3.4.23 \(\int x^m \sqrt {d+c^2 d x^2} (a+b \sinh ^{-1}(c x))^2 \, dx\) [323]

Optimal. Leaf size=199 \[ -\frac {2 b c x^{2+m} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{(2+m)^2 \sqrt {1+c^2 x^2}}+\frac {x^{1+m} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2+m}+\frac {2 b^2 c^2 x^{3+m} \sqrt {d+c^2 d x^2} \, _2F_1\left (\frac {1}{2},\frac {3+m}{2};\frac {5+m}{2};-c^2 x^2\right )}{(2+m)^2 (3+m) \sqrt {1+c^2 x^2}}+\frac {d \text {Int}\left (\frac {x^m \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {d+c^2 d x^2}},x\right )}{2+m} \]

[Out]

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

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Rubi [A]
time = 0.22, 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 x^m \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 0.14, size = 0, normalized size = 0.00 \begin {gather*} \int x^m \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^2,x)

[Out]

int(x^m*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^2,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*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^2,x, algorithm="maxima")

[Out]

integrate(sqrt(c^2*d*x^2 + d)*(b*arcsinh(c*x) + a)^2*x^m, 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*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{m} \sqrt {d \left (c^{2} x^{2} + 1\right )} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^m\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,\sqrt {d\,c^2\,x^2+d} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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