\(\int \frac {(a+b x^2)^2}{\sqrt {x} (c+d x^2)^2} \, dx\) [429]

Optimal result

Integrand size = 24, antiderivative size = 312 \[ \int \frac {\left (a+b x^2\right )^2}{\sqrt {x} \left (c+d x^2\right )^2} \, dx=\frac {2 b^2 \sqrt {x}}{d^2}+\frac {(b c-a d)^2 \sqrt {x}}{2 c d^2 \left (c+d x^2\right )}+\frac {(b c-a d) (5 b c+3 a d) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{7/4} d^{9/4}}-\frac {(b c-a d) (5 b c+3 a d) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{7/4} d^{9/4}}+\frac {(b c-a d) (5 b c+3 a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{7/4} d^{9/4}}-\frac {(b c-a d) (5 b c+3 a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{7/4} d^{9/4}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {474, 470, 335, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {\left (a+b x^2\right )^2}{\sqrt {x} \left (c+d x^2\right )^2} \, dx=\frac {(b c-a d) (3 a d+5 b c) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{7/4} d^{9/4}}-\frac {(b c-a d) (3 a d+5 b c) \arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt {2} c^{7/4} d^{9/4}}+\frac {(b c-a d) (3 a d+5 b c) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} c^{7/4} d^{9/4}}-\frac {(b c-a d) (3 a d+5 b c) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} c^{7/4} d^{9/4}}+\frac {\sqrt {x} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}+\frac {2 b^2 \sqrt {x}}{d^2} \]

[In]

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

[Out]

(2*b^2*Sqrt[x])/d^2 + ((b*c - a*d)^2*Sqrt[x])/(2*c*d^2*(c + d*x^2)) + ((b*c - a*d)*(5*b*c + 3*a*d)*ArcTan[1 -
(Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(4*Sqrt[2]*c^(7/4)*d^(9/4)) - ((b*c - a*d)*(5*b*c + 3*a*d)*ArcTan[1 + (Sqr
t[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(4*Sqrt[2]*c^(7/4)*d^(9/4)) + ((b*c - a*d)*(5*b*c + 3*a*d)*Log[Sqrt[c] - Sqrt[
2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*c^(7/4)*d^(9/4)) - ((b*c - a*d)*(5*b*c + 3*a*d)*Log[Sqrt[c
] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*c^(7/4)*d^(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 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 470

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

Rule 474

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

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

Mathematica [A] (verified)

Time = 0.67 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.65 \[ \int \frac {\left (a+b x^2\right )^2}{\sqrt {x} \left (c+d x^2\right )^2} \, dx=\frac {\frac {4 c^{3/4} \sqrt [4]{d} \sqrt {x} \left (-2 a b c d+a^2 d^2+b^2 c \left (5 c+4 d x^2\right )\right )}{c+d x^2}+\sqrt {2} \left (5 b^2 c^2-2 a b c d-3 a^2 d^2\right ) \arctan \left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )-\sqrt {2} \left (5 b^2 c^2-2 a b c d-3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{8 c^{7/4} d^{9/4}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 2.75 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.53

method	result	size
risch	\(\frac {2 b^{2} \sqrt {x}}{d^{2}}+\frac {\left (2 a d -2 b c \right ) \left (\frac {\left (a d -b c \right ) \sqrt {x}}{4 c \left (d \,x^{2}+c \right )}+\frac {\left (3 a d +5 b c \right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{32 c^{2}}\right )}{d^{2}}\)	\(166\)
derivativedivides	\(\frac {2 b^{2} \sqrt {x}}{d^{2}}+\frac {\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \sqrt {x}}{2 c \left (d \,x^{2}+c \right )}+\frac {\left (3 a^{2} d^{2}+2 a b c d -5 b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{16 c^{2}}}{d^{2}}\)	\(185\)
default	\(\frac {2 b^{2} \sqrt {x}}{d^{2}}+\frac {\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \sqrt {x}}{2 c \left (d \,x^{2}+c \right )}+\frac {\left (3 a^{2} d^{2}+2 a b c d -5 b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{16 c^{2}}}{d^{2}}\)	\(185\)

[In]

int((b*x^2+a)^2/(d*x^2+c)^2/x^(1/2),x,method=_RETURNVERBOSE)

[Out]

2*b^2*x^(1/2)/d^2+1/d^2*(2*a*d-2*b*c)*(1/4*(a*d-b*c)/c*x^(1/2)/(d*x^2+c)+1/32*(3*a*d+5*b*c)/c^2*(c/d)^(1/4)*2^
(1/2)*(ln((x+(c/d)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2))/(x-(c/d)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2)))+2*arctan(2^
(1/2)/(c/d)^(1/4)*x^(1/2)+1)+2*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.28 (sec) , antiderivative size = 1220, normalized size of antiderivative = 3.91 \[ \int \frac {\left (a+b x^2\right )^2}{\sqrt {x} \left (c+d x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

integrate((b*x^2+a)^2/(d*x^2+c)^2/x^(1/2),x, algorithm="fricas")

[Out]

1/8*((c*d^3*x^2 + c^2*d^2)*(-(625*b^8*c^8 - 1000*a*b^7*c^7*d - 900*a^2*b^6*c^6*d^2 + 1640*a^3*b^5*c^5*d^3 + 64
6*a^4*b^4*c^4*d^4 - 984*a^5*b^3*c^3*d^5 - 324*a^6*b^2*c^2*d^6 + 216*a^7*b*c*d^7 + 81*a^8*d^8)/(c^7*d^9))^(1/4)
*log(c^2*d^2*(-(625*b^8*c^8 - 1000*a*b^7*c^7*d - 900*a^2*b^6*c^6*d^2 + 1640*a^3*b^5*c^5*d^3 + 646*a^4*b^4*c^4*
d^4 - 984*a^5*b^3*c^3*d^5 - 324*a^6*b^2*c^2*d^6 + 216*a^7*b*c*d^7 + 81*a^8*d^8)/(c^7*d^9))^(1/4) - (5*b^2*c^2
- 2*a*b*c*d - 3*a^2*d^2)*sqrt(x)) - (-I*c*d^3*x^2 - I*c^2*d^2)*(-(625*b^8*c^8 - 1000*a*b^7*c^7*d - 900*a^2*b^6
*c^6*d^2 + 1640*a^3*b^5*c^5*d^3 + 646*a^4*b^4*c^4*d^4 - 984*a^5*b^3*c^3*d^5 - 324*a^6*b^2*c^2*d^6 + 216*a^7*b*
c*d^7 + 81*a^8*d^8)/(c^7*d^9))^(1/4)*log(I*c^2*d^2*(-(625*b^8*c^8 - 1000*a*b^7*c^7*d - 900*a^2*b^6*c^6*d^2 + 1
640*a^3*b^5*c^5*d^3 + 646*a^4*b^4*c^4*d^4 - 984*a^5*b^3*c^3*d^5 - 324*a^6*b^2*c^2*d^6 + 216*a^7*b*c*d^7 + 81*a
^8*d^8)/(c^7*d^9))^(1/4) - (5*b^2*c^2 - 2*a*b*c*d - 3*a^2*d^2)*sqrt(x)) - (I*c*d^3*x^2 + I*c^2*d^2)*(-(625*b^8
*c^8 - 1000*a*b^7*c^7*d - 900*a^2*b^6*c^6*d^2 + 1640*a^3*b^5*c^5*d^3 + 646*a^4*b^4*c^4*d^4 - 984*a^5*b^3*c^3*d
^5 - 324*a^6*b^2*c^2*d^6 + 216*a^7*b*c*d^7 + 81*a^8*d^8)/(c^7*d^9))^(1/4)*log(-I*c^2*d^2*(-(625*b^8*c^8 - 1000
*a*b^7*c^7*d - 900*a^2*b^6*c^6*d^2 + 1640*a^3*b^5*c^5*d^3 + 646*a^4*b^4*c^4*d^4 - 984*a^5*b^3*c^3*d^5 - 324*a^
6*b^2*c^2*d^6 + 216*a^7*b*c*d^7 + 81*a^8*d^8)/(c^7*d^9))^(1/4) - (5*b^2*c^2 - 2*a*b*c*d - 3*a^2*d^2)*sqrt(x))
- (c*d^3*x^2 + c^2*d^2)*(-(625*b^8*c^8 - 1000*a*b^7*c^7*d - 900*a^2*b^6*c^6*d^2 + 1640*a^3*b^5*c^5*d^3 + 646*a
^4*b^4*c^4*d^4 - 984*a^5*b^3*c^3*d^5 - 324*a^6*b^2*c^2*d^6 + 216*a^7*b*c*d^7 + 81*a^8*d^8)/(c^7*d^9))^(1/4)*lo
g(-c^2*d^2*(-(625*b^8*c^8 - 1000*a*b^7*c^7*d - 900*a^2*b^6*c^6*d^2 + 1640*a^3*b^5*c^5*d^3 + 646*a^4*b^4*c^4*d^
4 - 984*a^5*b^3*c^3*d^5 - 324*a^6*b^2*c^2*d^6 + 216*a^7*b*c*d^7 + 81*a^8*d^8)/(c^7*d^9))^(1/4) - (5*b^2*c^2 -
2*a*b*c*d - 3*a^2*d^2)*sqrt(x)) + 4*(4*b^2*c*d*x^2 + 5*b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(x))/(c*d^3*x^2 + c^
2*d^2)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1248 vs. \(2 (298) = 596\).

Time = 24.80 (sec) , antiderivative size = 1248, normalized size of antiderivative = 4.00 \[ \int \frac {\left (a+b x^2\right )^2}{\sqrt {x} \left (c+d x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

integrate((b*x**2+a)**2/(d*x**2+c)**2/x**(1/2),x)

[Out]

Piecewise((zoo*(-2*a**2/(7*x**(7/2)) - 4*a*b/(3*x**(3/2)) + 2*b**2*sqrt(x)), Eq(c, 0) & Eq(d, 0)), ((2*a**2*sq
rt(x) + 4*a*b*x**(5/2)/5 + 2*b**2*x**(9/2)/9)/c**2, Eq(d, 0)), ((-2*a**2/(7*x**(7/2)) - 4*a*b/(3*x**(3/2)) + 2
*b**2*sqrt(x))/d**2, Eq(c, 0)), (4*a**2*c*d**2*sqrt(x)/(8*c**3*d**2 + 8*c**2*d**3*x**2) - 3*a**2*c*d**2*(-c/d)
**(1/4)*log(sqrt(x) - (-c/d)**(1/4))/(8*c**3*d**2 + 8*c**2*d**3*x**2) + 3*a**2*c*d**2*(-c/d)**(1/4)*log(sqrt(x
) + (-c/d)**(1/4))/(8*c**3*d**2 + 8*c**2*d**3*x**2) + 6*a**2*c*d**2*(-c/d)**(1/4)*atan(sqrt(x)/(-c/d)**(1/4))/
(8*c**3*d**2 + 8*c**2*d**3*x**2) - 3*a**2*d**3*x**2*(-c/d)**(1/4)*log(sqrt(x) - (-c/d)**(1/4))/(8*c**3*d**2 +
8*c**2*d**3*x**2) + 3*a**2*d**3*x**2*(-c/d)**(1/4)*log(sqrt(x) + (-c/d)**(1/4))/(8*c**3*d**2 + 8*c**2*d**3*x**
2) + 6*a**2*d**3*x**2*(-c/d)**(1/4)*atan(sqrt(x)/(-c/d)**(1/4))/(8*c**3*d**2 + 8*c**2*d**3*x**2) - 8*a*b*c**2*
d*sqrt(x)/(8*c**3*d**2 + 8*c**2*d**3*x**2) - 2*a*b*c**2*d*(-c/d)**(1/4)*log(sqrt(x) - (-c/d)**(1/4))/(8*c**3*d
**2 + 8*c**2*d**3*x**2) + 2*a*b*c**2*d*(-c/d)**(1/4)*log(sqrt(x) + (-c/d)**(1/4))/(8*c**3*d**2 + 8*c**2*d**3*x
**2) + 4*a*b*c**2*d*(-c/d)**(1/4)*atan(sqrt(x)/(-c/d)**(1/4))/(8*c**3*d**2 + 8*c**2*d**3*x**2) - 2*a*b*c*d**2*
x**2*(-c/d)**(1/4)*log(sqrt(x) - (-c/d)**(1/4))/(8*c**3*d**2 + 8*c**2*d**3*x**2) + 2*a*b*c*d**2*x**2*(-c/d)**(
1/4)*log(sqrt(x) + (-c/d)**(1/4))/(8*c**3*d**2 + 8*c**2*d**3*x**2) + 4*a*b*c*d**2*x**2*(-c/d)**(1/4)*atan(sqrt
(x)/(-c/d)**(1/4))/(8*c**3*d**2 + 8*c**2*d**3*x**2) + 20*b**2*c**3*sqrt(x)/(8*c**3*d**2 + 8*c**2*d**3*x**2) +
5*b**2*c**3*(-c/d)**(1/4)*log(sqrt(x) - (-c/d)**(1/4))/(8*c**3*d**2 + 8*c**2*d**3*x**2) - 5*b**2*c**3*(-c/d)**
(1/4)*log(sqrt(x) + (-c/d)**(1/4))/(8*c**3*d**2 + 8*c**2*d**3*x**2) - 10*b**2*c**3*(-c/d)**(1/4)*atan(sqrt(x)/
(-c/d)**(1/4))/(8*c**3*d**2 + 8*c**2*d**3*x**2) + 16*b**2*c**2*d*x**(5/2)/(8*c**3*d**2 + 8*c**2*d**3*x**2) + 5
*b**2*c**2*d*x**2*(-c/d)**(1/4)*log(sqrt(x) - (-c/d)**(1/4))/(8*c**3*d**2 + 8*c**2*d**3*x**2) - 5*b**2*c**2*d*
x**2*(-c/d)**(1/4)*log(sqrt(x) + (-c/d)**(1/4))/(8*c**3*d**2 + 8*c**2*d**3*x**2) - 10*b**2*c**2*d*x**2*(-c/d)*
*(1/4)*atan(sqrt(x)/(-c/d)**(1/4))/(8*c**3*d**2 + 8*c**2*d**3*x**2), True))

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.05 \[ \int \frac {\left (a+b x^2\right )^2}{\sqrt {x} \left (c+d x^2\right )^2} \, dx=\frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {x}}{2 \, {\left (c d^{3} x^{2} + c^{2} d^{2}\right )}} + \frac {2 \, b^{2} \sqrt {x}}{d^{2}} - \frac {\frac {2 \, \sqrt {2} {\left (5 \, b^{2} c^{2} - 2 \, a b c d - 3 \, a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} + 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {2 \, \sqrt {2} {\left (5 \, b^{2} c^{2} - 2 \, a b c d - 3 \, a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} - 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {\sqrt {2} {\left (5 \, b^{2} c^{2} - 2 \, a b c d - 3 \, a^{2} d^{2}\right )} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (5 \, b^{2} c^{2} - 2 \, a b c d - 3 \, a^{2} d^{2}\right )} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}}}{16 \, c d^{2}} \]

[In]

integrate((b*x^2+a)^2/(d*x^2+c)^2/x^(1/2),x, algorithm="maxima")

[Out]

1/2*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(x)/(c*d^3*x^2 + c^2*d^2) + 2*b^2*sqrt(x)/d^2 - 1/16*(2*sqrt(2)*(5*b^2
*c^2 - 2*a*b*c*d - 3*a^2*d^2)*arctan(1/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/4) + 2*sqrt(d)*sqrt(x))/sqrt(sqrt(c)*sq
rt(d)))/(sqrt(c)*sqrt(sqrt(c)*sqrt(d))) + 2*sqrt(2)*(5*b^2*c^2 - 2*a*b*c*d - 3*a^2*d^2)*arctan(-1/2*sqrt(2)*(s
qrt(2)*c^(1/4)*d^(1/4) - 2*sqrt(d)*sqrt(x))/sqrt(sqrt(c)*sqrt(d)))/(sqrt(c)*sqrt(sqrt(c)*sqrt(d))) + sqrt(2)*(
5*b^2*c^2 - 2*a*b*c*d - 3*a^2*d^2)*log(sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x + sqrt(c))/(c^(3/4)*d^(1/4)
) - sqrt(2)*(5*b^2*c^2 - 2*a*b*c*d - 3*a^2*d^2)*log(-sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x + sqrt(c))/(c
^(3/4)*d^(1/4)))/(c*d^2)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.24 \[ \int \frac {\left (a+b x^2\right )^2}{\sqrt {x} \left (c+d x^2\right )^2} \, dx=\frac {2 \, b^{2} \sqrt {x}}{d^{2}} - \frac {\sqrt {2} {\left (5 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d - 3 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{8 \, c^{2} d^{3}} - \frac {\sqrt {2} {\left (5 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d - 3 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{8 \, c^{2} d^{3}} - \frac {\sqrt {2} {\left (5 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d - 3 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{16 \, c^{2} d^{3}} + \frac {\sqrt {2} {\left (5 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d - 3 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{16 \, c^{2} d^{3}} + \frac {b^{2} c^{2} \sqrt {x} - 2 \, a b c d \sqrt {x} + a^{2} d^{2} \sqrt {x}}{2 \, {\left (d x^{2} + c\right )} c d^{2}} \]

[In]

integrate((b*x^2+a)^2/(d*x^2+c)^2/x^(1/2),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 5.79 (sec) , antiderivative size = 1267, normalized size of antiderivative = 4.06 \[ \int \frac {\left (a+b x^2\right )^2}{\sqrt {x} \left (c+d x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

int((a + b*x^2)^2/(x^(1/2)*(c + d*x^2)^2),x)

[Out]

(2*b^2*x^(1/2))/d^2 + (x^(1/2)*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))/(2*c*(c*d^2 + d^3*x^2)) + (atan(((((x^(1/2)*(9
*a^4*d^4 + 25*b^4*c^4 - 26*a^2*b^2*c^2*d^2 - 20*a*b^3*c^3*d + 12*a^3*b*c*d^3))/(c^2*d) - ((a*d - b*c)*(3*a*d +
5*b*c)*(24*a^2*d^3 - 40*b^2*c^2*d + 16*a*b*c*d^2))/(8*(-c)^(7/4)*d^(9/4)))*(a*d - b*c)*(3*a*d + 5*b*c)*1i)/(8
*(-c)^(7/4)*d^(9/4)) + (((x^(1/2)*(9*a^4*d^4 + 25*b^4*c^4 - 26*a^2*b^2*c^2*d^2 - 20*a*b^3*c^3*d + 12*a^3*b*c*d
^3))/(c^2*d) + ((a*d - b*c)*(3*a*d + 5*b*c)*(24*a^2*d^3 - 40*b^2*c^2*d + 16*a*b*c*d^2))/(8*(-c)^(7/4)*d^(9/4))
)*(a*d - b*c)*(3*a*d + 5*b*c)*1i)/(8*(-c)^(7/4)*d^(9/4)))/((((x^(1/2)*(9*a^4*d^4 + 25*b^4*c^4 - 26*a^2*b^2*c^2
*d^2 - 20*a*b^3*c^3*d + 12*a^3*b*c*d^3))/(c^2*d) - ((a*d - b*c)*(3*a*d + 5*b*c)*(24*a^2*d^3 - 40*b^2*c^2*d + 1
6*a*b*c*d^2))/(8*(-c)^(7/4)*d^(9/4)))*(a*d - b*c)*(3*a*d + 5*b*c))/(8*(-c)^(7/4)*d^(9/4)) - (((x^(1/2)*(9*a^4*
d^4 + 25*b^4*c^4 - 26*a^2*b^2*c^2*d^2 - 20*a*b^3*c^3*d + 12*a^3*b*c*d^3))/(c^2*d) + ((a*d - b*c)*(3*a*d + 5*b*
c)*(24*a^2*d^3 - 40*b^2*c^2*d + 16*a*b*c*d^2))/(8*(-c)^(7/4)*d^(9/4)))*(a*d - b*c)*(3*a*d + 5*b*c))/(8*(-c)^(7
/4)*d^(9/4))))*(a*d - b*c)*(3*a*d + 5*b*c)*1i)/(4*(-c)^(7/4)*d^(9/4)) + (atan(((((x^(1/2)*(9*a^4*d^4 + 25*b^4*
c^4 - 26*a^2*b^2*c^2*d^2 - 20*a*b^3*c^3*d + 12*a^3*b*c*d^3))/(c^2*d) - ((a*d - b*c)*(3*a*d + 5*b*c)*(24*a^2*d^
3 - 40*b^2*c^2*d + 16*a*b*c*d^2)*1i)/(8*(-c)^(7/4)*d^(9/4)))*(a*d - b*c)*(3*a*d + 5*b*c))/(8*(-c)^(7/4)*d^(9/4
)) + (((x^(1/2)*(9*a^4*d^4 + 25*b^4*c^4 - 26*a^2*b^2*c^2*d^2 - 20*a*b^3*c^3*d + 12*a^3*b*c*d^3))/(c^2*d) + ((a
*d - b*c)*(3*a*d + 5*b*c)*(24*a^2*d^3 - 40*b^2*c^2*d + 16*a*b*c*d^2)*1i)/(8*(-c)^(7/4)*d^(9/4)))*(a*d - b*c)*(
3*a*d + 5*b*c))/(8*(-c)^(7/4)*d^(9/4)))/((((x^(1/2)*(9*a^4*d^4 + 25*b^4*c^4 - 26*a^2*b^2*c^2*d^2 - 20*a*b^3*c^
3*d + 12*a^3*b*c*d^3))/(c^2*d) - ((a*d - b*c)*(3*a*d + 5*b*c)*(24*a^2*d^3 - 40*b^2*c^2*d + 16*a*b*c*d^2)*1i)/(
8*(-c)^(7/4)*d^(9/4)))*(a*d - b*c)*(3*a*d + 5*b*c)*1i)/(8*(-c)^(7/4)*d^(9/4)) - (((x^(1/2)*(9*a^4*d^4 + 25*b^4
*c^4 - 26*a^2*b^2*c^2*d^2 - 20*a*b^3*c^3*d + 12*a^3*b*c*d^3))/(c^2*d) + ((a*d - b*c)*(3*a*d + 5*b*c)*(24*a^2*d
^3 - 40*b^2*c^2*d + 16*a*b*c*d^2)*1i)/(8*(-c)^(7/4)*d^(9/4)))*(a*d - b*c)*(3*a*d + 5*b*c)*1i)/(8*(-c)^(7/4)*d^
(9/4))))*(a*d - b*c)*(3*a*d + 5*b*c))/(4*(-c)^(7/4)*d^(9/4))