3.87 \(\int x^3 (d+e x^2)^2 (a+b \sec ^{-1}(c x)) \, dx\)

Optimal. Leaf size=242 \[ \frac{1}{4} d^2 x^4 \left (a+b \sec ^{-1}(c x)\right )+\frac{1}{3} d e x^6 \left (a+b \sec ^{-1}(c x)\right )+\frac{1}{8} e^2 x^8 \left (a+b \sec ^{-1}(c x)\right )-\frac{b x \left (c^2 x^2-1\right )^{3/2} \left (6 c^4 d^2+16 c^2 d e+9 e^2\right )}{72 c^7 \sqrt{c^2 x^2}}-\frac{b x \sqrt{c^2 x^2-1} \left (6 c^4 d^2+8 c^2 d e+3 e^2\right )}{24 c^7 \sqrt{c^2 x^2}}-\frac{b e x \left (c^2 x^2-1\right )^{5/2} \left (8 c^2 d+9 e\right )}{120 c^7 \sqrt{c^2 x^2}}-\frac{b e^2 x \left (c^2 x^2-1\right )^{7/2}}{56 c^7 \sqrt{c^2 x^2}} \]

[Out]

-(b*(6*c^4*d^2 + 8*c^2*d*e + 3*e^2)*x*Sqrt[-1 + c^2*x^2])/(24*c^7*Sqrt[c^2*x^2]) - (b*(6*c^4*d^2 + 16*c^2*d*e
+ 9*e^2)*x*(-1 + c^2*x^2)^(3/2))/(72*c^7*Sqrt[c^2*x^2]) - (b*e*(8*c^2*d + 9*e)*x*(-1 + c^2*x^2)^(5/2))/(120*c^
7*Sqrt[c^2*x^2]) - (b*e^2*x*(-1 + c^2*x^2)^(7/2))/(56*c^7*Sqrt[c^2*x^2]) + (d^2*x^4*(a + b*ArcSec[c*x]))/4 + (
d*e*x^6*(a + b*ArcSec[c*x]))/3 + (e^2*x^8*(a + b*ArcSec[c*x]))/8

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Rubi [A]  time = 0.223569, antiderivative size = 242, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {266, 43, 5238, 12, 1251, 771} \[ \frac{1}{4} d^2 x^4 \left (a+b \sec ^{-1}(c x)\right )+\frac{1}{3} d e x^6 \left (a+b \sec ^{-1}(c x)\right )+\frac{1}{8} e^2 x^8 \left (a+b \sec ^{-1}(c x)\right )-\frac{b x \left (c^2 x^2-1\right )^{3/2} \left (6 c^4 d^2+16 c^2 d e+9 e^2\right )}{72 c^7 \sqrt{c^2 x^2}}-\frac{b x \sqrt{c^2 x^2-1} \left (6 c^4 d^2+8 c^2 d e+3 e^2\right )}{24 c^7 \sqrt{c^2 x^2}}-\frac{b e x \left (c^2 x^2-1\right )^{5/2} \left (8 c^2 d+9 e\right )}{120 c^7 \sqrt{c^2 x^2}}-\frac{b e^2 x \left (c^2 x^2-1\right )^{7/2}}{56 c^7 \sqrt{c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*(6*c^4*d^2 + 8*c^2*d*e + 3*e^2)*x*Sqrt[-1 + c^2*x^2])/(24*c^7*Sqrt[c^2*x^2]) - (b*(6*c^4*d^2 + 16*c^2*d*e
+ 9*e^2)*x*(-1 + c^2*x^2)^(3/2))/(72*c^7*Sqrt[c^2*x^2]) - (b*e*(8*c^2*d + 9*e)*x*(-1 + c^2*x^2)^(5/2))/(120*c^
7*Sqrt[c^2*x^2]) - (b*e^2*x*(-1 + c^2*x^2)^(7/2))/(56*c^7*Sqrt[c^2*x^2]) + (d^2*x^4*(a + b*ArcSec[c*x]))/4 + (
d*e*x^6*(a + b*ArcSec[c*x]))/3 + (e^2*x^8*(a + b*ArcSec[c*x]))/8

Rule 266

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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]))

Rule 12

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

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 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]))

Rubi steps

\begin{align*} \int x^3 \left (d+e x^2\right )^2 \left (a+b \sec ^{-1}(c x)\right ) \, dx &=\frac{1}{4} d^2 x^4 \left (a+b \sec ^{-1}(c x)\right )+\frac{1}{3} d e x^6 \left (a+b \sec ^{-1}(c x)\right )+\frac{1}{8} e^2 x^8 \left (a+b \sec ^{-1}(c x)\right )-\frac{(b c x) \int \frac{x^3 \left (6 d^2+8 d e x^2+3 e^2 x^4\right )}{24 \sqrt{-1+c^2 x^2}} \, dx}{\sqrt{c^2 x^2}}\\ &=\frac{1}{4} d^2 x^4 \left (a+b \sec ^{-1}(c x)\right )+\frac{1}{3} d e x^6 \left (a+b \sec ^{-1}(c x)\right )+\frac{1}{8} e^2 x^8 \left (a+b \sec ^{-1}(c x)\right )-\frac{(b c x) \int \frac{x^3 \left (6 d^2+8 d e x^2+3 e^2 x^4\right )}{\sqrt{-1+c^2 x^2}} \, dx}{24 \sqrt{c^2 x^2}}\\ &=\frac{1}{4} d^2 x^4 \left (a+b \sec ^{-1}(c x)\right )+\frac{1}{3} d e x^6 \left (a+b \sec ^{-1}(c x)\right )+\frac{1}{8} e^2 x^8 \left (a+b \sec ^{-1}(c x)\right )-\frac{(b c x) \operatorname{Subst}\left (\int \frac{x \left (6 d^2+8 d e x+3 e^2 x^2\right )}{\sqrt{-1+c^2 x}} \, dx,x,x^2\right )}{48 \sqrt{c^2 x^2}}\\ &=\frac{1}{4} d^2 x^4 \left (a+b \sec ^{-1}(c x)\right )+\frac{1}{3} d e x^6 \left (a+b \sec ^{-1}(c x)\right )+\frac{1}{8} e^2 x^8 \left (a+b \sec ^{-1}(c x)\right )-\frac{(b c x) \operatorname{Subst}\left (\int \left (\frac{6 c^4 d^2+8 c^2 d e+3 e^2}{c^6 \sqrt{-1+c^2 x}}+\frac{\left (6 c^4 d^2+16 c^2 d e+9 e^2\right ) \sqrt{-1+c^2 x}}{c^6}+\frac{e \left (8 c^2 d+9 e\right ) \left (-1+c^2 x\right )^{3/2}}{c^6}+\frac{3 e^2 \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 (6 c^4 d^2+8 c^2 d e+3 e^2\right ) x \sqrt{-1+c^2 x^2}}{24 c^7 \sqrt{c^2 x^2}}-\frac{b \left (6 c^4 d^2+16 c^2 d e+9 e^2\right ) x \left (-1+c^2 x^2\right )^{3/2}}{72 c^7 \sqrt{c^2 x^2}}-\frac{b e \left (8 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^2 x \left (-1+c^2 x^2\right )^{7/2}}{56 c^7 \sqrt{c^2 x^2}}+\frac{1}{4} d^2 x^4 \left (a+b \sec ^{-1}(c x)\right )+\frac{1}{3} d e x^6 \left (a+b \sec ^{-1}(c x)\right )+\frac{1}{8} e^2 x^8 \left (a+b \sec ^{-1}(c x)\right )\\ \end{align*}

Mathematica [A]  time = 0.274735, size = 162, normalized size = 0.67 \[ \frac{1}{24} a x^4 \left (6 d^2+8 d e x^2+3 e^2 x^4\right )-\frac{b x \sqrt{1-\frac{1}{c^2 x^2}} \left (3 c^6 \left (70 d^2 x^2+56 d e x^4+15 e^2 x^6\right )+c^4 \left (420 d^2+224 d e x^2+54 e^2 x^4\right )+8 c^2 e \left (56 d+9 e x^2\right )+144 e^2\right )}{2520 c^7}+\frac{1}{24} b x^4 \sec ^{-1}(c x) \left (6 d^2+8 d e x^2+3 e^2 x^4\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*x^4*(6*d^2 + 8*d*e*x^2 + 3*e^2*x^4))/24 - (b*Sqrt[1 - 1/(c^2*x^2)]*x*(144*e^2 + 8*c^2*e*(56*d + 9*e*x^2) +
c^4*(420*d^2 + 224*d*e*x^2 + 54*e^2*x^4) + 3*c^6*(70*d^2*x^2 + 56*d*e*x^4 + 15*e^2*x^6)))/(2520*c^7) + (b*x^4*
(6*d^2 + 8*d*e*x^2 + 3*e^2*x^4)*ArcSec[c*x])/24

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Maple [A]  time = 0.171, size = 214, normalized size = 0.9 \begin{align*}{\frac{1}{{c}^{4}} \left ({\frac{a}{{c}^{4}} \left ({\frac{{e}^{2}{c}^{8}{x}^{8}}{8}}+{\frac{{c}^{8}ed{x}^{6}}{3}}+{\frac{{x}^{4}{c}^{8}{d}^{2}}{4}} \right ) }+{\frac{b}{{c}^{4}} \left ({\frac{{\rm arcsec} \left (cx\right ){e}^{2}{c}^{8}{x}^{8}}{8}}+{\frac{{\rm arcsec} \left (cx\right ){c}^{8}ed{x}^{6}}{3}}+{\frac{{\rm arcsec} \left (cx\right ){c}^{8}{x}^{4}{d}^{2}}{4}}-{\frac{ \left ({c}^{2}{x}^{2}-1 \right ) \left ( 45\,{c}^{6}{e}^{2}{x}^{6}+168\,{c}^{6}de{x}^{4}+210\,{c}^{6}{d}^{2}{x}^{2}+54\,{c}^{4}{e}^{2}{x}^{4}+224\,{c}^{4}de{x}^{2}+420\,{d}^{2}{c}^{4}+72\,{c}^{2}{e}^{2}{x}^{2}+448\,{c}^{2}ed+144\,{e}^{2} \right ) }{2520\,cx}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^4*(a/c^4*(1/8*e^2*c^8*x^8+1/3*c^8*e*d*x^6+1/4*x^4*c^8*d^2)+b/c^4*(1/8*arcsec(c*x)*e^2*c^8*x^8+1/3*arcsec(c
*x)*c^8*e*d*x^6+1/4*arcsec(c*x)*c^8*x^4*d^2-1/2520*(c^2*x^2-1)*(45*c^6*e^2*x^6+168*c^6*d*e*x^4+210*c^6*d^2*x^2
+54*c^4*e^2*x^4+224*c^4*d*e*x^2+420*c^4*d^2+72*c^2*e^2*x^2+448*c^2*d*e+144*e^2)/((c^2*x^2-1)/c^2/x^2)^(1/2)/c/
x))

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Maxima [A]  time = 0.98607, size = 346, normalized size = 1.43 \begin{align*} \frac{1}{8} \, a e^{2} x^{8} + \frac{1}{3} \, a d e x^{6} + \frac{1}{4} \, a d^{2} x^{4} + \frac{1}{12} \,{\left (3 \, x^{4} \operatorname{arcsec}\left (c x\right ) - \frac{c^{2} x^{3}{\left (-\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{3}{2}} + 3 \, x \sqrt{-\frac{1}{c^{2} x^{2}} + 1}}{c^{3}}\right )} b d^{2} + \frac{1}{45} \,{\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 e + \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^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*a*e^2*x^8 + 1/3*a*d*e*x^6 + 1/4*a*d^2*x^4 + 1/12*(3*x^4*arcsec(c*x) - (c^2*x^3*(-1/(c^2*x^2) + 1)^(3/2) +
3*x*sqrt(-1/(c^2*x^2) + 1))/c^3)*b*d^2 + 1/45*(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*e + 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^2

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Fricas [A]  time = 2.18912, size = 431, normalized size = 1.78 \begin{align*} \frac{315 \, a c^{8} e^{2} x^{8} + 840 \, a c^{8} d e x^{6} + 630 \, a c^{8} d^{2} x^{4} + 105 \,{\left (3 \, b c^{8} e^{2} x^{8} + 8 \, b c^{8} d e x^{6} + 6 \, b c^{8} d^{2} x^{4}\right )} \operatorname{arcsec}\left (c x\right ) -{\left (45 \, b c^{6} e^{2} x^{6} + 420 \, b c^{4} d^{2} + 448 \, b c^{2} d e + 6 \,{\left (28 \, b c^{6} d e + 9 \, b c^{4} e^{2}\right )} x^{4} + 144 \, b e^{2} + 2 \,{\left (105 \, b c^{6} d^{2} + 112 \, b c^{4} d e + 36 \, b c^{2} e^{2}\right )} x^{2}\right )} \sqrt{c^{2} x^{2} - 1}}{2520 \, c^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2520*(315*a*c^8*e^2*x^8 + 840*a*c^8*d*e*x^6 + 630*a*c^8*d^2*x^4 + 105*(3*b*c^8*e^2*x^8 + 8*b*c^8*d*e*x^6 + 6
*b*c^8*d^2*x^4)*arcsec(c*x) - (45*b*c^6*e^2*x^6 + 420*b*c^4*d^2 + 448*b*c^2*d*e + 6*(28*b*c^6*d*e + 9*b*c^4*e^
2)*x^4 + 144*b*e^2 + 2*(105*b*c^6*d^2 + 112*b*c^4*d*e + 36*b*c^2*e^2)*x^2)*sqrt(c^2*x^2 - 1))/c^8

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \left (a + b \operatorname{asec}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**3*(a + b*asec(c*x))*(d + e*x**2)**2, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (e x^{2} + d\right )}^{2}{\left (b \operatorname{arcsec}\left (c x\right ) + a\right )} x^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(e*x^2+d)^2*(a+b*arcsec(c*x)),x, algorithm="giac")

[Out]

integrate((e*x^2 + d)^2*(b*arcsec(c*x) + a)*x^3, x)