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

Optimal. Leaf size=453 \[ \frac {x \left (a+b x^2\right ) \left (16 a^2 d^2-16 a b c d+b^2 c^2\right )}{5 b^4 e \left (c+d x^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}-\frac {\sqrt {c} \left (a+b x^2\right ) \left (16 a^2 d^2-16 a b c d+b^2 c^2\right ) E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{5 b^4 \sqrt {d} e \left (c+d x^2\right ) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}-\frac {c^{3/2} \left (a+b x^2\right ) (7 b c-8 a d) F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{5 b^3 \sqrt {d} e \left (c+d x^2\right ) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}+\frac {x \left (a+b x^2\right ) (7 b c-8 a d)}{5 b^3 e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}+\frac {6 d x^3 \left (a+b x^2\right )}{5 b^2 e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}-\frac {x^3 \left (c+d x^2\right )}{b e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \]

[Out]

1/5*(-8*a*d+7*b*c)*x*(b*x^2+a)/b^3/e/(e*(b*x^2+a)/(d*x^2+c))^(1/2)+6/5*d*x^3*(b*x^2+a)/b^2/e/(e*(b*x^2+a)/(d*x
^2+c))^(1/2)+1/5*(16*a^2*d^2-16*a*b*c*d+b^2*c^2)*x*(b*x^2+a)/b^4/e/(d*x^2+c)/(e*(b*x^2+a)/(d*x^2+c))^(1/2)-x^3
*(d*x^2+c)/b/e/(e*(b*x^2+a)/(d*x^2+c))^(1/2)-1/5*c^(3/2)*(-8*a*d+7*b*c)*(b*x^2+a)*(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))/b^3/e/(d*x^2+c)/d^(1/2)/(c*(b*x^2
+a)/a/(d*x^2+c))^(1/2)/(e*(b*x^2+a)/(d*x^2+c))^(1/2)-1/5*(16*a^2*d^2-16*a*b*c*d+b^2*c^2)*(b*x^2+a)*(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)/b^4/e/(d
*x^2+c)/d^(1/2)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(e*(b*x^2+a)/(d*x^2+c))^(1/2)

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Rubi [A]  time = 0.68, antiderivative size = 453, 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 = {6719, 467, 581, 582, 531, 418, 492, 411} \[ \frac {x \left (a+b x^2\right ) \left (16 a^2 d^2-16 a b c d+b^2 c^2\right )}{5 b^4 e \left (c+d x^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}-\frac {\sqrt {c} \left (a+b x^2\right ) \left (16 a^2 d^2-16 a b c d+b^2 c^2\right ) E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{5 b^4 \sqrt {d} e \left (c+d x^2\right ) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}-\frac {c^{3/2} \left (a+b x^2\right ) (7 b c-8 a d) F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{5 b^3 \sqrt {d} e \left (c+d x^2\right ) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}+\frac {6 d x^3 \left (a+b x^2\right )}{5 b^2 e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}+\frac {x \left (a+b x^2\right ) (7 b c-8 a d)}{5 b^3 e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}-\frac {x^3 \left (c+d x^2\right )}{b e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((7*b*c - 8*a*d)*x*(a + b*x^2))/(5*b^3*e*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]) + (6*d*x^3*(a + b*x^2))/(5*b^2*e*S
qrt[(e*(a + b*x^2))/(c + d*x^2)]) + ((b^2*c^2 - 16*a*b*c*d + 16*a^2*d^2)*x*(a + b*x^2))/(5*b^4*e*Sqrt[(e*(a +
b*x^2))/(c + d*x^2)]*(c + d*x^2)) - (x^3*(c + d*x^2))/(b*e*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]) - (Sqrt[c]*(b^2*
c^2 - 16*a*b*c*d + 16*a^2*d^2)*(a + b*x^2)*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(5*b^4*Sqr
t[d]*e*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(c + d*x^2)) - (c^(3/2)*(7*b*c
- 8*a*d)*(a + b*x^2)*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(5*b^3*Sqrt[d]*e*Sqrt[(c*(a + b*
x^2))/(a*(c + d*x^2))]*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(c + d*x^2))

Rule 411

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

Rule 418

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

Rule 467

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 492

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(x*Sqrt[a + b*x^2])/(b*Sqr
t[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 531

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 581

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 582

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 6719

Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a*v^m*w^n)^FracPart[p])/(v^(m*F
racPart[p])*w^(n*FracPart[p])), Int[u*v^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] &&  !IntegerQ[p] &&  !
FreeQ[v, x] &&  !FreeQ[w, x]

Rubi steps

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

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Mathematica [C]  time = 0.50, size = 271, normalized size = 0.60 \[ \frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (i c \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \left (8 a^2 d^2-9 a b c d+b^2 c^2\right ) F\left (i \sinh ^{-1}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i c \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \left (16 a^2 d^2-16 a b c d+b^2 c^2\right ) E\left (i \sinh ^{-1}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+d x \sqrt {\frac {b}{a}} \left (c+d x^2\right ) \left (-8 a^2 d+a b \left (7 c-2 d x^2\right )+b^2 x^2 \left (2 c+d x^2\right )\right )\right )}{5 b^3 d e^2 \sqrt {\frac {b}{a}} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(Sqrt[b/a]*d*x*(c + d*x^2)*(-8*a^2*d + a*b*(7*c - 2*d*x^2) + b^2*x^2*(2*c +
d*x^2)) - I*c*(b^2*c^2 - 16*a*b*c*d + 16*a^2*d^2)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh
[Sqrt[b/a]*x], (a*d)/(b*c)] + I*c*(b^2*c^2 - 9*a*b*c*d + 8*a^2*d^2)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*El
lipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/(5*b^3*Sqrt[b/a]*d*e^2*(a + b*x^2))

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fricas [F]  time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (d^{2} x^{8} + 2 \, c d x^{6} + c^{2} x^{4}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{b^{2} e^{2} x^{4} + 2 \, a b e^{2} x^{2} + a^{2} e^{2}}, x\right ) \]

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]

integral((d^2*x^8 + 2*c*d*x^6 + c^2*x^4)*sqrt((b*e*x^2 + a*e)/(d*x^2 + c))/(b^2*e^2*x^4 + 2*a*b*e^2*x^2 + a^2*
e^2), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4}}{\left (\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}\right )^{\frac {3}{2}}}\,{d x} \]

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)*e/(d*x^2 + c))^(3/2), x)

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maple [A]  time = 0.06, size = 935, normalized size = 2.06 \[ \frac {\left (b \,x^{2}+a \right ) \left (\sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {-\frac {b}{a}}\, b^{2} d^{3} x^{7}-2 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {-\frac {b}{a}}\, a b \,d^{3} x^{5}+3 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {-\frac {b}{a}}\, b^{2} c \,d^{2} x^{5}-3 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {-\frac {b}{a}}\, a^{2} d^{3} x^{3}-5 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {-\frac {b}{a}}\, a^{2} d^{3} x^{3}+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}+2 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {-\frac {b}{a}}\, b^{2} c^{2} d \,x^{3}-3 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \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^{2} c \,d^{2} x +16 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, a^{2} c \,d^{2} \EllipticE \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {\frac {a d}{b c}}\right )-8 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, a^{2} c \,d^{2} \EllipticF \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {\frac {a d}{b c}}\right )+2 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {-\frac {b}{a}}\, a b \,c^{2} d 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 -16 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, a b \,c^{2} d \EllipticE \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {\frac {a d}{b c}}\right )+9 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, a b \,c^{2} d \EllipticF \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {\frac {a d}{b c}}\right )+\sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, b^{2} c^{3} \EllipticE \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {\frac {a d}{b c}}\right )-\sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, b^{2} c^{3} \EllipticF \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {\frac {a d}{b c}}\right )\right )}{5 \left (\frac {\left (b \,x^{2}+a \right ) e}{d \,x^{2}+c}\right )^{\frac {3}{2}} \left (d \,x^{2}+c \right )^{2} \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, b^{3} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*(b*x^2+a)*(((d*x^2+c)*(b*x^2+a))^(1/2)*(-1/a*b)^(1/2)*x^7*b^2*d^3-2*((d*x^2+c)*(b*x^2+a))^(1/2)*(-1/a*b)^(
1/2)*x^5*a*b*d^3+3*((d*x^2+c)*(b*x^2+a))^(1/2)*(-1/a*b)^(1/2)*x^5*b^2*c*d^2-3*((d*x^2+c)*(b*x^2+a))^(1/2)*(-1/
a*b)^(1/2)*a^2*d^3*x^3+2*((d*x^2+c)*(b*x^2+a))^(1/2)*(-1/a*b)^(1/2)*x^3*b^2*c^2*d-5*(b*d*x^4+a*d*x^2+b*c*x^2+a
*c)^(1/2)*(-1/a*b)^(1/2)*x^3*a^2*d^3+5*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(-1/a*b)^(1/2)*x^3*a*b*c*d^2-8*((d*
x^2+c)*(b*x^2+a))^(1/2)*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticF((-1/a*b)^(1/2)*x,(a/b/c*d)^(1/2))*a^
2*c*d^2+9*((d*x^2+c)*(b*x^2+a))^(1/2)*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticF((-1/a*b)^(1/2)*x,(a/b/
c*d)^(1/2))*a*b*c^2*d-((d*x^2+c)*(b*x^2+a))^(1/2)*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticF((-1/a*b)^(
1/2)*x,(a/b/c*d)^(1/2))*b^2*c^3+16*((d*x^2+c)*(b*x^2+a))^(1/2)*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*Ellipti
cE((-1/a*b)^(1/2)*x,(a/b/c*d)^(1/2))*a^2*c*d^2-16*((d*x^2+c)*(b*x^2+a))^(1/2)*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c
)^(1/2)*EllipticE((-1/a*b)^(1/2)*x,(a/b/c*d)^(1/2))*a*b*c^2*d+((d*x^2+c)*(b*x^2+a))^(1/2)*((b*x^2+a)/a)^(1/2)*
((d*x^2+c)/c)^(1/2)*EllipticE((-1/a*b)^(1/2)*x,(a/b/c*d)^(1/2))*b^2*c^3-3*((d*x^2+c)*(b*x^2+a))^(1/2)*(-1/a*b)
^(1/2)*x*a^2*c*d^2+2*((d*x^2+c)*(b*x^2+a))^(1/2)*(-1/a*b)^(1/2)*x*a*b*c^2*d-5*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1
/2)*(-1/a*b)^(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)*(-1/a*b)^(1/2)*a*b*c^2*d*x)/b^3/d/((b*x^2
+a)/(d*x^2+c)*e)^(3/2)/(d*x^2+c)^2/(-1/a*b)^(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 \[ \int \frac {x^{4}}{\left (\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}\right )^{\frac {3}{2}}}\,{d x} \]

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]

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

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

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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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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