3.5.24 \(\int \frac {A+B x}{x^{3/2} (a+c x^2)^2} \, dx\)

Optimal. Leaf size=304 \[ -\frac {\left (3 \sqrt {a} B+5 A \sqrt {c}\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{8 \sqrt {2} a^{9/4} \sqrt [4]{c}}+\frac {\left (3 \sqrt {a} B+5 A \sqrt {c}\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{8 \sqrt {2} a^{9/4} \sqrt [4]{c}}-\frac {\left (3 \sqrt {a} B-5 A \sqrt {c}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{9/4} \sqrt [4]{c}}+\frac {\left (3 \sqrt {a} B-5 A \sqrt {c}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt {2} a^{9/4} \sqrt [4]{c}}-\frac {5 A}{2 a^2 \sqrt {x}}+\frac {A+B x}{2 a \sqrt {x} \left (a+c x^2\right )} \]

________________________________________________________________________________________

Rubi [A]  time = 0.30, antiderivative size = 304, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {823, 829, 827, 1168, 1162, 617, 204, 1165, 628} \begin {gather*} -\frac {\left (3 \sqrt {a} B+5 A \sqrt {c}\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{8 \sqrt {2} a^{9/4} \sqrt [4]{c}}+\frac {\left (3 \sqrt {a} B+5 A \sqrt {c}\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{8 \sqrt {2} a^{9/4} \sqrt [4]{c}}-\frac {\left (3 \sqrt {a} B-5 A \sqrt {c}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{9/4} \sqrt [4]{c}}+\frac {\left (3 \sqrt {a} B-5 A \sqrt {c}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt {2} a^{9/4} \sqrt [4]{c}}-\frac {5 A}{2 a^2 \sqrt {x}}+\frac {A+B x}{2 a \sqrt {x} \left (a+c x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/(x^(3/2)*(a + c*x^2)^2),x]

[Out]

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

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 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 823

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(
m + 1)*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), x] + Di
st[1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Simp[f*(c^2*d^2*(2*p + 3) + a*c*e^2*
(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 827

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2, Subst[Int[(e*f
- d*g + g*x^2)/(c*d^2 + a*e^2 - 2*c*d*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0]

Rule 829

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Simp[((e*f - d*g)*(d
+ e*x)^(m + 1))/((m + 1)*(c*d^2 + a*e^2)), x] + Dist[1/(c*d^2 + a*e^2), Int[((d + e*x)^(m + 1)*Simp[c*d*f + a*
e*g - c*(e*f - d*g)*x, x])/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2, 0] &&
FractionQ[m] && LtQ[m, -1]

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]

Rule 1168

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a*c, 2]}, Dist[(d*q + a*e)/(2*a*c),
Int[(q + c*x^2)/(a + c*x^4), x], x] + Dist[(d*q - a*e)/(2*a*c), Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ
[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[-(a*c)]

Rubi steps

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

________________________________________________________________________________________

Mathematica [C]  time = 0.46, size = 256, normalized size = 0.84 \begin {gather*} \frac {\sqrt [4]{a} \left (\frac {8 a^{3/4} A}{\sqrt {x} \left (a+c x^2\right )}+\frac {8 a^{3/4} B \sqrt {x}}{a+c x^2}-\frac {3 \sqrt {2} B \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{\sqrt [4]{c}}+\frac {3 \sqrt {2} B \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{\sqrt [4]{c}}-\frac {6 \sqrt {2} B \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt [4]{c}}+\frac {6 \sqrt {2} B \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt [4]{c}}\right )-\frac {40 A \, _2F_1\left (-\frac {1}{4},1;\frac {3}{4};-\frac {c x^2}{a}\right )}{\sqrt {x}}}{16 a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/(x^(3/2)*(a + c*x^2)^2),x]

[Out]

((-40*A*Hypergeometric2F1[-1/4, 1, 3/4, -((c*x^2)/a)])/Sqrt[x] + a^(1/4)*((8*a^(3/4)*A)/(Sqrt[x]*(a + c*x^2))
+ (8*a^(3/4)*B*Sqrt[x])/(a + c*x^2) - (6*Sqrt[2]*B*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)])/c^(1/4) + (6
*Sqrt[2]*B*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)])/c^(1/4) - (3*Sqrt[2]*B*Log[Sqrt[a] - Sqrt[2]*a^(1/4)
*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/c^(1/4) + (3*Sqrt[2]*B*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*
x])/c^(1/4)))/(16*a^2)

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 1.00, size = 183, normalized size = 0.60 \begin {gather*} -\frac {\left (3 \sqrt {a} B-5 A \sqrt {c}\right ) \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {c} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}}\right )}{4 \sqrt {2} a^{9/4} \sqrt [4]{c}}+\frac {\left (3 \sqrt {a} B+5 A \sqrt {c}\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}}{\sqrt {a}+\sqrt {c} x}\right )}{4 \sqrt {2} a^{9/4} \sqrt [4]{c}}+\frac {-4 a A+a B x-5 A c x^2}{2 a^2 \sqrt {x} \left (a+c x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(A + B*x)/(x^(3/2)*(a + c*x^2)^2),x]

[Out]

(-4*a*A + a*B*x - 5*A*c*x^2)/(2*a^2*Sqrt[x]*(a + c*x^2)) - ((3*Sqrt[a]*B - 5*A*Sqrt[c])*ArcTan[(Sqrt[a] - Sqrt
[c]*x)/(Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x])])/(4*Sqrt[2]*a^(9/4)*c^(1/4)) + ((3*Sqrt[a]*B + 5*A*Sqrt[c])*ArcTanh[
(Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[c]*x)])/(4*Sqrt[2]*a^(9/4)*c^(1/4))

________________________________________________________________________________________

fricas [B]  time = 0.45, size = 877, normalized size = 2.88 \begin {gather*} -\frac {{\left (a^{2} c x^{3} + a^{3} x\right )} \sqrt {\frac {a^{4} \sqrt {-\frac {81 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{9} c}} + 30 \, A B}{a^{4}}} \log \left (-{\left (81 \, B^{4} a^{2} - 625 \, A^{4} c^{2}\right )} \sqrt {x} + {\left (5 \, A a^{7} c \sqrt {-\frac {81 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{9} c}} + 27 \, B^{3} a^{4} - 75 \, A^{2} B a^{3} c\right )} \sqrt {\frac {a^{4} \sqrt {-\frac {81 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{9} c}} + 30 \, A B}{a^{4}}}\right ) - {\left (a^{2} c x^{3} + a^{3} x\right )} \sqrt {\frac {a^{4} \sqrt {-\frac {81 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{9} c}} + 30 \, A B}{a^{4}}} \log \left (-{\left (81 \, B^{4} a^{2} - 625 \, A^{4} c^{2}\right )} \sqrt {x} - {\left (5 \, A a^{7} c \sqrt {-\frac {81 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{9} c}} + 27 \, B^{3} a^{4} - 75 \, A^{2} B a^{3} c\right )} \sqrt {\frac {a^{4} \sqrt {-\frac {81 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{9} c}} + 30 \, A B}{a^{4}}}\right ) - {\left (a^{2} c x^{3} + a^{3} x\right )} \sqrt {-\frac {a^{4} \sqrt {-\frac {81 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{9} c}} - 30 \, A B}{a^{4}}} \log \left (-{\left (81 \, B^{4} a^{2} - 625 \, A^{4} c^{2}\right )} \sqrt {x} + {\left (5 \, A a^{7} c \sqrt {-\frac {81 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{9} c}} - 27 \, B^{3} a^{4} + 75 \, A^{2} B a^{3} c\right )} \sqrt {-\frac {a^{4} \sqrt {-\frac {81 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{9} c}} - 30 \, A B}{a^{4}}}\right ) + {\left (a^{2} c x^{3} + a^{3} x\right )} \sqrt {-\frac {a^{4} \sqrt {-\frac {81 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{9} c}} - 30 \, A B}{a^{4}}} \log \left (-{\left (81 \, B^{4} a^{2} - 625 \, A^{4} c^{2}\right )} \sqrt {x} - {\left (5 \, A a^{7} c \sqrt {-\frac {81 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{9} c}} - 27 \, B^{3} a^{4} + 75 \, A^{2} B a^{3} c\right )} \sqrt {-\frac {a^{4} \sqrt {-\frac {81 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{9} c}} - 30 \, A B}{a^{4}}}\right ) + 4 \, {\left (5 \, A c x^{2} - B a x + 4 \, A a\right )} \sqrt {x}}{8 \, {\left (a^{2} c x^{3} + a^{3} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*((a^2*c*x^3 + a^3*x)*sqrt((a^4*sqrt(-(81*B^4*a^2 - 450*A^2*B^2*a*c + 625*A^4*c^2)/(a^9*c)) + 30*A*B)/a^4)
*log(-(81*B^4*a^2 - 625*A^4*c^2)*sqrt(x) + (5*A*a^7*c*sqrt(-(81*B^4*a^2 - 450*A^2*B^2*a*c + 625*A^4*c^2)/(a^9*
c)) + 27*B^3*a^4 - 75*A^2*B*a^3*c)*sqrt((a^4*sqrt(-(81*B^4*a^2 - 450*A^2*B^2*a*c + 625*A^4*c^2)/(a^9*c)) + 30*
A*B)/a^4)) - (a^2*c*x^3 + a^3*x)*sqrt((a^4*sqrt(-(81*B^4*a^2 - 450*A^2*B^2*a*c + 625*A^4*c^2)/(a^9*c)) + 30*A*
B)/a^4)*log(-(81*B^4*a^2 - 625*A^4*c^2)*sqrt(x) - (5*A*a^7*c*sqrt(-(81*B^4*a^2 - 450*A^2*B^2*a*c + 625*A^4*c^2
)/(a^9*c)) + 27*B^3*a^4 - 75*A^2*B*a^3*c)*sqrt((a^4*sqrt(-(81*B^4*a^2 - 450*A^2*B^2*a*c + 625*A^4*c^2)/(a^9*c)
) + 30*A*B)/a^4)) - (a^2*c*x^3 + a^3*x)*sqrt(-(a^4*sqrt(-(81*B^4*a^2 - 450*A^2*B^2*a*c + 625*A^4*c^2)/(a^9*c))
- 30*A*B)/a^4)*log(-(81*B^4*a^2 - 625*A^4*c^2)*sqrt(x) + (5*A*a^7*c*sqrt(-(81*B^4*a^2 - 450*A^2*B^2*a*c + 625
*A^4*c^2)/(a^9*c)) - 27*B^3*a^4 + 75*A^2*B*a^3*c)*sqrt(-(a^4*sqrt(-(81*B^4*a^2 - 450*A^2*B^2*a*c + 625*A^4*c^2
)/(a^9*c)) - 30*A*B)/a^4)) + (a^2*c*x^3 + a^3*x)*sqrt(-(a^4*sqrt(-(81*B^4*a^2 - 450*A^2*B^2*a*c + 625*A^4*c^2)
/(a^9*c)) - 30*A*B)/a^4)*log(-(81*B^4*a^2 - 625*A^4*c^2)*sqrt(x) - (5*A*a^7*c*sqrt(-(81*B^4*a^2 - 450*A^2*B^2*
a*c + 625*A^4*c^2)/(a^9*c)) - 27*B^3*a^4 + 75*A^2*B*a^3*c)*sqrt(-(a^4*sqrt(-(81*B^4*a^2 - 450*A^2*B^2*a*c + 62
5*A^4*c^2)/(a^9*c)) - 30*A*B)/a^4)) + 4*(5*A*c*x^2 - B*a*x + 4*A*a)*sqrt(x))/(a^2*c*x^3 + a^3*x)

________________________________________________________________________________________

giac [A]  time = 0.22, size = 281, normalized size = 0.92 \begin {gather*} -\frac {5 \, A c x^{2} - B a x + 4 \, A a}{2 \, {\left (c x^{\frac {5}{2}} + a \sqrt {x}\right )} a^{2}} + \frac {\sqrt {2} {\left (3 \, \left (a c^{3}\right )^{\frac {1}{4}} B a c - 5 \, \left (a c^{3}\right )^{\frac {3}{4}} A\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{8 \, a^{3} c^{2}} + \frac {\sqrt {2} {\left (3 \, \left (a c^{3}\right )^{\frac {1}{4}} B a c - 5 \, \left (a c^{3}\right )^{\frac {3}{4}} A\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{8 \, a^{3} c^{2}} + \frac {\sqrt {2} {\left (3 \, \left (a c^{3}\right )^{\frac {1}{4}} B a c + 5 \, \left (a c^{3}\right )^{\frac {3}{4}} A\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{c}}\right )}{16 \, a^{3} c^{2}} - \frac {\sqrt {2} {\left (3 \, \left (a c^{3}\right )^{\frac {1}{4}} B a c + 5 \, \left (a c^{3}\right )^{\frac {3}{4}} A\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{c}}\right )}{16 \, a^{3} c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(5*A*c*x^2 - B*a*x + 4*A*a)/((c*x^(5/2) + a*sqrt(x))*a^2) + 1/8*sqrt(2)*(3*(a*c^3)^(1/4)*B*a*c - 5*(a*c^3
)^(3/4)*A)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/c)^(1/4) + 2*sqrt(x))/(a/c)^(1/4))/(a^3*c^2) + 1/8*sqrt(2)*(3*(a*c^3
)^(1/4)*B*a*c - 5*(a*c^3)^(3/4)*A)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a/c)^(1/4) - 2*sqrt(x))/(a/c)^(1/4))/(a^3*c^2
) + 1/16*sqrt(2)*(3*(a*c^3)^(1/4)*B*a*c + 5*(a*c^3)^(3/4)*A)*log(sqrt(2)*sqrt(x)*(a/c)^(1/4) + x + sqrt(a/c))/
(a^3*c^2) - 1/16*sqrt(2)*(3*(a*c^3)^(1/4)*B*a*c + 5*(a*c^3)^(3/4)*A)*log(-sqrt(2)*sqrt(x)*(a/c)^(1/4) + x + sq
rt(a/c))/(a^3*c^2)

________________________________________________________________________________________

maple [A]  time = 0.08, size = 314, normalized size = 1.03 \begin {gather*} -\frac {A c \,x^{\frac {3}{2}}}{2 \left (c \,x^{2}+a \right ) a^{2}}+\frac {B \sqrt {x}}{2 \left (c \,x^{2}+a \right ) a}-\frac {5 \sqrt {2}\, A \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )}{8 \left (\frac {a}{c}\right )^{\frac {1}{4}} a^{2}}-\frac {5 \sqrt {2}\, A \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )}{8 \left (\frac {a}{c}\right )^{\frac {1}{4}} a^{2}}-\frac {5 \sqrt {2}\, A \ln \left (\frac {x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{c}}}{x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{c}}}\right )}{16 \left (\frac {a}{c}\right )^{\frac {1}{4}} a^{2}}+\frac {3 \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, B \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )}{8 a^{2}}+\frac {3 \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, B \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )}{8 a^{2}}+\frac {3 \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, B \ln \left (\frac {x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{c}}}{x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{c}}}\right )}{16 a^{2}}-\frac {2 A}{a^{2} \sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/x^(3/2)/(c*x^2+a)^2,x)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 1.19, size = 280, normalized size = 0.92 \begin {gather*} -\frac {5 \, A c x^{2} - B a x + 4 \, A a}{2 \, {\left (a^{2} c x^{\frac {5}{2}} + a^{3} \sqrt {x}\right )}} + \frac {\frac {2 \, \sqrt {2} {\left (3 \, B a \sqrt {c} - 5 \, A \sqrt {a} c\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} + 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {2 \, \sqrt {2} {\left (3 \, B a \sqrt {c} - 5 \, A \sqrt {a} c\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} - 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {\sqrt {2} {\left (3 \, B a \sqrt {c} + 5 \, A \sqrt {a} c\right )} \log \left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (3 \, B a \sqrt {c} + 5 \, A \sqrt {a} c\right )} \log \left (-\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}}}{16 \, a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^(3/2)/(c*x^2+a)^2,x, algorithm="maxima")

[Out]

-1/2*(5*A*c*x^2 - B*a*x + 4*A*a)/(a^2*c*x^(5/2) + a^3*sqrt(x)) + 1/16*(2*sqrt(2)*(3*B*a*sqrt(c) - 5*A*sqrt(a)*
c)*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4)*c^(1/4) + 2*sqrt(c)*sqrt(x))/sqrt(sqrt(a)*sqrt(c)))/(sqrt(a)*sqrt(sqrt(
a)*sqrt(c))*sqrt(c)) + 2*sqrt(2)*(3*B*a*sqrt(c) - 5*A*sqrt(a)*c)*arctan(-1/2*sqrt(2)*(sqrt(2)*a^(1/4)*c^(1/4)
- 2*sqrt(c)*sqrt(x))/sqrt(sqrt(a)*sqrt(c)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(c))*sqrt(c)) + sqrt(2)*(3*B*a*sqrt(c) +
5*A*sqrt(a)*c)*log(sqrt(2)*a^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x + sqrt(a))/(a^(3/4)*c^(3/4)) - sqrt(2)*(3*B*a*
sqrt(c) + 5*A*sqrt(a)*c)*log(-sqrt(2)*a^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x + sqrt(a))/(a^(3/4)*c^(3/4)))/a^2

________________________________________________________________________________________

mupad [B]  time = 0.25, size = 634, normalized size = 2.09 \begin {gather*} 2\,\mathrm {atanh}\left (\frac {1600\,A^2\,a^7\,c^4\,\sqrt {x}\,\sqrt {\frac {15\,A\,B}{32\,a^4}-\frac {25\,A^2\,\sqrt {-a^9\,c}}{64\,a^9}+\frac {9\,B^2\,\sqrt {-a^9\,c}}{64\,a^8\,c}}}{1000\,A^3\,a^5\,c^4-216\,B^3\,a^2\,c^2\,\sqrt {-a^9\,c}-360\,A\,B^2\,a^6\,c^3+600\,A^2\,B\,a\,c^3\,\sqrt {-a^9\,c}}-\frac {576\,B^2\,a^8\,c^3\,\sqrt {x}\,\sqrt {\frac {15\,A\,B}{32\,a^4}-\frac {25\,A^2\,\sqrt {-a^9\,c}}{64\,a^9}+\frac {9\,B^2\,\sqrt {-a^9\,c}}{64\,a^8\,c}}}{1000\,A^3\,a^5\,c^4-216\,B^3\,a^2\,c^2\,\sqrt {-a^9\,c}-360\,A\,B^2\,a^6\,c^3+600\,A^2\,B\,a\,c^3\,\sqrt {-a^9\,c}}\right )\,\sqrt {\frac {9\,B^2\,a\,\sqrt {-a^9\,c}-25\,A^2\,c\,\sqrt {-a^9\,c}+30\,A\,B\,a^5\,c}{64\,a^9\,c}}+2\,\mathrm {atanh}\left (\frac {1600\,A^2\,a^7\,c^4\,\sqrt {x}\,\sqrt {\frac {25\,A^2\,\sqrt {-a^9\,c}}{64\,a^9}+\frac {15\,A\,B}{32\,a^4}-\frac {9\,B^2\,\sqrt {-a^9\,c}}{64\,a^8\,c}}}{1000\,A^3\,a^5\,c^4+216\,B^3\,a^2\,c^2\,\sqrt {-a^9\,c}-360\,A\,B^2\,a^6\,c^3-600\,A^2\,B\,a\,c^3\,\sqrt {-a^9\,c}}-\frac {576\,B^2\,a^8\,c^3\,\sqrt {x}\,\sqrt {\frac {25\,A^2\,\sqrt {-a^9\,c}}{64\,a^9}+\frac {15\,A\,B}{32\,a^4}-\frac {9\,B^2\,\sqrt {-a^9\,c}}{64\,a^8\,c}}}{1000\,A^3\,a^5\,c^4+216\,B^3\,a^2\,c^2\,\sqrt {-a^9\,c}-360\,A\,B^2\,a^6\,c^3-600\,A^2\,B\,a\,c^3\,\sqrt {-a^9\,c}}\right )\,\sqrt {\frac {25\,A^2\,c\,\sqrt {-a^9\,c}-9\,B^2\,a\,\sqrt {-a^9\,c}+30\,A\,B\,a^5\,c}{64\,a^9\,c}}-\frac {\frac {2\,A}{a}-\frac {B\,x}{2\,a}+\frac {5\,A\,c\,x^2}{2\,a^2}}{a\,\sqrt {x}+c\,x^{5/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)/(x^(3/2)*(a + c*x^2)^2),x)

[Out]

2*atanh((1600*A^2*a^7*c^4*x^(1/2)*((15*A*B)/(32*a^4) - (25*A^2*(-a^9*c)^(1/2))/(64*a^9) + (9*B^2*(-a^9*c)^(1/2
))/(64*a^8*c))^(1/2))/(1000*A^3*a^5*c^4 - 216*B^3*a^2*c^2*(-a^9*c)^(1/2) - 360*A*B^2*a^6*c^3 + 600*A^2*B*a*c^3
*(-a^9*c)^(1/2)) - (576*B^2*a^8*c^3*x^(1/2)*((15*A*B)/(32*a^4) - (25*A^2*(-a^9*c)^(1/2))/(64*a^9) + (9*B^2*(-a
^9*c)^(1/2))/(64*a^8*c))^(1/2))/(1000*A^3*a^5*c^4 - 216*B^3*a^2*c^2*(-a^9*c)^(1/2) - 360*A*B^2*a^6*c^3 + 600*A
^2*B*a*c^3*(-a^9*c)^(1/2)))*((9*B^2*a*(-a^9*c)^(1/2) - 25*A^2*c*(-a^9*c)^(1/2) + 30*A*B*a^5*c)/(64*a^9*c))^(1/
2) + 2*atanh((1600*A^2*a^7*c^4*x^(1/2)*((25*A^2*(-a^9*c)^(1/2))/(64*a^9) + (15*A*B)/(32*a^4) - (9*B^2*(-a^9*c)
^(1/2))/(64*a^8*c))^(1/2))/(1000*A^3*a^5*c^4 + 216*B^3*a^2*c^2*(-a^9*c)^(1/2) - 360*A*B^2*a^6*c^3 - 600*A^2*B*
a*c^3*(-a^9*c)^(1/2)) - (576*B^2*a^8*c^3*x^(1/2)*((25*A^2*(-a^9*c)^(1/2))/(64*a^9) + (15*A*B)/(32*a^4) - (9*B^
2*(-a^9*c)^(1/2))/(64*a^8*c))^(1/2))/(1000*A^3*a^5*c^4 + 216*B^3*a^2*c^2*(-a^9*c)^(1/2) - 360*A*B^2*a^6*c^3 -
600*A^2*B*a*c^3*(-a^9*c)^(1/2)))*((25*A^2*c*(-a^9*c)^(1/2) - 9*B^2*a*(-a^9*c)^(1/2) + 30*A*B*a^5*c)/(64*a^9*c)
)^(1/2) - ((2*A)/a - (B*x)/(2*a) + (5*A*c*x^2)/(2*a^2))/(a*x^(1/2) + c*x^(5/2))

________________________________________________________________________________________

sympy [A]  time = 149.46, size = 1435, normalized size = 4.72

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x**(3/2)/(c*x**2+a)**2,x)

[Out]

Piecewise((zoo*(-2*A/(9*x**(9/2)) - 2*B/(7*x**(7/2))), Eq(a, 0) & Eq(c, 0)), ((-2*A/(9*x**(9/2)) - 2*B/(7*x**(
7/2)))/c**2, Eq(a, 0)), ((-2*A/sqrt(x) + 2*B*sqrt(x))/a**2, Eq(c, 0)), (-16*(-1)**(1/4)*A*a**(5/4)*(1/c)**(1/4
)/(8*(-1)**(1/4)*a**(13/4)*sqrt(x)*(1/c)**(1/4) + 8*(-1)**(1/4)*a**(9/4)*c*x**(5/2)*(1/c)**(1/4)) - 20*(-1)**(
1/4)*A*a**(1/4)*c*x**2*(1/c)**(1/4)/(8*(-1)**(1/4)*a**(13/4)*sqrt(x)*(1/c)**(1/4) + 8*(-1)**(1/4)*a**(9/4)*c*x
**(5/2)*(1/c)**(1/4)) - 5*A*a*sqrt(x)*log(-(-1)**(1/4)*a**(1/4)*(1/c)**(1/4) + sqrt(x))/(8*(-1)**(1/4)*a**(13/
4)*sqrt(x)*(1/c)**(1/4) + 8*(-1)**(1/4)*a**(9/4)*c*x**(5/2)*(1/c)**(1/4)) + 5*A*a*sqrt(x)*log((-1)**(1/4)*a**(
1/4)*(1/c)**(1/4) + sqrt(x))/(8*(-1)**(1/4)*a**(13/4)*sqrt(x)*(1/c)**(1/4) + 8*(-1)**(1/4)*a**(9/4)*c*x**(5/2)
*(1/c)**(1/4)) + 10*A*a*sqrt(x)*atan((-1)**(3/4)*sqrt(x)/(a**(1/4)*(1/c)**(1/4)))/(8*(-1)**(1/4)*a**(13/4)*sqr
t(x)*(1/c)**(1/4) + 8*(-1)**(1/4)*a**(9/4)*c*x**(5/2)*(1/c)**(1/4)) - 5*A*c*x**(5/2)*log(-(-1)**(1/4)*a**(1/4)
*(1/c)**(1/4) + sqrt(x))/(8*(-1)**(1/4)*a**(13/4)*sqrt(x)*(1/c)**(1/4) + 8*(-1)**(1/4)*a**(9/4)*c*x**(5/2)*(1/
c)**(1/4)) + 5*A*c*x**(5/2)*log((-1)**(1/4)*a**(1/4)*(1/c)**(1/4) + sqrt(x))/(8*(-1)**(1/4)*a**(13/4)*sqrt(x)*
(1/c)**(1/4) + 8*(-1)**(1/4)*a**(9/4)*c*x**(5/2)*(1/c)**(1/4)) + 10*A*c*x**(5/2)*atan((-1)**(3/4)*sqrt(x)/(a**
(1/4)*(1/c)**(1/4)))/(8*(-1)**(1/4)*a**(13/4)*sqrt(x)*(1/c)**(1/4) + 8*(-1)**(1/4)*a**(9/4)*c*x**(5/2)*(1/c)**
(1/4)) + 4*(-1)**(1/4)*B*a**(5/4)*x*(1/c)**(1/4)/(8*(-1)**(1/4)*a**(13/4)*sqrt(x)*(1/c)**(1/4) + 8*(-1)**(1/4)
*a**(9/4)*c*x**(5/2)*(1/c)**(1/4)) - 3*I*B*a**(3/2)*sqrt(x)*sqrt(1/c)*log(-(-1)**(1/4)*a**(1/4)*(1/c)**(1/4) +
sqrt(x))/(8*(-1)**(1/4)*a**(13/4)*sqrt(x)*(1/c)**(1/4) + 8*(-1)**(1/4)*a**(9/4)*c*x**(5/2)*(1/c)**(1/4)) + 3*
I*B*a**(3/2)*sqrt(x)*sqrt(1/c)*log((-1)**(1/4)*a**(1/4)*(1/c)**(1/4) + sqrt(x))/(8*(-1)**(1/4)*a**(13/4)*sqrt(
x)*(1/c)**(1/4) + 8*(-1)**(1/4)*a**(9/4)*c*x**(5/2)*(1/c)**(1/4)) - 6*I*B*a**(3/2)*sqrt(x)*sqrt(1/c)*atan((-1)
**(3/4)*sqrt(x)/(a**(1/4)*(1/c)**(1/4)))/(8*(-1)**(1/4)*a**(13/4)*sqrt(x)*(1/c)**(1/4) + 8*(-1)**(1/4)*a**(9/4
)*c*x**(5/2)*(1/c)**(1/4)) - 3*I*B*sqrt(a)*c*x**(5/2)*sqrt(1/c)*log(-(-1)**(1/4)*a**(1/4)*(1/c)**(1/4) + sqrt(
x))/(8*(-1)**(1/4)*a**(13/4)*sqrt(x)*(1/c)**(1/4) + 8*(-1)**(1/4)*a**(9/4)*c*x**(5/2)*(1/c)**(1/4)) + 3*I*B*sq
rt(a)*c*x**(5/2)*sqrt(1/c)*log((-1)**(1/4)*a**(1/4)*(1/c)**(1/4) + sqrt(x))/(8*(-1)**(1/4)*a**(13/4)*sqrt(x)*(
1/c)**(1/4) + 8*(-1)**(1/4)*a**(9/4)*c*x**(5/2)*(1/c)**(1/4)) - 6*I*B*sqrt(a)*c*x**(5/2)*sqrt(1/c)*atan((-1)**
(3/4)*sqrt(x)/(a**(1/4)*(1/c)**(1/4)))/(8*(-1)**(1/4)*a**(13/4)*sqrt(x)*(1/c)**(1/4) + 8*(-1)**(1/4)*a**(9/4)*
c*x**(5/2)*(1/c)**(1/4)), True))

________________________________________________________________________________________