3.7.6 \(\int (d+e x^2)^4 (a+b \sinh ^{-1}(c x)) \, dx\) [606]

Optimal. Leaf size=312 \[ -\frac {b \left (315 c^8 d^4-420 c^6 d^3 e+378 c^4 d^2 e^2-180 c^2 d e^3+35 e^4\right ) \sqrt {1+c^2 x^2}}{315 c^9}-\frac {4 b e \left (105 c^6 d^3-189 c^4 d^2 e+135 c^2 d e^2-35 e^3\right ) \left (1+c^2 x^2\right )^{3/2}}{945 c^9}-\frac {2 b e^2 \left (63 c^4 d^2-90 c^2 d e+35 e^2\right ) \left (1+c^2 x^2\right )^{5/2}}{525 c^9}-\frac {4 b \left (9 c^2 d-7 e\right ) e^3 \left (1+c^2 x^2\right )^{7/2}}{441 c^9}-\frac {b e^4 \left (1+c^2 x^2\right )^{9/2}}{81 c^9}+d^4 x \left (a+b \sinh ^{-1}(c x)\right )+\frac {4}{3} d^3 e x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac {6}{5} d^2 e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac {4}{7} d e^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{9} e^4 x^9 \left (a+b \sinh ^{-1}(c x)\right ) \]

[Out]

-4/945*b*e*(105*c^6*d^3-189*c^4*d^2*e+135*c^2*d*e^2-35*e^3)*(c^2*x^2+1)^(3/2)/c^9-2/525*b*e^2*(63*c^4*d^2-90*c
^2*d*e+35*e^2)*(c^2*x^2+1)^(5/2)/c^9-4/441*b*(9*c^2*d-7*e)*e^3*(c^2*x^2+1)^(7/2)/c^9-1/81*b*e^4*(c^2*x^2+1)^(9
/2)/c^9+d^4*x*(a+b*arcsinh(c*x))+4/3*d^3*e*x^3*(a+b*arcsinh(c*x))+6/5*d^2*e^2*x^5*(a+b*arcsinh(c*x))+4/7*d*e^3
*x^7*(a+b*arcsinh(c*x))+1/9*e^4*x^9*(a+b*arcsinh(c*x))-1/315*b*(315*c^8*d^4-420*c^6*d^3*e+378*c^4*d^2*e^2-180*
c^2*d*e^3+35*e^4)*(c^2*x^2+1)^(1/2)/c^9

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Rubi [A]
time = 0.24, antiderivative size = 312, normalized size of antiderivative = 1.00, 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 = {200, 5792, 12, 1813, 1864} \begin {gather*} d^4 x \left (a+b \sinh ^{-1}(c x)\right )+\frac {4}{3} d^3 e x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac {6}{5} d^2 e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac {4}{7} d e^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{9} e^4 x^9 \left (a+b \sinh ^{-1}(c x)\right )-\frac {4 b e^3 \left (c^2 x^2+1\right )^{7/2} \left (9 c^2 d-7 e\right )}{441 c^9}-\frac {b e^4 \left (c^2 x^2+1\right )^{9/2}}{81 c^9}-\frac {2 b e^2 \left (c^2 x^2+1\right )^{5/2} \left (63 c^4 d^2-90 c^2 d e+35 e^2\right )}{525 c^9}-\frac {4 b e \left (c^2 x^2+1\right )^{3/2} \left (105 c^6 d^3-189 c^4 d^2 e+135 c^2 d e^2-35 e^3\right )}{945 c^9}-\frac {b \sqrt {c^2 x^2+1} \left (315 c^8 d^4-420 c^6 d^3 e+378 c^4 d^2 e^2-180 c^2 d e^3+35 e^4\right )}{315 c^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x^2)^4*(a + b*ArcSinh[c*x]),x]

[Out]

-1/315*(b*(315*c^8*d^4 - 420*c^6*d^3*e + 378*c^4*d^2*e^2 - 180*c^2*d*e^3 + 35*e^4)*Sqrt[1 + c^2*x^2])/c^9 - (4
*b*e*(105*c^6*d^3 - 189*c^4*d^2*e + 135*c^2*d*e^2 - 35*e^3)*(1 + c^2*x^2)^(3/2))/(945*c^9) - (2*b*e^2*(63*c^4*
d^2 - 90*c^2*d*e + 35*e^2)*(1 + c^2*x^2)^(5/2))/(525*c^9) - (4*b*(9*c^2*d - 7*e)*e^3*(1 + c^2*x^2)^(7/2))/(441
*c^9) - (b*e^4*(1 + c^2*x^2)^(9/2))/(81*c^9) + d^4*x*(a + b*ArcSinh[c*x]) + (4*d^3*e*x^3*(a + b*ArcSinh[c*x]))
/3 + (6*d^2*e^2*x^5*(a + b*ArcSinh[c*x]))/5 + (4*d*e^3*x^7*(a + b*ArcSinh[c*x]))/7 + (e^4*x^9*(a + b*ArcSinh[c
*x]))/9

Rule 12

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

Rule 200

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 1813

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 1864

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])

Rule 5792

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])

Rubi steps

\begin {align*} \int \left (d+e x^2\right )^4 \left (a+b \sinh ^{-1}(c x)\right ) \, dx &=d^4 x \left (a+b \sinh ^{-1}(c x)\right )+\frac {4}{3} d^3 e x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac {6}{5} d^2 e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac {4}{7} d e^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{9} e^4 x^9 \left (a+b \sinh ^{-1}(c x)\right )-(b c) \int \frac {x \left (315 d^4+420 d^3 e x^2+378 d^2 e^2 x^4+180 d e^3 x^6+35 e^4 x^8\right )}{315 \sqrt {1+c^2 x^2}} \, dx\\ &=d^4 x \left (a+b \sinh ^{-1}(c x)\right )+\frac {4}{3} d^3 e x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac {6}{5} d^2 e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac {4}{7} d e^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{9} e^4 x^9 \left (a+b \sinh ^{-1}(c x)\right )-\frac {1}{315} (b c) \int \frac {x \left (315 d^4+420 d^3 e x^2+378 d^2 e^2 x^4+180 d e^3 x^6+35 e^4 x^8\right )}{\sqrt {1+c^2 x^2}} \, dx\\ &=d^4 x \left (a+b \sinh ^{-1}(c x)\right )+\frac {4}{3} d^3 e x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac {6}{5} d^2 e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac {4}{7} d e^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{9} e^4 x^9 \left (a+b \sinh ^{-1}(c x)\right )-\frac {1}{630} (b c) \text {Subst}\left (\int \frac {315 d^4+420 d^3 e x+378 d^2 e^2 x^2+180 d e^3 x^3+35 e^4 x^4}{\sqrt {1+c^2 x}} \, dx,x,x^2\right )\\ &=d^4 x \left (a+b \sinh ^{-1}(c x)\right )+\frac {4}{3} d^3 e x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac {6}{5} d^2 e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac {4}{7} d e^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{9} e^4 x^9 \left (a+b \sinh ^{-1}(c x)\right )-\frac {1}{630} (b c) \text {Subst}\left (\int \left (\frac {315 c^8 d^4-420 c^6 d^3 e+378 c^4 d^2 e^2-180 c^2 d e^3+35 e^4}{c^8 \sqrt {1+c^2 x}}+\frac {4 e \left (105 c^6 d^3-189 c^4 d^2 e+135 c^2 d e^2-35 e^3\right ) \sqrt {1+c^2 x}}{c^8}+\frac {6 e^2 \left (63 c^4 d^2-90 c^2 d e+35 e^2\right ) \left (1+c^2 x\right )^{3/2}}{c^8}+\frac {20 \left (9 c^2 d-7 e\right ) e^3 \left (1+c^2 x\right )^{5/2}}{c^8}+\frac {35 e^4 \left (1+c^2 x\right )^{7/2}}{c^8}\right ) \, dx,x,x^2\right )\\ &=-\frac {b \left (315 c^8 d^4-420 c^6 d^3 e+378 c^4 d^2 e^2-180 c^2 d e^3+35 e^4\right ) \sqrt {1+c^2 x^2}}{315 c^9}-\frac {4 b e \left (105 c^6 d^3-189 c^4 d^2 e+135 c^2 d e^2-35 e^3\right ) \left (1+c^2 x^2\right )^{3/2}}{945 c^9}-\frac {2 b e^2 \left (63 c^4 d^2-90 c^2 d e+35 e^2\right ) \left (1+c^2 x^2\right )^{5/2}}{525 c^9}-\frac {4 b \left (9 c^2 d-7 e\right ) e^3 \left (1+c^2 x^2\right )^{7/2}}{441 c^9}-\frac {b e^4 \left (1+c^2 x^2\right )^{9/2}}{81 c^9}+d^4 x \left (a+b \sinh ^{-1}(c x)\right )+\frac {4}{3} d^3 e x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac {6}{5} d^2 e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac {4}{7} d e^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{9} e^4 x^9 \left (a+b \sinh ^{-1}(c x)\right )\\ \end {align*}

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Mathematica [A]
time = 0.19, size = 260, normalized size = 0.83 \begin {gather*} \frac {315 a x \left (315 d^4+420 d^3 e x^2+378 d^2 e^2 x^4+180 d e^3 x^6+35 e^4 x^8\right )-\frac {b \sqrt {1+c^2 x^2} \left (4480 e^4-320 c^2 e^3 \left (81 d+7 e x^2\right )+48 c^4 e^2 \left (1323 d^2+270 d e x^2+35 e^2 x^4\right )-8 c^6 e \left (11025 d^3+3969 d^2 e x^2+1215 d e^2 x^4+175 e^3 x^6\right )+c^8 \left (99225 d^4+44100 d^3 e x^2+23814 d^2 e^2 x^4+8100 d e^3 x^6+1225 e^4 x^8\right )\right )}{c^9}+315 b x \left (315 d^4+420 d^3 e x^2+378 d^2 e^2 x^4+180 d e^3 x^6+35 e^4 x^8\right ) \sinh ^{-1}(c x)}{99225} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x^2)^4*(a + b*ArcSinh[c*x]),x]

[Out]

(315*a*x*(315*d^4 + 420*d^3*e*x^2 + 378*d^2*e^2*x^4 + 180*d*e^3*x^6 + 35*e^4*x^8) - (b*Sqrt[1 + c^2*x^2]*(4480
*e^4 - 320*c^2*e^3*(81*d + 7*e*x^2) + 48*c^4*e^2*(1323*d^2 + 270*d*e*x^2 + 35*e^2*x^4) - 8*c^6*e*(11025*d^3 +
3969*d^2*e*x^2 + 1215*d*e^2*x^4 + 175*e^3*x^6) + c^8*(99225*d^4 + 44100*d^3*e*x^2 + 23814*d^2*e^2*x^4 + 8100*d
*e^3*x^6 + 1225*e^4*x^8)))/c^9 + 315*b*x*(315*d^4 + 420*d^3*e*x^2 + 378*d^2*e^2*x^4 + 180*d*e^3*x^6 + 35*e^4*x
^8)*ArcSinh[c*x])/99225

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Maple [A]
time = 0.64, size = 451, normalized size = 1.45

method	result	size
derivativedivides	\(\frac {\frac {a \left (d^{4} c^{9} x +\frac {4}{3} d^{3} c^{9} e \,x^{3}+\frac {6}{5} d^{2} c^{9} e^{2} x^{5}+\frac {4}{7} d \,c^{9} e^{3} x^{7}+\frac {1}{9} e^{4} c^{9} x^{9}\right )}{c^{8}}+\frac {b \left (\arcsinh \left (c x \right ) d^{4} c^{9} x +\frac {4 \arcsinh \left (c x \right ) d^{3} c^{9} e \,x^{3}}{3}+\frac {6 \arcsinh \left (c x \right ) d^{2} c^{9} e^{2} x^{5}}{5}+\frac {4 \arcsinh \left (c x \right ) d \,c^{9} e^{3} x^{7}}{7}+\frac {\arcsinh \left (c x \right ) e^{4} c^{9} x^{9}}{9}-d^{4} c^{8} \sqrt {c^{2} x^{2}+1}-\frac {4 d^{3} c^{6} e \left (\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {c^{2} x^{2}+1}}{3}\right )}{3}-\frac {6 d^{2} c^{4} e^{2} \left (\frac {\sqrt {c^{2} x^{2}+1}\, c^{4} x^{4}}{5}-\frac {4 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{15}+\frac {8 \sqrt {c^{2} x^{2}+1}}{15}\right )}{5}-\frac {4 d \,c^{2} e^{3} \left (\frac {\sqrt {c^{2} x^{2}+1}\, c^{6} x^{6}}{7}-\frac {6 \sqrt {c^{2} x^{2}+1}\, c^{4} x^{4}}{35}+\frac {8 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{35}-\frac {16 \sqrt {c^{2} x^{2}+1}}{35}\right )}{7}-\frac {e^{4} \left (\frac {\sqrt {c^{2} x^{2}+1}\, c^{8} x^{8}}{9}-\frac {8 \sqrt {c^{2} x^{2}+1}\, c^{6} x^{6}}{63}+\frac {16 \sqrt {c^{2} x^{2}+1}\, c^{4} x^{4}}{105}-\frac {64 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{315}+\frac {128 \sqrt {c^{2} x^{2}+1}}{315}\right )}{9}\right )}{c^{8}}}{c}\)	\(451\)
default	\(\frac {\frac {a \left (d^{4} c^{9} x +\frac {4}{3} d^{3} c^{9} e \,x^{3}+\frac {6}{5} d^{2} c^{9} e^{2} x^{5}+\frac {4}{7} d \,c^{9} e^{3} x^{7}+\frac {1}{9} e^{4} c^{9} x^{9}\right )}{c^{8}}+\frac {b \left (\arcsinh \left (c x \right ) d^{4} c^{9} x +\frac {4 \arcsinh \left (c x \right ) d^{3} c^{9} e \,x^{3}}{3}+\frac {6 \arcsinh \left (c x \right ) d^{2} c^{9} e^{2} x^{5}}{5}+\frac {4 \arcsinh \left (c x \right ) d \,c^{9} e^{3} x^{7}}{7}+\frac {\arcsinh \left (c x \right ) e^{4} c^{9} x^{9}}{9}-d^{4} c^{8} \sqrt {c^{2} x^{2}+1}-\frac {4 d^{3} c^{6} e \left (\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {c^{2} x^{2}+1}}{3}\right )}{3}-\frac {6 d^{2} c^{4} e^{2} \left (\frac {\sqrt {c^{2} x^{2}+1}\, c^{4} x^{4}}{5}-\frac {4 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{15}+\frac {8 \sqrt {c^{2} x^{2}+1}}{15}\right )}{5}-\frac {4 d \,c^{2} e^{3} \left (\frac {\sqrt {c^{2} x^{2}+1}\, c^{6} x^{6}}{7}-\frac {6 \sqrt {c^{2} x^{2}+1}\, c^{4} x^{4}}{35}+\frac {8 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{35}-\frac {16 \sqrt {c^{2} x^{2}+1}}{35}\right )}{7}-\frac {e^{4} \left (\frac {\sqrt {c^{2} x^{2}+1}\, c^{8} x^{8}}{9}-\frac {8 \sqrt {c^{2} x^{2}+1}\, c^{6} x^{6}}{63}+\frac {16 \sqrt {c^{2} x^{2}+1}\, c^{4} x^{4}}{105}-\frac {64 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{315}+\frac {128 \sqrt {c^{2} x^{2}+1}}{315}\right )}{9}\right )}{c^{8}}}{c}\)	\(451\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d)^4*(a+b*arcsinh(c*x)),x,method=_RETURNVERBOSE)

[Out]

1/c*(a/c^8*(d^4*c^9*x+4/3*d^3*c^9*e*x^3+6/5*d^2*c^9*e^2*x^5+4/7*d*c^9*e^3*x^7+1/9*e^4*c^9*x^9)+b/c^8*(arcsinh(
c*x)*d^4*c^9*x+4/3*arcsinh(c*x)*d^3*c^9*e*x^3+6/5*arcsinh(c*x)*d^2*c^9*e^2*x^5+4/7*arcsinh(c*x)*d*c^9*e^3*x^7+
1/9*arcsinh(c*x)*e^4*c^9*x^9-d^4*c^8*(c^2*x^2+1)^(1/2)-4/3*d^3*c^6*e*(1/3*c^2*x^2*(c^2*x^2+1)^(1/2)-2/3*(c^2*x
^2+1)^(1/2))-6/5*d^2*c^4*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))-4/7*d*c^2*e^3*(1/7*(c^2*x^2+1)^(1/2)*c^6*x^6-6/35*(c^2*x^2+1)^(1/2)*c^4*x^4+8/35*c^2*x^2*(c^2*x^2+1)^(1/2
)-16/35*(c^2*x^2+1)^(1/2))-1/9*e^4*(1/9*(c^2*x^2+1)^(1/2)*c^8*x^8-8/63*(c^2*x^2+1)^(1/2)*c^6*x^6+16/105*(c^2*x
^2+1)^(1/2)*c^4*x^4-64/315*c^2*x^2*(c^2*x^2+1)^(1/2)+128/315*(c^2*x^2+1)^(1/2))))

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Maxima [A]
time = 0.29, size = 411, normalized size = 1.32 \begin {gather*} \frac {1}{9} \, a x^{9} e^{4} + \frac {4}{7} \, a d x^{7} e^{3} + \frac {6}{5} \, a d^{2} x^{5} e^{2} + \frac {4}{3} \, a d^{3} x^{3} e + a d^{4} 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 )} b d^{3} e + \frac {{\left (c x \operatorname {arsinh}\left (c x\right ) - \sqrt {c^{2} x^{2} + 1}\right )} b d^{4}}{c} + \frac {2}{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^{2} e^{2} + \frac {4}{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 d e^{3} + \frac {1}{2835} \, {\left (315 \, x^{9} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {35 \, \sqrt {c^{2} x^{2} + 1} x^{8}}{c^{2}} - \frac {40 \, \sqrt {c^{2} x^{2} + 1} x^{6}}{c^{4}} + \frac {48 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{6}} - \frac {64 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{8}} + \frac {128 \, \sqrt {c^{2} x^{2} + 1}}{c^{10}}\right )} c\right )} b e^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^4*(a+b*arcsinh(c*x)),x, algorithm="maxima")

[Out]

1/9*a*x^9*e^4 + 4/7*a*d*x^7*e^3 + 6/5*a*d^2*x^5*e^2 + 4/3*a*d^3*x^3*e + a*d^4*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))*b*d^3*e + (c*x*arcsinh(c*x) - sqrt(c^2*x^2 + 1))*b*d^4/
c + 2/25*(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^2*e^2 + 4/245*(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*d*e^3 + 1/2835*(315*x^9*arcsinh(c*x) - (35*sqr
t(c^2*x^2 + 1)*x^8/c^2 - 40*sqrt(c^2*x^2 + 1)*x^6/c^4 + 48*sqrt(c^2*x^2 + 1)*x^4/c^6 - 64*sqrt(c^2*x^2 + 1)*x^
2/c^8 + 128*sqrt(c^2*x^2 + 1)/c^10)*c)*b*e^4

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1039 vs. \(2 (278) = 556\).
time = 0.38, size = 1039, normalized size = 3.33 \begin {gather*} \frac {11025 \, a c^{9} x^{9} \cosh \left (1\right )^{4} + 11025 \, a c^{9} x^{9} \sinh \left (1\right )^{4} + 56700 \, a c^{9} d x^{7} \cosh \left (1\right )^{3} + 119070 \, a c^{9} d^{2} x^{5} \cosh \left (1\right )^{2} + 132300 \, a c^{9} d^{3} x^{3} \cosh \left (1\right ) + 99225 \, a c^{9} d^{4} x + 6300 \, {\left (7 \, a c^{9} x^{9} \cosh \left (1\right ) + 9 \, a c^{9} d x^{7}\right )} \sinh \left (1\right )^{3} + 1890 \, {\left (35 \, a c^{9} x^{9} \cosh \left (1\right )^{2} + 90 \, a c^{9} d x^{7} \cosh \left (1\right ) + 63 \, a c^{9} d^{2} x^{5}\right )} \sinh \left (1\right )^{2} + 315 \, {\left (35 \, b c^{9} x^{9} \cosh \left (1\right )^{4} + 35 \, b c^{9} x^{9} \sinh \left (1\right )^{4} + 180 \, b c^{9} d x^{7} \cosh \left (1\right )^{3} + 378 \, b c^{9} d^{2} x^{5} \cosh \left (1\right )^{2} + 420 \, b c^{9} d^{3} x^{3} \cosh \left (1\right ) + 315 \, b c^{9} d^{4} x + 20 \, {\left (7 \, b c^{9} x^{9} \cosh \left (1\right ) + 9 \, b c^{9} d x^{7}\right )} \sinh \left (1\right )^{3} + 6 \, {\left (35 \, b c^{9} x^{9} \cosh \left (1\right )^{2} + 90 \, b c^{9} d x^{7} \cosh \left (1\right ) + 63 \, b c^{9} d^{2} x^{5}\right )} \sinh \left (1\right )^{2} + 4 \, {\left (35 \, b c^{9} x^{9} \cosh \left (1\right )^{3} + 135 \, b c^{9} d x^{7} \cosh \left (1\right )^{2} + 189 \, b c^{9} d^{2} x^{5} \cosh \left (1\right ) + 105 \, b c^{9} d^{3} x^{3}\right )} \sinh \left (1\right )\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + 1260 \, {\left (35 \, a c^{9} x^{9} \cosh \left (1\right )^{3} + 135 \, a c^{9} d x^{7} \cosh \left (1\right )^{2} + 189 \, a c^{9} d^{2} x^{5} \cosh \left (1\right ) + 105 \, a c^{9} d^{3} x^{3}\right )} \sinh \left (1\right ) - {\left (99225 \, b c^{8} d^{4} + 35 \, {\left (35 \, b c^{8} x^{8} - 40 \, b c^{6} x^{6} + 48 \, b c^{4} x^{4} - 64 \, b c^{2} x^{2} + 128 \, b\right )} \cosh \left (1\right )^{4} + 35 \, {\left (35 \, b c^{8} x^{8} - 40 \, b c^{6} x^{6} + 48 \, b c^{4} x^{4} - 64 \, b c^{2} x^{2} + 128 \, b\right )} \sinh \left (1\right )^{4} + 1620 \, {\left (5 \, b c^{8} d x^{6} - 6 \, b c^{6} d x^{4} + 8 \, b c^{4} d x^{2} - 16 \, b c^{2} d\right )} \cosh \left (1\right )^{3} + 20 \, {\left (405 \, b c^{8} d x^{6} - 486 \, b c^{6} d x^{4} + 648 \, b c^{4} d x^{2} - 1296 \, b c^{2} d + 7 \, {\left (35 \, b c^{8} x^{8} - 40 \, b c^{6} x^{6} + 48 \, b c^{4} x^{4} - 64 \, b c^{2} x^{2} + 128 \, b\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )^{3} + 7938 \, {\left (3 \, b c^{8} d^{2} x^{4} - 4 \, b c^{6} d^{2} x^{2} + 8 \, b c^{4} d^{2}\right )} \cosh \left (1\right )^{2} + 6 \, {\left (3969 \, b c^{8} d^{2} x^{4} - 5292 \, b c^{6} d^{2} x^{2} + 10584 \, b c^{4} d^{2} + 35 \, {\left (35 \, b c^{8} x^{8} - 40 \, b c^{6} x^{6} + 48 \, b c^{4} x^{4} - 64 \, b c^{2} x^{2} + 128 \, b\right )} \cosh \left (1\right )^{2} + 810 \, {\left (5 \, b c^{8} d x^{6} - 6 \, b c^{6} d x^{4} + 8 \, b c^{4} d x^{2} - 16 \, b c^{2} d\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )^{2} + 44100 \, {\left (b c^{8} d^{3} x^{2} - 2 \, b c^{6} d^{3}\right )} \cosh \left (1\right ) + 4 \, {\left (11025 \, b c^{8} d^{3} x^{2} - 22050 \, b c^{6} d^{3} + 35 \, {\left (35 \, b c^{8} x^{8} - 40 \, b c^{6} x^{6} + 48 \, b c^{4} x^{4} - 64 \, b c^{2} x^{2} + 128 \, b\right )} \cosh \left (1\right )^{3} + 1215 \, {\left (5 \, b c^{8} d x^{6} - 6 \, b c^{6} d x^{4} + 8 \, b c^{4} d x^{2} - 16 \, b c^{2} d\right )} \cosh \left (1\right )^{2} + 3969 \, {\left (3 \, b c^{8} d^{2} x^{4} - 4 \, b c^{6} d^{2} x^{2} + 8 \, b c^{4} d^{2}\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \sqrt {c^{2} x^{2} + 1}}{99225 \, c^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^4*(a+b*arcsinh(c*x)),x, algorithm="fricas")

[Out]

1/99225*(11025*a*c^9*x^9*cosh(1)^4 + 11025*a*c^9*x^9*sinh(1)^4 + 56700*a*c^9*d*x^7*cosh(1)^3 + 119070*a*c^9*d^
2*x^5*cosh(1)^2 + 132300*a*c^9*d^3*x^3*cosh(1) + 99225*a*c^9*d^4*x + 6300*(7*a*c^9*x^9*cosh(1) + 9*a*c^9*d*x^7
)*sinh(1)^3 + 1890*(35*a*c^9*x^9*cosh(1)^2 + 90*a*c^9*d*x^7*cosh(1) + 63*a*c^9*d^2*x^5)*sinh(1)^2 + 315*(35*b*
c^9*x^9*cosh(1)^4 + 35*b*c^9*x^9*sinh(1)^4 + 180*b*c^9*d*x^7*cosh(1)^3 + 378*b*c^9*d^2*x^5*cosh(1)^2 + 420*b*c
^9*d^3*x^3*cosh(1) + 315*b*c^9*d^4*x + 20*(7*b*c^9*x^9*cosh(1) + 9*b*c^9*d*x^7)*sinh(1)^3 + 6*(35*b*c^9*x^9*co
sh(1)^2 + 90*b*c^9*d*x^7*cosh(1) + 63*b*c^9*d^2*x^5)*sinh(1)^2 + 4*(35*b*c^9*x^9*cosh(1)^3 + 135*b*c^9*d*x^7*c
osh(1)^2 + 189*b*c^9*d^2*x^5*cosh(1) + 105*b*c^9*d^3*x^3)*sinh(1))*log(c*x + sqrt(c^2*x^2 + 1)) + 1260*(35*a*c
^9*x^9*cosh(1)^3 + 135*a*c^9*d*x^7*cosh(1)^2 + 189*a*c^9*d^2*x^5*cosh(1) + 105*a*c^9*d^3*x^3)*sinh(1) - (99225
*b*c^8*d^4 + 35*(35*b*c^8*x^8 - 40*b*c^6*x^6 + 48*b*c^4*x^4 - 64*b*c^2*x^2 + 128*b)*cosh(1)^4 + 35*(35*b*c^8*x
^8 - 40*b*c^6*x^6 + 48*b*c^4*x^4 - 64*b*c^2*x^2 + 128*b)*sinh(1)^4 + 1620*(5*b*c^8*d*x^6 - 6*b*c^6*d*x^4 + 8*b
*c^4*d*x^2 - 16*b*c^2*d)*cosh(1)^3 + 20*(405*b*c^8*d*x^6 - 486*b*c^6*d*x^4 + 648*b*c^4*d*x^2 - 1296*b*c^2*d +
7*(35*b*c^8*x^8 - 40*b*c^6*x^6 + 48*b*c^4*x^4 - 64*b*c^2*x^2 + 128*b)*cosh(1))*sinh(1)^3 + 7938*(3*b*c^8*d^2*x
^4 - 4*b*c^6*d^2*x^2 + 8*b*c^4*d^2)*cosh(1)^2 + 6*(3969*b*c^8*d^2*x^4 - 5292*b*c^6*d^2*x^2 + 10584*b*c^4*d^2 +
35*(35*b*c^8*x^8 - 40*b*c^6*x^6 + 48*b*c^4*x^4 - 64*b*c^2*x^2 + 128*b)*cosh(1)^2 + 810*(5*b*c^8*d*x^6 - 6*b*c
^6*d*x^4 + 8*b*c^4*d*x^2 - 16*b*c^2*d)*cosh(1))*sinh(1)^2 + 44100*(b*c^8*d^3*x^2 - 2*b*c^6*d^3)*cosh(1) + 4*(1
1025*b*c^8*d^3*x^2 - 22050*b*c^6*d^3 + 35*(35*b*c^8*x^8 - 40*b*c^6*x^6 + 48*b*c^4*x^4 - 64*b*c^2*x^2 + 128*b)*
cosh(1)^3 + 1215*(5*b*c^8*d*x^6 - 6*b*c^6*d*x^4 + 8*b*c^4*d*x^2 - 16*b*c^2*d)*cosh(1)^2 + 3969*(3*b*c^8*d^2*x^
4 - 4*b*c^6*d^2*x^2 + 8*b*c^4*d^2)*cosh(1))*sinh(1))*sqrt(c^2*x^2 + 1))/c^9

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Sympy [A]
time = 2.00, size = 593, normalized size = 1.90 \begin {gather*} \begin {cases} a d^{4} x + \frac {4 a d^{3} e x^{3}}{3} + \frac {6 a d^{2} e^{2} x^{5}}{5} + \frac {4 a d e^{3} x^{7}}{7} + \frac {a e^{4} x^{9}}{9} + b d^{4} x \operatorname {asinh}{\left (c x \right )} + \frac {4 b d^{3} e x^{3} \operatorname {asinh}{\left (c x \right )}}{3} + \frac {6 b d^{2} e^{2} x^{5} \operatorname {asinh}{\left (c x \right )}}{5} + \frac {4 b d e^{3} x^{7} \operatorname {asinh}{\left (c x \right )}}{7} + \frac {b e^{4} x^{9} \operatorname {asinh}{\left (c x \right )}}{9} - \frac {b d^{4} \sqrt {c^{2} x^{2} + 1}}{c} - \frac {4 b d^{3} e x^{2} \sqrt {c^{2} x^{2} + 1}}{9 c} - \frac {6 b d^{2} e^{2} x^{4} \sqrt {c^{2} x^{2} + 1}}{25 c} - \frac {4 b d e^{3} x^{6} \sqrt {c^{2} x^{2} + 1}}{49 c} - \frac {b e^{4} x^{8} \sqrt {c^{2} x^{2} + 1}}{81 c} + \frac {8 b d^{3} e \sqrt {c^{2} x^{2} + 1}}{9 c^{3}} + \frac {8 b d^{2} e^{2} x^{2} \sqrt {c^{2} x^{2} + 1}}{25 c^{3}} + \frac {24 b d e^{3} x^{4} \sqrt {c^{2} x^{2} + 1}}{245 c^{3}} + \frac {8 b e^{4} x^{6} \sqrt {c^{2} x^{2} + 1}}{567 c^{3}} - \frac {16 b d^{2} e^{2} \sqrt {c^{2} x^{2} + 1}}{25 c^{5}} - \frac {32 b d e^{3} x^{2} \sqrt {c^{2} x^{2} + 1}}{245 c^{5}} - \frac {16 b e^{4} x^{4} \sqrt {c^{2} x^{2} + 1}}{945 c^{5}} + \frac {64 b d e^{3} \sqrt {c^{2} x^{2} + 1}}{245 c^{7}} + \frac {64 b e^{4} x^{2} \sqrt {c^{2} x^{2} + 1}}{2835 c^{7}} - \frac {128 b e^{4} \sqrt {c^{2} x^{2} + 1}}{2835 c^{9}} & \text {for}\: c \neq 0 \\a \left (d^{4} x + \frac {4 d^{3} e x^{3}}{3} + \frac {6 d^{2} e^{2} x^{5}}{5} + \frac {4 d e^{3} x^{7}}{7} + \frac {e^{4} x^{9}}{9}\right ) & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x**2+d)**4*(a+b*asinh(c*x)),x)

[Out]

Piecewise((a*d**4*x + 4*a*d**3*e*x**3/3 + 6*a*d**2*e**2*x**5/5 + 4*a*d*e**3*x**7/7 + a*e**4*x**9/9 + b*d**4*x*
asinh(c*x) + 4*b*d**3*e*x**3*asinh(c*x)/3 + 6*b*d**2*e**2*x**5*asinh(c*x)/5 + 4*b*d*e**3*x**7*asinh(c*x)/7 + b
*e**4*x**9*asinh(c*x)/9 - b*d**4*sqrt(c**2*x**2 + 1)/c - 4*b*d**3*e*x**2*sqrt(c**2*x**2 + 1)/(9*c) - 6*b*d**2*
e**2*x**4*sqrt(c**2*x**2 + 1)/(25*c) - 4*b*d*e**3*x**6*sqrt(c**2*x**2 + 1)/(49*c) - b*e**4*x**8*sqrt(c**2*x**2
+ 1)/(81*c) + 8*b*d**3*e*sqrt(c**2*x**2 + 1)/(9*c**3) + 8*b*d**2*e**2*x**2*sqrt(c**2*x**2 + 1)/(25*c**3) + 24
*b*d*e**3*x**4*sqrt(c**2*x**2 + 1)/(245*c**3) + 8*b*e**4*x**6*sqrt(c**2*x**2 + 1)/(567*c**3) - 16*b*d**2*e**2*
sqrt(c**2*x**2 + 1)/(25*c**5) - 32*b*d*e**3*x**2*sqrt(c**2*x**2 + 1)/(245*c**5) - 16*b*e**4*x**4*sqrt(c**2*x**
2 + 1)/(945*c**5) + 64*b*d*e**3*sqrt(c**2*x**2 + 1)/(245*c**7) + 64*b*e**4*x**2*sqrt(c**2*x**2 + 1)/(2835*c**7
) - 128*b*e**4*sqrt(c**2*x**2 + 1)/(2835*c**9), Ne(c, 0)), (a*(d**4*x + 4*d**3*e*x**3/3 + 6*d**2*e**2*x**5/5 +
4*d*e**3*x**7/7 + e**4*x**9/9), 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)^4*(a+b*arcsinh(c*x)),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 )\,{\left (e\,x^2+d\right )}^4 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*asinh(c*x))*(d + e*x^2)^4,x)

[Out]

int((a + b*asinh(c*x))*(d + e*x^2)^4, x)

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