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

Optimal. Leaf size=153 \[ -\frac{2 b d \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{c}-\frac{2 b e x^2 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{9 c}+\frac{4 b e \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{9 c^3}+d x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{3} e x^3 \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{4 b^2 e x}{9 c^2}+2 b^2 d x+\frac{2}{27} b^2 e x^3 \]

[Out]

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

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Rubi [A]  time = 0.277206, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.389, Rules used = {5706, 5653, 5717, 8, 5661, 5758, 30} \[ -\frac{2 b d \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{c}-\frac{2 b e x^2 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{9 c}+\frac{4 b e \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{9 c^3}+d x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{3} e x^3 \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{4 b^2 e x}{9 c^2}+2 b^2 d x+\frac{2}{27} b^2 e x^3 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5706

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a
+ b*ArcSinh[c*x])^n, (d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[e, c^2*d] && IntegerQ[p] &&
(p > 0 || IGtQ[n, 0])

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 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 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

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 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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.213184, size = 164, normalized size = 1.07 \[ \frac{9 a^2 c^3 x \left (3 d+e x^2\right )-6 a b \sqrt{c^2 x^2+1} \left (c^2 \left (9 d+e x^2\right )-2 e\right )-6 b \sinh ^{-1}(c x) \left (b \sqrt{c^2 x^2+1} \left (c^2 \left (9 d+e x^2\right )-2 e\right )-3 a c^3 x \left (3 d+e x^2\right )\right )+2 b^2 c x \left (c^2 \left (27 d+e x^2\right )-6 e\right )+9 b^2 c^3 x \sinh ^{-1}(c x)^2 \left (3 d+e x^2\right )}{27 c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.046, size = 271, normalized size = 1.8 \begin{align*}{\frac{1}{c} \left ({\frac{{a}^{2}}{{c}^{2}} \left ({\frac{{c}^{3}{x}^{3}e}{3}}+{c}^{3}dx \right ) }+{\frac{{b}^{2}}{{c}^{2}} \left ({c}^{2}d \left ( \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}cx-2\,{\it Arcsinh} \left ( cx \right ) \sqrt{{c}^{2}{x}^{2}+1}+2\,cx \right ) +{\frac{e}{27} \left ( 9\, \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}{c}^{3}{x}^{3}-6\,{\it Arcsinh} \left ( cx \right ){c}^{2}{x}^{2}\sqrt{{c}^{2}{x}^{2}+1}+27\, \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}cx+2\,{c}^{3}{x}^{3}-42\,{\it Arcsinh} \left ( cx \right ) \sqrt{{c}^{2}{x}^{2}+1}+42\,cx \right ) }-e \left ( \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}cx-2\,{\it Arcsinh} \left ( cx \right ) \sqrt{{c}^{2}{x}^{2}+1}+2\,cx \right ) \right ) }+2\,{\frac{ab \left ( 1/3\,{\it Arcsinh} \left ( cx \right ){c}^{3}{x}^{3}e+{\it Arcsinh} \left ( cx \right ){c}^{3}dx-1/3\,e \left ( 1/3\,{c}^{2}{x}^{2}\sqrt{{c}^{2}{x}^{2}+1}-2/3\,\sqrt{{c}^{2}{x}^{2}+1} \right ) -{c}^{2}d\sqrt{{c}^{2}{x}^{2}+1} \right ) }{{c}^{2}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c*(a^2/c^2*(1/3*c^3*x^3*e+c^3*d*x)+b^2/c^2*(c^2*d*(arcsinh(c*x)^2*c*x-2*arcsinh(c*x)*(c^2*x^2+1)^(1/2)+2*c*x
)+1/27*e*(9*arcsinh(c*x)^2*c^3*x^3-6*arcsinh(c*x)*c^2*x^2*(c^2*x^2+1)^(1/2)+27*arcsinh(c*x)^2*c*x+2*c^3*x^3-42
*arcsinh(c*x)*(c^2*x^2+1)^(1/2)+42*c*x)-e*(arcsinh(c*x)^2*c*x-2*arcsinh(c*x)*(c^2*x^2+1)^(1/2)+2*c*x))+2*a*b/c
^2*(1/3*arcsinh(c*x)*c^3*x^3*e+arcsinh(c*x)*c^3*d*x-1/3*e*(1/3*c^2*x^2*(c^2*x^2+1)^(1/2)-2/3*(c^2*x^2+1)^(1/2)
)-c^2*d*(c^2*x^2+1)^(1/2)))

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Maxima [A]  time = 1.17482, size = 294, normalized size = 1.92 \begin{align*} \frac{1}{3} \, b^{2} e x^{3} \operatorname{arsinh}\left (c x\right )^{2} + \frac{1}{3} \, a^{2} e x^{3} + b^{2} d x \operatorname{arsinh}\left (c x\right )^{2} + \frac{2}{9} \,{\left (3 \, x^{3} \operatorname{arsinh}\left (c x\right ) - c{\left (\frac{\sqrt{c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac{2 \, \sqrt{c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} a b e - \frac{2}{27} \,{\left (3 \, c{\left (\frac{\sqrt{c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac{2 \, \sqrt{c^{2} x^{2} + 1}}{c^{4}}\right )} \operatorname{arsinh}\left (c x\right ) - \frac{c^{2} x^{3} - 6 \, x}{c^{2}}\right )} b^{2} e + 2 \, b^{2} d{\left (x - \frac{\sqrt{c^{2} x^{2} + 1} \operatorname{arsinh}\left (c x\right )}{c}\right )} + a^{2} d x + \frac{2 \,{\left (c x \operatorname{arsinh}\left (c x\right ) - \sqrt{c^{2} x^{2} + 1}\right )} a b d}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*b^2*e*x^3*arcsinh(c*x)^2 + 1/3*a^2*e*x^3 + b^2*d*x*arcsinh(c*x)^2 + 2/9*(3*x^3*arcsinh(c*x) - c*(sqrt(c^2*
x^2 + 1)*x^2/c^2 - 2*sqrt(c^2*x^2 + 1)/c^4))*a*b*e - 2/27*(3*c*(sqrt(c^2*x^2 + 1)*x^2/c^2 - 2*sqrt(c^2*x^2 + 1
)/c^4)*arcsinh(c*x) - (c^2*x^3 - 6*x)/c^2)*b^2*e + 2*b^2*d*(x - sqrt(c^2*x^2 + 1)*arcsinh(c*x)/c) + a^2*d*x +
2*(c*x*arcsinh(c*x) - sqrt(c^2*x^2 + 1))*a*b*d/c

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Fricas [A]  time = 2.79202, size = 454, normalized size = 2.97 \begin{align*} \frac{{\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{3} e x^{3} + 9 \,{\left (b^{2} c^{3} e x^{3} + 3 \, b^{2} c^{3} d x\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )^{2} + 3 \,{\left (9 \,{\left (a^{2} + 2 \, b^{2}\right )} c^{3} d - 4 \, b^{2} c e\right )} x + 6 \,{\left (3 \, a b c^{3} e x^{3} + 9 \, a b c^{3} d x -{\left (b^{2} c^{2} e x^{2} + 9 \, b^{2} c^{2} d - 2 \, b^{2} e\right )} \sqrt{c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) - 6 \,{\left (a b c^{2} e x^{2} + 9 \, a b c^{2} d - 2 \, a b e\right )} \sqrt{c^{2} x^{2} + 1}}{27 \, c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.62599, size = 279, normalized size = 1.82 \begin{align*} \begin{cases} a^{2} d x + \frac{a^{2} e x^{3}}{3} + 2 a b d x \operatorname{asinh}{\left (c x \right )} + \frac{2 a b e x^{3} \operatorname{asinh}{\left (c x \right )}}{3} - \frac{2 a b d \sqrt{c^{2} x^{2} + 1}}{c} - \frac{2 a b e x^{2} \sqrt{c^{2} x^{2} + 1}}{9 c} + \frac{4 a b e \sqrt{c^{2} x^{2} + 1}}{9 c^{3}} + b^{2} d x \operatorname{asinh}^{2}{\left (c x \right )} + 2 b^{2} d x + \frac{b^{2} e x^{3} \operatorname{asinh}^{2}{\left (c x \right )}}{3} + \frac{2 b^{2} e x^{3}}{27} - \frac{2 b^{2} d \sqrt{c^{2} x^{2} + 1} \operatorname{asinh}{\left (c x \right )}}{c} - \frac{2 b^{2} e x^{2} \sqrt{c^{2} x^{2} + 1} \operatorname{asinh}{\left (c x \right )}}{9 c} - \frac{4 b^{2} e x}{9 c^{2}} + \frac{4 b^{2} e \sqrt{c^{2} x^{2} + 1} \operatorname{asinh}{\left (c x \right )}}{9 c^{3}} & \text{for}\: c \neq 0 \\a^{2} \left (d x + \frac{e x^{3}}{3}\right ) & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**2*d*x + a**2*e*x**3/3 + 2*a*b*d*x*asinh(c*x) + 2*a*b*e*x**3*asinh(c*x)/3 - 2*a*b*d*sqrt(c**2*x**
2 + 1)/c - 2*a*b*e*x**2*sqrt(c**2*x**2 + 1)/(9*c) + 4*a*b*e*sqrt(c**2*x**2 + 1)/(9*c**3) + b**2*d*x*asinh(c*x)
**2 + 2*b**2*d*x + b**2*e*x**3*asinh(c*x)**2/3 + 2*b**2*e*x**3/27 - 2*b**2*d*sqrt(c**2*x**2 + 1)*asinh(c*x)/c
- 2*b**2*e*x**2*sqrt(c**2*x**2 + 1)*asinh(c*x)/(9*c) - 4*b**2*e*x/(9*c**2) + 4*b**2*e*sqrt(c**2*x**2 + 1)*asin
h(c*x)/(9*c**3), Ne(c, 0)), (a**2*(d*x + e*x**3/3), True))

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Giac [B]  time = 2.07283, size = 373, normalized size = 2.44 \begin{align*} 2 \,{\left (x \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) - \frac{\sqrt{c^{2} x^{2} + 1}}{c}\right )} a b d +{\left (x \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )^{2} + 2 \, c{\left (\frac{x}{c} - \frac{\sqrt{c^{2} x^{2} + 1} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )}{c^{2}}\right )}\right )} b^{2} d + a^{2} d x + \frac{1}{27} \,{\left (9 \, a^{2} x^{3} + 6 \,{\left (3 \, x^{3} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) - \frac{{\left (c^{2} x^{2} + 1\right )}^{\frac{3}{2}} - 3 \, \sqrt{c^{2} x^{2} + 1}}{c^{3}}\right )} a b +{\left (9 \, x^{3} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )^{2} + 2 \, c{\left (\frac{c^{2} x^{3} - 6 \, x}{c^{3}} - \frac{3 \,{\left ({\left (c^{2} x^{2} + 1\right )}^{\frac{3}{2}} - 3 \, \sqrt{c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )}{c^{4}}\right )}\right )} b^{2}\right )} e \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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