\(\int \frac {x^2 (a+b \sec ^{-1}(c x))}{(d+e x^2)^{5/2}} \, dx\) [158]

Optimal result

Integrand size = 23, antiderivative size = 276 \[ \int \frac {x^2 \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx=-\frac {b c x^2 \sqrt {-1+c^2 x^2}}{3 d \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}+\frac {x^3 \left (a+b \sec ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {b c^2 x \sqrt {1-c^2 x^2} \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{3 d e \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}-\frac {b x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{3 d e \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \]

[Out]

1/3*x^3*(a+b*arcsec(c*x))/d/(e*x^2+d)^(3/2)-1/3*b*c*x^2*(c^2*x^2-1)^(1/2)/d/(c^2*d+e)/(c^2*x^2)^(1/2)/(e*x^2+d
)^(1/2)+1/3*b*c^2*x*EllipticE(c*x,(-e/c^2/d)^(1/2))*(-c^2*x^2+1)^(1/2)*(e*x^2+d)^(1/2)/d/e/(c^2*d+e)/(c^2*x^2)
^(1/2)/(c^2*x^2-1)^(1/2)/(1+e*x^2/d)^(1/2)-1/3*b*x*EllipticF(c*x,(-e/c^2/d)^(1/2))*(-c^2*x^2+1)^(1/2)*(1+e*x^2
/d)^(1/2)/d/e/(c^2*x^2)^(1/2)/(c^2*x^2-1)^(1/2)/(e*x^2+d)^(1/2)

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {270, 5346, 12, 482, 434, 438, 437, 435, 432, 430} \[ \int \frac {x^2 \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\frac {x^3 \left (a+b \sec ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}-\frac {b x \sqrt {1-c^2 x^2} \sqrt {\frac {e x^2}{d}+1} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{3 d e \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}+\frac {b c^2 x \sqrt {1-c^2 x^2} \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{3 d e \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \left (c^2 d+e\right ) \sqrt {\frac {e x^2}{d}+1}}-\frac {b c x^2 \sqrt {c^2 x^2-1}}{3 d \sqrt {c^2 x^2} \left (c^2 d+e\right ) \sqrt {d+e x^2}} \]

[In]

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

[Out]

-1/3*(b*c*x^2*Sqrt[-1 + c^2*x^2])/(d*(c^2*d + e)*Sqrt[c^2*x^2]*Sqrt[d + e*x^2]) + (x^3*(a + b*ArcSec[c*x]))/(3
*d*(d + e*x^2)^(3/2)) + (b*c^2*x*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/(3*d*
e*(c^2*d + e)*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[1 + (e*x^2)/d]) - (b*x*Sqrt[1 - c^2*x^2]*Sqrt[1 + (e*x^2)/
d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(3*d*e*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])

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] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*
c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 430

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]
))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && Gt
Q[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])

Rule 432

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*
x^2], Int[1/(Sqrt[a + b*x^2]*Sqrt[1 + (d/c)*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] &&  !GtQ[c, 0]

Rule 434

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Dist[b/d, Int[Sqrt[c + d*x^2]/Sqrt[a + b
*x^2], x], x] - Dist[(b*c - a*d)/d, Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d}, x]
&& PosQ[d/c] && NegQ[b/a]

Rule 435

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*Ell
ipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0
]

Rule 437

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[a + b*x^2]/Sqrt[1 + (b/a)*x^2]
, Int[Sqrt[1 + (b/a)*x^2]/Sqrt[c + d*x^2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &&  !GtQ
[a, 0]

Rule 438

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2]
, Int[Sqrt[a + b*x^2]/Sqrt[1 + (d/c)*x^2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] &&  !GtQ[c, 0]

Rule 482

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[e^(n - 1
)*(e*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(n*(b*c - a*d)*(p + 1))), x] - Dist[e^n/(n*(b*c -
a*d)*(p + 1)), Int[(e*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(m - n + 1) + d*(m + n*(p + q + 1)
+ 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && GeQ[n
, m - n + 1] && GtQ[m - n + 1, 0] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 5346

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*} \text {integral}& = \frac {x^3 \left (a+b \sec ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}-\frac {(b c x) \int \frac {x^2}{3 d \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}} \, dx}{\sqrt {c^2 x^2}} \\ & = \frac {x^3 \left (a+b \sec ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}-\frac {(b c x) \int \frac {x^2}{\sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}} \, dx}{3 d \sqrt {c^2 x^2}} \\ & = -\frac {b c x^2 \sqrt {-1+c^2 x^2}}{3 d \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}+\frac {x^3 \left (a+b \sec ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {(b c x) \int \frac {\sqrt {-1+c^2 x^2}}{\sqrt {d+e x^2}} \, dx}{3 d \left (c^2 d+e\right ) \sqrt {c^2 x^2}} \\ & = -\frac {b c x^2 \sqrt {-1+c^2 x^2}}{3 d \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}+\frac {x^3 \left (a+b \sec ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}-\frac {(b c x) \int \frac {1}{\sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{3 d e \sqrt {c^2 x^2}}+\frac {\left (b c^3 x\right ) \int \frac {\sqrt {d+e x^2}}{\sqrt {-1+c^2 x^2}} \, dx}{3 d e \left (c^2 d+e\right ) \sqrt {c^2 x^2}} \\ & = -\frac {b c x^2 \sqrt {-1+c^2 x^2}}{3 d \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}+\frac {x^3 \left (a+b \sec ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {\left (b c^3 x \sqrt {1-c^2 x^2}\right ) \int \frac {\sqrt {d+e x^2}}{\sqrt {1-c^2 x^2}} \, dx}{3 d e \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2}}-\frac {\left (b c x \sqrt {1+\frac {e x^2}{d}}\right ) \int \frac {1}{\sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}} \, dx}{3 d e \sqrt {c^2 x^2} \sqrt {d+e x^2}} \\ & = -\frac {b c x^2 \sqrt {-1+c^2 x^2}}{3 d \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}+\frac {x^3 \left (a+b \sec ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {\left (b c^3 x \sqrt {1-c^2 x^2} \sqrt {d+e x^2}\right ) \int \frac {\sqrt {1+\frac {e x^2}{d}}}{\sqrt {1-c^2 x^2}} \, dx}{3 d e \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}-\frac {\left (b c x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}}\right ) \int \frac {1}{\sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}}} \, dx}{3 d e \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \\ & = -\frac {b c x^2 \sqrt {-1+c^2 x^2}}{3 d \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}+\frac {x^3 \left (a+b \sec ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {b c^2 x \sqrt {1-c^2 x^2} \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{3 d e \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}-\frac {b x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{3 d e \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.67 \[ \int \frac {x^2 \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\frac {x^2 \left (a \left (c^2 d+e\right ) x-b c \sqrt {1-\frac {1}{c^2 x^2}} \left (d+e x^2\right )+b \left (c^2 d+e\right ) x \sec ^{-1}(c x)\right )}{3 d \left (c^2 d+e\right ) \left (d+e x^2\right )^{3/2}}+\frac {b c \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {1+\frac {e x^2}{d}} E\left (\arcsin \left (\sqrt {-\frac {e}{d}} x\right )|-\frac {c^2 d}{e}\right )}{3 d \sqrt {-\frac {e}{d}} \left (c^2 d+e\right ) \sqrt {1-c^2 x^2} \sqrt {d+e x^2}} \]

[In]

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

[Out]

(x^2*(a*(c^2*d + e)*x - b*c*Sqrt[1 - 1/(c^2*x^2)]*(d + e*x^2) + b*(c^2*d + e)*x*ArcSec[c*x]))/(3*d*(c^2*d + e)
*(d + e*x^2)^(3/2)) + (b*c*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[1 + (e*x^2)/d]*EllipticE[ArcSin[Sqrt[-(e/d)]*x], -((c^
2*d)/e)])/(3*d*Sqrt[-(e/d)]*(c^2*d + e)*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])

Maple [F]

\[\int \frac {x^{2} \left (a +b \,\operatorname {arcsec}\left (c x \right )\right )}{\left (e \,x^{2}+d \right )^{\frac {5}{2}}}d x\]

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.12 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.04 \[ \int \frac {x^2 \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\frac {{\left ({\left (b c^{3} d^{2} e + b c d e^{2}\right )} x^{3} \operatorname {arcsec}\left (c x\right ) + {\left (a c^{3} d^{2} e + a c d e^{2}\right )} x^{3} - {\left (b c d e^{2} x^{3} + b c d^{2} e x\right )} \sqrt {c^{2} x^{2} - 1}\right )} \sqrt {e x^{2} + d} - {\left ({\left (b c^{4} d e^{2} x^{4} + 2 \, b c^{4} d^{2} e x^{2} + b c^{4} d^{3}\right )} E(\arcsin \left (c x\right )\,|\,-\frac {e}{c^{2} d}) - {\left (b c^{4} d^{3} + {\left (b c^{4} d e^{2} + b e^{3}\right )} x^{4} + b d^{2} e + 2 \, {\left (b c^{4} d^{2} e + b d e^{2}\right )} x^{2}\right )} F(\arcsin \left (c x\right )\,|\,-\frac {e}{c^{2} d})\right )} \sqrt {-d}}{3 \, {\left (c^{3} d^{5} e + c d^{4} e^{2} + {\left (c^{3} d^{3} e^{3} + c d^{2} e^{4}\right )} x^{4} + 2 \, {\left (c^{3} d^{4} e^{2} + c d^{3} e^{3}\right )} x^{2}\right )}} \]

[In]

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

[Out]

1/3*(((b*c^3*d^2*e + b*c*d*e^2)*x^3*arcsec(c*x) + (a*c^3*d^2*e + a*c*d*e^2)*x^3 - (b*c*d*e^2*x^3 + b*c*d^2*e*x
)*sqrt(c^2*x^2 - 1))*sqrt(e*x^2 + d) - ((b*c^4*d*e^2*x^4 + 2*b*c^4*d^2*e*x^2 + b*c^4*d^3)*elliptic_e(arcsin(c*
x), -e/(c^2*d)) - (b*c^4*d^3 + (b*c^4*d*e^2 + b*e^3)*x^4 + b*d^2*e + 2*(b*c^4*d^2*e + b*d*e^2)*x^2)*elliptic_f
(arcsin(c*x), -e/(c^2*d)))*sqrt(-d))/(c^3*d^5*e + c*d^4*e^2 + (c^3*d^3*e^3 + c*d^2*e^4)*x^4 + 2*(c^3*d^4*e^2 +
c*d^3*e^3)*x^2)

Sympy [F(-1)]

Timed out. \[ \int \frac {x^2 \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {x^2 \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arcsec}\left (c x\right ) + a\right )} x^{2}}{{\left (e x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]

[In]

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

[Out]

-1/3*a*(x/((e*x^2 + d)^(3/2)*e) - x/(sqrt(e*x^2 + d)*d*e)) + b*integrate(x^2*arctan(sqrt(c*x + 1)*sqrt(c*x - 1
))/((e^2*x^4 + 2*d*e*x^2 + d^2)*sqrt(e*x^2 + d)), x)

Giac [F]

\[ \int \frac {x^2 \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arcsec}\left (c x\right ) + a\right )} x^{2}}{{\left (e x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2 \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\int \frac {x^2\,\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^{5/2}} \,d x \]

[In]

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

[Out]

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