3.292 \(\int \frac{x^3 (a+b \sinh ^{-1}(c x))^2}{\sqrt{d+c^2 d x^2}} \, dx\)

Optimal. Leaf size=265 \[ \frac{4 a b x \sqrt{c^2 x^2+1}}{3 c^3 \sqrt{c^2 d x^2+d}}-\frac{2 b x^3 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{9 c \sqrt{c^2 d x^2+d}}+\frac{x^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^2 d}-\frac{2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^4 d}+\frac{2 b^2 \left (c^2 x^2+1\right )^2}{27 c^4 \sqrt{c^2 d x^2+d}}-\frac{14 b^2 \left (c^2 x^2+1\right )}{9 c^4 \sqrt{c^2 d x^2+d}}+\frac{4 b^2 x \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)}{3 c^3 \sqrt{c^2 d x^2+d}} \]

[Out]

(4*a*b*x*Sqrt[1 + c^2*x^2])/(3*c^3*Sqrt[d + c^2*d*x^2]) - (14*b^2*(1 + c^2*x^2))/(9*c^4*Sqrt[d + c^2*d*x^2]) +
(2*b^2*(1 + c^2*x^2)^2)/(27*c^4*Sqrt[d + c^2*d*x^2]) + (4*b^2*x*Sqrt[1 + c^2*x^2]*ArcSinh[c*x])/(3*c^3*Sqrt[d
+ c^2*d*x^2]) - (2*b*x^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(9*c*Sqrt[d + c^2*d*x^2]) - (2*Sqrt[d + c^2*
d*x^2]*(a + b*ArcSinh[c*x])^2)/(3*c^4*d) + (x^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(3*c^2*d)

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Rubi [A]  time = 0.329653, antiderivative size = 265, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5758, 5717, 5653, 261, 5661, 266, 43} \[ \frac{4 a b x \sqrt{c^2 x^2+1}}{3 c^3 \sqrt{c^2 d x^2+d}}-\frac{2 b x^3 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{9 c \sqrt{c^2 d x^2+d}}+\frac{x^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^2 d}-\frac{2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^4 d}+\frac{2 b^2 \left (c^2 x^2+1\right )^2}{27 c^4 \sqrt{c^2 d x^2+d}}-\frac{14 b^2 \left (c^2 x^2+1\right )}{9 c^4 \sqrt{c^2 d x^2+d}}+\frac{4 b^2 x \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)}{3 c^3 \sqrt{c^2 d x^2+d}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*a*b*x*Sqrt[1 + c^2*x^2])/(3*c^3*Sqrt[d + c^2*d*x^2]) - (14*b^2*(1 + c^2*x^2))/(9*c^4*Sqrt[d + c^2*d*x^2]) +
(2*b^2*(1 + c^2*x^2)^2)/(27*c^4*Sqrt[d + c^2*d*x^2]) + (4*b^2*x*Sqrt[1 + c^2*x^2]*ArcSinh[c*x])/(3*c^3*Sqrt[d
+ c^2*d*x^2]) - (2*b*x^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(9*c*Sqrt[d + c^2*d*x^2]) - (2*Sqrt[d + c^2*
d*x^2]*(a + b*ArcSinh[c*x])^2)/(3*c^4*d) + (x^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(3*c^2*d)

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 5717

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

Rule 5653

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

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -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 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{x^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{d+c^2 d x^2}} \, dx &=\frac{x^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^2 d}-\frac{2 \int \frac{x \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{d+c^2 d x^2}} \, dx}{3 c^2}-\frac{\left (2 b \sqrt{1+c^2 x^2}\right ) \int x^2 \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{3 c \sqrt{d+c^2 d x^2}}\\ &=-\frac{2 b x^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c \sqrt{d+c^2 d x^2}}-\frac{2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^4 d}+\frac{x^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^2 d}+\frac{\left (2 b^2 \sqrt{1+c^2 x^2}\right ) \int \frac{x^3}{\sqrt{1+c^2 x^2}} \, dx}{9 \sqrt{d+c^2 d x^2}}+\frac{\left (4 b \sqrt{1+c^2 x^2}\right ) \int \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{3 c^3 \sqrt{d+c^2 d x^2}}\\ &=\frac{4 a b x \sqrt{1+c^2 x^2}}{3 c^3 \sqrt{d+c^2 d x^2}}-\frac{2 b x^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c \sqrt{d+c^2 d x^2}}-\frac{2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^4 d}+\frac{x^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^2 d}+\frac{\left (b^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{1+c^2 x}} \, dx,x,x^2\right )}{9 \sqrt{d+c^2 d x^2}}+\frac{\left (4 b^2 \sqrt{1+c^2 x^2}\right ) \int \sinh ^{-1}(c x) \, dx}{3 c^3 \sqrt{d+c^2 d x^2}}\\ &=\frac{4 a b x \sqrt{1+c^2 x^2}}{3 c^3 \sqrt{d+c^2 d x^2}}+\frac{4 b^2 x \sqrt{1+c^2 x^2} \sinh ^{-1}(c x)}{3 c^3 \sqrt{d+c^2 d x^2}}-\frac{2 b x^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c \sqrt{d+c^2 d x^2}}-\frac{2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^4 d}+\frac{x^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^2 d}+\frac{\left (b^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{c^2 \sqrt{1+c^2 x}}+\frac{\sqrt{1+c^2 x}}{c^2}\right ) \, dx,x,x^2\right )}{9 \sqrt{d+c^2 d x^2}}-\frac{\left (4 b^2 \sqrt{1+c^2 x^2}\right ) \int \frac{x}{\sqrt{1+c^2 x^2}} \, dx}{3 c^2 \sqrt{d+c^2 d x^2}}\\ &=\frac{4 a b x \sqrt{1+c^2 x^2}}{3 c^3 \sqrt{d+c^2 d x^2}}-\frac{14 b^2 \left (1+c^2 x^2\right )}{9 c^4 \sqrt{d+c^2 d x^2}}+\frac{2 b^2 \left (1+c^2 x^2\right )^2}{27 c^4 \sqrt{d+c^2 d x^2}}+\frac{4 b^2 x \sqrt{1+c^2 x^2} \sinh ^{-1}(c x)}{3 c^3 \sqrt{d+c^2 d x^2}}-\frac{2 b x^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c \sqrt{d+c^2 d x^2}}-\frac{2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^4 d}+\frac{x^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^2 d}\\ \end{align*}

Mathematica [A]  time = 0.280226, size = 176, normalized size = 0.66 \[ \frac{9 a^2 \left (c^4 x^4-c^2 x^2-2\right )-6 a b c x \left (c^2 x^2-6\right ) \sqrt{c^2 x^2+1}-6 b \sinh ^{-1}(c x) \left (a \left (-3 c^4 x^4+3 c^2 x^2+6\right )+b c x \sqrt{c^2 x^2+1} \left (c^2 x^2-6\right )\right )+2 b^2 \left (c^4 x^4-19 c^2 x^2-20\right )+9 b^2 \left (c^4 x^4-c^2 x^2-2\right ) \sinh ^{-1}(c x)^2}{27 c^4 \sqrt{c^2 d x^2+d}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*a*b*c*x*(-6 + c^2*x^2)*Sqrt[1 + c^2*x^2] + 2*b^2*(-20 - 19*c^2*x^2 + c^4*x^4) + 9*a^2*(-2 - c^2*x^2 + c^4*
x^4) - 6*b*(b*c*x*(-6 + c^2*x^2)*Sqrt[1 + c^2*x^2] + a*(6 + 3*c^2*x^2 - 3*c^4*x^4))*ArcSinh[c*x] + 9*b^2*(-2 -
c^2*x^2 + c^4*x^4)*ArcSinh[c*x]^2)/(27*c^4*Sqrt[d + c^2*d*x^2])

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Maple [B]  time = 0.267, size = 706, normalized size = 2.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^(1/2),x)

[Out]

a^2*(1/3*x^2/c^2/d*(c^2*d*x^2+d)^(1/2)-2/3/d/c^4*(c^2*d*x^2+d)^(1/2))+b^2*(1/216*(d*(c^2*x^2+1))^(1/2)*(4*c^4*
x^4+4*c^3*x^3*(c^2*x^2+1)^(1/2)+5*c^2*x^2+3*c*x*(c^2*x^2+1)^(1/2)+1)*(9*arcsinh(c*x)^2-6*arcsinh(c*x)+2)/c^4/d
/(c^2*x^2+1)-3/8*(d*(c^2*x^2+1))^(1/2)*(c^2*x^2+c*x*(c^2*x^2+1)^(1/2)+1)*(arcsinh(c*x)^2-2*arcsinh(c*x)+2)/c^4
/d/(c^2*x^2+1)-3/8*(d*(c^2*x^2+1))^(1/2)*(c^2*x^2-c*x*(c^2*x^2+1)^(1/2)+1)*(arcsinh(c*x)^2+2*arcsinh(c*x)+2)/c
^4/d/(c^2*x^2+1)+1/216*(d*(c^2*x^2+1))^(1/2)*(4*c^4*x^4-4*c^3*x^3*(c^2*x^2+1)^(1/2)+5*c^2*x^2-3*c*x*(c^2*x^2+1
)^(1/2)+1)*(9*arcsinh(c*x)^2+6*arcsinh(c*x)+2)/c^4/d/(c^2*x^2+1))+2*a*b*(1/72*(d*(c^2*x^2+1))^(1/2)*(4*c^4*x^4
+4*c^3*x^3*(c^2*x^2+1)^(1/2)+5*c^2*x^2+3*c*x*(c^2*x^2+1)^(1/2)+1)*(-1+3*arcsinh(c*x))/c^4/d/(c^2*x^2+1)-3/8*(d
*(c^2*x^2+1))^(1/2)*(c^2*x^2+c*x*(c^2*x^2+1)^(1/2)+1)*(-1+arcsinh(c*x))/c^4/d/(c^2*x^2+1)-3/8*(d*(c^2*x^2+1))^
(1/2)*(c^2*x^2-c*x*(c^2*x^2+1)^(1/2)+1)*(1+arcsinh(c*x))/c^4/d/(c^2*x^2+1)+1/72*(d*(c^2*x^2+1))^(1/2)*(4*c^4*x
^4-4*c^3*x^3*(c^2*x^2+1)^(1/2)+5*c^2*x^2-3*c*x*(c^2*x^2+1)^(1/2)+1)*(1+3*arcsinh(c*x))/c^4/d/(c^2*x^2+1))

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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^3*(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 [A]  time = 3.12222, size = 536, normalized size = 2.02 \begin{align*} \frac{9 \,{\left (b^{2} c^{4} x^{4} - b^{2} c^{2} x^{2} - 2 \, b^{2}\right )} \sqrt{c^{2} d x^{2} + d} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )^{2} + 6 \,{\left (3 \, a b c^{4} x^{4} - 3 \, a b c^{2} x^{2} - 6 \, a b -{\left (b^{2} c^{3} x^{3} - 6 \, b^{2} c x\right )} \sqrt{c^{2} x^{2} + 1}\right )} \sqrt{c^{2} d x^{2} + d} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) +{\left ({\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{4} x^{4} -{\left (9 \, a^{2} + 38 \, b^{2}\right )} c^{2} x^{2} - 18 \, a^{2} - 40 \, b^{2} - 6 \,{\left (a b c^{3} x^{3} - 6 \, a b c x\right )} \sqrt{c^{2} x^{2} + 1}\right )} \sqrt{c^{2} d x^{2} + d}}{27 \,{\left (c^{6} d x^{2} + c^{4} d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^(1/2),x, algorithm="fricas")

[Out]

1/27*(9*(b^2*c^4*x^4 - b^2*c^2*x^2 - 2*b^2)*sqrt(c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 + 1))^2 + 6*(3*a*b*c^4*
x^4 - 3*a*b*c^2*x^2 - 6*a*b - (b^2*c^3*x^3 - 6*b^2*c*x)*sqrt(c^2*x^2 + 1))*sqrt(c^2*d*x^2 + d)*log(c*x + sqrt(
c^2*x^2 + 1)) + ((9*a^2 + 2*b^2)*c^4*x^4 - (9*a^2 + 38*b^2)*c^2*x^2 - 18*a^2 - 40*b^2 - 6*(a*b*c^3*x^3 - 6*a*b
*c*x)*sqrt(c^2*x^2 + 1))*sqrt(c^2*d*x^2 + d))/(c^6*d*x^2 + c^4*d)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3} \left (a + b \operatorname{asinh}{\left (c x \right )}\right )^{2}}{\sqrt{d \left (c^{2} x^{2} + 1\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(a+b*asinh(c*x))**2/(c**2*d*x**2+d)**(1/2),x)

[Out]

Integral(x**3*(a + b*asinh(c*x))**2/sqrt(d*(c**2*x**2 + 1)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{2} x^{3}}{\sqrt{c^{2} d x^{2} + d}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^(1/2),x, algorithm="giac")

[Out]

integrate((b*arcsinh(c*x) + a)^2*x^3/sqrt(c^2*d*x^2 + d), x)