\(\int \frac {x^{3/2} (c+d x^2)^3}{(a+b x^2)^2} \, dx\) [454]

Optimal result

Integrand size = 24, antiderivative size = 386 \[ \int \frac {x^{3/2} \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\frac {d \left (497 b^2 c^2-1098 a b c d+585 a^2 d^2\right ) \sqrt {x}}{90 b^4}+\frac {d (113 b c-117 a d) \sqrt {x} \left (c+d x^2\right )}{90 b^3}+\frac {13 d \sqrt {x} \left (c+d x^2\right )^2}{18 b^2}-\frac {\sqrt {x} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}-\frac {(b c-13 a d) (b c-a d)^2 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{3/4} b^{17/4}}+\frac {(b c-13 a d) (b c-a d)^2 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{3/4} b^{17/4}}-\frac {(b c-13 a d) (b c-a d)^2 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{3/4} b^{17/4}}+\frac {(b c-13 a d) (b c-a d)^2 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{3/4} b^{17/4}} \]

[Out]

-1/8*(-13*a*d+b*c)*(-a*d+b*c)^2*arctan(1-b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(3/4)/b^(17/4)*2^(1/2)+1/8*(-13*a*
d+b*c)*(-a*d+b*c)^2*arctan(1+b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(3/4)/b^(17/4)*2^(1/2)-1/16*(-13*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^(3/4)/b^(17/4)*2^(1/2)+1/16*(-13*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^(3/4)/b^(17/4)*2^(1/2)+1/90*d*(585*a^2*d^2-1
098*a*b*c*d+497*b^2*c^2)*x^(1/2)/b^4+1/90*d*(-117*a*d+113*b*c)*(d*x^2+c)*x^(1/2)/b^3+13/18*d*(d*x^2+c)^2*x^(1/
2)/b^2-1/2*(d*x^2+c)^3*x^(1/2)/b/(b*x^2+a)

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 386, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {477, 478, 542, 396, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {x^{3/2} \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ) (b c-13 a d) (b c-a d)^2}{4 \sqrt {2} a^{3/4} b^{17/4}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right ) (b c-13 a d) (b c-a d)^2}{4 \sqrt {2} a^{3/4} b^{17/4}}-\frac {(b c-13 a d) (b c-a d)^2 \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{3/4} b^{17/4}}+\frac {(b c-13 a d) (b c-a d)^2 \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{3/4} b^{17/4}}+\frac {d \sqrt {x} \left (585 a^2 d^2-1098 a b c d+497 b^2 c^2\right )}{90 b^4}+\frac {d \sqrt {x} \left (c+d x^2\right ) (113 b c-117 a d)}{90 b^3}-\frac {\sqrt {x} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac {13 d \sqrt {x} \left (c+d x^2\right )^2}{18 b^2} \]

[In]

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

[Out]

(d*(497*b^2*c^2 - 1098*a*b*c*d + 585*a^2*d^2)*Sqrt[x])/(90*b^4) + (d*(113*b*c - 117*a*d)*Sqrt[x]*(c + d*x^2))/
(90*b^3) + (13*d*Sqrt[x]*(c + d*x^2)^2)/(18*b^2) - (Sqrt[x]*(c + d*x^2)^3)/(2*b*(a + b*x^2)) - ((b*c - 13*a*d)
*(b*c - a*d)^2*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(3/4)*b^(17/4)) + ((b*c - 13*a*d)*(
b*c - a*d)^2*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(3/4)*b^(17/4)) - ((b*c - 13*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^(3/4)*b^(17/4)) + ((b*c -
13*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^(3/4)*b^(17/4))

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 217

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[
{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b
]]))

Rule 396

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[d*x*((a + b*x^n)^(p + 1)/(b*(n*(
p + 1) + 1))), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]

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 542

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[
f*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/(b*(n*(p + q + 1) + 1))), x] + Dist[1/(b*(n*(p + q + 1) + 1)), Int[(a +
b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*(b*e - a*f + b*e*n*(p + q + 1)) + (d*(b*e - a*f) + f*n*q*(b*c - a*d) + b*
d*e*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && GtQ[q, 0] && NeQ[n*(p + q + 1) + 1
, 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^4 \left (c+d x^4\right )^3}{\left (a+b x^4\right )^2} \, dx,x,\sqrt {x}\right ) \\ & = -\frac {\sqrt {x} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac {\text {Subst}\left (\int \frac {\left (c+d x^4\right )^2 \left (c+13 d x^4\right )}{a+b x^4} \, dx,x,\sqrt {x}\right )}{2 b} \\ & = \frac {13 d \sqrt {x} \left (c+d x^2\right )^2}{18 b^2}-\frac {\sqrt {x} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac {\text {Subst}\left (\int \frac {\left (c+d x^4\right ) \left (c (9 b c-13 a d)+d (113 b c-117 a d) x^4\right )}{a+b x^4} \, dx,x,\sqrt {x}\right )}{18 b^2} \\ & = \frac {d (113 b c-117 a d) \sqrt {x} \left (c+d x^2\right )}{90 b^3}+\frac {13 d \sqrt {x} \left (c+d x^2\right )^2}{18 b^2}-\frac {\sqrt {x} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac {\text {Subst}\left (\int \frac {c \left (45 b^2 c^2-178 a b c d+117 a^2 d^2\right )+d \left (497 b^2 c^2-1098 a b c d+585 a^2 d^2\right ) x^4}{a+b x^4} \, dx,x,\sqrt {x}\right )}{90 b^3} \\ & = \frac {d \left (497 b^2 c^2-1098 a b c d+585 a^2 d^2\right ) \sqrt {x}}{90 b^4}+\frac {d (113 b c-117 a d) \sqrt {x} \left (c+d x^2\right )}{90 b^3}+\frac {13 d \sqrt {x} \left (c+d x^2\right )^2}{18 b^2}-\frac {\sqrt {x} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac {\left ((b c-13 a d) (b c-a d)^2\right ) \text {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,\sqrt {x}\right )}{2 b^4} \\ & = \frac {d \left (497 b^2 c^2-1098 a b c d+585 a^2 d^2\right ) \sqrt {x}}{90 b^4}+\frac {d (113 b c-117 a d) \sqrt {x} \left (c+d x^2\right )}{90 b^3}+\frac {13 d \sqrt {x} \left (c+d x^2\right )^2}{18 b^2}-\frac {\sqrt {x} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac {\left ((b c-13 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 \sqrt {a} b^4}+\frac {\left ((b c-13 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 \sqrt {a} b^4} \\ & = \frac {d \left (497 b^2 c^2-1098 a b c d+585 a^2 d^2\right ) \sqrt {x}}{90 b^4}+\frac {d (113 b c-117 a d) \sqrt {x} \left (c+d x^2\right )}{90 b^3}+\frac {13 d \sqrt {x} \left (c+d x^2\right )^2}{18 b^2}-\frac {\sqrt {x} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac {\left ((b c-13 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 \sqrt {a} b^{9/2}}+\frac {\left ((b c-13 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 \sqrt {a} b^{9/2}}-\frac {\left ((b c-13 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} a^{3/4} b^{17/4}}-\frac {\left ((b c-13 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} a^{3/4} b^{17/4}} \\ & = \frac {d \left (497 b^2 c^2-1098 a b c d+585 a^2 d^2\right ) \sqrt {x}}{90 b^4}+\frac {d (113 b c-117 a d) \sqrt {x} \left (c+d x^2\right )}{90 b^3}+\frac {13 d \sqrt {x} \left (c+d x^2\right )^2}{18 b^2}-\frac {\sqrt {x} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}-\frac {(b c-13 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} a^{3/4} b^{17/4}}+\frac {(b c-13 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} a^{3/4} b^{17/4}}+\frac {\left ((b c-13 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} a^{3/4} b^{17/4}}-\frac {\left ((b c-13 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} a^{3/4} b^{17/4}} \\ & = \frac {d \left (497 b^2 c^2-1098 a b c d+585 a^2 d^2\right ) \sqrt {x}}{90 b^4}+\frac {d (113 b c-117 a d) \sqrt {x} \left (c+d x^2\right )}{90 b^3}+\frac {13 d \sqrt {x} \left (c+d x^2\right )^2}{18 b^2}-\frac {\sqrt {x} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}-\frac {(b c-13 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} a^{3/4} b^{17/4}}+\frac {(b c-13 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} a^{3/4} b^{17/4}}-\frac {(b c-13 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} a^{3/4} b^{17/4}}+\frac {(b c-13 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} a^{3/4} b^{17/4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.50 (sec) , antiderivative size = 256, normalized size of antiderivative = 0.66 \[ \int \frac {x^{3/2} \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\frac {\frac {4 \sqrt [4]{b} \sqrt {x} \left (585 a^3 d^3+9 a^2 b d^2 \left (-135 c+52 d x^2\right )+a b^2 d \left (675 c^2-972 c d x^2-52 d^2 x^4\right )+b^3 \left (-45 c^3+540 c^2 d x^2+108 c d^2 x^4+20 d^3 x^6\right )\right )}{a+b x^2}+\frac {45 \sqrt {2} (b c-a d)^2 (-b c+13 a d) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{a^{3/4}}+\frac {45 \sqrt {2} (b c-13 a d) (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{a^{3/4}}}{360 b^{17/4}} \]

[In]

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

[Out]

((4*b^(1/4)*Sqrt[x]*(585*a^3*d^3 + 9*a^2*b*d^2*(-135*c + 52*d*x^2) + a*b^2*d*(675*c^2 - 972*c*d*x^2 - 52*d^2*x
^4) + b^3*(-45*c^3 + 540*c^2*d*x^2 + 108*c*d^2*x^4 + 20*d^3*x^6)))/(a + b*x^2) + (45*Sqrt[2]*(b*c - a*d)^2*(-(
b*c) + 13*a*d)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/a^(3/4) + (45*Sqrt[2]*(b*c - 1
3*a*d)*(b*c - a*d)^2*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/a^(3/4))/(360*b^(17/4))

Maple [A] (verified)

Time = 2.78 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.60

method	result	size
risch	\(\frac {2 d \left (5 b^{2} d^{2} x^{4}-18 x^{2} a b \,d^{2}+27 x^{2} b^{2} c d +135 a^{2} d^{2}-270 a b c d +135 b^{2} c^{2}\right ) \sqrt {x}}{45 b^{4}}-\frac {\left (2 a^{2} d^{2}-4 a b c d +2 b^{2} c^{2}\right ) \left (\frac {\left (-\frac {a d}{4}+\frac {b c}{4}\right ) \sqrt {x}}{b \,x^{2}+a}+\frac {\left (13 a d -b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a}\right )}{b^{4}}\)	\(230\)
derivativedivides	\(\frac {2 d \left (\frac {b^{2} d^{2} x^{\frac {9}{2}}}{9}-\frac {2 a b \,d^{2} x^{\frac {5}{2}}}{5}+\frac {3 b^{2} c d \,x^{\frac {5}{2}}}{5}+3 a^{2} d^{2} \sqrt {x}-6 a b c d \sqrt {x}+3 b^{2} c^{2} \sqrt {x}\right )}{b^{4}}-\frac {2 \left (\frac {\left (-\frac {1}{4} a^{3} d^{3}+\frac {3}{4} a^{2} b c \,d^{2}-\frac {3}{4} a \,b^{2} c^{2} d +\frac {1}{4} b^{3} c^{3}\right ) \sqrt {x}}{b \,x^{2}+a}+\frac {\left (13 a^{3} d^{3}-27 a^{2} b c \,d^{2}+15 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a}\right )}{b^{4}}\)	\(269\)
default	\(\frac {2 d \left (\frac {b^{2} d^{2} x^{\frac {9}{2}}}{9}-\frac {2 a b \,d^{2} x^{\frac {5}{2}}}{5}+\frac {3 b^{2} c d \,x^{\frac {5}{2}}}{5}+3 a^{2} d^{2} \sqrt {x}-6 a b c d \sqrt {x}+3 b^{2} c^{2} \sqrt {x}\right )}{b^{4}}-\frac {2 \left (\frac {\left (-\frac {1}{4} a^{3} d^{3}+\frac {3}{4} a^{2} b c \,d^{2}-\frac {3}{4} a \,b^{2} c^{2} d +\frac {1}{4} b^{3} c^{3}\right ) \sqrt {x}}{b \,x^{2}+a}+\frac {\left (13 a^{3} d^{3}-27 a^{2} b c \,d^{2}+15 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a}\right )}{b^{4}}\)	\(269\)

[In]

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

[Out]

2/45*d*(5*b^2*d^2*x^4-18*a*b*d^2*x^2+27*b^2*c*d*x^2+135*a^2*d^2-270*a*b*c*d+135*b^2*c^2)*x^(1/2)/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^(1/2)/(b*x^2+a)+1/32*(13*a*d-b*c)*(a/b)^(1/4)/a*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 = 0.28 (sec) , antiderivative size = 1764, normalized size of antiderivative = 4.57 \[ \int \frac {x^{3/2} \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/360*(45*(b^5*x^2 + a*b^4)*(-(b^12*c^12 - 60*a*b^11*c^11*d + 1458*a^2*b^10*c^10*d^2 - 18412*a^3*b^9*c^9*d^3
+ 130239*a^4*b^8*c^8*d^4 - 535032*a^5*b^7*c^7*d^5 + 1365756*a^6*b^6*c^6*d^6 - 2272824*a^7*b^5*c^5*d^7 + 252020
7*a^8*b^4*c^4*d^8 - 1853644*a^9*b^3*c^3*d^9 + 871026*a^10*b^2*c^2*d^10 - 237276*a^11*b*c*d^11 + 28561*a^12*d^1
2)/(a^3*b^17))^(1/4)*log(a*b^4*(-(b^12*c^12 - 60*a*b^11*c^11*d + 1458*a^2*b^10*c^10*d^2 - 18412*a^3*b^9*c^9*d^
3 + 130239*a^4*b^8*c^8*d^4 - 535032*a^5*b^7*c^7*d^5 + 1365756*a^6*b^6*c^6*d^6 - 2272824*a^7*b^5*c^5*d^7 + 2520
207*a^8*b^4*c^4*d^8 - 1853644*a^9*b^3*c^3*d^9 + 871026*a^10*b^2*c^2*d^10 - 237276*a^11*b*c*d^11 + 28561*a^12*d
^12)/(a^3*b^17))^(1/4) - (b^3*c^3 - 15*a*b^2*c^2*d + 27*a^2*b*c*d^2 - 13*a^3*d^3)*sqrt(x)) + 45*(I*b^5*x^2 + I
*a*b^4)*(-(b^12*c^12 - 60*a*b^11*c^11*d + 1458*a^2*b^10*c^10*d^2 - 18412*a^3*b^9*c^9*d^3 + 130239*a^4*b^8*c^8*
d^4 - 535032*a^5*b^7*c^7*d^5 + 1365756*a^6*b^6*c^6*d^6 - 2272824*a^7*b^5*c^5*d^7 + 2520207*a^8*b^4*c^4*d^8 - 1
853644*a^9*b^3*c^3*d^9 + 871026*a^10*b^2*c^2*d^10 - 237276*a^11*b*c*d^11 + 28561*a^12*d^12)/(a^3*b^17))^(1/4)*
log(I*a*b^4*(-(b^12*c^12 - 60*a*b^11*c^11*d + 1458*a^2*b^10*c^10*d^2 - 18412*a^3*b^9*c^9*d^3 + 130239*a^4*b^8*
c^8*d^4 - 535032*a^5*b^7*c^7*d^5 + 1365756*a^6*b^6*c^6*d^6 - 2272824*a^7*b^5*c^5*d^7 + 2520207*a^8*b^4*c^4*d^8
- 1853644*a^9*b^3*c^3*d^9 + 871026*a^10*b^2*c^2*d^10 - 237276*a^11*b*c*d^11 + 28561*a^12*d^12)/(a^3*b^17))^(1
/4) - (b^3*c^3 - 15*a*b^2*c^2*d + 27*a^2*b*c*d^2 - 13*a^3*d^3)*sqrt(x)) + 45*(-I*b^5*x^2 - I*a*b^4)*(-(b^12*c^
12 - 60*a*b^11*c^11*d + 1458*a^2*b^10*c^10*d^2 - 18412*a^3*b^9*c^9*d^3 + 130239*a^4*b^8*c^8*d^4 - 535032*a^5*b
^7*c^7*d^5 + 1365756*a^6*b^6*c^6*d^6 - 2272824*a^7*b^5*c^5*d^7 + 2520207*a^8*b^4*c^4*d^8 - 1853644*a^9*b^3*c^3
*d^9 + 871026*a^10*b^2*c^2*d^10 - 237276*a^11*b*c*d^11 + 28561*a^12*d^12)/(a^3*b^17))^(1/4)*log(-I*a*b^4*(-(b^
12*c^12 - 60*a*b^11*c^11*d + 1458*a^2*b^10*c^10*d^2 - 18412*a^3*b^9*c^9*d^3 + 130239*a^4*b^8*c^8*d^4 - 535032*
a^5*b^7*c^7*d^5 + 1365756*a^6*b^6*c^6*d^6 - 2272824*a^7*b^5*c^5*d^7 + 2520207*a^8*b^4*c^4*d^8 - 1853644*a^9*b^
3*c^3*d^9 + 871026*a^10*b^2*c^2*d^10 - 237276*a^11*b*c*d^11 + 28561*a^12*d^12)/(a^3*b^17))^(1/4) - (b^3*c^3 -
15*a*b^2*c^2*d + 27*a^2*b*c*d^2 - 13*a^3*d^3)*sqrt(x)) - 45*(b^5*x^2 + a*b^4)*(-(b^12*c^12 - 60*a*b^11*c^11*d
+ 1458*a^2*b^10*c^10*d^2 - 18412*a^3*b^9*c^9*d^3 + 130239*a^4*b^8*c^8*d^4 - 535032*a^5*b^7*c^7*d^5 + 1365756*a
^6*b^6*c^6*d^6 - 2272824*a^7*b^5*c^5*d^7 + 2520207*a^8*b^4*c^4*d^8 - 1853644*a^9*b^3*c^3*d^9 + 871026*a^10*b^2
*c^2*d^10 - 237276*a^11*b*c*d^11 + 28561*a^12*d^12)/(a^3*b^17))^(1/4)*log(-a*b^4*(-(b^12*c^12 - 60*a*b^11*c^11
*d + 1458*a^2*b^10*c^10*d^2 - 18412*a^3*b^9*c^9*d^3 + 130239*a^4*b^8*c^8*d^4 - 535032*a^5*b^7*c^7*d^5 + 136575
6*a^6*b^6*c^6*d^6 - 2272824*a^7*b^5*c^5*d^7 + 2520207*a^8*b^4*c^4*d^8 - 1853644*a^9*b^3*c^3*d^9 + 871026*a^10*
b^2*c^2*d^10 - 237276*a^11*b*c*d^11 + 28561*a^12*d^12)/(a^3*b^17))^(1/4) - (b^3*c^3 - 15*a*b^2*c^2*d + 27*a^2*
b*c*d^2 - 13*a^3*d^3)*sqrt(x)) - 4*(20*b^3*d^3*x^6 - 45*b^3*c^3 + 675*a*b^2*c^2*d - 1215*a^2*b*c*d^2 + 585*a^3
*d^3 + 4*(27*b^3*c*d^2 - 13*a*b^2*d^3)*x^4 + 36*(15*b^3*c^2*d - 27*a*b^2*c*d^2 + 13*a^2*b*d^3)*x^2)*sqrt(x))/(
b^5*x^2 + a*b^4)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1833 vs. \(2 (369) = 738\).

Time = 152.06 (sec) , antiderivative size = 1833, normalized size of antiderivative = 4.75 \[ \int \frac {x^{3/2} \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Piecewise((zoo*(-2*c**3/(3*x**(3/2)) + 6*c**2*d*sqrt(x) + 6*c*d**2*x**(5/2)/5 + 2*d**3*x**(9/2)/9), Eq(a, 0) &
Eq(b, 0)), ((2*c**3*x**(5/2)/5 + 2*c**2*d*x**(9/2)/3 + 6*c*d**2*x**(13/2)/13 + 2*d**3*x**(17/2)/17)/a**2, Eq(
b, 0)), ((-2*c**3/(3*x**(3/2)) + 6*c**2*d*sqrt(x) + 6*c*d**2*x**(5/2)/5 + 2*d**3*x**(9/2)/9)/b**2, Eq(a, 0)),
(2340*a**4*d**3*sqrt(x)/(360*a**2*b**4 + 360*a*b**5*x**2) + 585*a**4*d**3*(-a/b)**(1/4)*log(sqrt(x) - (-a/b)**
(1/4))/(360*a**2*b**4 + 360*a*b**5*x**2) - 585*a**4*d**3*(-a/b)**(1/4)*log(sqrt(x) + (-a/b)**(1/4))/(360*a**2*
b**4 + 360*a*b**5*x**2) - 1170*a**4*d**3*(-a/b)**(1/4)*atan(sqrt(x)/(-a/b)**(1/4))/(360*a**2*b**4 + 360*a*b**5
*x**2) - 4860*a**3*b*c*d**2*sqrt(x)/(360*a**2*b**4 + 360*a*b**5*x**2) - 1215*a**3*b*c*d**2*(-a/b)**(1/4)*log(s
qrt(x) - (-a/b)**(1/4))/(360*a**2*b**4 + 360*a*b**5*x**2) + 1215*a**3*b*c*d**2*(-a/b)**(1/4)*log(sqrt(x) + (-a
/b)**(1/4))/(360*a**2*b**4 + 360*a*b**5*x**2) + 2430*a**3*b*c*d**2*(-a/b)**(1/4)*atan(sqrt(x)/(-a/b)**(1/4))/(
360*a**2*b**4 + 360*a*b**5*x**2) + 1872*a**3*b*d**3*x**(5/2)/(360*a**2*b**4 + 360*a*b**5*x**2) + 585*a**3*b*d*
*3*x**2*(-a/b)**(1/4)*log(sqrt(x) - (-a/b)**(1/4))/(360*a**2*b**4 + 360*a*b**5*x**2) - 585*a**3*b*d**3*x**2*(-
a/b)**(1/4)*log(sqrt(x) + (-a/b)**(1/4))/(360*a**2*b**4 + 360*a*b**5*x**2) - 1170*a**3*b*d**3*x**2*(-a/b)**(1/
4)*atan(sqrt(x)/(-a/b)**(1/4))/(360*a**2*b**4 + 360*a*b**5*x**2) + 2700*a**2*b**2*c**2*d*sqrt(x)/(360*a**2*b**
4 + 360*a*b**5*x**2) + 675*a**2*b**2*c**2*d*(-a/b)**(1/4)*log(sqrt(x) - (-a/b)**(1/4))/(360*a**2*b**4 + 360*a*
b**5*x**2) - 675*a**2*b**2*c**2*d*(-a/b)**(1/4)*log(sqrt(x) + (-a/b)**(1/4))/(360*a**2*b**4 + 360*a*b**5*x**2)
- 1350*a**2*b**2*c**2*d*(-a/b)**(1/4)*atan(sqrt(x)/(-a/b)**(1/4))/(360*a**2*b**4 + 360*a*b**5*x**2) - 3888*a*
*2*b**2*c*d**2*x**(5/2)/(360*a**2*b**4 + 360*a*b**5*x**2) - 1215*a**2*b**2*c*d**2*x**2*(-a/b)**(1/4)*log(sqrt(
x) - (-a/b)**(1/4))/(360*a**2*b**4 + 360*a*b**5*x**2) + 1215*a**2*b**2*c*d**2*x**2*(-a/b)**(1/4)*log(sqrt(x) +
(-a/b)**(1/4))/(360*a**2*b**4 + 360*a*b**5*x**2) + 2430*a**2*b**2*c*d**2*x**2*(-a/b)**(1/4)*atan(sqrt(x)/(-a/
b)**(1/4))/(360*a**2*b**4 + 360*a*b**5*x**2) - 208*a**2*b**2*d**3*x**(9/2)/(360*a**2*b**4 + 360*a*b**5*x**2) -
180*a*b**3*c**3*sqrt(x)/(360*a**2*b**4 + 360*a*b**5*x**2) - 45*a*b**3*c**3*(-a/b)**(1/4)*log(sqrt(x) - (-a/b)
**(1/4))/(360*a**2*b**4 + 360*a*b**5*x**2) + 45*a*b**3*c**3*(-a/b)**(1/4)*log(sqrt(x) + (-a/b)**(1/4))/(360*a*
*2*b**4 + 360*a*b**5*x**2) + 90*a*b**3*c**3*(-a/b)**(1/4)*atan(sqrt(x)/(-a/b)**(1/4))/(360*a**2*b**4 + 360*a*b
**5*x**2) + 2160*a*b**3*c**2*d*x**(5/2)/(360*a**2*b**4 + 360*a*b**5*x**2) + 675*a*b**3*c**2*d*x**2*(-a/b)**(1/
4)*log(sqrt(x) - (-a/b)**(1/4))/(360*a**2*b**4 + 360*a*b**5*x**2) - 675*a*b**3*c**2*d*x**2*(-a/b)**(1/4)*log(s
qrt(x) + (-a/b)**(1/4))/(360*a**2*b**4 + 360*a*b**5*x**2) - 1350*a*b**3*c**2*d*x**2*(-a/b)**(1/4)*atan(sqrt(x)
/(-a/b)**(1/4))/(360*a**2*b**4 + 360*a*b**5*x**2) + 432*a*b**3*c*d**2*x**(9/2)/(360*a**2*b**4 + 360*a*b**5*x**
2) + 80*a*b**3*d**3*x**(13/2)/(360*a**2*b**4 + 360*a*b**5*x**2) - 45*b**4*c**3*x**2*(-a/b)**(1/4)*log(sqrt(x)
- (-a/b)**(1/4))/(360*a**2*b**4 + 360*a*b**5*x**2) + 45*b**4*c**3*x**2*(-a/b)**(1/4)*log(sqrt(x) + (-a/b)**(1/
4))/(360*a**2*b**4 + 360*a*b**5*x**2) + 90*b**4*c**3*x**2*(-a/b)**(1/4)*atan(sqrt(x)/(-a/b)**(1/4))/(360*a**2*
b**4 + 360*a*b**5*x**2), True))

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 445, normalized size of antiderivative = 1.15 \[ \int \frac {x^{3/2} \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=-\frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {x}}{2 \, {\left (b^{5} x^{2} + a b^{4}\right )}} + \frac {2 \, {\left (5 \, b^{2} d^{3} x^{\frac {9}{2}} + 9 \, {\left (3 \, b^{2} c d^{2} - 2 \, a b d^{3}\right )} x^{\frac {5}{2}} + 135 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} \sqrt {x}\right )}}{45 \, b^{4}} + \frac {\frac {2 \, \sqrt {2} {\left (b^{3} c^{3} - 15 \, a b^{2} c^{2} d + 27 \, a^{2} b c d^{2} - 13 \, a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} {\left (b^{3} c^{3} - 15 \, a b^{2} c^{2} d + 27 \, a^{2} b c d^{2} - 13 \, a^{3} d^{3}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} {\left (b^{3} c^{3} - 15 \, a b^{2} c^{2} d + 27 \, a^{2} b c d^{2} - 13 \, a^{3} d^{3}\right )} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (b^{3} c^{3} - 15 \, a b^{2} c^{2} d + 27 \, a^{2} b c d^{2} - 13 \, a^{3} d^{3}\right )} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}}}{16 \, b^{4}} \]

[In]

integrate(x^(3/2)*(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)*sqrt(x)/(b^5*x^2 + a*b^4) + 2/45*(5*b^2*d^3*x^(9/2) +
9*(3*b^2*c*d^2 - 2*a*b*d^3)*x^(5/2) + 135*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*sqrt(x))/b^4 + 1/16*(2*sqrt(2)*
(b^3*c^3 - 15*a*b^2*c^2*d + 27*a^2*b*c*d^2 - 13*a^3*d^3)*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) + 2*sqrt(
b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))) + 2*sqrt(2)*(b^3*c^3 - 15*a*b^2*c^2*d + 27*
a^2*b*c*d^2 - 13*a^3*d^3)*arctan(-1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) - 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(
b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))) + sqrt(2)*(b^3*c^3 - 15*a*b^2*c^2*d + 27*a^2*b*c*d^2 - 13*a^3*d^3)*log(sq
rt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(3/4)*b^(1/4)) - sqrt(2)*(b^3*c^3 - 15*a*b^2*c^2*d + 2
7*a^2*b*c*d^2 - 13*a^3*d^3)*log(-sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(3/4)*b^(1/4)))/b^4

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 552, normalized size of antiderivative = 1.43 \[ \int \frac {x^{3/2} \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 15 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 27 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - 13 \, \left (a b^{3}\right )^{\frac {1}{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^{5}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 15 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 27 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - 13 \, \left (a b^{3}\right )^{\frac {1}{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^{5}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 15 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 27 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - 13 \, \left (a b^{3}\right )^{\frac {1}{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^{5}} - \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 15 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 27 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - 13 \, \left (a b^{3}\right )^{\frac {1}{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^{5}} - \frac {b^{3} c^{3} \sqrt {x} - 3 \, a b^{2} c^{2} d \sqrt {x} + 3 \, a^{2} b c d^{2} \sqrt {x} - a^{3} d^{3} \sqrt {x}}{2 \, {\left (b x^{2} + a\right )} b^{4}} + \frac {2 \, {\left (5 \, b^{16} d^{3} x^{\frac {9}{2}} + 27 \, b^{16} c d^{2} x^{\frac {5}{2}} - 18 \, a b^{15} d^{3} x^{\frac {5}{2}} + 135 \, b^{16} c^{2} d \sqrt {x} - 270 \, a b^{15} c d^{2} \sqrt {x} + 135 \, a^{2} b^{14} d^{3} \sqrt {x}\right )}}{45 \, b^{18}} \]

[In]

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

[Out]

1/8*sqrt(2)*((a*b^3)^(1/4)*b^3*c^3 - 15*(a*b^3)^(1/4)*a*b^2*c^2*d + 27*(a*b^3)^(1/4)*a^2*b*c*d^2 - 13*(a*b^3)^
(1/4)*a^3*d^3)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/(a*b^5) + 1/8*sqrt(2)*((a*b^3
)^(1/4)*b^3*c^3 - 15*(a*b^3)^(1/4)*a*b^2*c^2*d + 27*(a*b^3)^(1/4)*a^2*b*c*d^2 - 13*(a*b^3)^(1/4)*a^3*d^3)*arct
an(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(a*b^5) + 1/16*sqrt(2)*((a*b^3)^(1/4)*b^3*c^3 -
15*(a*b^3)^(1/4)*a*b^2*c^2*d + 27*(a*b^3)^(1/4)*a^2*b*c*d^2 - 13*(a*b^3)^(1/4)*a^3*d^3)*log(sqrt(2)*sqrt(x)*(
a/b)^(1/4) + x + sqrt(a/b))/(a*b^5) - 1/16*sqrt(2)*((a*b^3)^(1/4)*b^3*c^3 - 15*(a*b^3)^(1/4)*a*b^2*c^2*d + 27*
(a*b^3)^(1/4)*a^2*b*c*d^2 - 13*(a*b^3)^(1/4)*a^3*d^3)*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a*b^5
) - 1/2*(b^3*c^3*sqrt(x) - 3*a*b^2*c^2*d*sqrt(x) + 3*a^2*b*c*d^2*sqrt(x) - a^3*d^3*sqrt(x))/((b*x^2 + a)*b^4)
+ 2/45*(5*b^16*d^3*x^(9/2) + 27*b^16*c*d^2*x^(5/2) - 18*a*b^15*d^3*x^(5/2) + 135*b^16*c^2*d*sqrt(x) - 270*a*b^
15*c*d^2*sqrt(x) + 135*a^2*b^14*d^3*sqrt(x))/b^18

Mupad [B] (verification not implemented)

Time = 5.43 (sec) , antiderivative size = 1691, normalized size of antiderivative = 4.38 \[ \int \frac {x^{3/2} \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

x^(1/2)*((6*c^2*d)/b^2 + (2*a*((4*a*d^3)/b^3 - (6*c*d^2)/b^2))/b - (2*a^2*d^3)/b^4) - x^(5/2)*((4*a*d^3)/(5*b^
3) - (6*c*d^2)/(5*b^2)) + (2*d^3*x^(9/2))/(9*b^2) + (x^(1/2)*((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) + (atan(((((x^(1/2)*(169*a^6*d^6 + b^6*c^6 + 279*a^2*b^4*c^4*d^2 - 836*a
^3*b^3*c^3*d^3 + 1119*a^4*b^2*c^2*d^4 - 30*a*b^5*c^5*d - 702*a^5*b*c*d^5))/b^5 + ((a*d - b*c)^2*(13*a*d - b*c)
*(13*a^4*d^3 - a*b^3*c^3 + 15*a^2*b^2*c^2*d - 27*a^3*b*c*d^2))/((-a)^(3/4)*b^(21/4)))*(a*d - b*c)^2*(13*a*d -
b*c)*1i)/(8*(-a)^(3/4)*b^(17/4)) + (((x^(1/2)*(169*a^6*d^6 + b^6*c^6 + 279*a^2*b^4*c^4*d^2 - 836*a^3*b^3*c^3*d
^3 + 1119*a^4*b^2*c^2*d^4 - 30*a*b^5*c^5*d - 702*a^5*b*c*d^5))/b^5 - ((a*d - b*c)^2*(13*a*d - b*c)*(13*a^4*d^3
- a*b^3*c^3 + 15*a^2*b^2*c^2*d - 27*a^3*b*c*d^2))/((-a)^(3/4)*b^(21/4)))*(a*d - b*c)^2*(13*a*d - b*c)*1i)/(8*
(-a)^(3/4)*b^(17/4)))/((((x^(1/2)*(169*a^6*d^6 + b^6*c^6 + 279*a^2*b^4*c^4*d^2 - 836*a^3*b^3*c^3*d^3 + 1119*a^
4*b^2*c^2*d^4 - 30*a*b^5*c^5*d - 702*a^5*b*c*d^5))/b^5 + ((a*d - b*c)^2*(13*a*d - b*c)*(13*a^4*d^3 - a*b^3*c^3
+ 15*a^2*b^2*c^2*d - 27*a^3*b*c*d^2))/((-a)^(3/4)*b^(21/4)))*(a*d - b*c)^2*(13*a*d - b*c))/(8*(-a)^(3/4)*b^(1
7/4)) - (((x^(1/2)*(169*a^6*d^6 + b^6*c^6 + 279*a^2*b^4*c^4*d^2 - 836*a^3*b^3*c^3*d^3 + 1119*a^4*b^2*c^2*d^4 -
30*a*b^5*c^5*d - 702*a^5*b*c*d^5))/b^5 - ((a*d - b*c)^2*(13*a*d - b*c)*(13*a^4*d^3 - a*b^3*c^3 + 15*a^2*b^2*c
^2*d - 27*a^3*b*c*d^2))/((-a)^(3/4)*b^(21/4)))*(a*d - b*c)^2*(13*a*d - b*c))/(8*(-a)^(3/4)*b^(17/4))))*(a*d -
b*c)^2*(13*a*d - b*c)*1i)/(4*(-a)^(3/4)*b^(17/4)) - (atan(((((x^(1/2)*(169*a^6*d^6 + b^6*c^6 + 279*a^2*b^4*c^4
*d^2 - 836*a^3*b^3*c^3*d^3 + 1119*a^4*b^2*c^2*d^4 - 30*a*b^5*c^5*d - 702*a^5*b*c*d^5))/b^5 - ((a*d - b*c)^2*(1
3*a*d - b*c)*(13*a^4*d^3 - a*b^3*c^3 + 15*a^2*b^2*c^2*d - 27*a^3*b*c*d^2)*1i)/((-a)^(3/4)*b^(21/4)))*(a*d - b*
c)^2*(13*a*d - b*c))/(8*(-a)^(3/4)*b^(17/4)) + (((x^(1/2)*(169*a^6*d^6 + b^6*c^6 + 279*a^2*b^4*c^4*d^2 - 836*a
^3*b^3*c^3*d^3 + 1119*a^4*b^2*c^2*d^4 - 30*a*b^5*c^5*d - 702*a^5*b*c*d^5))/b^5 + ((a*d - b*c)^2*(13*a*d - b*c)
*(13*a^4*d^3 - a*b^3*c^3 + 15*a^2*b^2*c^2*d - 27*a^3*b*c*d^2)*1i)/((-a)^(3/4)*b^(21/4)))*(a*d - b*c)^2*(13*a*d
- b*c))/(8*(-a)^(3/4)*b^(17/4)))/((((x^(1/2)*(169*a^6*d^6 + b^6*c^6 + 279*a^2*b^4*c^4*d^2 - 836*a^3*b^3*c^3*d
^3 + 1119*a^4*b^2*c^2*d^4 - 30*a*b^5*c^5*d - 702*a^5*b*c*d^5))/b^5 - ((a*d - b*c)^2*(13*a*d - b*c)*(13*a^4*d^3
- a*b^3*c^3 + 15*a^2*b^2*c^2*d - 27*a^3*b*c*d^2)*1i)/((-a)^(3/4)*b^(21/4)))*(a*d - b*c)^2*(13*a*d - b*c)*1i)/
(8*(-a)^(3/4)*b^(17/4)) - (((x^(1/2)*(169*a^6*d^6 + b^6*c^6 + 279*a^2*b^4*c^4*d^2 - 836*a^3*b^3*c^3*d^3 + 1119
*a^4*b^2*c^2*d^4 - 30*a*b^5*c^5*d - 702*a^5*b*c*d^5))/b^5 + ((a*d - b*c)^2*(13*a*d - b*c)*(13*a^4*d^3 - a*b^3*
c^3 + 15*a^2*b^2*c^2*d - 27*a^3*b*c*d^2)*1i)/((-a)^(3/4)*b^(21/4)))*(a*d - b*c)^2*(13*a*d - b*c)*1i)/(8*(-a)^(
3/4)*b^(17/4))))*(a*d - b*c)^2*(13*a*d - b*c))/(4*(-a)^(3/4)*b^(17/4))