3.472 \(\int x^2 (d+e x^2)^2 (a+b \cosh ^{-1}(c x)) \, dx\)

Optimal. Leaf size=260 \[ \frac {1}{3} d^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac {b e \left (1-c^2 x^2\right )^3 \left (14 c^2 d+15 e\right )}{175 c^7 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b e^2 \left (1-c^2 x^2\right )^4}{49 c^7 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b \left (1-c^2 x^2\right )^2 \left (35 c^4 d^2+84 c^2 d e+45 e^2\right )}{315 c^7 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b \left (1-c^2 x^2\right ) \left (35 c^4 d^2+42 c^2 d e+15 e^2\right )}{105 c^7 \sqrt {c x-1} \sqrt {c x+1}} \]

[Out]

1/3*d^2*x^3*(a+b*arccosh(c*x))+2/5*d*e*x^5*(a+b*arccosh(c*x))+1/7*e^2*x^7*(a+b*arccosh(c*x))+1/105*b*(35*c^4*d
^2+42*c^2*d*e+15*e^2)*(-c^2*x^2+1)/c^7/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/315*b*(35*c^4*d^2+84*c^2*d*e+45*e^2)*(-c^
2*x^2+1)^2/c^7/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/175*b*e*(14*c^2*d+15*e)*(-c^2*x^2+1)^3/c^7/(c*x-1)^(1/2)/(c*x+1)^
(1/2)-1/49*b*e^2*(-c^2*x^2+1)^4/c^7/(c*x-1)^(1/2)/(c*x+1)^(1/2)

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Rubi [A]  time = 0.32, antiderivative size = 260, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {270, 5790, 12, 520, 1251, 771} \[ \frac {1}{3} d^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac {b \left (1-c^2 x^2\right )^2 \left (35 c^4 d^2+84 c^2 d e+45 e^2\right )}{315 c^7 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b \left (1-c^2 x^2\right ) \left (35 c^4 d^2+42 c^2 d e+15 e^2\right )}{105 c^7 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b e \left (1-c^2 x^2\right )^3 \left (14 c^2 d+15 e\right )}{175 c^7 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b e^2 \left (1-c^2 x^2\right )^4}{49 c^7 \sqrt {c x-1} \sqrt {c x+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*(35*c^4*d^2 + 42*c^2*d*e + 15*e^2)*(1 - c^2*x^2))/(105*c^7*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*(35*c^4*d^2 +
84*c^2*d*e + 45*e^2)*(1 - c^2*x^2)^2)/(315*c^7*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*e*(14*c^2*d + 15*e)*(1 - c^
2*x^2)^3)/(175*c^7*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*e^2*(1 - c^2*x^2)^4)/(49*c^7*Sqrt[-1 + c*x]*Sqrt[1 + c*x
]) + (d^2*x^3*(a + b*ArcCosh[c*x]))/3 + (2*d*e*x^5*(a + b*ArcCosh[c*x]))/5 + (e^2*x^7*(a + b*ArcCosh[c*x]))/7

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 520

Int[(u_.)*((c_) + (d_.)*(x_)^(n_.) + (e_.)*(x_)^(n2_.))^(q_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b
2_.)*(x_)^(non2_.))^(p_.), x_Symbol] :> Dist[((a1 + b1*x^(n/2))^FracPart[p]*(a2 + b2*x^(n/2))^FracPart[p])/(a1
*a2 + b1*b2*x^n)^FracPart[p], Int[u*(a1*a2 + b1*b2*x^n)^p*(c + d*x^n + e*x^(2*n))^q, x], x] /; FreeQ[{a1, b1,
a2, b2, c, d, e, n, p, q}, x] && EqQ[non2, n/2] && EqQ[n2, 2*n] && EqQ[a2*b1 + a1*b2, 0]

Rule 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rule 1251

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 5790

Int[((a_.) + ArcCosh[(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*ArcCosh[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/(Sqrt[
1 + c*x]*Sqrt[-1 + c*x]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] &
& (GtQ[p, 0] || (IGtQ[(m - 1)/2, 0] && LeQ[m + p, 0]))

Rubi steps

\begin {align*} \int x^2 \left (d+e x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=\frac {1}{3} d^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \cosh ^{-1}(c x)\right )-(b c) \int \frac {x^3 \left (35 d^2+42 d e x^2+15 e^2 x^4\right )}{105 \sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=\frac {1}{3} d^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{105} (b c) \int \frac {x^3 \left (35 d^2+42 d e x^2+15 e^2 x^4\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=\frac {1}{3} d^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b c \sqrt {-1+c^2 x^2}\right ) \int \frac {x^3 \left (35 d^2+42 d e x^2+15 e^2 x^4\right )}{\sqrt {-1+c^2 x^2}} \, dx}{105 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {1}{3} d^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b c \sqrt {-1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {x \left (35 d^2+42 d e x+15 e^2 x^2\right )}{\sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{210 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {1}{3} d^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b c \sqrt {-1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {35 c^4 d^2+42 c^2 d e+15 e^2}{c^6 \sqrt {-1+c^2 x}}+\frac {\left (35 c^4 d^2+84 c^2 d e+45 e^2\right ) \sqrt {-1+c^2 x}}{c^6}+\frac {3 e \left (14 c^2 d+15 e\right ) \left (-1+c^2 x\right )^{3/2}}{c^6}+\frac {15 e^2 \left (-1+c^2 x\right )^{5/2}}{c^6}\right ) \, dx,x,x^2\right )}{210 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b \left (35 c^4 d^2+42 c^2 d e+15 e^2\right ) \left (1-c^2 x^2\right )}{105 c^7 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b \left (35 c^4 d^2+84 c^2 d e+45 e^2\right ) \left (1-c^2 x^2\right )^2}{315 c^7 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b e \left (14 c^2 d+15 e\right ) \left (1-c^2 x^2\right )^3}{175 c^7 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b e^2 \left (1-c^2 x^2\right )^4}{49 c^7 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {1}{3} d^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \cosh ^{-1}(c x)\right )\\ \end {align*}

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Mathematica [A]  time = 0.26, size = 163, normalized size = 0.63 \[ \frac {105 a x^3 \left (35 d^2+42 d e x^2+15 e^2 x^4\right )-\frac {b \sqrt {c x-1} \sqrt {c x+1} \left (c^6 \left (1225 d^2 x^2+882 d e x^4+225 e^2 x^6\right )+2 c^4 \left (1225 d^2+588 d e x^2+135 e^2 x^4\right )+24 c^2 e \left (98 d+15 e x^2\right )+720 e^2\right )}{c^7}+105 b x^3 \cosh ^{-1}(c x) \left (35 d^2+42 d e x^2+15 e^2 x^4\right )}{11025} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(105*a*x^3*(35*d^2 + 42*d*e*x^2 + 15*e^2*x^4) - (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(720*e^2 + 24*c^2*e*(98*d + 15
*e*x^2) + 2*c^4*(1225*d^2 + 588*d*e*x^2 + 135*e^2*x^4) + c^6*(1225*d^2*x^2 + 882*d*e*x^4 + 225*e^2*x^6)))/c^7
+ 105*b*x^3*(35*d^2 + 42*d*e*x^2 + 15*e^2*x^4)*ArcCosh[c*x])/11025

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fricas [A]  time = 0.56, size = 198, normalized size = 0.76 \[ \frac {1575 \, a c^{7} e^{2} x^{7} + 4410 \, a c^{7} d e x^{5} + 3675 \, a c^{7} d^{2} x^{3} + 105 \, {\left (15 \, b c^{7} e^{2} x^{7} + 42 \, b c^{7} d e x^{5} + 35 \, b c^{7} d^{2} x^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (225 \, b c^{6} e^{2} x^{6} + 2450 \, b c^{4} d^{2} + 2352 \, b c^{2} d e + 18 \, {\left (49 \, b c^{6} d e + 15 \, b c^{4} e^{2}\right )} x^{4} + 720 \, b e^{2} + {\left (1225 \, b c^{6} d^{2} + 1176 \, b c^{4} d e + 360 \, b c^{2} e^{2}\right )} x^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{11025 \, c^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11025*(1575*a*c^7*e^2*x^7 + 4410*a*c^7*d*e*x^5 + 3675*a*c^7*d^2*x^3 + 105*(15*b*c^7*e^2*x^7 + 42*b*c^7*d*e*x
^5 + 35*b*c^7*d^2*x^3)*log(c*x + sqrt(c^2*x^2 - 1)) - (225*b*c^6*e^2*x^6 + 2450*b*c^4*d^2 + 2352*b*c^2*d*e + 1
8*(49*b*c^6*d*e + 15*b*c^4*e^2)*x^4 + 720*b*e^2 + (1225*b*c^6*d^2 + 1176*b*c^4*d*e + 360*b*c^2*e^2)*x^2)*sqrt(
c^2*x^2 - 1))/c^7

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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*(e*x^2+d)^2*(a+b*arccosh(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 = 195, normalized size = 0.75 \[ \frac {\frac {a \left (\frac {1}{7} e^{2} c^{7} x^{7}+\frac {2}{5} c^{7} d e \,x^{5}+\frac {1}{3} x^{3} c^{7} d^{2}\right )}{c^{4}}+\frac {b \left (\frac {\mathrm {arccosh}\left (c x \right ) e^{2} c^{7} x^{7}}{7}+\frac {2 \,\mathrm {arccosh}\left (c x \right ) c^{7} d e \,x^{5}}{5}+\frac {\mathrm {arccosh}\left (c x \right ) c^{7} x^{3} d^{2}}{3}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (225 c^{6} e^{2} x^{6}+882 c^{6} d e \,x^{4}+1225 c^{6} d^{2} x^{2}+270 c^{4} e^{2} x^{4}+1176 c^{4} d e \,x^{2}+2450 d^{2} c^{4}+360 c^{2} e^{2} x^{2}+2352 c^{2} d e +720 e^{2}\right )}{11025}\right )}{c^{4}}}{c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^3*(a/c^4*(1/7*e^2*c^7*x^7+2/5*c^7*d*e*x^5+1/3*x^3*c^7*d^2)+b/c^4*(1/7*arccosh(c*x)*e^2*c^7*x^7+2/5*arccosh
(c*x)*c^7*d*e*x^5+1/3*arccosh(c*x)*c^7*x^3*d^2-1/11025*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(225*c^6*e^2*x^6+882*c^6*d*
e*x^4+1225*c^6*d^2*x^2+270*c^4*e^2*x^4+1176*c^4*d*e*x^2+2450*c^4*d^2+360*c^2*e^2*x^2+2352*c^2*d*e+720*e^2)))

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maxima [A]  time = 0.36, size = 247, normalized size = 0.95 \[ \frac {1}{7} \, a e^{2} x^{7} + \frac {2}{5} \, a d e x^{5} + \frac {1}{3} \, a d^{2} x^{3} + \frac {1}{9} \, {\left (3 \, x^{3} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} b d^{2} + \frac {2}{75} \, {\left (15 \, x^{5} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1}}{c^{6}}\right )} c\right )} b d e + \frac {1}{245} \, {\left (35 \, x^{7} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {5 \, \sqrt {c^{2} x^{2} - 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {c^{2} x^{2} - 1}}{c^{8}}\right )} c\right )} b e^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*a*e^2*x^7 + 2/5*a*d*e*x^5 + 1/3*a*d^2*x^3 + 1/9*(3*x^3*arccosh(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 + 2/75*(15*x^5*arccosh(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 + 1/245*(35*x^7*arccosh(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^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 6.04, size = 340, normalized size = 1.31 \[ \begin {cases} \frac {a d^{2} x^{3}}{3} + \frac {2 a d e x^{5}}{5} + \frac {a e^{2} x^{7}}{7} + \frac {b d^{2} x^{3} \operatorname {acosh}{\left (c x \right )}}{3} + \frac {2 b d e x^{5} \operatorname {acosh}{\left (c x \right )}}{5} + \frac {b e^{2} x^{7} \operatorname {acosh}{\left (c x \right )}}{7} - \frac {b d^{2} x^{2} \sqrt {c^{2} x^{2} - 1}}{9 c} - \frac {2 b d e x^{4} \sqrt {c^{2} x^{2} - 1}}{25 c} - \frac {b e^{2} x^{6} \sqrt {c^{2} x^{2} - 1}}{49 c} - \frac {2 b d^{2} \sqrt {c^{2} x^{2} - 1}}{9 c^{3}} - \frac {8 b d e x^{2} \sqrt {c^{2} x^{2} - 1}}{75 c^{3}} - \frac {6 b e^{2} x^{4} \sqrt {c^{2} x^{2} - 1}}{245 c^{3}} - \frac {16 b d e \sqrt {c^{2} x^{2} - 1}}{75 c^{5}} - \frac {8 b e^{2} x^{2} \sqrt {c^{2} x^{2} - 1}}{245 c^{5}} - \frac {16 b e^{2} \sqrt {c^{2} x^{2} - 1}}{245 c^{7}} & \text {for}\: c \neq 0 \\\left (a + \frac {i \pi b}{2}\right ) \left (\frac {d^{2} x^{3}}{3} + \frac {2 d e x^{5}}{5} + \frac {e^{2} x^{7}}{7}\right ) & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*d**2*x**3/3 + 2*a*d*e*x**5/5 + a*e**2*x**7/7 + b*d**2*x**3*acosh(c*x)/3 + 2*b*d*e*x**5*acosh(c*x)
/5 + b*e**2*x**7*acosh(c*x)/7 - b*d**2*x**2*sqrt(c**2*x**2 - 1)/(9*c) - 2*b*d*e*x**4*sqrt(c**2*x**2 - 1)/(25*c
) - b*e**2*x**6*sqrt(c**2*x**2 - 1)/(49*c) - 2*b*d**2*sqrt(c**2*x**2 - 1)/(9*c**3) - 8*b*d*e*x**2*sqrt(c**2*x*
*2 - 1)/(75*c**3) - 6*b*e**2*x**4*sqrt(c**2*x**2 - 1)/(245*c**3) - 16*b*d*e*sqrt(c**2*x**2 - 1)/(75*c**5) - 8*
b*e**2*x**2*sqrt(c**2*x**2 - 1)/(245*c**5) - 16*b*e**2*sqrt(c**2*x**2 - 1)/(245*c**7), Ne(c, 0)), ((a + I*pi*b
/2)*(d**2*x**3/3 + 2*d*e*x**5/5 + e**2*x**7/7), True))

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