3.5.76 \(\int \frac {1}{\sqrt {x} (a+b x^2) (c+d x^2)^2} \, dx\) [476]

Optimal. Leaf size=536 \[ -\frac {d \sqrt {x}}{2 c (b c-a d) \left (c+d x^2\right )}-\frac {b^{7/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{3/4} (b c-a d)^2}+\frac {b^{7/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{3/4} (b c-a d)^2}+\frac {d^{3/4} (7 b c-3 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{7/4} (b c-a d)^2}-\frac {d^{3/4} (7 b c-3 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{7/4} (b c-a d)^2}-\frac {b^{7/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{3/4} (b c-a d)^2}+\frac {b^{7/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{3/4} (b c-a d)^2}+\frac {d^{3/4} (7 b c-3 a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{7/4} (b c-a d)^2}-\frac {d^{3/4} (7 b c-3 a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{7/4} (b c-a d)^2} \]

[Out]

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

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Rubi [A]
time = 0.36, antiderivative size = 536, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {477, 425, 536, 217, 1179, 642, 1176, 631, 210} \begin {gather*} -\frac {b^{7/4} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{3/4} (b c-a d)^2}+\frac {b^{7/4} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} a^{3/4} (b c-a d)^2}-\frac {b^{7/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{3/4} (b c-a d)^2}+\frac {b^{7/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{3/4} (b c-a d)^2}+\frac {d^{3/4} (7 b c-3 a d) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{7/4} (b c-a d)^2}-\frac {d^{3/4} (7 b c-3 a d) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt {2} c^{7/4} (b c-a d)^2}+\frac {d^{3/4} (7 b c-3 a d) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} c^{7/4} (b c-a d)^2}-\frac {d^{3/4} (7 b c-3 a d) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} c^{7/4} (b c-a d)^2}-\frac {d \sqrt {x}}{2 c \left (c+d x^2\right ) (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[x]*(a + b*x^2)*(c + d*x^2)^2),x]

[Out]

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

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 425

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*x*(a + b*x^n)^(p + 1)*
((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - a*d))), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1
)*(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c,
d, n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomi
alQ[a, b, c, d, n, p, q, x]

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 536

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f
)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a
, b, c, d, e, f, n}, x]

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

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Mathematica [A]
time = 1.01, size = 273, normalized size = 0.51 \begin {gather*} \frac {\frac {4 d (-b c+a d) \sqrt {x}}{c \left (c+d x^2\right )}-\frac {4 \sqrt {2} b^{7/4} \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{a^{3/4}}+\frac {\sqrt {2} d^{3/4} (7 b c-3 a d) \tan ^{-1}\left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )}{c^{7/4}}+\frac {4 \sqrt {2} b^{7/4} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{a^{3/4}}+\frac {\sqrt {2} d^{3/4} (-7 b c+3 a d) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{c^{7/4}}}{8 (b c-a d)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[x]*(a + b*x^2)*(c + d*x^2)^2),x]

[Out]

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

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Maple [A]
time = 0.09, size = 273, normalized size = 0.51

method	result	size
derivativedivides	\(\frac {b^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (a d -b c \right )^{2} a}+\frac {2 d \left (\frac {\left (a d -b c \right ) \sqrt {x}}{4 c \left (d \,x^{2}+c \right )}+\frac {\left (3 a d -7 b c \right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{32 c^{2}}\right )}{\left (a d -b c \right )^{2}}\)	\(273\)
default	\(\frac {b^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (a d -b c \right )^{2} a}+\frac {2 d \left (\frac {\left (a d -b c \right ) \sqrt {x}}{4 c \left (d \,x^{2}+c \right )}+\frac {\left (3 a d -7 b c \right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{32 c^{2}}\right )}{\left (a d -b c \right )^{2}}\)	\(273\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*b^2/(a*d-b*c)^2*(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))+2*d/
(a*d-b*c)^2*(1/4*(a*d-b*c)/c*x^(1/2)/(d*x^2+c)+1/32*(3*a*d-7*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)))

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Maxima [A]
time = 0.56, size = 489, normalized size = 0.91 \begin {gather*} -\frac {d \sqrt {x}}{2 \, {\left (b c^{3} - a c^{2} d + {\left (b c^{2} d - a c d^{2}\right )} x^{2}\right )}} + \frac {\frac {2 \, \sqrt {2} b^{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 {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} b^{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 {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} b^{\frac {7}{4}} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}}} - \frac {\sqrt {2} b^{\frac {7}{4}} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}}}}{4 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}} - \frac {\frac {2 \, \sqrt {2} {\left (7 \, b c d - 3 \, a 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 (7 \, b c d - 3 \, a 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 (7 \, b c d - 3 \, a 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 (7 \, b c d - 3 \, a 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 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*d*sqrt(x)/(b*c^3 - a*c^2*d + (b*c^2*d - a*c*d^2)*x^2) + 1/4*(2*sqrt(2)*b^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(a)*sqrt(sqrt(a)*sqrt(b))) + 2*sqrt(2)*b^2*arct
an(-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)*sqr
t(b))) + sqrt(2)*b^(7/4)*log(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/a^(3/4) - sqrt(2)*b^(7/4)*
log(-sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/a^(3/4))/(b^2*c^2 - 2*a*b*c*d + a^2*d^2) - 1/16*(2
*sqrt(2)*(7*b*c*d - 3*a*d^2)*arctan(1/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/4) + 2*sqrt(d)*sqrt(x))/sqrt(sqrt(c)*sqr
t(d)))/(sqrt(c)*sqrt(sqrt(c)*sqrt(d))) + 2*sqrt(2)*(7*b*c*d - 3*a*d^2)*arctan(-1/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^
(1/4) - 2*sqrt(d)*sqrt(x))/sqrt(sqrt(c)*sqrt(d)))/(sqrt(c)*sqrt(sqrt(c)*sqrt(d))) + sqrt(2)*(7*b*c*d - 3*a*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)*(7*b*c*d - 3*a*d^2)*l
og(-sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x + sqrt(c))/(c^(3/4)*d^(1/4)))/(b^2*c^3 - 2*a*b*c^2*d + a^2*c*d
^2)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 3310 vs. \(2 (393) = 786\).
time = 12.91, size = 3310, normalized size = 6.18 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(4*(b*c^3 - a*c^2*d + (b*c^2*d - a*c*d^2)*x^2)*(-(2401*b^4*c^4*d^3 - 4116*a*b^3*c^3*d^4 + 2646*a^2*b^2*c^2
*d^5 - 756*a^3*b*c*d^6 + 81*a^4*d^7)/(b^8*c^15 - 8*a*b^7*c^14*d + 28*a^2*b^6*c^13*d^2 - 56*a^3*b^5*c^12*d^3 +
70*a^4*b^4*c^11*d^4 - 56*a^5*b^3*c^10*d^5 + 28*a^6*b^2*c^9*d^6 - 8*a^7*b*c^8*d^7 + a^8*c^7*d^8))^(1/4)*arctan(
((b^6*c^11 - 6*a*b^5*c^10*d + 15*a^2*b^4*c^9*d^2 - 20*a^3*b^3*c^8*d^3 + 15*a^4*b^2*c^7*d^4 - 6*a^5*b*c^6*d^5 +
a^6*c^5*d^6)*sqrt((49*b^2*c^2*d^2 - 42*a*b*c*d^3 + 9*a^2*d^4)*x + (b^4*c^8 - 4*a*b^3*c^7*d + 6*a^2*b^2*c^6*d^
2 - 4*a^3*b*c^5*d^3 + a^4*c^4*d^4)*sqrt(-(2401*b^4*c^4*d^3 - 4116*a*b^3*c^3*d^4 + 2646*a^2*b^2*c^2*d^5 - 756*a
^3*b*c*d^6 + 81*a^4*d^7)/(b^8*c^15 - 8*a*b^7*c^14*d + 28*a^2*b^6*c^13*d^2 - 56*a^3*b^5*c^12*d^3 + 70*a^4*b^4*c
^11*d^4 - 56*a^5*b^3*c^10*d^5 + 28*a^6*b^2*c^9*d^6 - 8*a^7*b*c^8*d^7 + a^8*c^7*d^8)))*(-(2401*b^4*c^4*d^3 - 41
16*a*b^3*c^3*d^4 + 2646*a^2*b^2*c^2*d^5 - 756*a^3*b*c*d^6 + 81*a^4*d^7)/(b^8*c^15 - 8*a*b^7*c^14*d + 28*a^2*b^
6*c^13*d^2 - 56*a^3*b^5*c^12*d^3 + 70*a^4*b^4*c^11*d^4 - 56*a^5*b^3*c^10*d^5 + 28*a^6*b^2*c^9*d^6 - 8*a^7*b*c^
8*d^7 + a^8*c^7*d^8))^(3/4) + (7*b^7*c^12*d - 45*a*b^6*c^11*d^2 + 123*a^2*b^5*c^10*d^3 - 185*a^3*b^4*c^9*d^4 +
165*a^4*b^3*c^8*d^5 - 87*a^5*b^2*c^7*d^6 + 25*a^6*b*c^6*d^7 - 3*a^7*c^5*d^8)*sqrt(x)*(-(2401*b^4*c^4*d^3 - 41
16*a*b^3*c^3*d^4 + 2646*a^2*b^2*c^2*d^5 - 756*a^3*b*c*d^6 + 81*a^4*d^7)/(b^8*c^15 - 8*a*b^7*c^14*d + 28*a^2*b^
6*c^13*d^2 - 56*a^3*b^5*c^12*d^3 + 70*a^4*b^4*c^11*d^4 - 56*a^5*b^3*c^10*d^5 + 28*a^6*b^2*c^9*d^6 - 8*a^7*b*c^
8*d^7 + a^8*c^7*d^8))^(3/4))/(2401*b^4*c^4*d^3 - 4116*a*b^3*c^3*d^4 + 2646*a^2*b^2*c^2*d^5 - 756*a^3*b*c*d^6 +
81*a^4*d^7)) + 16*(-b^7/(a^3*b^8*c^8 - 8*a^4*b^7*c^7*d + 28*a^5*b^6*c^6*d^2 - 56*a^6*b^5*c^5*d^3 + 70*a^7*b^4
*c^4*d^4 - 56*a^8*b^3*c^3*d^5 + 28*a^9*b^2*c^2*d^6 - 8*a^10*b*c*d^7 + a^11*d^8))^(1/4)*(b*c^3 - a*c^2*d + (b*c
^2*d - a*c*d^2)*x^2)*arctan(((a^2*b^6*c^6 - 6*a^3*b^5*c^5*d + 15*a^4*b^4*c^4*d^2 - 20*a^5*b^3*c^3*d^3 + 15*a^6
*b^2*c^2*d^4 - 6*a^7*b*c*d^5 + a^8*d^6)*(-b^7/(a^3*b^8*c^8 - 8*a^4*b^7*c^7*d + 28*a^5*b^6*c^6*d^2 - 56*a^6*b^5
*c^5*d^3 + 70*a^7*b^4*c^4*d^4 - 56*a^8*b^3*c^3*d^5 + 28*a^9*b^2*c^2*d^6 - 8*a^10*b*c*d^7 + a^11*d^8))^(3/4)*sq
rt(b^4*x + (a^2*b^4*c^4 - 4*a^3*b^3*c^3*d + 6*a^4*b^2*c^2*d^2 - 4*a^5*b*c*d^3 + a^6*d^4)*sqrt(-b^7/(a^3*b^8*c^
8 - 8*a^4*b^7*c^7*d + 28*a^5*b^6*c^6*d^2 - 56*a^6*b^5*c^5*d^3 + 70*a^7*b^4*c^4*d^4 - 56*a^8*b^3*c^3*d^5 + 28*a
^9*b^2*c^2*d^6 - 8*a^10*b*c*d^7 + a^11*d^8))) - (a^2*b^8*c^6 - 6*a^3*b^7*c^5*d + 15*a^4*b^6*c^4*d^2 - 20*a^5*b
^5*c^3*d^3 + 15*a^6*b^4*c^2*d^4 - 6*a^7*b^3*c*d^5 + a^8*b^2*d^6)*(-b^7/(a^3*b^8*c^8 - 8*a^4*b^7*c^7*d + 28*a^5
*b^6*c^6*d^2 - 56*a^6*b^5*c^5*d^3 + 70*a^7*b^4*c^4*d^4 - 56*a^8*b^3*c^3*d^5 + 28*a^9*b^2*c^2*d^6 - 8*a^10*b*c*
d^7 + a^11*d^8))^(3/4)*sqrt(x))/b^7) + 4*(-b^7/(a^3*b^8*c^8 - 8*a^4*b^7*c^7*d + 28*a^5*b^6*c^6*d^2 - 56*a^6*b^
5*c^5*d^3 + 70*a^7*b^4*c^4*d^4 - 56*a^8*b^3*c^3*d^5 + 28*a^9*b^2*c^2*d^6 - 8*a^10*b*c*d^7 + a^11*d^8))^(1/4)*(
b*c^3 - a*c^2*d + (b*c^2*d - a*c*d^2)*x^2)*log(b^2*sqrt(x) + (-b^7/(a^3*b^8*c^8 - 8*a^4*b^7*c^7*d + 28*a^5*b^6
*c^6*d^2 - 56*a^6*b^5*c^5*d^3 + 70*a^7*b^4*c^4*d^4 - 56*a^8*b^3*c^3*d^5 + 28*a^9*b^2*c^2*d^6 - 8*a^10*b*c*d^7
+ a^11*d^8))^(1/4)*(a*b^2*c^2 - 2*a^2*b*c*d + a^3*d^2)) - 4*(-b^7/(a^3*b^8*c^8 - 8*a^4*b^7*c^7*d + 28*a^5*b^6*
c^6*d^2 - 56*a^6*b^5*c^5*d^3 + 70*a^7*b^4*c^4*d^4 - 56*a^8*b^3*c^3*d^5 + 28*a^9*b^2*c^2*d^6 - 8*a^10*b*c*d^7 +
a^11*d^8))^(1/4)*(b*c^3 - a*c^2*d + (b*c^2*d - a*c*d^2)*x^2)*log(b^2*sqrt(x) - (-b^7/(a^3*b^8*c^8 - 8*a^4*b^7
*c^7*d + 28*a^5*b^6*c^6*d^2 - 56*a^6*b^5*c^5*d^3 + 70*a^7*b^4*c^4*d^4 - 56*a^8*b^3*c^3*d^5 + 28*a^9*b^2*c^2*d^
6 - 8*a^10*b*c*d^7 + a^11*d^8))^(1/4)*(a*b^2*c^2 - 2*a^2*b*c*d + a^3*d^2)) + (b*c^3 - a*c^2*d + (b*c^2*d - a*c
*d^2)*x^2)*(-(2401*b^4*c^4*d^3 - 4116*a*b^3*c^3*d^4 + 2646*a^2*b^2*c^2*d^5 - 756*a^3*b*c*d^6 + 81*a^4*d^7)/(b^
8*c^15 - 8*a*b^7*c^14*d + 28*a^2*b^6*c^13*d^2 - 56*a^3*b^5*c^12*d^3 + 70*a^4*b^4*c^11*d^4 - 56*a^5*b^3*c^10*d^
5 + 28*a^6*b^2*c^9*d^6 - 8*a^7*b*c^8*d^7 + a^8*c^7*d^8))^(1/4)*log(-(7*b*c*d - 3*a*d^2)*sqrt(x) + (b^2*c^4 - 2
*a*b*c^3*d + a^2*c^2*d^2)*(-(2401*b^4*c^4*d^3 - 4116*a*b^3*c^3*d^4 + 2646*a^2*b^2*c^2*d^5 - 756*a^3*b*c*d^6 +
81*a^4*d^7)/(b^8*c^15 - 8*a*b^7*c^14*d + 28*a^2*b^6*c^13*d^2 - 56*a^3*b^5*c^12*d^3 + 70*a^4*b^4*c^11*d^4 - 56*
a^5*b^3*c^10*d^5 + 28*a^6*b^2*c^9*d^6 - 8*a^7*b*c^8*d^7 + a^8*c^7*d^8))^(1/4)) - (b*c^3 - a*c^2*d + (b*c^2*d -
a*c*d^2)*x^2)*(-(2401*b^4*c^4*d^3 - 4116*a*b^3*c^3*d^4 + 2646*a^2*b^2*c^2*d^5 - 756*a^3*b*c*d^6 + 81*a^4*d^7)
/(b^8*c^15 - 8*a*b^7*c^14*d + 28*a^2*b^6*c^13*d^2 - 56*a^3*b^5*c^12*d^3 + 70*a^4*b^4*c^11*d^4 - 56*a^5*b^3*c^1
0*d^5 + 28*a^6*b^2*c^9*d^6 - 8*a^7*b*c^8*d^7 + a^8*c^7*d^8))^(1/4)*log(-(7*b*c*d - 3*a*d^2)*sqrt(x) - (b^2*c^4
- 2*a*b*c^3*d + a^2*c^2*d^2)*(-(2401*b^4*c^4*d^3 - 4116*a*b^3*c^3*d^4 + 2646*a^2*b^2*c^2*d^5 - 756*a^3*b*c*d^
6 + 81*a^4*d^7)/(b^8*c^15 - 8*a*b^7*c^14*d + 28*a^2*b^6*c^13*d^2 - 56*a^3*b^5*c^12*d^3 + 70*a^4*b^4*c^11*d^4 -
56*a^5*b^3*c^10*d^5 + 28*a^6*b^2*c^9*d^6 - 8*a...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]
time = 1.78, size = 673, normalized size = 1.26 \begin {gather*} \frac {\left (a b^{3}\right )^{\frac {1}{4}} b \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 )}{\sqrt {2} a b^{2} c^{2} - 2 \, \sqrt {2} a^{2} b c d + \sqrt {2} a^{3} d^{2}} + \frac {\left (a b^{3}\right )^{\frac {1}{4}} b \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 )}{\sqrt {2} a b^{2} c^{2} - 2 \, \sqrt {2} a^{2} b c d + \sqrt {2} a^{3} d^{2}} + \frac {\left (a b^{3}\right )^{\frac {1}{4}} b \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{2 \, {\left (\sqrt {2} a b^{2} c^{2} - 2 \, \sqrt {2} a^{2} b c d + \sqrt {2} a^{3} d^{2}\right )}} - \frac {\left (a b^{3}\right )^{\frac {1}{4}} b \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{2 \, {\left (\sqrt {2} a b^{2} c^{2} - 2 \, \sqrt {2} a^{2} b c d + \sqrt {2} a^{3} d^{2}\right )}} - \frac {{\left (7 \, \left (c d^{3}\right )^{\frac {1}{4}} b c - 3 \, \left (c d^{3}\right )^{\frac {1}{4}} a d\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 )}{4 \, {\left (\sqrt {2} b^{2} c^{4} - 2 \, \sqrt {2} a b c^{3} d + \sqrt {2} a^{2} c^{2} d^{2}\right )}} - \frac {{\left (7 \, \left (c d^{3}\right )^{\frac {1}{4}} b c - 3 \, \left (c d^{3}\right )^{\frac {1}{4}} a d\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 )}{4 \, {\left (\sqrt {2} b^{2} c^{4} - 2 \, \sqrt {2} a b c^{3} d + \sqrt {2} a^{2} c^{2} d^{2}\right )}} - \frac {{\left (7 \, \left (c d^{3}\right )^{\frac {1}{4}} b c - 3 \, \left (c d^{3}\right )^{\frac {1}{4}} a d\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{8 \, {\left (\sqrt {2} b^{2} c^{4} - 2 \, \sqrt {2} a b c^{3} d + \sqrt {2} a^{2} c^{2} d^{2}\right )}} + \frac {{\left (7 \, \left (c d^{3}\right )^{\frac {1}{4}} b c - 3 \, \left (c d^{3}\right )^{\frac {1}{4}} a d\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{8 \, {\left (\sqrt {2} b^{2} c^{4} - 2 \, \sqrt {2} a b c^{3} d + \sqrt {2} a^{2} c^{2} d^{2}\right )}} - \frac {d \sqrt {x}}{2 \, {\left (b c^{2} - a c d\right )} {\left (d x^{2} + c\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*b^3)^(1/4)*b*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/(sqrt(2)*a*b^2*c^2 - 2*sqrt(
2)*a^2*b*c*d + sqrt(2)*a^3*d^2) + (a*b^3)^(1/4)*b*arctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^
(1/4))/(sqrt(2)*a*b^2*c^2 - 2*sqrt(2)*a^2*b*c*d + sqrt(2)*a^3*d^2) + 1/2*(a*b^3)^(1/4)*b*log(sqrt(2)*sqrt(x)*(
a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2)*a*b^2*c^2 - 2*sqrt(2)*a^2*b*c*d + sqrt(2)*a^3*d^2) - 1/2*(a*b^3)^(1/4)*b*
log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2)*a*b^2*c^2 - 2*sqrt(2)*a^2*b*c*d + sqrt(2)*a^3*d^2)
- 1/4*(7*(c*d^3)^(1/4)*b*c - 3*(c*d^3)^(1/4)*a*d)*arctan(1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) + 2*sqrt(x))/(c/d)^(
1/4))/(sqrt(2)*b^2*c^4 - 2*sqrt(2)*a*b*c^3*d + sqrt(2)*a^2*c^2*d^2) - 1/4*(7*(c*d^3)^(1/4)*b*c - 3*(c*d^3)^(1/
4)*a*d)*arctan(-1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) - 2*sqrt(x))/(c/d)^(1/4))/(sqrt(2)*b^2*c^4 - 2*sqrt(2)*a*b*c^
3*d + sqrt(2)*a^2*c^2*d^2) - 1/8*(7*(c*d^3)^(1/4)*b*c - 3*(c*d^3)^(1/4)*a*d)*log(sqrt(2)*sqrt(x)*(c/d)^(1/4) +
x + sqrt(c/d))/(sqrt(2)*b^2*c^4 - 2*sqrt(2)*a*b*c^3*d + sqrt(2)*a^2*c^2*d^2) + 1/8*(7*(c*d^3)^(1/4)*b*c - 3*(
c*d^3)^(1/4)*a*d)*log(-sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(sqrt(2)*b^2*c^4 - 2*sqrt(2)*a*b*c^3*d + s
qrt(2)*a^2*c^2*d^2) - 1/2*d*sqrt(x)/((b*c^2 - a*c*d)*(d*x^2 + c))

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Mupad [B]
time = 1.42, size = 2500, normalized size = 4.66 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

atan(((-b^7/(16*a^11*d^8 + 16*a^3*b^8*c^8 - 128*a^4*b^7*c^7*d + 448*a^5*b^6*c^6*d^2 - 896*a^6*b^5*c^5*d^3 + 11
20*a^7*b^4*c^4*d^4 - 896*a^8*b^3*c^3*d^5 + 448*a^9*b^2*c^2*d^6 - 128*a^10*b*c*d^7))^(1/4)*((-b^7/(16*a^11*d^8
+ 16*a^3*b^8*c^8 - 128*a^4*b^7*c^7*d + 448*a^5*b^6*c^6*d^2 - 896*a^6*b^5*c^5*d^3 + 1120*a^7*b^4*c^4*d^4 - 896*
a^8*b^3*c^3*d^5 + 448*a^9*b^2*c^2*d^6 - 128*a^10*b*c*d^7))^(1/4)*((2*(81*a^4*b^7*d^10 + 448*b^11*c^4*d^6 - 214
5*a*b^10*c^3*d^7 - 675*a^3*b^8*c*d^9 + 1971*a^2*b^9*c^2*d^8))/(b^3*c^7 - a^3*c^4*d^3 + 3*a^2*b*c^5*d^2 - 3*a*b
^2*c^6*d) + (-b^7/(16*a^11*d^8 + 16*a^3*b^8*c^8 - 128*a^4*b^7*c^7*d + 448*a^5*b^6*c^6*d^2 - 896*a^6*b^5*c^5*d^
3 + 1120*a^7*b^4*c^4*d^4 - 896*a^8*b^3*c^3*d^5 + 448*a^9*b^2*c^2*d^6 - 128*a^10*b*c*d^7))^(3/4)*((2*(-b^7/(16*
a^11*d^8 + 16*a^3*b^8*c^8 - 128*a^4*b^7*c^7*d + 448*a^5*b^6*c^6*d^2 - 896*a^6*b^5*c^5*d^3 + 1120*a^7*b^4*c^4*d
^4 - 896*a^8*b^3*c^3*d^5 + 448*a^9*b^2*c^2*d^6 - 128*a^10*b*c*d^7))^(1/4)*(28672*a^2*b^13*c^13*d^5 - 4096*a*b^
14*c^14*d^4 - 78848*a^3*b^12*c^12*d^6 + 90112*a^4*b^11*c^11*d^7 + 28672*a^5*b^10*c^10*d^8 - 229376*a^6*b^9*c^9
*d^9 + 329728*a^7*b^8*c^8*d^10 - 253952*a^8*b^7*c^7*d^11 + 114688*a^9*b^6*c^6*d^12 - 28672*a^10*b^5*c^5*d^13 +
3072*a^11*b^4*c^4*d^14))/(b^3*c^7 - a^3*c^4*d^3 + 3*a^2*b*c^5*d^2 - 3*a*b^2*c^6*d) + (x^(1/2)*(4096*b^17*c^15
*d^4 - 32768*a*b^16*c^14*d^5 + 114688*a^2*b^15*c^13*d^6 - 216832*a^3*b^14*c^12*d^7 + 175616*a^4*b^13*c^11*d^8
+ 210176*a^5*b^12*c^10*d^9 - 907264*a^6*b^11*c^9*d^10 + 1511936*a^7*b^10*c^8*d^11 - 1580032*a^8*b^9*c^7*d^12 +
1114624*a^9*b^8*c^6*d^13 - 530432*a^10*b^7*c^5*d^14 + 163072*a^11*b^6*c^4*d^15 - 29184*a^12*b^5*c^3*d^16 + 23
04*a^13*b^4*c^2*d^17))/(b^6*c^10 + a^6*c^4*d^6 - 6*a^5*b*c^5*d^5 + 15*a^2*b^4*c^8*d^2 - 20*a^3*b^3*c^7*d^3 + 1
5*a^4*b^2*c^6*d^4 - 6*a*b^5*c^9*d))) + (x^(1/2)*(81*a^4*b^9*d^11 + 3185*b^13*c^4*d^7 - 4788*a*b^12*c^3*d^8 - 7
56*a^3*b^10*c*d^10 + 2790*a^2*b^11*c^2*d^9))/(b^6*c^10 + a^6*c^4*d^6 - 6*a^5*b*c^5*d^5 + 15*a^2*b^4*c^8*d^2 -
20*a^3*b^3*c^7*d^3 + 15*a^4*b^2*c^6*d^4 - 6*a*b^5*c^9*d))*1i - (-b^7/(16*a^11*d^8 + 16*a^3*b^8*c^8 - 128*a^4*b
^7*c^7*d + 448*a^5*b^6*c^6*d^2 - 896*a^6*b^5*c^5*d^3 + 1120*a^7*b^4*c^4*d^4 - 896*a^8*b^3*c^3*d^5 + 448*a^9*b^
2*c^2*d^6 - 128*a^10*b*c*d^7))^(1/4)*((-b^7/(16*a^11*d^8 + 16*a^3*b^8*c^8 - 128*a^4*b^7*c^7*d + 448*a^5*b^6*c^
6*d^2 - 896*a^6*b^5*c^5*d^3 + 1120*a^7*b^4*c^4*d^4 - 896*a^8*b^3*c^3*d^5 + 448*a^9*b^2*c^2*d^6 - 128*a^10*b*c*
d^7))^(1/4)*((2*(81*a^4*b^7*d^10 + 448*b^11*c^4*d^6 - 2145*a*b^10*c^3*d^7 - 675*a^3*b^8*c*d^9 + 1971*a^2*b^9*c
^2*d^8))/(b^3*c^7 - a^3*c^4*d^3 + 3*a^2*b*c^5*d^2 - 3*a*b^2*c^6*d) + (-b^7/(16*a^11*d^8 + 16*a^3*b^8*c^8 - 128
*a^4*b^7*c^7*d + 448*a^5*b^6*c^6*d^2 - 896*a^6*b^5*c^5*d^3 + 1120*a^7*b^4*c^4*d^4 - 896*a^8*b^3*c^3*d^5 + 448*
a^9*b^2*c^2*d^6 - 128*a^10*b*c*d^7))^(3/4)*((2*(-b^7/(16*a^11*d^8 + 16*a^3*b^8*c^8 - 128*a^4*b^7*c^7*d + 448*a
^5*b^6*c^6*d^2 - 896*a^6*b^5*c^5*d^3 + 1120*a^7*b^4*c^4*d^4 - 896*a^8*b^3*c^3*d^5 + 448*a^9*b^2*c^2*d^6 - 128*
a^10*b*c*d^7))^(1/4)*(28672*a^2*b^13*c^13*d^5 - 4096*a*b^14*c^14*d^4 - 78848*a^3*b^12*c^12*d^6 + 90112*a^4*b^1
1*c^11*d^7 + 28672*a^5*b^10*c^10*d^8 - 229376*a^6*b^9*c^9*d^9 + 329728*a^7*b^8*c^8*d^10 - 253952*a^8*b^7*c^7*d
^11 + 114688*a^9*b^6*c^6*d^12 - 28672*a^10*b^5*c^5*d^13 + 3072*a^11*b^4*c^4*d^14))/(b^3*c^7 - a^3*c^4*d^3 + 3*
a^2*b*c^5*d^2 - 3*a*b^2*c^6*d) - (x^(1/2)*(4096*b^17*c^15*d^4 - 32768*a*b^16*c^14*d^5 + 114688*a^2*b^15*c^13*d
^6 - 216832*a^3*b^14*c^12*d^7 + 175616*a^4*b^13*c^11*d^8 + 210176*a^5*b^12*c^10*d^9 - 907264*a^6*b^11*c^9*d^10
+ 1511936*a^7*b^10*c^8*d^11 - 1580032*a^8*b^9*c^7*d^12 + 1114624*a^9*b^8*c^6*d^13 - 530432*a^10*b^7*c^5*d^14
+ 163072*a^11*b^6*c^4*d^15 - 29184*a^12*b^5*c^3*d^16 + 2304*a^13*b^4*c^2*d^17))/(b^6*c^10 + a^6*c^4*d^6 - 6*a^
5*b*c^5*d^5 + 15*a^2*b^4*c^8*d^2 - 20*a^3*b^3*c^7*d^3 + 15*a^4*b^2*c^6*d^4 - 6*a*b^5*c^9*d))) - (x^(1/2)*(81*a
^4*b^9*d^11 + 3185*b^13*c^4*d^7 - 4788*a*b^12*c^3*d^8 - 756*a^3*b^10*c*d^10 + 2790*a^2*b^11*c^2*d^9))/(b^6*c^1
0 + a^6*c^4*d^6 - 6*a^5*b*c^5*d^5 + 15*a^2*b^4*c^8*d^2 - 20*a^3*b^3*c^7*d^3 + 15*a^4*b^2*c^6*d^4 - 6*a*b^5*c^9
*d))*1i)/((-b^7/(16*a^11*d^8 + 16*a^3*b^8*c^8 - 128*a^4*b^7*c^7*d + 448*a^5*b^6*c^6*d^2 - 896*a^6*b^5*c^5*d^3
+ 1120*a^7*b^4*c^4*d^4 - 896*a^8*b^3*c^3*d^5 + 448*a^9*b^2*c^2*d^6 - 128*a^10*b*c*d^7))^(1/4)*((-b^7/(16*a^11*
d^8 + 16*a^3*b^8*c^8 - 128*a^4*b^7*c^7*d + 448*a^5*b^6*c^6*d^2 - 896*a^6*b^5*c^5*d^3 + 1120*a^7*b^4*c^4*d^4 -
896*a^8*b^3*c^3*d^5 + 448*a^9*b^2*c^2*d^6 - 128*a^10*b*c*d^7))^(1/4)*((2*(81*a^4*b^7*d^10 + 448*b^11*c^4*d^6 -
2145*a*b^10*c^3*d^7 - 675*a^3*b^8*c*d^9 + 1971*a^2*b^9*c^2*d^8))/(b^3*c^7 - a^3*c^4*d^3 + 3*a^2*b*c^5*d^2 - 3
*a*b^2*c^6*d) + (-b^7/(16*a^11*d^8 + 16*a^3*b^8*c^8 - 128*a^4*b^7*c^7*d + 448*a^5*b^6*c^6*d^2 - 896*a^6*b^5*c^
5*d^3 + 1120*a^7*b^4*c^4*d^4 - 896*a^8*b^3*c^3*d^5 + 448*a^9*b^2*c^2*d^6 - 128*a^10*b*c*d^7))^(3/4)*((2*(-b^7/
(16*a^11*d^8 + 16*a^3*b^8*c^8 - 128*a^4*b^7*c^7*d + 448*a^5*b^6*c^6*d^2 - 896*a^6*b^5*c^5*d^3 + 1120*a^7*b^4*c
^4*d^4 - 896*a^8*b^3*c^3*d^5 + 448*a^9*b^2*c^2*...

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