3.4.22 \(\int x^m (d+c^2 d x^2)^{3/2} (a+b \sinh ^{-1}(c x))^2 \, dx\) [322]

Optimal. Leaf size=488 \[ \frac {2 b^2 c^2 d x^{3+m} \sqrt {d+c^2 d x^2}}{(4+m)^3}-\frac {6 b c d x^{2+m} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{(2+m)^2 (4+m) \sqrt {1+c^2 x^2}}-\frac {2 b c d x^{2+m} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{\left (8+6 m+m^2\right ) \sqrt {1+c^2 x^2}}-\frac {2 b c^3 d x^{4+m} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{(4+m)^2 \sqrt {1+c^2 x^2}}+\frac {3 d x^{1+m} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{8+6 m+m^2}+\frac {x^{1+m} \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{4+m}+\frac {6 b^2 c^2 d x^{3+m} \sqrt {d+c^2 d x^2} \, _2F_1\left (\frac {1}{2},\frac {3+m}{2};\frac {5+m}{2};-c^2 x^2\right )}{(2+m)^2 (3+m) (4+m) \sqrt {1+c^2 x^2}}+\frac {2 b^2 c^2 d (10+3 m) x^{3+m} \sqrt {d+c^2 d x^2} \, _2F_1\left (\frac {1}{2},\frac {3+m}{2};\frac {5+m}{2};-c^2 x^2\right )}{(2+m) (3+m) (4+m)^3 \sqrt {1+c^2 x^2}}+\frac {3 d^2 \text {Int}\left (\frac {x^m \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {d+c^2 d x^2}},x\right )}{8+6 m+m^2} \]

[Out]

x^(1+m)*(c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(c*x))^2/(4+m)+2*b^2*c^2*d*x^(3+m)*(c^2*d*x^2+d)^(1/2)/(4+m)^3+3*d*x^(
1+m)*(a+b*arcsinh(c*x))^2*(c^2*d*x^2+d)^(1/2)/(m^2+6*m+8)-6*b*c*d*x^(2+m)*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/
2)/(2+m)^2/(4+m)/(c^2*x^2+1)^(1/2)-2*b*c*d*x^(2+m)*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/(m^2+6*m+8)/(c^2*x^2
+1)^(1/2)-2*b*c^3*d*x^(4+m)*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/(4+m)^2/(c^2*x^2+1)^(1/2)+2*b^2*c^2*d*(10+3
*m)*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)/(4+m)^3/(m^2+5*m+6)/(c^2*x^2+
1)^(1/2)+6*b^2*c^2*d*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/(m^2
+7*m+12)/(c^2*x^2+1)^(1/2)+3*d^2*Unintegrable(x^m*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^(1/2),x)/(m^2+6*m+8)

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Rubi [A]
time = 0.42, 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 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(2*b^2*c^2*d*x^(3 + m)*Sqrt[d + c^2*d*x^2])/(4 + m)^3 - (6*b*c*d*x^(2 + m)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[
c*x]))/((2 + m)^2*(4 + m)*Sqrt[1 + c^2*x^2]) - (2*b*c*d*x^(2 + m)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/((
8 + 6*m + m^2)*Sqrt[1 + c^2*x^2]) - (2*b*c^3*d*x^(4 + m)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/((4 + m)^2*
Sqrt[1 + c^2*x^2]) + (3*d*x^(1 + m)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(8 + 6*m + m^2) + (x^(1 + m)*(
d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2)/(4 + m) + (6*b^2*c^2*d*x^(3 + m)*Sqrt[d + c^2*d*x^2]*Hypergeometr
ic2F1[1/2, (3 + m)/2, (5 + m)/2, -(c^2*x^2)])/((2 + m)^2*(3 + m)*(4 + m)*Sqrt[1 + c^2*x^2]) + (2*b^2*c^2*d*(10
+ 3*m)*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)*(3 +
m)*(4 + m)^3*Sqrt[1 + c^2*x^2]) + (3*d^2*Defer[Int][(x^m*(a + b*ArcSinh[c*x])^2)/Sqrt[d + c^2*d*x^2], x])/(8 +
6*m + m^2)

Rubi steps

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

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Mathematica [A]
time = 0.20, size = 0, normalized size = 0.00 \begin {gather*} \int x^m \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3004 deep

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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)^(3/2)*(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 = -0.00 \begin {gather*} \int x^m\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^{3/2} \,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)^(3/2),x)

[Out]

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

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