3.149 \(\int (c e+d e x) (a+b \sinh ^{-1}(c+d x))^4 \, dx\)

Optimal. Leaf size=195 \[ -\frac{3 b^3 e (c+d x) \sqrt{(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )}{2 d}+\frac{3 b^2 e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^2}{2 d}+\frac{3 b^2 e \left (a+b \sinh ^{-1}(c+d x)\right )^2}{4 d}-\frac{b e (c+d x) \sqrt{(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{d}+\frac{e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^4}{2 d}+\frac{e \left (a+b \sinh ^{-1}(c+d x)\right )^4}{4 d}+\frac{3 b^4 e (c+d x)^2}{4 d} \]

[Out]

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

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Rubi [A]  time = 0.320859, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {5865, 12, 5661, 5758, 5675, 30} \[ -\frac{3 b^3 e (c+d x) \sqrt{(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )}{2 d}+\frac{3 b^2 e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^2}{2 d}+\frac{3 b^2 e \left (a+b \sinh ^{-1}(c+d x)\right )^2}{4 d}-\frac{b e (c+d x) \sqrt{(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{d}+\frac{e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^4}{2 d}+\frac{e \left (a+b \sinh ^{-1}(c+d x)\right )^4}{4 d}+\frac{3 b^4 e (c+d x)^2}{4 d} \]

Antiderivative was successfully verified.

[In]

Int[(c*e + d*e*x)*(a + b*ArcSinh[c + d*x])^4,x]

[Out]

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

Rule 5865

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
]

Rule 12

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

Rule 5661

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 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 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 30

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

Rubi steps

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

Mathematica [A]  time = 0.310301, size = 300, normalized size = 1.54 \[ \frac{e \left (\left (6 a^2 b^2+2 a^4+3 b^4\right ) (c+d x)^2-2 a b \left (2 a^2+3 b^2\right ) (c+d x) \sqrt{(c+d x)^2+1}+3 b^2 \sinh ^{-1}(c+d x)^2 \left (4 a^2 (c+d x)^2+2 a^2-4 a b (c+d x) \sqrt{(c+d x)^2+1}+2 b^2 (c+d x)^2+b^2\right )+2 a b \left (2 a^2+3 b^2\right ) \sinh ^{-1}(c+d x)-2 b (c+d x) \sinh ^{-1}(c+d x) \left (6 a^2 b \sqrt{(c+d x)^2+1}-4 a^3 (c+d x)-6 a b^2 (c+d x)+3 b^3 \sqrt{(c+d x)^2+1}\right )+4 b^3 \sinh ^{-1}(c+d x)^3 \left (2 a (c+d x)^2+a-b \sqrt{(c+d x)^2+1} (c+d x)\right )+b^4 \left (2 (c+d x)^2+1\right ) \sinh ^{-1}(c+d x)^4\right )}{4 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*e + d*e*x)*(a + b*ArcSinh[c + d*x])^4,x]

[Out]

(e*((2*a^4 + 6*a^2*b^2 + 3*b^4)*(c + d*x)^2 - 2*a*b*(2*a^2 + 3*b^2)*(c + d*x)*Sqrt[1 + (c + d*x)^2] + 2*a*b*(2
*a^2 + 3*b^2)*ArcSinh[c + d*x] - 2*b*(c + d*x)*(-4*a^3*(c + d*x) - 6*a*b^2*(c + d*x) + 6*a^2*b*Sqrt[1 + (c + d
*x)^2] + 3*b^3*Sqrt[1 + (c + d*x)^2])*ArcSinh[c + d*x] + 3*b^2*(2*a^2 + b^2 + 4*a^2*(c + d*x)^2 + 2*b^2*(c + d
*x)^2 - 4*a*b*(c + d*x)*Sqrt[1 + (c + d*x)^2])*ArcSinh[c + d*x]^2 + 4*b^3*(a + 2*a*(c + d*x)^2 - b*(c + d*x)*S
qrt[1 + (c + d*x)^2])*ArcSinh[c + d*x]^3 + b^4*(1 + 2*(c + d*x)^2)*ArcSinh[c + d*x]^4))/(4*d)

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Maple [B]  time = 0.069, size = 371, normalized size = 1.9 \begin{align*}{\frac{1}{d} \left ({\frac{ \left ( dx+c \right ) ^{2}e{a}^{4}}{2}}+e{b}^{4} \left ({\frac{ \left ({\it Arcsinh} \left ( dx+c \right ) \right ) ^{4} \left ( 1+ \left ( dx+c \right ) ^{2} \right ) }{2}}- \left ({\it Arcsinh} \left ( dx+c \right ) \right ) ^{3} \left ( dx+c \right ) \sqrt{1+ \left ( dx+c \right ) ^{2}}-{\frac{ \left ({\it Arcsinh} \left ( dx+c \right ) \right ) ^{4}}{4}}+{\frac{3\, \left ({\it Arcsinh} \left ( dx+c \right ) \right ) ^{2} \left ( 1+ \left ( dx+c \right ) ^{2} \right ) }{2}}-{\frac{3\,{\it Arcsinh} \left ( dx+c \right ) \left ( dx+c \right ) }{2}\sqrt{1+ \left ( dx+c \right ) ^{2}}}-{\frac{3\, \left ({\it Arcsinh} \left ( dx+c \right ) \right ) ^{2}}{4}}+{\frac{3\, \left ( dx+c \right ) ^{2}}{4}}+{\frac{3}{4}} \right ) +4\,ea{b}^{3} \left ( 1/2\, \left ({\it Arcsinh} \left ( dx+c \right ) \right ) ^{3} \left ( 1+ \left ( dx+c \right ) ^{2} \right ) -3/4\, \left ({\it Arcsinh} \left ( dx+c \right ) \right ) ^{2}\sqrt{1+ \left ( dx+c \right ) ^{2}} \left ( dx+c \right ) -1/4\, \left ({\it Arcsinh} \left ( dx+c \right ) \right ) ^{3}+3/4\, \left ( 1+ \left ( dx+c \right ) ^{2} \right ){\it Arcsinh} \left ( dx+c \right ) -3/8\, \left ( dx+c \right ) \sqrt{1+ \left ( dx+c \right ) ^{2}}-3/8\,{\it Arcsinh} \left ( dx+c \right ) \right ) +6\,e{a}^{2}{b}^{2} \left ( 1/2\, \left ({\it Arcsinh} \left ( dx+c \right ) \right ) ^{2} \left ( 1+ \left ( dx+c \right ) ^{2} \right ) -1/2\,{\it Arcsinh} \left ( dx+c \right ) \sqrt{1+ \left ( dx+c \right ) ^{2}} \left ( dx+c \right ) -1/4\, \left ({\it Arcsinh} \left ( dx+c \right ) \right ) ^{2}+1/4\, \left ( dx+c \right ) ^{2}+1/4 \right ) +4\,e{a}^{3}b \left ( 1/2\,{\it Arcsinh} \left ( dx+c \right ) \left ( dx+c \right ) ^{2}-1/4\, \left ( dx+c \right ) \sqrt{1+ \left ( dx+c \right ) ^{2}}+1/4\,{\it Arcsinh} \left ( dx+c \right ) \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*e*x+c*e)*(a+b*arcsinh(d*x+c))^4,x)

[Out]

1/d*(1/2*(d*x+c)^2*e*a^4+e*b^4*(1/2*arcsinh(d*x+c)^4*(1+(d*x+c)^2)-arcsinh(d*x+c)^3*(d*x+c)*(1+(d*x+c)^2)^(1/2
)-1/4*arcsinh(d*x+c)^4+3/2*arcsinh(d*x+c)^2*(1+(d*x+c)^2)-3/2*arcsinh(d*x+c)*(1+(d*x+c)^2)^(1/2)*(d*x+c)-3/4*a
rcsinh(d*x+c)^2+3/4*(d*x+c)^2+3/4)+4*e*a*b^3*(1/2*arcsinh(d*x+c)^3*(1+(d*x+c)^2)-3/4*arcsinh(d*x+c)^2*(1+(d*x+
c)^2)^(1/2)*(d*x+c)-1/4*arcsinh(d*x+c)^3+3/4*(1+(d*x+c)^2)*arcsinh(d*x+c)-3/8*(d*x+c)*(1+(d*x+c)^2)^(1/2)-3/8*
arcsinh(d*x+c))+6*e*a^2*b^2*(1/2*arcsinh(d*x+c)^2*(1+(d*x+c)^2)-1/2*arcsinh(d*x+c)*(1+(d*x+c)^2)^(1/2)*(d*x+c)
-1/4*arcsinh(d*x+c)^2+1/4*(d*x+c)^2+1/4)+4*e*a^3*b*(1/2*arcsinh(d*x+c)*(d*x+c)^2-1/4*(d*x+c)*(1+(d*x+c)^2)^(1/
2)+1/4*arcsinh(d*x+c)))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)*(a+b*arcsinh(d*x+c))^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.96975, size = 1287, normalized size = 6.6 \begin{align*} \frac{{\left (2 \, a^{4} + 6 \, a^{2} b^{2} + 3 \, b^{4}\right )} d^{2} e x^{2} + 2 \,{\left (2 \, a^{4} + 6 \, a^{2} b^{2} + 3 \, b^{4}\right )} c d e x +{\left (2 \, b^{4} d^{2} e x^{2} + 4 \, b^{4} c d e x +{\left (2 \, b^{4} c^{2} + b^{4}\right )} e\right )} \log \left (d x + c + \sqrt{d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{4} + 4 \,{\left (2 \, a b^{3} d^{2} e x^{2} + 4 \, a b^{3} c d e x +{\left (2 \, a b^{3} c^{2} + a b^{3}\right )} e -{\left (b^{4} d e x + b^{4} c e\right )} \sqrt{d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )} \log \left (d x + c + \sqrt{d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{3} + 3 \,{\left (2 \,{\left (2 \, a^{2} b^{2} + b^{4}\right )} d^{2} e x^{2} + 4 \,{\left (2 \, a^{2} b^{2} + b^{4}\right )} c d e x +{\left (2 \, a^{2} b^{2} + b^{4} + 2 \,{\left (2 \, a^{2} b^{2} + b^{4}\right )} c^{2}\right )} e - 4 \,{\left (a b^{3} d e x + a b^{3} c e\right )} \sqrt{d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )} \log \left (d x + c + \sqrt{d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{2} + 2 \,{\left (2 \,{\left (2 \, a^{3} b + 3 \, a b^{3}\right )} d^{2} e x^{2} + 4 \,{\left (2 \, a^{3} b + 3 \, a b^{3}\right )} c d e x +{\left (2 \, a^{3} b + 3 \, a b^{3} + 2 \,{\left (2 \, a^{3} b + 3 \, a b^{3}\right )} c^{2}\right )} e - 3 \,{\left ({\left (2 \, a^{2} b^{2} + b^{4}\right )} d e x +{\left (2 \, a^{2} b^{2} + b^{4}\right )} c e\right )} \sqrt{d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )} \log \left (d x + c + \sqrt{d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) - 2 \,{\left ({\left (2 \, a^{3} b + 3 \, a b^{3}\right )} d e x +{\left (2 \, a^{3} b + 3 \, a b^{3}\right )} c e\right )} \sqrt{d^{2} x^{2} + 2 \, c d x + c^{2} + 1}}{4 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)*(a+b*arcsinh(d*x+c))^4,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 6.31193, size = 1027, normalized size = 5.27 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)*(a+b*asinh(d*x+c))**4,x)

[Out]

Piecewise((a**4*c*e*x + a**4*d*e*x**2/2 + 2*a**3*b*c**2*e*asinh(c + d*x)/d + 4*a**3*b*c*e*x*asinh(c + d*x) - a
**3*b*c*e*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/d + 2*a**3*b*d*e*x**2*asinh(c + d*x) - a**3*b*e*x*sqrt(c**2 + 2
*c*d*x + d**2*x**2 + 1) + a**3*b*e*asinh(c + d*x)/d + 3*a**2*b**2*c**2*e*asinh(c + d*x)**2/d + 6*a**2*b**2*c*e
*x*asinh(c + d*x)**2 + 3*a**2*b**2*c*e*x - 3*a**2*b**2*c*e*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)
/d + 3*a**2*b**2*d*e*x**2*asinh(c + d*x)**2 + 3*a**2*b**2*d*e*x**2/2 - 3*a**2*b**2*e*x*sqrt(c**2 + 2*c*d*x + d
**2*x**2 + 1)*asinh(c + d*x) + 3*a**2*b**2*e*asinh(c + d*x)**2/(2*d) + 2*a*b**3*c**2*e*asinh(c + d*x)**3/d + 3
*a*b**3*c**2*e*asinh(c + d*x)/d + 4*a*b**3*c*e*x*asinh(c + d*x)**3 + 6*a*b**3*c*e*x*asinh(c + d*x) - 3*a*b**3*
c*e*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)**2/d - 3*a*b**3*c*e*sqrt(c**2 + 2*c*d*x + d**2*x**2 +
1)/(2*d) + 2*a*b**3*d*e*x**2*asinh(c + d*x)**3 + 3*a*b**3*d*e*x**2*asinh(c + d*x) - 3*a*b**3*e*x*sqrt(c**2 + 2
*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)**2 - 3*a*b**3*e*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/2 + a*b**3*e*asi
nh(c + d*x)**3/d + 3*a*b**3*e*asinh(c + d*x)/(2*d) + b**4*c**2*e*asinh(c + d*x)**4/(2*d) + 3*b**4*c**2*e*asinh
(c + d*x)**2/(2*d) + b**4*c*e*x*asinh(c + d*x)**4 + 3*b**4*c*e*x*asinh(c + d*x)**2 + 3*b**4*c*e*x/2 - b**4*c*e
*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)**3/d - 3*b**4*c*e*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*as
inh(c + d*x)/(2*d) + b**4*d*e*x**2*asinh(c + d*x)**4/2 + 3*b**4*d*e*x**2*asinh(c + d*x)**2/2 + 3*b**4*d*e*x**2
/4 - b**4*e*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)**3 - 3*b**4*e*x*sqrt(c**2 + 2*c*d*x + d**2*x
**2 + 1)*asinh(c + d*x)/2 + b**4*e*asinh(c + d*x)**4/(4*d) + 3*b**4*e*asinh(c + d*x)**2/(4*d), Ne(d, 0)), (c*e
*x*(a + b*asinh(c))**4, True))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d e x + c e\right )}{\left (b \operatorname{arsinh}\left (d x + c\right ) + a\right )}^{4}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)*(a+b*arcsinh(d*x+c))^4,x, algorithm="giac")

[Out]

integrate((d*e*x + c*e)*(b*arcsinh(d*x + c) + a)^4, x)