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

Optimal. Leaf size=180 \[ -\frac{2 b c x^3 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{9 \sqrt{c^2 x^2+1}}-\frac{2 b x \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{3 c \sqrt{c^2 x^2+1}}+\frac{\left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^2 d}+\frac{2 b^2 \left (c^2 x^2+1\right ) \sqrt{c^2 d x^2+d}}{27 c^2}+\frac{4 b^2 \sqrt{c^2 d x^2+d}}{9 c^2} \]

[Out]

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

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Rubi [A]  time = 0.152233, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {5717, 5679, 444, 43} \[ -\frac{2 b c x^3 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{9 \sqrt{c^2 x^2+1}}-\frac{2 b x \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{3 c \sqrt{c^2 x^2+1}}+\frac{\left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^2 d}+\frac{2 b^2 \left (c^2 x^2+1\right ) \sqrt{c^2 d x^2+d}}{27 c^2}+\frac{4 b^2 \sqrt{c^2 d x^2+d}}{9 c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 5679

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(d + e*x^2
)^p, x]}, Dist[a + b*ArcSinh[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 + c^2*x^2], x], x], x]] /;
FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]

Rule 444

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

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

Mathematica [A]  time = 0.27076, size = 166, normalized size = 0.92 \[ \frac{\sqrt{c^2 d x^2+d} \left (-6 a b c x \sqrt{c^2 x^2+1} \left (c^2 x^2+3\right )+6 b \sinh ^{-1}(c x) \left (3 a \left (c^2 x^2+1\right )^2-b c x \sqrt{c^2 x^2+1} \left (c^2 x^2+3\right )\right )+9 \left (a c^2 x^2+a\right )^2+2 b^2 \left (c^4 x^4+8 c^2 x^2+7\right )+9 \left (b c^2 x^2+b\right )^2 \sinh ^{-1}(c x)^2\right )}{27 c^2 \left (c^2 x^2+1\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d + c^2*d*x^2]*(-6*a*b*c*x*Sqrt[1 + c^2*x^2]*(3 + c^2*x^2) + 9*(a + a*c^2*x^2)^2 + 2*b^2*(7 + 8*c^2*x^2
+ c^4*x^4) + 6*b*(3*a*(1 + c^2*x^2)^2 - b*c*x*Sqrt[1 + c^2*x^2]*(3 + c^2*x^2))*ArcSinh[c*x] + 9*(b + b*c^2*x^2
)^2*ArcSinh[c*x]^2))/(27*c^2*(1 + c^2*x^2))

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Maple [B]  time = 0.177, size = 657, normalized size = 3.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a^2/c^2/d*(c^2*d*x^2+d)^(3/2)+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^2/(c^2*x^2+1)+1/8*(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^2/(c^2*x^2+1)+1/8*(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^2/(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^2/(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^2/(c^2*x^2+1)+1/8*(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^2/(c^2*x^2+1)+1/8*(d*(c^2*x^2+1))^(1/2)*(c^2*x^2-c*x*(c^2*x^2+1)^(1/2)+1)*(1+arcsi
nh(c*x))/c^2/(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^2/(c^2*x^2+1))

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Maxima [A]  time = 1.26318, size = 247, normalized size = 1.37 \begin{align*} \frac{2}{27} \, b^{2}{\left (\frac{\sqrt{c^{2} x^{2} + 1} d^{\frac{3}{2}} x^{2} + \frac{7 \, \sqrt{c^{2} x^{2} + 1} d^{\frac{3}{2}}}{c^{2}}}{d} - \frac{3 \,{\left (c^{2} d^{\frac{3}{2}} x^{3} + 3 \, d^{\frac{3}{2}} x\right )} \operatorname{arsinh}\left (c x\right )}{c d}\right )} + \frac{{\left (c^{2} d x^{2} + d\right )}^{\frac{3}{2}} b^{2} \operatorname{arsinh}\left (c x\right )^{2}}{3 \, c^{2} d} + \frac{2 \,{\left (c^{2} d x^{2} + d\right )}^{\frac{3}{2}} a b \operatorname{arsinh}\left (c x\right )}{3 \, c^{2} d} - \frac{2 \,{\left (c^{2} d^{\frac{3}{2}} x^{3} + 3 \, d^{\frac{3}{2}} x\right )} a b}{9 \, c d} + \frac{{\left (c^{2} d x^{2} + d\right )}^{\frac{3}{2}} a^{2}}{3 \, c^{2} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/27*b^2*((sqrt(c^2*x^2 + 1)*d^(3/2)*x^2 + 7*sqrt(c^2*x^2 + 1)*d^(3/2)/c^2)/d - 3*(c^2*d^(3/2)*x^3 + 3*d^(3/2)
*x)*arcsinh(c*x)/(c*d)) + 1/3*(c^2*d*x^2 + d)^(3/2)*b^2*arcsinh(c*x)^2/(c^2*d) + 2/3*(c^2*d*x^2 + d)^(3/2)*a*b
*arcsinh(c*x)/(c^2*d) - 2/9*(c^2*d^(3/2)*x^3 + 3*d^(3/2)*x)*a*b/(c*d) + 1/3*(c^2*d*x^2 + d)^(3/2)*a^2/(c^2*d)

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Fricas [A]  time = 2.71232, size = 531, normalized size = 2.95 \begin{align*} \frac{9 \,{\left (b^{2} c^{4} x^{4} + 2 \, b^{2} c^{2} x^{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} + 6 \, a b c^{2} x^{2} + 3 \, a b -{\left (b^{2} c^{3} x^{3} + 3 \, 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} + 2 \,{\left (9 \, a^{2} + 8 \, b^{2}\right )} c^{2} x^{2} + 9 \, a^{2} + 14 \, b^{2} - 6 \,{\left (a b c^{3} x^{3} + 3 \, a b c x\right )} \sqrt{c^{2} x^{2} + 1}\right )} \sqrt{c^{2} d x^{2} + d}}{27 \,{\left (c^{4} x^{2} + c^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(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 + 2*b^2*c^2*x^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 + 6*a*b*c^2*x^2 + 3*a*b - (b^2*c^3*x^3 + 3*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 + 2*(9*a^2 + 8*b^2)*c^2*x^2 + 9*a^2 + 14*b^2 - 6*(a*b*c^3*x^3 + 3*a*b
*c*x)*sqrt(c^2*x^2 + 1))*sqrt(c^2*d*x^2 + d))/(c^4*x^2 + c^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \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*(a+b*asinh(c*x))**2*(c**2*d*x**2+d)**(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError