3.1.13 \(\int (d+e x)^2 (a+b \sinh ^{-1}(c x))^2 \, dx\) [13]

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

[Out]

2*b^2*d^2*x-4/9*b^2*e^2*x/c^2+1/2*b^2*d*e*x^2+2/27*b^2*e^2*x^3-1/3*d^3*(a+b*arcsinh(c*x))^2/e+1/2*d*e*(a+b*arc
sinh(c*x))^2/c^2+1/3*(e*x+d)^3*(a+b*arcsinh(c*x))^2/e-2*b*d^2*(a+b*arcsinh(c*x))*(c^2*x^2+1)^(1/2)/c+4/9*b*e^2
*(a+b*arcsinh(c*x))*(c^2*x^2+1)^(1/2)/c^3-b*d*e*x*(a+b*arcsinh(c*x))*(c^2*x^2+1)^(1/2)/c-2/9*b*e^2*x^2*(a+b*ar
csinh(c*x))*(c^2*x^2+1)^(1/2)/c

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Rubi [A]
time = 0.32, antiderivative size = 239, normalized size of antiderivative = 1.00, 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 = {5828, 5838, 5783, 5798, 8, 5812, 30} \begin {gather*} -\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 {2 b e^2 x^2 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{9 c}+\frac {4 b e^2 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{9 c^3}-\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 \end {gather*}

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 8

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

Rule 30

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

Rule 5783

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

Rule 5798

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/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)
^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 5812

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

Rule 5828

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 5838

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]))

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*}

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Mathematica [A]
time = 0.19, size = 248, normalized size = 1.04 \begin {gather*} \frac {18 a^2 c^3 x \left (3 d^2+3 d e x+e^2 x^2\right )-6 a b \sqrt {1+c^2 x^2} \left (-4 e^2+c^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )\right )+b^2 c x \left (-24 e^2+c^2 \left (108 d^2+27 d e x+4 e^2 x^2\right )\right )-6 b \left (-3 a \left (3 c d e+2 c^3 x \left (3 d^2+3 d e x+e^2 x^2\right )\right )+b \sqrt {1+c^2 x^2} \left (-4 e^2+c^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )\right )\right ) \sinh ^{-1}(c x)+9 b^2 c \left (6 c^2 d^2 x+2 c^2 e^2 x^3+3 d \left (e+2 c^2 e x^2\right )\right ) \sinh ^{-1}(c x)^2}{54 c^3} \end {gather*}

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)

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Maple [A]
time = 2.28, size = 361, normalized size = 1.51 \[\frac {\frac {\left (c e x +c d \right )^{3} a^{2}}{3 c^{2} e}+\frac {b^{2} \left (-\frac {2 e^{2} \arcsinh \left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{9}+\frac {e \left (18 \arcsinh \left (c x \right )^{2} x c e +54 \arcsinh \left (c x \right )^{2} c d +4 c e x +27 c d \right ) \left (c^{2} x^{2}+1\right )}{54}-\frac {\arcsinh \left (c x \right ) \left (3 c^{2} d e x +6 c^{2} d^{2}-2 e^{2}\right ) \sqrt {c^{2} x^{2}+1}}{3}+\frac {\left (27 \arcsinh \left (c x \right )^{2} c^{2} d^{2}-9 \arcsinh \left (c x \right )^{2} e^{2}+54 c^{2} d^{2}-14 e^{2}\right ) x c}{27}-\frac {d e c \left (2 \arcsinh \left (c x \right )^{2}+1\right )}{4}\right )}{c^{2}}+\frac {2 a b \left (\frac {\arcsinh \left (c x \right ) c^{3} d^{3}}{3 e}+\arcsinh \left (c x \right ) c^{3} d^{2} x +e \arcsinh \left (c x \right ) c^{3} d \,x^{2}+\frac {e^{2} \arcsinh \left (c x \right ) c^{3} x^{3}}{3}-\frac {c^{3} d^{3} \arcsinh \left (c x \right )+3 d^{2} c^{2} e \sqrt {c^{2} x^{2}+1}+3 d c \,e^{2} \left (\frac {\sqrt {c^{2} x^{2}+1}\, c x}{2}-\frac {\arcsinh \left (c x \right )}{2}\right )+e^{3} \left (\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {c^{2} x^{2}+1}}{3}\right )}{3 e}\right )}{c^{2}}}{c}\]

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*(-2/9*e^2*arcsinh(c*x)*(c^2*x^2+1)^(3/2)+1/54*e*(18*arcsinh(c*x)^2*x*
c*e+54*arcsinh(c*x)^2*c*d+4*c*e*x+27*c*d)*(c^2*x^2+1)-1/3*arcsinh(c*x)*(3*c^2*d*e*x+6*c^2*d^2-2*e^2)*(c^2*x^2+
1)^(1/2)+1/27*(27*arcsinh(c*x)^2*c^2*d^2-9*arcsinh(c*x)^2*e^2+54*c^2*d^2-14*e^2)*x*c-1/4*d*e*c*(2*arcsinh(c*x)
^2+1))+2*a*b/c^2*(1/3*e^2*arcsinh(c*x)*x^3*c^3+e*arcsinh(c*x)*x^2*c^3*d+arcsinh(c*x)*x*c^3*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^2*x^2+1)^(1/2)*
c*x-1/2*arcsinh(c*x))+3*d^2*c^2*e*(c^2*x^2+1)^(1/2)+c^3*d^3*arcsinh(c*x))))

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Maxima [A]
time = 0.30, size = 378, normalized size = 1.58 \begin {gather*} \frac {1}{3} \, b^{2} x^{3} \operatorname {arsinh}\left (c x\right )^{2} e^{2} + b^{2} d x^{2} \operatorname {arsinh}\left (c x\right )^{2} e + b^{2} d^{2} x \operatorname {arsinh}\left (c x\right )^{2} + \frac {1}{3} \, a^{2} x^{3} e^{2} + a^{2} d x^{2} e + 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 + {\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 (c x\right )}{c^{3}}\right )}\right )} a b d e + \frac {1}{2} \, {\left (c^{2} {\left (\frac {x^{2}}{c^{2}} - \frac {\log \left (c x + \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 (c x\right )}{c^{3}}\right )} \operatorname {arsinh}\left (c x\right )\right )} b^{2} d e + \frac {2 \, {\left (c x \operatorname {arsinh}\left (c x\right ) - \sqrt {c^{2} x^{2} + 1}\right )} a b d^{2}}{c} + \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} \end {gather*}

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 616 vs. \(2 (213) = 426\).
time = 0.34, size = 616, normalized size = 2.58 \begin {gather*} \frac {27 \, {\left (2 \, a^{2} + b^{2}\right )} c^{3} d x^{2} \cosh \left (1\right ) + 54 \, {\left (a^{2} + 2 \, b^{2}\right )} c^{3} d^{2} x + 2 \, {\left ({\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{3} x^{3} - 12 \, b^{2} c x\right )} \cosh \left (1\right )^{2} + 9 \, {\left (2 \, b^{2} c^{3} x^{3} \cosh \left (1\right )^{2} + 2 \, b^{2} c^{3} x^{3} \sinh \left (1\right )^{2} + 6 \, b^{2} c^{3} d^{2} x + 3 \, {\left (2 \, b^{2} c^{3} d x^{2} + b^{2} c d\right )} \cosh \left (1\right ) + {\left (4 \, b^{2} c^{3} x^{3} \cosh \left (1\right ) + 6 \, b^{2} c^{3} d x^{2} + 3 \, b^{2} c d\right )} \sinh \left (1\right )\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 2 \, {\left ({\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{3} x^{3} - 12 \, b^{2} c x\right )} \sinh \left (1\right )^{2} + 6 \, {\left (6 \, a b c^{3} x^{3} \cosh \left (1\right )^{2} + 6 \, a b c^{3} x^{3} \sinh \left (1\right )^{2} + 18 \, a b c^{3} d^{2} x + 9 \, {\left (2 \, a b c^{3} d x^{2} + a b c d\right )} \cosh \left (1\right ) + 3 \, {\left (4 \, a b c^{3} x^{3} \cosh \left (1\right ) + 6 \, a b c^{3} d x^{2} + 3 \, a b c d\right )} \sinh \left (1\right ) - {\left (9 \, b^{2} c^{2} d x \cosh \left (1\right ) + 18 \, b^{2} c^{2} d^{2} + 2 \, {\left (b^{2} c^{2} x^{2} - 2 \, b^{2}\right )} \cosh \left (1\right )^{2} + 2 \, {\left (b^{2} c^{2} x^{2} - 2 \, b^{2}\right )} \sinh \left (1\right )^{2} + {\left (9 \, b^{2} c^{2} d x + 4 \, {\left (b^{2} c^{2} x^{2} - 2 \, b^{2}\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + {\left (27 \, {\left (2 \, a^{2} + b^{2}\right )} c^{3} d x^{2} + 4 \, {\left ({\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{3} x^{3} - 12 \, b^{2} c x\right )} \cosh \left (1\right )\right )} \sinh \left (1\right ) - 6 \, {\left (9 \, a b c^{2} d x \cosh \left (1\right ) + 18 \, a b c^{2} d^{2} + 2 \, {\left (a b c^{2} x^{2} - 2 \, a b\right )} \cosh \left (1\right )^{2} + 2 \, {\left (a b c^{2} x^{2} - 2 \, a b\right )} \sinh \left (1\right )^{2} + {\left (9 \, a b c^{2} d x + 4 \, {\left (a b c^{2} x^{2} - 2 \, a b\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \sqrt {c^{2} x^{2} + 1}}{54 \, c^{3}} \end {gather*}

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*(27*(2*a^2 + b^2)*c^3*d*x^2*cosh(1) + 54*(a^2 + 2*b^2)*c^3*d^2*x + 2*((9*a^2 + 2*b^2)*c^3*x^3 - 12*b^2*c*
x)*cosh(1)^2 + 9*(2*b^2*c^3*x^3*cosh(1)^2 + 2*b^2*c^3*x^3*sinh(1)^2 + 6*b^2*c^3*d^2*x + 3*(2*b^2*c^3*d*x^2 + b
^2*c*d)*cosh(1) + (4*b^2*c^3*x^3*cosh(1) + 6*b^2*c^3*d*x^2 + 3*b^2*c*d)*sinh(1))*log(c*x + sqrt(c^2*x^2 + 1))^
2 + 2*((9*a^2 + 2*b^2)*c^3*x^3 - 12*b^2*c*x)*sinh(1)^2 + 6*(6*a*b*c^3*x^3*cosh(1)^2 + 6*a*b*c^3*x^3*sinh(1)^2
+ 18*a*b*c^3*d^2*x + 9*(2*a*b*c^3*d*x^2 + a*b*c*d)*cosh(1) + 3*(4*a*b*c^3*x^3*cosh(1) + 6*a*b*c^3*d*x^2 + 3*a*
b*c*d)*sinh(1) - (9*b^2*c^2*d*x*cosh(1) + 18*b^2*c^2*d^2 + 2*(b^2*c^2*x^2 - 2*b^2)*cosh(1)^2 + 2*(b^2*c^2*x^2
- 2*b^2)*sinh(1)^2 + (9*b^2*c^2*d*x + 4*(b^2*c^2*x^2 - 2*b^2)*cosh(1))*sinh(1))*sqrt(c^2*x^2 + 1))*log(c*x + s
qrt(c^2*x^2 + 1)) + (27*(2*a^2 + b^2)*c^3*d*x^2 + 4*((9*a^2 + 2*b^2)*c^3*x^3 - 12*b^2*c*x)*cosh(1))*sinh(1) -
6*(9*a*b*c^2*d*x*cosh(1) + 18*a*b*c^2*d^2 + 2*(a*b*c^2*x^2 - 2*a*b)*cosh(1)^2 + 2*(a*b*c^2*x^2 - 2*a*b)*sinh(1
)^2 + (9*a*b*c^2*d*x + 4*(a*b*c^2*x^2 - 2*a*b)*cosh(1))*sinh(1))*sqrt(c^2*x^2 + 1))/c^3

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Sympy [A]
time = 0.31, size = 454, normalized size = 1.90 \begin {gather*} \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 {gather*}

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))

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

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]

Exception raised: RuntimeError >> An error occurred running a Giac command:INPUT:sage2OUTPUT:sym2poly/r2sym(co
nst gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d+e\,x\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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