3.4.21 \(\int x^m (d+c^2 d x^2)^{5/2} (a+b \sinh ^{-1}(c x))^2 \, dx\) [321]

Optimal. Leaf size=936 \[ \frac {10 b^2 c^2 d^2 x^{3+m} \sqrt {d+c^2 d x^2}}{(4+m)^3 (6+m)}+\frac {2 b^2 c^2 d^2 \left (52+15 m+m^2\right ) x^{3+m} \sqrt {d+c^2 d x^2}}{(4+m)^2 (6+m)^3}+\frac {2 b^2 c^4 d^2 x^{5+m} \sqrt {d+c^2 d x^2}}{(6+m)^3}-\frac {30 b c d^2 x^{2+m} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{(2+m)^2 (4+m) (6+m) \sqrt {1+c^2 x^2}}-\frac {10 b c d^2 x^{2+m} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{(6+m) \left (8+6 m+m^2\right ) \sqrt {1+c^2 x^2}}-\frac {2 b c d^2 x^{2+m} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{\left (12+8 m+m^2\right ) \sqrt {1+c^2 x^2}}-\frac {10 b c^3 d^2 x^{4+m} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{(4+m)^2 (6+m) \sqrt {1+c^2 x^2}}-\frac {4 b c^3 d^2 x^{4+m} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{(4+m) (6+m) \sqrt {1+c^2 x^2}}-\frac {2 b c^5 d^2 x^{6+m} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{(6+m)^2 \sqrt {1+c^2 x^2}}+\frac {15 d^2 x^{1+m} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{(6+m) \left (8+6 m+m^2\right )}+\frac {5 d x^{1+m} \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{(4+m) (6+m)}+\frac {x^{1+m} \left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{6+m}+\frac {30 b^2 c^2 d^2 x^{3+m} \sqrt {d+c^2 d x^2} \, _2F_1\left (\frac {1}{2},\frac {3+m}{2};\frac {5+m}{2};-c^2 x^2\right )}{(2+m)^2 (3+m) (4+m) (6+m) \sqrt {1+c^2 x^2}}+\frac {10 b^2 c^2 d^2 (10+3 m) x^{3+m} \sqrt {d+c^2 d x^2} \, _2F_1\left (\frac {1}{2},\frac {3+m}{2};\frac {5+m}{2};-c^2 x^2\right )}{(2+m) (3+m) (4+m)^3 (6+m) \sqrt {1+c^2 x^2}}+\frac {2 b^2 c^2 d^2 \left (264+130 m+15 m^2\right ) x^{3+m} \sqrt {d+c^2 d x^2} \, _2F_1\left (\frac {1}{2},\frac {3+m}{2};\frac {5+m}{2};-c^2 x^2\right )}{(2+m) (3+m) (4+m)^2 (6+m)^3 \sqrt {1+c^2 x^2}}+\frac {15 d^3 \text {Int}\left (\frac {x^m \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {d+c^2 d x^2}},x\right )}{(6+m) \left (8+6 m+m^2\right )} \]

[Out]

5*d*x^(1+m)*(c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(c*x))^2/(4+m)/(6+m)+x^(1+m)*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x)
)^2/(6+m)+10*b^2*c^2*d^2*x^(3+m)*(c^2*d*x^2+d)^(1/2)/(4+m)^3/(6+m)+2*b^2*c^2*d^2*(m^2+15*m+52)*x^(3+m)*(c^2*d*
x^2+d)^(1/2)/(4+m)^2/(6+m)^3+2*b^2*c^4*d^2*x^(5+m)*(c^2*d*x^2+d)^(1/2)/(6+m)^3+15*d^2*x^(1+m)*(a+b*arcsinh(c*x
))^2*(c^2*d*x^2+d)^(1/2)/(6+m)/(m^2+6*m+8)-30*b*c*d^2*x^(2+m)*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/(2+m)^2/(
4+m)/(6+m)/(c^2*x^2+1)^(1/2)-10*b*c*d^2*x^(2+m)*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/(6+m)/(m^2+6*m+8)/(c^2*
x^2+1)^(1/2)-2*b*c*d^2*x^(2+m)*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/(m^2+8*m+12)/(c^2*x^2+1)^(1/2)-10*b*c^3*
d^2*x^(4+m)*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/(4+m)^2/(6+m)/(c^2*x^2+1)^(1/2)-4*b*c^3*d^2*x^(4+m)*(a+b*ar
csinh(c*x))*(c^2*d*x^2+d)^(1/2)/(4+m)/(6+m)/(c^2*x^2+1)^(1/2)-2*b*c^5*d^2*x^(6+m)*(a+b*arcsinh(c*x))*(c^2*d*x^
2+d)^(1/2)/(6+m)^2/(c^2*x^2+1)^(1/2)+10*b^2*c^2*d^2*(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/(6+m)/(m^2+5*m+6)/(c^2*x^2+1)^(1/2)+30*b^2*c^2*d^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/(6+m)/(m^2+7*m+12)/(c^2*x^2+1)^(1/2)+2*b^2*c^2*d
^2*(15*m^2+130*m+264)*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)^2/(6+
m)^3/(m^2+5*m+6)/(c^2*x^2+1)^(1/2)+15*d^3*Unintegrable(x^m*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^(1/2),x)/(6+m)/(
m^2+6*m+8)

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Rubi [A]
time = 0.88, 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 )^{5/2} \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)^(5/2)*(a + b*ArcSinh[c*x])^2,x]

[Out]

(10*b^2*c^2*d^2*x^(3 + m)*Sqrt[d + c^2*d*x^2])/((4 + m)^3*(6 + m)) + (2*b^2*c^2*d^2*(52 + 15*m + m^2)*x^(3 + m
)*Sqrt[d + c^2*d*x^2])/((4 + m)^2*(6 + m)^3) + (2*b^2*c^4*d^2*x^(5 + m)*Sqrt[d + c^2*d*x^2])/(6 + m)^3 - (30*b
*c*d^2*x^(2 + m)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/((2 + m)^2*(4 + m)*(6 + m)*Sqrt[1 + c^2*x^2]) - (10
*b*c*d^2*x^(2 + m)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/((6 + m)*(8 + 6*m + m^2)*Sqrt[1 + c^2*x^2]) - (2*
b*c*d^2*x^(2 + m)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/((12 + 8*m + m^2)*Sqrt[1 + c^2*x^2]) - (10*b*c^3*d
^2*x^(4 + m)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/((4 + m)^2*(6 + m)*Sqrt[1 + c^2*x^2]) - (4*b*c^3*d^2*x^
(4 + m)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/((4 + m)*(6 + m)*Sqrt[1 + c^2*x^2]) - (2*b*c^5*d^2*x^(6 + m)
*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/((6 + m)^2*Sqrt[1 + c^2*x^2]) + (15*d^2*x^(1 + m)*Sqrt[d + c^2*d*x^
2]*(a + b*ArcSinh[c*x])^2)/((6 + m)*(8 + 6*m + m^2)) + (5*d*x^(1 + m)*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x
])^2)/((4 + m)*(6 + m)) + (x^(1 + m)*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2)/(6 + m) + (30*b^2*c^2*d^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)*(4 +
m)*(6 + m)*Sqrt[1 + c^2*x^2]) + (10*b^2*c^2*d^2*(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*(6 + m)*Sqrt[1 + c^2*x^2]) + (2*b^2*c^2*d^2*(
264 + 130*m + 15*m^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)*(3 + m)*(4 + m)^2*(6 + m)^3*Sqrt[1 + c^2*x^2]) + (15*d^3*Defer[Int][(x^m*(a + b*ArcSinh[c*x])^2)/Sqrt
[d + c^2*d*x^2], x])/((6 + m)*(8 + 6*m + m^2))

Rubi steps

\begin {align*} \int x^m \left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx &=\int x^m \left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx\\ \end {align*}

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Mathematica [A]
time = 1.96, size = 0, normalized size = 0.00 \begin {gather*} \int x^m \left (d+c^2 d x^2\right )^{5/2} \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)^(5/2)*(a + b*ArcSinh[c*x])^2,x]

[Out]

Integrate[x^m*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2, x]

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Maple [A]
time = 180.00, size = 0, normalized size = 0.00 \[\int x^{m} \left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \arcsinh \left (c x \right )\right )^{2}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))^2,x)

[Out]

int(x^m*(c^2*d*x^2+d)^(5/2)*(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)^(5/2)*(a+b*arcsinh(c*x))^2,x, algorithm="maxima")

[Out]

integrate((c^2*d*x^2 + d)^(5/2)*(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)^(5/2)*(a+b*arcsinh(c*x))^2,x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m*(c**2*d*x**2+d)**(5/2)*(a+b*asinh(c*x))**2,x)

[Out]

Timed out

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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)^(5/2)*(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 )}^{5/2} \,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)^(5/2),x)

[Out]

int(x^m*(a + b*asinh(c*x))^2*(d + c^2*d*x^2)^(5/2), x)

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