3.3.83 \(\int x^4 (\frac {e (a+b x^2)}{c+d x^2})^{3/2} \, dx\) [283]

Optimal. Leaf size=391 \[ -\frac {\left (16 a c-\frac {16 b c^2}{d}-\frac {a^2 d}{b}\right ) e x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{5 d^2}-\frac {e x^3 \left (a+b x^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{d}-\frac {(8 b c-7 a d) e x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{5 d^3}+\frac {6 b e x^3 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{5 d^2}-\frac {\sqrt {c} \left (16 b^2 c^2-16 a b c d+a^2 d^2\right ) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{5 b d^{7/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {c^{3/2} (8 b c-7 a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{5 d^{7/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \]

[Out]

-1/5*(16*a*c-16*b*c^2/d-a^2*d/b)*e*x*(e*(b*x^2+a)/(d*x^2+c))^(1/2)/d^2-e*x^3*(b*x^2+a)*(e*(b*x^2+a)/(d*x^2+c))
^(1/2)/d-1/5*(-7*a*d+8*b*c)*e*x*(d*x^2+c)*(e*(b*x^2+a)/(d*x^2+c))^(1/2)/d^3+6/5*b*e*x^3*(d*x^2+c)*(e*(b*x^2+a)
/(d*x^2+c))^(1/2)/d^2+1/5*c^(3/2)*(-7*a*d+8*b*c)*e*(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),(1-b*c/a/d)^(1/2))*(e*(b*x^2+a)/(d*x^2+c))^(1/2)/d^(7/2)/(c*(b*x^2+a)/a/(d*x^2+c))^
(1/2)-1/5*(a^2*d^2-16*a*b*c*d+16*b^2*c^2)*e*(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),(1-b*c/a/d)^(1/2))*c^(1/2)*(e*(b*x^2+a)/(d*x^2+c))^(1/2)/b/d^(7/2)/(c*(b*x^2+a)/a/(d*x^2+c
))^(1/2)

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Rubi [A]
time = 0.36, antiderivative size = 391, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {1986, 478, 595, 596, 545, 429, 506, 422} \begin {gather*} -\frac {\sqrt {c} e \left (a^2 d^2-16 a b c d+16 b^2 c^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} E\left (\text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{5 b d^{7/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {e x \left (-\frac {a^2 d}{b}+16 a c-\frac {16 b c^2}{d}\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{5 d^2}+\frac {c^{3/2} e (8 b c-7 a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} F\left (\text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{5 d^{7/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {e x \left (c+d x^2\right ) (8 b c-7 a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{5 d^3}+\frac {6 b e x^3 \left (c+d x^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{5 d^2}-\frac {e x^3 \left (a+b x^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/5*((16*a*c - (16*b*c^2)/d - (a^2*d)/b)*e*x*Sqrt[(e*(a + b*x^2))/(c + d*x^2)])/d^2 - (e*x^3*(a + b*x^2)*Sqrt
[(e*(a + b*x^2))/(c + d*x^2)])/d - ((8*b*c - 7*a*d)*e*x*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(c + d*x^2))/(5*d^3)
+ (6*b*e*x^3*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(c + d*x^2))/(5*d^2) - (Sqrt[c]*(16*b^2*c^2 - 16*a*b*c*d + a^2
*d^2)*e*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(5*b*d^(7/2
)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]) + (c^(3/2)*(8*b*c - 7*a*d)*e*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*Ellipt
icF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(5*d^(7/2)*Sqrt[(c*(a + b*x^2))/(a*(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 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*} \int x^4 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2} \, dx &=\frac {\left (e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}\right ) \int \frac {x^4 \left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^{3/2}} \, dx}{\sqrt {a+b x^2}}\\ &=-\frac {e x^3 \left (a+b x^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{d}+\frac {\left (e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}\right ) \int \frac {x^2 \sqrt {a+b x^2} \left (3 a+6 b x^2\right )}{\sqrt {c+d x^2}} \, dx}{d \sqrt {a+b x^2}}\\ &=-\frac {e x^3 \left (a+b x^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{d}+\frac {6 b e x^3 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{5 d^2}+\frac {\left (e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}\right ) \int \frac {x^2 \left (-3 a (6 b c-5 a d)-3 b (8 b c-7 a d) x^2\right )}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{5 d^2 \sqrt {a+b x^2}}\\ &=-\frac {e x^3 \left (a+b x^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{d}-\frac {(8 b c-7 a d) e x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{5 d^3}+\frac {6 b e x^3 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{5 d^2}-\frac {\left (e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}\right ) \int \frac {-3 a b c (8 b c-7 a d)-3 b \left (16 b^2 c^2-16 a b c d+a^2 d^2\right ) x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{15 b d^3 \sqrt {a+b x^2}}\\ &=-\frac {e x^3 \left (a+b x^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{d}-\frac {(8 b c-7 a d) e x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{5 d^3}+\frac {6 b e x^3 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{5 d^2}+\frac {\left (a c (8 b c-7 a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}\right ) \int \frac {1}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{5 d^3 \sqrt {a+b x^2}}+\frac {\left (\left (16 b^2 c^2-16 a b c d+a^2 d^2\right ) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}\right ) \int \frac {x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{5 d^3 \sqrt {a+b x^2}}\\ &=\frac {\left (16 b^2 c^2-16 a b c d+a^2 d^2\right ) e x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{5 b d^3}-\frac {e x^3 \left (a+b x^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{d}-\frac {(8 b c-7 a d) e x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{5 d^3}+\frac {6 b e x^3 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{5 d^2}+\frac {c^{3/2} (8 b c-7 a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{5 d^{7/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {\left (c \left (16 b^2 c^2-16 a b c d+a^2 d^2\right ) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}\right ) \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{5 b d^3 \sqrt {a+b x^2}}\\ &=\frac {\left (16 b^2 c^2-16 a b c d+a^2 d^2\right ) e x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{5 b d^3}-\frac {e x^3 \left (a+b x^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{d}-\frac {(8 b c-7 a d) e x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{5 d^3}+\frac {6 b e x^3 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{5 d^2}-\frac {\sqrt {c} \left (16 b^2 c^2-16 a b c d+a^2 d^2\right ) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{5 b d^{7/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {c^{3/2} (8 b c-7 a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{5 d^{7/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 4.76, size = 290, normalized size = 0.74 \begin {gather*} \frac {e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (\sqrt {\frac {b}{a}} d x \left (a^2 d \left (7 c+2 d x^2\right )+b^2 x^2 \left (-8 c^2-2 c d x^2+d^2 x^4\right )+a b \left (-8 c^2+5 c d x^2+3 d^2 x^4\right )\right )-i c \left (16 b^2 c^2-16 a b c d+a^2 d^2\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \sinh ^{-1}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+8 i c \left (2 b^2 c^2-3 a b c d+a^2 d^2\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} F\left (i \sinh ^{-1}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )\right )}{5 \sqrt {\frac {b}{a}} d^4 \left (a+b x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(e*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(Sqrt[b/a]*d*x*(a^2*d*(7*c + 2*d*x^2) + b^2*x^2*(-8*c^2 - 2*c*d*x^2 + d^2
*x^4) + a*b*(-8*c^2 + 5*c*d*x^2 + 3*d^2*x^4)) - I*c*(16*b^2*c^2 - 16*a*b*c*d + a^2*d^2)*Sqrt[1 + (b*x^2)/a]*Sq
rt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + (8*I)*c*(2*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*S
qrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/(5*Sqrt[b/a]*d^4*(a +
b*x^2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(932\) vs. \(2(427)=854\).
time = 0.11, size = 933, normalized size = 2.39

method	result	size
risch	\(\frac {x \left (b d \,x^{2}+2 a d -3 b c \right ) \left (d \,x^{2}+c \right ) e \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}}{5 d^{3}}+\frac {\left (-\frac {2 \left (a^{2} d^{2}-11 a b c d +11 b^{2} c^{2}\right ) a c e \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\EllipticF \left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d e +b c e}{c e b}}\right )-\EllipticE \left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d e +b c e}{c e b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {d e b \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}\, \left (a d e +b c e +e \left (a d -b c \right )\right )}-\frac {c \left (7 a^{2} d^{2}-13 a b c d +5 b^{2} c^{2}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \EllipticF \left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d e +b c e}{c e b}}\right )}{d \sqrt {-\frac {b}{a}}\, \sqrt {d e b \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}+\frac {5 c^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {\left (d e b \,x^{2}+a d e \right ) x}{c \left (a d -b c \right ) e \sqrt {\left (x^{2}+\frac {c}{d}\right ) \left (d e b \,x^{2}+a d e \right )}}+\frac {\left (\frac {1}{c}-\frac {a d}{c \left (a d -b c \right )}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \EllipticF \left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d e +b c e}{c e b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {d e b \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}+\frac {2 b d a e \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\EllipticF \left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d e +b c e}{c e b}}\right )-\EllipticE \left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d e +b c e}{c e b}}\right )\right )}{\left (a d -b c \right ) \sqrt {-\frac {b}{a}}\, \sqrt {d e b \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}\, \left (a d e +b c e +e \left (a d -b c \right )\right )}\right )}{d}\right ) e \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \sqrt {\left (d \,x^{2}+c \right ) e \left (b \,x^{2}+a \right )}}{5 d^{3} \left (b \,x^{2}+a \right )}\)	\(775\)
default	\(\frac {\left (\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}\right )^{\frac {3}{2}} \left (d \,x^{2}+c \right ) \left (\sqrt {-\frac {b}{a}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, b^{2} d^{3} x^{7}+3 \sqrt {-\frac {b}{a}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, a b \,d^{3} x^{5}-2 \sqrt {-\frac {b}{a}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, b^{2} c \,d^{2} x^{5}+5 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {-\frac {b}{a}}\, a b c \,d^{2} x^{3}-5 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {-\frac {b}{a}}\, b^{2} c^{2} d \,x^{3}+2 \sqrt {-\frac {b}{a}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, a^{2} d^{3} x^{3}-3 \sqrt {-\frac {b}{a}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, b^{2} c^{2} d \,x^{3}-8 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \EllipticF \left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, a^{2} c \,d^{2}+24 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \EllipticF \left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, a b \,c^{2} d -16 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \EllipticF \left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, b^{2} c^{3}+\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \EllipticE \left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, a^{2} c \,d^{2}-16 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \EllipticE \left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, a b \,c^{2} d +16 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \EllipticE \left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, b^{2} c^{3}+5 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {-\frac {b}{a}}\, a^{2} c \,d^{2} x -5 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {-\frac {b}{a}}\, a b \,c^{2} d x +2 \sqrt {-\frac {b}{a}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, a^{2} c \,d^{2} x -3 \sqrt {-\frac {b}{a}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, a b \,c^{2} d x \right )}{5 d^{4} \left (b \,x^{2}+a \right )^{2} \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}}\)	\(933\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*(e*(b*x^2+a)/(d*x^2+c))^(3/2)*(d*x^2+c)*((-b/a)^(1/2)*((d*x^2+c)*(b*x^2+a))^(1/2)*b^2*d^3*x^7+3*(-b/a)^(1/
2)*((d*x^2+c)*(b*x^2+a))^(1/2)*a*b*d^3*x^5-2*(-b/a)^(1/2)*((d*x^2+c)*(b*x^2+a))^(1/2)*b^2*c*d^2*x^5+5*(b*d*x^4
+a*d*x^2+b*c*x^2+a*c)^(1/2)*(-b/a)^(1/2)*a*b*c*d^2*x^3-5*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(-b/a)^(1/2)*b^2*
c^2*d*x^3+2*(-b/a)^(1/2)*((d*x^2+c)*(b*x^2+a))^(1/2)*a^2*d^3*x^3-3*(-b/a)^(1/2)*((d*x^2+c)*(b*x^2+a))^(1/2)*b^
2*c^2*d*x^3-8*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*((d*x^2+c)*(b*
x^2+a))^(1/2)*a^2*c*d^2+24*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*(
(d*x^2+c)*(b*x^2+a))^(1/2)*a*b*c^2*d-16*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(a*d/
b/c)^(1/2))*((d*x^2+c)*(b*x^2+a))^(1/2)*b^2*c^3+((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticE(x*(-b/a)^(1/
2),(a*d/b/c)^(1/2))*((d*x^2+c)*(b*x^2+a))^(1/2)*a^2*c*d^2-16*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticE
(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*((d*x^2+c)*(b*x^2+a))^(1/2)*a*b*c^2*d+16*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1
/2)*EllipticE(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*((d*x^2+c)*(b*x^2+a))^(1/2)*b^2*c^3+5*(b*d*x^4+a*d*x^2+b*c*x^2+a
*c)^(1/2)*(-b/a)^(1/2)*a^2*c*d^2*x-5*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(-b/a)^(1/2)*a*b*c^2*d*x+2*(-b/a)^(1/
2)*((d*x^2+c)*(b*x^2+a))^(1/2)*a^2*c*d^2*x-3*(-b/a)^(1/2)*((d*x^2+c)*(b*x^2+a))^(1/2)*a*b*c^2*d*x)/d^4/(b*x^2+
a)^2/(-b/a)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> Symbolic function elliptic_ec takes exactly 1 arguments (2 given)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^4\,{\left (\frac {e\,\left (b\,x^2+a\right )}{d\,x^2+c}\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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