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

Optimal. Leaf size=221 \[ d^2 e x^3 \left (a+b \sinh ^{-1}(c x)\right )+d^3 x \left (a+b \sinh ^{-1}(c x)\right )+\frac{3}{5} d e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{7} e^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )-\frac{b e \left (c^2 x^2+1\right )^{3/2} \left (35 c^4 d^2-42 c^2 d e+15 e^2\right )}{105 c^7}-\frac{b \sqrt{c^2 x^2+1} \left (-35 c^4 d^2 e+35 c^6 d^3+21 c^2 d e^2-5 e^3\right )}{35 c^7}-\frac{3 b e^2 \left (c^2 x^2+1\right )^{5/2} \left (7 c^2 d-5 e\right )}{175 c^7}-\frac{b e^3 \left (c^2 x^2+1\right )^{7/2}}{49 c^7} \]

[Out]

-(b*(35*c^6*d^3 - 35*c^4*d^2*e + 21*c^2*d*e^2 - 5*e^3)*Sqrt[1 + c^2*x^2])/(35*c^7) - (b*e*(35*c^4*d^2 - 42*c^2
*d*e + 15*e^2)*(1 + c^2*x^2)^(3/2))/(105*c^7) - (3*b*(7*c^2*d - 5*e)*e^2*(1 + c^2*x^2)^(5/2))/(175*c^7) - (b*e
^3*(1 + c^2*x^2)^(7/2))/(49*c^7) + d^3*x*(a + b*ArcSinh[c*x]) + d^2*e*x^3*(a + b*ArcSinh[c*x]) + (3*d*e^2*x^5*
(a + b*ArcSinh[c*x]))/5 + (e^3*x^7*(a + b*ArcSinh[c*x]))/7

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Rubi [A]  time = 0.260728, antiderivative size = 221, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {194, 5704, 12, 1799, 1850} \[ d^2 e x^3 \left (a+b \sinh ^{-1}(c x)\right )+d^3 x \left (a+b \sinh ^{-1}(c x)\right )+\frac{3}{5} d e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{7} e^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )-\frac{b e \left (c^2 x^2+1\right )^{3/2} \left (35 c^4 d^2-42 c^2 d e+15 e^2\right )}{105 c^7}-\frac{b \sqrt{c^2 x^2+1} \left (-35 c^4 d^2 e+35 c^6 d^3+21 c^2 d e^2-5 e^3\right )}{35 c^7}-\frac{3 b e^2 \left (c^2 x^2+1\right )^{5/2} \left (7 c^2 d-5 e\right )}{175 c^7}-\frac{b e^3 \left (c^2 x^2+1\right )^{7/2}}{49 c^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*(35*c^6*d^3 - 35*c^4*d^2*e + 21*c^2*d*e^2 - 5*e^3)*Sqrt[1 + c^2*x^2])/(35*c^7) - (b*e*(35*c^4*d^2 - 42*c^2
*d*e + 15*e^2)*(1 + c^2*x^2)^(3/2))/(105*c^7) - (3*b*(7*c^2*d - 5*e)*e^2*(1 + c^2*x^2)^(5/2))/(175*c^7) - (b*e
^3*(1 + c^2*x^2)^(7/2))/(49*c^7) + d^3*x*(a + b*ArcSinh[c*x]) + d^2*e*x^3*(a + b*ArcSinh[c*x]) + (3*d*e^2*x^5*
(a + b*ArcSinh[c*x]))/5 + (e^3*x^7*(a + b*ArcSinh[c*x]))/7

Rule 194

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rule 5704

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(d + e*x^2
)^p, x]}, Dist[a + b*ArcSinh[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 + c^2*x^2], x], x], x]] /;
FreeQ[{a, b, c, d, e}, x] && NeQ[e, c^2*d] && (IGtQ[p, 0] || ILtQ[p + 1/2, 0])

Rule 12

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

Rule 1799

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)*SubstFor[x^2,
Pq, x]*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]

Rule 1850

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rubi steps

\begin{align*} \int \left (d+e x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right ) \, dx &=d^3 x \left (a+b \sinh ^{-1}(c x)\right )+d^2 e x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac{3}{5} d e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{7} e^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )-(b c) \int \frac{x \left (35 d^3+35 d^2 e x^2+21 d e^2 x^4+5 e^3 x^6\right )}{35 \sqrt{1+c^2 x^2}} \, dx\\ &=d^3 x \left (a+b \sinh ^{-1}(c x)\right )+d^2 e x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac{3}{5} d e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{7} e^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )-\frac{1}{35} (b c) \int \frac{x \left (35 d^3+35 d^2 e x^2+21 d e^2 x^4+5 e^3 x^6\right )}{\sqrt{1+c^2 x^2}} \, dx\\ &=d^3 x \left (a+b \sinh ^{-1}(c x)\right )+d^2 e x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac{3}{5} d e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{7} e^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )-\frac{1}{70} (b c) \operatorname{Subst}\left (\int \frac{35 d^3+35 d^2 e x+21 d e^2 x^2+5 e^3 x^3}{\sqrt{1+c^2 x}} \, dx,x,x^2\right )\\ &=d^3 x \left (a+b \sinh ^{-1}(c x)\right )+d^2 e x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac{3}{5} d e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{7} e^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )-\frac{1}{70} (b c) \operatorname{Subst}\left (\int \left (\frac{35 c^6 d^3-35 c^4 d^2 e+21 c^2 d e^2-5 e^3}{c^6 \sqrt{1+c^2 x}}+\frac{e \left (35 c^4 d^2-42 c^2 d e+15 e^2\right ) \sqrt{1+c^2 x}}{c^6}+\frac{3 \left (7 c^2 d-5 e\right ) e^2 \left (1+c^2 x\right )^{3/2}}{c^6}+\frac{5 e^3 \left (1+c^2 x\right )^{5/2}}{c^6}\right ) \, dx,x,x^2\right )\\ &=-\frac{b \left (35 c^6 d^3-35 c^4 d^2 e+21 c^2 d e^2-5 e^3\right ) \sqrt{1+c^2 x^2}}{35 c^7}-\frac{b e \left (35 c^4 d^2-42 c^2 d e+15 e^2\right ) \left (1+c^2 x^2\right )^{3/2}}{105 c^7}-\frac{3 b \left (7 c^2 d-5 e\right ) e^2 \left (1+c^2 x^2\right )^{5/2}}{175 c^7}-\frac{b e^3 \left (1+c^2 x^2\right )^{7/2}}{49 c^7}+d^3 x \left (a+b \sinh ^{-1}(c x)\right )+d^2 e x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac{3}{5} d e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{7} e^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )\\ \end{align*}

Mathematica [A]  time = 0.272513, size = 187, normalized size = 0.85 \[ a \left (d^2 e x^3+d^3 x+\frac{3}{5} d e^2 x^5+\frac{e^3 x^7}{7}\right )-\frac{b \sqrt{c^2 x^2+1} \left (c^6 \left (1225 d^2 e x^2+3675 d^3+441 d e^2 x^4+75 e^3 x^6\right )-2 c^4 e \left (1225 d^2+294 d e x^2+45 e^2 x^4\right )+24 c^2 e^2 \left (49 d+5 e x^2\right )-240 e^3\right )}{3675 c^7}+b \sinh ^{-1}(c x) \left (d^2 e x^3+d^3 x+\frac{3}{5} d e^2 x^5+\frac{e^3 x^7}{7}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

a*(d^3*x + d^2*e*x^3 + (3*d*e^2*x^5)/5 + (e^3*x^7)/7) - (b*Sqrt[1 + c^2*x^2]*(-240*e^3 + 24*c^2*e^2*(49*d + 5*
e*x^2) - 2*c^4*e*(1225*d^2 + 294*d*e*x^2 + 45*e^2*x^4) + c^6*(3675*d^3 + 1225*d^2*e*x^2 + 441*d*e^2*x^4 + 75*e
^3*x^6)))/(3675*c^7) + b*(d^3*x + d^2*e*x^3 + (3*d*e^2*x^5)/5 + (e^3*x^7)/7)*ArcSinh[c*x]

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Maple [A]  time = 0.005, size = 316, normalized size = 1.4 \begin{align*}{\frac{1}{c} \left ({\frac{a}{{c}^{6}} \left ({\frac{{e}^{3}{c}^{7}{x}^{7}}{7}}+{\frac{3\,{c}^{7}d{e}^{2}{x}^{5}}{5}}+{c}^{7}{d}^{2}e{x}^{3}+x{c}^{7}{d}^{3} \right ) }+{\frac{b}{{c}^{6}} \left ({\frac{{\it Arcsinh} \left ( cx \right ){e}^{3}{c}^{7}{x}^{7}}{7}}+{\frac{3\,{\it Arcsinh} \left ( cx \right ){c}^{7}d{e}^{2}{x}^{5}}{5}}+{\it Arcsinh} \left ( cx \right ){c}^{7}{d}^{2}e{x}^{3}+{\it Arcsinh} \left ( cx \right ){c}^{7}x{d}^{3}-{\frac{{e}^{3}}{7} \left ({\frac{{c}^{6}{x}^{6}}{7}\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{6\,{c}^{4}{x}^{4}}{35}\sqrt{{c}^{2}{x}^{2}+1}}+{\frac{8\,{c}^{2}{x}^{2}}{35}\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{16}{35}\sqrt{{c}^{2}{x}^{2}+1}} \right ) }-{\frac{3\,{c}^{2}d{e}^{2}}{5} \left ({\frac{{c}^{4}{x}^{4}}{5}\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{4\,{c}^{2}{x}^{2}}{15}\sqrt{{c}^{2}{x}^{2}+1}}+{\frac{8}{15}\sqrt{{c}^{2}{x}^{2}+1}} \right ) }-{c}^{4}{d}^{2}e \left ({\frac{{c}^{2}{x}^{2}}{3}\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{2}{3}\sqrt{{c}^{2}{x}^{2}+1}} \right ) -{d}^{3}{c}^{6}\sqrt{{c}^{2}{x}^{2}+1} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c*(a/c^6*(1/7*e^3*c^7*x^7+3/5*c^7*d*e^2*x^5+c^7*d^2*e*x^3+x*c^7*d^3)+b/c^6*(1/7*arcsinh(c*x)*e^3*c^7*x^7+3/5
*arcsinh(c*x)*c^7*d*e^2*x^5+arcsinh(c*x)*c^7*d^2*e*x^3+arcsinh(c*x)*c^7*x*d^3-1/7*e^3*(1/7*c^6*x^6*(c^2*x^2+1)
^(1/2)-6/35*c^4*x^4*(c^2*x^2+1)^(1/2)+8/35*c^2*x^2*(c^2*x^2+1)^(1/2)-16/35*(c^2*x^2+1)^(1/2))-3/5*c^2*d*e^2*(1
/5*c^4*x^4*(c^2*x^2+1)^(1/2)-4/15*c^2*x^2*(c^2*x^2+1)^(1/2)+8/15*(c^2*x^2+1)^(1/2))-c^4*d^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^3*c^6*(c^2*x^2+1)^(1/2)))

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Maxima [A]  time = 1.24061, size = 387, normalized size = 1.75 \begin{align*} \frac{1}{7} \, a e^{3} x^{7} + \frac{3}{5} \, a d e^{2} x^{5} + a d^{2} e x^{3} + \frac{1}{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 )} b d^{2} e + \frac{1}{25} \,{\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 )} b d e^{2} + \frac{1}{245} \,{\left (35 \, x^{7} \operatorname{arsinh}\left (c x\right ) -{\left (\frac{5 \, \sqrt{c^{2} x^{2} + 1} x^{6}}{c^{2}} - \frac{6 \, \sqrt{c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac{8 \, \sqrt{c^{2} x^{2} + 1} x^{2}}{c^{6}} - \frac{16 \, \sqrt{c^{2} x^{2} + 1}}{c^{8}}\right )} c\right )} b e^{3} + a d^{3} x + \frac{{\left (c x \operatorname{arsinh}\left (c x\right ) - \sqrt{c^{2} x^{2} + 1}\right )} b d^{3}}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*a*e^3*x^7 + 3/5*a*d*e^2*x^5 + a*d^2*e*x^3 + 1/3*(3*x^3*arcsinh(c*x) - c*(sqrt(c^2*x^2 + 1)*x^2/c^2 - 2*sqr
t(c^2*x^2 + 1)/c^4))*b*d^2*e + 1/25*(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)*b*d*e^2 + 1/245*(35*x^7*arcsinh(c*x) - (5*sqrt(c^2*x^2 + 1)*x^6/c^2 - 6*
sqrt(c^2*x^2 + 1)*x^4/c^4 + 8*sqrt(c^2*x^2 + 1)*x^2/c^6 - 16*sqrt(c^2*x^2 + 1)/c^8)*c)*b*e^3 + a*d^3*x + (c*x*
arcsinh(c*x) - sqrt(c^2*x^2 + 1))*b*d^3/c

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Fricas [A]  time = 2.54864, size = 559, normalized size = 2.53 \begin{align*} \frac{525 \, a c^{7} e^{3} x^{7} + 2205 \, a c^{7} d e^{2} x^{5} + 3675 \, a c^{7} d^{2} e x^{3} + 3675 \, a c^{7} d^{3} x + 105 \,{\left (5 \, b c^{7} e^{3} x^{7} + 21 \, b c^{7} d e^{2} x^{5} + 35 \, b c^{7} d^{2} e x^{3} + 35 \, b c^{7} d^{3} x\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) -{\left (75 \, b c^{6} e^{3} x^{6} + 3675 \, b c^{6} d^{3} - 2450 \, b c^{4} d^{2} e + 1176 \, b c^{2} d e^{2} + 9 \,{\left (49 \, b c^{6} d e^{2} - 10 \, b c^{4} e^{3}\right )} x^{4} - 240 \, b e^{3} +{\left (1225 \, b c^{6} d^{2} e - 588 \, b c^{4} d e^{2} + 120 \, b c^{2} e^{3}\right )} x^{2}\right )} \sqrt{c^{2} x^{2} + 1}}{3675 \, c^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3675*(525*a*c^7*e^3*x^7 + 2205*a*c^7*d*e^2*x^5 + 3675*a*c^7*d^2*e*x^3 + 3675*a*c^7*d^3*x + 105*(5*b*c^7*e^3*
x^7 + 21*b*c^7*d*e^2*x^5 + 35*b*c^7*d^2*e*x^3 + 35*b*c^7*d^3*x)*log(c*x + sqrt(c^2*x^2 + 1)) - (75*b*c^6*e^3*x
^6 + 3675*b*c^6*d^3 - 2450*b*c^4*d^2*e + 1176*b*c^2*d*e^2 + 9*(49*b*c^6*d*e^2 - 10*b*c^4*e^3)*x^4 - 240*b*e^3
+ (1225*b*c^6*d^2*e - 588*b*c^4*d*e^2 + 120*b*c^2*e^3)*x^2)*sqrt(c^2*x^2 + 1))/c^7

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Sympy [A]  time = 8.74923, size = 389, normalized size = 1.76 \begin{align*} \begin{cases} a d^{3} x + a d^{2} e x^{3} + \frac{3 a d e^{2} x^{5}}{5} + \frac{a e^{3} x^{7}}{7} + b d^{3} x \operatorname{asinh}{\left (c x \right )} + b d^{2} e x^{3} \operatorname{asinh}{\left (c x \right )} + \frac{3 b d e^{2} x^{5} \operatorname{asinh}{\left (c x \right )}}{5} + \frac{b e^{3} x^{7} \operatorname{asinh}{\left (c x \right )}}{7} - \frac{b d^{3} \sqrt{c^{2} x^{2} + 1}}{c} - \frac{b d^{2} e x^{2} \sqrt{c^{2} x^{2} + 1}}{3 c} - \frac{3 b d e^{2} x^{4} \sqrt{c^{2} x^{2} + 1}}{25 c} - \frac{b e^{3} x^{6} \sqrt{c^{2} x^{2} + 1}}{49 c} + \frac{2 b d^{2} e \sqrt{c^{2} x^{2} + 1}}{3 c^{3}} + \frac{4 b d e^{2} x^{2} \sqrt{c^{2} x^{2} + 1}}{25 c^{3}} + \frac{6 b e^{3} x^{4} \sqrt{c^{2} x^{2} + 1}}{245 c^{3}} - \frac{8 b d e^{2} \sqrt{c^{2} x^{2} + 1}}{25 c^{5}} - \frac{8 b e^{3} x^{2} \sqrt{c^{2} x^{2} + 1}}{245 c^{5}} + \frac{16 b e^{3} \sqrt{c^{2} x^{2} + 1}}{245 c^{7}} & \text{for}\: c \neq 0 \\a \left (d^{3} x + d^{2} e x^{3} + \frac{3 d e^{2} x^{5}}{5} + \frac{e^{3} x^{7}}{7}\right ) & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*d**3*x + a*d**2*e*x**3 + 3*a*d*e**2*x**5/5 + a*e**3*x**7/7 + b*d**3*x*asinh(c*x) + b*d**2*e*x**3*
asinh(c*x) + 3*b*d*e**2*x**5*asinh(c*x)/5 + b*e**3*x**7*asinh(c*x)/7 - b*d**3*sqrt(c**2*x**2 + 1)/c - b*d**2*e
*x**2*sqrt(c**2*x**2 + 1)/(3*c) - 3*b*d*e**2*x**4*sqrt(c**2*x**2 + 1)/(25*c) - b*e**3*x**6*sqrt(c**2*x**2 + 1)
/(49*c) + 2*b*d**2*e*sqrt(c**2*x**2 + 1)/(3*c**3) + 4*b*d*e**2*x**2*sqrt(c**2*x**2 + 1)/(25*c**3) + 6*b*e**3*x
**4*sqrt(c**2*x**2 + 1)/(245*c**3) - 8*b*d*e**2*sqrt(c**2*x**2 + 1)/(25*c**5) - 8*b*e**3*x**2*sqrt(c**2*x**2 +
1)/(245*c**5) + 16*b*e**3*sqrt(c**2*x**2 + 1)/(245*c**7), Ne(c, 0)), (a*(d**3*x + d**2*e*x**3 + 3*d*e**2*x**5
/5 + e**3*x**7/7), True))

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Giac [A]  time = 1.8443, size = 396, normalized size = 1.79 \begin{align*}{\left (x \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) - \frac{\sqrt{c^{2} x^{2} + 1}}{c}\right )} b d^{3} + a d^{3} x + \frac{1}{245} \,{\left (35 \, a x^{7} +{\left (35 \, x^{7} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) - \frac{5 \,{\left (c^{2} x^{2} + 1\right )}^{\frac{7}{2}} - 21 \,{\left (c^{2} x^{2} + 1\right )}^{\frac{5}{2}} + 35 \,{\left (c^{2} x^{2} + 1\right )}^{\frac{3}{2}} - 35 \, \sqrt{c^{2} x^{2} + 1}}{c^{7}}\right )} b\right )} e^{3} + \frac{1}{25} \,{\left (15 \, a d x^{5} +{\left (15 \, x^{5} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) - \frac{3 \,{\left (c^{2} x^{2} + 1\right )}^{\frac{5}{2}} - 10 \,{\left (c^{2} x^{2} + 1\right )}^{\frac{3}{2}} + 15 \, \sqrt{c^{2} x^{2} + 1}}{c^{5}}\right )} b d\right )} e^{2} + \frac{1}{3} \,{\left (3 \, a d^{2} x^{3} +{\left (3 \, x^{3} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) - \frac{{\left (c^{2} x^{2} + 1\right )}^{\frac{3}{2}} - 3 \, \sqrt{c^{2} x^{2} + 1}}{c^{3}}\right )} b d^{2}\right )} e \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x*log(c*x + sqrt(c^2*x^2 + 1)) - sqrt(c^2*x^2 + 1)/c)*b*d^3 + a*d^3*x + 1/245*(35*a*x^7 + (35*x^7*log(c*x + s
qrt(c^2*x^2 + 1)) - (5*(c^2*x^2 + 1)^(7/2) - 21*(c^2*x^2 + 1)^(5/2) + 35*(c^2*x^2 + 1)^(3/2) - 35*sqrt(c^2*x^2
+ 1))/c^7)*b)*e^3 + 1/25*(15*a*d*x^5 + (15*x^5*log(c*x + sqrt(c^2*x^2 + 1)) - (3*(c^2*x^2 + 1)^(5/2) - 10*(c^
2*x^2 + 1)^(3/2) + 15*sqrt(c^2*x^2 + 1))/c^5)*b*d)*e^2 + 1/3*(3*a*d^2*x^3 + (3*x^3*log(c*x + sqrt(c^2*x^2 + 1)
) - ((c^2*x^2 + 1)^(3/2) - 3*sqrt(c^2*x^2 + 1))/c^3)*b*d^2)*e