\(\int \frac {(c+d x^2)^3}{x^{5/2} (a+b x^2)^2} \, dx\) [458]

Optimal result

Integrand size = 24, antiderivative size = 367 \[ \int \frac {\left (c+d x^2\right )^3}{x^{5/2} \left (a+b x^2\right )^2} \, dx=-\frac {c^2 (7 b c-3 a d)}{6 a^2 b x^{3/2}}-\frac {d^2 (b c-5 a d) \sqrt {x}}{2 a b^2}+\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{3/2} \left (a+b x^2\right )}+\frac {(b c-a d)^2 (7 b c+5 a d) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{11/4} b^{9/4}}-\frac {(b c-a d)^2 (7 b c+5 a d) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{11/4} b^{9/4}}+\frac {(b c-a d)^2 (7 b c+5 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{11/4} b^{9/4}}-\frac {(b c-a d)^2 (7 b c+5 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{11/4} b^{9/4}} \]

[Out]

-1/6*c^2*(-3*a*d+7*b*c)/a^2/b/x^(3/2)+1/2*(-a*d+b*c)*(d*x^2+c)^2/a/b/x^(3/2)/(b*x^2+a)+1/8*(-a*d+b*c)^2*(5*a*d
+7*b*c)*arctan(1-b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(11/4)/b^(9/4)*2^(1/2)-1/8*(-a*d+b*c)^2*(5*a*d+7*b*c)*arct
an(1+b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(11/4)/b^(9/4)*2^(1/2)+1/16*(-a*d+b*c)^2*(5*a*d+7*b*c)*ln(a^(1/2)+x*b^
(1/2)-a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(11/4)/b^(9/4)*2^(1/2)-1/16*(-a*d+b*c)^2*(5*a*d+7*b*c)*ln(a^(1/2)+x*b
^(1/2)+a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(11/4)/b^(9/4)*2^(1/2)-1/2*d^2*(-5*a*d+b*c)*x^(1/2)/a/b^2

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 367, 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, 479, 584, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {\left (c+d x^2\right )^3}{x^{5/2} \left (a+b x^2\right )^2} \, dx=\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ) (b c-a d)^2 (5 a d+7 b c)}{4 \sqrt {2} a^{11/4} b^{9/4}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right ) (b c-a d)^2 (5 a d+7 b c)}{4 \sqrt {2} a^{11/4} b^{9/4}}+\frac {(b c-a d)^2 (5 a d+7 b c) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{11/4} b^{9/4}}-\frac {(b c-a d)^2 (5 a d+7 b c) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{11/4} b^{9/4}}-\frac {c^2 (7 b c-3 a d)}{6 a^2 b x^{3/2}}-\frac {d^2 \sqrt {x} (b c-5 a d)}{2 a b^2}+\frac {\left (c+d x^2\right )^2 (b c-a d)}{2 a b x^{3/2} \left (a+b x^2\right )} \]

[In]

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

[Out]

-1/6*(c^2*(7*b*c - 3*a*d))/(a^2*b*x^(3/2)) - (d^2*(b*c - 5*a*d)*Sqrt[x])/(2*a*b^2) + ((b*c - a*d)*(c + d*x^2)^
2)/(2*a*b*x^(3/2)*(a + b*x^2)) + ((b*c - a*d)^2*(7*b*c + 5*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])
/(4*Sqrt[2]*a^(11/4)*b^(9/4)) - ((b*c - a*d)^2*(7*b*c + 5*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/
(4*Sqrt[2]*a^(11/4)*b^(9/4)) + ((b*c - a*d)^2*(7*b*c + 5*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] +
Sqrt[b]*x])/(8*Sqrt[2]*a^(11/4)*b^(9/4)) - ((b*c - a*d)^2*(7*b*c + 5*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4
)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(11/4)*b^(9/4))

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 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 479

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

Rule 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 {\left (c+d x^4\right )^3}{x^4 \left (a+b x^4\right )^2} \, dx,x,\sqrt {x}\right ) \\ & = \frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{3/2} \left (a+b x^2\right )}-\frac {\text {Subst}\left (\int \frac {\left (c+d x^4\right ) \left (-c (7 b c-3 a d)+d (b c-5 a d) x^4\right )}{x^4 \left (a+b x^4\right )} \, dx,x,\sqrt {x}\right )}{2 a b} \\ & = \frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{3/2} \left (a+b x^2\right )}-\frac {\text {Subst}\left (\int \left (\frac {d^2 (b c-5 a d)}{b}+\frac {c^2 (-7 b c+3 a d)}{a x^4}+\frac {(-b c+a d)^2 (7 b c+5 a d)}{a b \left (a+b x^4\right )}\right ) \, dx,x,\sqrt {x}\right )}{2 a b} \\ & = -\frac {c^2 (7 b c-3 a d)}{6 a^2 b x^{3/2}}-\frac {d^2 (b c-5 a d) \sqrt {x}}{2 a b^2}+\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{3/2} \left (a+b x^2\right )}-\frac {\left ((b c-a d)^2 (7 b c+5 a d)\right ) \text {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,\sqrt {x}\right )}{2 a^2 b^2} \\ & = -\frac {c^2 (7 b c-3 a d)}{6 a^2 b x^{3/2}}-\frac {d^2 (b c-5 a d) \sqrt {x}}{2 a b^2}+\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{3/2} \left (a+b x^2\right )}-\frac {\left ((b c-a d)^2 (7 b c+5 a d)\right ) \text {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 a^{5/2} b^2}-\frac {\left ((b c-a d)^2 (7 b c+5 a d)\right ) \text {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 a^{5/2} b^2} \\ & = -\frac {c^2 (7 b c-3 a d)}{6 a^2 b x^{3/2}}-\frac {d^2 (b c-5 a d) \sqrt {x}}{2 a b^2}+\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{3/2} \left (a+b x^2\right )}-\frac {\left ((b c-a d)^2 (7 b c+5 a d)\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 a^{5/2} b^{5/2}}-\frac {\left ((b c-a d)^2 (7 b c+5 a d)\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 a^{5/2} b^{5/2}}+\frac {\left ((b c-a d)^2 (7 b c+5 a d)\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^{11/4} b^{9/4}}+\frac {\left ((b c-a d)^2 (7 b c+5 a d)\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^{11/4} b^{9/4}} \\ & = -\frac {c^2 (7 b c-3 a d)}{6 a^2 b x^{3/2}}-\frac {d^2 (b c-5 a d) \sqrt {x}}{2 a b^2}+\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{3/2} \left (a+b x^2\right )}+\frac {(b c-a d)^2 (7 b c+5 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{11/4} b^{9/4}}-\frac {(b c-a d)^2 (7 b c+5 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{11/4} b^{9/4}}-\frac {\left ((b c-a d)^2 (7 b c+5 a d)\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^{11/4} b^{9/4}}+\frac {\left ((b c-a d)^2 (7 b c+5 a d)\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^{11/4} b^{9/4}} \\ & = -\frac {c^2 (7 b c-3 a d)}{6 a^2 b x^{3/2}}-\frac {d^2 (b c-5 a d) \sqrt {x}}{2 a b^2}+\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{3/2} \left (a+b x^2\right )}+\frac {(b c-a d)^2 (7 b c+5 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{11/4} b^{9/4}}-\frac {(b c-a d)^2 (7 b c+5 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{11/4} b^{9/4}}+\frac {(b c-a d)^2 (7 b c+5 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{11/4} b^{9/4}}-\frac {(b c-a d)^2 (7 b c+5 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{11/4} b^{9/4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.48 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.62 \[ \int \frac {\left (c+d x^2\right )^3}{x^{5/2} \left (a+b x^2\right )^2} \, dx=\frac {\frac {4 a^{3/4} \sqrt [4]{b} \left (-7 b^3 c^3 x^2+15 a^3 d^3 x^2+3 a^2 b d^2 x^2 \left (-3 c+4 d x^2\right )+a b^2 c^2 \left (-4 c+9 d x^2\right )\right )}{x^{3/2} \left (a+b x^2\right )}+3 \sqrt {2} (b c-a d)^2 (7 b c+5 a d) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )-3 \sqrt {2} (b c-a d)^2 (7 b c+5 a d) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{24 a^{11/4} b^{9/4}} \]

[In]

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

[Out]

((4*a^(3/4)*b^(1/4)*(-7*b^3*c^3*x^2 + 15*a^3*d^3*x^2 + 3*a^2*b*d^2*x^2*(-3*c + 4*d*x^2) + a*b^2*c^2*(-4*c + 9*
d*x^2)))/(x^(3/2)*(a + b*x^2)) + 3*Sqrt[2]*(b*c - a*d)^2*(7*b*c + 5*a*d)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]
*a^(1/4)*b^(1/4)*Sqrt[x])] - 3*Sqrt[2]*(b*c - a*d)^2*(7*b*c + 5*a*d)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])
/(Sqrt[a] + Sqrt[b]*x)])/(24*a^(11/4)*b^(9/4))

Maple [A] (verified)

Time = 2.78 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.55

method	result	size
risch	\(\frac {2 a^{2} d^{3} x^{2}-\frac {2 b^{2} c^{3}}{3}}{b^{2} x^{\frac {3}{2}} a^{2}}-\frac {\left (2 a^{2} d^{2}-4 a b c d +2 b^{2} c^{2}\right ) \left (\frac {\left (-\frac {a d}{4}+\frac {b c}{4}\right ) \sqrt {x}}{b \,x^{2}+a}+\frac {\left (5 a d +7 b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a}\right )}{a^{2} b^{2}}\)	\(201\)
derivativedivides	\(\frac {2 \sqrt {x}\, d^{3}}{b^{2}}-\frac {2 c^{3}}{3 a^{2} x^{\frac {3}{2}}}-\frac {2 \left (\frac {\left (-\frac {1}{4} a^{3} d^{3}+\frac {3}{4} a^{2} b c \,d^{2}-\frac {3}{4} a \,b^{2} c^{2} d +\frac {1}{4} b^{3} c^{3}\right ) \sqrt {x}}{b \,x^{2}+a}+\frac {\left (5 a^{3} d^{3}-3 a^{2} b c \,d^{2}-9 a \,b^{2} c^{2} d +7 b^{3} c^{3}\right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a}\right )}{a^{2} b^{2}}\)	\(225\)
default	\(\frac {2 \sqrt {x}\, d^{3}}{b^{2}}-\frac {2 c^{3}}{3 a^{2} x^{\frac {3}{2}}}-\frac {2 \left (\frac {\left (-\frac {1}{4} a^{3} d^{3}+\frac {3}{4} a^{2} b c \,d^{2}-\frac {3}{4} a \,b^{2} c^{2} d +\frac {1}{4} b^{3} c^{3}\right ) \sqrt {x}}{b \,x^{2}+a}+\frac {\left (5 a^{3} d^{3}-3 a^{2} b c \,d^{2}-9 a \,b^{2} c^{2} d +7 b^{3} c^{3}\right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a}\right )}{a^{2} b^{2}}\)	\(225\)

[In]

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

[Out]

2/3*(3*a^2*d^3*x^2-b^2*c^3)/b^2/x^(3/2)/a^2-1/a^2/b^2*(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*(5*a*d+7*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 = 0.39 (sec) , antiderivative size = 1778, normalized size of antiderivative = 4.84 \[ \int \frac {\left (c+d x^2\right )^3}{x^{5/2} \left (a+b x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/24*(3*(a^2*b^3*x^4 + a^3*b^2*x^2)*(-(2401*b^12*c^12 - 12348*a*b^11*c^11*d + 19698*a^2*b^10*c^10*d^2 + 2324*
a^3*b^9*c^9*d^3 - 37665*a^4*b^8*c^8*d^4 + 27144*a^5*b^7*c^7*d^5 + 19068*a^6*b^6*c^6*d^6 - 28728*a^7*b^5*c^5*d^
7 + 1071*a^8*b^4*c^4*d^8 + 11060*a^9*b^3*c^3*d^9 - 3150*a^10*b^2*c^2*d^10 - 1500*a^11*b*c*d^11 + 625*a^12*d^12
)/(a^11*b^9))^(1/4)*log(a^3*b^2*(-(2401*b^12*c^12 - 12348*a*b^11*c^11*d + 19698*a^2*b^10*c^10*d^2 + 2324*a^3*b
^9*c^9*d^3 - 37665*a^4*b^8*c^8*d^4 + 27144*a^5*b^7*c^7*d^5 + 19068*a^6*b^6*c^6*d^6 - 28728*a^7*b^5*c^5*d^7 + 1
071*a^8*b^4*c^4*d^8 + 11060*a^9*b^3*c^3*d^9 - 3150*a^10*b^2*c^2*d^10 - 1500*a^11*b*c*d^11 + 625*a^12*d^12)/(a^
11*b^9))^(1/4) + (7*b^3*c^3 - 9*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 5*a^3*d^3)*sqrt(x)) + 3*(I*a^2*b^3*x^4 + I*a^3*b
^2*x^2)*(-(2401*b^12*c^12 - 12348*a*b^11*c^11*d + 19698*a^2*b^10*c^10*d^2 + 2324*a^3*b^9*c^9*d^3 - 37665*a^4*b
^8*c^8*d^4 + 27144*a^5*b^7*c^7*d^5 + 19068*a^6*b^6*c^6*d^6 - 28728*a^7*b^5*c^5*d^7 + 1071*a^8*b^4*c^4*d^8 + 11
060*a^9*b^3*c^3*d^9 - 3150*a^10*b^2*c^2*d^10 - 1500*a^11*b*c*d^11 + 625*a^12*d^12)/(a^11*b^9))^(1/4)*log(I*a^3
*b^2*(-(2401*b^12*c^12 - 12348*a*b^11*c^11*d + 19698*a^2*b^10*c^10*d^2 + 2324*a^3*b^9*c^9*d^3 - 37665*a^4*b^8*
c^8*d^4 + 27144*a^5*b^7*c^7*d^5 + 19068*a^6*b^6*c^6*d^6 - 28728*a^7*b^5*c^5*d^7 + 1071*a^8*b^4*c^4*d^8 + 11060
*a^9*b^3*c^3*d^9 - 3150*a^10*b^2*c^2*d^10 - 1500*a^11*b*c*d^11 + 625*a^12*d^12)/(a^11*b^9))^(1/4) + (7*b^3*c^3
- 9*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 5*a^3*d^3)*sqrt(x)) + 3*(-I*a^2*b^3*x^4 - I*a^3*b^2*x^2)*(-(2401*b^12*c^12
- 12348*a*b^11*c^11*d + 19698*a^2*b^10*c^10*d^2 + 2324*a^3*b^9*c^9*d^3 - 37665*a^4*b^8*c^8*d^4 + 27144*a^5*b^7
*c^7*d^5 + 19068*a^6*b^6*c^6*d^6 - 28728*a^7*b^5*c^5*d^7 + 1071*a^8*b^4*c^4*d^8 + 11060*a^9*b^3*c^3*d^9 - 3150
*a^10*b^2*c^2*d^10 - 1500*a^11*b*c*d^11 + 625*a^12*d^12)/(a^11*b^9))^(1/4)*log(-I*a^3*b^2*(-(2401*b^12*c^12 -
12348*a*b^11*c^11*d + 19698*a^2*b^10*c^10*d^2 + 2324*a^3*b^9*c^9*d^3 - 37665*a^4*b^8*c^8*d^4 + 27144*a^5*b^7*c
^7*d^5 + 19068*a^6*b^6*c^6*d^6 - 28728*a^7*b^5*c^5*d^7 + 1071*a^8*b^4*c^4*d^8 + 11060*a^9*b^3*c^3*d^9 - 3150*a
^10*b^2*c^2*d^10 - 1500*a^11*b*c*d^11 + 625*a^12*d^12)/(a^11*b^9))^(1/4) + (7*b^3*c^3 - 9*a*b^2*c^2*d - 3*a^2*
b*c*d^2 + 5*a^3*d^3)*sqrt(x)) - 3*(a^2*b^3*x^4 + a^3*b^2*x^2)*(-(2401*b^12*c^12 - 12348*a*b^11*c^11*d + 19698*
a^2*b^10*c^10*d^2 + 2324*a^3*b^9*c^9*d^3 - 37665*a^4*b^8*c^8*d^4 + 27144*a^5*b^7*c^7*d^5 + 19068*a^6*b^6*c^6*d
^6 - 28728*a^7*b^5*c^5*d^7 + 1071*a^8*b^4*c^4*d^8 + 11060*a^9*b^3*c^3*d^9 - 3150*a^10*b^2*c^2*d^10 - 1500*a^11
*b*c*d^11 + 625*a^12*d^12)/(a^11*b^9))^(1/4)*log(-a^3*b^2*(-(2401*b^12*c^12 - 12348*a*b^11*c^11*d + 19698*a^2*
b^10*c^10*d^2 + 2324*a^3*b^9*c^9*d^3 - 37665*a^4*b^8*c^8*d^4 + 27144*a^5*b^7*c^7*d^5 + 19068*a^6*b^6*c^6*d^6 -
28728*a^7*b^5*c^5*d^7 + 1071*a^8*b^4*c^4*d^8 + 11060*a^9*b^3*c^3*d^9 - 3150*a^10*b^2*c^2*d^10 - 1500*a^11*b*c
*d^11 + 625*a^12*d^12)/(a^11*b^9))^(1/4) + (7*b^3*c^3 - 9*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 5*a^3*d^3)*sqrt(x)) -
4*(12*a^2*b*d^3*x^4 - 4*a*b^2*c^3 - (7*b^3*c^3 - 9*a*b^2*c^2*d + 9*a^2*b*c*d^2 - 15*a^3*d^3)*x^2)*sqrt(x))/(a^
2*b^3*x^4 + a^3*b^2*x^2)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2008 vs. \(2 (340) = 680\).

Time = 113.35 (sec) , antiderivative size = 2008, normalized size of antiderivative = 5.47 \[ \int \frac {\left (c+d x^2\right )^3}{x^{5/2} \left (a+b x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Piecewise((zoo*(-2*c**3/(11*x**(11/2)) - 6*c**2*d/(7*x**(7/2)) - 2*c*d**2/x**(3/2) + 2*d**3*sqrt(x)), Eq(a, 0)
& 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)/a**2, Eq(b,
0)), ((-2*c**3/(11*x**(11/2)) - 6*c**2*d/(7*x**(7/2)) - 2*c*d**2/x**(3/2) + 2*d**3*sqrt(x))/b**2, Eq(a, 0)),
(15*a**4*d**3*x**(3/2)*(-a/b)**(1/4)*log(sqrt(x) - (-a/b)**(1/4))/(24*a**4*b**2*x**(3/2) + 24*a**3*b**3*x**(7/
2)) - 15*a**4*d**3*x**(3/2)*(-a/b)**(1/4)*log(sqrt(x) + (-a/b)**(1/4))/(24*a**4*b**2*x**(3/2) + 24*a**3*b**3*x
**(7/2)) - 30*a**4*d**3*x**(3/2)*(-a/b)**(1/4)*atan(sqrt(x)/(-a/b)**(1/4))/(24*a**4*b**2*x**(3/2) + 24*a**3*b*
*3*x**(7/2)) + 60*a**4*d**3*x**2/(24*a**4*b**2*x**(3/2) + 24*a**3*b**3*x**(7/2)) - 9*a**3*b*c*d**2*x**(3/2)*(-
a/b)**(1/4)*log(sqrt(x) - (-a/b)**(1/4))/(24*a**4*b**2*x**(3/2) + 24*a**3*b**3*x**(7/2)) + 9*a**3*b*c*d**2*x**
(3/2)*(-a/b)**(1/4)*log(sqrt(x) + (-a/b)**(1/4))/(24*a**4*b**2*x**(3/2) + 24*a**3*b**3*x**(7/2)) + 18*a**3*b*c
*d**2*x**(3/2)*(-a/b)**(1/4)*atan(sqrt(x)/(-a/b)**(1/4))/(24*a**4*b**2*x**(3/2) + 24*a**3*b**3*x**(7/2)) - 36*
a**3*b*c*d**2*x**2/(24*a**4*b**2*x**(3/2) + 24*a**3*b**3*x**(7/2)) + 15*a**3*b*d**3*x**(7/2)*(-a/b)**(1/4)*log
(sqrt(x) - (-a/b)**(1/4))/(24*a**4*b**2*x**(3/2) + 24*a**3*b**3*x**(7/2)) - 15*a**3*b*d**3*x**(7/2)*(-a/b)**(1
/4)*log(sqrt(x) + (-a/b)**(1/4))/(24*a**4*b**2*x**(3/2) + 24*a**3*b**3*x**(7/2)) - 30*a**3*b*d**3*x**(7/2)*(-a
/b)**(1/4)*atan(sqrt(x)/(-a/b)**(1/4))/(24*a**4*b**2*x**(3/2) + 24*a**3*b**3*x**(7/2)) + 48*a**3*b*d**3*x**4/(
24*a**4*b**2*x**(3/2) + 24*a**3*b**3*x**(7/2)) - 16*a**2*b**2*c**3/(24*a**4*b**2*x**(3/2) + 24*a**3*b**3*x**(7
/2)) - 27*a**2*b**2*c**2*d*x**(3/2)*(-a/b)**(1/4)*log(sqrt(x) - (-a/b)**(1/4))/(24*a**4*b**2*x**(3/2) + 24*a**
3*b**3*x**(7/2)) + 27*a**2*b**2*c**2*d*x**(3/2)*(-a/b)**(1/4)*log(sqrt(x) + (-a/b)**(1/4))/(24*a**4*b**2*x**(3
/2) + 24*a**3*b**3*x**(7/2)) + 54*a**2*b**2*c**2*d*x**(3/2)*(-a/b)**(1/4)*atan(sqrt(x)/(-a/b)**(1/4))/(24*a**4
*b**2*x**(3/2) + 24*a**3*b**3*x**(7/2)) + 36*a**2*b**2*c**2*d*x**2/(24*a**4*b**2*x**(3/2) + 24*a**3*b**3*x**(7
/2)) - 9*a**2*b**2*c*d**2*x**(7/2)*(-a/b)**(1/4)*log(sqrt(x) - (-a/b)**(1/4))/(24*a**4*b**2*x**(3/2) + 24*a**3
*b**3*x**(7/2)) + 9*a**2*b**2*c*d**2*x**(7/2)*(-a/b)**(1/4)*log(sqrt(x) + (-a/b)**(1/4))/(24*a**4*b**2*x**(3/2
) + 24*a**3*b**3*x**(7/2)) + 18*a**2*b**2*c*d**2*x**(7/2)*(-a/b)**(1/4)*atan(sqrt(x)/(-a/b)**(1/4))/(24*a**4*b
**2*x**(3/2) + 24*a**3*b**3*x**(7/2)) + 21*a*b**3*c**3*x**(3/2)*(-a/b)**(1/4)*log(sqrt(x) - (-a/b)**(1/4))/(24
*a**4*b**2*x**(3/2) + 24*a**3*b**3*x**(7/2)) - 21*a*b**3*c**3*x**(3/2)*(-a/b)**(1/4)*log(sqrt(x) + (-a/b)**(1/
4))/(24*a**4*b**2*x**(3/2) + 24*a**3*b**3*x**(7/2)) - 42*a*b**3*c**3*x**(3/2)*(-a/b)**(1/4)*atan(sqrt(x)/(-a/b
)**(1/4))/(24*a**4*b**2*x**(3/2) + 24*a**3*b**3*x**(7/2)) - 28*a*b**3*c**3*x**2/(24*a**4*b**2*x**(3/2) + 24*a*
*3*b**3*x**(7/2)) - 27*a*b**3*c**2*d*x**(7/2)*(-a/b)**(1/4)*log(sqrt(x) - (-a/b)**(1/4))/(24*a**4*b**2*x**(3/2
) + 24*a**3*b**3*x**(7/2)) + 27*a*b**3*c**2*d*x**(7/2)*(-a/b)**(1/4)*log(sqrt(x) + (-a/b)**(1/4))/(24*a**4*b**
2*x**(3/2) + 24*a**3*b**3*x**(7/2)) + 54*a*b**3*c**2*d*x**(7/2)*(-a/b)**(1/4)*atan(sqrt(x)/(-a/b)**(1/4))/(24*
a**4*b**2*x**(3/2) + 24*a**3*b**3*x**(7/2)) + 21*b**4*c**3*x**(7/2)*(-a/b)**(1/4)*log(sqrt(x) - (-a/b)**(1/4))
/(24*a**4*b**2*x**(3/2) + 24*a**3*b**3*x**(7/2)) - 21*b**4*c**3*x**(7/2)*(-a/b)**(1/4)*log(sqrt(x) + (-a/b)**(
1/4))/(24*a**4*b**2*x**(3/2) + 24*a**3*b**3*x**(7/2)) - 42*b**4*c**3*x**(7/2)*(-a/b)**(1/4)*atan(sqrt(x)/(-a/b
)**(1/4))/(24*a**4*b**2*x**(3/2) + 24*a**3*b**3*x**(7/2)), True))

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 415, normalized size of antiderivative = 1.13 \[ \int \frac {\left (c+d x^2\right )^3}{x^{5/2} \left (a+b x^2\right )^2} \, dx=\frac {2 \, d^{3} \sqrt {x}}{b^{2}} - \frac {4 \, a b^{2} c^{3} + {\left (7 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} - 3 \, a^{3} d^{3}\right )} x^{2}}{6 \, {\left (a^{2} b^{3} x^{\frac {7}{2}} + a^{3} b^{2} x^{\frac {3}{2}}\right )}} - \frac {\frac {2 \, \sqrt {2} {\left (7 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 5 \, a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} {\left (7 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 5 \, a^{3} d^{3}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} {\left (7 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 5 \, a^{3} d^{3}\right )} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (7 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 5 \, a^{3} d^{3}\right )} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}}}{16 \, a^{2} b^{2}} \]

[In]

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

[Out]

2*d^3*sqrt(x)/b^2 - 1/6*(4*a*b^2*c^3 + (7*b^3*c^3 - 9*a*b^2*c^2*d + 9*a^2*b*c*d^2 - 3*a^3*d^3)*x^2)/(a^2*b^3*x
^(7/2) + a^3*b^2*x^(3/2)) - 1/16*(2*sqrt(2)*(7*b^3*c^3 - 9*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 5*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)*(7*b^3*c^3 - 9*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 5*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)*(7*b^3*c^3 - 9*a*b^2*
c^2*d - 3*a^2*b*c*d^2 + 5*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)
) - sqrt(2)*(7*b^3*c^3 - 9*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 5*a^3*d^3)*log(-sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqr
t(b)*x + sqrt(a))/(a^(3/4)*b^(1/4)))/(a^2*b^2)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 501, normalized size of antiderivative = 1.37 \[ \int \frac {\left (c+d x^2\right )^3}{x^{5/2} \left (a+b x^2\right )^2} \, dx=\frac {2 \, d^{3} \sqrt {x}}{b^{2}} - \frac {2 \, c^{3}}{3 \, a^{2} x^{\frac {3}{2}}} - \frac {\sqrt {2} {\left (7 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 9 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d - 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} + 5 \, \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^{3} b^{3}} - \frac {\sqrt {2} {\left (7 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 9 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d - 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} + 5 \, \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^{3} b^{3}} - \frac {\sqrt {2} {\left (7 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 9 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d - 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} + 5 \, \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^{3} b^{3}} + \frac {\sqrt {2} {\left (7 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 9 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d - 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} + 5 \, \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^{3} b^{3}} - \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 )} a^{2} b^{2}} \]

[In]

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

[Out]

2*d^3*sqrt(x)/b^2 - 2/3*c^3/(a^2*x^(3/2)) - 1/8*sqrt(2)*(7*(a*b^3)^(1/4)*b^3*c^3 - 9*(a*b^3)^(1/4)*a*b^2*c^2*d
- 3*(a*b^3)^(1/4)*a^2*b*c*d^2 + 5*(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^3*b^3) - 1/8*sqrt(2)*(7*(a*b^3)^(1/4)*b^3*c^3 - 9*(a*b^3)^(1/4)*a*b^2*c^2*d - 3*(a*b^3)^(1/4)
*a^2*b*c*d^2 + 5*(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^
3*b^3) - 1/16*sqrt(2)*(7*(a*b^3)^(1/4)*b^3*c^3 - 9*(a*b^3)^(1/4)*a*b^2*c^2*d - 3*(a*b^3)^(1/4)*a^2*b*c*d^2 + 5
*(a*b^3)^(1/4)*a^3*d^3)*log(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^3*b^3) + 1/16*sqrt(2)*(7*(a*b^3)^(
1/4)*b^3*c^3 - 9*(a*b^3)^(1/4)*a*b^2*c^2*d - 3*(a*b^3)^(1/4)*a^2*b*c*d^2 + 5*(a*b^3)^(1/4)*a^3*d^3)*log(-sqrt(
2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^3*b^3) - 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)*a^2*b^2)

Mupad [B] (verification not implemented)

Time = 5.49 (sec) , antiderivative size = 1759, normalized size of antiderivative = 4.79 \[ \int \frac {\left (c+d x^2\right )^3}{x^{5/2} \left (a+b x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

((x^2*(3*a^3*d^3 - 7*b^3*c^3 + 9*a*b^2*c^2*d - 9*a^2*b*c*d^2))/(6*a^2) - (2*b^2*c^3)/(3*a))/(b^3*x^(7/2) + a*b
^2*x^(3/2)) + (2*d^3*x^(1/2))/b^2 - (atan((((x^(1/2)*(1568*a^6*b^15*c^6 + 800*a^12*b^9*d^6 - 4032*a^7*b^14*c^5
*d - 960*a^11*b^10*c*d^5 + 1248*a^8*b^13*c^4*d^2 + 3968*a^9*b^12*c^3*d^3 - 2592*a^10*b^11*c^2*d^4) - ((a*d - b
*c)^2*(5*a*d + 7*b*c)*(1792*a^9*b^14*c^3 + 1280*a^12*b^11*d^3 - 2304*a^10*b^13*c^2*d - 768*a^11*b^12*c*d^2))/(
8*(-a)^(11/4)*b^(9/4)))*(a*d - b*c)^2*(5*a*d + 7*b*c)*1i)/(8*(-a)^(11/4)*b^(9/4)) + ((x^(1/2)*(1568*a^6*b^15*c
^6 + 800*a^12*b^9*d^6 - 4032*a^7*b^14*c^5*d - 960*a^11*b^10*c*d^5 + 1248*a^8*b^13*c^4*d^2 + 3968*a^9*b^12*c^3*
d^3 - 2592*a^10*b^11*c^2*d^4) + ((a*d - b*c)^2*(5*a*d + 7*b*c)*(1792*a^9*b^14*c^3 + 1280*a^12*b^11*d^3 - 2304*
a^10*b^13*c^2*d - 768*a^11*b^12*c*d^2))/(8*(-a)^(11/4)*b^(9/4)))*(a*d - b*c)^2*(5*a*d + 7*b*c)*1i)/(8*(-a)^(11
/4)*b^(9/4)))/(((x^(1/2)*(1568*a^6*b^15*c^6 + 800*a^12*b^9*d^6 - 4032*a^7*b^14*c^5*d - 960*a^11*b^10*c*d^5 + 1
248*a^8*b^13*c^4*d^2 + 3968*a^9*b^12*c^3*d^3 - 2592*a^10*b^11*c^2*d^4) - ((a*d - b*c)^2*(5*a*d + 7*b*c)*(1792*
a^9*b^14*c^3 + 1280*a^12*b^11*d^3 - 2304*a^10*b^13*c^2*d - 768*a^11*b^12*c*d^2))/(8*(-a)^(11/4)*b^(9/4)))*(a*d
- b*c)^2*(5*a*d + 7*b*c))/(8*(-a)^(11/4)*b^(9/4)) - ((x^(1/2)*(1568*a^6*b^15*c^6 + 800*a^12*b^9*d^6 - 4032*a^
7*b^14*c^5*d - 960*a^11*b^10*c*d^5 + 1248*a^8*b^13*c^4*d^2 + 3968*a^9*b^12*c^3*d^3 - 2592*a^10*b^11*c^2*d^4) +
((a*d - b*c)^2*(5*a*d + 7*b*c)*(1792*a^9*b^14*c^3 + 1280*a^12*b^11*d^3 - 2304*a^10*b^13*c^2*d - 768*a^11*b^12
*c*d^2))/(8*(-a)^(11/4)*b^(9/4)))*(a*d - b*c)^2*(5*a*d + 7*b*c))/(8*(-a)^(11/4)*b^(9/4))))*(a*d - b*c)^2*(5*a*
d + 7*b*c)*1i)/(4*(-a)^(11/4)*b^(9/4)) - (atan((((x^(1/2)*(1568*a^6*b^15*c^6 + 800*a^12*b^9*d^6 - 4032*a^7*b^1
4*c^5*d - 960*a^11*b^10*c*d^5 + 1248*a^8*b^13*c^4*d^2 + 3968*a^9*b^12*c^3*d^3 - 2592*a^10*b^11*c^2*d^4) - ((a*
d - b*c)^2*(5*a*d + 7*b*c)*(1792*a^9*b^14*c^3 + 1280*a^12*b^11*d^3 - 2304*a^10*b^13*c^2*d - 768*a^11*b^12*c*d^
2)*1i)/(8*(-a)^(11/4)*b^(9/4)))*(a*d - b*c)^2*(5*a*d + 7*b*c))/(8*(-a)^(11/4)*b^(9/4)) + ((x^(1/2)*(1568*a^6*b
^15*c^6 + 800*a^12*b^9*d^6 - 4032*a^7*b^14*c^5*d - 960*a^11*b^10*c*d^5 + 1248*a^8*b^13*c^4*d^2 + 3968*a^9*b^12
*c^3*d^3 - 2592*a^10*b^11*c^2*d^4) + ((a*d - b*c)^2*(5*a*d + 7*b*c)*(1792*a^9*b^14*c^3 + 1280*a^12*b^11*d^3 -
2304*a^10*b^13*c^2*d - 768*a^11*b^12*c*d^2)*1i)/(8*(-a)^(11/4)*b^(9/4)))*(a*d - b*c)^2*(5*a*d + 7*b*c))/(8*(-a
)^(11/4)*b^(9/4)))/(((x^(1/2)*(1568*a^6*b^15*c^6 + 800*a^12*b^9*d^6 - 4032*a^7*b^14*c^5*d - 960*a^11*b^10*c*d^
5 + 1248*a^8*b^13*c^4*d^2 + 3968*a^9*b^12*c^3*d^3 - 2592*a^10*b^11*c^2*d^4) - ((a*d - b*c)^2*(5*a*d + 7*b*c)*(
1792*a^9*b^14*c^3 + 1280*a^12*b^11*d^3 - 2304*a^10*b^13*c^2*d - 768*a^11*b^12*c*d^2)*1i)/(8*(-a)^(11/4)*b^(9/4
)))*(a*d - b*c)^2*(5*a*d + 7*b*c)*1i)/(8*(-a)^(11/4)*b^(9/4)) - ((x^(1/2)*(1568*a^6*b^15*c^6 + 800*a^12*b^9*d^
6 - 4032*a^7*b^14*c^5*d - 960*a^11*b^10*c*d^5 + 1248*a^8*b^13*c^4*d^2 + 3968*a^9*b^12*c^3*d^3 - 2592*a^10*b^11
*c^2*d^4) + ((a*d - b*c)^2*(5*a*d + 7*b*c)*(1792*a^9*b^14*c^3 + 1280*a^12*b^11*d^3 - 2304*a^10*b^13*c^2*d - 76
8*a^11*b^12*c*d^2)*1i)/(8*(-a)^(11/4)*b^(9/4)))*(a*d - b*c)^2*(5*a*d + 7*b*c)*1i)/(8*(-a)^(11/4)*b^(9/4))))*(a
*d - b*c)^2*(5*a*d + 7*b*c))/(4*(-a)^(11/4)*b^(9/4))