3.1.10 \(\int x^4 (d+c^2 d x^2)^2 (a+b \sinh ^{-1}(c x)) \, dx\) [10]

Optimal. Leaf size=181 \[ -\frac {8 b d^2 \sqrt {1+c^2 x^2}}{315 c^5}-\frac {4 b d^2 \left (1+c^2 x^2\right )^{3/2}}{945 c^5}-\frac {b d^2 \left (1+c^2 x^2\right )^{5/2}}{525 c^5}+\frac {10 b d^2 \left (1+c^2 x^2\right )^{7/2}}{441 c^5}-\frac {b d^2 \left (1+c^2 x^2\right )^{9/2}}{81 c^5}+\frac {1}{5} d^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac {2}{7} c^2 d^2 x^7 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{9} c^4 d^2 x^9 \left (a+b \sinh ^{-1}(c x)\right ) \]

[Out]

-4/945*b*d^2*(c^2*x^2+1)^(3/2)/c^5-1/525*b*d^2*(c^2*x^2+1)^(5/2)/c^5+10/441*b*d^2*(c^2*x^2+1)^(7/2)/c^5-1/81*b
*d^2*(c^2*x^2+1)^(9/2)/c^5+1/5*d^2*x^5*(a+b*arcsinh(c*x))+2/7*c^2*d^2*x^7*(a+b*arcsinh(c*x))+1/9*c^4*d^2*x^9*(
a+b*arcsinh(c*x))-8/315*b*d^2*(c^2*x^2+1)^(1/2)/c^5

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Rubi [A]
time = 0.13, antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {276, 5803, 12, 1265, 911, 1167} \begin {gather*} \frac {1}{9} c^4 d^2 x^9 \left (a+b \sinh ^{-1}(c x)\right )+\frac {2}{7} c^2 d^2 x^7 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{5} d^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )-\frac {b d^2 \left (c^2 x^2+1\right )^{9/2}}{81 c^5}+\frac {10 b d^2 \left (c^2 x^2+1\right )^{7/2}}{441 c^5}-\frac {b d^2 \left (c^2 x^2+1\right )^{5/2}}{525 c^5}-\frac {4 b d^2 \left (c^2 x^2+1\right )^{3/2}}{945 c^5}-\frac {8 b d^2 \sqrt {c^2 x^2+1}}{315 c^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-8*b*d^2*Sqrt[1 + c^2*x^2])/(315*c^5) - (4*b*d^2*(1 + c^2*x^2)^(3/2))/(945*c^5) - (b*d^2*(1 + c^2*x^2)^(5/2))
/(525*c^5) + (10*b*d^2*(1 + c^2*x^2)^(7/2))/(441*c^5) - (b*d^2*(1 + c^2*x^2)^(9/2))/(81*c^5) + (d^2*x^5*(a + b
*ArcSinh[c*x]))/5 + (2*c^2*d^2*x^7*(a + b*ArcSinh[c*x]))/7 + (c^4*d^2*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 276

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

Rule 911

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> With[{q = Denominator[m]}, Dist[q/e, Subst[Int[x^(q*(m + 1) - 1)*((e*f - d*g)/e + g*(x^q/e))^n*((c*d^2 - b*d
*e + a*e^2)/e^2 - (2*c*d - b*e)*(x^q/e^2) + c*(x^(2*q)/e^2))^p, x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, b, c
, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegersQ[n,
p] && FractionQ[m]

Rule 1167

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(
d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]

Rule 1265

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2,
Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] &&
IntegerQ[(m - 1)/2]

Rule 5803

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u =
IntHide[(f*x)^m*(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, f, m}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]

Rubi steps

\begin {align*} \int x^4 \left (d+c^2 d x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right ) \, dx &=\frac {1}{5} d^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac {2}{7} c^2 d^2 x^7 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{9} c^4 d^2 x^9 \left (a+b \sinh ^{-1}(c x)\right )-(b c) \int \frac {d^2 x^5 \left (63+90 c^2 x^2+35 c^4 x^4\right )}{315 \sqrt {1+c^2 x^2}} \, dx\\ &=\frac {1}{5} d^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac {2}{7} c^2 d^2 x^7 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{9} c^4 d^2 x^9 \left (a+b \sinh ^{-1}(c x)\right )-\frac {1}{315} \left (b c d^2\right ) \int \frac {x^5 \left (63+90 c^2 x^2+35 c^4 x^4\right )}{\sqrt {1+c^2 x^2}} \, dx\\ &=\frac {1}{5} d^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac {2}{7} c^2 d^2 x^7 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{9} c^4 d^2 x^9 \left (a+b \sinh ^{-1}(c x)\right )-\frac {1}{630} \left (b c d^2\right ) \text {Subst}\left (\int \frac {x^2 \left (63+90 c^2 x+35 c^4 x^2\right )}{\sqrt {1+c^2 x}} \, dx,x,x^2\right )\\ &=\frac {1}{5} d^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac {2}{7} c^2 d^2 x^7 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{9} c^4 d^2 x^9 \left (a+b \sinh ^{-1}(c x)\right )-\frac {\left (b d^2\right ) \text {Subst}\left (\int \left (-\frac {1}{c^2}+\frac {x^2}{c^2}\right )^2 \left (8+20 x^2+35 x^4\right ) \, dx,x,\sqrt {1+c^2 x^2}\right )}{315 c}\\ &=\frac {1}{5} d^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac {2}{7} c^2 d^2 x^7 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{9} c^4 d^2 x^9 \left (a+b \sinh ^{-1}(c x)\right )-\frac {\left (b d^2\right ) \text {Subst}\left (\int \left (\frac {8}{c^4}+\frac {4 x^2}{c^4}+\frac {3 x^4}{c^4}-\frac {50 x^6}{c^4}+\frac {35 x^8}{c^4}\right ) \, dx,x,\sqrt {1+c^2 x^2}\right )}{315 c}\\ &=-\frac {8 b d^2 \sqrt {1+c^2 x^2}}{315 c^5}-\frac {4 b d^2 \left (1+c^2 x^2\right )^{3/2}}{945 c^5}-\frac {b d^2 \left (1+c^2 x^2\right )^{5/2}}{525 c^5}+\frac {10 b d^2 \left (1+c^2 x^2\right )^{7/2}}{441 c^5}-\frac {b d^2 \left (1+c^2 x^2\right )^{9/2}}{81 c^5}+\frac {1}{5} d^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac {2}{7} c^2 d^2 x^7 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{9} c^4 d^2 x^9 \left (a+b \sinh ^{-1}(c x)\right )\\ \end {align*}

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Mathematica [A]
time = 0.07, size = 119, normalized size = 0.66 \begin {gather*} \frac {d^2 \left (315 a c^5 x^5 \left (63+90 c^2 x^2+35 c^4 x^4\right )-b \sqrt {1+c^2 x^2} \left (2104-1052 c^2 x^2+789 c^4 x^4+2650 c^6 x^6+1225 c^8 x^8\right )+315 b c^5 x^5 \left (63+90 c^2 x^2+35 c^4 x^4\right ) \sinh ^{-1}(c x)\right )}{99225 c^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(d^2*(315*a*c^5*x^5*(63 + 90*c^2*x^2 + 35*c^4*x^4) - b*Sqrt[1 + c^2*x^2]*(2104 - 1052*c^2*x^2 + 789*c^4*x^4 +
2650*c^6*x^6 + 1225*c^8*x^8) + 315*b*c^5*x^5*(63 + 90*c^2*x^2 + 35*c^4*x^4)*ArcSinh[c*x]))/(99225*c^5)

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Maple [A]
time = 0.99, size = 167, normalized size = 0.92

method	result	size
derivativedivides	\(\frac {d^{2} a \left (\frac {1}{9} c^{9} x^{9}+\frac {2}{7} c^{7} x^{7}+\frac {1}{5} c^{5} x^{5}\right )+d^{2} b \left (\frac {\arcsinh \left (c x \right ) c^{9} x^{9}}{9}+\frac {2 \arcsinh \left (c x \right ) c^{7} x^{7}}{7}+\frac {\arcsinh \left (c x \right ) c^{5} x^{5}}{5}-\frac {c^{8} x^{8} \sqrt {c^{2} x^{2}+1}}{81}-\frac {106 c^{6} x^{6} \sqrt {c^{2} x^{2}+1}}{3969}-\frac {263 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{33075}+\frac {1052 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{99225}-\frac {2104 \sqrt {c^{2} x^{2}+1}}{99225}\right )}{c^{5}}\)	\(167\)
default	\(\frac {d^{2} a \left (\frac {1}{9} c^{9} x^{9}+\frac {2}{7} c^{7} x^{7}+\frac {1}{5} c^{5} x^{5}\right )+d^{2} b \left (\frac {\arcsinh \left (c x \right ) c^{9} x^{9}}{9}+\frac {2 \arcsinh \left (c x \right ) c^{7} x^{7}}{7}+\frac {\arcsinh \left (c x \right ) c^{5} x^{5}}{5}-\frac {c^{8} x^{8} \sqrt {c^{2} x^{2}+1}}{81}-\frac {106 c^{6} x^{6} \sqrt {c^{2} x^{2}+1}}{3969}-\frac {263 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{33075}+\frac {1052 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{99225}-\frac {2104 \sqrt {c^{2} x^{2}+1}}{99225}\right )}{c^{5}}\)	\(167\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^5*(d^2*a*(1/9*c^9*x^9+2/7*c^7*x^7+1/5*c^5*x^5)+d^2*b*(1/9*arcsinh(c*x)*c^9*x^9+2/7*arcsinh(c*x)*c^7*x^7+1/
5*arcsinh(c*x)*c^5*x^5-1/81*c^8*x^8*(c^2*x^2+1)^(1/2)-106/3969*c^6*x^6*(c^2*x^2+1)^(1/2)-263/33075*c^4*x^4*(c^
2*x^2+1)^(1/2)+1052/99225*c^2*x^2*(c^2*x^2+1)^(1/2)-2104/99225*(c^2*x^2+1)^(1/2)))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 319 vs. \(2 (155) = 310\).
time = 0.26, size = 319, normalized size = 1.76 \begin {gather*} \frac {1}{9} \, a c^{4} d^{2} x^{9} + \frac {2}{7} \, a c^{2} d^{2} x^{7} + \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 c^{4} d^{2} + \frac {1}{5} \, a d^{2} x^{5} + \frac {2}{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 c^{2} d^{2} + \frac {1}{75} \, {\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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*a*c^4*d^2*x^9 + 2/7*a*c^2*d^2*x^7 + 1/2835*(315*x^9*arcsinh(c*x) - (35*sqrt(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*c^4*d^2 + 1/5*a*d^2*x^5 + 2/245*(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*c^2*d^2 + 1/75*(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

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Fricas [A]
time = 0.36, size = 165, normalized size = 0.91 \begin {gather*} \frac {11025 \, a c^{9} d^{2} x^{9} + 28350 \, a c^{7} d^{2} x^{7} + 19845 \, a c^{5} d^{2} x^{5} + 315 \, {\left (35 \, b c^{9} d^{2} x^{9} + 90 \, b c^{7} d^{2} x^{7} + 63 \, b c^{5} d^{2} x^{5}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - {\left (1225 \, b c^{8} d^{2} x^{8} + 2650 \, b c^{6} d^{2} x^{6} + 789 \, b c^{4} d^{2} x^{4} - 1052 \, b c^{2} d^{2} x^{2} + 2104 \, b d^{2}\right )} \sqrt {c^{2} x^{2} + 1}}{99225 \, c^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/99225*(11025*a*c^9*d^2*x^9 + 28350*a*c^7*d^2*x^7 + 19845*a*c^5*d^2*x^5 + 315*(35*b*c^9*d^2*x^9 + 90*b*c^7*d^
2*x^7 + 63*b*c^5*d^2*x^5)*log(c*x + sqrt(c^2*x^2 + 1)) - (1225*b*c^8*d^2*x^8 + 2650*b*c^6*d^2*x^6 + 789*b*c^4*
d^2*x^4 - 1052*b*c^2*d^2*x^2 + 2104*b*d^2)*sqrt(c^2*x^2 + 1))/c^5

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Sympy [A]
time = 1.45, size = 230, normalized size = 1.27 \begin {gather*} \begin {cases} \frac {a c^{4} d^{2} x^{9}}{9} + \frac {2 a c^{2} d^{2} x^{7}}{7} + \frac {a d^{2} x^{5}}{5} + \frac {b c^{4} d^{2} x^{9} \operatorname {asinh}{\left (c x \right )}}{9} - \frac {b c^{3} d^{2} x^{8} \sqrt {c^{2} x^{2} + 1}}{81} + \frac {2 b c^{2} d^{2} x^{7} \operatorname {asinh}{\left (c x \right )}}{7} - \frac {106 b c d^{2} x^{6} \sqrt {c^{2} x^{2} + 1}}{3969} + \frac {b d^{2} x^{5} \operatorname {asinh}{\left (c x \right )}}{5} - \frac {263 b d^{2} x^{4} \sqrt {c^{2} x^{2} + 1}}{33075 c} + \frac {1052 b d^{2} x^{2} \sqrt {c^{2} x^{2} + 1}}{99225 c^{3}} - \frac {2104 b d^{2} \sqrt {c^{2} x^{2} + 1}}{99225 c^{5}} & \text {for}\: c \neq 0 \\\frac {a d^{2} x^{5}}{5} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*c**4*d**2*x**9/9 + 2*a*c**2*d**2*x**7/7 + a*d**2*x**5/5 + b*c**4*d**2*x**9*asinh(c*x)/9 - b*c**3*
d**2*x**8*sqrt(c**2*x**2 + 1)/81 + 2*b*c**2*d**2*x**7*asinh(c*x)/7 - 106*b*c*d**2*x**6*sqrt(c**2*x**2 + 1)/396
9 + b*d**2*x**5*asinh(c*x)/5 - 263*b*d**2*x**4*sqrt(c**2*x**2 + 1)/(33075*c) + 1052*b*d**2*x**2*sqrt(c**2*x**2
+ 1)/(99225*c**3) - 2104*b*d**2*sqrt(c**2*x**2 + 1)/(99225*c**5), Ne(c, 0)), (a*d**2*x**5/5, 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(x^4*(c^2*d*x^2+d)^2*(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 = -0.01 \begin {gather*} \int x^4\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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