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

Optimal. Leaf size=297 \[ \frac{2 a^{3/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (3 \sqrt{a} B+5 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15 c^{3/4} \sqrt{e x} \sqrt{a+c x^2}}-\frac{4 a^{5/4} B \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 c^{3/4} \sqrt{e x} \sqrt{a+c x^2}}+\frac{2 \sqrt{e x} \sqrt{a+c x^2} (5 A+3 B x)}{15 e}+\frac{4 a B x \sqrt{a+c x^2}}{5 \sqrt{c} \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )} \]

[Out]

(2*Sqrt[e*x]*(5*A + 3*B*x)*Sqrt[a + c*x^2])/(15*e) + (4*a*B*x*Sqrt[a + c*x^2])/(
5*Sqrt[c]*Sqrt[e*x]*(Sqrt[a] + Sqrt[c]*x)) - (4*a^(5/4)*B*Sqrt[x]*(Sqrt[a] + Sqr
t[c]*x)*Sqrt[(a + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticE[2*ArcTan[(c^(1/4)*Sq
rt[x])/a^(1/4)], 1/2])/(5*c^(3/4)*Sqrt[e*x]*Sqrt[a + c*x^2]) + (2*a^(3/4)*(3*Sqr
t[a]*B + 5*A*Sqrt[c])*Sqrt[x]*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + c*x^2)/(Sqrt[a] +
Sqrt[c]*x)^2]*EllipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(15*c^(3/4)*S
qrt[e*x]*Sqrt[a + c*x^2])

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Rubi [A]  time = 0.645078, antiderivative size = 297, 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 \[ \frac{2 a^{3/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (3 \sqrt{a} B+5 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15 c^{3/4} \sqrt{e x} \sqrt{a+c x^2}}-\frac{4 a^{5/4} B \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 c^{3/4} \sqrt{e x} \sqrt{a+c x^2}}+\frac{2 \sqrt{e x} \sqrt{a+c x^2} (5 A+3 B x)}{15 e}+\frac{4 a B x \sqrt{a+c x^2}}{5 \sqrt{c} \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[e*x]*(5*A + 3*B*x)*Sqrt[a + c*x^2])/(15*e) + (4*a*B*x*Sqrt[a + c*x^2])/(
5*Sqrt[c]*Sqrt[e*x]*(Sqrt[a] + Sqrt[c]*x)) - (4*a^(5/4)*B*Sqrt[x]*(Sqrt[a] + Sqr
t[c]*x)*Sqrt[(a + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticE[2*ArcTan[(c^(1/4)*Sq
rt[x])/a^(1/4)], 1/2])/(5*c^(3/4)*Sqrt[e*x]*Sqrt[a + c*x^2]) + (2*a^(3/4)*(3*Sqr
t[a]*B + 5*A*Sqrt[c])*Sqrt[x]*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + c*x^2)/(Sqrt[a] +
Sqrt[c]*x)^2]*EllipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(15*c^(3/4)*S
qrt[e*x]*Sqrt[a + c*x^2])

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Rubi in Sympy [A]  time = 79.7745, size = 280, normalized size = 0.94 \[ - \frac{4 B a^{\frac{5}{4}} \sqrt{x} \sqrt{\frac{a + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{5 c^{\frac{3}{4}} \sqrt{e x} \sqrt{a + c x^{2}}} + \frac{4 B a x \sqrt{a + c x^{2}}}{5 \sqrt{c} \sqrt{e x} \left (\sqrt{a} + \sqrt{c} x\right )} + \frac{2 a^{\frac{3}{4}} \sqrt{x} \sqrt{\frac{a + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) \left (5 A \sqrt{c} + 3 B \sqrt{a}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{15 c^{\frac{3}{4}} \sqrt{e x} \sqrt{a + c x^{2}}} + \frac{4 \sqrt{e x} \left (\frac{5 A}{2} + \frac{3 B x}{2}\right ) \sqrt{a + c x^{2}}}{15 e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4*B*a**(5/4)*sqrt(x)*sqrt((a + c*x**2)/(sqrt(a) + sqrt(c)*x)**2)*(sqrt(a) + sqr
t(c)*x)*elliptic_e(2*atan(c**(1/4)*sqrt(x)/a**(1/4)), 1/2)/(5*c**(3/4)*sqrt(e*x)
*sqrt(a + c*x**2)) + 4*B*a*x*sqrt(a + c*x**2)/(5*sqrt(c)*sqrt(e*x)*(sqrt(a) + sq
rt(c)*x)) + 2*a**(3/4)*sqrt(x)*sqrt((a + c*x**2)/(sqrt(a) + sqrt(c)*x)**2)*(sqrt
(a) + sqrt(c)*x)*(5*A*sqrt(c) + 3*B*sqrt(a))*elliptic_f(2*atan(c**(1/4)*sqrt(x)/
a**(1/4)), 1/2)/(15*c**(3/4)*sqrt(e*x)*sqrt(a + c*x**2)) + 4*sqrt(e*x)*(5*A/2 +
3*B*x/2)*sqrt(a + c*x**2)/(15*e)

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Mathematica [C]  time = 0.750518, size = 227, normalized size = 0.76 \[ \frac{-12 a^{3/2} B \sqrt{c} x^{3/2} \sqrt{\frac{a}{c x^2}+1} E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )+2 \sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} \left (a+c x^2\right ) (6 a B+c x (5 A+3 B x))+4 a \sqrt{c} x^{3/2} \sqrt{\frac{a}{c x^2}+1} \left (3 \sqrt{a} B+5 i A \sqrt{c}\right ) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )}{15 c \sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} \sqrt{e x} \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[(I*Sqrt[a])/Sqrt[c]]*(a + c*x^2)*(6*a*B + c*x*(5*A + 3*B*x)) - 12*a^(3/2
)*B*Sqrt[c]*Sqrt[1 + a/(c*x^2)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqr
t[c]]/Sqrt[x]], -1] + 4*a*(3*Sqrt[a]*B + (5*I)*A*Sqrt[c])*Sqrt[c]*Sqrt[1 + a/(c*
x^2)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[c]]/Sqrt[x]], -1])/(15*S
qrt[(I*Sqrt[a])/Sqrt[c]]*c*Sqrt[e*x]*Sqrt[a + c*x^2])

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Maple [A]  time = 0.036, size = 312, normalized size = 1.1 \[{\frac{2}{15\,c} \left ( 5\,A{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ) \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}\sqrt{-ac}a+6\,B{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ) \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{a}^{2}-3\,B{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ) \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{a}^{2}+3\,B{c}^{2}{x}^{4}+5\,A{c}^{2}{x}^{3}+3\,aBc{x}^{2}+5\,aAcx \right ){\frac{1}{\sqrt{c{x}^{2}+a}}}{\frac{1}{\sqrt{ex}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15/(c*x^2+a)^(1/2)/c*(5*A*EllipticF(((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2),1/
2*2^(1/2))*((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)*(-a*c)^(1/2)*a+6*B*EllipticE(((c*x
+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2),1/2*2^(1/2))*((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)*a^2-3*B*EllipticF(((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2),1/2*2^(1/2))*((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)*a^2+3*B*c^2*x^4+5*A*c^2*x^3+3*a*B*c*x^2+5*a*A*c*x)/
(e*x)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 7.9665, size = 97, normalized size = 0.33 \[ \frac{A \sqrt{a} \sqrt{x} \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt{e} \Gamma \left (\frac{5}{4}\right )} + \frac{B \sqrt{a} x^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt{e} \Gamma \left (\frac{7}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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