3.1055 \(\int \frac{A+B x}{\sqrt{e x} \sqrt{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=300 \[ \frac{\sqrt [4]{a} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \left (\frac{A \sqrt{c}}{\sqrt{a}}+B\right ) \sqrt{\frac{a+b x+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right ),\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{c^{3/4} \sqrt{e x} \sqrt{a+b x+c x^2}}-\frac{2 \sqrt [4]{a} B \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+b x+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{c^{3/4} \sqrt{e x} \sqrt{a+b x+c x^2}}+\frac{2 B x \sqrt{a+b x+c x^2}}{\sqrt{c} \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )} \]

[Out]

(2*B*x*Sqrt[a + b*x + c*x^2])/(Sqrt[c]*Sqrt[e*x]*(Sqrt[a] + Sqrt[c]*x)) - (2*a^(1/4)*B*Sqrt[x]*(Sqrt[a] + Sqrt
[c]*x)*Sqrt[(a + b*x + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticE[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], (2 - b/(
Sqrt[a]*Sqrt[c]))/4])/(c^(3/4)*Sqrt[e*x]*Sqrt[a + b*x + c*x^2]) + (a^(1/4)*(B + (A*Sqrt[c])/Sqrt[a])*Sqrt[x]*(
Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + b*x + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1
/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(c^(3/4)*Sqrt[e*x]*Sqrt[a + b*x + c*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.218873, antiderivative size = 300, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {841, 839, 1197, 1103, 1195} \[ \frac{\sqrt [4]{a} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \left (\frac{A \sqrt{c}}{\sqrt{a}}+B\right ) \sqrt{\frac{a+b x+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{c^{3/4} \sqrt{e x} \sqrt{a+b x+c x^2}}-\frac{2 \sqrt [4]{a} B \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+b x+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{c^{3/4} \sqrt{e x} \sqrt{a+b x+c x^2}}+\frac{2 B x \sqrt{a+b x+c x^2}}{\sqrt{c} \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*B*x*Sqrt[a + b*x + c*x^2])/(Sqrt[c]*Sqrt[e*x]*(Sqrt[a] + Sqrt[c]*x)) - (2*a^(1/4)*B*Sqrt[x]*(Sqrt[a] + Sqrt
[c]*x)*Sqrt[(a + b*x + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticE[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], (2 - b/(
Sqrt[a]*Sqrt[c]))/4])/(c^(3/4)*Sqrt[e*x]*Sqrt[a + b*x + c*x^2]) + (a^(1/4)*(B + (A*Sqrt[c])/Sqrt[a])*Sqrt[x]*(
Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + b*x + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1
/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(c^(3/4)*Sqrt[e*x]*Sqrt[a + b*x + c*x^2])

Rule 841

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

Rule 839

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

Rule 1197

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 2]}, Dist[(
e + d*q)/q, Int[1/Sqrt[a + b*x^2 + c*x^4], x], x] - Dist[e/q, Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x]
/; NeQ[e + d*q, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]

Rule 1103

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[((1 + q^2*x^2)*Sqrt[(
a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]*EllipticF[2*ArcTan[q*x], 1/2 - (b*q^2)/(4*c)])/(2*q*Sqrt[a + b*x^2 + c
*x^4]), x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]

Rule 1195

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, -Simp[
(d*x*Sqrt[a + b*x^2 + c*x^4])/(a*(1 + q^2*x^2)), x] + Simp[(d*(1 + q^2*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q
^2*x^2)^2)]*EllipticE[2*ArcTan[q*x], 1/2 - (b*q^2)/(4*c)])/(q*Sqrt[a + b*x^2 + c*x^4]), x] /; EqQ[e + d*q^2, 0
]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]

Rubi steps

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

Mathematica [C]  time = 1.84535, size = 444, normalized size = 1.48 \[ -\frac{x^2 \left (-\frac{i \sqrt{\frac{4 a}{x \left (\sqrt{b^2-4 a c}+b\right )}+2} \sqrt{\frac{-x \sqrt{b^2-4 a c}+2 a+b x}{b x-x \sqrt{b^2-4 a c}}} \left (B \sqrt{b^2-4 a c}+2 A c-b B\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{a}{\sqrt{b^2-4 a c}+b}}}{\sqrt{x}}\right ),\frac{\sqrt{b^2-4 a c}+b}{b-\sqrt{b^2-4 a c}}\right )}{\sqrt{x}}-\frac{4 B \sqrt{\frac{a}{\sqrt{b^2-4 a c}+b}} (a+x (b+c x))}{x^2}+\frac{i B \left (\sqrt{b^2-4 a c}-b\right ) \sqrt{\frac{4 a}{x \left (\sqrt{b^2-4 a c}+b\right )}+2} \sqrt{\frac{-x \sqrt{b^2-4 a c}+2 a+b x}{b x-x \sqrt{b^2-4 a c}}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{a}{b+\sqrt{b^2-4 a c}}}}{\sqrt{x}}\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )}{\sqrt{x}}\right )}{2 c \sqrt{e x} \sqrt{\frac{a}{\sqrt{b^2-4 a c}+b}} \sqrt{a+x (b+c x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(x^2*((-4*B*Sqrt[a/(b + Sqrt[b^2 - 4*a*c])]*(a + x*(b + c*x)))/x^2 + (I*B*(-b + Sqrt[b^2 - 4*a*c])*Sqrt[2 + (
4*a)/((b + Sqrt[b^2 - 4*a*c])*x)]*Sqrt[(2*a + b*x - Sqrt[b^2 - 4*a*c]*x)/(b*x - Sqrt[b^2 - 4*a*c]*x)]*Elliptic
E[I*ArcSinh[(Sqrt[2]*Sqrt[a/(b + Sqrt[b^2 - 4*a*c])])/Sqrt[x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c]
)])/Sqrt[x] - (I*(-(b*B) + 2*A*c + B*Sqrt[b^2 - 4*a*c])*Sqrt[2 + (4*a)/((b + Sqrt[b^2 - 4*a*c])*x)]*Sqrt[(2*a
+ b*x - Sqrt[b^2 - 4*a*c]*x)/(b*x - Sqrt[b^2 - 4*a*c]*x)]*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[a/(b + Sqrt[b^2 -
4*a*c])])/Sqrt[x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])])/Sqrt[x]))/(2*c*Sqrt[a/(b + Sqrt[b^2 - 4*
a*c])]*Sqrt[e*x]*Sqrt[a + x*(b + c*x)])

________________________________________________________________________________________

Maple [A]  time = 0.036, size = 538, normalized size = 1.8 \begin{align*}{\frac{1}{{c}^{2}}\sqrt{{ \left ( 2\,cx+\sqrt{-4\,ac+{b}^{2}}+b \right ) \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) ^{-1}}}\sqrt{{ \left ( -2\,cx+\sqrt{-4\,ac+{b}^{2}}-b \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}}\sqrt{-{cx \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) ^{-1}}} \left ( A{\it EllipticF} \left ( \sqrt{{ \left ( 2\,cx+\sqrt{-4\,ac+{b}^{2}}+b \right ) \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) ^{-1}}},{\frac{\sqrt{2}}{2}\sqrt{{ \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}}} \right ) c\sqrt{-4\,ac+{b}^{2}}+A{\it EllipticF} \left ( \sqrt{{ \left ( 2\,cx+\sqrt{-4\,ac+{b}^{2}}+b \right ) \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) ^{-1}}},{\frac{\sqrt{2}}{2}\sqrt{{ \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}}} \right ) cb-2\,B{\it EllipticF} \left ( \sqrt{{\frac{2\,cx+\sqrt{-4\,ac+{b}^{2}}+b}{b+\sqrt{-4\,ac+{b}^{2}}}}},1/2\,\sqrt{2}\sqrt{{\frac{b+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}} \right ) ac-B{\it EllipticE} \left ( \sqrt{{ \left ( 2\,cx+\sqrt{-4\,ac+{b}^{2}}+b \right ) \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) ^{-1}}},{\frac{\sqrt{2}}{2}\sqrt{{ \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}}} \right ) \sqrt{-4\,ac+{b}^{2}}b+4\,B{\it EllipticE} \left ( \sqrt{{\frac{2\,cx+\sqrt{-4\,ac+{b}^{2}}+b}{b+\sqrt{-4\,ac+{b}^{2}}}}},1/2\,\sqrt{2}\sqrt{{\frac{b+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}} \right ) ac-B{\it EllipticE} \left ( \sqrt{{ \left ( 2\,cx+\sqrt{-4\,ac+{b}^{2}}+b \right ) \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) ^{-1}}},{\frac{\sqrt{2}}{2}\sqrt{{ \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}}} \right ){b}^{2} \right ){\frac{1}{\sqrt{c{x}^{2}+bx+a}}}{\frac{1}{\sqrt{ex}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(c*x^2+b*x+a)^(1/2)*((2*c*x+(-4*a*c+b^2)^(1/2)+b)/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*((-2*c*x+(-4*a*c+b^2)^(1/2)-
b)/(-4*a*c+b^2)^(1/2))^(1/2)*(-c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*(A*EllipticF(((2*c*x+(-4*a*c+b^2)^(1/2)+b)/(b
+(-4*a*c+b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2))*c*(-4*a*c+b^2)^(1/2
)+A*EllipticF(((2*c*x+(-4*a*c+b^2)^(1/2)+b)/(b+(-4*a*c+b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(-4*a*c+b^2)^(1/2))/
(-4*a*c+b^2)^(1/2))^(1/2))*c*b-2*B*EllipticF(((2*c*x+(-4*a*c+b^2)^(1/2)+b)/(b+(-4*a*c+b^2)^(1/2)))^(1/2),1/2*2
^(1/2)*((b+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2))*a*c-B*EllipticE(((2*c*x+(-4*a*c+b^2)^(1/2)+b)/(b+(-4
*a*c+b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2))*(-4*a*c+b^2)^(1/2)*b+4*
B*EllipticE(((2*c*x+(-4*a*c+b^2)^(1/2)+b)/(b+(-4*a*c+b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(-4*a*c+b^2)^(1/2))/(-
4*a*c+b^2)^(1/2))^(1/2))*a*c-B*EllipticE(((2*c*x+(-4*a*c+b^2)^(1/2)+b)/(b+(-4*a*c+b^2)^(1/2)))^(1/2),1/2*2^(1/
2)*((b+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2))*b^2)/(e*x)^(1/2)/c^2

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{B x + A}{\sqrt{c x^{2} + b x + a} \sqrt{e x}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((B*x + A)/(sqrt(c*x^2 + b*x + a)*sqrt(e*x)), x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{c x^{2} + b x + a}{\left (B x + A\right )} \sqrt{e x}}{c e x^{3} + b e x^{2} + a e x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(c*x^2 + b*x + a)*(B*x + A)*sqrt(e*x)/(c*e*x^3 + b*e*x^2 + a*e*x), x)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{A + B x}{\sqrt{e x} \sqrt{a + b x + c x^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((A + B*x)/(sqrt(e*x)*sqrt(a + b*x + c*x**2)), x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{B x + A}{\sqrt{c x^{2} + b x + a} \sqrt{e x}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((B*x + A)/(sqrt(c*x^2 + b*x + a)*sqrt(e*x)), x)