\(\int \frac {x^{5/2} (c+d x^2)^3}{(a+b x^2)^2} \, dx\) [453]

Optimal result

Integrand size = 24, antiderivative size = 374 \[ \int \frac {x^{5/2} \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\frac {d \left (7 b^2 c^2-11 a b c d+5 a^2 d^2\right ) x^{3/2}}{2 b^4}+\frac {3 d^2 (11 b c-5 a d) x^{7/2}}{14 b^3}+\frac {15 d^3 x^{11/2}}{22 b^2}-\frac {x^{3/2} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}-\frac {3 (b c-5 a d) (b c-a d)^2 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} \sqrt [4]{a} b^{19/4}}+\frac {3 (b c-5 a d) (b c-a d)^2 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} \sqrt [4]{a} b^{19/4}}+\frac {3 (b c-5 a d) (b c-a d)^2 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} \sqrt [4]{a} b^{19/4}}-\frac {3 (b c-5 a d) (b c-a d)^2 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} \sqrt [4]{a} b^{19/4}} \]

[Out]

1/2*d*(5*a^2*d^2-11*a*b*c*d+7*b^2*c^2)*x^(3/2)/b^4+3/14*d^2*(-5*a*d+11*b*c)*x^(7/2)/b^3+15/22*d^3*x^(11/2)/b^2
-1/2*x^(3/2)*(d*x^2+c)^3/b/(b*x^2+a)-3/8*(-5*a*d+b*c)*(-a*d+b*c)^2*arctan(1-b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a
^(1/4)/b^(19/4)*2^(1/2)+3/8*(-5*a*d+b*c)*(-a*d+b*c)^2*arctan(1+b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(1/4)/b^(19/
4)*2^(1/2)+3/16*(-5*a*d+b*c)*(-a*d+b*c)^2*ln(a^(1/2)+x*b^(1/2)-a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(1/4)/b^(19/
4)*2^(1/2)-3/16*(-5*a*d+b*c)*(-a*d+b*c)^2*ln(a^(1/2)+x*b^(1/2)+a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(1/4)/b^(19/
4)*2^(1/2)

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {477, 478, 584, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {x^{5/2} \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\frac {d x^{3/2} \left (5 a^2 d^2-11 a b c d+7 b^2 c^2\right )}{2 b^4}-\frac {3 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ) (b c-5 a d) (b c-a d)^2}{4 \sqrt {2} \sqrt [4]{a} b^{19/4}}+\frac {3 \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right ) (b c-5 a d) (b c-a d)^2}{4 \sqrt {2} \sqrt [4]{a} b^{19/4}}+\frac {3 (b c-5 a d) (b c-a d)^2 \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} \sqrt [4]{a} b^{19/4}}-\frac {3 (b c-5 a d) (b c-a d)^2 \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} \sqrt [4]{a} b^{19/4}}+\frac {3 d^2 x^{7/2} (11 b c-5 a d)}{14 b^3}-\frac {x^{3/2} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac {15 d^3 x^{11/2}}{22 b^2} \]

[In]

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

[Out]

(d*(7*b^2*c^2 - 11*a*b*c*d + 5*a^2*d^2)*x^(3/2))/(2*b^4) + (3*d^2*(11*b*c - 5*a*d)*x^(7/2))/(14*b^3) + (15*d^3
*x^(11/2))/(22*b^2) - (x^(3/2)*(c + d*x^2)^3)/(2*b*(a + b*x^2)) - (3*(b*c - 5*a*d)*(b*c - a*d)^2*ArcTan[1 - (S
qrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(1/4)*b^(19/4)) + (3*(b*c - 5*a*d)*(b*c - a*d)^2*ArcTan[1 + (Sq
rt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(1/4)*b^(19/4)) + (3*(b*c - 5*a*d)*(b*c - a*d)^2*Log[Sqrt[a] - S
qrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(1/4)*b^(19/4)) - (3*(b*c - 5*a*d)*(b*c - a*d)^2*Log
[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(1/4)*b^(19/4))

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 303

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 477

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 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 584

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_))^
(r_.), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^r, x], x] /; FreeQ[{a,
b, c, d, e, f, g, m, n}, x] && IGtQ[p, -2] && IGtQ[q, 0] && IGtQ[r, 0]

Rule 631

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 642

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 1176

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 1179

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*} \text {integral}& = 2 \text {Subst}\left (\int \frac {x^6 \left (c+d x^4\right )^3}{\left (a+b x^4\right )^2} \, dx,x,\sqrt {x}\right ) \\ & = -\frac {x^{3/2} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac {\text {Subst}\left (\int \frac {x^2 \left (c+d x^4\right )^2 \left (3 c+15 d x^4\right )}{a+b x^4} \, dx,x,\sqrt {x}\right )}{2 b} \\ & = -\frac {x^{3/2} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac {\text {Subst}\left (\int \left (\frac {3 d \left (7 b^2 c^2-11 a b c d+5 a^2 d^2\right ) x^2}{b^3}+\frac {3 d^2 (11 b c-5 a d) x^6}{b^2}+\frac {15 d^3 x^{10}}{b}+\frac {3 \left (b^3 c^3-7 a b^2 c^2 d+11 a^2 b c d^2-5 a^3 d^3\right ) x^2}{b^3 \left (a+b x^4\right )}\right ) \, dx,x,\sqrt {x}\right )}{2 b} \\ & = \frac {d \left (7 b^2 c^2-11 a b c d+5 a^2 d^2\right ) x^{3/2}}{2 b^4}+\frac {3 d^2 (11 b c-5 a d) x^{7/2}}{14 b^3}+\frac {15 d^3 x^{11/2}}{22 b^2}-\frac {x^{3/2} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac {\left (3 (b c-5 a d) (b c-a d)^2\right ) \text {Subst}\left (\int \frac {x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{2 b^4} \\ & = \frac {d \left (7 b^2 c^2-11 a b c d+5 a^2 d^2\right ) x^{3/2}}{2 b^4}+\frac {3 d^2 (11 b c-5 a d) x^{7/2}}{14 b^3}+\frac {15 d^3 x^{11/2}}{22 b^2}-\frac {x^{3/2} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}-\frac {\left (3 (b c-5 a d) (b c-a d)^2\right ) \text {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 b^{9/2}}+\frac {\left (3 (b c-5 a d) (b c-a d)^2\right ) \text {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 b^{9/2}} \\ & = \frac {d \left (7 b^2 c^2-11 a b c d+5 a^2 d^2\right ) x^{3/2}}{2 b^4}+\frac {3 d^2 (11 b c-5 a d) x^{7/2}}{14 b^3}+\frac {15 d^3 x^{11/2}}{22 b^2}-\frac {x^{3/2} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac {\left (3 (b c-5 a d) (b c-a d)^2\right ) \text {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^5}+\frac {\left (3 (b c-5 a d) (b c-a d)^2\right ) \text {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^5}+\frac {\left (3 (b c-5 a d) (b c-a d)^2\right ) \text {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^{19/4}}+\frac {\left (3 (b c-5 a d) (b c-a d)^2\right ) \text {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^{19/4}} \\ & = \frac {d \left (7 b^2 c^2-11 a b c d+5 a^2 d^2\right ) x^{3/2}}{2 b^4}+\frac {3 d^2 (11 b c-5 a d) x^{7/2}}{14 b^3}+\frac {15 d^3 x^{11/2}}{22 b^2}-\frac {x^{3/2} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac {3 (b c-5 a d) (b c-a d)^2 \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^{19/4}}-\frac {3 (b c-5 a d) (b c-a d)^2 \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^{19/4}}+\frac {\left (3 (b c-5 a d) (b c-a d)^2\right ) \text {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^{19/4}}-\frac {\left (3 (b c-5 a d) (b c-a d)^2\right ) \text {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^{19/4}} \\ & = \frac {d \left (7 b^2 c^2-11 a b c d+5 a^2 d^2\right ) x^{3/2}}{2 b^4}+\frac {3 d^2 (11 b c-5 a d) x^{7/2}}{14 b^3}+\frac {15 d^3 x^{11/2}}{22 b^2}-\frac {x^{3/2} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}-\frac {3 (b c-5 a d) (b c-a d)^2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} \sqrt [4]{a} b^{19/4}}+\frac {3 (b c-5 a d) (b c-a d)^2 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} \sqrt [4]{a} b^{19/4}}+\frac {3 (b c-5 a d) (b c-a d)^2 \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^{19/4}}-\frac {3 (b c-5 a d) (b c-a d)^2 \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^{19/4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.52 (sec) , antiderivative size = 257, normalized size of antiderivative = 0.69 \[ \int \frac {x^{5/2} \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\frac {\frac {4 b^{3/4} x^{3/2} \left (385 a^3 d^3+11 a^2 b d^2 \left (-77 c+20 d x^2\right )+a b^2 d \left (539 c^2-484 c d x^2-60 d^2 x^4\right )+b^3 \left (-77 c^3+308 c^2 d x^2+132 c d^2 x^4+28 d^3 x^6\right )\right )}{a+b x^2}+\frac {231 \sqrt {2} (b c-a d)^2 (-b c+5 a d) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{\sqrt [4]{a}}+\frac {231 \sqrt {2} (b c-a d)^2 (-b c+5 a d) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{\sqrt [4]{a}}}{616 b^{19/4}} \]

[In]

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

[Out]

((4*b^(3/4)*x^(3/2)*(385*a^3*d^3 + 11*a^2*b*d^2*(-77*c + 20*d*x^2) + a*b^2*d*(539*c^2 - 484*c*d*x^2 - 60*d^2*x
^4) + b^3*(-77*c^3 + 308*c^2*d*x^2 + 132*c*d^2*x^4 + 28*d^3*x^6)))/(a + b*x^2) + (231*Sqrt[2]*(b*c - a*d)^2*(-
(b*c) + 5*a*d)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/a^(1/4) + (231*Sqrt[2]*(b*c -
a*d)^2*(-(b*c) + 5*a*d)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/a^(1/4))/(616*b^(19/
4))

Maple [A] (verified)

Time = 3.02 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.61

method	result	size
risch	\(\frac {2 d \,x^{\frac {3}{2}} \left (7 b^{2} d^{2} x^{4}-22 x^{2} a b \,d^{2}+33 x^{2} b^{2} c d +77 a^{2} d^{2}-154 a b c d +77 b^{2} c^{2}\right )}{77 b^{4}}-\frac {\left (2 a^{2} d^{2}-4 a b c d +2 b^{2} c^{2}\right ) \left (\frac {\left (-\frac {a d}{4}+\frac {b c}{4}\right ) x^{\frac {3}{2}}}{b \,x^{2}+a}+\frac {\left (\frac {15 a d}{4}-\frac {3 b c}{4}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{b^{4}}\)	\(230\)
derivativedivides	\(\frac {2 d \left (\frac {b^{2} d^{2} x^{\frac {11}{2}}}{11}+\frac {\left (-2 a b \,d^{2}+3 b^{2} c d \right ) x^{\frac {7}{2}}}{7}+\frac {\left (3 a^{2} d^{2}-6 a b c d +3 b^{2} c^{2}\right ) x^{\frac {3}{2}}}{3}\right )}{b^{4}}-\frac {2 \left (\frac {\left (-\frac {1}{4} a^{3} d^{3}+\frac {3}{4} a^{2} b c \,d^{2}-\frac {3}{4} a \,b^{2} c^{2} d +\frac {1}{4} b^{3} c^{3}\right ) x^{\frac {3}{2}}}{b \,x^{2}+a}+\frac {\left (\frac {15}{4} a^{3} d^{3}-\frac {33}{4} a^{2} b c \,d^{2}+\frac {21}{4} a \,b^{2} c^{2} d -\frac {3}{4} b^{3} c^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{b^{4}}\)	\(266\)
default	\(\frac {2 d \left (\frac {b^{2} d^{2} x^{\frac {11}{2}}}{11}+\frac {\left (-2 a b \,d^{2}+3 b^{2} c d \right ) x^{\frac {7}{2}}}{7}+\frac {\left (3 a^{2} d^{2}-6 a b c d +3 b^{2} c^{2}\right ) x^{\frac {3}{2}}}{3}\right )}{b^{4}}-\frac {2 \left (\frac {\left (-\frac {1}{4} a^{3} d^{3}+\frac {3}{4} a^{2} b c \,d^{2}-\frac {3}{4} a \,b^{2} c^{2} d +\frac {1}{4} b^{3} c^{3}\right ) x^{\frac {3}{2}}}{b \,x^{2}+a}+\frac {\left (\frac {15}{4} a^{3} d^{3}-\frac {33}{4} a^{2} b c \,d^{2}+\frac {21}{4} a \,b^{2} c^{2} d -\frac {3}{4} b^{3} c^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{b^{4}}\)	\(266\)

[In]

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

[Out]

2/77*d*x^(3/2)*(7*b^2*d^2*x^4-22*a*b*d^2*x^2+33*b^2*c*d*x^2+77*a^2*d^2-154*a*b*c*d+77*b^2*c^2)/b^4-1/b^4*(2*a^
2*d^2-4*a*b*c*d+2*b^2*c^2)*((-1/4*a*d+1/4*b*c)*x^(3/2)/(b*x^2+a)+1/8*(15/4*a*d-3/4*b*c)/b/(a/b)^(1/4)*2^(1/2)*
(ln((x-(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2))/(x+(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2)))+2*arctan(2^(1/2)/
(a/b)^(1/4)*x^(1/2)+1)+2*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.29 (sec) , antiderivative size = 2104, normalized size of antiderivative = 5.63 \[ \int \frac {x^{5/2} \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/616*(231*(b^5*x^2 + a*b^4)*(-(b^12*c^12 - 28*a*b^11*c^11*d + 338*a^2*b^10*c^10*d^2 - 2316*a^3*b^9*c^9*d^3 +
10015*a^4*b^8*c^8*d^4 - 28856*a^5*b^7*c^7*d^5 + 57148*a^6*b^6*c^6*d^6 - 78968*a^7*b^5*c^5*d^7 + 76111*a^8*b^4
*c^4*d^8 - 50220*a^9*b^3*c^3*d^9 + 21650*a^10*b^2*c^2*d^10 - 5500*a^11*b*c*d^11 + 625*a^12*d^12)/(a*b^19))^(1/
4)*log(27*a*b^14*(-(b^12*c^12 - 28*a*b^11*c^11*d + 338*a^2*b^10*c^10*d^2 - 2316*a^3*b^9*c^9*d^3 + 10015*a^4*b^
8*c^8*d^4 - 28856*a^5*b^7*c^7*d^5 + 57148*a^6*b^6*c^6*d^6 - 78968*a^7*b^5*c^5*d^7 + 76111*a^8*b^4*c^4*d^8 - 50
220*a^9*b^3*c^3*d^9 + 21650*a^10*b^2*c^2*d^10 - 5500*a^11*b*c*d^11 + 625*a^12*d^12)/(a*b^19))^(3/4) - 27*(b^9*
c^9 - 21*a*b^8*c^8*d + 180*a^2*b^7*c^7*d^2 - 820*a^3*b^6*c^6*d^3 + 2190*a^4*b^5*c^5*d^4 - 3606*a^5*b^4*c^4*d^5
+ 3716*a^6*b^3*c^3*d^6 - 2340*a^7*b^2*c^2*d^7 + 825*a^8*b*c*d^8 - 125*a^9*d^9)*sqrt(x)) + 231*(-I*b^5*x^2 - I
*a*b^4)*(-(b^12*c^12 - 28*a*b^11*c^11*d + 338*a^2*b^10*c^10*d^2 - 2316*a^3*b^9*c^9*d^3 + 10015*a^4*b^8*c^8*d^4
- 28856*a^5*b^7*c^7*d^5 + 57148*a^6*b^6*c^6*d^6 - 78968*a^7*b^5*c^5*d^7 + 76111*a^8*b^4*c^4*d^8 - 50220*a^9*b
^3*c^3*d^9 + 21650*a^10*b^2*c^2*d^10 - 5500*a^11*b*c*d^11 + 625*a^12*d^12)/(a*b^19))^(1/4)*log(27*I*a*b^14*(-(
b^12*c^12 - 28*a*b^11*c^11*d + 338*a^2*b^10*c^10*d^2 - 2316*a^3*b^9*c^9*d^3 + 10015*a^4*b^8*c^8*d^4 - 28856*a^
5*b^7*c^7*d^5 + 57148*a^6*b^6*c^6*d^6 - 78968*a^7*b^5*c^5*d^7 + 76111*a^8*b^4*c^4*d^8 - 50220*a^9*b^3*c^3*d^9
+ 21650*a^10*b^2*c^2*d^10 - 5500*a^11*b*c*d^11 + 625*a^12*d^12)/(a*b^19))^(3/4) - 27*(b^9*c^9 - 21*a*b^8*c^8*d
+ 180*a^2*b^7*c^7*d^2 - 820*a^3*b^6*c^6*d^3 + 2190*a^4*b^5*c^5*d^4 - 3606*a^5*b^4*c^4*d^5 + 3716*a^6*b^3*c^3*
d^6 - 2340*a^7*b^2*c^2*d^7 + 825*a^8*b*c*d^8 - 125*a^9*d^9)*sqrt(x)) + 231*(I*b^5*x^2 + I*a*b^4)*(-(b^12*c^12
- 28*a*b^11*c^11*d + 338*a^2*b^10*c^10*d^2 - 2316*a^3*b^9*c^9*d^3 + 10015*a^4*b^8*c^8*d^4 - 28856*a^5*b^7*c^7*
d^5 + 57148*a^6*b^6*c^6*d^6 - 78968*a^7*b^5*c^5*d^7 + 76111*a^8*b^4*c^4*d^8 - 50220*a^9*b^3*c^3*d^9 + 21650*a^
10*b^2*c^2*d^10 - 5500*a^11*b*c*d^11 + 625*a^12*d^12)/(a*b^19))^(1/4)*log(-27*I*a*b^14*(-(b^12*c^12 - 28*a*b^1
1*c^11*d + 338*a^2*b^10*c^10*d^2 - 2316*a^3*b^9*c^9*d^3 + 10015*a^4*b^8*c^8*d^4 - 28856*a^5*b^7*c^7*d^5 + 5714
8*a^6*b^6*c^6*d^6 - 78968*a^7*b^5*c^5*d^7 + 76111*a^8*b^4*c^4*d^8 - 50220*a^9*b^3*c^3*d^9 + 21650*a^10*b^2*c^2
*d^10 - 5500*a^11*b*c*d^11 + 625*a^12*d^12)/(a*b^19))^(3/4) - 27*(b^9*c^9 - 21*a*b^8*c^8*d + 180*a^2*b^7*c^7*d
^2 - 820*a^3*b^6*c^6*d^3 + 2190*a^4*b^5*c^5*d^4 - 3606*a^5*b^4*c^4*d^5 + 3716*a^6*b^3*c^3*d^6 - 2340*a^7*b^2*c
^2*d^7 + 825*a^8*b*c*d^8 - 125*a^9*d^9)*sqrt(x)) - 231*(b^5*x^2 + a*b^4)*(-(b^12*c^12 - 28*a*b^11*c^11*d + 338
*a^2*b^10*c^10*d^2 - 2316*a^3*b^9*c^9*d^3 + 10015*a^4*b^8*c^8*d^4 - 28856*a^5*b^7*c^7*d^5 + 57148*a^6*b^6*c^6*
d^6 - 78968*a^7*b^5*c^5*d^7 + 76111*a^8*b^4*c^4*d^8 - 50220*a^9*b^3*c^3*d^9 + 21650*a^10*b^2*c^2*d^10 - 5500*a
^11*b*c*d^11 + 625*a^12*d^12)/(a*b^19))^(1/4)*log(-27*a*b^14*(-(b^12*c^12 - 28*a*b^11*c^11*d + 338*a^2*b^10*c^
10*d^2 - 2316*a^3*b^9*c^9*d^3 + 10015*a^4*b^8*c^8*d^4 - 28856*a^5*b^7*c^7*d^5 + 57148*a^6*b^6*c^6*d^6 - 78968*
a^7*b^5*c^5*d^7 + 76111*a^8*b^4*c^4*d^8 - 50220*a^9*b^3*c^3*d^9 + 21650*a^10*b^2*c^2*d^10 - 5500*a^11*b*c*d^11
+ 625*a^12*d^12)/(a*b^19))^(3/4) - 27*(b^9*c^9 - 21*a*b^8*c^8*d + 180*a^2*b^7*c^7*d^2 - 820*a^3*b^6*c^6*d^3 +
2190*a^4*b^5*c^5*d^4 - 3606*a^5*b^4*c^4*d^5 + 3716*a^6*b^3*c^3*d^6 - 2340*a^7*b^2*c^2*d^7 + 825*a^8*b*c*d^8 -
125*a^9*d^9)*sqrt(x)) - 4*(28*b^3*d^3*x^7 + 12*(11*b^3*c*d^2 - 5*a*b^2*d^3)*x^5 + 44*(7*b^3*c^2*d - 11*a*b^2*
c*d^2 + 5*a^2*b*d^3)*x^3 - 77*(b^3*c^3 - 7*a*b^2*c^2*d + 11*a^2*b*c*d^2 - 5*a^3*d^3)*x)*sqrt(x))/(b^5*x^2 + a*
b^4)

Sympy [F(-1)]

Timed out. \[ \int \frac {x^{5/2} \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 337, normalized size of antiderivative = 0.90 \[ \int \frac {x^{5/2} \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=-\frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{\frac {3}{2}}}{2 \, {\left (b^{5} x^{2} + a b^{4}\right )}} + \frac {3 \, {\left (b^{3} c^{3} - 7 \, a b^{2} c^{2} d + 11 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{16 \, b^{4}} + \frac {2 \, {\left (7 \, b^{2} d^{3} x^{\frac {11}{2}} + 11 \, {\left (3 \, b^{2} c d^{2} - 2 \, a b d^{3}\right )} x^{\frac {7}{2}} + 77 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x^{\frac {3}{2}}\right )}}{77 \, b^{4}} \]

[In]

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

[Out]

-1/2*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*x^(3/2)/(b^5*x^2 + a*b^4) + 3/16*(b^3*c^3 - 7*a*b^2*c
^2*d + 11*a^2*b*c*d^2 - 5*a^3*d^3)*(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*sqrt(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)*sqrt(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 + sqrt(a))/(a^(1/4)*b^(3/4)))/b^4 + 2/77*(7*b^2*d^3*x^(11/2) + 11*(3*b^2*c*d^2 - 2*a*b*d^3)*x^(7/2) + 77*
(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*x^(3/2))/b^4

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 552, normalized size of antiderivative = 1.48 \[ \int \frac {x^{5/2} \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=-\frac {b^{3} c^{3} x^{\frac {3}{2}} - 3 \, a b^{2} c^{2} d x^{\frac {3}{2}} + 3 \, a^{2} b c d^{2} x^{\frac {3}{2}} - a^{3} d^{3} x^{\frac {3}{2}}}{2 \, {\left (b x^{2} + a\right )} b^{4}} + \frac {3 \, \sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} - 7 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d + 11 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{2} b c d^{2} - 5 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \, a b^{7}} + \frac {3 \, \sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} - 7 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d + 11 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{2} b c d^{2} - 5 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{3} d^{3}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \, a b^{7}} - \frac {3 \, \sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} - 7 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d + 11 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{2} b c d^{2} - 5 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{3} d^{3}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{16 \, a b^{7}} + \frac {3 \, \sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} - 7 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d + 11 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{2} b c d^{2} - 5 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{3} d^{3}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{16 \, a b^{7}} + \frac {2 \, {\left (7 \, b^{20} d^{3} x^{\frac {11}{2}} + 33 \, b^{20} c d^{2} x^{\frac {7}{2}} - 22 \, a b^{19} d^{3} x^{\frac {7}{2}} + 77 \, b^{20} c^{2} d x^{\frac {3}{2}} - 154 \, a b^{19} c d^{2} x^{\frac {3}{2}} + 77 \, a^{2} b^{18} d^{3} x^{\frac {3}{2}}\right )}}{77 \, b^{22}} \]

[In]

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

[Out]

-1/2*(b^3*c^3*x^(3/2) - 3*a*b^2*c^2*d*x^(3/2) + 3*a^2*b*c*d^2*x^(3/2) - a^3*d^3*x^(3/2))/((b*x^2 + a)*b^4) + 3
/8*sqrt(2)*((a*b^3)^(3/4)*b^3*c^3 - 7*(a*b^3)^(3/4)*a*b^2*c^2*d + 11*(a*b^3)^(3/4)*a^2*b*c*d^2 - 5*(a*b^3)^(3/
4)*a^3*d^3)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/(a*b^7) + 3/8*sqrt(2)*((a*b^3)^(
3/4)*b^3*c^3 - 7*(a*b^3)^(3/4)*a*b^2*c^2*d + 11*(a*b^3)^(3/4)*a^2*b*c*d^2 - 5*(a*b^3)^(3/4)*a^3*d^3)*arctan(-1
/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(a*b^7) - 3/16*sqrt(2)*((a*b^3)^(3/4)*b^3*c^3 - 7*(a
*b^3)^(3/4)*a*b^2*c^2*d + 11*(a*b^3)^(3/4)*a^2*b*c*d^2 - 5*(a*b^3)^(3/4)*a^3*d^3)*log(sqrt(2)*sqrt(x)*(a/b)^(1
/4) + x + sqrt(a/b))/(a*b^7) + 3/16*sqrt(2)*((a*b^3)^(3/4)*b^3*c^3 - 7*(a*b^3)^(3/4)*a*b^2*c^2*d + 11*(a*b^3)^
(3/4)*a^2*b*c*d^2 - 5*(a*b^3)^(3/4)*a^3*d^3)*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a*b^7) + 2/77*
(7*b^20*d^3*x^(11/2) + 33*b^20*c*d^2*x^(7/2) - 22*a*b^19*d^3*x^(7/2) + 77*b^20*c^2*d*x^(3/2) - 154*a*b^19*c*d^
2*x^(3/2) + 77*a^2*b^18*d^3*x^(3/2))/b^22

Mupad [B] (verification not implemented)

Time = 5.24 (sec) , antiderivative size = 681, normalized size of antiderivative = 1.82 \[ \int \frac {x^{5/2} \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=x^{3/2}\,\left (\frac {2\,c^2\,d}{b^2}+\frac {2\,a\,\left (\frac {4\,a\,d^3}{b^3}-\frac {6\,c\,d^2}{b^2}\right )}{3\,b}-\frac {2\,a^2\,d^3}{3\,b^4}\right )-x^{7/2}\,\left (\frac {4\,a\,d^3}{7\,b^3}-\frac {6\,c\,d^2}{7\,b^2}\right )+\frac {2\,d^3\,x^{11/2}}{11\,b^2}+\frac {x^{3/2}\,\left (\frac {a^3\,d^3}{2}-\frac {3\,a^2\,b\,c\,d^2}{2}+\frac {3\,a\,b^2\,c^2\,d}{2}-\frac {b^3\,c^3}{2}\right )}{b^5\,x^2+a\,b^4}-\frac {3\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {x}\,{\left (a\,d-b\,c\right )}^2\,\left (5\,a\,d-b\,c\right )\,\left (25\,a^7\,d^6-110\,a^6\,b\,c\,d^5+191\,a^5\,b^2\,c^2\,d^4-164\,a^4\,b^3\,c^3\,d^3+71\,a^3\,b^4\,c^4\,d^2-14\,a^2\,b^5\,c^5\,d+a\,b^6\,c^6\right )}{{\left (-a\right )}^{1/4}\,\left (125\,a^{10}\,d^9-825\,a^9\,b\,c\,d^8+2340\,a^8\,b^2\,c^2\,d^7-3716\,a^7\,b^3\,c^3\,d^6+3606\,a^6\,b^4\,c^4\,d^5-2190\,a^5\,b^5\,c^5\,d^4+820\,a^4\,b^6\,c^6\,d^3-180\,a^3\,b^7\,c^7\,d^2+21\,a^2\,b^8\,c^8\,d-a\,b^9\,c^9\right )}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (5\,a\,d-b\,c\right )}{4\,{\left (-a\right )}^{1/4}\,b^{19/4}}-\frac {\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {x}\,{\left (a\,d-b\,c\right )}^2\,\left (5\,a\,d-b\,c\right )\,\left (25\,a^7\,d^6-110\,a^6\,b\,c\,d^5+191\,a^5\,b^2\,c^2\,d^4-164\,a^4\,b^3\,c^3\,d^3+71\,a^3\,b^4\,c^4\,d^2-14\,a^2\,b^5\,c^5\,d+a\,b^6\,c^6\right )\,1{}\mathrm {i}}{{\left (-a\right )}^{1/4}\,\left (125\,a^{10}\,d^9-825\,a^9\,b\,c\,d^8+2340\,a^8\,b^2\,c^2\,d^7-3716\,a^7\,b^3\,c^3\,d^6+3606\,a^6\,b^4\,c^4\,d^5-2190\,a^5\,b^5\,c^5\,d^4+820\,a^4\,b^6\,c^6\,d^3-180\,a^3\,b^7\,c^7\,d^2+21\,a^2\,b^8\,c^8\,d-a\,b^9\,c^9\right )}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (5\,a\,d-b\,c\right )\,3{}\mathrm {i}}{4\,{\left (-a\right )}^{1/4}\,b^{19/4}} \]

[In]

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

[Out]

x^(3/2)*((2*c^2*d)/b^2 + (2*a*((4*a*d^3)/b^3 - (6*c*d^2)/b^2))/(3*b) - (2*a^2*d^3)/(3*b^4)) - x^(7/2)*((4*a*d^
3)/(7*b^3) - (6*c*d^2)/(7*b^2)) + (2*d^3*x^(11/2))/(11*b^2) + (x^(3/2)*((a^3*d^3)/2 - (b^3*c^3)/2 + (3*a*b^2*c
^2*d)/2 - (3*a^2*b*c*d^2)/2))/(a*b^4 + b^5*x^2) - (3*atan((b^(1/4)*x^(1/2)*(a*d - b*c)^2*(5*a*d - b*c)*(25*a^7
*d^6 + a*b^6*c^6 - 14*a^2*b^5*c^5*d + 71*a^3*b^4*c^4*d^2 - 164*a^4*b^3*c^3*d^3 + 191*a^5*b^2*c^2*d^4 - 110*a^6
*b*c*d^5))/((-a)^(1/4)*(125*a^10*d^9 - a*b^9*c^9 + 21*a^2*b^8*c^8*d - 180*a^3*b^7*c^7*d^2 + 820*a^4*b^6*c^6*d^
3 - 2190*a^5*b^5*c^5*d^4 + 3606*a^6*b^4*c^4*d^5 - 3716*a^7*b^3*c^3*d^6 + 2340*a^8*b^2*c^2*d^7 - 825*a^9*b*c*d^
8)))*(a*d - b*c)^2*(5*a*d - b*c))/(4*(-a)^(1/4)*b^(19/4)) - (atan((b^(1/4)*x^(1/2)*(a*d - b*c)^2*(5*a*d - b*c)
*(25*a^7*d^6 + a*b^6*c^6 - 14*a^2*b^5*c^5*d + 71*a^3*b^4*c^4*d^2 - 164*a^4*b^3*c^3*d^3 + 191*a^5*b^2*c^2*d^4 -
110*a^6*b*c*d^5)*1i)/((-a)^(1/4)*(125*a^10*d^9 - a*b^9*c^9 + 21*a^2*b^8*c^8*d - 180*a^3*b^7*c^7*d^2 + 820*a^4
*b^6*c^6*d^3 - 2190*a^5*b^5*c^5*d^4 + 3606*a^6*b^4*c^4*d^5 - 3716*a^7*b^3*c^3*d^6 + 2340*a^8*b^2*c^2*d^7 - 825
*a^9*b*c*d^8)))*(a*d - b*c)^2*(5*a*d - b*c)*3i)/(4*(-a)^(1/4)*b^(19/4))