3.1.12 \(\int (d+e x)^3 (a+b \sinh ^{-1}(c x))^2 \, dx\) [12]

Optimal. Leaf size=368 \[ 2 b^2 d^3 x-\frac {4 b^2 d e^2 x}{3 c^2}+\frac {3}{4} b^2 d^2 e x^2-\frac {3 b^2 e^3 x^2}{32 c^2}+\frac {2}{9} b^2 d e^2 x^3+\frac {1}{32} b^2 e^3 x^4-\frac {2 b d^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c}+\frac {4 b d e^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^3}-\frac {3 b d^2 e x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 c}+\frac {3 b e^3 x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{16 c^3}-\frac {2 b d e^2 x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c}-\frac {b e^3 x^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 c}-\frac {d^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{4 e}+\frac {3 d^2 e \left (a+b \sinh ^{-1}(c x)\right )^2}{4 c^2}-\frac {3 e^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{32 c^4}+\frac {(d+e x)^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{4 e} \]

[Out]

2*b^2*d^3*x-4/3*b^2*d*e^2*x/c^2+3/4*b^2*d^2*e*x^2-3/32*b^2*e^3*x^2/c^2+2/9*b^2*d*e^2*x^3+1/32*b^2*e^3*x^4-1/4*
d^4*(a+b*arcsinh(c*x))^2/e+3/4*d^2*e*(a+b*arcsinh(c*x))^2/c^2-3/32*e^3*(a+b*arcsinh(c*x))^2/c^4+1/4*(e*x+d)^4*
(a+b*arcsinh(c*x))^2/e-2*b*d^3*(a+b*arcsinh(c*x))*(c^2*x^2+1)^(1/2)/c+4/3*b*d*e^2*(a+b*arcsinh(c*x))*(c^2*x^2+
1)^(1/2)/c^3-3/2*b*d^2*e*x*(a+b*arcsinh(c*x))*(c^2*x^2+1)^(1/2)/c+3/16*b*e^3*x*(a+b*arcsinh(c*x))*(c^2*x^2+1)^
(1/2)/c^3-2/3*b*d*e^2*x^2*(a+b*arcsinh(c*x))*(c^2*x^2+1)^(1/2)/c-1/8*b*e^3*x^3*(a+b*arcsinh(c*x))*(c^2*x^2+1)^
(1/2)/c

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Rubi [A]
time = 0.49, antiderivative size = 368, normalized size of antiderivative = 1.00, number of steps used = 18, 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 {3 e^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{32 c^4}-\frac {2 b d^3 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{c}-\frac {3 b d^2 e x \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{2 c}+\frac {3 d^2 e \left (a+b \sinh ^{-1}(c x)\right )^2}{4 c^2}-\frac {2 b d e^2 x^2 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{3 c}-\frac {b e^3 x^3 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{8 c}+\frac {4 b d e^2 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^3}+\frac {3 b e^3 x \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{16 c^3}-\frac {d^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{4 e}+\frac {(d+e x)^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{4 e}-\frac {4 b^2 d e^2 x}{3 c^2}-\frac {3 b^2 e^3 x^2}{32 c^2}+2 b^2 d^3 x+\frac {3}{4} b^2 d^2 e x^2+\frac {2}{9} b^2 d e^2 x^3+\frac {1}{32} b^2 e^3 x^4 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^3*(a + b*ArcSinh[c*x])^2,x]

[Out]

2*b^2*d^3*x - (4*b^2*d*e^2*x)/(3*c^2) + (3*b^2*d^2*e*x^2)/4 - (3*b^2*e^3*x^2)/(32*c^2) + (2*b^2*d*e^2*x^3)/9 +
(b^2*e^3*x^4)/32 - (2*b*d^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/c + (4*b*d*e^2*Sqrt[1 + c^2*x^2]*(a + b*A
rcSinh[c*x]))/(3*c^3) - (3*b*d^2*e*x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(2*c) + (3*b*e^3*x*Sqrt[1 + c^2*x
^2]*(a + b*ArcSinh[c*x]))/(16*c^3) - (2*b*d*e^2*x^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(3*c) - (b*e^3*x^3
*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(8*c) - (d^4*(a + b*ArcSinh[c*x])^2)/(4*e) + (3*d^2*e*(a + b*ArcSinh[
c*x])^2)/(4*c^2) - (3*e^3*(a + b*ArcSinh[c*x])^2)/(32*c^4) + ((d + e*x)^4*(a + b*ArcSinh[c*x])^2)/(4*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)^3 \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx &=\frac {(d+e x)^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{4 e}-\frac {(b c) \int \frac {(d+e x)^4 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx}{2 e}\\ &=\frac {(d+e x)^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{4 e}-\frac {(b c) \int \left (\frac {d^4 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}}+\frac {4 d^3 e x \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}}+\frac {6 d^2 e^2 x^2 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}}+\frac {4 d e^3 x^3 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}}+\frac {e^4 x^4 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}}\right ) \, dx}{2 e}\\ &=\frac {(d+e x)^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{4 e}-\left (2 b c d^3\right ) \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx-\frac {\left (b c d^4\right ) \int \frac {a+b \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{2 e}-\left (3 b c d^2 e\right ) \int \frac {x^2 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx-\left (2 b c d e^2\right ) \int \frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx-\frac {1}{2} \left (b c e^3\right ) \int \frac {x^4 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx\\ &=-\frac {2 b d^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c}-\frac {3 b d^2 e x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 c}-\frac {2 b d e^2 x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c}-\frac {b e^3 x^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 c}-\frac {d^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{4 e}+\frac {(d+e x)^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{4 e}+\left (2 b^2 d^3\right ) \int 1 \, dx+\frac {1}{2} \left (3 b^2 d^2 e\right ) \int x \, dx+\frac {\left (3 b d^2 e\right ) \int \frac {a+b \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{2 c}+\frac {1}{3} \left (2 b^2 d e^2\right ) \int x^2 \, dx+\frac {\left (4 b d e^2\right ) \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx}{3 c}+\frac {1}{8} \left (b^2 e^3\right ) \int x^3 \, dx+\frac {\left (3 b e^3\right ) \int \frac {x^2 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx}{8 c}\\ &=2 b^2 d^3 x+\frac {3}{4} b^2 d^2 e x^2+\frac {2}{9} b^2 d e^2 x^3+\frac {1}{32} b^2 e^3 x^4-\frac {2 b d^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c}+\frac {4 b d e^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^3}-\frac {3 b d^2 e x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 c}+\frac {3 b e^3 x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{16 c^3}-\frac {2 b d e^2 x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c}-\frac {b e^3 x^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 c}-\frac {d^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{4 e}+\frac {3 d^2 e \left (a+b \sinh ^{-1}(c x)\right )^2}{4 c^2}+\frac {(d+e x)^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{4 e}-\frac {\left (4 b^2 d e^2\right ) \int 1 \, dx}{3 c^2}-\frac {\left (3 b e^3\right ) \int \frac {a+b \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{16 c^3}-\frac {\left (3 b^2 e^3\right ) \int x \, dx}{16 c^2}\\ &=2 b^2 d^3 x-\frac {4 b^2 d e^2 x}{3 c^2}+\frac {3}{4} b^2 d^2 e x^2-\frac {3 b^2 e^3 x^2}{32 c^2}+\frac {2}{9} b^2 d e^2 x^3+\frac {1}{32} b^2 e^3 x^4-\frac {2 b d^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c}+\frac {4 b d e^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^3}-\frac {3 b d^2 e x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 c}+\frac {3 b e^3 x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{16 c^3}-\frac {2 b d e^2 x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c}-\frac {b e^3 x^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 c}-\frac {d^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{4 e}+\frac {3 d^2 e \left (a+b \sinh ^{-1}(c x)\right )^2}{4 c^2}-\frac {3 e^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{32 c^4}+\frac {(d+e x)^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{4 e}\\ \end {align*}

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Mathematica [A]
time = 0.28, size = 354, normalized size = 0.96 \begin {gather*} \frac {c \left (72 a^2 c^3 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )-6 a b \sqrt {1+c^2 x^2} \left (-e^2 (64 d+9 e x)+c^2 \left (96 d^3+72 d^2 e x+32 d e^2 x^2+6 e^3 x^3\right )\right )+b^2 c x \left (-3 e^2 (128 d+9 e x)+c^2 \left (576 d^3+216 d^2 e x+64 d e^2 x^2+9 e^3 x^3\right )\right )\right )-6 b \left (-3 a \left (24 c^2 d^2 e-3 e^3+8 c^4 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )\right )+b c \sqrt {1+c^2 x^2} \left (-e^2 (64 d+9 e x)+c^2 \left (96 d^3+72 d^2 e x+32 d e^2 x^2+6 e^3 x^3\right )\right )\right ) \sinh ^{-1}(c x)+9 b^2 \left (24 c^2 d^2 e-3 e^3+8 c^4 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )\right ) \sinh ^{-1}(c x)^2}{288 c^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^3*(a + b*ArcSinh[c*x])^2,x]

[Out]

(c*(72*a^2*c^3*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) - 6*a*b*Sqrt[1 + c^2*x^2]*(-(e^2*(64*d + 9*e*x))
+ c^2*(96*d^3 + 72*d^2*e*x + 32*d*e^2*x^2 + 6*e^3*x^3)) + b^2*c*x*(-3*e^2*(128*d + 9*e*x) + c^2*(576*d^3 + 216
*d^2*e*x + 64*d*e^2*x^2 + 9*e^3*x^3))) - 6*b*(-3*a*(24*c^2*d^2*e - 3*e^3 + 8*c^4*x*(4*d^3 + 6*d^2*e*x + 4*d*e^
2*x^2 + e^3*x^3)) + b*c*Sqrt[1 + c^2*x^2]*(-(e^2*(64*d + 9*e*x)) + c^2*(96*d^3 + 72*d^2*e*x + 32*d*e^2*x^2 + 6
*e^3*x^3)))*ArcSinh[c*x] + 9*b^2*(24*c^2*d^2*e - 3*e^3 + 8*c^4*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3))*
ArcSinh[c*x]^2)/(288*c^4)

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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \left (e x +d \right )^{3} \left (a +b \arcsinh \left (c x \right )\right )^{2}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^3*(a+b*arcsinh(c*x))^2,x)

[Out]

int((e*x+d)^3*(a+b*arcsinh(c*x))^2,x)

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Maxima [A]
time = 0.30, size = 586, normalized size = 1.59 \begin {gather*} \frac {1}{4} \, b^{2} x^{4} \operatorname {arsinh}\left (c x\right )^{2} e^{3} + b^{2} d x^{3} \operatorname {arsinh}\left (c x\right )^{2} e^{2} + \frac {3}{2} \, b^{2} d^{2} x^{2} \operatorname {arsinh}\left (c x\right )^{2} e + b^{2} d^{3} x \operatorname {arsinh}\left (c x\right )^{2} + \frac {1}{4} \, a^{2} x^{4} e^{3} + a^{2} d x^{3} e^{2} + \frac {3}{2} \, a^{2} d^{2} x^{2} e + 2 \, b^{2} d^{3} {\left (x - \frac {\sqrt {c^{2} x^{2} + 1} \operatorname {arsinh}\left (c x\right )}{c}\right )} + a^{2} d^{3} x + \frac {3}{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 (c x\right )}{c^{3}}\right )}\right )} a b d^{2} e + \frac {3}{4} \, {\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^{2} e + \frac {2 \, {\left (c x \operatorname {arsinh}\left (c x\right ) - \sqrt {c^{2} x^{2} + 1}\right )} a b d^{3}}{c} + \frac {2}{3} \, {\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^{2} - \frac {2}{9} \, {\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^{2} + \frac {1}{16} \, {\left (8 \, x^{4} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {2 \, \sqrt {c^{2} x^{2} + 1} x^{3}}{c^{2}} - \frac {3 \, \sqrt {c^{2} x^{2} + 1} x}{c^{4}} + \frac {3 \, \operatorname {arsinh}\left (c x\right )}{c^{5}}\right )} c\right )} a b e^{3} + \frac {1}{32} \, {\left ({\left (\frac {x^{4}}{c^{2}} - \frac {3 \, x^{2}}{c^{4}} + \frac {3 \, \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2}}{c^{6}}\right )} c^{2} - 2 \, {\left (\frac {2 \, \sqrt {c^{2} x^{2} + 1} x^{3}}{c^{2}} - \frac {3 \, \sqrt {c^{2} x^{2} + 1} x}{c^{4}} + \frac {3 \, \operatorname {arsinh}\left (c x\right )}{c^{5}}\right )} c \operatorname {arsinh}\left (c x\right )\right )} b^{2} e^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(a+b*arcsinh(c*x))^2,x, algorithm="maxima")

[Out]

1/4*b^2*x^4*arcsinh(c*x)^2*e^3 + b^2*d*x^3*arcsinh(c*x)^2*e^2 + 3/2*b^2*d^2*x^2*arcsinh(c*x)^2*e + b^2*d^3*x*a
rcsinh(c*x)^2 + 1/4*a^2*x^4*e^3 + a^2*d*x^3*e^2 + 3/2*a^2*d^2*x^2*e + 2*b^2*d^3*(x - sqrt(c^2*x^2 + 1)*arcsinh
(c*x)/c) + a^2*d^3*x + 3/2*(2*x^2*arcsinh(c*x) - c*(sqrt(c^2*x^2 + 1)*x/c^2 - arcsinh(c*x)/c^3))*a*b*d^2*e + 3
/4*(c^2*(x^2/c^2 - log(c*x + sqrt(c^2*x^2 + 1))^2/c^4) - 2*c*(sqrt(c^2*x^2 + 1)*x/c^2 - arcsinh(c*x)/c^3)*arcs
inh(c*x))*b^2*d^2*e + 2*(c*x*arcsinh(c*x) - sqrt(c^2*x^2 + 1))*a*b*d^3/c + 2/3*(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^2 - 2/9*(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 + 1/16*(8*x^4*arcsinh(c*x) - (2*sqrt(c^2*x^2 + 1)*x
^3/c^2 - 3*sqrt(c^2*x^2 + 1)*x/c^4 + 3*arcsinh(c*x)/c^5)*c)*a*b*e^3 + 1/32*((x^4/c^2 - 3*x^2/c^4 + 3*log(c*x +
sqrt(c^2*x^2 + 1))^2/c^6)*c^2 - 2*(2*sqrt(c^2*x^2 + 1)*x^3/c^2 - 3*sqrt(c^2*x^2 + 1)*x/c^4 + 3*arcsinh(c*x)/c
^5)*c*arcsinh(c*x))*b^2*e^3

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1185 vs. \(2 (321) = 642\).
time = 0.36, size = 1185, normalized size = 3.22 \begin {gather*} \frac {216 \, {\left (2 \, a^{2} + b^{2}\right )} c^{4} d^{2} x^{2} \cosh \left (1\right ) + 288 \, {\left (a^{2} + 2 \, b^{2}\right )} c^{4} d^{3} x + 9 \, {\left ({\left (8 \, a^{2} + b^{2}\right )} c^{4} x^{4} - 3 \, b^{2} c^{2} x^{2}\right )} \cosh \left (1\right )^{3} + 9 \, {\left ({\left (8 \, a^{2} + b^{2}\right )} c^{4} x^{4} - 3 \, b^{2} c^{2} x^{2}\right )} \sinh \left (1\right )^{3} + 32 \, {\left ({\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{4} d x^{3} - 12 \, b^{2} c^{2} d x\right )} \cosh \left (1\right )^{2} + 9 \, {\left (32 \, b^{2} c^{4} d x^{3} \cosh \left (1\right )^{2} + 32 \, b^{2} c^{4} d^{3} x + {\left (8 \, b^{2} c^{4} x^{4} - 3 \, b^{2}\right )} \cosh \left (1\right )^{3} + {\left (8 \, b^{2} c^{4} x^{4} - 3 \, b^{2}\right )} \sinh \left (1\right )^{3} + {\left (32 \, b^{2} c^{4} d x^{3} + 3 \, {\left (8 \, b^{2} c^{4} x^{4} - 3 \, b^{2}\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )^{2} + 24 \, {\left (2 \, b^{2} c^{4} d^{2} x^{2} + b^{2} c^{2} d^{2}\right )} \cosh \left (1\right ) + {\left (64 \, b^{2} c^{4} d x^{3} \cosh \left (1\right ) + 48 \, b^{2} c^{4} d^{2} x^{2} + 24 \, b^{2} c^{2} d^{2} + 3 \, {\left (8 \, b^{2} c^{4} x^{4} - 3 \, b^{2}\right )} \cosh \left (1\right )^{2}\right )} \sinh \left (1\right )\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + {\left (32 \, {\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{4} d x^{3} - 384 \, b^{2} c^{2} d x + 27 \, {\left ({\left (8 \, a^{2} + b^{2}\right )} c^{4} x^{4} - 3 \, b^{2} c^{2} x^{2}\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )^{2} + 6 \, {\left (96 \, a b c^{4} d x^{3} \cosh \left (1\right )^{2} + 96 \, a b c^{4} d^{3} x + 3 \, {\left (8 \, a b c^{4} x^{4} - 3 \, a b\right )} \cosh \left (1\right )^{3} + 3 \, {\left (8 \, a b c^{4} x^{4} - 3 \, a b\right )} \sinh \left (1\right )^{3} + 3 \, {\left (32 \, a b c^{4} d x^{3} + 3 \, {\left (8 \, a b c^{4} x^{4} - 3 \, a b\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )^{2} + 72 \, {\left (2 \, a b c^{4} d^{2} x^{2} + a b c^{2} d^{2}\right )} \cosh \left (1\right ) + 3 \, {\left (64 \, a b c^{4} d x^{3} \cosh \left (1\right ) + 48 \, a b c^{4} d^{2} x^{2} + 24 \, a b c^{2} d^{2} + 3 \, {\left (8 \, a b c^{4} x^{4} - 3 \, a b\right )} \cosh \left (1\right )^{2}\right )} \sinh \left (1\right ) - {\left (72 \, b^{2} c^{3} d^{2} x \cosh \left (1\right ) + 96 \, b^{2} c^{3} d^{3} + 3 \, {\left (2 \, b^{2} c^{3} x^{3} - 3 \, b^{2} c x\right )} \cosh \left (1\right )^{3} + 3 \, {\left (2 \, b^{2} c^{3} x^{3} - 3 \, b^{2} c x\right )} \sinh \left (1\right )^{3} + 32 \, {\left (b^{2} c^{3} d x^{2} - 2 \, b^{2} c d\right )} \cosh \left (1\right )^{2} + {\left (32 \, b^{2} c^{3} d x^{2} - 64 \, b^{2} c d + 9 \, {\left (2 \, b^{2} c^{3} x^{3} - 3 \, b^{2} c x\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )^{2} + {\left (72 \, b^{2} c^{3} d^{2} x + 9 \, {\left (2 \, b^{2} c^{3} x^{3} - 3 \, b^{2} c x\right )} \cosh \left (1\right )^{2} + 64 \, {\left (b^{2} c^{3} d x^{2} - 2 \, b^{2} c d\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 (216 \, {\left (2 \, a^{2} + b^{2}\right )} c^{4} d^{2} x^{2} + 27 \, {\left ({\left (8 \, a^{2} + b^{2}\right )} c^{4} x^{4} - 3 \, b^{2} c^{2} x^{2}\right )} \cosh \left (1\right )^{2} + 64 \, {\left ({\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{4} d x^{3} - 12 \, b^{2} c^{2} d x\right )} \cosh \left (1\right )\right )} \sinh \left (1\right ) - 6 \, {\left (72 \, a b c^{3} d^{2} x \cosh \left (1\right ) + 96 \, a b c^{3} d^{3} + 3 \, {\left (2 \, a b c^{3} x^{3} - 3 \, a b c x\right )} \cosh \left (1\right )^{3} + 3 \, {\left (2 \, a b c^{3} x^{3} - 3 \, a b c x\right )} \sinh \left (1\right )^{3} + 32 \, {\left (a b c^{3} d x^{2} - 2 \, a b c d\right )} \cosh \left (1\right )^{2} + {\left (32 \, a b c^{3} d x^{2} - 64 \, a b c d + 9 \, {\left (2 \, a b c^{3} x^{3} - 3 \, a b c x\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )^{2} + {\left (72 \, a b c^{3} d^{2} x + 9 \, {\left (2 \, a b c^{3} x^{3} - 3 \, a b c x\right )} \cosh \left (1\right )^{2} + 64 \, {\left (a b c^{3} d x^{2} - 2 \, a b c d\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \sqrt {c^{2} x^{2} + 1}}{288 \, c^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(a+b*arcsinh(c*x))^2,x, algorithm="fricas")

[Out]

1/288*(216*(2*a^2 + b^2)*c^4*d^2*x^2*cosh(1) + 288*(a^2 + 2*b^2)*c^4*d^3*x + 9*((8*a^2 + b^2)*c^4*x^4 - 3*b^2*
c^2*x^2)*cosh(1)^3 + 9*((8*a^2 + b^2)*c^4*x^4 - 3*b^2*c^2*x^2)*sinh(1)^3 + 32*((9*a^2 + 2*b^2)*c^4*d*x^3 - 12*
b^2*c^2*d*x)*cosh(1)^2 + 9*(32*b^2*c^4*d*x^3*cosh(1)^2 + 32*b^2*c^4*d^3*x + (8*b^2*c^4*x^4 - 3*b^2)*cosh(1)^3
+ (8*b^2*c^4*x^4 - 3*b^2)*sinh(1)^3 + (32*b^2*c^4*d*x^3 + 3*(8*b^2*c^4*x^4 - 3*b^2)*cosh(1))*sinh(1)^2 + 24*(2
*b^2*c^4*d^2*x^2 + b^2*c^2*d^2)*cosh(1) + (64*b^2*c^4*d*x^3*cosh(1) + 48*b^2*c^4*d^2*x^2 + 24*b^2*c^2*d^2 + 3*
(8*b^2*c^4*x^4 - 3*b^2)*cosh(1)^2)*sinh(1))*log(c*x + sqrt(c^2*x^2 + 1))^2 + (32*(9*a^2 + 2*b^2)*c^4*d*x^3 - 3
84*b^2*c^2*d*x + 27*((8*a^2 + b^2)*c^4*x^4 - 3*b^2*c^2*x^2)*cosh(1))*sinh(1)^2 + 6*(96*a*b*c^4*d*x^3*cosh(1)^2
+ 96*a*b*c^4*d^3*x + 3*(8*a*b*c^4*x^4 - 3*a*b)*cosh(1)^3 + 3*(8*a*b*c^4*x^4 - 3*a*b)*sinh(1)^3 + 3*(32*a*b*c^
4*d*x^3 + 3*(8*a*b*c^4*x^4 - 3*a*b)*cosh(1))*sinh(1)^2 + 72*(2*a*b*c^4*d^2*x^2 + a*b*c^2*d^2)*cosh(1) + 3*(64*
a*b*c^4*d*x^3*cosh(1) + 48*a*b*c^4*d^2*x^2 + 24*a*b*c^2*d^2 + 3*(8*a*b*c^4*x^4 - 3*a*b)*cosh(1)^2)*sinh(1) - (
72*b^2*c^3*d^2*x*cosh(1) + 96*b^2*c^3*d^3 + 3*(2*b^2*c^3*x^3 - 3*b^2*c*x)*cosh(1)^3 + 3*(2*b^2*c^3*x^3 - 3*b^2
*c*x)*sinh(1)^3 + 32*(b^2*c^3*d*x^2 - 2*b^2*c*d)*cosh(1)^2 + (32*b^2*c^3*d*x^2 - 64*b^2*c*d + 9*(2*b^2*c^3*x^3
- 3*b^2*c*x)*cosh(1))*sinh(1)^2 + (72*b^2*c^3*d^2*x + 9*(2*b^2*c^3*x^3 - 3*b^2*c*x)*cosh(1)^2 + 64*(b^2*c^3*d
*x^2 - 2*b^2*c*d)*cosh(1))*sinh(1))*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1)) + (216*(2*a^2 + b^2)*c^4*d
^2*x^2 + 27*((8*a^2 + b^2)*c^4*x^4 - 3*b^2*c^2*x^2)*cosh(1)^2 + 64*((9*a^2 + 2*b^2)*c^4*d*x^3 - 12*b^2*c^2*d*x
)*cosh(1))*sinh(1) - 6*(72*a*b*c^3*d^2*x*cosh(1) + 96*a*b*c^3*d^3 + 3*(2*a*b*c^3*x^3 - 3*a*b*c*x)*cosh(1)^3 +
3*(2*a*b*c^3*x^3 - 3*a*b*c*x)*sinh(1)^3 + 32*(a*b*c^3*d*x^2 - 2*a*b*c*d)*cosh(1)^2 + (32*a*b*c^3*d*x^2 - 64*a*
b*c*d + 9*(2*a*b*c^3*x^3 - 3*a*b*c*x)*cosh(1))*sinh(1)^2 + (72*a*b*c^3*d^2*x + 9*(2*a*b*c^3*x^3 - 3*a*b*c*x)*c
osh(1)^2 + 64*(a*b*c^3*d*x^2 - 2*a*b*c*d)*cosh(1))*sinh(1))*sqrt(c^2*x^2 + 1))/c^4

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 743 vs. \(2 (364) = 728\).
time = 0.47, size = 743, normalized size = 2.02 \begin {gather*} \begin {cases} a^{2} d^{3} x + \frac {3 a^{2} d^{2} e x^{2}}{2} + a^{2} d e^{2} x^{3} + \frac {a^{2} e^{3} x^{4}}{4} + 2 a b d^{3} x \operatorname {asinh}{\left (c x \right )} + 3 a b d^{2} e x^{2} \operatorname {asinh}{\left (c x \right )} + 2 a b d e^{2} x^{3} \operatorname {asinh}{\left (c x \right )} + \frac {a b e^{3} x^{4} \operatorname {asinh}{\left (c x \right )}}{2} - \frac {2 a b d^{3} \sqrt {c^{2} x^{2} + 1}}{c} - \frac {3 a b d^{2} e x \sqrt {c^{2} x^{2} + 1}}{2 c} - \frac {2 a b d e^{2} x^{2} \sqrt {c^{2} x^{2} + 1}}{3 c} - \frac {a b e^{3} x^{3} \sqrt {c^{2} x^{2} + 1}}{8 c} + \frac {3 a b d^{2} e \operatorname {asinh}{\left (c x \right )}}{2 c^{2}} + \frac {4 a b d e^{2} \sqrt {c^{2} x^{2} + 1}}{3 c^{3}} + \frac {3 a b e^{3} x \sqrt {c^{2} x^{2} + 1}}{16 c^{3}} - \frac {3 a b e^{3} \operatorname {asinh}{\left (c x \right )}}{16 c^{4}} + b^{2} d^{3} x \operatorname {asinh}^{2}{\left (c x \right )} + 2 b^{2} d^{3} x + \frac {3 b^{2} d^{2} e x^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{2} + \frac {3 b^{2} d^{2} e x^{2}}{4} + b^{2} d e^{2} x^{3} \operatorname {asinh}^{2}{\left (c x \right )} + \frac {2 b^{2} d e^{2} x^{3}}{9} + \frac {b^{2} e^{3} x^{4} \operatorname {asinh}^{2}{\left (c x \right )}}{4} + \frac {b^{2} e^{3} x^{4}}{32} - \frac {2 b^{2} d^{3} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{c} - \frac {3 b^{2} d^{2} e x \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{2 c} - \frac {2 b^{2} d e^{2} x^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{3 c} - \frac {b^{2} e^{3} x^{3} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{8 c} + \frac {3 b^{2} d^{2} e \operatorname {asinh}^{2}{\left (c x \right )}}{4 c^{2}} - \frac {4 b^{2} d e^{2} x}{3 c^{2}} - \frac {3 b^{2} e^{3} x^{2}}{32 c^{2}} + \frac {4 b^{2} d e^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{3 c^{3}} + \frac {3 b^{2} e^{3} x \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{16 c^{3}} - \frac {3 b^{2} e^{3} \operatorname {asinh}^{2}{\left (c x \right )}}{32 c^{4}} & \text {for}\: c \neq 0 \\a^{2} \left (d^{3} x + \frac {3 d^{2} e x^{2}}{2} + d e^{2} x^{3} + \frac {e^{3} x^{4}}{4}\right ) & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**3*(a+b*asinh(c*x))**2,x)

[Out]

Piecewise((a**2*d**3*x + 3*a**2*d**2*e*x**2/2 + a**2*d*e**2*x**3 + a**2*e**3*x**4/4 + 2*a*b*d**3*x*asinh(c*x)
+ 3*a*b*d**2*e*x**2*asinh(c*x) + 2*a*b*d*e**2*x**3*asinh(c*x) + a*b*e**3*x**4*asinh(c*x)/2 - 2*a*b*d**3*sqrt(c
**2*x**2 + 1)/c - 3*a*b*d**2*e*x*sqrt(c**2*x**2 + 1)/(2*c) - 2*a*b*d*e**2*x**2*sqrt(c**2*x**2 + 1)/(3*c) - a*b
*e**3*x**3*sqrt(c**2*x**2 + 1)/(8*c) + 3*a*b*d**2*e*asinh(c*x)/(2*c**2) + 4*a*b*d*e**2*sqrt(c**2*x**2 + 1)/(3*
c**3) + 3*a*b*e**3*x*sqrt(c**2*x**2 + 1)/(16*c**3) - 3*a*b*e**3*asinh(c*x)/(16*c**4) + b**2*d**3*x*asinh(c*x)*
*2 + 2*b**2*d**3*x + 3*b**2*d**2*e*x**2*asinh(c*x)**2/2 + 3*b**2*d**2*e*x**2/4 + b**2*d*e**2*x**3*asinh(c*x)**
2 + 2*b**2*d*e**2*x**3/9 + b**2*e**3*x**4*asinh(c*x)**2/4 + b**2*e**3*x**4/32 - 2*b**2*d**3*sqrt(c**2*x**2 + 1
)*asinh(c*x)/c - 3*b**2*d**2*e*x*sqrt(c**2*x**2 + 1)*asinh(c*x)/(2*c) - 2*b**2*d*e**2*x**2*sqrt(c**2*x**2 + 1)
*asinh(c*x)/(3*c) - b**2*e**3*x**3*sqrt(c**2*x**2 + 1)*asinh(c*x)/(8*c) + 3*b**2*d**2*e*asinh(c*x)**2/(4*c**2)
- 4*b**2*d*e**2*x/(3*c**2) - 3*b**2*e**3*x**2/(32*c**2) + 4*b**2*d*e**2*sqrt(c**2*x**2 + 1)*asinh(c*x)/(3*c**
3) + 3*b**2*e**3*x*sqrt(c**2*x**2 + 1)*asinh(c*x)/(16*c**3) - 3*b**2*e**3*asinh(c*x)**2/(32*c**4), Ne(c, 0)),
(a**2*(d**3*x + 3*d**2*e*x**2/2 + d*e**2*x**3 + e**3*x**4/4), 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)^3*(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 )}^3 \,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)^3,x)

[Out]

int((a + b*asinh(c*x))^2*(d + e*x)^3, x)

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