3.259 \(\int x^2 \sqrt{d+c^2 d x^2} (a+b \sinh ^{-1}(c x))^2 \, dx\)

Optimal. Leaf size=291 \[ -\frac{b c x^4 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{8 \sqrt{c^2 x^2+1}}+\frac{1}{4} x^3 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{b x^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{8 c \sqrt{c^2 x^2+1}}+\frac{x \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{8 c^2}-\frac{\sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^3}{24 b c^3 \sqrt{c^2 x^2+1}}+\frac{1}{32} b^2 x^3 \sqrt{c^2 d x^2+d}+\frac{b^2 x \sqrt{c^2 d x^2+d}}{64 c^2}-\frac{b^2 \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)}{64 c^3 \sqrt{c^2 x^2+1}} \]

[Out]

(b^2*x*Sqrt[d + c^2*d*x^2])/(64*c^2) + (b^2*x^3*Sqrt[d + c^2*d*x^2])/32 - (b^2*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x
])/(64*c^3*Sqrt[1 + c^2*x^2]) - (b*x^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(8*c*Sqrt[1 + c^2*x^2]) - (b*
c*x^4*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(8*Sqrt[1 + c^2*x^2]) + (x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[
c*x])^2)/(8*c^2) + (x^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/4 - (Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*
x])^3)/(24*b*c^3*Sqrt[1 + c^2*x^2])

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Rubi [A]  time = 0.372315, antiderivative size = 291, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {5742, 5758, 5675, 5661, 321, 215} \[ -\frac{b c x^4 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{8 \sqrt{c^2 x^2+1}}+\frac{1}{4} x^3 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{b x^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{8 c \sqrt{c^2 x^2+1}}+\frac{x \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{8 c^2}-\frac{\sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^3}{24 b c^3 \sqrt{c^2 x^2+1}}+\frac{1}{32} b^2 x^3 \sqrt{c^2 d x^2+d}+\frac{b^2 x \sqrt{c^2 d x^2+d}}{64 c^2}-\frac{b^2 \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)}{64 c^3 \sqrt{c^2 x^2+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b^2*x*Sqrt[d + c^2*d*x^2])/(64*c^2) + (b^2*x^3*Sqrt[d + c^2*d*x^2])/32 - (b^2*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x
])/(64*c^3*Sqrt[1 + c^2*x^2]) - (b*x^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(8*c*Sqrt[1 + c^2*x^2]) - (b*
c*x^4*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(8*Sqrt[1 + c^2*x^2]) + (x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[
c*x])^2)/(8*c^2) + (x^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/4 - (Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*
x])^3)/(24*b*c^3*Sqrt[1 + c^2*x^2])

Rule 5742

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(
(f*x)^(m + 1)*Sqrt[d + e*x^2]*(a + b*ArcSinh[c*x])^n)/(f*(m + 2)), x] + (Dist[Sqrt[d + e*x^2]/((m + 2)*Sqrt[1
+ c^2*x^2]), Int[((f*x)^m*(a + b*ArcSinh[c*x])^n)/Sqrt[1 + c^2*x^2], x], x] - Dist[(b*c*n*Sqrt[d + e*x^2])/(f*
(m + 2)*Sqrt[1 + c^2*x^2]), Int[(f*x)^(m + 1)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f
, m}, x] && EqQ[e, c^2*d] && GtQ[n, 0] &&  !LtQ[m, -1] && (RationalQ[m] || EqQ[n, 1])

Rule 5758

Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp
[(f*(f*x)^(m - 1)*Sqrt[d + e*x^2]*(a + b*ArcSinh[c*x])^n)/(e*m), x] + (-Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)
^(m - 2)*(a + b*ArcSinh[c*x])^n)/Sqrt[d + e*x^2], x], x] - Dist[(b*f*n*Sqrt[1 + c^2*x^2])/(c*m*Sqrt[d + e*x^2]
), Int[(f*x)^(m - 1)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] &&
GtQ[n, 0] && GtQ[m, 1] && IntegerQ[m]

Rule 5675

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(a + b*ArcSinh[c*x]
)^(n + 1)/(b*c*Sqrt[d]*(n + 1)), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c^2*d] && GtQ[d, 0] && NeQ[n, -1
]

Rule 5661

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcS
inh[c*x])^n)/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcSinh[c*x])^(n - 1))/Sqrt
[1 + c^2*x^2], x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rubi steps

\begin{align*} \int x^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx &=\frac{1}{4} x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{\sqrt{d+c^2 d x^2} \int \frac{x^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{1+c^2 x^2}} \, dx}{4 \sqrt{1+c^2 x^2}}-\frac{\left (b c \sqrt{d+c^2 d x^2}\right ) \int x^3 \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{2 \sqrt{1+c^2 x^2}}\\ &=-\frac{b c x^4 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 \sqrt{1+c^2 x^2}}+\frac{x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{8 c^2}+\frac{1}{4} x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{\sqrt{d+c^2 d x^2} \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{1+c^2 x^2}} \, dx}{8 c^2 \sqrt{1+c^2 x^2}}-\frac{\left (b \sqrt{d+c^2 d x^2}\right ) \int x \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{4 c \sqrt{1+c^2 x^2}}+\frac{\left (b^2 c^2 \sqrt{d+c^2 d x^2}\right ) \int \frac{x^4}{\sqrt{1+c^2 x^2}} \, dx}{8 \sqrt{1+c^2 x^2}}\\ &=\frac{1}{32} b^2 x^3 \sqrt{d+c^2 d x^2}-\frac{b x^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 c \sqrt{1+c^2 x^2}}-\frac{b c x^4 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 \sqrt{1+c^2 x^2}}+\frac{x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{8 c^2}+\frac{1}{4} x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{\sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{24 b c^3 \sqrt{1+c^2 x^2}}-\frac{\left (3 b^2 \sqrt{d+c^2 d x^2}\right ) \int \frac{x^2}{\sqrt{1+c^2 x^2}} \, dx}{32 \sqrt{1+c^2 x^2}}+\frac{\left (b^2 \sqrt{d+c^2 d x^2}\right ) \int \frac{x^2}{\sqrt{1+c^2 x^2}} \, dx}{8 \sqrt{1+c^2 x^2}}\\ &=\frac{b^2 x \sqrt{d+c^2 d x^2}}{64 c^2}+\frac{1}{32} b^2 x^3 \sqrt{d+c^2 d x^2}-\frac{b x^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 c \sqrt{1+c^2 x^2}}-\frac{b c x^4 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 \sqrt{1+c^2 x^2}}+\frac{x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{8 c^2}+\frac{1}{4} x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{\sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{24 b c^3 \sqrt{1+c^2 x^2}}+\frac{\left (3 b^2 \sqrt{d+c^2 d x^2}\right ) \int \frac{1}{\sqrt{1+c^2 x^2}} \, dx}{64 c^2 \sqrt{1+c^2 x^2}}-\frac{\left (b^2 \sqrt{d+c^2 d x^2}\right ) \int \frac{1}{\sqrt{1+c^2 x^2}} \, dx}{16 c^2 \sqrt{1+c^2 x^2}}\\ &=\frac{b^2 x \sqrt{d+c^2 d x^2}}{64 c^2}+\frac{1}{32} b^2 x^3 \sqrt{d+c^2 d x^2}-\frac{b^2 \sqrt{d+c^2 d x^2} \sinh ^{-1}(c x)}{64 c^3 \sqrt{1+c^2 x^2}}-\frac{b x^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 c \sqrt{1+c^2 x^2}}-\frac{b c x^4 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 \sqrt{1+c^2 x^2}}+\frac{x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{8 c^2}+\frac{1}{4} x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{\sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{24 b c^3 \sqrt{1+c^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 1.3118, size = 207, normalized size = 0.71 \[ -\frac{-96 a^2 c x \left (2 c^2 x^2+1\right ) \sqrt{c^2 d x^2+d}+96 a^2 \sqrt{d} \log \left (\sqrt{d} \sqrt{c^2 d x^2+d}+c d x\right )+\frac{12 a b \sqrt{c^2 d x^2+d} \left (8 \sinh ^{-1}(c x)^2-4 \sinh \left (4 \sinh ^{-1}(c x)\right ) \sinh ^{-1}(c x)+\cosh \left (4 \sinh ^{-1}(c x)\right )\right )}{\sqrt{c^2 x^2+1}}+\frac{b^2 \sqrt{c^2 d x^2+d} \left (32 \sinh ^{-1}(c x)^3-3 \left (8 \sinh ^{-1}(c x)^2+1\right ) \sinh \left (4 \sinh ^{-1}(c x)\right )+12 \sinh ^{-1}(c x) \cosh \left (4 \sinh ^{-1}(c x)\right )\right )}{\sqrt{c^2 x^2+1}}}{768 c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(-96*a^2*c*x*(1 + 2*c^2*x^2)*Sqrt[d + c^2*d*x^2] + 96*a^2*Sqrt[d]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]] +
(12*a*b*Sqrt[d + c^2*d*x^2]*(8*ArcSinh[c*x]^2 + Cosh[4*ArcSinh[c*x]] - 4*ArcSinh[c*x]*Sinh[4*ArcSinh[c*x]]))/S
qrt[1 + c^2*x^2] + (b^2*Sqrt[d + c^2*d*x^2]*(32*ArcSinh[c*x]^3 + 12*ArcSinh[c*x]*Cosh[4*ArcSinh[c*x]] - 3*(1 +
8*ArcSinh[c*x]^2)*Sinh[4*ArcSinh[c*x]]))/Sqrt[1 + c^2*x^2])/(768*c^3)

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Maple [B]  time = 0.241, size = 701, normalized size = 2.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*a^2*x*(c^2*d*x^2+d)^(3/2)/c^2/d-1/8*a^2/c^2*x*(c^2*d*x^2+d)^(1/2)-1/8*a^2/c^2*d*ln(x*c^2*d/(c^2*d)^(1/2)+(
c^2*d*x^2+d)^(1/2))/(c^2*d)^(1/2)+1/4*b^2*(d*(c^2*x^2+1))^(1/2)*c^2/(c^2*x^2+1)*arcsinh(c*x)^2*x^5+3/8*b^2*(d*
(c^2*x^2+1))^(1/2)/(c^2*x^2+1)*arcsinh(c*x)^2*x^3-1/8*b^2*(d*(c^2*x^2+1))^(1/2)*c/(c^2*x^2+1)^(1/2)*arcsinh(c*
x)*x^4-1/8*b^2*(d*(c^2*x^2+1))^(1/2)/c/(c^2*x^2+1)^(1/2)*arcsinh(c*x)*x^2+1/8*b^2*(d*(c^2*x^2+1))^(1/2)/c^2/(c
^2*x^2+1)*arcsinh(c*x)^2*x-1/64*b^2*(d*(c^2*x^2+1))^(1/2)/c^3/(c^2*x^2+1)^(1/2)*arcsinh(c*x)-1/24*b^2*(d*(c^2*
x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/c^3*arcsinh(c*x)^3+1/32*b^2*(d*(c^2*x^2+1))^(1/2)*c^2/(c^2*x^2+1)*x^5+3/64*b^2
*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)*x^3+1/64*b^2*(d*(c^2*x^2+1))^(1/2)/c^2/(c^2*x^2+1)*x+1/2*a*b*(d*(c^2*x^2+1)
)^(1/2)*c^2/(c^2*x^2+1)*arcsinh(c*x)*x^5-1/8*a*b*(d*(c^2*x^2+1))^(1/2)*c/(c^2*x^2+1)^(1/2)*x^4+3/4*a*b*(d*(c^2
*x^2+1))^(1/2)/(c^2*x^2+1)*arcsinh(c*x)*x^3-1/8*a*b*(d*(c^2*x^2+1))^(1/2)/c/(c^2*x^2+1)^(1/2)*x^2+1/4*a*b*(d*(
c^2*x^2+1))^(1/2)/c^2/(c^2*x^2+1)*arcsinh(c*x)*x-1/64*a*b*(d*(c^2*x^2+1))^(1/2)/c^3/(c^2*x^2+1)^(1/2)-1/8*a*b*
(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/c^3*arcsinh(c*x)^2

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b^{2} x^{2} \operatorname{arsinh}\left (c x\right )^{2} + 2 \, a b x^{2} \operatorname{arsinh}\left (c x\right ) + a^{2} x^{2}\right )} \sqrt{c^{2} d x^{2} + d}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{c^{2} d x^{2} + d}{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{2} x^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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