∫ x5∕2 ___________________________

3.497 \(\int \frac {x^{5/2}}{(a+b x^2)^2 (c+d x^2)^3} \, dx\)

Optimal. Leaf size=703 \[ -\frac {3 \sqrt [4]{d} \left (-a^2 d^2+18 a b c d+15 b^2 c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{5/4} (b c-a d)^4}+\frac {3 \sqrt [4]{d} \left (-a^2 d^2+18 a b c d+15 b^2 c^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{5/4} (b c-a d)^4}+\frac {3 \sqrt [4]{d} \left (-a^2 d^2+18 a b c d+15 b^2 c^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{5/4} (b c-a d)^4}-\frac {3 \sqrt [4]{d} \left (-a^2 d^2+18 a b c d+15 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt {2} c^{5/4} (b c-a d)^4}+\frac {3 b^{5/4} (3 a d+b c) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} \sqrt [4]{a} (b c-a d)^4}-\frac {3 b^{5/4} (3 a d+b c) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} \sqrt [4]{a} (b c-a d)^4}-\frac {3 b^{5/4} (3 a d+b c) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} \sqrt [4]{a} (b c-a d)^4}+\frac {3 b^{5/4} (3 a d+b c) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt {2} \sqrt [4]{a} (b c-a d)^4}-\frac {3 d x^{3/2} (a d+7 b c)}{16 c \left (c+d x^2\right ) (b c-a d)^3}-\frac {x^{3/2}}{2 \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}-\frac {3 d x^{3/2}}{4 \left (c+d x^2\right )^2 (b c-a d)^2} \]

[Out]

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

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

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

4*d^(1/4)*(-a^2*d^2+18*a*b*c*d+15*b^2*c^2)*arctan(1-d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/c^(5/4)/(-a*d+b*c)^4*2^(1

/2)-3/64*d^(1/4)*(-a^2*d^2+18*a*b*c*d+15*b^2*c^2)*arctan(1+d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/c^(5/4)/(-a*d+b*c)

^4*2^(1/2)+3/16*b^(5/4)*(3*a*d+b*c)*ln(a^(1/2)+x*b^(1/2)-a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(1/4)/(-a*d+b*c)^4

*2^(1/2)-3/16*b^(5/4)*(3*a*d+b*c)*ln(a^(1/2)+x*b^(1/2)+a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(1/4)/(-a*d+b*c)^4*2

^(1/2)-3/128*d^(1/4)*(-a^2*d^2+18*a*b*c*d+15*b^2*c^2)*ln(c^(1/2)+x*d^(1/2)-c^(1/4)*d^(1/4)*2^(1/2)*x^(1/2))/c^

(5/4)/(-a*d+b*c)^4*2^(1/2)+3/128*d^(1/4)*(-a^2*d^2+18*a*b*c*d+15*b^2*c^2)*ln(c^(1/2)+x*d^(1/2)+c^(1/4)*d^(1/4)

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

________________________________________________________________________________________

Rubi [A] time = 1.02, antiderivative size = 703, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {466, 471, 579, 584, 297, 1162, 617, 204, 1165, 628} \[ -\frac {3 \sqrt [4]{d} \left (-a^2 d^2+18 a b c d+15 b^2 c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{5/4} (b c-a d)^4}+\frac {3 \sqrt [4]{d} \left (-a^2 d^2+18 a b c d+15 b^2 c^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{5/4} (b c-a d)^4}+\frac {3 \sqrt [4]{d} \left (-a^2 d^2+18 a b c d+15 b^2 c^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{5/4} (b c-a d)^4}-\frac {3 \sqrt [4]{d} \left (-a^2 d^2+18 a b c d+15 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt {2} c^{5/4} (b c-a d)^4}+\frac {3 b^{5/4} (3 a d+b c) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} \sqrt [4]{a} (b c-a d)^4}-\frac {3 b^{5/4} (3 a d+b c) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} \sqrt [4]{a} (b c-a d)^4}-\frac {3 b^{5/4} (3 a d+b c) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} \sqrt [4]{a} (b c-a d)^4}+\frac {3 b^{5/4} (3 a d+b c) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt {2} \sqrt [4]{a} (b c-a d)^4}-\frac {3 d x^{3/2} (a d+7 b c)}{16 c \left (c+d x^2\right ) (b c-a d)^3}-\frac {x^{3/2}}{2 \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}-\frac {3 d x^{3/2}}{4 \left (c+d x^2\right )^2 (b c-a d)^2} \]

Antiderivative was successfully veriﬁed.

[In]

Int[x^(5/2)/((a + b*x^2)^2*(c + d*x^2)^3),x]

[Out]

(-3*d*x^(3/2))/(4*(b*c - a*d)^2*(c + d*x^2)^2) - x^(3/2)/(2*(b*c - a*d)*(a + b*x^2)*(c + d*x^2)^2) - (3*d*(7*b

*c + a*d)*x^(3/2))/(16*c*(b*c - a*d)^3*(c + d*x^2)) - (3*b^(5/4)*(b*c + 3*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqr

t[x])/a^(1/4)])/(4*Sqrt[2]*a^(1/4)*(b*c - a*d)^4) + (3*b^(5/4)*(b*c + 3*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[

x])/a^(1/4)])/(4*Sqrt[2]*a^(1/4)*(b*c - a*d)^4) + (3*d^(1/4)*(15*b^2*c^2 + 18*a*b*c*d - a^2*d^2)*ArcTan[1 - (S

qrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(5/4)*(b*c - a*d)^4) - (3*d^(1/4)*(15*b^2*c^2 + 18*a*b*c*d - a

^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(5/4)*(b*c - a*d)^4) + (3*b^(5/4)*(b*c +

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

/4)*(b*c + 3*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(1/4)*(b*c - a*d)^4

) - (3*d^(1/4)*(15*b^2*c^2 + 18*a*b*c*d - a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])

/(64*Sqrt[2]*c^(5/4)*(b*c - a*d)^4) + (3*d^(1/4)*(15*b^2*c^2 + 18*a*b*c*d - a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(

1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(5/4)*(b*c - a*d)^4)

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /

; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 297

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]},

Dist[1/(2*s), Int[(r + s*x^2)/(a + b*x^4), x], x] - Dist[1/(2*s), Int[(r - s*x^2)/(a + b*x^4), x], x]] /; Free

Q[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ,

b]]))

Rule 466

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = Deno

minator[m]}, Dist[k/e, Subst[Int[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/e^n)^p*(c + (d*x^(k*n))/e^n)^q, x], x, (e*

x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && FractionQ[m] && Intege

rQ[p]

Rule 471

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 + 1))/(n*(b*c - a*d)*(p + 1)), x] - Dist[e^n/(n*(b*c -

a*d)*(p + 1)), Int[(e*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(m - n + 1) + d*(m + n*(p + q + 1)

+ 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && GeQ[n

, m - n + 1] && GtQ[m - n + 1, 0] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 579

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_)*((e_) + (f_.)*(x_)^(n_)), x

_Symbol] :> -Simp[((b*e - a*f)*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*g*n*(b*c - a*d)*(p +

1)), x] + Dist[1/(a*n*(b*c - a*d)*(p + 1)), Int[(g*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f)*(

m + 1) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d,

e, f, g, m, q}, x] && IGtQ[n, 0] && LtQ[p, -1]

Rule 584

Int[(((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n_)))/((c_) + (d_.)*(x_)^(n_)), x_Sy

mbol] :> Int[ExpandIntegrand[((g*x)^m*(a + b*x^n)^p*(e + f*x^n))/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e,

f, g, m, p}, x] && IGtQ[n, 0]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub

st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /;

FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +

c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S

imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},

x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[

(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /

; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rubi steps

\begin {align*} \int \frac {x^{5/2}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx &=2 \operatorname {Subst}\left (\int \frac {x^6}{\left (a+b x^4\right )^2 \left (c+d x^4\right )^3} \, dx,x,\sqrt {x}\right )\\ &=-\frac {x^{3/2}}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {\operatorname {Subst}\left (\int \frac {x^2 \left (3 c-9 d x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )^3} \, dx,x,\sqrt {x}\right )}{2 (b c-a d)}\\ &=-\frac {3 d x^{3/2}}{4 (b c-a d)^2 \left (c+d x^2\right )^2}-\frac {x^{3/2}}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {\operatorname {Subst}\left (\int \frac {x^2 \left (12 c (2 b c+a d)-60 b c d x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )^2} \, dx,x,\sqrt {x}\right )}{16 c (b c-a d)^2}\\ &=-\frac {3 d x^{3/2}}{4 (b c-a d)^2 \left (c+d x^2\right )^2}-\frac {x^{3/2}}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}-\frac {3 d (7 b c+a d) x^{3/2}}{16 c (b c-a d)^3 \left (c+d x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {x^2 \left (12 c \left (8 b^2 c^2+17 a b c d-a^2 d^2\right )-12 b c d (7 b c+a d) x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt {x}\right )}{64 c^2 (b c-a d)^3}\\ &=-\frac {3 d x^{3/2}}{4 (b c-a d)^2 \left (c+d x^2\right )^2}-\frac {x^{3/2}}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}-\frac {3 d (7 b c+a d) x^{3/2}}{16 c (b c-a d)^3 \left (c+d x^2\right )}+\frac {\operatorname {Subst}\left (\int \left (\frac {96 b^2 c^2 (b c+3 a d) x^2}{(b c-a d) \left (a+b x^4\right )}-\frac {12 c d \left (15 b^2 c^2+18 a b c d-a^2 d^2\right ) x^2}{(b c-a d) \left (c+d x^4\right )}\right ) \, dx,x,\sqrt {x}\right )}{64 c^2 (b c-a d)^3}\\ &=-\frac {3 d x^{3/2}}{4 (b c-a d)^2 \left (c+d x^2\right )^2}-\frac {x^{3/2}}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}-\frac {3 d (7 b c+a d) x^{3/2}}{16 c (b c-a d)^3 \left (c+d x^2\right )}+\frac {\left (3 b^2 (b c+3 a d)\right ) \operatorname {Subst}\left (\int \frac {x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{2 (b c-a d)^4}-\frac {\left (3 d \left (15 b^2 c^2+18 a b c d-a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{16 c (b c-a d)^4}\\ &=-\frac {3 d x^{3/2}}{4 (b c-a d)^2 \left (c+d x^2\right )^2}-\frac {x^{3/2}}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}-\frac {3 d (7 b c+a d) x^{3/2}}{16 c (b c-a d)^3 \left (c+d x^2\right )}-\frac {\left (3 b^{3/2} (b c+3 a d)\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 (b c-a d)^4}+\frac {\left (3 b^{3/2} (b c+3 a d)\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 (b c-a d)^4}+\frac {\left (3 \sqrt {d} \left (15 b^2 c^2+18 a b c d-a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{32 c (b c-a d)^4}-\frac {\left (3 \sqrt {d} \left (15 b^2 c^2+18 a b c d-a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{32 c (b c-a d)^4}\\ &=-\frac {3 d x^{3/2}}{4 (b c-a d)^2 \left (c+d x^2\right )^2}-\frac {x^{3/2}}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}-\frac {3 d (7 b c+a d) x^{3/2}}{16 c (b c-a d)^3 \left (c+d x^2\right )}+\frac {(3 b (b c+3 a d)) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{8 (b c-a d)^4}+\frac {(3 b (b c+3 a d)) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{8 (b c-a d)^4}+\frac {\left (3 b^{5/4} (b c+3 a d)\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {x}\right )}{8 \sqrt {2} \sqrt [4]{a} (b c-a d)^4}+\frac {\left (3 b^{5/4} (b c+3 a d)\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {x}\right )}{8 \sqrt {2} \sqrt [4]{a} (b c-a d)^4}-\frac {\left (3 \left (15 b^2 c^2+18 a b c d-a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx,x,\sqrt {x}\right )}{64 c (b c-a d)^4}-\frac {\left (3 \left (15 b^2 c^2+18 a b c d-a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx,x,\sqrt {x}\right )}{64 c (b c-a d)^4}-\frac {\left (3 \sqrt [4]{d} \left (15 b^2 c^2+18 a b c d-a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{d}}+2 x}{-\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx,x,\sqrt {x}\right )}{64 \sqrt {2} c^{5/4} (b c-a d)^4}-\frac {\left (3 \sqrt [4]{d} \left (15 b^2 c^2+18 a b c d-a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{d}}-2 x}{-\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx,x,\sqrt {x}\right )}{64 \sqrt {2} c^{5/4} (b c-a d)^4}\\ &=-\frac {3 d x^{3/2}}{4 (b c-a d)^2 \left (c+d x^2\right )^2}-\frac {x^{3/2}}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}-\frac {3 d (7 b c+a d) x^{3/2}}{16 c (b c-a d)^3 \left (c+d x^2\right )}+\frac {3 b^{5/4} (b c+3 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} \sqrt [4]{a} (b c-a d)^4}-\frac {3 b^{5/4} (b c+3 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} \sqrt [4]{a} (b c-a d)^4}-\frac {3 \sqrt [4]{d} \left (15 b^2 c^2+18 a b c d-a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{5/4} (b c-a d)^4}+\frac {3 \sqrt [4]{d} \left (15 b^2 c^2+18 a b c d-a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{5/4} (b c-a d)^4}+\frac {\left (3 b^{5/4} (b c+3 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} \sqrt [4]{a} (b c-a d)^4}-\frac {\left (3 b^{5/4} (b c+3 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} \sqrt [4]{a} (b c-a d)^4}-\frac {\left (3 \sqrt [4]{d} \left (15 b^2 c^2+18 a b c d-a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{5/4} (b c-a d)^4}+\frac {\left (3 \sqrt [4]{d} \left (15 b^2 c^2+18 a b c d-a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{5/4} (b c-a d)^4}\\ &=-\frac {3 d x^{3/2}}{4 (b c-a d)^2 \left (c+d x^2\right )^2}-\frac {x^{3/2}}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}-\frac {3 d (7 b c+a d) x^{3/2}}{16 c (b c-a d)^3 \left (c+d x^2\right )}-\frac {3 b^{5/4} (b c+3 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} \sqrt [4]{a} (b c-a d)^4}+\frac {3 b^{5/4} (b c+3 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} \sqrt [4]{a} (b c-a d)^4}+\frac {3 \sqrt [4]{d} \left (15 b^2 c^2+18 a b c d-a^2 d^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{5/4} (b c-a d)^4}-\frac {3 \sqrt [4]{d} \left (15 b^2 c^2+18 a b c d-a^2 d^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{5/4} (b c-a d)^4}+\frac {3 b^{5/4} (b c+3 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} \sqrt [4]{a} (b c-a d)^4}-\frac {3 b^{5/4} (b c+3 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} \sqrt [4]{a} (b c-a d)^4}-\frac {3 \sqrt [4]{d} \left (15 b^2 c^2+18 a b c d-a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{5/4} (b c-a d)^4}+\frac {3 \sqrt [4]{d} \left (15 b^2 c^2+18 a b c d-a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{5/4} (b c-a d)^4}\\ \end {align*}

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Mathematica [A] time = 2.01, size = 604, normalized size = 0.86 \[ \frac {\frac {3 \sqrt {2} \sqrt [4]{d} \left (a^2 d^2-18 a b c d-15 b^2 c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{c^{5/4}}+\frac {3 \sqrt {2} \sqrt [4]{d} \left (-a^2 d^2+18 a b c d+15 b^2 c^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{c^{5/4}}+\frac {6 \sqrt {2} \sqrt [4]{d} \left (-a^2 d^2+18 a b c d+15 b^2 c^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{c^{5/4}}+\frac {6 \sqrt {2} \sqrt [4]{d} \left (a^2 d^2-18 a b c d-15 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{c^{5/4}}+\frac {24 \sqrt {2} b^{5/4} (3 a d+b c) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{\sqrt [4]{a}}-\frac {24 \sqrt {2} b^{5/4} (3 a d+b c) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{\sqrt [4]{a}}-\frac {48 \sqrt {2} b^{5/4} (3 a d+b c) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt [4]{a}}+\frac {48 \sqrt {2} b^{5/4} (3 a d+b c) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt [4]{a}}-\frac {64 b^2 x^{3/2} (b c-a d)}{a+b x^2}+\frac {8 d x^{3/2} (a d-b c) (3 a d+13 b c)}{c \left (c+d x^2\right )}-\frac {32 d x^{3/2} (b c-a d)^2}{\left (c+d x^2\right )^2}}{128 (b c-a d)^4} \]

Antiderivative was successfully veriﬁed.

[In]

Integrate[x^(5/2)/((a + b*x^2)^2*(c + d*x^2)^3),x]

[Out]

((-64*b^2*(b*c - a*d)*x^(3/2))/(a + b*x^2) - (32*d*(b*c - a*d)^2*x^(3/2))/(c + d*x^2)^2 + (8*d*(-(b*c) + a*d)*

(13*b*c + 3*a*d)*x^(3/2))/(c*(c + d*x^2)) - (48*Sqrt[2]*b^(5/4)*(b*c + 3*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt

[x])/a^(1/4)])/a^(1/4) + (48*Sqrt[2]*b^(5/4)*(b*c + 3*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/a^(1

/4) + (6*Sqrt[2]*d^(1/4)*(15*b^2*c^2 + 18*a*b*c*d - a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/c^

(5/4) + (6*Sqrt[2]*d^(1/4)*(-15*b^2*c^2 - 18*a*b*c*d + a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])

/c^(5/4) + (24*Sqrt[2]*b^(5/4)*(b*c + 3*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/a^(1/

4) - (24*Sqrt[2]*b^(5/4)*(b*c + 3*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/a^(1/4) + (

3*Sqrt[2]*d^(1/4)*(-15*b^2*c^2 - 18*a*b*c*d + a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]

*x])/c^(5/4) + (3*Sqrt[2]*d^(1/4)*(15*b^2*c^2 + 18*a*b*c*d - a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sq

rt[x] + Sqrt[d]*x])/c^(5/4))/(128*(b*c - a*d)^4)

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fricas [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^(5/2)/(b*x^2+a)^2/(d*x^2+c)^3,x, algorithm="fricas")

[Out]

Timed out

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giac [B] time = 2.57, size = 1238, normalized size = 1.76 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)/(b*x^2+a)^2/(d*x^2+c)^3,x, algorithm="giac")

[Out]

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

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

sqrt(2)*a^2*b^4*c^3*d + 6*sqrt(2)*a^3*b^3*c^2*d^2 - 4*sqrt(2)*a^4*b^2*c*d^3 + sqrt(2)*a^5*b*d^4) + 3/4*((a*b^3

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

*a*b^5*c^4 - 4*sqrt(2)*a^2*b^4*c^3*d + 6*sqrt(2)*a^3*b^3*c^2*d^2 - 4*sqrt(2)*a^4*b^2*c*d^3 + sqrt(2)*a^5*b*d^4

) - 3/32*(15*(c*d^3)^(3/4)*b^2*c^2 + 18*(c*d^3)^(3/4)*a*b*c*d - (c*d^3)^(3/4)*a^2*d^2)*arctan(1/2*sqrt(2)*(sqr

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

c^4*d^4 - 4*sqrt(2)*a^3*b*c^3*d^5 + sqrt(2)*a^4*c^2*d^6) - 3/32*(15*(c*d^3)^(3/4)*b^2*c^2 + 18*(c*d^3)^(3/4)*a

*b*c*d - (c*d^3)^(3/4)*a^2*d^2)*arctan(-1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) - 2*sqrt(x))/(c/d)^(1/4))/(sqrt(2)*b^

4*c^6*d^2 - 4*sqrt(2)*a*b^3*c^5*d^3 + 6*sqrt(2)*a^2*b^2*c^4*d^4 - 4*sqrt(2)*a^3*b*c^3*d^5 + sqrt(2)*a^4*c^2*d^

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

*b^5*c^4 - 4*sqrt(2)*a^2*b^4*c^3*d + 6*sqrt(2)*a^3*b^3*c^2*d^2 - 4*sqrt(2)*a^4*b^2*c*d^3 + sqrt(2)*a^5*b*d^4)

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

^5*c^4 - 4*sqrt(2)*a^2*b^4*c^3*d + 6*sqrt(2)*a^3*b^3*c^2*d^2 - 4*sqrt(2)*a^4*b^2*c*d^3 + sqrt(2)*a^5*b*d^4) +

3/64*(15*(c*d^3)^(3/4)*b^2*c^2 + 18*(c*d^3)^(3/4)*a*b*c*d - (c*d^3)^(3/4)*a^2*d^2)*log(sqrt(2)*sqrt(x)*(c/d)^(

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

^3*b*c^3*d^5 + sqrt(2)*a^4*c^2*d^6) - 3/64*(15*(c*d^3)^(3/4)*b^2*c^2 + 18*(c*d^3)^(3/4)*a*b*c*d - (c*d^3)^(3/4

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

6*sqrt(2)*a^2*b^2*c^4*d^4 - 4*sqrt(2)*a^3*b*c^3*d^5 + sqrt(2)*a^4*c^2*d^6) - 1/16*(13*b*c*d^2*x^(7/2) + 3*a*d^

3*x^(7/2) + 17*b*c^2*d*x^(3/2) - a*c*d^2*x^(3/2))/((b^3*c^4 - 3*a*b^2*c^3*d + 3*a^2*b*c^2*d^2 - a^3*c*d^3)*(d*

x^2 + c)^2)

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maple [A] time = 0.03, size = 1067, normalized size = 1.52 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/2)/(b*x^2+a)^2/(d*x^2+c)^3,x)

[Out]

1/2*b^2/(a*d-b*c)^4*x^(3/2)/(b*x^2+a)*a*d-1/2*b^3/(a*d-b*c)^4*x^(3/2)/(b*x^2+a)*c+9/16*b/(a*d-b*c)^4/(a/b)^(1/

4)*2^(1/2)*a*d*ln((x-(a/b)^(1/4)*2^(1/2)*x^(1/2)+(a/b)^(1/2))/(x+(a/b)^(1/4)*2^(1/2)*x^(1/2)+(a/b)^(1/2)))+9/8

*b/(a*d-b*c)^4/(a/b)^(1/4)*2^(1/2)*a*d*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)+9/8*b/(a*d-b*c)^4/(a/b)^(1/4)*2^(

1/2)*a*d*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)+3/16*b^2/(a*d-b*c)^4/(a/b)^(1/4)*2^(1/2)*c*ln((x-(a/b)^(1/4)*2^

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

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

*x^(1/2)-1)+3/16*d^4/(a*d-b*c)^4/(d*x^2+c)^2/c*x^(7/2)*a^2+5/8*d^3/(a*d-b*c)^4/(d*x^2+c)^2*x^(7/2)*a*b-13/16*d

^2/(a*d-b*c)^4/(d*x^2+c)^2*c*x^(7/2)*b^2-1/16*d^3/(a*d-b*c)^4/(d*x^2+c)^2*x^(3/2)*a^2+9/8*d^2/(a*d-b*c)^4/(d*x

^2+c)^2*x^(3/2)*a*b*c-17/16*d/(a*d-b*c)^4/(d*x^2+c)^2*x^(3/2)*b^2*c^2+3/64*d^2/(a*d-b*c)^4/c/(c/d)^(1/4)*2^(1/

2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)*a^2-27/32*d/(a*d-b*c)^4/(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4

)*x^(1/2)+1)*a*b-45/64/(a*d-b*c)^4*c/(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)*b^2+3/64*d^2/(a

*d-b*c)^4/c/(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*a^2-27/32*d/(a*d-b*c)^4/(c/d)^(1/4)*2^(1

/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*a*b-45/64/(a*d-b*c)^4*c/(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/

4)*x^(1/2)-1)*b^2+3/128*d^2/(a*d-b*c)^4/c/(c/d)^(1/4)*2^(1/2)*ln((x-(c/d)^(1/4)*2^(1/2)*x^(1/2)+(c/d)^(1/2))/(

x+(c/d)^(1/4)*2^(1/2)*x^(1/2)+(c/d)^(1/2)))*a^2-27/64*d/(a*d-b*c)^4/(c/d)^(1/4)*2^(1/2)*ln((x-(c/d)^(1/4)*2^(1

/2)*x^(1/2)+(c/d)^(1/2))/(x+(c/d)^(1/4)*2^(1/2)*x^(1/2)+(c/d)^(1/2)))*a*b-45/128/(a*d-b*c)^4*c/(c/d)^(1/4)*2^(

1/2)*ln((x-(c/d)^(1/4)*2^(1/2)*x^(1/2)+(c/d)^(1/2))/(x+(c/d)^(1/4)*2^(1/2)*x^(1/2)+(c/d)^(1/2)))*b^2

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maxima [A] time = 2.78, size = 791, normalized size = 1.13 \[ \frac {3 \, {\left (b^{3} c + 3 \, a b^{2} d\right )} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{16 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )}} - \frac {3 \, {\left (15 \, b^{2} c^{2} d + 18 \, a b c d^{2} - a^{2} d^{3}\right )} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} + 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {\sqrt {c} \sqrt {d}} \sqrt {d}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} - 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {\sqrt {c} \sqrt {d}} \sqrt {d}} - \frac {\sqrt {2} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}}\right )}}{128 \, {\left (b^{4} c^{5} - 4 \, a b^{3} c^{4} d + 6 \, a^{2} b^{2} c^{3} d^{2} - 4 \, a^{3} b c^{2} d^{3} + a^{4} c d^{4}\right )}} - \frac {3 \, {\left (7 \, b^{2} c d^{2} + a b d^{3}\right )} x^{\frac {11}{2}} + 3 \, {\left (11 \, b^{2} c^{2} d + 4 \, a b c d^{2} + a^{2} d^{3}\right )} x^{\frac {7}{2}} + {\left (8 \, b^{2} c^{3} + 17 \, a b c^{2} d - a^{2} c d^{2}\right )} x^{\frac {3}{2}}}{16 \, {\left (a b^{3} c^{6} - 3 \, a^{2} b^{2} c^{5} d + 3 \, a^{3} b c^{4} d^{2} - a^{4} c^{3} d^{3} + {\left (b^{4} c^{4} d^{2} - 3 \, a b^{3} c^{3} d^{3} + 3 \, a^{2} b^{2} c^{2} d^{4} - a^{3} b c d^{5}\right )} x^{6} + {\left (2 \, b^{4} c^{5} d - 5 \, a b^{3} c^{4} d^{2} + 3 \, a^{2} b^{2} c^{3} d^{3} + a^{3} b c^{2} d^{4} - a^{4} c d^{5}\right )} x^{4} + {\left (b^{4} c^{6} - a b^{3} c^{5} d - 3 \, a^{2} b^{2} c^{4} d^{2} + 5 \, a^{3} b c^{3} d^{3} - 2 \, a^{4} c^{2} d^{4}\right )} x^{2}\right )}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)/(b*x^2+a)^2/(d*x^2+c)^3,x, algorithm="maxima")

[Out]

3/16*(b^3*c + 3*a*b^2*d)*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) + 2*sqrt(b)*sqrt(x))/sqrt(sqrt

(a)*sqrt(b)))/(sqrt(sqrt(a)*sqrt(b))*sqrt(b)) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) - 2*sqr

t(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(sqrt(a)*sqrt(b))*sqrt(b)) - sqrt(2)*log(sqrt(2)*a^(1/4)*b^(1/4)*sqr

t(x) + sqrt(b)*x + sqrt(a))/(a^(1/4)*b^(3/4)) + sqrt(2)*log(-sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqr

t(a))/(a^(1/4)*b^(3/4)))/(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4) - 3/128*(15*b

^2*c^2*d + 18*a*b*c*d^2 - a^2*d^3)*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/4) + 2*sqrt(d)*sqrt(x))

/sqrt(sqrt(c)*sqrt(d)))/(sqrt(sqrt(c)*sqrt(d))*sqrt(d)) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/

4) - 2*sqrt(d)*sqrt(x))/sqrt(sqrt(c)*sqrt(d)))/(sqrt(sqrt(c)*sqrt(d))*sqrt(d)) - sqrt(2)*log(sqrt(2)*c^(1/4)*d

^(1/4)*sqrt(x) + sqrt(d)*x + sqrt(c))/(c^(1/4)*d^(3/4)) + sqrt(2)*log(-sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(

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

- 1/16*(3*(7*b^2*c*d^2 + a*b*d^3)*x^(11/2) + 3*(11*b^2*c^2*d + 4*a*b*c*d^2 + a^2*d^3)*x^(7/2) + (8*b^2*c^3 +

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

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

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

^4)*x^2)

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mupad [B] time = 7.63, size = 44169, normalized size = 62.83 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/2)/((a + b*x^2)^2*(c + d*x^2)^3),x)

[Out]

2*atan((((((864*a*b^27*c^23*d^4 - (27*a^24*b^4*d^27)/16 + (1863*a^23*b^5*c*d^26)/16 - 5184*a^2*b^26*c^22*d^5 -

(132597*a^3*b^25*c^21*d^6)/16 + (2587113*a^4*b^24*c^20*d^7)/16 - (4585005*a^5*b^23*c^19*d^8)/8 + (5105997*a^6

*b^22*c^18*d^9)/8 + (22410891*a^7*b^21*c^17*d^10)/16 - (93270447*a^8*b^20*c^16*d^11)/16 + (13320261*a^9*b^19*c

^15*d^12)/2 + (12854835*a^10*b^18*c^14*d^13)/2 - (279642213*a^11*b^17*c^13*d^14)/8 + (501573033*a^12*b^16*c^12

*d^15)/8 - (274240863*a^13*b^15*c^11*d^16)/4 + (196146927*a^14*b^14*c^10*d^17)/4 - (166924665*a^15*b^13*c^9*d^

18)/8 + (14462037*a^16*b^12*c^8*d^19)/8 + (8300637*a^17*b^11*c^7*d^20)/2 - (6325749*a^18*b^10*c^6*d^21)/2 + (1

9723743*a^19*b^9*c^5*d^22)/16 - (4658715*a^20*b^8*c^4*d^23)/16 + (327267*a^21*b^7*c^3*d^24)/8 - (24867*a^22*b^

6*c^2*d^25)/8)*1i)/(b^21*c^23 - a^21*c^2*d^21 + 21*a^20*b*c^3*d^20 + 210*a^2*b^19*c^21*d^2 - 1330*a^3*b^18*c^2

0*d^3 + 5985*a^4*b^17*c^19*d^4 - 20349*a^5*b^16*c^18*d^5 + 54264*a^6*b^15*c^17*d^6 - 116280*a^7*b^14*c^16*d^7

+ 203490*a^8*b^13*c^15*d^8 - 293930*a^9*b^12*c^14*d^9 + 352716*a^10*b^11*c^13*d^10 - 352716*a^11*b^10*c^12*d^1

1 + 293930*a^12*b^9*c^11*d^12 - 203490*a^13*b^8*c^10*d^13 + 116280*a^14*b^7*c^9*d^14 - 54264*a^15*b^6*c^8*d^15

+ 20349*a^16*b^5*c^7*d^16 - 5985*a^17*b^4*c^6*d^17 + 1330*a^18*b^3*c^5*d^18 - 210*a^19*b^2*c^4*d^19 - 21*a*b^

20*c^22*d) - (9*x^(1/2)*(-(81*a^8*d^9 + 4100625*b^8*c^8*d + 19683000*a*b^7*c^7*d^2 + 34335900*a^2*b^6*c^6*d^3

+ 24406920*a^3*b^5*c^5*d^4 + 3888486*a^4*b^4*c^4*d^5 - 1627128*a^5*b^3*c^3*d^6 + 152604*a^6*b^2*c^2*d^7 - 5832

*a^7*b*c*d^8)/(16777216*b^16*c^21 + 16777216*a^16*c^5*d^16 - 268435456*a^15*b*c^6*d^15 + 2013265920*a^2*b^14*c

^19*d^2 - 9395240960*a^3*b^13*c^18*d^3 + 30534533120*a^4*b^12*c^17*d^4 - 73282879488*a^5*b^11*c^16*d^5 + 13435

1945728*a^6*b^10*c^15*d^6 - 191931351040*a^7*b^9*c^14*d^7 + 215922769920*a^8*b^8*c^13*d^8 - 191931351040*a^9*b

^7*c^12*d^9 + 134351945728*a^10*b^6*c^11*d^10 - 73282879488*a^11*b^5*c^10*d^11 + 30534533120*a^12*b^4*c^9*d^12

- 9395240960*a^13*b^3*c^8*d^13 + 2013265920*a^14*b^2*c^7*d^14 - 268435456*a*b^15*c^20*d))^(1/4)*(16777216*a*b

^26*c^23*d^4 + 262144*a^23*b^4*c*d^26 - 167772160*a^2*b^25*c^22*d^5 + 612630528*a^3*b^24*c^21*d^6 - 533725184*

a^4*b^23*c^20*d^7 - 2827485184*a^5*b^22*c^19*d^8 + 8081375232*a^6*b^21*c^18*d^9 + 6940786688*a^7*b^20*c^17*d^1

0 - 89661636608*a^8*b^19*c^16*d^11 + 273093230592*a^9*b^18*c^15*d^12 - 518906707968*a^10*b^17*c^14*d^13 + 7246

29454848*a^11*b^16*c^13*d^14 - 805866307584*a^12*b^15*c^12*d^15 + 754870910976*a^13*b^14*c^11*d^16 - 615914668

032*a^14*b^13*c^10*d^17 + 437990719488*a^15*b^12*c^9*d^18 - 263356153856*a^16*b^11*c^8*d^19 + 127919980544*a^1

7*b^10*c^7*d^20 - 47752151040*a^18*b^9*c^6*d^21 + 12955418624*a^19*b^8*c^5*d^22 - 2370830336*a^20*b^7*c^4*d^23

+ 259522560*a^21*b^6*c^3*d^24 - 13631488*a^22*b^5*c^2*d^25))/(65536*(b^18*c^20 + a^18*c^2*d^18 - 18*a^17*b*c^

3*d^17 + 153*a^2*b^16*c^18*d^2 - 816*a^3*b^15*c^17*d^3 + 3060*a^4*b^14*c^16*d^4 - 8568*a^5*b^13*c^15*d^5 + 185

64*a^6*b^12*c^14*d^6 - 31824*a^7*b^11*c^13*d^7 + 43758*a^8*b^10*c^12*d^8 - 48620*a^9*b^9*c^11*d^9 + 43758*a^10

*b^8*c^10*d^10 - 31824*a^11*b^7*c^9*d^11 + 18564*a^12*b^6*c^8*d^12 - 8568*a^13*b^5*c^7*d^13 + 3060*a^14*b^4*c^

6*d^14 - 816*a^15*b^3*c^5*d^15 + 153*a^16*b^2*c^4*d^16 - 18*a*b^17*c^19*d)))*(-(81*a^8*d^9 + 4100625*b^8*c^8*d

+ 19683000*a*b^7*c^7*d^2 + 34335900*a^2*b^6*c^6*d^3 + 24406920*a^3*b^5*c^5*d^4 + 3888486*a^4*b^4*c^4*d^5 - 16

27128*a^5*b^3*c^3*d^6 + 152604*a^6*b^2*c^2*d^7 - 5832*a^7*b*c*d^8)/(16777216*b^16*c^21 + 16777216*a^16*c^5*d^1

6 - 268435456*a^15*b*c^6*d^15 + 2013265920*a^2*b^14*c^19*d^2 - 9395240960*a^3*b^13*c^18*d^3 + 30534533120*a^4*

b^12*c^17*d^4 - 73282879488*a^5*b^11*c^16*d^5 + 134351945728*a^6*b^10*c^15*d^6 - 191931351040*a^7*b^9*c^14*d^7

+ 215922769920*a^8*b^8*c^13*d^8 - 191931351040*a^9*b^7*c^12*d^9 + 134351945728*a^10*b^6*c^11*d^10 - 732828794

88*a^11*b^5*c^10*d^11 + 30534533120*a^12*b^4*c^9*d^12 - 9395240960*a^13*b^3*c^8*d^13 + 2013265920*a^14*b^2*c^7

*d^14 - 268435456*a*b^15*c^20*d))^(3/4) - (9*x^(1/2)*(729*a^11*b^8*d^15 + 4100625*a*b^18*c^10*d^5 + 367902*a^1

0*b^9*c*d^14 + 45453150*a^2*b^17*c^9*d^6 + 206135685*a^3*b^16*c^8*d^7 + 505671336*a^4*b^15*c^7*d^8 + 754592274

*a^5*b^14*c^6*d^9 + 718242228*a^6*b^13*c^5*d^10 + 406721250*a^7*b^12*c^4*d^11 + 89841960*a^8*b^11*c^3*d^12 - 1

3218147*a^9*b^10*c^2*d^13))/(65536*(b^18*c^20 + a^18*c^2*d^18 - 18*a^17*b*c^3*d^17 + 153*a^2*b^16*c^18*d^2 - 8

16*a^3*b^15*c^17*d^3 + 3060*a^4*b^14*c^16*d^4 - 8568*a^5*b^13*c^15*d^5 + 18564*a^6*b^12*c^14*d^6 - 31824*a^7*b

^11*c^13*d^7 + 43758*a^8*b^10*c^12*d^8 - 48620*a^9*b^9*c^11*d^9 + 43758*a^10*b^8*c^10*d^10 - 31824*a^11*b^7*c^

9*d^11 + 18564*a^12*b^6*c^8*d^12 - 8568*a^13*b^5*c^7*d^13 + 3060*a^14*b^4*c^6*d^14 - 816*a^15*b^3*c^5*d^15 + 1

53*a^16*b^2*c^4*d^16 - 18*a*b^17*c^19*d)))*(-(81*a^8*d^9 + 4100625*b^8*c^8*d + 19683000*a*b^7*c^7*d^2 + 343359

00*a^2*b^6*c^6*d^3 + 24406920*a^3*b^5*c^5*d^4 + 3888486*a^4*b^4*c^4*d^5 - 1627128*a^5*b^3*c^3*d^6 + 152604*a^6

*b^2*c^2*d^7 - 5832*a^7*b*c*d^8)/(16777216*b^16*c^21 + 16777216*a^16*c^5*d^16 - 268435456*a^15*b*c^6*d^15 + 20

13265920*a^2*b^14*c^19*d^2 - 9395240960*a^3*b^13*c^18*d^3 + 30534533120*a^4*b^12*c^17*d^4 - 73282879488*a^5*b^

11*c^16*d^5 + 134351945728*a^6*b^10*c^15*d^6 - 191931351040*a^7*b^9*c^14*d^7 + 215922769920*a^8*b^8*c^13*d^8 -

191931351040*a^9*b^7*c^12*d^9 + 134351945728*a^10*b^6*c^11*d^10 - 73282879488*a^11*b^5*c^10*d^11 + 3053453312

0*a^12*b^4*c^9*d^12 - 9395240960*a^13*b^3*c^8*d^13 + 2013265920*a^14*b^2*c^7*d^14 - 268435456*a*b^15*c^20*d))^

(1/4) - ((((864*a*b^27*c^23*d^4 - (27*a^24*b^4*d^27)/16 + (1863*a^23*b^5*c*d^26)/16 - 5184*a^2*b^26*c^22*d^5 -