3.379 \(\int \frac {A+B x^2}{\sqrt {x} (a+b x^2)^2} \, dx\)

Optimal. Leaf size=261 \[ -\frac {(a B+3 A b) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{7/4} b^{5/4}}+\frac {(a B+3 A b) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{7/4} b^{5/4}}-\frac {(a B+3 A b) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{7/4} b^{5/4}}+\frac {(a B+3 A b) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt {2} a^{7/4} b^{5/4}}+\frac {\sqrt {x} (A b-a B)}{2 a b \left (a+b x^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.18, antiderivative size = 261, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {457, 329, 211, 1165, 628, 1162, 617, 204} \[ -\frac {(a B+3 A b) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{7/4} b^{5/4}}+\frac {(a B+3 A b) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{7/4} b^{5/4}}-\frac {(a B+3 A b) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{7/4} b^{5/4}}+\frac {(a B+3 A b) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt {2} a^{7/4} b^{5/4}}+\frac {\sqrt {x} (A b-a B)}{2 a b \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x^2)/(Sqrt[x]*(a + b*x^2)^2),x]

[Out]

((A*b - a*B)*Sqrt[x])/(2*a*b*(a + b*x^2)) - ((3*A*b + a*B)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*S
qrt[2]*a^(7/4)*b^(5/4)) + ((3*A*b + a*B)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(7/4)*b^(
5/4)) - ((3*A*b + a*B)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(7/4)*b^(5/4))
+ ((3*A*b + a*B)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(7/4)*b^(5/4))

Rule 204

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

Rule 211

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 329

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 457

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> -Simp[((b*c - a*d
)*(e*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*b*e*n*(p + 1)), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*b
*n*(p + 1)), Int[(e*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &
& LtQ[p, -1] && (( !IntegerQ[p + 1/2] && NeQ[p, -5/4]) ||  !RationalQ[m] || (IGtQ[n, 0] && ILtQ[p + 1/2, 0] &&
LeQ[-1, m, -(n*(p + 1))]))

Rule 617

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 628

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 1162

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 1165

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

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Mathematica [A]  time = 0.23, size = 203, normalized size = 0.78 \[ \frac {\frac {(a B+3 A b) \left (8 a^{3/4} \sqrt [4]{b} \sqrt {x}-3 \sqrt {2} \left (a+b x^2\right ) \left (\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )-\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )+2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )-2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )\right )\right )}{a^{7/4} \sqrt [4]{b}}-32 B \sqrt {x}}{48 b \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x^2)/(Sqrt[x]*(a + b*x^2)^2),x]

[Out]

(-32*B*Sqrt[x] + ((3*A*b + a*B)*(8*a^(3/4)*b^(1/4)*Sqrt[x] - 3*Sqrt[2]*(a + b*x^2)*(2*ArcTan[1 - (Sqrt[2]*b^(1
/4)*Sqrt[x])/a^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4
)*Sqrt[x] + Sqrt[b]*x] - Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])))/(a^(7/4)*b^(1/4)))/(48*
b*(a + b*x^2))

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fricas [B]  time = 0.54, size = 717, normalized size = 2.75 \[ \frac {4 \, {\left (a b^{2} x^{2} + a^{2} b\right )} \left (-\frac {B^{4} a^{4} + 12 \, A B^{3} a^{3} b + 54 \, A^{2} B^{2} a^{2} b^{2} + 108 \, A^{3} B a b^{3} + 81 \, A^{4} b^{4}}{a^{7} b^{5}}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {a^{4} b^{2} \sqrt {-\frac {B^{4} a^{4} + 12 \, A B^{3} a^{3} b + 54 \, A^{2} B^{2} a^{2} b^{2} + 108 \, A^{3} B a b^{3} + 81 \, A^{4} b^{4}}{a^{7} b^{5}}} + {\left (B^{2} a^{2} + 6 \, A B a b + 9 \, A^{2} b^{2}\right )} x} a^{5} b^{4} \left (-\frac {B^{4} a^{4} + 12 \, A B^{3} a^{3} b + 54 \, A^{2} B^{2} a^{2} b^{2} + 108 \, A^{3} B a b^{3} + 81 \, A^{4} b^{4}}{a^{7} b^{5}}\right )^{\frac {3}{4}} - {\left (B a^{6} b^{4} + 3 \, A a^{5} b^{5}\right )} \sqrt {x} \left (-\frac {B^{4} a^{4} + 12 \, A B^{3} a^{3} b + 54 \, A^{2} B^{2} a^{2} b^{2} + 108 \, A^{3} B a b^{3} + 81 \, A^{4} b^{4}}{a^{7} b^{5}}\right )^{\frac {3}{4}}}{B^{4} a^{4} + 12 \, A B^{3} a^{3} b + 54 \, A^{2} B^{2} a^{2} b^{2} + 108 \, A^{3} B a b^{3} + 81 \, A^{4} b^{4}}\right ) + {\left (a b^{2} x^{2} + a^{2} b\right )} \left (-\frac {B^{4} a^{4} + 12 \, A B^{3} a^{3} b + 54 \, A^{2} B^{2} a^{2} b^{2} + 108 \, A^{3} B a b^{3} + 81 \, A^{4} b^{4}}{a^{7} b^{5}}\right )^{\frac {1}{4}} \log \left (a^{2} b \left (-\frac {B^{4} a^{4} + 12 \, A B^{3} a^{3} b + 54 \, A^{2} B^{2} a^{2} b^{2} + 108 \, A^{3} B a b^{3} + 81 \, A^{4} b^{4}}{a^{7} b^{5}}\right )^{\frac {1}{4}} + {\left (B a + 3 \, A b\right )} \sqrt {x}\right ) - {\left (a b^{2} x^{2} + a^{2} b\right )} \left (-\frac {B^{4} a^{4} + 12 \, A B^{3} a^{3} b + 54 \, A^{2} B^{2} a^{2} b^{2} + 108 \, A^{3} B a b^{3} + 81 \, A^{4} b^{4}}{a^{7} b^{5}}\right )^{\frac {1}{4}} \log \left (-a^{2} b \left (-\frac {B^{4} a^{4} + 12 \, A B^{3} a^{3} b + 54 \, A^{2} B^{2} a^{2} b^{2} + 108 \, A^{3} B a b^{3} + 81 \, A^{4} b^{4}}{a^{7} b^{5}}\right )^{\frac {1}{4}} + {\left (B a + 3 \, A b\right )} \sqrt {x}\right ) - 4 \, {\left (B a - A b\right )} \sqrt {x}}{8 \, {\left (a b^{2} x^{2} + a^{2} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(4*(a*b^2*x^2 + a^2*b)*(-(B^4*a^4 + 12*A*B^3*a^3*b + 54*A^2*B^2*a^2*b^2 + 108*A^3*B*a*b^3 + 81*A^4*b^4)/(a
^7*b^5))^(1/4)*arctan((sqrt(a^4*b^2*sqrt(-(B^4*a^4 + 12*A*B^3*a^3*b + 54*A^2*B^2*a^2*b^2 + 108*A^3*B*a*b^3 + 8
1*A^4*b^4)/(a^7*b^5)) + (B^2*a^2 + 6*A*B*a*b + 9*A^2*b^2)*x)*a^5*b^4*(-(B^4*a^4 + 12*A*B^3*a^3*b + 54*A^2*B^2*
a^2*b^2 + 108*A^3*B*a*b^3 + 81*A^4*b^4)/(a^7*b^5))^(3/4) - (B*a^6*b^4 + 3*A*a^5*b^5)*sqrt(x)*(-(B^4*a^4 + 12*A
*B^3*a^3*b + 54*A^2*B^2*a^2*b^2 + 108*A^3*B*a*b^3 + 81*A^4*b^4)/(a^7*b^5))^(3/4))/(B^4*a^4 + 12*A*B^3*a^3*b +
54*A^2*B^2*a^2*b^2 + 108*A^3*B*a*b^3 + 81*A^4*b^4)) + (a*b^2*x^2 + a^2*b)*(-(B^4*a^4 + 12*A*B^3*a^3*b + 54*A^2
*B^2*a^2*b^2 + 108*A^3*B*a*b^3 + 81*A^4*b^4)/(a^7*b^5))^(1/4)*log(a^2*b*(-(B^4*a^4 + 12*A*B^3*a^3*b + 54*A^2*B
^2*a^2*b^2 + 108*A^3*B*a*b^3 + 81*A^4*b^4)/(a^7*b^5))^(1/4) + (B*a + 3*A*b)*sqrt(x)) - (a*b^2*x^2 + a^2*b)*(-(
B^4*a^4 + 12*A*B^3*a^3*b + 54*A^2*B^2*a^2*b^2 + 108*A^3*B*a*b^3 + 81*A^4*b^4)/(a^7*b^5))^(1/4)*log(-a^2*b*(-(B
^4*a^4 + 12*A*B^3*a^3*b + 54*A^2*B^2*a^2*b^2 + 108*A^3*B*a*b^3 + 81*A^4*b^4)/(a^7*b^5))^(1/4) + (B*a + 3*A*b)*
sqrt(x)) - 4*(B*a - A*b)*sqrt(x))/(a*b^2*x^2 + a^2*b)

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giac [A]  time = 0.38, size = 273, normalized size = 1.05 \[ \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} B a + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \, a^{2} b^{2}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} B a + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \, a^{2} b^{2}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} B a + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{16 \, a^{2} b^{2}} - \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} B a + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{16 \, a^{2} b^{2}} - \frac {B a \sqrt {x} - A b \sqrt {x}}{2 \, {\left (b x^{2} + a\right )} a b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*sqrt(2)*((a*b^3)^(1/4)*B*a + 3*(a*b^3)^(1/4)*A*b)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/
b)^(1/4))/(a^2*b^2) + 1/8*sqrt(2)*((a*b^3)^(1/4)*B*a + 3*(a*b^3)^(1/4)*A*b)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)
^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(a^2*b^2) + 1/16*sqrt(2)*((a*b^3)^(1/4)*B*a + 3*(a*b^3)^(1/4)*A*b)*log(sqrt(2
)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^2*b^2) - 1/16*sqrt(2)*((a*b^3)^(1/4)*B*a + 3*(a*b^3)^(1/4)*A*b)*log(
-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^2*b^2) - 1/2*(B*a*sqrt(x) - A*b*sqrt(x))/((b*x^2 + a)*a*b)

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maple [A]  time = 0.01, size = 305, normalized size = 1.17 \[ \frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, A \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{8 a^{2}}+\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, A \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{8 a^{2}}+\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, A \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}\right )}{16 a^{2}}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, B \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{8 a b}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, B \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{8 a b}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, B \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}\right )}{16 a b}+\frac {\left (A b -B a \right ) \sqrt {x}}{2 \left (b \,x^{2}+a \right ) a b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x^2+A)/(b*x^2+a)^2/x^(1/2),x)

[Out]

1/2*(A*b-B*a)*x^(1/2)/a/b/(b*x^2+a)+3/8/a^2*(a/b)^(1/4)*2^(1/2)*A*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)+3/8/a^
2*(a/b)^(1/4)*2^(1/2)*A*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)+3/16/a^2*(a/b)^(1/4)*2^(1/2)*A*ln((x+(a/b)^(1/4)
*2^(1/2)*x^(1/2)+(a/b)^(1/2))/(x-(a/b)^(1/4)*2^(1/2)*x^(1/2)+(a/b)^(1/2)))+1/8/a/b*(a/b)^(1/4)*2^(1/2)*B*arcta
n(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)+1/8/a/b*(a/b)^(1/4)*2^(1/2)*B*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)+1/16/a/b*
(a/b)^(1/4)*2^(1/2)*B*ln((x+(a/b)^(1/4)*2^(1/2)*x^(1/2)+(a/b)^(1/2))/(x-(a/b)^(1/4)*2^(1/2)*x^(1/2)+(a/b)^(1/2
)))

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maxima [A]  time = 2.46, size = 241, normalized size = 0.92 \[ -\frac {{\left (B a - A b\right )} \sqrt {x}}{2 \, {\left (a b^{2} x^{2} + a^{2} b\right )}} + \frac {\frac {2 \, \sqrt {2} {\left (B a + 3 \, A b\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} {\left (B a + 3 \, A b\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} {\left (B a + 3 \, A b\right )} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (B a + 3 \, A b\right )} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}}}{16 \, a b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(B*a - A*b)*sqrt(x)/(a*b^2*x^2 + a^2*b) + 1/16*(2*sqrt(2)*(B*a + 3*A*b)*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*a + 3*A*
b)*arctan(-1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) - 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt
(a)*sqrt(b))) + sqrt(2)*(B*a + 3*A*b)*log(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(3/4)*b^(1
/4)) - sqrt(2)*(B*a + 3*A*b)*log(-sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(3/4)*b^(1/4)))/(a
*b)

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mupad [B]  time = 0.43, size = 750, normalized size = 2.87 \[ \frac {\mathrm {atan}\left (\frac {\frac {\left (3\,A\,b+B\,a\right )\,\left (\frac {\sqrt {x}\,\left (9\,A^2\,b^3+6\,A\,B\,a\,b^2+B^2\,a^2\,b\right )}{a^2}-\frac {\left (3\,A\,b+B\,a\right )\,\left (24\,A\,b^3+8\,B\,a\,b^2\right )\,1{}\mathrm {i}}{8\,{\left (-a\right )}^{7/4}\,b^{5/4}}\right )}{8\,{\left (-a\right )}^{7/4}\,b^{5/4}}+\frac {\left (3\,A\,b+B\,a\right )\,\left (\frac {\sqrt {x}\,\left (9\,A^2\,b^3+6\,A\,B\,a\,b^2+B^2\,a^2\,b\right )}{a^2}+\frac {\left (3\,A\,b+B\,a\right )\,\left (24\,A\,b^3+8\,B\,a\,b^2\right )\,1{}\mathrm {i}}{8\,{\left (-a\right )}^{7/4}\,b^{5/4}}\right )}{8\,{\left (-a\right )}^{7/4}\,b^{5/4}}}{\frac {\left (3\,A\,b+B\,a\right )\,\left (\frac {\sqrt {x}\,\left (9\,A^2\,b^3+6\,A\,B\,a\,b^2+B^2\,a^2\,b\right )}{a^2}-\frac {\left (3\,A\,b+B\,a\right )\,\left (24\,A\,b^3+8\,B\,a\,b^2\right )\,1{}\mathrm {i}}{8\,{\left (-a\right )}^{7/4}\,b^{5/4}}\right )\,1{}\mathrm {i}}{8\,{\left (-a\right )}^{7/4}\,b^{5/4}}-\frac {\left (3\,A\,b+B\,a\right )\,\left (\frac {\sqrt {x}\,\left (9\,A^2\,b^3+6\,A\,B\,a\,b^2+B^2\,a^2\,b\right )}{a^2}+\frac {\left (3\,A\,b+B\,a\right )\,\left (24\,A\,b^3+8\,B\,a\,b^2\right )\,1{}\mathrm {i}}{8\,{\left (-a\right )}^{7/4}\,b^{5/4}}\right )\,1{}\mathrm {i}}{8\,{\left (-a\right )}^{7/4}\,b^{5/4}}}\right )\,\left (3\,A\,b+B\,a\right )}{4\,{\left (-a\right )}^{7/4}\,b^{5/4}}+\frac {\sqrt {x}\,\left (A\,b-B\,a\right )}{2\,a\,b\,\left (b\,x^2+a\right )}+\frac {\mathrm {atan}\left (\frac {\frac {\left (3\,A\,b+B\,a\right )\,\left (\frac {\sqrt {x}\,\left (9\,A^2\,b^3+6\,A\,B\,a\,b^2+B^2\,a^2\,b\right )}{a^2}-\frac {\left (3\,A\,b+B\,a\right )\,\left (24\,A\,b^3+8\,B\,a\,b^2\right )}{8\,{\left (-a\right )}^{7/4}\,b^{5/4}}\right )\,1{}\mathrm {i}}{8\,{\left (-a\right )}^{7/4}\,b^{5/4}}+\frac {\left (3\,A\,b+B\,a\right )\,\left (\frac {\sqrt {x}\,\left (9\,A^2\,b^3+6\,A\,B\,a\,b^2+B^2\,a^2\,b\right )}{a^2}+\frac {\left (3\,A\,b+B\,a\right )\,\left (24\,A\,b^3+8\,B\,a\,b^2\right )}{8\,{\left (-a\right )}^{7/4}\,b^{5/4}}\right )\,1{}\mathrm {i}}{8\,{\left (-a\right )}^{7/4}\,b^{5/4}}}{\frac {\left (3\,A\,b+B\,a\right )\,\left (\frac {\sqrt {x}\,\left (9\,A^2\,b^3+6\,A\,B\,a\,b^2+B^2\,a^2\,b\right )}{a^2}-\frac {\left (3\,A\,b+B\,a\right )\,\left (24\,A\,b^3+8\,B\,a\,b^2\right )}{8\,{\left (-a\right )}^{7/4}\,b^{5/4}}\right )}{8\,{\left (-a\right )}^{7/4}\,b^{5/4}}-\frac {\left (3\,A\,b+B\,a\right )\,\left (\frac {\sqrt {x}\,\left (9\,A^2\,b^3+6\,A\,B\,a\,b^2+B^2\,a^2\,b\right )}{a^2}+\frac {\left (3\,A\,b+B\,a\right )\,\left (24\,A\,b^3+8\,B\,a\,b^2\right )}{8\,{\left (-a\right )}^{7/4}\,b^{5/4}}\right )}{8\,{\left (-a\right )}^{7/4}\,b^{5/4}}}\right )\,\left (3\,A\,b+B\,a\right )\,1{}\mathrm {i}}{4\,{\left (-a\right )}^{7/4}\,b^{5/4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x^2)/(x^(1/2)*(a + b*x^2)^2),x)

[Out]

(atan((((3*A*b + B*a)*((x^(1/2)*(9*A^2*b^3 + B^2*a^2*b + 6*A*B*a*b^2))/a^2 - ((3*A*b + B*a)*(24*A*b^3 + 8*B*a*
b^2))/(8*(-a)^(7/4)*b^(5/4)))*1i)/(8*(-a)^(7/4)*b^(5/4)) + ((3*A*b + B*a)*((x^(1/2)*(9*A^2*b^3 + B^2*a^2*b + 6
*A*B*a*b^2))/a^2 + ((3*A*b + B*a)*(24*A*b^3 + 8*B*a*b^2))/(8*(-a)^(7/4)*b^(5/4)))*1i)/(8*(-a)^(7/4)*b^(5/4)))/
(((3*A*b + B*a)*((x^(1/2)*(9*A^2*b^3 + B^2*a^2*b + 6*A*B*a*b^2))/a^2 - ((3*A*b + B*a)*(24*A*b^3 + 8*B*a*b^2))/
(8*(-a)^(7/4)*b^(5/4))))/(8*(-a)^(7/4)*b^(5/4)) - ((3*A*b + B*a)*((x^(1/2)*(9*A^2*b^3 + B^2*a^2*b + 6*A*B*a*b^
2))/a^2 + ((3*A*b + B*a)*(24*A*b^3 + 8*B*a*b^2))/(8*(-a)^(7/4)*b^(5/4))))/(8*(-a)^(7/4)*b^(5/4))))*(3*A*b + B*
a)*1i)/(4*(-a)^(7/4)*b^(5/4)) + (atan((((3*A*b + B*a)*((x^(1/2)*(9*A^2*b^3 + B^2*a^2*b + 6*A*B*a*b^2))/a^2 - (
(3*A*b + B*a)*(24*A*b^3 + 8*B*a*b^2)*1i)/(8*(-a)^(7/4)*b^(5/4))))/(8*(-a)^(7/4)*b^(5/4)) + ((3*A*b + B*a)*((x^
(1/2)*(9*A^2*b^3 + B^2*a^2*b + 6*A*B*a*b^2))/a^2 + ((3*A*b + B*a)*(24*A*b^3 + 8*B*a*b^2)*1i)/(8*(-a)^(7/4)*b^(
5/4))))/(8*(-a)^(7/4)*b^(5/4)))/(((3*A*b + B*a)*((x^(1/2)*(9*A^2*b^3 + B^2*a^2*b + 6*A*B*a*b^2))/a^2 - ((3*A*b
+ B*a)*(24*A*b^3 + 8*B*a*b^2)*1i)/(8*(-a)^(7/4)*b^(5/4)))*1i)/(8*(-a)^(7/4)*b^(5/4)) - ((3*A*b + B*a)*((x^(1/
2)*(9*A^2*b^3 + B^2*a^2*b + 6*A*B*a*b^2))/a^2 + ((3*A*b + B*a)*(24*A*b^3 + 8*B*a*b^2)*1i)/(8*(-a)^(7/4)*b^(5/4
)))*1i)/(8*(-a)^(7/4)*b^(5/4))))*(3*A*b + B*a))/(4*(-a)^(7/4)*b^(5/4)) + (x^(1/2)*(A*b - B*a))/(2*a*b*(a + b*x
^2))

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sympy [A]  time = 78.54, size = 959, normalized size = 3.67 \[ \begin {cases} \tilde {\infty } \left (- \frac {2 A}{7 x^{\frac {7}{2}}} - \frac {2 B}{3 x^{\frac {3}{2}}}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {2 A \sqrt {x} + \frac {2 B x^{\frac {5}{2}}}{5}}{a^{2}} & \text {for}\: b = 0 \\\frac {- \frac {2 A}{7 x^{\frac {7}{2}}} - \frac {2 B}{3 x^{\frac {3}{2}}}}{b^{2}} & \text {for}\: a = 0 \\- \frac {3 \sqrt [4]{-1} A a^{\frac {5}{4}} b \sqrt [4]{\frac {1}{b}} \log {\left (- \sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac {1}{b}} + \sqrt {x} \right )}}{8 a^{3} b + 8 a^{2} b^{2} x^{2}} + \frac {3 \sqrt [4]{-1} A a^{\frac {5}{4}} b \sqrt [4]{\frac {1}{b}} \log {\left (\sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac {1}{b}} + \sqrt {x} \right )}}{8 a^{3} b + 8 a^{2} b^{2} x^{2}} - \frac {6 \sqrt [4]{-1} A a^{\frac {5}{4}} b \sqrt [4]{\frac {1}{b}} \operatorname {atan}{\left (\frac {\left (-1\right )^{\frac {3}{4}} \sqrt {x}}{\sqrt [4]{a} \sqrt [4]{\frac {1}{b}}} \right )}}{8 a^{3} b + 8 a^{2} b^{2} x^{2}} - \frac {3 \sqrt [4]{-1} A \sqrt [4]{a} b^{2} x^{2} \sqrt [4]{\frac {1}{b}} \log {\left (- \sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac {1}{b}} + \sqrt {x} \right )}}{8 a^{3} b + 8 a^{2} b^{2} x^{2}} + \frac {3 \sqrt [4]{-1} A \sqrt [4]{a} b^{2} x^{2} \sqrt [4]{\frac {1}{b}} \log {\left (\sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac {1}{b}} + \sqrt {x} \right )}}{8 a^{3} b + 8 a^{2} b^{2} x^{2}} - \frac {6 \sqrt [4]{-1} A \sqrt [4]{a} b^{2} x^{2} \sqrt [4]{\frac {1}{b}} \operatorname {atan}{\left (\frac {\left (-1\right )^{\frac {3}{4}} \sqrt {x}}{\sqrt [4]{a} \sqrt [4]{\frac {1}{b}}} \right )}}{8 a^{3} b + 8 a^{2} b^{2} x^{2}} + \frac {4 A a b \sqrt {x}}{8 a^{3} b + 8 a^{2} b^{2} x^{2}} - \frac {\sqrt [4]{-1} B a^{\frac {9}{4}} \sqrt [4]{\frac {1}{b}} \log {\left (- \sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac {1}{b}} + \sqrt {x} \right )}}{8 a^{3} b + 8 a^{2} b^{2} x^{2}} + \frac {\sqrt [4]{-1} B a^{\frac {9}{4}} \sqrt [4]{\frac {1}{b}} \log {\left (\sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac {1}{b}} + \sqrt {x} \right )}}{8 a^{3} b + 8 a^{2} b^{2} x^{2}} - \frac {2 \sqrt [4]{-1} B a^{\frac {9}{4}} \sqrt [4]{\frac {1}{b}} \operatorname {atan}{\left (\frac {\left (-1\right )^{\frac {3}{4}} \sqrt {x}}{\sqrt [4]{a} \sqrt [4]{\frac {1}{b}}} \right )}}{8 a^{3} b + 8 a^{2} b^{2} x^{2}} - \frac {\sqrt [4]{-1} B a^{\frac {5}{4}} b x^{2} \sqrt [4]{\frac {1}{b}} \log {\left (- \sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac {1}{b}} + \sqrt {x} \right )}}{8 a^{3} b + 8 a^{2} b^{2} x^{2}} + \frac {\sqrt [4]{-1} B a^{\frac {5}{4}} b x^{2} \sqrt [4]{\frac {1}{b}} \log {\left (\sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac {1}{b}} + \sqrt {x} \right )}}{8 a^{3} b + 8 a^{2} b^{2} x^{2}} - \frac {2 \sqrt [4]{-1} B a^{\frac {5}{4}} b x^{2} \sqrt [4]{\frac {1}{b}} \operatorname {atan}{\left (\frac {\left (-1\right )^{\frac {3}{4}} \sqrt {x}}{\sqrt [4]{a} \sqrt [4]{\frac {1}{b}}} \right )}}{8 a^{3} b + 8 a^{2} b^{2} x^{2}} - \frac {4 B a^{2} \sqrt {x}}{8 a^{3} b + 8 a^{2} b^{2} x^{2}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x**2+A)/(b*x**2+a)**2/x**(1/2),x)

[Out]

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

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