b \sinh ^{-1}(c x)\right )}{4 d^3 \left (1+c^2 x^2\right )^2}+\frac {(3-m) x^{1+m} \left (a+b \sinh ^{-1}(c x)\right )}{8 d^3 \left (1+c^2 x^2\right )}-\frac {b c (3-m) x^{2+m} \, _2F_1\left (\frac {3}{2},\frac {2+m}{2};\frac {4+m}{2};-c^2 x^2\right )}{8 d^3 (2+m)}-\frac {b c x^{2+m} \, _2F_1\left (\frac {5}{2},\frac {2+m}{2};\frac {4+m}{2};-c^2 x^2\right )}{4 d^3 (2+m)}+\frac {(1-m) (3-m) \text {Int}\left (\frac {x^m \left (a+b \sinh ^{-1}(c x)\right )}{d+c^2 d x^2},x\right )}{8 d^2} \]

[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

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Rubi [A]
time = 0.17, 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 = {} \begin {gather*} \int \frac {x^m \left (a+b \sinh ^{-1}(c x)\right )}{\left (d+c^2 d x^2\right )^3} \, dx \end {gather*}

Verification is not applicable to the result.

[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*} \int \frac {x^m \left (a+b \sinh ^{-1}(c x)\right )}{\left (d+c^2 d x^2\right )^3} \, dx &=\frac {x^{1+m} \left (a+b \sinh ^{-1}(c x)\right )}{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 \left (a+b \sinh ^{-1}(c x)\right )}{\left (d+c^2 d x^2\right )^2} \, dx}{4 d}\\ &=\frac {x^{1+m} \left (a+b \sinh ^{-1}(c x)\right )}{4 d^3 \left (1+c^2 x^2\right )^2}+\frac {(3-m) x^{1+m} \left (a+b \sinh ^{-1}(c x)\right )}{8 d^3 \left (1+c^2 x^2\right )}-\frac {b c x^{2+m} \, _2F_1\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 \left (a+b \sinh ^{-1}(c x)\right )}{d+c^2 d x^2} \, dx}{8 d^2}\\ &=\frac {x^{1+m} \left (a+b \sinh ^{-1}(c x)\right )}{4 d^3 \left (1+c^2 x^2\right )^2}+\frac {(3-m) x^{1+m} \left (a+b \sinh ^{-1}(c x)\right )}{8 d^3 \left (1+c^2 x^2\right )}-\frac {b c (3-m) x^{2+m} \, _2F_1\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} \, _2F_1\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 \left (a+b \sinh ^{-1}(c x)\right )}{d+c^2 d x^2} \, dx}{8 d^2}\\ \end {align*}

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Mathematica [A]
time = 4.14, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^m \left (a+b \sinh ^{-1}(c x)\right )}{\left (d+c^2 d x^2\right )^3} \, dx \end {gather*}

Verification is not applicable to the result.

[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]

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Maple [A]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {x^{m} \left (a +b \arcsinh \left (c x \right )\right )}{\left (c^{2} d \,x^{2}+d \right )^{3}}\, dx\]

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)^3,x)

[Out]

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

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)^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)

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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}} \end {gather*}

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)**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

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)^3,x, algorithm="giac")

[Out]

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

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 \end {gather*}

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)^3,x)

[Out]

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

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