3.7.14 \(\int (d+e x^2)^2 (a+b \sinh ^{-1}(c x))^2 \, dx\) [614]

Optimal. Leaf size=329 \[ 2 b^2 d^2 x-\frac {8 b^2 d e x}{9 c^2}+\frac {16 b^2 e^2 x}{75 c^4}+\frac {4}{27} b^2 d e x^3-\frac {8 b^2 e^2 x^3}{225 c^2}+\frac {2}{125} b^2 e^2 x^5-\frac {2 b d^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c}+\frac {8 b d e \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c^3}-\frac {16 b e^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{75 c^5}-\frac {4 b d e x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c}+\frac {8 b e^2 x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{75 c^3}-\frac {2 b e^2 x^4 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{25 c}+d^2 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {2}{3} d e x^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{5} e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )^2 \]

[Out]

2*b^2*d^2*x-8/9*b^2*d*e*x/c^2+16/75*b^2*e^2*x/c^4+4/27*b^2*d*e*x^3-8/225*b^2*e^2*x^3/c^2+2/125*b^2*e^2*x^5+d^2
*x*(a+b*arcsinh(c*x))^2+2/3*d*e*x^3*(a+b*arcsinh(c*x))^2+1/5*e^2*x^5*(a+b*arcsinh(c*x))^2-2*b*d^2*(a+b*arcsinh
(c*x))*(c^2*x^2+1)^(1/2)/c+8/9*b*d*e*(a+b*arcsinh(c*x))*(c^2*x^2+1)^(1/2)/c^3-16/75*b*e^2*(a+b*arcsinh(c*x))*(
c^2*x^2+1)^(1/2)/c^5-4/9*b*d*e*x^2*(a+b*arcsinh(c*x))*(c^2*x^2+1)^(1/2)/c+8/75*b*e^2*x^2*(a+b*arcsinh(c*x))*(c
^2*x^2+1)^(1/2)/c^3-2/25*b*e^2*x^4*(a+b*arcsinh(c*x))*(c^2*x^2+1)^(1/2)/c

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Rubi [A]
time = 0.41, antiderivative size = 329, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {5793, 5772, 5798, 8, 5776, 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 {4 b d e x^2 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{9 c}-\frac {2 b e^2 x^4 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{25 c}-\frac {16 b e^2 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{75 c^5}+\frac {8 b d e \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{9 c^3}+\frac {8 b e^2 x^2 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{75 c^3}+d^2 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {2}{3} d e x^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{5} e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {16 b^2 e^2 x}{75 c^4}-\frac {8 b^2 d e x}{9 c^2}-\frac {8 b^2 e^2 x^3}{225 c^2}+2 b^2 d^2 x+\frac {4}{27} b^2 d e x^3+\frac {2}{125} b^2 e^2 x^5 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2*b^2*d^2*x - (8*b^2*d*e*x)/(9*c^2) + (16*b^2*e^2*x)/(75*c^4) + (4*b^2*d*e*x^3)/27 - (8*b^2*e^2*x^3)/(225*c^2)
+ (2*b^2*e^2*x^5)/125 - (2*b*d^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/c + (8*b*d*e*Sqrt[1 + c^2*x^2]*(a +
b*ArcSinh[c*x]))/(9*c^3) - (16*b*e^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(75*c^5) - (4*b*d*e*x^2*Sqrt[1 +
c^2*x^2]*(a + b*ArcSinh[c*x]))/(9*c) + (8*b*e^2*x^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(75*c^3) - (2*b*e^
2*x^4*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(25*c) + d^2*x*(a + b*ArcSinh[c*x])^2 + (2*d*e*x^3*(a + b*ArcSin
h[c*x])^2)/3 + (e^2*x^5*(a + b*ArcSinh[c*x])^2)/5

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 5772

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 5776

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 5793

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 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
]

Rubi steps

\begin {align*} \int \left (d+e x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx &=\int \left (d^2 \left (a+b \sinh ^{-1}(c x)\right )^2+2 d e x^2 \left (a+b \sinh ^{-1}(c x)\right )^2+e^2 x^4 \left (a+b \sinh ^{-1}(c x)\right )^2\right ) \, dx\\ &=d^2 \int \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx+(2 d e) \int x^2 \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx+e^2 \int x^4 \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx\\ &=d^2 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {2}{3} d e x^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{5} e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )^2-\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 {1}{3} (4 b c d e) \int \frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx-\frac {1}{5} \left (2 b c e^2\right ) \int \frac {x^5 \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 {4 b d e x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c}-\frac {2 b e^2 x^4 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{25 c}+d^2 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {2}{3} d e x^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{5} e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )^2+\left (2 b^2 d^2\right ) \int 1 \, dx+\frac {1}{9} \left (4 b^2 d e\right ) \int x^2 \, dx+\frac {(8 b d e) \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx}{9 c}+\frac {1}{25} \left (2 b^2 e^2\right ) \int x^4 \, dx+\frac {\left (8 b e^2\right ) \int \frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx}{25 c}\\ &=2 b^2 d^2 x+\frac {4}{27} b^2 d e x^3+\frac {2}{125} b^2 e^2 x^5-\frac {2 b d^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c}+\frac {8 b d e \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c^3}-\frac {4 b d e x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c}+\frac {8 b e^2 x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{75 c^3}-\frac {2 b e^2 x^4 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{25 c}+d^2 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {2}{3} d e x^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{5} e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {\left (8 b^2 d e\right ) \int 1 \, dx}{9 c^2}-\frac {\left (16 b e^2\right ) \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx}{75 c^3}-\frac {\left (8 b^2 e^2\right ) \int x^2 \, dx}{75 c^2}\\ &=2 b^2 d^2 x-\frac {8 b^2 d e x}{9 c^2}+\frac {4}{27} b^2 d e x^3-\frac {8 b^2 e^2 x^3}{225 c^2}+\frac {2}{125} b^2 e^2 x^5-\frac {2 b d^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c}+\frac {8 b d e \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c^3}-\frac {16 b e^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{75 c^5}-\frac {4 b d e x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c}+\frac {8 b e^2 x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{75 c^3}-\frac {2 b e^2 x^4 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{25 c}+d^2 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {2}{3} d e x^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{5} e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {\left (16 b^2 e^2\right ) \int 1 \, dx}{75 c^4}\\ &=2 b^2 d^2 x-\frac {8 b^2 d e x}{9 c^2}+\frac {16 b^2 e^2 x}{75 c^4}+\frac {4}{27} b^2 d e x^3-\frac {8 b^2 e^2 x^3}{225 c^2}+\frac {2}{125} b^2 e^2 x^5-\frac {2 b d^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c}+\frac {8 b d e \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c^3}-\frac {16 b e^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{75 c^5}-\frac {4 b d e x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c}+\frac {8 b e^2 x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{75 c^3}-\frac {2 b e^2 x^4 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{25 c}+d^2 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {2}{3} d e x^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{5} e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )^2\\ \end {align*}

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Mathematica [A]
time = 0.23, size = 289, normalized size = 0.88 \begin {gather*} \frac {225 a^2 c^5 x \left (15 d^2+10 d e x^2+3 e^2 x^4\right )-30 a b \sqrt {1+c^2 x^2} \left (24 e^2-4 c^2 e \left (25 d+3 e x^2\right )+c^4 \left (225 d^2+50 d e x^2+9 e^2 x^4\right )\right )+2 b^2 c x \left (360 e^2-60 c^2 e \left (25 d+e x^2\right )+c^4 \left (3375 d^2+250 d e x^2+27 e^2 x^4\right )\right )-30 b \left (-15 a c^5 x \left (15 d^2+10 d e x^2+3 e^2 x^4\right )+b \sqrt {1+c^2 x^2} \left (24 e^2-4 c^2 e \left (25 d+3 e x^2\right )+c^4 \left (225 d^2+50 d e x^2+9 e^2 x^4\right )\right )\right ) \sinh ^{-1}(c x)+225 b^2 c^5 x \left (15 d^2+10 d e x^2+3 e^2 x^4\right ) \sinh ^{-1}(c x)^2}{3375 c^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(225*a^2*c^5*x*(15*d^2 + 10*d*e*x^2 + 3*e^2*x^4) - 30*a*b*Sqrt[1 + c^2*x^2]*(24*e^2 - 4*c^2*e*(25*d + 3*e*x^2)
+ c^4*(225*d^2 + 50*d*e*x^2 + 9*e^2*x^4)) + 2*b^2*c*x*(360*e^2 - 60*c^2*e*(25*d + e*x^2) + c^4*(3375*d^2 + 25
0*d*e*x^2 + 27*e^2*x^4)) - 30*b*(-15*a*c^5*x*(15*d^2 + 10*d*e*x^2 + 3*e^2*x^4) + b*Sqrt[1 + c^2*x^2]*(24*e^2 -
4*c^2*e*(25*d + 3*e*x^2) + c^4*(225*d^2 + 50*d*e*x^2 + 9*e^2*x^4)))*ArcSinh[c*x] + 225*b^2*c^5*x*(15*d^2 + 10
*d*e*x^2 + 3*e^2*x^4)*ArcSinh[c*x]^2)/(3375*c^5)

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Maple [A]
time = 1.75, size = 438, normalized size = 1.33 \[\frac {\frac {a^{2} \left (d^{2} c^{5} x +\frac {2}{3} d \,c^{5} e \,x^{3}+\frac {1}{5} e^{2} c^{5} x^{5}\right )}{c^{4}}+\frac {b^{2} \left (-\frac {2 e^{2} \arcsinh \left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{25}+\frac {e^{2} x c \left (25 \arcsinh \left (c x \right )^{2}+2\right ) \left (c^{2} x^{2}+1\right )^{2}}{125}-\frac {4 e \arcsinh \left (c x \right ) \left (5 c^{2} d -3 e \right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{45}+\frac {2 \left (1125 \arcsinh \left (c x \right )^{2} c^{2} d -675 \arcsinh \left (c x \right )^{2} e +250 c^{2} d -114 e \right ) e x c \left (c^{2} x^{2}+1\right )}{3375}-\frac {2 \arcsinh \left (c x \right ) \left (15 c^{4} d^{2}-10 c^{2} d e +3 e^{2}\right ) \sqrt {c^{2} x^{2}+1}}{15}+\frac {\left (3375 \arcsinh \left (c x \right )^{2} c^{4} d^{2}-2250 \arcsinh \left (c x \right )^{2} c^{2} d e +6750 c^{4} d^{2}+675 \arcsinh \left (c x \right )^{2} e^{2}-3500 c^{2} d e +894 e^{2}\right ) x c}{3375}\right )}{c^{4}}+\frac {2 a b \left (\arcsinh \left (c x \right ) d^{2} c^{5} x +\frac {2 \arcsinh \left (c x \right ) d \,c^{5} e \,x^{3}}{3}+\frac {\arcsinh \left (c x \right ) e^{2} c^{5} x^{5}}{5}-d^{2} c^{4} \sqrt {c^{2} x^{2}+1}-\frac {2 d \,c^{2} e \left (\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {c^{2} x^{2}+1}}{3}\right )}{3}-\frac {e^{2} \left (\frac {\sqrt {c^{2} x^{2}+1}\, c^{4} x^{4}}{5}-\frac {4 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{15}+\frac {8 \sqrt {c^{2} x^{2}+1}}{15}\right )}{5}\right )}{c^{4}}}{c}\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c*(a^2/c^4*(d^2*c^5*x+2/3*d*c^5*e*x^3+1/5*e^2*c^5*x^5)+1/c^4*b^2*(-2/25*e^2*arcsinh(c*x)*(c^2*x^2+1)^(5/2)+1
/125*e^2*x*c*(25*arcsinh(c*x)^2+2)*(c^2*x^2+1)^2-4/45*e*arcsinh(c*x)*(5*c^2*d-3*e)*(c^2*x^2+1)^(3/2)+2/3375*(1
125*arcsinh(c*x)^2*c^2*d-675*arcsinh(c*x)^2*e+250*c^2*d-114*e)*e*x*c*(c^2*x^2+1)-2/15*arcsinh(c*x)*(15*c^4*d^2
-10*c^2*d*e+3*e^2)*(c^2*x^2+1)^(1/2)+1/3375*(3375*arcsinh(c*x)^2*c^4*d^2-2250*arcsinh(c*x)^2*c^2*d*e+6750*c^4*
d^2+675*arcsinh(c*x)^2*e^2-3500*c^2*d*e+894*e^2)*x*c)+2*a*b/c^4*(1/5*arcsinh(c*x)*e^2*x^5*c^5+2/3*arcsinh(c*x)
*c^5*d*e*x^3+arcsinh(c*x)*x*c^5*d^2-1/5*e^2*(1/5*(c^2*x^2+1)^(1/2)*c^4*x^4-4/15*c^2*x^2*(c^2*x^2+1)^(1/2)+8/15
*(c^2*x^2+1)^(1/2))-2/3*d*c^2*e*(1/3*c^2*x^2*(c^2*x^2+1)^(1/2)-2/3*(c^2*x^2+1)^(1/2))-d^2*c^4*(c^2*x^2+1)^(1/2
)))

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Maxima [A]
time = 0.27, size = 429, normalized size = 1.30 \begin {gather*} \frac {1}{5} \, b^{2} x^{5} \operatorname {arsinh}\left (c x\right )^{2} e^{2} + \frac {2}{3} \, b^{2} d x^{3} \operatorname {arsinh}\left (c x\right )^{2} e + \frac {1}{5} \, a^{2} x^{5} e^{2} + b^{2} d^{2} x \operatorname {arsinh}\left (c x\right )^{2} + \frac {2}{3} \, a^{2} d x^{3} 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 + \frac {4}{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 d e - \frac {4}{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} 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}{75} \, {\left (15 \, x^{5} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{2}} - \frac {4 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} a b e^{2} - \frac {2}{1125} \, {\left (15 \, {\left (\frac {3 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{2}} - \frac {4 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1}}{c^{6}}\right )} c \operatorname {arsinh}\left (c x\right ) - \frac {9 \, c^{4} x^{5} - 20 \, c^{2} x^{3} + 120 \, x}{c^{4}}\right )} b^{2} e^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*b^2*x^5*arcsinh(c*x)^2*e^2 + 2/3*b^2*d*x^3*arcsinh(c*x)^2*e + 1/5*a^2*x^5*e^2 + b^2*d^2*x*arcsinh(c*x)^2 +
2/3*a^2*d*x^3*e + 2*b^2*d^2*(x - sqrt(c^2*x^2 + 1)*arcsinh(c*x)/c) + a^2*d^2*x + 4/9*(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*d*e - 4/27*(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*d*e + 2*(c*x*arcsinh(c*x) - sqrt(c^2*x^2 + 1))*a*b*
d^2/c + 2/75*(15*x^5*arcsinh(c*x) - (3*sqrt(c^2*x^2 + 1)*x^4/c^2 - 4*sqrt(c^2*x^2 + 1)*x^2/c^4 + 8*sqrt(c^2*x^
2 + 1)/c^6)*c)*a*b*e^2 - 2/1125*(15*(3*sqrt(c^2*x^2 + 1)*x^4/c^2 - 4*sqrt(c^2*x^2 + 1)*x^2/c^4 + 8*sqrt(c^2*x^
2 + 1)/c^6)*c*arcsinh(c*x) - (9*c^4*x^5 - 20*c^2*x^3 + 120*x)/c^4)*b^2*e^2

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 767 vs. \(2 (291) = 582\).
time = 0.41, size = 767, normalized size = 2.33 \begin {gather*} \frac {3375 \, {\left (a^{2} + 2 \, b^{2}\right )} c^{5} d^{2} x + 3 \, {\left (9 \, {\left (25 \, a^{2} + 2 \, b^{2}\right )} c^{5} x^{5} - 40 \, b^{2} c^{3} x^{3} + 240 \, b^{2} c x\right )} \cosh \left (1\right )^{2} + 225 \, {\left (3 \, b^{2} c^{5} x^{5} \cosh \left (1\right )^{2} + 3 \, b^{2} c^{5} x^{5} \sinh \left (1\right )^{2} + 10 \, b^{2} c^{5} d x^{3} \cosh \left (1\right ) + 15 \, b^{2} c^{5} d^{2} x + 2 \, {\left (3 \, b^{2} c^{5} x^{5} \cosh \left (1\right ) + 5 \, b^{2} c^{5} d x^{3}\right )} \sinh \left (1\right )\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 3 \, {\left (9 \, {\left (25 \, a^{2} + 2 \, b^{2}\right )} c^{5} x^{5} - 40 \, b^{2} c^{3} x^{3} + 240 \, b^{2} c x\right )} \sinh \left (1\right )^{2} + 250 \, {\left ({\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{5} d x^{3} - 12 \, b^{2} c^{3} d x\right )} \cosh \left (1\right ) + 30 \, {\left (45 \, a b c^{5} x^{5} \cosh \left (1\right )^{2} + 45 \, a b c^{5} x^{5} \sinh \left (1\right )^{2} + 150 \, a b c^{5} d x^{3} \cosh \left (1\right ) + 225 \, a b c^{5} d^{2} x + 30 \, {\left (3 \, a b c^{5} x^{5} \cosh \left (1\right ) + 5 \, a b c^{5} d x^{3}\right )} \sinh \left (1\right ) - {\left (225 \, b^{2} c^{4} d^{2} + 3 \, {\left (3 \, b^{2} c^{4} x^{4} - 4 \, b^{2} c^{2} x^{2} + 8 \, b^{2}\right )} \cosh \left (1\right )^{2} + 3 \, {\left (3 \, b^{2} c^{4} x^{4} - 4 \, b^{2} c^{2} x^{2} + 8 \, b^{2}\right )} \sinh \left (1\right )^{2} + 50 \, {\left (b^{2} c^{4} d x^{2} - 2 \, b^{2} c^{2} d\right )} \cosh \left (1\right ) + 2 \, {\left (25 \, b^{2} c^{4} d x^{2} - 50 \, b^{2} c^{2} d + 3 \, {\left (3 \, b^{2} c^{4} x^{4} - 4 \, b^{2} c^{2} x^{2} + 8 \, 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 ) + 2 \, {\left (125 \, {\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{5} d x^{3} - 1500 \, b^{2} c^{3} d x + 3 \, {\left (9 \, {\left (25 \, a^{2} + 2 \, b^{2}\right )} c^{5} x^{5} - 40 \, b^{2} c^{3} x^{3} + 240 \, b^{2} c x\right )} \cosh \left (1\right )\right )} \sinh \left (1\right ) - 30 \, {\left (225 \, a b c^{4} d^{2} + 3 \, {\left (3 \, a b c^{4} x^{4} - 4 \, a b c^{2} x^{2} + 8 \, a b\right )} \cosh \left (1\right )^{2} + 3 \, {\left (3 \, a b c^{4} x^{4} - 4 \, a b c^{2} x^{2} + 8 \, a b\right )} \sinh \left (1\right )^{2} + 50 \, {\left (a b c^{4} d x^{2} - 2 \, a b c^{2} d\right )} \cosh \left (1\right ) + 2 \, {\left (25 \, a b c^{4} d x^{2} - 50 \, a b c^{2} d + 3 \, {\left (3 \, a b c^{4} x^{4} - 4 \, a b c^{2} x^{2} + 8 \, a b\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \sqrt {c^{2} x^{2} + 1}}{3375 \, c^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3375*(3375*(a^2 + 2*b^2)*c^5*d^2*x + 3*(9*(25*a^2 + 2*b^2)*c^5*x^5 - 40*b^2*c^3*x^3 + 240*b^2*c*x)*cosh(1)^2
+ 225*(3*b^2*c^5*x^5*cosh(1)^2 + 3*b^2*c^5*x^5*sinh(1)^2 + 10*b^2*c^5*d*x^3*cosh(1) + 15*b^2*c^5*d^2*x + 2*(3
*b^2*c^5*x^5*cosh(1) + 5*b^2*c^5*d*x^3)*sinh(1))*log(c*x + sqrt(c^2*x^2 + 1))^2 + 3*(9*(25*a^2 + 2*b^2)*c^5*x^
5 - 40*b^2*c^3*x^3 + 240*b^2*c*x)*sinh(1)^2 + 250*((9*a^2 + 2*b^2)*c^5*d*x^3 - 12*b^2*c^3*d*x)*cosh(1) + 30*(4
5*a*b*c^5*x^5*cosh(1)^2 + 45*a*b*c^5*x^5*sinh(1)^2 + 150*a*b*c^5*d*x^3*cosh(1) + 225*a*b*c^5*d^2*x + 30*(3*a*b
*c^5*x^5*cosh(1) + 5*a*b*c^5*d*x^3)*sinh(1) - (225*b^2*c^4*d^2 + 3*(3*b^2*c^4*x^4 - 4*b^2*c^2*x^2 + 8*b^2)*cos
h(1)^2 + 3*(3*b^2*c^4*x^4 - 4*b^2*c^2*x^2 + 8*b^2)*sinh(1)^2 + 50*(b^2*c^4*d*x^2 - 2*b^2*c^2*d)*cosh(1) + 2*(2
5*b^2*c^4*d*x^2 - 50*b^2*c^2*d + 3*(3*b^2*c^4*x^4 - 4*b^2*c^2*x^2 + 8*b^2)*cosh(1))*sinh(1))*sqrt(c^2*x^2 + 1)
)*log(c*x + sqrt(c^2*x^2 + 1)) + 2*(125*(9*a^2 + 2*b^2)*c^5*d*x^3 - 1500*b^2*c^3*d*x + 3*(9*(25*a^2 + 2*b^2)*c
^5*x^5 - 40*b^2*c^3*x^3 + 240*b^2*c*x)*cosh(1))*sinh(1) - 30*(225*a*b*c^4*d^2 + 3*(3*a*b*c^4*x^4 - 4*a*b*c^2*x
^2 + 8*a*b)*cosh(1)^2 + 3*(3*a*b*c^4*x^4 - 4*a*b*c^2*x^2 + 8*a*b)*sinh(1)^2 + 50*(a*b*c^4*d*x^2 - 2*a*b*c^2*d)
*cosh(1) + 2*(25*a*b*c^4*d*x^2 - 50*a*b*c^2*d + 3*(3*a*b*c^4*x^4 - 4*a*b*c^2*x^2 + 8*a*b)*cosh(1))*sinh(1))*sq
rt(c^2*x^2 + 1))/c^5

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Sympy [A]
time = 0.58, size = 595, normalized size = 1.81 \begin {gather*} \begin {cases} a^{2} d^{2} x + \frac {2 a^{2} d e x^{3}}{3} + \frac {a^{2} e^{2} x^{5}}{5} + 2 a b d^{2} x \operatorname {asinh}{\left (c x \right )} + \frac {4 a b d e x^{3} \operatorname {asinh}{\left (c x \right )}}{3} + \frac {2 a b e^{2} x^{5} \operatorname {asinh}{\left (c x \right )}}{5} - \frac {2 a b d^{2} \sqrt {c^{2} x^{2} + 1}}{c} - \frac {4 a b d e x^{2} \sqrt {c^{2} x^{2} + 1}}{9 c} - \frac {2 a b e^{2} x^{4} \sqrt {c^{2} x^{2} + 1}}{25 c} + \frac {8 a b d e \sqrt {c^{2} x^{2} + 1}}{9 c^{3}} + \frac {8 a b e^{2} x^{2} \sqrt {c^{2} x^{2} + 1}}{75 c^{3}} - \frac {16 a b e^{2} \sqrt {c^{2} x^{2} + 1}}{75 c^{5}} + b^{2} d^{2} x \operatorname {asinh}^{2}{\left (c x \right )} + 2 b^{2} d^{2} x + \frac {2 b^{2} d e x^{3} \operatorname {asinh}^{2}{\left (c x \right )}}{3} + \frac {4 b^{2} d e x^{3}}{27} + \frac {b^{2} e^{2} x^{5} \operatorname {asinh}^{2}{\left (c x \right )}}{5} + \frac {2 b^{2} e^{2} x^{5}}{125} - \frac {2 b^{2} d^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{c} - \frac {4 b^{2} d e x^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{9 c} - \frac {2 b^{2} e^{2} x^{4} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{25 c} - \frac {8 b^{2} d e x}{9 c^{2}} - \frac {8 b^{2} e^{2} x^{3}}{225 c^{2}} + \frac {8 b^{2} d e \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{9 c^{3}} + \frac {8 b^{2} e^{2} x^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{75 c^{3}} + \frac {16 b^{2} e^{2} x}{75 c^{4}} - \frac {16 b^{2} e^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{75 c^{5}} & \text {for}\: c \neq 0 \\a^{2} \left (d^{2} x + \frac {2 d e x^{3}}{3} + \frac {e^{2} x^{5}}{5}\right ) & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**2*d**2*x + 2*a**2*d*e*x**3/3 + a**2*e**2*x**5/5 + 2*a*b*d**2*x*asinh(c*x) + 4*a*b*d*e*x**3*asinh
(c*x)/3 + 2*a*b*e**2*x**5*asinh(c*x)/5 - 2*a*b*d**2*sqrt(c**2*x**2 + 1)/c - 4*a*b*d*e*x**2*sqrt(c**2*x**2 + 1)
/(9*c) - 2*a*b*e**2*x**4*sqrt(c**2*x**2 + 1)/(25*c) + 8*a*b*d*e*sqrt(c**2*x**2 + 1)/(9*c**3) + 8*a*b*e**2*x**2
*sqrt(c**2*x**2 + 1)/(75*c**3) - 16*a*b*e**2*sqrt(c**2*x**2 + 1)/(75*c**5) + b**2*d**2*x*asinh(c*x)**2 + 2*b**
2*d**2*x + 2*b**2*d*e*x**3*asinh(c*x)**2/3 + 4*b**2*d*e*x**3/27 + b**2*e**2*x**5*asinh(c*x)**2/5 + 2*b**2*e**2
*x**5/125 - 2*b**2*d**2*sqrt(c**2*x**2 + 1)*asinh(c*x)/c - 4*b**2*d*e*x**2*sqrt(c**2*x**2 + 1)*asinh(c*x)/(9*c
) - 2*b**2*e**2*x**4*sqrt(c**2*x**2 + 1)*asinh(c*x)/(25*c) - 8*b**2*d*e*x/(9*c**2) - 8*b**2*e**2*x**3/(225*c**
2) + 8*b**2*d*e*sqrt(c**2*x**2 + 1)*asinh(c*x)/(9*c**3) + 8*b**2*e**2*x**2*sqrt(c**2*x**2 + 1)*asinh(c*x)/(75*
c**3) + 16*b**2*e**2*x/(75*c**4) - 16*b**2*e**2*sqrt(c**2*x**2 + 1)*asinh(c*x)/(75*c**5), Ne(c, 0)), (a**2*(d*
*2*x + 2*d*e*x**3/3 + e**2*x**5/5), 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^2+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 (e\,x^2+d\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)^2,x)

[Out]

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

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