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

Optimal. Leaf size=239 \[ -\frac{2 b d^2 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{c}-\frac{b d e x \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{c}+\frac{d e \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^2}+\frac{4 b e^2 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{9 c^3}-\frac{2 b e^2 x^2 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{9 c}-\frac{d^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 e}+\frac{(d+e x)^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 e}-\frac{4 b^2 e^2 x}{9 c^2}+2 b^2 d^2 x+\frac{1}{2} b^2 d e x^2+\frac{2}{27} b^2 e^2 x^3 \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.511892, antiderivative size = 239, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 7, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.389, Rules used = {5801, 5821, 5675, 5717, 8, 5758, 30} \[ -\frac{2 b d^2 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{c}-\frac{b d e x \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{c}+\frac{d e \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^2}+\frac{4 b e^2 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{9 c^3}-\frac{2 b e^2 x^2 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{9 c}-\frac{d^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 e}+\frac{(d+e x)^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 e}-\frac{4 b^2 e^2 x}{9 c^2}+2 b^2 d^2 x+\frac{1}{2} b^2 d e x^2+\frac{2}{27} b^2 e^2 x^3 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5801

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

Rule 5821

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

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

Mathematica [A]  time = 0.367827, size = 248, normalized size = 1.04 \[ \frac{18 a^2 c^3 x \left (3 d^2+3 d e x+e^2 x^2\right )-6 a b \sqrt{c^2 x^2+1} \left (c^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )-4 e^2\right )-6 b \sinh ^{-1}(c x) \left (b \sqrt{c^2 x^2+1} \left (c^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )-4 e^2\right )-3 a \left (2 c^3 x \left (3 d^2+3 d e x+e^2 x^2\right )+3 c d e\right )\right )+b^2 c x \left (c^2 \left (108 d^2+27 d e x+4 e^2 x^2\right )-24 e^2\right )+9 b^2 c \sinh ^{-1}(c x)^2 \left (6 c^2 d^2 x+3 d \left (2 c^2 e x^2+e\right )+2 c^2 e^2 x^3\right )}{54 c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(18*a^2*c^3*x*(3*d^2 + 3*d*e*x + e^2*x^2) - 6*a*b*Sqrt[1 + c^2*x^2]*(-4*e^2 + c^2*(18*d^2 + 9*d*e*x + 2*e^2*x^
2)) + b^2*c*x*(-24*e^2 + c^2*(108*d^2 + 27*d*e*x + 4*e^2*x^2)) - 6*b*(-3*a*(3*c*d*e + 2*c^3*x*(3*d^2 + 3*d*e*x
+ e^2*x^2)) + b*Sqrt[1 + c^2*x^2]*(-4*e^2 + c^2*(18*d^2 + 9*d*e*x + 2*e^2*x^2)))*ArcSinh[c*x] + 9*b^2*c*(6*c^
2*d^2*x + 2*c^2*e^2*x^3 + 3*d*(e + 2*c^2*e*x^2))*ArcSinh[c*x]^2)/(54*c^3)

________________________________________________________________________________________

Maple [A]  time = 0.061, size = 410, normalized size = 1.7 \begin{align*}{\frac{1}{c} \left ({\frac{ \left ( cex+cd \right ) ^{3}{a}^{2}}{3\,{c}^{2}e}}+{\frac{{b}^{2}}{{c}^{2}} \left ({c}^{2}{d}^{2} \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{cde}{2} \left ( 2\, \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}{c}^{2}{x}^{2}-2\,{\it Arcsinh} \left ( cx \right ) \sqrt{{c}^{2}{x}^{2}+1}cx+ \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}+{c}^{2}{x}^{2}+1 \right ) }+{\frac{{e}^{2}}{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}^{2} \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}{{c}^{2}} \left ( 1/3\,{e}^{2}{\it Arcsinh} \left ( cx \right ){c}^{3}{x}^{3}+e{\it Arcsinh} \left ( cx \right ){c}^{3}{x}^{2}d+{\it Arcsinh} \left ( cx \right ){c}^{3}x{d}^{2}+1/3\,{\frac{{c}^{3}{d}^{3}{\it Arcsinh} \left ( cx \right ) }{e}}-1/3\,{\frac{{e}^{3} \left ( 1/3\,{c}^{2}{x}^{2}\sqrt{{c}^{2}{x}^{2}+1}-2/3\,\sqrt{{c}^{2}{x}^{2}+1} \right ) +3\,cd{e}^{2} \left ( 1/2\,cx\sqrt{{c}^{2}{x}^{2}+1}-1/2\,{\it Arcsinh} \left ( cx \right ) \right ) +3\,{c}^{2}{d}^{2}e\sqrt{{c}^{2}{x}^{2}+1}+{c}^{3}{d}^{3}{\it Arcsinh} \left ( cx \right ) }{e}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.38478, size = 552, normalized size = 2.31 \begin{align*} \frac{1}{3} \, b^{2} e^{2} x^{3} \operatorname{arsinh}\left (c x\right )^{2} + b^{2} d e x^{2} \operatorname{arsinh}\left (c x\right )^{2} + \frac{1}{3} \, a^{2} e^{2} x^{3} + b^{2} d^{2} x \operatorname{arsinh}\left (c x\right )^{2} + a^{2} d e x^{2} +{\left (2 \, x^{2} \operatorname{arsinh}\left (c x\right ) - c{\left (\frac{\sqrt{c^{2} x^{2} + 1} x}{c^{2}} - \frac{\operatorname{arsinh}\left (\frac{c^{2} x}{\sqrt{c^{2}}}\right )}{\sqrt{c^{2}} c^{2}}\right )}\right )} a b d e + \frac{1}{2} \,{\left (c^{2}{\left (\frac{x^{2}}{c^{2}} - \frac{\log \left (\frac{c^{2} x}{\sqrt{c^{2}}} + \sqrt{c^{2} x^{2} + 1}\right )^{2}}{c^{4}}\right )} - 2 \, c{\left (\frac{\sqrt{c^{2} x^{2} + 1} x}{c^{2}} - \frac{\operatorname{arsinh}\left (\frac{c^{2} x}{\sqrt{c^{2}}}\right )}{\sqrt{c^{2}} c^{2}}\right )} \operatorname{arsinh}\left (c x\right )\right )} b^{2} d e + \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^{2} - \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} + 2 \, b^{2} d^{2}{\left (x - \frac{\sqrt{c^{2} x^{2} + 1} \operatorname{arsinh}\left (c x\right )}{c}\right )} + a^{2} d^{2} x + \frac{2 \,{\left (c x \operatorname{arsinh}\left (c x\right ) - \sqrt{c^{2} x^{2} + 1}\right )} a b d^{2}}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 2.57816, size = 687, normalized size = 2.87 \begin{align*} \frac{2 \,{\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{3} e^{2} x^{3} + 27 \,{\left (2 \, a^{2} + b^{2}\right )} c^{3} d e x^{2} + 9 \,{\left (2 \, b^{2} c^{3} e^{2} x^{3} + 6 \, b^{2} c^{3} d e x^{2} + 6 \, b^{2} c^{3} d^{2} x + 3 \, b^{2} c d e\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )^{2} + 6 \,{\left (9 \,{\left (a^{2} + 2 \, b^{2}\right )} c^{3} d^{2} - 4 \, b^{2} c e^{2}\right )} x + 6 \,{\left (6 \, a b c^{3} e^{2} x^{3} + 18 \, a b c^{3} d e x^{2} + 18 \, a b c^{3} d^{2} x + 9 \, a b c d e -{\left (2 \, b^{2} c^{2} e^{2} x^{2} + 9 \, b^{2} c^{2} d e x + 18 \, b^{2} c^{2} d^{2} - 4 \, b^{2} e^{2}\right )} \sqrt{c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) - 6 \,{\left (2 \, a b c^{2} e^{2} x^{2} + 9 \, a b c^{2} d e x + 18 \, a b c^{2} d^{2} - 4 \, a b e^{2}\right )} \sqrt{c^{2} x^{2} + 1}}{54 \, c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/54*(2*(9*a^2 + 2*b^2)*c^3*e^2*x^3 + 27*(2*a^2 + b^2)*c^3*d*e*x^2 + 9*(2*b^2*c^3*e^2*x^3 + 6*b^2*c^3*d*e*x^2
+ 6*b^2*c^3*d^2*x + 3*b^2*c*d*e)*log(c*x + sqrt(c^2*x^2 + 1))^2 + 6*(9*(a^2 + 2*b^2)*c^3*d^2 - 4*b^2*c*e^2)*x
+ 6*(6*a*b*c^3*e^2*x^3 + 18*a*b*c^3*d*e*x^2 + 18*a*b*c^3*d^2*x + 9*a*b*c*d*e - (2*b^2*c^2*e^2*x^2 + 9*b^2*c^2*
d*e*x + 18*b^2*c^2*d^2 - 4*b^2*e^2)*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1)) - 6*(2*a*b*c^2*e^2*x^2 + 9
*a*b*c^2*d*e*x + 18*a*b*c^2*d^2 - 4*a*b*e^2)*sqrt(c^2*x^2 + 1))/c^3

________________________________________________________________________________________

Sympy [A]  time = 2.32498, size = 454, normalized size = 1.9 \begin{align*} \begin{cases} a^{2} d^{2} x + a^{2} d e x^{2} + \frac{a^{2} e^{2} x^{3}}{3} + 2 a b d^{2} x \operatorname{asinh}{\left (c x \right )} + 2 a b d e x^{2} \operatorname{asinh}{\left (c x \right )} + \frac{2 a b e^{2} x^{3} \operatorname{asinh}{\left (c x \right )}}{3} - \frac{2 a b d^{2} \sqrt{c^{2} x^{2} + 1}}{c} - \frac{a b d e x \sqrt{c^{2} x^{2} + 1}}{c} - \frac{2 a b e^{2} x^{2} \sqrt{c^{2} x^{2} + 1}}{9 c} + \frac{a b d e \operatorname{asinh}{\left (c x \right )}}{c^{2}} + \frac{4 a b e^{2} \sqrt{c^{2} x^{2} + 1}}{9 c^{3}} + b^{2} d^{2} x \operatorname{asinh}^{2}{\left (c x \right )} + 2 b^{2} d^{2} x + b^{2} d e x^{2} \operatorname{asinh}^{2}{\left (c x \right )} + \frac{b^{2} d e x^{2}}{2} + \frac{b^{2} e^{2} x^{3} \operatorname{asinh}^{2}{\left (c x \right )}}{3} + \frac{2 b^{2} e^{2} x^{3}}{27} - \frac{2 b^{2} d^{2} \sqrt{c^{2} x^{2} + 1} \operatorname{asinh}{\left (c x \right )}}{c} - \frac{b^{2} d e x \sqrt{c^{2} x^{2} + 1} \operatorname{asinh}{\left (c x \right )}}{c} - \frac{2 b^{2} e^{2} x^{2} \sqrt{c^{2} x^{2} + 1} \operatorname{asinh}{\left (c x \right )}}{9 c} + \frac{b^{2} d e \operatorname{asinh}^{2}{\left (c x \right )}}{2 c^{2}} - \frac{4 b^{2} e^{2} x}{9 c^{2}} + \frac{4 b^{2} e^{2} \sqrt{c^{2} x^{2} + 1} \operatorname{asinh}{\left (c x \right )}}{9 c^{3}} & \text{for}\: c \neq 0 \\a^{2} \left (d^{2} x + d e x^{2} + \frac{e^{2} x^{3}}{3}\right ) & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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