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

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

[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 = 0.15, 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 = {} \[ \int x^m \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx \]

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]

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

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 = 0.43, size = 0, normalized size = 0.00 \[ \int x^m \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx \]

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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fricas [A]  time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{2} c^{2} d x^{2} + a^{2} d + {\left (b^{2} c^{2} d x^{2} + b^{2} d\right )} \operatorname {arsinh}\left (c x\right )^{2} + 2 \, {\left (a b c^{2} d x^{2} + a b d\right )} \operatorname {arsinh}\left (c x\right )\right )} \sqrt {c^{2} d x^{2} + d} x^{m}, x\right ) \]

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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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [A]  time = 1.42, 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 \[ \int {\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{m}\,{d x} \]

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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mupad [A]  time = 0.00, size = -1, normalized size = -0.00 \[ \int x^m\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^{3/2} \,d x \]

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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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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]

Timed out

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