3.2.29 \(\int (c e+d e x)^2 (a+b \sinh ^{-1}(c+d x))^2 \, dx\) [129]

Optimal. Leaf size=136 \[ -\frac {4}{9} b^2 e^2 x+\frac {2 b^2 e^2 (c+d x)^3}{27 d}+\frac {4 b e^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )}{9 d}-\frac {2 b e^2 (c+d x)^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )}{9 d}+\frac {e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^2}{3 d} \]

[Out]

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

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Rubi [A]
time = 0.14, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {5859, 12, 5776, 5812, 5798, 8, 30} \begin {gather*} \frac {e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^2}{3 d}-\frac {2 b e^2 \sqrt {(c+d x)^2+1} (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )}{9 d}+\frac {4 b e^2 \sqrt {(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )}{9 d}+\frac {2 b^2 e^2 (c+d x)^3}{27 d}-\frac {4}{9} b^2 e^2 x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 8

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 30

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

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 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 5859

Int[((a_.) + ArcSinh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[
Int[((d*e - c*f)/d + f*(x/d))^m*(a + b*ArcSinh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x
]

Rubi steps

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

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Mathematica [A]
time = 0.11, size = 147, normalized size = 1.08 \begin {gather*} \frac {e^2 \left (-12 b^2 (c+d x)+\left (9 a^2+2 b^2\right ) (c+d x)^3+6 a b \left (2-(c+d x)^2\right ) \sqrt {1+(c+d x)^2}+6 b \left (3 a (c+d x)^3+2 b \sqrt {1+(c+d x)^2}-b (c+d x)^2 \sqrt {1+(c+d x)^2}\right ) \sinh ^{-1}(c+d x)+9 b^2 (c+d x)^3 \sinh ^{-1}(c+d x)^2\right )}{27 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(e^2*(-12*b^2*(c + d*x) + (9*a^2 + 2*b^2)*(c + d*x)^3 + 6*a*b*(2 - (c + d*x)^2)*Sqrt[1 + (c + d*x)^2] + 6*b*(3
*a*(c + d*x)^3 + 2*b*Sqrt[1 + (c + d*x)^2] - b*(c + d*x)^2*Sqrt[1 + (c + d*x)^2])*ArcSinh[c + d*x] + 9*b^2*(c
+ d*x)^3*ArcSinh[c + d*x]^2))/(27*d)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(307\) vs. \(2(122)=244\).
time = 3.14, size = 308, normalized size = 2.26

method	result	size
default	\(\frac {e^{2} \left (d x +c \right )^{3} a^{2}}{3 d}+\frac {e^{2} b^{2} \left (9 \arcsinh \left (d x +c \right )^{2} x^{3} d^{3}+27 \arcsinh \left (d x +c \right )^{2} x^{2} c \,d^{2}-6 \arcsinh \left (d x +c \right ) \sqrt {d^{2} x^{2}+2 c d x +c^{2}+1}\, x^{2} d^{2}+27 \arcsinh \left (d x +c \right )^{2} x \,c^{2} d +2 d^{3} x^{3}-12 \arcsinh \left (d x +c \right ) \sqrt {d^{2} x^{2}+2 c d x +c^{2}+1}\, x c d +9 \arcsinh \left (d x +c \right )^{2} c^{3}+6 x^{2} c \,d^{2}-6 \arcsinh \left (d x +c \right ) \sqrt {d^{2} x^{2}+2 c d x +c^{2}+1}\, c^{2}+6 x \,c^{2} d +2 c^{3}+12 \arcsinh \left (d x +c \right ) \sqrt {d^{2} x^{2}+2 c d x +c^{2}+1}-12 d x -12 c \right )}{27 d}+\frac {2 a b \,e^{2} \left (\frac {\left (d x +c \right )^{3} \arcsinh \left (d x +c \right )}{3}-\frac {\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{9}+\frac {2 \sqrt {1+\left (d x +c \right )^{2}}}{9}\right )}{d}\)	\(308\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*e^2*(d*x+c)^3*a^2/d+1/27*e^2*b^2*(9*arcsinh(d*x+c)^2*x^3*d^3+27*arcsinh(d*x+c)^2*x^2*c*d^2-6*arcsinh(d*x+c
)*(d^2*x^2+2*c*d*x+c^2+1)^(1/2)*x^2*d^2+27*arcsinh(d*x+c)^2*x*c^2*d+2*d^3*x^3-12*arcsinh(d*x+c)*(d^2*x^2+2*c*d
*x+c^2+1)^(1/2)*x*c*d+9*arcsinh(d*x+c)^2*c^3+6*x^2*c*d^2-6*arcsinh(d*x+c)*(d^2*x^2+2*c*d*x+c^2+1)^(1/2)*c^2+6*
x*c^2*d+2*c^3+12*arcsinh(d*x+c)*(d^2*x^2+2*c*d*x+c^2+1)^(1/2)-12*d*x-12*c)/d+2*a*b*e^2/d*(1/3*(d*x+c)^3*arcsin
h(d*x+c)-1/9*(d*x+c)^2*(1+(d*x+c)^2)^(1/2)+2/9*(1+(d*x+c)^2)^(1/2))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a^2*d^2*x^3*e^2 + a^2*c*d*x^2*e^2 + (2*x^2*arcsinh(d*x + c) - d*(3*c^2*arcsinh(2*(d^2*x + c*d)/sqrt(-4*c^2
*d^2 + 4*(c^2 + 1)*d^2))/d^3 + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*x/d^2 - (c^2 + 1)*arcsinh(2*(d^2*x + c*d)/sqr
t(-4*c^2*d^2 + 4*(c^2 + 1)*d^2))/d^3 - 3*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*c/d^3))*a*b*c*d*e^2 + 1/9*(6*x^3*ar
csinh(d*x + c) - d*(2*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*x^2/d^2 - 15*c^3*arcsinh(2*(d^2*x + c*d)/sqrt(-4*c^2*d
^2 + 4*(c^2 + 1)*d^2))/d^4 - 5*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*c*x/d^3 + 9*(c^2 + 1)*c*arcsinh(2*(d^2*x + c*
d)/sqrt(-4*c^2*d^2 + 4*(c^2 + 1)*d^2))/d^4 + 15*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*c^2/d^4 - 4*sqrt(d^2*x^2 + 2
*c*d*x + c^2 + 1)*(c^2 + 1)/d^4))*a*b*d^2*e^2 + a^2*c^2*x*e^2 + 2*((d*x + c)*arcsinh(d*x + c) - sqrt((d*x + c)
^2 + 1))*a*b*c^2*e^2/d + 1/3*(b^2*d^2*x^3*e^2 + 3*b^2*c*d*x^2*e^2 + 3*b^2*c^2*x*e^2)*log(d*x + c + sqrt(d^2*x^
2 + 2*c*d*x + c^2 + 1))^2 - integrate(2/3*(b^2*d^5*x^5*e^2 + 5*b^2*c*d^4*x^4*e^2 + (10*b^2*c^2*d^3 + b^2*d^3)*
x^3*e^2 + 3*(3*b^2*c^3*d^2 + b^2*c*d^2)*x^2*e^2 + 3*(b^2*c^4*d + b^2*c^2*d)*x*e^2 + (b^2*d^4*x^4*e^2 + 4*b^2*c
*d^3*x^3*e^2 + 6*b^2*c^2*d^2*x^2*e^2 + 3*b^2*c^3*d*x*e^2)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))*log(d*x + c + sqr
t(d^2*x^2 + 2*c*d*x + c^2 + 1))/(d^3*x^3 + 3*c*d^2*x^2 + c^3 + (3*c^2*d + d)*x + (d^2*x^2 + 2*c*d*x + c^2 + 1)
^(3/2) + c), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 784 vs. \(2 (117) = 234\).
time = 0.39, size = 784, normalized size = 5.76 \begin {gather*} \frac {{\left ({\left (9 \, a^{2} + 2 \, b^{2}\right )} d^{3} x^{3} + 3 \, {\left (9 \, a^{2} + 2 \, b^{2}\right )} c d^{2} x^{2} + 3 \, {\left ({\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{2} - 4 \, b^{2}\right )} d x\right )} \cosh \left (1\right )^{2} + 9 \, {\left ({\left (b^{2} d^{3} x^{3} + 3 \, b^{2} c d^{2} x^{2} + 3 \, b^{2} c^{2} d x + b^{2} c^{3}\right )} \cosh \left (1\right )^{2} + 2 \, {\left (b^{2} d^{3} x^{3} + 3 \, b^{2} c d^{2} x^{2} + 3 \, b^{2} c^{2} d x + b^{2} c^{3}\right )} \cosh \left (1\right ) \sinh \left (1\right ) + {\left (b^{2} d^{3} x^{3} + 3 \, b^{2} c d^{2} x^{2} + 3 \, b^{2} c^{2} d x + b^{2} c^{3}\right )} \sinh \left (1\right )^{2}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{2} + 2 \, {\left ({\left (9 \, a^{2} + 2 \, b^{2}\right )} d^{3} x^{3} + 3 \, {\left (9 \, a^{2} + 2 \, b^{2}\right )} c d^{2} x^{2} + 3 \, {\left ({\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{2} - 4 \, b^{2}\right )} d x\right )} \cosh \left (1\right ) \sinh \left (1\right ) + {\left ({\left (9 \, a^{2} + 2 \, b^{2}\right )} d^{3} x^{3} + 3 \, {\left (9 \, a^{2} + 2 \, b^{2}\right )} c d^{2} x^{2} + 3 \, {\left ({\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{2} - 4 \, b^{2}\right )} d x\right )} \sinh \left (1\right )^{2} + 6 \, {\left (3 \, {\left (a b d^{3} x^{3} + 3 \, a b c d^{2} x^{2} + 3 \, a b c^{2} d x + a b c^{3}\right )} \cosh \left (1\right )^{2} + 6 \, {\left (a b d^{3} x^{3} + 3 \, a b c d^{2} x^{2} + 3 \, a b c^{2} d x + a b c^{3}\right )} \cosh \left (1\right ) \sinh \left (1\right ) + 3 \, {\left (a b d^{3} x^{3} + 3 \, a b c d^{2} x^{2} + 3 \, a b c^{2} d x + a b c^{3}\right )} \sinh \left (1\right )^{2} - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} {\left ({\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} - 2 \, b^{2}\right )} \cosh \left (1\right )^{2} + 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} - 2 \, b^{2}\right )} \cosh \left (1\right ) \sinh \left (1\right ) + {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} - 2 \, b^{2}\right )} \sinh \left (1\right )^{2}\right )}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) - 6 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} {\left ({\left (a b d^{2} x^{2} + 2 \, a b c d x + a b c^{2} - 2 \, a b\right )} \cosh \left (1\right )^{2} + 2 \, {\left (a b d^{2} x^{2} + 2 \, a b c d x + a b c^{2} - 2 \, a b\right )} \cosh \left (1\right ) \sinh \left (1\right ) + {\left (a b d^{2} x^{2} + 2 \, a b c d x + a b c^{2} - 2 \, a b\right )} \sinh \left (1\right )^{2}\right )}}{27 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 610 vs. \(2 (126) = 252\).
time = 0.34, size = 610, normalized size = 4.49 \begin {gather*} \begin {cases} a^{2} c^{2} e^{2} x + a^{2} c d e^{2} x^{2} + \frac {a^{2} d^{2} e^{2} x^{3}}{3} + \frac {2 a b c^{3} e^{2} \operatorname {asinh}{\left (c + d x \right )}}{3 d} + 2 a b c^{2} e^{2} x \operatorname {asinh}{\left (c + d x \right )} - \frac {2 a b c^{2} e^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{9 d} + 2 a b c d e^{2} x^{2} \operatorname {asinh}{\left (c + d x \right )} - \frac {4 a b c e^{2} x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{9} + \frac {2 a b d^{2} e^{2} x^{3} \operatorname {asinh}{\left (c + d x \right )}}{3} - \frac {2 a b d e^{2} x^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{9} + \frac {4 a b e^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{9 d} + \frac {b^{2} c^{3} e^{2} \operatorname {asinh}^{2}{\left (c + d x \right )}}{3 d} + b^{2} c^{2} e^{2} x \operatorname {asinh}^{2}{\left (c + d x \right )} + \frac {2 b^{2} c^{2} e^{2} x}{9} - \frac {2 b^{2} c^{2} e^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}{\left (c + d x \right )}}{9 d} + b^{2} c d e^{2} x^{2} \operatorname {asinh}^{2}{\left (c + d x \right )} + \frac {2 b^{2} c d e^{2} x^{2}}{9} - \frac {4 b^{2} c e^{2} x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}{\left (c + d x \right )}}{9} + \frac {b^{2} d^{2} e^{2} x^{3} \operatorname {asinh}^{2}{\left (c + d x \right )}}{3} + \frac {2 b^{2} d^{2} e^{2} x^{3}}{27} - \frac {2 b^{2} d e^{2} x^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}{\left (c + d x \right )}}{9} - \frac {4 b^{2} e^{2} x}{9} + \frac {4 b^{2} e^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}{\left (c + d x \right )}}{9 d} & \text {for}\: d \neq 0 \\c^{2} e^{2} x \left (a + b \operatorname {asinh}{\left (c \right )}\right )^{2} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (c\,e+d\,e\,x\right )}^2\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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