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

Optimal. Leaf size=202 \[ \frac {1}{9} c^6 d^3 x^9 \left (a+b \sinh ^{-1}(c x)\right )+\frac {3}{7} c^4 d^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )+\frac {3}{5} c^2 d^3 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{3} d^3 x^3 \left (a+b \sinh ^{-1}(c x)\right )-\frac {b d^3 \left (c^2 x^2+1\right )^{9/2}}{81 c^3}+\frac {b d^3 \left (c^2 x^2+1\right )^{7/2}}{441 c^3}+\frac {2 b d^3 \left (c^2 x^2+1\right )^{5/2}}{525 c^3}+\frac {8 b d^3 \left (c^2 x^2+1\right )^{3/2}}{945 c^3}+\frac {16 b d^3 \sqrt {c^2 x^2+1}}{315 c^3} \]

[Out]

8/945*b*d^3*(c^2*x^2+1)^(3/2)/c^3+2/525*b*d^3*(c^2*x^2+1)^(5/2)/c^3+1/441*b*d^3*(c^2*x^2+1)^(7/2)/c^3-1/81*b*d
^3*(c^2*x^2+1)^(9/2)/c^3+1/3*d^3*x^3*(a+b*arcsinh(c*x))+3/5*c^2*d^3*x^5*(a+b*arcsinh(c*x))+3/7*c^4*d^3*x^7*(a+
b*arcsinh(c*x))+1/9*c^6*d^3*x^9*(a+b*arcsinh(c*x))+16/315*b*d^3*(c^2*x^2+1)^(1/2)/c^3

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Rubi [A]  time = 0.25, antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {270, 5730, 12, 1799, 1620} \[ \frac {1}{9} c^6 d^3 x^9 \left (a+b \sinh ^{-1}(c x)\right )+\frac {3}{7} c^4 d^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )+\frac {3}{5} c^2 d^3 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{3} d^3 x^3 \left (a+b \sinh ^{-1}(c x)\right )-\frac {b d^3 \left (c^2 x^2+1\right )^{9/2}}{81 c^3}+\frac {b d^3 \left (c^2 x^2+1\right )^{7/2}}{441 c^3}+\frac {2 b d^3 \left (c^2 x^2+1\right )^{5/2}}{525 c^3}+\frac {8 b d^3 \left (c^2 x^2+1\right )^{3/2}}{945 c^3}+\frac {16 b d^3 \sqrt {c^2 x^2+1}}{315 c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(16*b*d^3*Sqrt[1 + c^2*x^2])/(315*c^3) + (8*b*d^3*(1 + c^2*x^2)^(3/2))/(945*c^3) + (2*b*d^3*(1 + c^2*x^2)^(5/2
))/(525*c^3) + (b*d^3*(1 + c^2*x^2)^(7/2))/(441*c^3) - (b*d^3*(1 + c^2*x^2)^(9/2))/(81*c^3) + (d^3*x^3*(a + b*
ArcSinh[c*x]))/3 + (3*c^2*d^3*x^5*(a + b*ArcSinh[c*x]))/5 + (3*c^4*d^3*x^7*(a + b*ArcSinh[c*x]))/7 + (c^6*d^3*
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 270

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 1620

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

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 5730

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^2 \left (d+c^2 d x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right ) \, dx &=\frac {1}{3} d^3 x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac {3}{5} c^2 d^3 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac {3}{7} c^4 d^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{9} c^6 d^3 x^9 \left (a+b \sinh ^{-1}(c x)\right )-(b c) \int \frac {d^3 x^3 \left (105+189 c^2 x^2+135 c^4 x^4+35 c^6 x^6\right )}{315 \sqrt {1+c^2 x^2}} \, dx\\ &=\frac {1}{3} d^3 x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac {3}{5} c^2 d^3 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac {3}{7} c^4 d^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{9} c^6 d^3 x^9 \left (a+b \sinh ^{-1}(c x)\right )-\frac {1}{315} \left (b c d^3\right ) \int \frac {x^3 \left (105+189 c^2 x^2+135 c^4 x^4+35 c^6 x^6\right )}{\sqrt {1+c^2 x^2}} \, dx\\ &=\frac {1}{3} d^3 x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac {3}{5} c^2 d^3 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac {3}{7} c^4 d^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{9} c^6 d^3 x^9 \left (a+b \sinh ^{-1}(c x)\right )-\frac {1}{630} \left (b c d^3\right ) \operatorname {Subst}\left (\int \frac {x \left (105+189 c^2 x+135 c^4 x^2+35 c^6 x^3\right )}{\sqrt {1+c^2 x}} \, dx,x,x^2\right )\\ &=\frac {1}{3} d^3 x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac {3}{5} c^2 d^3 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac {3}{7} c^4 d^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{9} c^6 d^3 x^9 \left (a+b \sinh ^{-1}(c x)\right )-\frac {1}{630} \left (b c d^3\right ) \operatorname {Subst}\left (\int \left (-\frac {16}{c^2 \sqrt {1+c^2 x}}-\frac {8 \sqrt {1+c^2 x}}{c^2}-\frac {6 \left (1+c^2 x\right )^{3/2}}{c^2}-\frac {5 \left (1+c^2 x\right )^{5/2}}{c^2}+\frac {35 \left (1+c^2 x\right )^{7/2}}{c^2}\right ) \, dx,x,x^2\right )\\ &=\frac {16 b d^3 \sqrt {1+c^2 x^2}}{315 c^3}+\frac {8 b d^3 \left (1+c^2 x^2\right )^{3/2}}{945 c^3}+\frac {2 b d^3 \left (1+c^2 x^2\right )^{5/2}}{525 c^3}+\frac {b d^3 \left (1+c^2 x^2\right )^{7/2}}{441 c^3}-\frac {b d^3 \left (1+c^2 x^2\right )^{9/2}}{81 c^3}+\frac {1}{3} d^3 x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac {3}{5} c^2 d^3 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac {3}{7} c^4 d^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{9} c^6 d^3 x^9 \left (a+b \sinh ^{-1}(c x)\right )\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 135, normalized size = 0.67 \[ \frac {d^3 \left (315 a c^3 x^3 \left (35 c^6 x^6+135 c^4 x^4+189 c^2 x^2+105\right )-b \sqrt {c^2 x^2+1} \left (1225 c^8 x^8+4675 c^6 x^6+6297 c^4 x^4+2629 c^2 x^2-5258\right )+315 b c^3 x^3 \left (35 c^6 x^6+135 c^4 x^4+189 c^2 x^2+105\right ) \sinh ^{-1}(c x)\right )}{99225 c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d^3*(315*a*c^3*x^3*(105 + 189*c^2*x^2 + 135*c^4*x^4 + 35*c^6*x^6) - b*Sqrt[1 + c^2*x^2]*(-5258 + 2629*c^2*x^2
+ 6297*c^4*x^4 + 4675*c^6*x^6 + 1225*c^8*x^8) + 315*b*c^3*x^3*(105 + 189*c^2*x^2 + 135*c^4*x^4 + 35*c^6*x^6)*
ArcSinh[c*x]))/(99225*c^3)

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fricas [A]  time = 0.61, size = 189, normalized size = 0.94 \[ \frac {11025 \, a c^{9} d^{3} x^{9} + 42525 \, a c^{7} d^{3} x^{7} + 59535 \, a c^{5} d^{3} x^{5} + 33075 \, a c^{3} d^{3} x^{3} + 315 \, {\left (35 \, b c^{9} d^{3} x^{9} + 135 \, b c^{7} d^{3} x^{7} + 189 \, b c^{5} d^{3} x^{5} + 105 \, b c^{3} d^{3} x^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - {\left (1225 \, b c^{8} d^{3} x^{8} + 4675 \, b c^{6} d^{3} x^{6} + 6297 \, b c^{4} d^{3} x^{4} + 2629 \, b c^{2} d^{3} x^{2} - 5258 \, b d^{3}\right )} \sqrt {c^{2} x^{2} + 1}}{99225 \, c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/99225*(11025*a*c^9*d^3*x^9 + 42525*a*c^7*d^3*x^7 + 59535*a*c^5*d^3*x^5 + 33075*a*c^3*d^3*x^3 + 315*(35*b*c^9
*d^3*x^9 + 135*b*c^7*d^3*x^7 + 189*b*c^5*d^3*x^5 + 105*b*c^3*d^3*x^3)*log(c*x + sqrt(c^2*x^2 + 1)) - (1225*b*c
^8*d^3*x^8 + 4675*b*c^6*d^3*x^6 + 6297*b*c^4*d^3*x^4 + 2629*b*c^2*d^3*x^2 - 5258*b*d^3)*sqrt(c^2*x^2 + 1))/c^3

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(c^2*d*x^2+d)^3*(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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maple [A]  time = 0.01, size = 187, normalized size = 0.93 \[ \frac {d^{3} a \left (\frac {1}{9} c^{9} x^{9}+\frac {3}{7} c^{7} x^{7}+\frac {3}{5} c^{5} x^{5}+\frac {1}{3} c^{3} x^{3}\right )+d^{3} b \left (\frac {\arcsinh \left (c x \right ) c^{9} x^{9}}{9}+\frac {3 \arcsinh \left (c x \right ) c^{7} x^{7}}{7}+\frac {3 \arcsinh \left (c x \right ) c^{5} x^{5}}{5}+\frac {\arcsinh \left (c x \right ) c^{3} x^{3}}{3}-\frac {c^{8} x^{8} \sqrt {c^{2} x^{2}+1}}{81}-\frac {187 c^{6} x^{6} \sqrt {c^{2} x^{2}+1}}{3969}-\frac {2099 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{33075}-\frac {2629 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{99225}+\frac {5258 \sqrt {c^{2} x^{2}+1}}{99225}\right )}{c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^3*(d^3*a*(1/9*c^9*x^9+3/7*c^7*x^7+3/5*c^5*x^5+1/3*c^3*x^3)+d^3*b*(1/9*arcsinh(c*x)*c^9*x^9+3/7*arcsinh(c*x
)*c^7*x^7+3/5*arcsinh(c*x)*c^5*x^5+1/3*arcsinh(c*x)*c^3*x^3-1/81*c^8*x^8*(c^2*x^2+1)^(1/2)-187/3969*c^6*x^6*(c
^2*x^2+1)^(1/2)-2099/33075*c^4*x^4*(c^2*x^2+1)^(1/2)-2629/99225*c^2*x^2*(c^2*x^2+1)^(1/2)+5258/99225*(c^2*x^2+
1)^(1/2)))

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maxima [B]  time = 0.64, size = 388, normalized size = 1.92 \[ \frac {1}{9} \, a c^{6} d^{3} x^{9} + \frac {3}{7} \, a c^{4} d^{3} x^{7} + \frac {1}{2835} \, {\left (315 \, x^{9} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {35 \, \sqrt {c^{2} x^{2} + 1} x^{8}}{c^{2}} - \frac {40 \, \sqrt {c^{2} x^{2} + 1} x^{6}}{c^{4}} + \frac {48 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{6}} - \frac {64 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{8}} + \frac {128 \, \sqrt {c^{2} x^{2} + 1}}{c^{10}}\right )} c\right )} b c^{6} d^{3} + \frac {3}{5} \, a c^{2} d^{3} x^{5} + \frac {3}{245} \, {\left (35 \, x^{7} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {5 \, \sqrt {c^{2} x^{2} + 1} x^{6}}{c^{2}} - \frac {6 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{6}} - \frac {16 \, \sqrt {c^{2} x^{2} + 1}}{c^{8}}\right )} c\right )} b c^{4} d^{3} + \frac {1}{25} \, {\left (15 \, x^{5} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{2}} - \frac {4 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} b c^{2} d^{3} + \frac {1}{3} \, a d^{3} x^{3} + \frac {1}{9} \, {\left (3 \, x^{3} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b d^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*a*c^6*d^3*x^9 + 3/7*a*c^4*d^3*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^6*d^3 + 3/5*a*c^2*d^3*x^5 + 3/245*(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^4*d^3 + 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*c^2*d^3 + 1
/3*a*d^3*x^3 + 1/9*(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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int x^2\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^3 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 16.75, size = 265, normalized size = 1.31 \[ \begin {cases} \frac {a c^{6} d^{3} x^{9}}{9} + \frac {3 a c^{4} d^{3} x^{7}}{7} + \frac {3 a c^{2} d^{3} x^{5}}{5} + \frac {a d^{3} x^{3}}{3} + \frac {b c^{6} d^{3} x^{9} \operatorname {asinh}{\left (c x \right )}}{9} - \frac {b c^{5} d^{3} x^{8} \sqrt {c^{2} x^{2} + 1}}{81} + \frac {3 b c^{4} d^{3} x^{7} \operatorname {asinh}{\left (c x \right )}}{7} - \frac {187 b c^{3} d^{3} x^{6} \sqrt {c^{2} x^{2} + 1}}{3969} + \frac {3 b c^{2} d^{3} x^{5} \operatorname {asinh}{\left (c x \right )}}{5} - \frac {2099 b c d^{3} x^{4} \sqrt {c^{2} x^{2} + 1}}{33075} + \frac {b d^{3} x^{3} \operatorname {asinh}{\left (c x \right )}}{3} - \frac {2629 b d^{3} x^{2} \sqrt {c^{2} x^{2} + 1}}{99225 c} + \frac {5258 b d^{3} \sqrt {c^{2} x^{2} + 1}}{99225 c^{3}} & \text {for}\: c \neq 0 \\\frac {a d^{3} x^{3}}{3} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*c**6*d**3*x**9/9 + 3*a*c**4*d**3*x**7/7 + 3*a*c**2*d**3*x**5/5 + a*d**3*x**3/3 + b*c**6*d**3*x**9
*asinh(c*x)/9 - b*c**5*d**3*x**8*sqrt(c**2*x**2 + 1)/81 + 3*b*c**4*d**3*x**7*asinh(c*x)/7 - 187*b*c**3*d**3*x*
*6*sqrt(c**2*x**2 + 1)/3969 + 3*b*c**2*d**3*x**5*asinh(c*x)/5 - 2099*b*c*d**3*x**4*sqrt(c**2*x**2 + 1)/33075 +
b*d**3*x**3*asinh(c*x)/3 - 2629*b*d**3*x**2*sqrt(c**2*x**2 + 1)/(99225*c) + 5258*b*d**3*sqrt(c**2*x**2 + 1)/(
99225*c**3), Ne(c, 0)), (a*d**3*x**3/3, True))

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