3.317 \(\int \frac {1}{x^4 (\frac {e (a+b x^2)}{c+d x^2})^{3/2}} \, dx\)

Optimal. Leaf size=444 \[ \frac {\left (a+b x^2\right ) (8 b c-7 a d)}{3 a^3 e x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}-\frac {d x \left (a+b x^2\right ) (8 b c-7 a d)}{3 a^3 e \left (c+d x^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}-\frac {\sqrt {c} \sqrt {d} \left (a+b x^2\right ) (4 b c-3 a d) F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 a^3 e \left (c+d x^2\right ) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}+\frac {\sqrt {c} \sqrt {d} \left (a+b x^2\right ) (8 b c-7 a d) E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 a^3 e \left (c+d x^2\right ) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}-\frac {\left (a+b x^2\right ) (4 b c-3 a d)}{3 a^2 b e x^3 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}+\frac {b c-a d}{a b e x^3 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \]

[Out]

(-a*d+b*c)/a/b/e/x^3/(e*(b*x^2+a)/(d*x^2+c))^(1/2)-1/3*(-3*a*d+4*b*c)*(b*x^2+a)/a^2/b/e/x^3/(e*(b*x^2+a)/(d*x^
2+c))^(1/2)+1/3*(-7*a*d+8*b*c)*(b*x^2+a)/a^3/e/x/(e*(b*x^2+a)/(d*x^2+c))^(1/2)-1/3*d*(-7*a*d+8*b*c)*x*(b*x^2+a
)/a^3/e/(d*x^2+c)/(e*(b*x^2+a)/(d*x^2+c))^(1/2)+1/3*(-7*a*d+8*b*c)*(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)*d^(1/2)/a^3/e/(d*x^2+c)/(c*(b*
x^2+a)/a/(d*x^2+c))^(1/2)/(e*(b*x^2+a)/(d*x^2+c))^(1/2)-1/3*(-3*a*d+4*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))*c^(1/2)*d^(1/2)/a^3/e/(d*x^2+c
)/(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 = 0.65, antiderivative size = 444, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {6719, 468, 583, 531, 418, 492, 411} \[ \frac {\left (a+b x^2\right ) (8 b c-7 a d)}{3 a^3 e x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}-\frac {\left (a+b x^2\right ) (4 b c-3 a d)}{3 a^2 b e x^3 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}-\frac {d x \left (a+b x^2\right ) (8 b c-7 a d)}{3 a^3 e \left (c+d x^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}-\frac {\sqrt {c} \sqrt {d} \left (a+b x^2\right ) (4 b c-3 a d) F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 a^3 e \left (c+d x^2\right ) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}+\frac {\sqrt {c} \sqrt {d} \left (a+b x^2\right ) (8 b c-7 a d) E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 a^3 e \left (c+d x^2\right ) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}+\frac {b c-a d}{a b e x^3 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^4*((e*(a + b*x^2))/(c + d*x^2))^(3/2)),x]

[Out]

(b*c - a*d)/(a*b*e*x^3*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]) - ((4*b*c - 3*a*d)*(a + b*x^2))/(3*a^2*b*e*x^3*Sqrt[
(e*(a + b*x^2))/(c + d*x^2)]) + ((8*b*c - 7*a*d)*(a + b*x^2))/(3*a^3*e*x*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]) -
(d*(8*b*c - 7*a*d)*x*(a + b*x^2))/(3*a^3*e*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(c + d*x^2)) + (Sqrt[c]*Sqrt[d]*(
8*b*c - 7*a*d)*(a + b*x^2)*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*a^3*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)) - (Sqrt[c]*Sqrt[d]*(4*b*c - 3*a*d)*(a + b*
x^2)*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*a^3*e*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*S
qrt[(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 468

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[((c*b -
a*d)*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1))/(a*b*e*n*(p + 1)), x] + Dist[1/(a*b*n*(p + 1)), I
nt[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 2)*Simp[c*(c*b*n*(p + 1) + (c*b - a*d)*(m + 1)) + d*(c*b*n*(p
+ 1) + (c*b - a*d)*(m + n*(q - 1) + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[n, 0] && LtQ[p, -1] && GtQ[q, 1] && 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 583

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[(e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*c*g*(m + 1)), x] + Dist[1/(a*c*
g^n*(m + 1)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e
*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] &&
IGtQ[n, 0] && LtQ[m, -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 {1}{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 {\left (c+d x^2\right )^{3/2}}{x^4 \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 {b c-a d}{a b e x^3 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}-\frac {\sqrt {a+b x^2} \int \frac {-c (4 b c-3 a d)-d (3 b c-2 a d) x^2}{x^4 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{a b e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}}\\ &=\frac {b c-a d}{a b e x^3 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}-\frac {(4 b c-3 a d) \left (a+b x^2\right )}{3 a^2 b e x^3 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}+\frac {\sqrt {a+b x^2} \int \frac {-b c^2 (8 b c-7 a d)-b c d (4 b c-3 a d) x^2}{x^2 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{3 a^2 b c e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}}\\ &=\frac {b c-a d}{a b e x^3 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}-\frac {(4 b c-3 a d) \left (a+b x^2\right )}{3 a^2 b e x^3 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}+\frac {(8 b c-7 a d) \left (a+b x^2\right )}{3 a^3 e x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}-\frac {\sqrt {a+b x^2} \int \frac {a b c^2 d (4 b c-3 a d)+b^2 c^2 d (8 b c-7 a d) x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{3 a^3 b c^2 e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}}\\ &=\frac {b c-a d}{a b e x^3 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}-\frac {(4 b c-3 a d) \left (a+b x^2\right )}{3 a^2 b e x^3 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}+\frac {(8 b c-7 a d) \left (a+b x^2\right )}{3 a^3 e x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}-\frac {\left (b d (8 b c-7 a d) \sqrt {a+b x^2}\right ) \int \frac {x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{3 a^3 e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}}-\frac {\left (d (4 b c-3 a d) \sqrt {a+b x^2}\right ) \int \frac {1}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{3 a^2 e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}}\\ &=\frac {b c-a d}{a b e x^3 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}-\frac {(4 b c-3 a d) \left (a+b x^2\right )}{3 a^2 b e x^3 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}+\frac {(8 b c-7 a d) \left (a+b x^2\right )}{3 a^3 e x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}-\frac {d (8 b c-7 a d) x \left (a+b x^2\right )}{3 a^3 e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}-\frac {\sqrt {c} \sqrt {d} (4 b c-3 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 )}{3 a^3 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 d (8 b c-7 a d) \sqrt {a+b x^2}\right ) \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{3 a^3 e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}}\\ &=\frac {b c-a d}{a b e x^3 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}-\frac {(4 b c-3 a d) \left (a+b x^2\right )}{3 a^2 b e x^3 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}+\frac {(8 b c-7 a d) \left (a+b x^2\right )}{3 a^3 e x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}-\frac {d (8 b c-7 a d) x \left (a+b x^2\right )}{3 a^3 e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}+\frac {\sqrt {c} \sqrt {d} (8 b c-7 a d) \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 )}{3 a^3 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 {\sqrt {c} \sqrt {d} (4 b c-3 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 )}{3 a^3 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 = 0.48, size = 266, normalized size = 0.60 \[ \frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (-i x^3 \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \left (3 a^2 d^2-11 a b c d+8 b^2 c^2\right ) F\left (i \sinh ^{-1}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-\sqrt {\frac {b}{a}} \left (c+d x^2\right ) \left (a^2 \left (c+4 d x^2\right )+a b \left (7 d x^4-4 c x^2\right )-8 b^2 c x^4\right )-i b c x^3 \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} (7 a d-8 b c) E\left (i \sinh ^{-1}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )\right )}{3 a^3 e^2 x^3 \sqrt {\frac {b}{a}} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(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]*(c + d*x^2)*(-8*b^2*c*x^4 + a^2*(c + 4*d*x^2) + a*b*(-4*c*x^2
+ 7*d*x^4))) - I*b*c*(-8*b*c + 7*a*d)*x^3*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a
]*x], (a*d)/(b*c)] - I*(8*b^2*c^2 - 11*a*b*c*d + 3*a^2*d^2)*x^3*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*Ellipt
icF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/(3*a^3*Sqrt[b/a]*e^2*x^3*(a + b*x^2))

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fricas [F]  time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (d^{2} x^{4} + 2 \, c d x^{2} + c^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{b^{2} e^{2} x^{8} + 2 \, a b e^{2} x^{6} + a^{2} e^{2} x^{4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(e*(b*x^2+a)/(d*x^2+c))^(3/2),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}\right )^{\frac {3}{2}} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(e*(b*x^2+a)/(d*x^2+c))^(3/2),x, algorithm="giac")

[Out]

integrate(1/(((b*x^2 + a)*e/(d*x^2 + c))^(3/2)*x^4), x)

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maple [A]  time = 0.04, size = 866, normalized size = 1.95 \[ -\frac {\left (b \,x^{2}+a \right ) \left (4 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {-\frac {b}{a}}\, a b \,d^{2} x^{6}+3 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {-\frac {b}{a}}\, a b \,d^{2} x^{6}-5 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {-\frac {b}{a}}\, b^{2} c d \,x^{6}-3 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {-\frac {b}{a}}\, b^{2} c d \,x^{6}+4 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {-\frac {b}{a}}\, a^{2} d^{2} x^{4}-3 \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} d^{2} x^{3} \EllipticF \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {\frac {a d}{b c}}\right )+3 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {-\frac {b}{a}}\, a b c d \,x^{4}-7 \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 d \,x^{3} \EllipticE \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {\frac {a d}{b c}}\right )+11 \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 d \,x^{3} \EllipticF \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {\frac {a d}{b c}}\right )-5 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {-\frac {b}{a}}\, b^{2} c^{2} x^{4}-3 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {-\frac {b}{a}}\, b^{2} c^{2} x^{4}+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}}\, b^{2} c^{2} x^{3} \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}}\, b^{2} c^{2} x^{3} \EllipticF \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {\frac {a d}{b c}}\right )+5 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {-\frac {b}{a}}\, a^{2} c d \,x^{2}-4 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {-\frac {b}{a}}\, a b \,c^{2} x^{2}+\sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {-\frac {b}{a}}\, a^{2} c^{2}\right )}{3 \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}\, a^{3} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^4/((b*x^2+a)/(d*x^2+c)*e)^(3/2),x)

[Out]

-1/3*(b*x^2+a)*(4*((d*x^2+c)*(b*x^2+a))^(1/2)*(-1/a*b)^(1/2)*a*b*d^2*x^6-5*((d*x^2+c)*(b*x^2+a))^(1/2)*(-1/a*b
)^(1/2)*b^2*c*d*x^6+3*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(-1/a*b)^(1/2)*a*b*d^2*x^6-3*(b*d*x^4+a*d*x^2+b*c*x^
2+a*c)^(1/2)*(-1/a*b)^(1/2)*b^2*c*d*x^6-3*((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))*x^3*a^2*d^2+11*((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))*x^3*a*b*c*d-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))*x^3*b^2*c^2-7*((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))*x^3*a*b*c*d+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)*EllipticE((-1/a*b)^(1/2)*x,(a/b/c*d)^(1/2))*x
^3*b^2*c^2+4*((d*x^2+c)*(b*x^2+a))^(1/2)*(-1/a*b)^(1/2)*a^2*d^2*x^4-5*((d*x^2+c)*(b*x^2+a))^(1/2)*(-1/a*b)^(1/
2)*b^2*c^2*x^4+3*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(-1/a*b)^(1/2)*a*b*c*d*x^4-3*(b*d*x^4+a*d*x^2+b*c*x^2+a*c
)^(1/2)*(-1/a*b)^(1/2)*x^4*b^2*c^2+5*((d*x^2+c)*(b*x^2+a))^(1/2)*(-1/a*b)^(1/2)*a^2*c*d*x^2-4*((d*x^2+c)*(b*x^
2+a))^(1/2)*(-1/a*b)^(1/2)*a*b*c^2*x^2+((d*x^2+c)*(b*x^2+a))^(1/2)*(-1/a*b)^(1/2)*a^2*c^2)/((b*x^2+a)/(d*x^2+c
)*e)^(3/2)/(d*x^2+c)^2/a^3/x^3/(-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 \[ \int \frac {1}{\left (\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}\right )^{\frac {3}{2}} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(e*(b*x^2+a)/(d*x^2+c))^(3/2),x, algorithm="maxima")

[Out]

integrate(1/(((b*x^2 + a)*e/(d*x^2 + c))^(3/2)*x^4), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4*((e*(a + b*x^2))/(c + d*x^2))^(3/2)),x)

[Out]

int(1/(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(1/x**4/(e*(b*x**2+a)/(d*x**2+c))**(3/2),x)

[Out]

Timed out

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