3.76 \(\int x^5 (d+e x^2) (a+b \sec ^{-1}(c x)) \, dx\)

Optimal. Leaf size=196 \[ \frac {1}{6} d x^6 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{8} e x^8 \left (a+b \sec ^{-1}(c x)\right )-\frac {b x \left (c^2 x^2-1\right )^{5/2} \left (4 c^2 d+9 e\right )}{120 c^7 \sqrt {c^2 x^2}}-\frac {b x \left (c^2 x^2-1\right )^{3/2} \left (8 c^2 d+9 e\right )}{72 c^7 \sqrt {c^2 x^2}}-\frac {b x \sqrt {c^2 x^2-1} \left (4 c^2 d+3 e\right )}{24 c^7 \sqrt {c^2 x^2}}-\frac {b e x \left (c^2 x^2-1\right )^{7/2}}{56 c^7 \sqrt {c^2 x^2}} \]

[Out]

1/6*d*x^6*(a+b*arcsec(c*x))+1/8*e*x^8*(a+b*arcsec(c*x))-1/72*b*(8*c^2*d+9*e)*x*(c^2*x^2-1)^(3/2)/c^7/(c^2*x^2)
^(1/2)-1/120*b*(4*c^2*d+9*e)*x*(c^2*x^2-1)^(5/2)/c^7/(c^2*x^2)^(1/2)-1/56*b*e*x*(c^2*x^2-1)^(7/2)/c^7/(c^2*x^2
)^(1/2)-1/24*b*(4*c^2*d+3*e)*x*(c^2*x^2-1)^(1/2)/c^7/(c^2*x^2)^(1/2)

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Rubi [A]  time = 0.15, antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {14, 5238, 12, 446, 77} \[ \frac {1}{6} d x^6 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{8} e x^8 \left (a+b \sec ^{-1}(c x)\right )-\frac {b x \left (c^2 x^2-1\right )^{5/2} \left (4 c^2 d+9 e\right )}{120 c^7 \sqrt {c^2 x^2}}-\frac {b x \left (c^2 x^2-1\right )^{3/2} \left (8 c^2 d+9 e\right )}{72 c^7 \sqrt {c^2 x^2}}-\frac {b x \sqrt {c^2 x^2-1} \left (4 c^2 d+3 e\right )}{24 c^7 \sqrt {c^2 x^2}}-\frac {b e x \left (c^2 x^2-1\right )^{7/2}}{56 c^7 \sqrt {c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[x^5*(d + e*x^2)*(a + b*ArcSec[c*x]),x]

[Out]

-(b*(4*c^2*d + 3*e)*x*Sqrt[-1 + c^2*x^2])/(24*c^7*Sqrt[c^2*x^2]) - (b*(8*c^2*d + 9*e)*x*(-1 + c^2*x^2)^(3/2))/
(72*c^7*Sqrt[c^2*x^2]) - (b*(4*c^2*d + 9*e)*x*(-1 + c^2*x^2)^(5/2))/(120*c^7*Sqrt[c^2*x^2]) - (b*e*x*(-1 + c^2
*x^2)^(7/2))/(56*c^7*Sqrt[c^2*x^2]) + (d*x^6*(a + b*ArcSec[c*x]))/6 + (e*x^8*(a + b*ArcSec[c*x]))/8

Rule 12

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 5238

Int[((a_.) + ArcSec[(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*ArcSec[c*x], u, x] - Dist[(b*c*x)/Sqrt[c^2*x^2], Int[SimplifyI
ntegrand[u/(x*Sqrt[c^2*x^2 - 1]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && ((IGtQ[p, 0] &&  !(ILtQ
[(m - 1)/2, 0] && GtQ[m + 2*p + 3, 0])) || (IGtQ[(m + 1)/2, 0] &&  !(ILtQ[p, 0] && GtQ[m + 2*p + 3, 0])) || (I
LtQ[(m + 2*p + 1)/2, 0] &&  !ILtQ[(m - 1)/2, 0]))

Rubi steps

\begin {align*} \int x^5 \left (d+e x^2\right ) \left (a+b \sec ^{-1}(c x)\right ) \, dx &=\frac {1}{6} d x^6 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{8} e x^8 \left (a+b \sec ^{-1}(c x)\right )-\frac {(b c x) \int \frac {x^5 \left (4 d+3 e x^2\right )}{24 \sqrt {-1+c^2 x^2}} \, dx}{\sqrt {c^2 x^2}}\\ &=\frac {1}{6} d x^6 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{8} e x^8 \left (a+b \sec ^{-1}(c x)\right )-\frac {(b c x) \int \frac {x^5 \left (4 d+3 e x^2\right )}{\sqrt {-1+c^2 x^2}} \, dx}{24 \sqrt {c^2 x^2}}\\ &=\frac {1}{6} d x^6 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{8} e x^8 \left (a+b \sec ^{-1}(c x)\right )-\frac {(b c x) \operatorname {Subst}\left (\int \frac {x^2 (4 d+3 e x)}{\sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{48 \sqrt {c^2 x^2}}\\ &=\frac {1}{6} d x^6 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{8} e x^8 \left (a+b \sec ^{-1}(c x)\right )-\frac {(b c x) \operatorname {Subst}\left (\int \left (\frac {4 c^2 d+3 e}{c^6 \sqrt {-1+c^2 x}}+\frac {\left (8 c^2 d+9 e\right ) \sqrt {-1+c^2 x}}{c^6}+\frac {\left (4 c^2 d+9 e\right ) \left (-1+c^2 x\right )^{3/2}}{c^6}+\frac {3 e \left (-1+c^2 x\right )^{5/2}}{c^6}\right ) \, dx,x,x^2\right )}{48 \sqrt {c^2 x^2}}\\ &=-\frac {b \left (4 c^2 d+3 e\right ) x \sqrt {-1+c^2 x^2}}{24 c^7 \sqrt {c^2 x^2}}-\frac {b \left (8 c^2 d+9 e\right ) x \left (-1+c^2 x^2\right )^{3/2}}{72 c^7 \sqrt {c^2 x^2}}-\frac {b \left (4 c^2 d+9 e\right ) x \left (-1+c^2 x^2\right )^{5/2}}{120 c^7 \sqrt {c^2 x^2}}-\frac {b e x \left (-1+c^2 x^2\right )^{7/2}}{56 c^7 \sqrt {c^2 x^2}}+\frac {1}{6} d x^6 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{8} e x^8 \left (a+b \sec ^{-1}(c x)\right )\\ \end {align*}

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Mathematica [A]  time = 0.24, size = 118, normalized size = 0.60 \[ \frac {1}{24} a x^6 \left (4 d+3 e x^2\right )-\frac {b x \sqrt {1-\frac {1}{c^2 x^2}} \left (c^6 \left (84 d x^4+45 e x^6\right )+2 c^4 \left (56 d x^2+27 e x^4\right )+8 c^2 \left (28 d+9 e x^2\right )+144 e\right )}{2520 c^7}+\frac {1}{24} b x^6 \sec ^{-1}(c x) \left (4 d+3 e x^2\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^5*(d + e*x^2)*(a + b*ArcSec[c*x]),x]

[Out]

(a*x^6*(4*d + 3*e*x^2))/24 - (b*Sqrt[1 - 1/(c^2*x^2)]*x*(144*e + 8*c^2*(28*d + 9*e*x^2) + 2*c^4*(56*d*x^2 + 27
*e*x^4) + c^6*(84*d*x^4 + 45*e*x^6)))/(2520*c^7) + (b*x^6*(4*d + 3*e*x^2)*ArcSec[c*x])/24

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fricas [A]  time = 1.98, size = 128, normalized size = 0.65 \[ \frac {315 \, a c^{8} e x^{8} + 420 \, a c^{8} d x^{6} + 105 \, {\left (3 \, b c^{8} e x^{8} + 4 \, b c^{8} d x^{6}\right )} \operatorname {arcsec}\left (c x\right ) - {\left (45 \, b c^{6} e x^{6} + 6 \, {\left (14 \, b c^{6} d + 9 \, b c^{4} e\right )} x^{4} + 224 \, b c^{2} d + 8 \, {\left (14 \, b c^{4} d + 9 \, b c^{2} e\right )} x^{2} + 144 \, b e\right )} \sqrt {c^{2} x^{2} - 1}}{2520 \, c^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(e*x^2+d)*(a+b*arcsec(c*x)),x, algorithm="fricas")

[Out]

1/2520*(315*a*c^8*e*x^8 + 420*a*c^8*d*x^6 + 105*(3*b*c^8*e*x^8 + 4*b*c^8*d*x^6)*arcsec(c*x) - (45*b*c^6*e*x^6
+ 6*(14*b*c^6*d + 9*b*c^4*e)*x^4 + 224*b*c^2*d + 8*(14*b*c^4*d + 9*b*c^2*e)*x^2 + 144*b*e)*sqrt(c^2*x^2 - 1))/
c^8

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(e*x^2+d)*(a+b*arcsec(c*x)),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.05, size = 152, normalized size = 0.78 \[ \frac {\frac {a \left (\frac {1}{8} e \,c^{8} x^{8}+\frac {1}{6} c^{8} d \,x^{6}\right )}{c^{2}}+\frac {b \left (\frac {\mathrm {arcsec}\left (c x \right ) e \,c^{8} x^{8}}{8}+\frac {\mathrm {arcsec}\left (c x \right ) c^{8} x^{6} d}{6}-\frac {\left (c^{2} x^{2}-1\right ) \left (45 e \,c^{6} x^{6}+84 x^{4} c^{6} d +54 c^{4} e \,x^{4}+112 c^{4} d \,x^{2}+72 c^{2} e \,x^{2}+224 c^{2} d +144 e \right )}{2520 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{2}}}{c^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(e*x^2+d)*(a+b*arcsec(c*x)),x)

[Out]

1/c^6*(a/c^2*(1/8*e*c^8*x^8+1/6*c^8*d*x^6)+b/c^2*(1/8*arcsec(c*x)*e*c^8*x^8+1/6*arcsec(c*x)*c^8*x^6*d-1/2520*(
c^2*x^2-1)*(45*c^6*e*x^6+84*c^6*d*x^4+54*c^4*e*x^4+112*c^4*d*x^2+72*c^2*e*x^2+224*c^2*d+144*e)/((c^2*x^2-1)/c^
2/x^2)^(1/2)/c/x))

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maxima [A]  time = 0.33, size = 185, normalized size = 0.94 \[ \frac {1}{8} \, a e x^{8} + \frac {1}{6} \, a d x^{6} + \frac {1}{90} \, {\left (15 \, x^{6} \operatorname {arcsec}\left (c x\right ) - \frac {3 \, c^{4} x^{5} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} + 10 \, c^{2} x^{3} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 15 \, x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{5}}\right )} b d + \frac {1}{280} \, {\left (35 \, x^{8} \operatorname {arcsec}\left (c x\right ) - \frac {5 \, c^{6} x^{7} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {7}{2}} + 21 \, c^{4} x^{5} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} + 35 \, c^{2} x^{3} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 35 \, x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{7}}\right )} b e \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(e*x^2+d)*(a+b*arcsec(c*x)),x, algorithm="maxima")

[Out]

1/8*a*e*x^8 + 1/6*a*d*x^6 + 1/90*(15*x^6*arcsec(c*x) - (3*c^4*x^5*(-1/(c^2*x^2) + 1)^(5/2) + 10*c^2*x^3*(-1/(c
^2*x^2) + 1)^(3/2) + 15*x*sqrt(-1/(c^2*x^2) + 1))/c^5)*b*d + 1/280*(35*x^8*arcsec(c*x) - (5*c^6*x^7*(-1/(c^2*x
^2) + 1)^(7/2) + 21*c^4*x^5*(-1/(c^2*x^2) + 1)^(5/2) + 35*c^2*x^3*(-1/(c^2*x^2) + 1)^(3/2) + 35*x*sqrt(-1/(c^2
*x^2) + 1))/c^7)*b*e

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int x^5\,\left (e\,x^2+d\right )\,\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(d + e*x^2)*(a + b*acos(1/(c*x))),x)

[Out]

int(x^5*(d + e*x^2)*(a + b*acos(1/(c*x))), x)

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sympy [A]  time = 7.92, size = 364, normalized size = 1.86 \[ \frac {a d x^{6}}{6} + \frac {a e x^{8}}{8} + \frac {b d x^{6} \operatorname {asec}{\left (c x \right )}}{6} + \frac {b e x^{8} \operatorname {asec}{\left (c x \right )}}{8} - \frac {b d \left (\begin {cases} \frac {x^{4} \sqrt {c^{2} x^{2} - 1}}{5 c} + \frac {4 x^{2} \sqrt {c^{2} x^{2} - 1}}{15 c^{3}} + \frac {8 \sqrt {c^{2} x^{2} - 1}}{15 c^{5}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {i x^{4} \sqrt {- c^{2} x^{2} + 1}}{5 c} + \frac {4 i x^{2} \sqrt {- c^{2} x^{2} + 1}}{15 c^{3}} + \frac {8 i \sqrt {- c^{2} x^{2} + 1}}{15 c^{5}} & \text {otherwise} \end {cases}\right )}{6 c} - \frac {b e \left (\begin {cases} \frac {x^{6} \sqrt {c^{2} x^{2} - 1}}{7 c} + \frac {6 x^{4} \sqrt {c^{2} x^{2} - 1}}{35 c^{3}} + \frac {8 x^{2} \sqrt {c^{2} x^{2} - 1}}{35 c^{5}} + \frac {16 \sqrt {c^{2} x^{2} - 1}}{35 c^{7}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {i x^{6} \sqrt {- c^{2} x^{2} + 1}}{7 c} + \frac {6 i x^{4} \sqrt {- c^{2} x^{2} + 1}}{35 c^{3}} + \frac {8 i x^{2} \sqrt {- c^{2} x^{2} + 1}}{35 c^{5}} + \frac {16 i \sqrt {- c^{2} x^{2} + 1}}{35 c^{7}} & \text {otherwise} \end {cases}\right )}{8 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(e*x**2+d)*(a+b*asec(c*x)),x)

[Out]

a*d*x**6/6 + a*e*x**8/8 + b*d*x**6*asec(c*x)/6 + b*e*x**8*asec(c*x)/8 - b*d*Piecewise((x**4*sqrt(c**2*x**2 - 1
)/(5*c) + 4*x**2*sqrt(c**2*x**2 - 1)/(15*c**3) + 8*sqrt(c**2*x**2 - 1)/(15*c**5), Abs(c**2*x**2) > 1), (I*x**4
*sqrt(-c**2*x**2 + 1)/(5*c) + 4*I*x**2*sqrt(-c**2*x**2 + 1)/(15*c**3) + 8*I*sqrt(-c**2*x**2 + 1)/(15*c**5), Tr
ue))/(6*c) - b*e*Piecewise((x**6*sqrt(c**2*x**2 - 1)/(7*c) + 6*x**4*sqrt(c**2*x**2 - 1)/(35*c**3) + 8*x**2*sqr
t(c**2*x**2 - 1)/(35*c**5) + 16*sqrt(c**2*x**2 - 1)/(35*c**7), Abs(c**2*x**2) > 1), (I*x**6*sqrt(-c**2*x**2 +
1)/(7*c) + 6*I*x**4*sqrt(-c**2*x**2 + 1)/(35*c**3) + 8*I*x**2*sqrt(-c**2*x**2 + 1)/(35*c**5) + 16*I*sqrt(-c**2
*x**2 + 1)/(35*c**7), True))/(8*c)

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