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

Optimal. Leaf size=254 \[ \frac{\left (\sqrt{a} B-A \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{2 \sqrt{2} a^{3/4} c^{3/4}}-\frac{\left (\sqrt{a} B-A \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{2 \sqrt{2} a^{3/4} c^{3/4}}-\frac{\left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{3/4} c^{3/4}}+\frac{\left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} a^{3/4} c^{3/4}} \]

[Out]

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

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Rubi [A]  time = 0.185014, antiderivative size = 254, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35, Rules used = {827, 1168, 1162, 617, 204, 1165, 628} \[ \frac{\left (\sqrt{a} B-A \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{2 \sqrt{2} a^{3/4} c^{3/4}}-\frac{\left (\sqrt{a} B-A \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{2 \sqrt{2} a^{3/4} c^{3/4}}-\frac{\left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{3/4} c^{3/4}}+\frac{\left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} a^{3/4} c^{3/4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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)]

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 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 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 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 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]

Rubi steps

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

Mathematica [A]  time = 0.0605114, size = 262, normalized size = 1.03 \[ \frac{-\sqrt{2} \sqrt [4]{a} A \sqrt{c} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )+\sqrt{2} \sqrt [4]{a} A \sqrt{c} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )-2 \sqrt{2} \sqrt [4]{a} A \sqrt{c} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )+2 \sqrt{2} \sqrt [4]{a} A \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )-4 (-a)^{3/4} B \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{-a}}\right )+4 (-a)^{3/4} B \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{-a}}\right )}{4 a c^{3/4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[2]*a^(1/4)*A*Sqrt[c]*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)] + 2*Sqrt[2]*a^(1/4)*A*Sqrt[c]*ArcT
an[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)] - 4*(-a)^(3/4)*B*ArcTan[(c^(1/4)*Sqrt[x])/(-a)^(1/4)] + 4*(-a)^(3/4)
*B*ArcTanh[(c^(1/4)*Sqrt[x])/(-a)^(1/4)] - Sqrt[2]*a^(1/4)*A*Sqrt[c]*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*Sqr
t[x] + Sqrt[c]*x] + Sqrt[2]*a^(1/4)*A*Sqrt[c]*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(4*a
*c^(3/4))

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Maple [A]  time = 0.006, size = 268, normalized size = 1.1 \begin{align*}{\frac{A\sqrt{2}}{4\,a}\sqrt [4]{{\frac{a}{c}}}\ln \left ({ \left ( x+\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ) }+{\frac{A\sqrt{2}}{2\,a}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ) }+{\frac{A\sqrt{2}}{2\,a}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ) }+{\frac{B\sqrt{2}}{4\,c}\ln \left ({ \left ( x-\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ( x+\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{B\sqrt{2}}{2\,c}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{B\sqrt{2}}{2\,c}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.37221, size = 1575, normalized size = 6.2 \begin{align*} \frac{1}{2} \, \sqrt{-\frac{a c \sqrt{-\frac{B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{3} c^{3}}} + 2 \, A B}{a c}} \log \left (-{\left (B^{4} a^{2} - A^{4} c^{2}\right )} \sqrt{x} +{\left (B a^{3} c^{2} \sqrt{-\frac{B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{3} c^{3}}} - A B^{2} a^{2} c + A^{3} a c^{2}\right )} \sqrt{-\frac{a c \sqrt{-\frac{B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{3} c^{3}}} + 2 \, A B}{a c}}\right ) - \frac{1}{2} \, \sqrt{-\frac{a c \sqrt{-\frac{B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{3} c^{3}}} + 2 \, A B}{a c}} \log \left (-{\left (B^{4} a^{2} - A^{4} c^{2}\right )} \sqrt{x} -{\left (B a^{3} c^{2} \sqrt{-\frac{B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{3} c^{3}}} - A B^{2} a^{2} c + A^{3} a c^{2}\right )} \sqrt{-\frac{a c \sqrt{-\frac{B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{3} c^{3}}} + 2 \, A B}{a c}}\right ) - \frac{1}{2} \, \sqrt{\frac{a c \sqrt{-\frac{B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{3} c^{3}}} - 2 \, A B}{a c}} \log \left (-{\left (B^{4} a^{2} - A^{4} c^{2}\right )} \sqrt{x} +{\left (B a^{3} c^{2} \sqrt{-\frac{B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{3} c^{3}}} + A B^{2} a^{2} c - A^{3} a c^{2}\right )} \sqrt{\frac{a c \sqrt{-\frac{B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{3} c^{3}}} - 2 \, A B}{a c}}\right ) + \frac{1}{2} \, \sqrt{\frac{a c \sqrt{-\frac{B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{3} c^{3}}} - 2 \, A B}{a c}} \log \left (-{\left (B^{4} a^{2} - A^{4} c^{2}\right )} \sqrt{x} -{\left (B a^{3} c^{2} \sqrt{-\frac{B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{3} c^{3}}} + A B^{2} a^{2} c - A^{3} a c^{2}\right )} \sqrt{\frac{a c \sqrt{-\frac{B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{3} c^{3}}} - 2 \, A B}{a c}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(-(a*c*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^3*c^3)) + 2*A*B)/(a*c))*log(-(B^4*a^2 - A^4*c^2)*s
qrt(x) + (B*a^3*c^2*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^3*c^3)) - A*B^2*a^2*c + A^3*a*c^2)*sqrt(-(a*c
*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^3*c^3)) + 2*A*B)/(a*c))) - 1/2*sqrt(-(a*c*sqrt(-(B^4*a^2 - 2*A^2
*B^2*a*c + A^4*c^2)/(a^3*c^3)) + 2*A*B)/(a*c))*log(-(B^4*a^2 - A^4*c^2)*sqrt(x) - (B*a^3*c^2*sqrt(-(B^4*a^2 -
2*A^2*B^2*a*c + A^4*c^2)/(a^3*c^3)) - A*B^2*a^2*c + A^3*a*c^2)*sqrt(-(a*c*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4
*c^2)/(a^3*c^3)) + 2*A*B)/(a*c))) - 1/2*sqrt((a*c*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^3*c^3)) - 2*A*B
)/(a*c))*log(-(B^4*a^2 - A^4*c^2)*sqrt(x) + (B*a^3*c^2*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^3*c^3)) +
A*B^2*a^2*c - A^3*a*c^2)*sqrt((a*c*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^3*c^3)) - 2*A*B)/(a*c))) + 1/2
*sqrt((a*c*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^3*c^3)) - 2*A*B)/(a*c))*log(-(B^4*a^2 - A^4*c^2)*sqrt(
x) - (B*a^3*c^2*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^3*c^3)) + A*B^2*a^2*c - A^3*a*c^2)*sqrt((a*c*sqrt
(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^3*c^3)) - 2*A*B)/(a*c)))

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Sympy [A]  time = 29.4705, size = 196, normalized size = 0.77 \begin{align*} A \left (\begin{cases} \frac{\tilde{\infty }}{x^{\frac{3}{2}}} & \text{for}\: a = 0 \wedge c = 0 \\- \frac{2}{3 c x^{\frac{3}{2}}} & \text{for}\: a = 0 \\\frac{2 \sqrt{x}}{a} & \text{for}\: c = 0 \\- \frac{\sqrt [4]{-1} \sqrt [4]{\frac{1}{c}} \log{\left (- \sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac{1}{c}} + \sqrt{x} \right )}}{2 a^{\frac{3}{4}}} + \frac{\sqrt [4]{-1} \sqrt [4]{\frac{1}{c}} \log{\left (\sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac{1}{c}} + \sqrt{x} \right )}}{2 a^{\frac{3}{4}}} - \frac{\sqrt [4]{-1} \sqrt [4]{\frac{1}{c}} \operatorname{atan}{\left (\frac{\left (-1\right )^{\frac{3}{4}} \sqrt{x}}{\sqrt [4]{a} \sqrt [4]{\frac{1}{c}}} \right )}}{a^{\frac{3}{4}}} & \text{otherwise} \end{cases}\right ) + 2 B \operatorname{RootSum}{\left (256 t^{4} a c^{3} + 1, \left ( t \mapsto t \log{\left (64 t^{3} a c^{2} + \sqrt{x} \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*Piecewise((zoo/x**(3/2), Eq(a, 0) & Eq(c, 0)), (-2/(3*c*x**(3/2)), Eq(a, 0)), (2*sqrt(x)/a, Eq(c, 0)), (-(-1
)**(1/4)*(1/c)**(1/4)*log(-(-1)**(1/4)*a**(1/4)*(1/c)**(1/4) + sqrt(x))/(2*a**(3/4)) + (-1)**(1/4)*(1/c)**(1/4
)*log((-1)**(1/4)*a**(1/4)*(1/c)**(1/4) + sqrt(x))/(2*a**(3/4)) - (-1)**(1/4)*(1/c)**(1/4)*atan((-1)**(3/4)*sq
rt(x)/(a**(1/4)*(1/c)**(1/4)))/a**(3/4), True)) + 2*B*RootSum(256*_t**4*a*c**3 + 1, Lambda(_t, _t*log(64*_t**3
*a*c**2 + sqrt(x))))

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Giac [A]  time = 1.32138, size = 329, normalized size = 1.3 \begin{align*} \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} A c^{2} + \left (a c^{3}\right )^{\frac{3}{4}} B\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 )}{2 \, a c^{3}} + \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} A c^{2} + \left (a c^{3}\right )^{\frac{3}{4}} B\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 )}{2 \, a c^{3}} + \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} A c^{2} - \left (a c^{3}\right )^{\frac{3}{4}} B\right )} \log \left (\sqrt{2} \sqrt{x} \left (\frac{a}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{c}}\right )}{4 \, a c^{3}} - \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} A c^{2} - \left (a c^{3}\right )^{\frac{3}{4}} B\right )} \log \left (-\sqrt{2} \sqrt{x} \left (\frac{a}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{c}}\right )}{4 \, a c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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