3.4.42 \(\int \frac {(a+\frac {b}{c+d x^2})^{3/2}}{x^4} \, dx\) [342]

Optimal. Leaf size=388 \[ \frac {b \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{c x^3}-\frac {(8 b+a c) d^2 x \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{3 c^3}-\frac {(4 b+a c) \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{3 c^2 x^3}+\frac {(8 b+a c) d \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{3 c^3 x}+\frac {(8 b+a c) d^{3/2} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 c^{5/2} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}}-\frac {a (4 b+a c) d^{3/2} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 c^{3/2} (b+a c) \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}} \]

[Out]

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

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Rubi [A]
time = 0.41, antiderivative size = 388, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {1985, 1986, 479, 597, 545, 429, 506, 422} \begin {gather*} -\frac {a d^{3/2} (a c+4 b) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} F\left (\text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 c^{3/2} (a c+b) \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}+\frac {d^{3/2} (a c+8 b) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} E\left (\text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 c^{5/2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}-\frac {d^2 x (a c+8 b) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{3 c^3}+\frac {d (a c+8 b) \left (c+d x^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{3 c^3 x}-\frac {(a c+4 b) \left (c+d x^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{3 c^2 x^3}+\frac {b \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{c x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b/(c + d*x^2))^(3/2)/x^4,x]

[Out]

(b*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])/(c*x^3) - ((8*b + a*c)*d^2*x*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])/
(3*c^3) - ((4*b + a*c)*(c + d*x^2)*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])/(3*c^2*x^3) + ((8*b + a*c)*d*(c + d*
x^2)*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])/(3*c^3*x) + ((8*b + a*c)*d^(3/2)*Sqrt[(b + a*c + a*d*x^2)/(c + d*x
^2)]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], b/(b + a*c)])/(3*c^(5/2)*Sqrt[(c*(b + a*c + a*d*x^2))/((b + a*c)*(
c + d*x^2))]) - (a*(4*b + a*c)*d^(3/2)*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt
[c]], b/(b + a*c)])/(3*c^(3/2)*(b + a*c)*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 479

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)),
Int[(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 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 597

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

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Mathematica [C] Result contains complex when optimal does not.
time = 10.57, size = 329, normalized size = 0.85 \begin {gather*} -\frac {\sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \left (\sqrt {\frac {a d}{b+a c}} \left (a^2 c \left (c-d x^2\right ) \left (c+d x^2\right )^2+b^2 \left (c^2-4 c d x^2-8 d^2 x^4\right )+a b \left (2 c^3-3 c^2 d x^2-13 c d^2 x^4-8 d^3 x^6\right )\right )-i a c (8 b+a c) d^2 x^3 \sqrt {\frac {b+a c+a d x^2}{b+a c}} \sqrt {1+\frac {d x^2}{c}} E\left (i \sinh ^{-1}\left (\sqrt {\frac {a d}{b+a c}} x\right )|1+\frac {b}{a c}\right )+4 i a b c d^2 x^3 \sqrt {\frac {b+a c+a d x^2}{b+a c}} \sqrt {1+\frac {d x^2}{c}} F\left (i \sinh ^{-1}\left (\sqrt {\frac {a d}{b+a c}} x\right )|1+\frac {b}{a c}\right )\right )}{3 c^3 \sqrt {\frac {a d}{b+a c}} x^3 \left (b+a \left (c+d x^2\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*(Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(Sqrt[(a*d)/(b + a*c)]*(a^2*c*(c - d*x^2)*(c + d*x^2)^2 + b^2*(c^2
- 4*c*d*x^2 - 8*d^2*x^4) + a*b*(2*c^3 - 3*c^2*d*x^2 - 13*c*d^2*x^4 - 8*d^3*x^6)) - I*a*c*(8*b + a*c)*d^2*x^3*
Sqrt[(b + a*c + a*d*x^2)/(b + a*c)]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[(a*d)/(b + a*c)]*x], 1 + b/(a
*c)] + (4*I)*a*b*c*d^2*x^3*Sqrt[(b + a*c + a*d*x^2)/(b + a*c)]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[(a
*d)/(b + a*c)]*x], 1 + b/(a*c)]))/(c^3*Sqrt[(a*d)/(b + a*c)]*x^3*(b + a*(c + d*x^2)))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1038\) vs. \(2(424)=848\).
time = 0.07, size = 1039, normalized size = 2.68 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*((-a*d/(a*c+b))^(1/2)*((d*x^2+c)*(a*d*x^2+a*c+b))^(1/2)*a^2*c*d^3*x^6+3*(-a*d/(a*c+b))^(1/2)*(a*d^2*x^4+2*
a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*a*b*d^3*x^6+5*(-a*d/(a*c+b))^(1/2)*((d*x^2+c)*(a*d*x^2+a*c+b))^(1/2)*a*b*d^
3*x^6-((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)
)*((d*x^2+c)*(a*d*x^2+a*c+b))^(1/2)*a^2*c^2*d^2*x^3+(-a*d/(a*c+b))^(1/2)*((d*x^2+c)*(a*d*x^2+a*c+b))^(1/2)*a^2
*c^2*d^2*x^4+4*((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))*((d*x^2+c)*(a*d*x^2+a*c+b))^(1/2)*a*b*c*d^2*x^3-8*((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))*((d*x^2+c)*(a*d*x^2+a*c+b))^(1/2)*a*b*c*d^2*x^3+3*(-a
*d/(a*c+b))^(1/2)*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*a*b*c*d^2*x^4+10*(-a*d/(a*c+b))^(1/2)*((d*x^
2+c)*(a*d*x^2+a*c+b))^(1/2)*a*b*c*d^2*x^4+3*(-a*d/(a*c+b))^(1/2)*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/
2)*b^2*d^2*x^4-(-a*d/(a*c+b))^(1/2)*((d*x^2+c)*(a*d*x^2+a*c+b))^(1/2)*a^2*c^3*d*x^2+5*(-a*d/(a*c+b))^(1/2)*((d
*x^2+c)*(a*d*x^2+a*c+b))^(1/2)*b^2*d^2*x^4+3*(-a*d/(a*c+b))^(1/2)*((d*x^2+c)*(a*d*x^2+a*c+b))^(1/2)*a*b*c^2*d*
x^2-(-a*d/(a*c+b))^(1/2)*((d*x^2+c)*(a*d*x^2+a*c+b))^(1/2)*a^2*c^4+4*(-a*d/(a*c+b))^(1/2)*((d*x^2+c)*(a*d*x^2+
a*c+b))^(1/2)*b^2*c*d*x^2-2*(-a*d/(a*c+b))^(1/2)*((d*x^2+c)*(a*d*x^2+a*c+b))^(1/2)*a*b*c^3-(-a*d/(a*c+b))^(1/2
)*((d*x^2+c)*(a*d*x^2+a*c+b))^(1/2)*b^2*c^2)*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)/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)/c^3/(a*d*x^2+a*c+b)

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

[Out]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (\frac {a c + a d x^{2} + b}{c + d x^{2}}\right )^{\frac {3}{2}}}{x^{4}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/(c + d*x^2))^(3/2)/x^4,x)

[Out]

int((a + b/(c + d*x^2))^(3/2)/x^4, x)

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