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

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

[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]*E
llipticF[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])

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Rubi [A]  time = 0.531974, 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 \[ \frac{\sqrt [4]{a} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \left (\frac{A \sqrt{c}}{\sqrt{a}}+B\right ) \sqrt{\frac{a+b x+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{c^{3/4} \sqrt{e x} \sqrt{a+b x+c x^2}}-\frac{2 \sqrt [4]{a} B \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+b x+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{c^{3/4} \sqrt{e x} \sqrt{a+b x+c x^2}}+\frac{2 B x \sqrt{a+b x+c x^2}}{\sqrt{c} \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )} \]

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]*E
llipticF[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])

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Rubi in Sympy [A]  time = 71.0224, size = 279, normalized size = 0.93 \[ - \frac{2 B \sqrt [4]{a} \sqrt{x} \sqrt{\frac{a + b x + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2} - \frac{b}{4 \sqrt{a} \sqrt{c}}\right )}{c^{\frac{3}{4}} \sqrt{e x} \sqrt{a + b x + c x^{2}}} + \frac{2 B x \sqrt{a + b x + c x^{2}}}{\sqrt{c} \sqrt{e x} \left (\sqrt{a} + \sqrt{c} x\right )} + \frac{\sqrt{x} \sqrt{\frac{a + b x + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) \left (A \sqrt{c} + B \sqrt{a}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2} - \frac{b}{4 \sqrt{a} \sqrt{c}}\right )}{\sqrt [4]{a} c^{\frac{3}{4}} \sqrt{e x} \sqrt{a + b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/(e*x)**(1/2)/(c*x**2+b*x+a)**(1/2),x)

[Out]

-2*B*a**(1/4)*sqrt(x)*sqrt((a + b*x + c*x**2)/(sqrt(a) + sqrt(c)*x)**2)*(sqrt(a)
+ sqrt(c)*x)*elliptic_e(2*atan(c**(1/4)*sqrt(x)/a**(1/4)), 1/2 - b/(4*sqrt(a)*s
qrt(c)))/(c**(3/4)*sqrt(e*x)*sqrt(a + b*x + c*x**2)) + 2*B*x*sqrt(a + b*x + c*x*
*2)/(sqrt(c)*sqrt(e*x)*(sqrt(a) + sqrt(c)*x)) + sqrt(x)*sqrt((a + b*x + c*x**2)/
(sqrt(a) + sqrt(c)*x)**2)*(sqrt(a) + sqrt(c)*x)*(A*sqrt(c) + B*sqrt(a))*elliptic
_f(2*atan(c**(1/4)*sqrt(x)/a**(1/4)), 1/2 - b/(4*sqrt(a)*sqrt(c)))/(a**(1/4)*c**
(3/4)*sqrt(e*x)*sqrt(a + b*x + c*x**2))

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Mathematica [C]  time = 3.10452, size = 444, normalized size = 1.48 \[ -\frac{x^2 \left (-\frac{i \sqrt{\frac{4 a}{x \left (\sqrt{b^2-4 a c}+b\right )}+2} \sqrt{\frac{-x \sqrt{b^2-4 a c}+2 a+b x}{b x-x \sqrt{b^2-4 a c}}} \left (B \sqrt{b^2-4 a c}+2 A c-b B\right ) F\left (i \sinh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{a}{b+\sqrt{b^2-4 a c}}}}{\sqrt{x}}\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )}{\sqrt{x}}-\frac{4 B \sqrt{\frac{a}{\sqrt{b^2-4 a c}+b}} (a+x (b+c x))}{x^2}+\frac{i B \left (\sqrt{b^2-4 a c}-b\right ) \sqrt{\frac{4 a}{x \left (\sqrt{b^2-4 a c}+b\right )}+2} \sqrt{\frac{-x \sqrt{b^2-4 a c}+2 a+b x}{b x-x \sqrt{b^2-4 a c}}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{a}{b+\sqrt{b^2-4 a c}}}}{\sqrt{x}}\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )}{\sqrt{x}}\right )}{2 c \sqrt{e x} \sqrt{\frac{a}{\sqrt{b^2-4 a c}+b}} \sqrt{a+x (b+c x)}} \]

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)]*EllipticE[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)])

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Maple [A]  time = 0.063, size = 538, normalized size = 1.8 \[{\frac{1}{{c}^{2}}\sqrt{{1 \left ( b+2\,cx+\sqrt{-4\,ac+{b}^{2}} \right ) \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) ^{-1}}}\sqrt{{1 \left ( -b-2\,cx+\sqrt{-4\,ac+{b}^{2}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}}\sqrt{-{cx \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) ^{-1}}} \left ( A{\it EllipticF} \left ( \sqrt{{1 \left ( b+2\,cx+\sqrt{-4\,ac+{b}^{2}} \right ) \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) ^{-1}}},{\frac{\sqrt{2}}{2}\sqrt{{1 \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}}} \right ) c\sqrt{-4\,ac+{b}^{2}}+A{\it EllipticF} \left ( \sqrt{{1 \left ( b+2\,cx+\sqrt{-4\,ac+{b}^{2}} \right ) \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) ^{-1}}},{\frac{\sqrt{2}}{2}\sqrt{{1 \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}}} \right ) cb-2\,B{\it EllipticF} \left ( \sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{b+\sqrt{-4\,ac+{b}^{2}}}}},1/2\,\sqrt{2}\sqrt{{\frac{b+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}} \right ) ac-B{\it EllipticE} \left ( \sqrt{{1 \left ( b+2\,cx+\sqrt{-4\,ac+{b}^{2}} \right ) \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) ^{-1}}},{\frac{\sqrt{2}}{2}\sqrt{{1 \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}}} \right ) \sqrt{-4\,ac+{b}^{2}}b+4\,B{\it EllipticE} \left ( \sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{b+\sqrt{-4\,ac+{b}^{2}}}}},1/2\,\sqrt{2}\sqrt{{\frac{b+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}} \right ) ac-B{\it EllipticE} \left ( \sqrt{{1 \left ( b+2\,cx+\sqrt{-4\,ac+{b}^{2}} \right ) \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) ^{-1}}},{\frac{\sqrt{2}}{2}\sqrt{{1 \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}}} \right ){b}^{2} \right ){\frac{1}{\sqrt{c{x}^{2}+bx+a}}}{\frac{1}{\sqrt{ex}}}} \]

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)*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2
)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-c*x/(b+(-4*a*c+b^2)
^(1/2)))^(1/2)*(A*EllipticF(((b+2*c*x+(-4*a*c+b^2)^(1/2))/(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(((b+2*c*x+(-4*a*c+b^2)^(1/2))/(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*Ellip
ticF(((b+2*c*x+(-4*a*c+b^2)^(1/2))/(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(((b+2*c*x+(-4*a*
c+b^2)^(1/2))/(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(((b+2*c*x+(-4*a*c+
b^2)^(1/2))/(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(((b+2*c*x+(-4*a*c+b^2)^(1/2))/(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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{B x + A}{\sqrt{c x^{2} + b x + a} \sqrt{e x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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