\(\int x^2 (a+\frac {b}{c+d x^2})^{3/2} \, dx\) [339]

Optimal result

Integrand size = 21, antiderivative size = 331 \[ \int x^2 \left (a+\frac {b}{c+d x^2}\right )^{3/2} \, dx=\frac {(7 b-a c) x \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{3 d}+\frac {4 a x \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{3 d}-\frac {x \left (b+a c+a d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{d}-\frac {\sqrt {c} (7 b-a c) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 d^{3/2} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}}+\frac {\sqrt {c} (3 b-a c) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{3 d^{3/2} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}} \]

[Out]

1/3*(-a*c+7*b)*x*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)/d+4/3*a*x*(d*x^2+c)*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)/d-x*(
a*d*x^2+a*c+b)*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)/d-1/3*(-a*c+7*b)*(1/(1+d*x^2/c))^(1/2)*(1+d*x^2/c)^(1/2)*Elli
pticE(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)/d^(3/2)
/(c*(a*d*x^2+a*c+b)/(a*c+b)/(d*x^2+c))^(1/2)+1/3*(-a*c+3*b)*(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))*c^(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)

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {1985, 1986, 478, 542, 545, 429, 506, 422} \[ \int x^2 \left (a+\frac {b}{c+d x^2}\right )^{3/2} \, dx=\frac {\sqrt {c} (3 b-a c) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{3 d^{3/2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}-\frac {\sqrt {c} (7 b-a c) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 d^{3/2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}+\frac {4 a x \left (c+d x^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{3 d}-\frac {x \left (a c+a d x^2+b\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{d}+\frac {x (7 b-a c) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{3 d} \]

[In]

Int[x^2*(a + b/(c + d*x^2))^(3/2),x]

[Out]

((7*b - a*c)*x*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])/(3*d) + (4*a*x*(c + d*x^2)*Sqrt[(b + a*c + a*d*x^2)/(c +
d*x^2)])/(3*d) - (x*(b + a*c + a*d*x^2)*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])/d - (Sqrt[c]*(7*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*d^(3/2)*Sqrt[(c*(b +
a*c + a*d*x^2))/((b + a*c)*(c + d*x^2))]) + (Sqrt[c]*(3*b - a*c)*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*Ellipti
cF[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 478

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*n*(p + 1))), x] - Dist[e^n/(b*n*(p + 1)), Int[(e*x)^
(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1)*Simp[c*(m - n + 1) + d*(m + n*(q - 1) + 1)*x^n, x], x], x] /;
FreeQ[{a, b, c, d, e}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && 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 542

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

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 \left (\frac {b+a c+a d x^2}{c+d x^2}\right )^{3/2} \, 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 \left (b+a c+a d x^2\right )^{3/2}}{\left (c+d x^2\right )^{3/2}} \, dx}{\sqrt {b+a c+a d x^2}} \\ & = -\frac {x \left (b+a c+a d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{d}+\frac {\left (\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 (b+a c+4 a d x^2\right )}{\sqrt {c+d x^2}} \, dx}{d \sqrt {b+a c+a d x^2}} \\ & = \frac {4 a x \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{3 d}-\frac {x \left (b+a c+a d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{d}+\frac {\left (\sqrt {c+d x^2} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}\right ) \int \frac {(3 b-a c) (b+a c) d+a (7 b-a c) d^2 x^2}{\sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{3 d^2 \sqrt {b+a c+a d x^2}} \\ & = \frac {4 a x \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{3 d}-\frac {x \left (b+a c+a d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{d}+\frac {\left (a (7 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 ((3 b-a 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 {(7 b-a c) x \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{3 d}+\frac {4 a x \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{3 d}-\frac {x \left (b+a c+a d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{d}+\frac {\sqrt {c} (3 b-a c) \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 (7 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 d \sqrt {b+a c+a d x^2}} \\ & = \frac {(7 b-a c) x \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{3 d}+\frac {4 a x \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{3 d}-\frac {x \left (b+a c+a d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{d}-\frac {\sqrt {c} (7 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 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 {\sqrt {c} (3 b-a c) \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 = 10.42 (sec) , antiderivative size = 256, normalized size of antiderivative = 0.77 \[ \int x^2 \left (a+\frac {b}{c+d x^2}\right )^{3/2} \, dx=\frac {\sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \left (\sqrt {\frac {d}{c}} x \left (-3 b^2-2 a b \left (c+d x^2\right )+a^2 \left (c+d x^2\right )^2\right )+i \left (-7 b^2-6 a b c+a^2 c^2\right ) \sqrt {\frac {b+a c+a d x^2}{b+a c}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right )|\frac {a c}{b+a c}\right )+4 i b (b+a c) \sqrt {\frac {b+a c+a d x^2}{b+a c}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right ),\frac {a c}{b+a c}\right )\right )}{3 d \sqrt {\frac {d}{c}} \left (b+a \left (c+d x^2\right )\right )} \]

[In]

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

[Out]

(Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(Sqrt[d/c]*x*(-3*b^2 - 2*a*b*(c + d*x^2) + a^2*(c + d*x^2)^2) + I*(-7*b
^2 - 6*a*b*c + a^2*c^2)*Sqrt[(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)] + (4*I)*b*(b + a*c)*Sqrt[(b + a*c + a*d*x^2)/(b + a*c)]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*A
rcSinh[Sqrt[d/c]*x], (a*c)/(b + a*c)]))/(3*d*Sqrt[d/c]*(b + a*(c + d*x^2)))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(819\) vs. \(2(371)=742\).

Time = 9.35 (sec) , antiderivative size = 820, normalized size of antiderivative = 2.48

method	result	size
default	\(\frac {\left (\sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}\, \sqrt {-\frac {a d}{a c +b}}\, a^{2} d^{2} x^{5}+2 \sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}\, \sqrt {-\frac {a d}{a c +b}}\, a^{2} c d \,x^{3}+\sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}\, \sqrt {-\frac {a d}{a c +b}}\, a b d \,x^{3}-\sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}\, \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^{2} c^{2}-3 \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}}\, a b d \,x^{3}+\sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}\, \sqrt {-\frac {a d}{a c +b}}\, a^{2} c^{2} x -5 \sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}\, \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 ) a b c +7 \sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}\, \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 b c +\sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}\, \sqrt {-\frac {a d}{a c +b}}\, a b c x +3 \sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}\, \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^{2}-3 \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}}\, a b c x -3 \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}}\, b^{2} x \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}}\, \left (a d \,x^{2}+a c +b \right )}\)	\(820\)
risch	\(\text {Expression too large to display}\)	\(1213\)

[In]

int(x^2*(a+b/(d*x^2+c))^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/3*(((a*d*x^2+a*c+b)*(d*x^2+c))^(1/2)*(-a*d/(a*c+b))^(1/2)*a^2*d^2*x^5+2*((a*d*x^2+a*c+b)*(d*x^2+c))^(1/2)*(-
a*d/(a*c+b))^(1/2)*a^2*c*d*x^3+((a*d*x^2+a*c+b)*(d*x^2+c))^(1/2)*(-a*d/(a*c+b))^(1/2)*a*b*d*x^3-((a*d*x^2+a*c+
b)*(d*x^2+c))^(1/2)*((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^2*c^2-3*(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*b*d*x^3+((a*d
*x^2+a*c+b)*(d*x^2+c))^(1/2)*(-a*d/(a*c+b))^(1/2)*a^2*c^2*x-5*((a*d*x^2+a*c+b)*(d*x^2+c))^(1/2)*((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))*a*b*c+7*((a*d*x^2+
a*c+b)*(d*x^2+c))^(1/2)*((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*b*c+((a*d*x^2+a*c+b)*(d*x^2+c))^(1/2)*(-a*d/(a*c+b))^(1/2)*a*b*c*x+3*((a*d*x^2+a*c+b)*(d
*x^2+c))^(1/2)*((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^2-3*(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*b*c*x-3*(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)*b^2*x)*((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)

Fricas [A] (verification not implemented)

none

Time = 0.09 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.62 \[ \int x^2 \left (a+\frac {b}{c+d x^2}\right )^{3/2} \, dx=\frac {{\left (a^{2} c^{3} - 7 \, a b c^{2}\right )} \sqrt {a} x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a c + b}{a c}) - {\left (a^{2} c^{3} - 7 \, a b c^{2} + {\left (a^{2} c^{2} - 2 \, a b c - 3 \, b^{2}\right )} d\right )} \sqrt {a} x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a c + b}{a c}) + {\left (a^{2} c d^{2} x^{4} + 4 \, a b c d x^{2} - a^{2} c^{3} + 7 \, a b c^{2}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{3 \, a c d^{2} x} \]

[In]

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

[Out]

1/3*((a^2*c^3 - 7*a*b*c^2)*sqrt(a)*x*sqrt(-c/d)*elliptic_e(arcsin(sqrt(-c/d)/x), (a*c + b)/(a*c)) - (a^2*c^3 -
7*a*b*c^2 + (a^2*c^2 - 2*a*b*c - 3*b^2)*d)*sqrt(a)*x*sqrt(-c/d)*elliptic_f(arcsin(sqrt(-c/d)/x), (a*c + b)/(a
*c)) + (a^2*c*d^2*x^4 + 4*a*b*c*d*x^2 - a^2*c^3 + 7*a*b*c^2)*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c)))/(a*c*d^2*x
)

Sympy [F]

\[ \int x^2 \left (a+\frac {b}{c+d x^2}\right )^{3/2} \, dx=\int x^{2} \left (\frac {a c + a d x^{2} + b}{c + d x^{2}}\right )^{\frac {3}{2}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int x^2 \left (a+\frac {b}{c+d x^2}\right )^{3/2} \, dx=\int { {\left (a + \frac {b}{d x^{2} + c}\right )}^{\frac {3}{2}} x^{2} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int x^2 \left (a+\frac {b}{c+d x^2}\right )^{3/2} \, dx=\int { {\left (a + \frac {b}{d x^{2} + c}\right )}^{\frac {3}{2}} x^{2} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int x^2 \left (a+\frac {b}{c+d x^2}\right )^{3/2} \, dx=\int x^2\,{\left (a+\frac {b}{d\,x^2+c}\right )}^{3/2} \,d x \]

[In]

int(x^2*(a + b/(c + d*x^2))^(3/2),x)

[Out]

int(x^2*(a + b/(c + d*x^2))^(3/2), x)