3.461 \(\int x^4 (d+e x^2) (a+b \cosh ^{-1}(c x)) \, dx\)

Optimal. Leaf size=177 \[ \frac {1}{5} d x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{7} e x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac {8 b \sqrt {c x-1} \sqrt {c x+1} \left (49 c^2 d+30 e\right )}{3675 c^7}-\frac {4 b x^2 \sqrt {c x-1} \sqrt {c x+1} \left (49 c^2 d+30 e\right )}{3675 c^5}-\frac {b x^4 \sqrt {c x-1} \sqrt {c x+1} \left (49 c^2 d+30 e\right )}{1225 c^3}-\frac {b e x^6 \sqrt {c x-1} \sqrt {c x+1}}{49 c} \]

[Out]

1/5*d*x^5*(a+b*arccosh(c*x))+1/7*e*x^7*(a+b*arccosh(c*x))-8/3675*b*(49*c^2*d+30*e)*(c*x-1)^(1/2)*(c*x+1)^(1/2)
/c^7-4/3675*b*(49*c^2*d+30*e)*x^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^5-1/1225*b*(49*c^2*d+30*e)*x^4*(c*x-1)^(1/2)*(
c*x+1)^(1/2)/c^3-1/49*b*e*x^6*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c

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Rubi [A]  time = 0.14, antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {5786, 460, 100, 12, 74} \[ \frac {1}{5} d x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{7} e x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac {b x^4 \sqrt {c x-1} \sqrt {c x+1} \left (49 c^2 d+30 e\right )}{1225 c^3}-\frac {4 b x^2 \sqrt {c x-1} \sqrt {c x+1} \left (49 c^2 d+30 e\right )}{3675 c^5}-\frac {8 b \sqrt {c x-1} \sqrt {c x+1} \left (49 c^2 d+30 e\right )}{3675 c^7}-\frac {b e x^6 \sqrt {c x-1} \sqrt {c x+1}}{49 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-8*b*(49*c^2*d + 30*e)*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(3675*c^7) - (4*b*(49*c^2*d + 30*e)*x^2*Sqrt[-1 + c*x]*S
qrt[1 + c*x])/(3675*c^5) - (b*(49*c^2*d + 30*e)*x^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(1225*c^3) - (b*e*x^6*Sqrt[-
1 + c*x]*Sqrt[1 + c*x])/(49*c) + (d*x^5*(a + b*ArcCosh[c*x]))/5 + (e*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 74

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] &
& EqQ[a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]

Rule 100

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m - 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 1)), x] + Dist[1/(d*f*(m + n + p + 1)), I
nt[(a + b*x)^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1)
+ c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d,
e, f, n, p}, x] && GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]

Rule 460

Int[((e_.)*(x_))^(m_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b2_.)*(x_)^(non2_.))^(p_.)*((c_) + (d_.)
*(x_)^(n_)), x_Symbol] :> Simp[(d*(e*x)^(m + 1)*(a1 + b1*x^(n/2))^(p + 1)*(a2 + b2*x^(n/2))^(p + 1))/(b1*b2*e*
(m + n*(p + 1) + 1)), x] - Dist[(a1*a2*d*(m + 1) - b1*b2*c*(m + n*(p + 1) + 1))/(b1*b2*(m + n*(p + 1) + 1)), I
nt[(e*x)^m*(a1 + b1*x^(n/2))^p*(a2 + b2*x^(n/2))^p, x], x] /; FreeQ[{a1, b1, a2, b2, c, d, e, m, n, p}, x] &&
EqQ[non2, n/2] && EqQ[a2*b1 + a1*b2, 0] && NeQ[m + n*(p + 1) + 1, 0]

Rule 5786

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(d*(f*x)^(
m + 1)*(a + b*ArcCosh[c*x]))/(f*(m + 1)), x] + (-Dist[(b*c)/(f*(m + 1)*(m + 3)), Int[((f*x)^(m + 1)*(d*(m + 3)
+ e*(m + 1)*x^2))/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x] + Simp[(e*(f*x)^(m + 3)*(a + b*ArcCosh[c*x]))/(f^3*(
m + 3)), x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[c^2*d + e, 0] && NeQ[m, -1] && NeQ[m, -3]

Rubi steps

\begin {align*} \int x^4 \left (d+e x^2\right ) \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=\frac {1}{5} d x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{7} e x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{35} (b c) \int \frac {x^5 \left (7 d+5 e x^2\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=-\frac {b e x^6 \sqrt {-1+c x} \sqrt {1+c x}}{49 c}+\frac {1}{5} d x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{7} e x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{245} \left (b c \left (-49 d-\frac {30 e}{c^2}\right )\right ) \int \frac {x^5}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=-\frac {b \left (49 c^2 d+30 e\right ) x^4 \sqrt {-1+c x} \sqrt {1+c x}}{1225 c^3}-\frac {b e x^6 \sqrt {-1+c x} \sqrt {1+c x}}{49 c}+\frac {1}{5} d x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{7} e x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b \left (49 c^2 d+30 e\right )\right ) \int \frac {4 x^3}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{1225 c^3}\\ &=-\frac {b \left (49 c^2 d+30 e\right ) x^4 \sqrt {-1+c x} \sqrt {1+c x}}{1225 c^3}-\frac {b e x^6 \sqrt {-1+c x} \sqrt {1+c x}}{49 c}+\frac {1}{5} d x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{7} e x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (4 b \left (49 c^2 d+30 e\right )\right ) \int \frac {x^3}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{1225 c^3}\\ &=-\frac {4 b \left (49 c^2 d+30 e\right ) x^2 \sqrt {-1+c x} \sqrt {1+c x}}{3675 c^5}-\frac {b \left (49 c^2 d+30 e\right ) x^4 \sqrt {-1+c x} \sqrt {1+c x}}{1225 c^3}-\frac {b e x^6 \sqrt {-1+c x} \sqrt {1+c x}}{49 c}+\frac {1}{5} d x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{7} e x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (4 b \left (49 c^2 d+30 e\right )\right ) \int \frac {2 x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{3675 c^5}\\ &=-\frac {4 b \left (49 c^2 d+30 e\right ) x^2 \sqrt {-1+c x} \sqrt {1+c x}}{3675 c^5}-\frac {b \left (49 c^2 d+30 e\right ) x^4 \sqrt {-1+c x} \sqrt {1+c x}}{1225 c^3}-\frac {b e x^6 \sqrt {-1+c x} \sqrt {1+c x}}{49 c}+\frac {1}{5} d x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{7} e x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (8 b \left (49 c^2 d+30 e\right )\right ) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{3675 c^5}\\ &=-\frac {8 b \left (49 c^2 d+30 e\right ) \sqrt {-1+c x} \sqrt {1+c x}}{3675 c^7}-\frac {4 b \left (49 c^2 d+30 e\right ) x^2 \sqrt {-1+c x} \sqrt {1+c x}}{3675 c^5}-\frac {b \left (49 c^2 d+30 e\right ) x^4 \sqrt {-1+c x} \sqrt {1+c x}}{1225 c^3}-\frac {b e x^6 \sqrt {-1+c x} \sqrt {1+c x}}{49 c}+\frac {1}{5} d x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{7} e x^7 \left (a+b \cosh ^{-1}(c x)\right )\\ \end {align*}

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Mathematica [A]  time = 0.11, size = 122, normalized size = 0.69 \[ \frac {1}{35} a x^5 \left (7 d+5 e x^2\right )-\frac {b \sqrt {c x-1} \sqrt {c x+1} \left (3 c^6 \left (49 d x^4+25 e x^6\right )+2 c^4 \left (98 d x^2+45 e x^4\right )+8 c^2 \left (49 d+15 e x^2\right )+240 e\right )}{3675 c^7}+\frac {1}{35} b x^5 \cosh ^{-1}(c x) \left (7 d+5 e x^2\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*x^5*(7*d + 5*e*x^2))/35 - (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(240*e + 8*c^2*(49*d + 15*e*x^2) + 2*c^4*(98*d*x^
2 + 45*e*x^4) + 3*c^6*(49*d*x^4 + 25*e*x^6)))/(3675*c^7) + (b*x^5*(7*d + 5*e*x^2)*ArcCosh[c*x])/35

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fricas [A]  time = 0.69, size = 140, normalized size = 0.79 \[ \frac {525 \, a c^{7} e x^{7} + 735 \, a c^{7} d x^{5} + 105 \, {\left (5 \, b c^{7} e x^{7} + 7 \, b c^{7} d x^{5}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (75 \, b c^{6} e x^{6} + 3 \, {\left (49 \, b c^{6} d + 30 \, b c^{4} e\right )} x^{4} + 392 \, b c^{2} d + 4 \, {\left (49 \, b c^{4} d + 30 \, b c^{2} e\right )} x^{2} + 240 \, b e\right )} \sqrt {c^{2} x^{2} - 1}}{3675 \, c^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3675*(525*a*c^7*e*x^7 + 735*a*c^7*d*x^5 + 105*(5*b*c^7*e*x^7 + 7*b*c^7*d*x^5)*log(c*x + sqrt(c^2*x^2 - 1)) -
(75*b*c^6*e*x^6 + 3*(49*b*c^6*d + 30*b*c^4*e)*x^4 + 392*b*c^2*d + 4*(49*b*c^4*d + 30*b*c^2*e)*x^2 + 240*b*e)*
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^4*(e*x^2+d)*(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 = 0.03, size = 133, normalized size = 0.75 \[ \frac {\frac {a \left (\frac {1}{7} e \,c^{7} x^{7}+\frac {1}{5} c^{7} x^{5} d \right )}{c^{2}}+\frac {b \left (\frac {\mathrm {arccosh}\left (c x \right ) e \,c^{7} x^{7}}{7}+\frac {\mathrm {arccosh}\left (c x \right ) c^{7} x^{5} d}{5}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (75 c^{6} e \,x^{6}+147 c^{6} d \,x^{4}+90 c^{4} e \,x^{4}+196 c^{4} d \,x^{2}+120 c^{2} x^{2} e +392 c^{2} d +240 e \right )}{3675}\right )}{c^{2}}}{c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^5*(a/c^2*(1/7*e*c^7*x^7+1/5*c^7*x^5*d)+b/c^2*(1/7*arccosh(c*x)*e*c^7*x^7+1/5*arccosh(c*x)*c^7*x^5*d-1/3675
*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(75*c^6*e*x^6+147*c^6*d*x^4+90*c^4*e*x^4+196*c^4*d*x^2+120*c^2*e*x^2+392*c^2*d+24
0*e)))

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maxima [A]  time = 0.45, size = 178, normalized size = 1.01 \[ \frac {1}{7} \, a e x^{7} + \frac {1}{5} \, a d x^{5} + \frac {1}{75} \, {\left (15 \, x^{5} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1}}{c^{6}}\right )} c\right )} b d + \frac {1}{245} \, {\left (35 \, x^{7} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {5 \, \sqrt {c^{2} x^{2} - 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {c^{2} x^{2} - 1}}{c^{8}}\right )} c\right )} b e \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*a*e*x^7 + 1/5*a*d*x^5 + 1/75*(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 + 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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int x^4\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\left (e\,x^2+d\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 5.64, size = 230, normalized size = 1.30 \[ \begin {cases} \frac {a d x^{5}}{5} + \frac {a e x^{7}}{7} + \frac {b d x^{5} \operatorname {acosh}{\left (c x \right )}}{5} + \frac {b e x^{7} \operatorname {acosh}{\left (c x \right )}}{7} - \frac {b d x^{4} \sqrt {c^{2} x^{2} - 1}}{25 c} - \frac {b e x^{6} \sqrt {c^{2} x^{2} - 1}}{49 c} - \frac {4 b d x^{2} \sqrt {c^{2} x^{2} - 1}}{75 c^{3}} - \frac {6 b e x^{4} \sqrt {c^{2} x^{2} - 1}}{245 c^{3}} - \frac {8 b d \sqrt {c^{2} x^{2} - 1}}{75 c^{5}} - \frac {8 b e x^{2} \sqrt {c^{2} x^{2} - 1}}{245 c^{5}} - \frac {16 b e \sqrt {c^{2} x^{2} - 1}}{245 c^{7}} & \text {for}\: c \neq 0 \\\left (a + \frac {i \pi b}{2}\right ) \left (\frac {d x^{5}}{5} + \frac {e x^{7}}{7}\right ) & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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