\(\int x^2 \sqrt {a+\frac {b}{c+d x^2}} \, dx\) [326]

Optimal result

Integrand size = 21, antiderivative size = 282 \[ \int x^2 \sqrt {a+\frac {b}{c+d x^2}} \, dx=\frac {(b-a c) x \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{3 a d}+\frac {x \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{3 d}-\frac {\sqrt {c} (b-a c) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 a d^{3/2} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}}-\frac {c^{3/2} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{3 d^{3/2} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1985, 1986, 489, 545, 429, 506, 422} \[ \int x^2 \sqrt {a+\frac {b}{c+d x^2}} \, dx=-\frac {c^{3/2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{3 d^{3/2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}-\frac {\sqrt {c} (b-a c) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 a d^{3/2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}+\frac {x (b-a c) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{3 a d}+\frac {x \left (c+d x^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{3 d} \]

[In]

Int[x^2*Sqrt[a + b/(c + d*x^2)],x]

[Out]

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

Rule 422

Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sq
rt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c + d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /;
FreeQ[{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]

Rule 429

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

Rule 489

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/(b*(m + n*(p + q) + 1))), x] - Dist[e^n/(b*(m + n*(p +
q) + 1)), Int[(e*x)^(m - n)*(a + b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[a*c*(m - n + 1) + (a*d*(m - n + 1) - n*q*(b
*c - a*d))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && GtQ[q, 0] &&
GtQ[m - n + 1, 0] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 506

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt
[c + d*x^2])), x] - Dist[c/b, Int[Sqrt[a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b
*c - a*d, 0] && PosQ[b/a] && PosQ[d/c] &&  !SimplerSqrtQ[b/a, d/c]

Rule 545

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Dist[
e, Int[(a + b*x^n)^p*(c + d*x^n)^q, x], x] + Dist[f, Int[x^n*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, n, p, q}, x]

Rule 1985

Int[(u_.)*((a_) + (b_.)/((c_) + (d_.)*(x_)^(n_)))^(p_), x_Symbol] :> Int[u*((b + a*c + a*d*x^n)/(c + d*x^n))^p
, x] /; FreeQ[{a, b, c, d, n, p}, x]

Rule 1986

Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))^(q_.)*((c_) + (d_.)*(x_)^(n_))^(r_.))^(p_), x_Symbol] :> Dist[Simp
[(e*(a + b*x^n)^q*(c + d*x^n)^r)^p/((a + b*x^n)^(p*q)*(c + d*x^n)^(p*r))], Int[u*(a + b*x^n)^(p*q)*(c + d*x^n)
^(p*r), x], x] /; FreeQ[{a, b, c, d, e, n, p, q, r}, x]

Rubi steps \begin{align*} \text {integral}& = \int x^2 \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \, dx \\ & = \frac {\left (\sqrt {c+d x^2} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}\right ) \int \frac {x^2 \sqrt {b+a c+a d x^2}}{\sqrt {c+d x^2}} \, dx}{\sqrt {b+a c+a d x^2}} \\ & = \frac {x \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{3 d}-\frac {\left (\sqrt {c+d x^2} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}\right ) \int \frac {c (b+a c)-(b-a c) d x^2}{\sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{3 d \sqrt {b+a c+a d x^2}} \\ & = \frac {x \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{3 d}+\frac {\left ((b-a c) \sqrt {c+d x^2} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}\right ) \int \frac {x^2}{\sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{3 \sqrt {b+a c+a d x^2}}-\frac {\left (c (b+a c) \sqrt {c+d x^2} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}\right ) \int \frac {1}{\sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{3 d \sqrt {b+a c+a d x^2}} \\ & = \frac {(b-a c) x \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{3 a d}+\frac {x \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{3 d}-\frac {c^{3/2} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 d^{3/2} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}}-\frac {\left (c (b-a c) \sqrt {c+d x^2} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}\right ) \int \frac {\sqrt {b+a c+a d x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{3 a d \sqrt {b+a c+a d x^2}} \\ & = \frac {(b-a c) x \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{3 a d}+\frac {x \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{3 d}-\frac {\sqrt {c} (b-a c) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 a d^{3/2} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}}-\frac {c^{3/2} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 d^{3/2} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 9.20 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.86 \[ \int x^2 \sqrt {a+\frac {b}{c+d x^2}} \, dx=\frac {\sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \left (a \sqrt {\frac {d}{c}} x \left (c+d x^2\right ) \left (b+a \left (c+d x^2\right )\right )+i \left (-b^2+a^2 c^2\right ) \sqrt {\frac {b+a c+a d x^2}{b+a c}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right )|\frac {a c}{b+a c}\right )+i b (b+a c) \sqrt {\frac {b+a c+a d x^2}{b+a c}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right ),\frac {a c}{b+a c}\right )\right )}{3 a d \sqrt {\frac {d}{c}} \left (b+a \left (c+d x^2\right )\right )} \]

[In]

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

[Out]

(Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(a*Sqrt[d/c]*x*(c + d*x^2)*(b + a*(c + d*x^2)) + I*(-b^2 + a^2*c^2)*Sqr
t[(b + a*c + a*d*x^2)/(b + a*c)]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (a*c)/(b + a*c)] + I*b*
(b + a*c)*Sqrt[(b + a*c + a*d*x^2)/(b + a*c)]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (a*c)/(b +
a*c)]))/(3*a*d*Sqrt[d/c]*(b + a*(c + d*x^2)))

Maple [A] (verified)

Time = 5.92 (sec) , antiderivative size = 406, normalized size of antiderivative = 1.44

method	result	size
default	\(\frac {\left (\sqrt {-\frac {a d}{a c +b}}\, a \,d^{2} x^{5}+2 \sqrt {-\frac {a d}{a c +b}}\, a c d \,x^{3}+\sqrt {-\frac {a d}{a c +b}}\, b d \,x^{3}-\sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right ) a \,c^{2}+\sqrt {-\frac {a d}{a c +b}}\, a \,c^{2} x -2 \sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right ) b c +\sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right ) b c +\sqrt {-\frac {a d}{a c +b}}\, b c x \right ) \left (d \,x^{2}+c \right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}{3 d \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {-\frac {a d}{a c +b}}\, \sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}}\)	\(406\)
risch	\(\frac {x \left (d \,x^{2}+c \right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}{3 d}-\frac {\left (\frac {a \,c^{2} \sqrt {1+\frac {a d \,x^{2}}{a c +b}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )}{\sqrt {-\frac {a d}{a c +b}}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}}+\frac {b c \sqrt {1+\frac {a d \,x^{2}}{a c +b}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )}{\sqrt {-\frac {a d}{a c +b}}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}}-\frac {2 \left (a c d -b d \right ) \left (a \,c^{2}+b c \right ) \sqrt {1+\frac {a d \,x^{2}}{a c +b}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )-E\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )\right )}{\sqrt {-\frac {a d}{a c +b}}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \left (2 a c d +2 b d \right )}\right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}\, \sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}}{3 d \left (a d \,x^{2}+a c +b \right )}\)	\(525\)

[In]

int(x^2*(a+b/(d*x^2+c))^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/3*((-a*d/(a*c+b))^(1/2)*a*d^2*x^5+2*(-a*d/(a*c+b))^(1/2)*a*c*d*x^3+(-a*d/(a*c+b))^(1/2)*b*d*x^3-((a*d*x^2+a*
c+b)/(a*c+b))^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticE(x*(-a*d/(a*c+b))^(1/2),((a*c+b)/a/c)^(1/2))*a*c^2+(-a*d/(a*c
+b))^(1/2)*a*c^2*x-2*((a*d*x^2+a*c+b)/(a*c+b))^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticF(x*(-a*d/(a*c+b))^(1/2),((a*
c+b)/a/c)^(1/2))*b*c+((a*d*x^2+a*c+b)/(a*c+b))^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticE(x*(-a*d/(a*c+b))^(1/2),((a*
c+b)/a/c)^(1/2))*b*c+(-a*d/(a*c+b))^(1/2)*b*c*x)*(d*x^2+c)*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)/d/(a*d^2*x^4+2*a*
c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)/(-a*d/(a*c+b))^(1/2)/((a*d*x^2+a*c+b)*(d*x^2+c))^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.11 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.59 \[ \int x^2 \sqrt {a+\frac {b}{c+d x^2}} \, dx=\frac {{\left (a c^{2} - b c\right )} \sqrt {a} x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a c + b}{a c}) - {\left (a c^{2} - b c + {\left (a c + b\right )} d\right )} \sqrt {a} x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a c + b}{a c}) + {\left (a d^{2} x^{4} + b d x^{2} - a c^{2} + b c\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{3 \, a d^{2} x} \]

[In]

integrate(x^2*(a+b/(d*x^2+c))^(1/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int x^2 \sqrt {a+\frac {b}{c+d x^2}} \, dx=\int x^{2} \sqrt {\frac {a c + a d x^{2} + b}{c + d x^{2}}}\, dx \]

[In]

integrate(x**2*(a+b/(d*x**2+c))**(1/2),x)

[Out]

Integral(x**2*sqrt((a*c + a*d*x**2 + b)/(c + d*x**2)), x)

Maxima [F]

\[ \int x^2 \sqrt {a+\frac {b}{c+d x^2}} \, dx=\int { \sqrt {a + \frac {b}{d x^{2} + c}} x^{2} \,d x } \]

[In]

integrate(x^2*(a+b/(d*x^2+c))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(a + b/(d*x^2 + c))*x^2, x)

Giac [F]

\[ \int x^2 \sqrt {a+\frac {b}{c+d x^2}} \, dx=\int { \sqrt {a + \frac {b}{d x^{2} + c}} x^{2} \,d x } \]

[In]

integrate(x^2*(a+b/(d*x^2+c))^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(a + b/(d*x^2 + c))*x^2, x)

Mupad [F(-1)]

Timed out. \[ \int x^2 \sqrt {a+\frac {b}{c+d x^2}} \, dx=\int x^2\,\sqrt {a+\frac {b}{d\,x^2+c}} \,d x \]

[In]

int(x^2*(a + b/(c + d*x^2))^(1/2),x)

[Out]

int(x^2*(a + b/(c + d*x^2))^(1/2), x)