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

Optimal. Leaf size=287 \[ -\frac{\sqrt [4]{a} e \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (\sqrt{a} B-3 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{3 c^{5/4} \sqrt{e x} \sqrt{a+c x^2}}-\frac{2 \sqrt [4]{a} A e \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{c^{3/4} \sqrt{e x} \sqrt{a+c x^2}}+\frac{2 A e x \sqrt{a+c x^2}}{\sqrt{c} \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}+\frac{2 B \sqrt{e x} \sqrt{a+c x^2}}{3 c} \]

[Out]

(2*B*Sqrt[e*x]*Sqrt[a + c*x^2])/(3*c) + (2*A*e*x*Sqrt[a + c*x^2])/(Sqrt[c]*Sqrt[
e*x]*(Sqrt[a] + Sqrt[c]*x)) - (2*a^(1/4)*A*e*Sqrt[x]*(Sqrt[a] + Sqrt[c]*x)*Sqrt[
(a + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticE[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4
)], 1/2])/(c^(3/4)*Sqrt[e*x]*Sqrt[a + c*x^2]) - (a^(1/4)*(Sqrt[a]*B - 3*A*Sqrt[c
])*e*Sqrt[x]*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*Ell
ipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(3*c^(5/4)*Sqrt[e*x]*Sqrt[a +
c*x^2])

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Rubi [A]  time = 0.60994, antiderivative size = 287, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{\sqrt [4]{a} e \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (\sqrt{a} B-3 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{3 c^{5/4} \sqrt{e x} \sqrt{a+c x^2}}-\frac{2 \sqrt [4]{a} A e \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{c^{3/4} \sqrt{e x} \sqrt{a+c x^2}}+\frac{2 A e x \sqrt{a+c x^2}}{\sqrt{c} \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}+\frac{2 B \sqrt{e x} \sqrt{a+c x^2}}{3 c} \]

Antiderivative was successfully verified.

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

[Out]

(2*B*Sqrt[e*x]*Sqrt[a + c*x^2])/(3*c) + (2*A*e*x*Sqrt[a + c*x^2])/(Sqrt[c]*Sqrt[
e*x]*(Sqrt[a] + Sqrt[c]*x)) - (2*a^(1/4)*A*e*Sqrt[x]*(Sqrt[a] + Sqrt[c]*x)*Sqrt[
(a + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticE[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4
)], 1/2])/(c^(3/4)*Sqrt[e*x]*Sqrt[a + c*x^2]) - (a^(1/4)*(Sqrt[a]*B - 3*A*Sqrt[c
])*e*Sqrt[x]*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*Ell
ipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(3*c^(5/4)*Sqrt[e*x]*Sqrt[a +
c*x^2])

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Rubi in Sympy [A]  time = 38.2631, size = 267, normalized size = 0.93 \[ - \frac{2 A \sqrt [4]{a} e \sqrt{x} \sqrt{\frac{a + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{c^{\frac{3}{4}} \sqrt{e x} \sqrt{a + c x^{2}}} + \frac{2 A e x \sqrt{a + c x^{2}}}{\sqrt{c} \sqrt{e x} \left (\sqrt{a} + \sqrt{c} x\right )} + \frac{2 B \sqrt{e x} \sqrt{a + c x^{2}}}{3 c} + \frac{\sqrt [4]{a} e \sqrt{x} \sqrt{\frac{a + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) \left (3 A \sqrt{c} - B \sqrt{a}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{3 c^{\frac{5}{4}} \sqrt{e x} \sqrt{a + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A*a**(1/4)*e*sqrt(x)*sqrt((a + c*x**2)/(sqrt(a) + sqrt(c)*x)**2)*(sqrt(a) + s
qrt(c)*x)*elliptic_e(2*atan(c**(1/4)*sqrt(x)/a**(1/4)), 1/2)/(c**(3/4)*sqrt(e*x)
*sqrt(a + c*x**2)) + 2*A*e*x*sqrt(a + c*x**2)/(sqrt(c)*sqrt(e*x)*(sqrt(a) + sqrt
(c)*x)) + 2*B*sqrt(e*x)*sqrt(a + c*x**2)/(3*c) + a**(1/4)*e*sqrt(x)*sqrt((a + c*
x**2)/(sqrt(a) + sqrt(c)*x)**2)*(sqrt(a) + sqrt(c)*x)*(3*A*sqrt(c) - B*sqrt(a))*
elliptic_f(2*atan(c**(1/4)*sqrt(x)/a**(1/4)), 1/2)/(3*c**(5/4)*sqrt(e*x)*sqrt(a
+ c*x**2))

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Mathematica [C]  time = 0.902064, size = 216, normalized size = 0.75 \[ \frac{2 e \left (\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} \left (a+c x^2\right ) (3 A+B x)+\sqrt{a} x^{3/2} \sqrt{\frac{a}{c x^2}+1} \left (3 A \sqrt{c}-i \sqrt{a} B\right ) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )-3 \sqrt{a} A \sqrt{c} x^{3/2} \sqrt{\frac{a}{c x^2}+1} E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{3 c \sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} \sqrt{e x} \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*e*(Sqrt[(I*Sqrt[a])/Sqrt[c]]*(3*A + B*x)*(a + c*x^2) - 3*Sqrt[a]*A*Sqrt[c]*Sq
rt[1 + a/(c*x^2)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[c]]/Sqrt[x]]
, -1] + Sqrt[a]*((-I)*Sqrt[a]*B + 3*A*Sqrt[c])*Sqrt[1 + a/(c*x^2)]*x^(3/2)*Ellip
ticF[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[c]]/Sqrt[x]], -1]))/(3*Sqrt[(I*Sqrt[a])/Sqr
t[c]]*c*Sqrt[e*x]*Sqrt[a + c*x^2])

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Maple [A]  time = 0.016, size = 295, normalized size = 1. \[ -{\frac{1}{3\,{c}^{2}x}\sqrt{ex} \left ( 3\,A\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ) ac-6\,A\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ) ac+B\sqrt{{1 \left ( cx+\sqrt{-ac} \right ){\frac{1}{\sqrt{-ac}}}}}\sqrt{2}\sqrt{{1 \left ( -cx+\sqrt{-ac} \right ){\frac{1}{\sqrt{-ac}}}}}\sqrt{-{cx{\frac{1}{\sqrt{-ac}}}}}{\it EllipticF} \left ( \sqrt{{1 \left ( cx+\sqrt{-ac} \right ){\frac{1}{\sqrt{-ac}}}}},{\frac{\sqrt{2}}{2}} \right ) \sqrt{-ac}a-2\,B{c}^{2}{x}^{3}-2\,aBcx \right ){\frac{1}{\sqrt{c{x}^{2}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(e*x)^(1/2)/(c*x^2+a)^(1/2)*(3*A*((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2)*2^
(1/2)*((-c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2)*(-x*c/(-a*c)^(1/2))^(1/2)*Ellipti
cF(((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2),1/2*2^(1/2))*a*c-6*A*((c*x+(-a*c)^(1/
2))/(-a*c)^(1/2))^(1/2)*2^(1/2)*((-c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2)*(-x*c/(
-a*c)^(1/2))^(1/2)*EllipticE(((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2),1/2*2^(1/2)
)*a*c+B*((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2)*2^(1/2)*((-c*x+(-a*c)^(1/2))/(-a
*c)^(1/2))^(1/2)*(-x*c/(-a*c)^(1/2))^(1/2)*EllipticF(((c*x+(-a*c)^(1/2))/(-a*c)^
(1/2))^(1/2),1/2*2^(1/2))*(-a*c)^(1/2)*a-2*B*c^2*x^3-2*a*B*c*x)/x/c^2

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 7.14877, size = 92, normalized size = 0.32 \[ \frac{A \left (e x\right )^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt{a} e \Gamma \left (\frac{7}{4}\right )} + \frac{B \left (e x\right )^{\frac{5}{2}} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt{a} e^{2} \Gamma \left (\frac{9}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*(e*x)**(3/2)*gamma(3/4)*hyper((1/2, 3/4), (7/4,), c*x**2*exp_polar(I*pi)/a)/(2
*sqrt(a)*e*gamma(7/4)) + B*(e*x)**(5/2)*gamma(5/4)*hyper((1/2, 5/4), (9/4,), c*x
**2*exp_polar(I*pi)/a)/(2*sqrt(a)*e**2*gamma(9/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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