\(\int x^4 (a+\frac {b}{c+d x^2})^{3/2} \, dx\) [338]

Optimal result

Integrand size = 21, antiderivative size = 405 \[ \int x^4 \left (a+\frac {b}{c+d x^2}\right )^{3/2} \, dx=\frac {\left (b^2-14 a b c+a^2 c^2\right ) x \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{5 a d^2}+\frac {(7 b-a c) x \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{5 d^2}+\frac {6 a x^3 \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{5 d}-\frac {x^3 \left (b+a c+a d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{d}-\frac {\sqrt {c} \left (b^2-14 a b c+a^2 c^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{5 a d^{5/2} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}}-\frac {c^{3/2} (7 b-a c) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{5 d^{5/2} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}} \]

[Out]

1/5*(a^2*c^2-14*a*b*c+b^2)*x*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)/a/d^2+1/5*(-a*c+7*b)*x*(d*x^2+c)*((a*d*x^2+a*c+
b)/(d*x^2+c))^(1/2)/d^2+6/5*a*x^3*(d*x^2+c)*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)/d-x^3*(a*d*x^2+a*c+b)*((a*d*x^2+
a*c+b)/(d*x^2+c))^(1/2)/d-1/5*c^(3/2)*(-a*c+7*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))*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)/d^(5/2)/(c*(a*d*x^2+a*c+b)/(a*c+
b)/(d*x^2+c))^(1/2)-1/5*(a^2*c^2-14*a*b*c+b^2)*(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^(5/2)/(c*(a*d*x^2+a*c+
b)/(a*c+b)/(d*x^2+c))^(1/2)

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 405, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {1985, 1986, 478, 595, 596, 545, 429, 506, 422} \[ \int x^4 \left (a+\frac {b}{c+d x^2}\right )^{3/2} \, dx=-\frac {\sqrt {c} \left (a^2 c^2-14 a b c+b^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{5 a d^{5/2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}+\frac {x \left (a^2 c^2-14 a b c+b^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{5 a d^2}-\frac {c^{3/2} (7 b-a c) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{5 d^{5/2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}+\frac {x (7 b-a c) \left (c+d x^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{5 d^2}+\frac {6 a x^3 \left (c+d x^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{5 d}-\frac {x^3 \left (a c+a d x^2+b\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{d} \]

[In]

Int[x^4*(a + b/(c + d*x^2))^(3/2),x]

[Out]

((b^2 - 14*a*b*c + a^2*c^2)*x*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])/(5*a*d^2) + ((7*b - a*c)*x*(c + d*x^2)*Sq
rt[(b + a*c + a*d*x^2)/(c + d*x^2)])/(5*d^2) + (6*a*x^3*(c + d*x^2)*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])/(5*
d) - (x^3*(b + a*c + a*d*x^2)*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])/d - (Sqrt[c]*(b^2 - 14*a*b*c + a^2*c^2)*S
qrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], b/(b + a*c)])/(5*a*d^(5/2)*Sqrt[(c
*(b + a*c + a*d*x^2))/((b + a*c)*(c + d*x^2))]) - (c^(3/2)*(7*b - a*c)*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*E
llipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], b/(b + a*c)])/(5*d^(5/2)*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 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 595

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[f*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/(b*g*(m + n*(p + q + 1) + 1))), x] + Dis
t[1/(b*(m + n*(p + q + 1) + 1)), Int[(g*x)^m*(a + b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*((b*e - a*f)*(m + 1) + b
*e*n*(p + q + 1)) + (d*(b*e - a*f)*(m + 1) + f*n*q*(b*c - a*d) + b*e*d*n*(p + q + 1))*x^n, x], x], x] /; FreeQ
[{a, b, c, d, e, f, g, m, p}, x] && IGtQ[n, 0] && GtQ[q, 0] &&  !(EqQ[q, 1] && SimplerQ[e + f*x^n, c + d*x^n])

Rule 596

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

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^4 \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^4 \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^3 \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 {x^2 \sqrt {b+a c+a d x^2} \left (3 (b+a c)+6 a d x^2\right )}{\sqrt {c+d x^2}} \, dx}{d \sqrt {b+a c+a d x^2}} \\ & = \frac {6 a x^3 \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{5 d}-\frac {x^3 \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 {x^2 \left (3 (5 b-a c) (b+a c) d+3 a (7 b-a c) d^2 x^2\right )}{\sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{5 d^2 \sqrt {b+a c+a d x^2}} \\ & = \frac {(7 b-a c) x \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{5 d^2}+\frac {6 a x^3 \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{5 d}-\frac {x^3 \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 a c (7 b-a c) (b+a c) d^2-3 a \left (b^2-14 a b c+a^2 c^2\right ) d^3 x^2}{\sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{15 a d^4 \sqrt {b+a c+a d x^2}} \\ & = \frac {(7 b-a c) x \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{5 d^2}+\frac {6 a x^3 \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{5 d}-\frac {x^3 \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 (c (7 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}{5 d^2 \sqrt {b+a c+a d x^2}}+\frac {\left (\left (b^2-14 a b c+a^2 c^2\right ) \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}{5 d \sqrt {b+a c+a d x^2}} \\ & = \frac {\left (b^2-14 a b c+a^2 c^2\right ) x \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{5 a d^2}+\frac {(7 b-a c) x \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{5 d^2}+\frac {6 a x^3 \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{5 d}-\frac {x^3 \left (b+a c+a d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{d}-\frac {c^{3/2} (7 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 )}{5 d^{5/2} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}}-\frac {\left (c \left (b^2-14 a b c+a^2 c^2\right ) \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}{5 a d^2 \sqrt {b+a c+a d x^2}} \\ & = \frac {\left (b^2-14 a b c+a^2 c^2\right ) x \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{5 a d^2}+\frac {(7 b-a c) x \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{5 d^2}+\frac {6 a x^3 \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{5 d}-\frac {x^3 \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} \left (b^2-14 a b c+a^2 c^2\right ) \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 )}{5 a d^{5/2} \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} (7 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 )}{5 d^{5/2} \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 = 10.63 (sec) , antiderivative size = 312, normalized size of antiderivative = 0.77 \[ \int x^4 \left (a+\frac {b}{c+d x^2}\right )^{3/2} \, dx=\frac {\sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \left (a \sqrt {\frac {d}{c}} x \left (-a^2 \left (c-d x^2\right ) \left (c+d x^2\right )^2+b^2 \left (7 c+2 d x^2\right )+3 a b \left (2 c^2+3 c d x^2+d^2 x^4\right )\right )-i \left (b^3-13 a b^2 c-13 a^2 b c^2+a^3 c^3\right ) \sqrt {\frac {b+a c+a d x^2}{b+a c}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right )|\frac {a c}{b+a c}\right )+i b \left (b^2-6 a b c-7 a^2 c^2\right ) \sqrt {\frac {b+a c+a d x^2}{b+a c}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right ),\frac {a c}{b+a c}\right )\right )}{5 a c^2 \left (\frac {d}{c}\right )^{5/2} \left (b+a \left (c+d x^2\right )\right )} \]

[In]

Integrate[x^4*(a + b/(c + d*x^2))^(3/2),x]

[Out]

(Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(a*Sqrt[d/c]*x*(-(a^2*(c - d*x^2)*(c + d*x^2)^2) + b^2*(7*c + 2*d*x^2)
+ 3*a*b*(2*c^2 + 3*c*d*x^2 + d^2*x^4)) - I*(b^3 - 13*a*b^2*c - 13*a^2*b*c^2 + a^3*c^3)*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)] + I*b*(b^2 - 6*a*b*c - 7*a
^2*c^2)*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)]))/(5*a*c^2*(d/c)^(5/2)*(b + a*(c + d*x^2)))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1064\) vs. \(2(441)=882\).

Time = 11.57 (sec) , antiderivative size = 1065, normalized size of antiderivative = 2.63

method	result	size
risch	\(\text {Expression too large to display}\)	\(1065\)
default	\(\text {Expression too large to display}\)	\(1101\)

[In]

int(x^4*(a+b/(d*x^2+c))^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/5/d^2*x*(-a*d*x^2+a*c-2*b)*(d*x^2+c)*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)+1/5/d^2*(a^2*c^3/(-a*d/(a*c+b))^(1/2
)*(1+a*d/(a*c+b)*x^2)^(1/2)*(1+1/c*d*x^2)^(1/2)/(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*EllipticF(x*(-
a*d/(a*c+b))^(1/2),(-1+(2*a*c*d+b*d)/d/c/a)^(1/2))-7*b^2*c/(-a*d/(a*c+b))^(1/2)*(1+a*d/(a*c+b)*x^2)^(1/2)*(1+1
/c*d*x^2)^(1/2)/(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*EllipticF(x*(-a*d/(a*c+b))^(1/2),(-1+(2*a*c*d+
b*d)/d/c/a)^(1/2))-a*b*c^2/(-a*d/(a*c+b))^(1/2)*(1+a*d/(a*c+b)*x^2)^(1/2)*(1+1/c*d*x^2)^(1/2)/(a*d^2*x^4+2*a*c
*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*EllipticF(x*(-a*d/(a*c+b))^(1/2),(-1+(2*a*c*d+b*d)/d/c/a)^(1/2))+5*b^2*c^2*((a
*d^2*x^2+a*c*d+b*d)/c/b*x/d/((x^2+c/d)*(a*d^2*x^2+a*c*d+b*d))^(1/2)+(1/c-(a*c*d+b*d)/c/b/d)/(-a*d/(a*c+b))^(1/
2)*(1+a*d/(a*c+b)*x^2)^(1/2)*(1+1/c*d*x^2)^(1/2)/(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*EllipticF(x*(
-a*d/(a*c+b))^(1/2),(-1+(2*a*c*d+b*d)/d/c/a)^(1/2))+2*a*d/b/c*(a*c^2+b*c)/(-a*d/(a*c+b))^(1/2)*(1+a*d/(a*c+b)*
x^2)^(1/2)*(1+1/c*d*x^2)^(1/2)/(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)/(2*a*c*d+2*b*d)*(EllipticF(x*(-
a*d/(a*c+b))^(1/2),(-1+(2*a*c*d+b*d)/d/c/a)^(1/2))-EllipticE(x*(-a*d/(a*c+b))^(1/2),(-1+(2*a*c*d+b*d)/d/c/a)^(
1/2))))-2*(a^2*c^2*d-9*a*b*c*d+b^2*d)*(a*c^2+b*c)/(-a*d/(a*c+b))^(1/2)*(1+a*d/(a*c+b)*x^2)^(1/2)*(1+1/c*d*x^2)
^(1/2)/(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)/(2*a*c*d+2*b*d)*(EllipticF(x*(-a*d/(a*c+b))^(1/2),(-1+(
2*a*c*d+b*d)/d/c/a)^(1/2))-EllipticE(x*(-a*d/(a*c+b))^(1/2),(-1+(2*a*c*d+b*d)/d/c/a)^(1/2))))*((a*d*x^2+a*c+b)
/(d*x^2+c))^(1/2)*((a*d*x^2+a*c+b)*(d*x^2+c))^(1/2)/(a*d*x^2+a*c+b)

Fricas [A] (verification not implemented)

none

Time = 0.11 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.58 \[ \int x^4 \left (a+\frac {b}{c+d x^2}\right )^{3/2} \, dx=-\frac {{\left (a^{2} c^{3} - 14 \, a b c^{2} + b^{2} c\right )} \sqrt {a} x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a c + b}{a c}) - {\left (a^{2} c^{3} - 14 \, a b c^{2} + b^{2} c + {\left (a^{2} c^{2} - 6 \, a b c - 7 \, b^{2}\right )} d\right )} \sqrt {a} x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a c + b}{a c}) - {\left (a^{2} d^{3} x^{6} + 2 \, a b d^{2} x^{4} + a^{2} c^{3} - 14 \, a b c^{2} - {\left (7 \, a b c - b^{2}\right )} d x^{2} + b^{2} c\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{5 \, a d^{3} x} \]

[In]

integrate(x^4*(a+b/(d*x^2+c))^(3/2),x, algorithm="fricas")

[Out]

-1/5*((a^2*c^3 - 14*a*b*c^2 + b^2*c)*sqrt(a)*x*sqrt(-c/d)*elliptic_e(arcsin(sqrt(-c/d)/x), (a*c + b)/(a*c)) -
(a^2*c^3 - 14*a*b*c^2 + b^2*c + (a^2*c^2 - 6*a*b*c - 7*b^2)*d)*sqrt(a)*x*sqrt(-c/d)*elliptic_f(arcsin(sqrt(-c/
d)/x), (a*c + b)/(a*c)) - (a^2*d^3*x^6 + 2*a*b*d^2*x^4 + a^2*c^3 - 14*a*b*c^2 - (7*a*b*c - b^2)*d*x^2 + b^2*c)
*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c)))/(a*d^3*x)

Sympy [F]

\[ \int x^4 \left (a+\frac {b}{c+d x^2}\right )^{3/2} \, dx=\int x^{4} \left (\frac {a c + a d x^{2} + b}{c + d x^{2}}\right )^{\frac {3}{2}}\, dx \]

[In]

integrate(x**4*(a+b/(d*x**2+c))**(3/2),x)

[Out]

Integral(x**4*((a*c + a*d*x**2 + b)/(c + d*x**2))**(3/2), x)

Maxima [F]

\[ \int x^4 \left (a+\frac {b}{c+d x^2}\right )^{3/2} \, dx=\int { {\left (a + \frac {b}{d x^{2} + c}\right )}^{\frac {3}{2}} x^{4} \,d x } \]

[In]

integrate(x^4*(a+b/(d*x^2+c))^(3/2),x, algorithm="maxima")

[Out]

integrate((a + b/(d*x^2 + c))^(3/2)*x^4, x)

Giac [F]

\[ \int x^4 \left (a+\frac {b}{c+d x^2}\right )^{3/2} \, dx=\int { {\left (a + \frac {b}{d x^{2} + c}\right )}^{\frac {3}{2}} x^{4} \,d x } \]

[In]

integrate(x^4*(a+b/(d*x^2+c))^(3/2),x, algorithm="giac")

[Out]

integrate((a + b/(d*x^2 + c))^(3/2)*x^4, x)

Mupad [F(-1)]

Timed out. \[ \int x^4 \left (a+\frac {b}{c+d x^2}\right )^{3/2} \, dx=\int x^4\,{\left (a+\frac {b}{d\,x^2+c}\right )}^{3/2} \,d x \]

[In]

int(x^4*(a + b/(c + d*x^2))^(3/2),x)

[Out]

int(x^4*(a + b/(c + d*x^2))^(3/2), x)