3.470 \(\int \frac{A+B x}{\sqrt{e x} \left (a+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=290 \[ -\frac{\sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (\sqrt{a} B-A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 a^{5/4} c^{3/4} \sqrt{e x} \sqrt{a+c x^2}}+\frac{B \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{a^{3/4} c^{3/4} \sqrt{e x} \sqrt{a+c x^2}}+\frac{\sqrt{e x} (A+B x)}{a e \sqrt{a+c x^2}}-\frac{B x \sqrt{a+c x^2}}{a \sqrt{c} \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )} \]

[Out]

(Sqrt[e*x]*(A + B*x))/(a*e*Sqrt[a + c*x^2]) - (B*x*Sqrt[a + c*x^2])/(a*Sqrt[c]*S
qrt[e*x]*(Sqrt[a] + Sqrt[c]*x)) + (B*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]
)/(a^(3/4)*c^(3/4)*Sqrt[e*x]*Sqrt[a + c*x^2]) - ((Sqrt[a]*B - A*Sqrt[c])*Sqrt[x]
*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticF[2*Arc
Tan[(c^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(2*a^(5/4)*c^(3/4)*Sqrt[e*x]*Sqrt[a + c*x^
2])

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Rubi [A]  time = 0.65119, antiderivative size = 290, 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 \[ -\frac{\sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (\sqrt{a} B-A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 a^{5/4} c^{3/4} \sqrt{e x} \sqrt{a+c x^2}}+\frac{B \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{a^{3/4} c^{3/4} \sqrt{e x} \sqrt{a+c x^2}}+\frac{\sqrt{e x} (A+B x)}{a e \sqrt{a+c x^2}}-\frac{B x \sqrt{a+c x^2}}{a \sqrt{c} \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/(Sqrt[e*x]*(a + c*x^2)^(3/2)),x]

[Out]

(Sqrt[e*x]*(A + B*x))/(a*e*Sqrt[a + c*x^2]) - (B*x*Sqrt[a + c*x^2])/(a*Sqrt[c]*S
qrt[e*x]*(Sqrt[a] + Sqrt[c]*x)) + (B*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]
)/(a^(3/4)*c^(3/4)*Sqrt[e*x]*Sqrt[a + c*x^2]) - ((Sqrt[a]*B - A*Sqrt[c])*Sqrt[x]
*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticF[2*Arc
Tan[(c^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(2*a^(5/4)*c^(3/4)*Sqrt[e*x]*Sqrt[a + c*x^
2])

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Rubi in Sympy [A]  time = 82.3915, size = 260, normalized size = 0.9 \[ - \frac{B x \sqrt{a + c x^{2}}}{a \sqrt{c} \sqrt{e x} \left (\sqrt{a} + \sqrt{c} x\right )} + \frac{B \sqrt{x} \sqrt{\frac{a + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{a^{\frac{3}{4}} c^{\frac{3}{4}} \sqrt{e x} \sqrt{a + c x^{2}}} + \frac{\sqrt{e x} \left (A + B x\right )}{a e \sqrt{a + c x^{2}}} + \frac{\sqrt{x} \sqrt{\frac{a + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) \left (A \sqrt{c} - B \sqrt{a}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{2 a^{\frac{5}{4}} c^{\frac{3}{4}} \sqrt{e x} \sqrt{a + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/(e*x)**(1/2)/(c*x**2+a)**(3/2),x)

[Out]

-B*x*sqrt(a + c*x**2)/(a*sqrt(c)*sqrt(e*x)*(sqrt(a) + sqrt(c)*x)) + B*sqrt(x)*sq
rt((a + c*x**2)/(sqrt(a) + sqrt(c)*x)**2)*(sqrt(a) + sqrt(c)*x)*elliptic_e(2*ata
n(c**(1/4)*sqrt(x)/a**(1/4)), 1/2)/(a**(3/4)*c**(3/4)*sqrt(e*x)*sqrt(a + c*x**2)
) + sqrt(e*x)*(A + B*x)/(a*e*sqrt(a + c*x**2)) + sqrt(x)*sqrt((a + 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)/(2*a**(5/4)*c**(3/4)*sqrt(e*x)*sqrt(a + c*
x**2))

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Mathematica [C]  time = 0.664508, size = 211, normalized size = 0.73 \[ \frac{i \sqrt{c} x^{3/2} \sqrt{\frac{a}{c x^2}+1} \left (A \sqrt{c}+i \sqrt{a} B\right ) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )+\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} (A c x-a B)+\sqrt{a} B \sqrt{c} x^{3/2} \sqrt{\frac{a}{c x^2}+1} E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )}{a c \sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} \sqrt{e x} \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/(Sqrt[e*x]*(a + c*x^2)^(3/2)),x]

[Out]

(Sqrt[(I*Sqrt[a])/Sqrt[c]]*(-(a*B) + A*c*x) + Sqrt[a]*B*Sqrt[c]*Sqrt[1 + a/(c*x^
2)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[c]]/Sqrt[x]], -1] + I*(I*S
qrt[a]*B + A*Sqrt[c])*Sqrt[c]*Sqrt[1 + a/(c*x^2)]*x^(3/2)*EllipticF[I*ArcSinh[Sq
rt[(I*Sqrt[a])/Sqrt[c]]/Sqrt[x]], -1])/(a*Sqrt[(I*Sqrt[a])/Sqrt[c]]*c*Sqrt[e*x]*
Sqrt[a + c*x^2])

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Maple [A]  time = 0.047, size = 288, normalized size = 1. \[{\frac{1}{2\,ac} \left ( A\sqrt{{1 \left ( cx+\sqrt{-ac} \right ){\frac{1}{\sqrt{-ac}}}}}\sqrt{2}\sqrt{{1 \left ( -cx+\sqrt{-ac} \right ){\frac{1}{\sqrt{-ac}}}}}\sqrt{-{cx{\frac{1}{\sqrt{-ac}}}}}{\it EllipticF} \left ( \sqrt{{1 \left ( cx+\sqrt{-ac} \right ){\frac{1}{\sqrt{-ac}}}}},{\frac{\sqrt{2}}{2}} \right ) \sqrt{-ac}+B\sqrt{{1 \left ( cx+\sqrt{-ac} \right ){\frac{1}{\sqrt{-ac}}}}}\sqrt{2}\sqrt{{1 \left ( -cx+\sqrt{-ac} \right ){\frac{1}{\sqrt{-ac}}}}}\sqrt{-{cx{\frac{1}{\sqrt{-ac}}}}}{\it EllipticF} \left ( \sqrt{{1 \left ( cx+\sqrt{-ac} \right ){\frac{1}{\sqrt{-ac}}}}},{\frac{\sqrt{2}}{2}} \right ) a-2\,B\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ) a+2\,Bc{x}^{2}+2\,Acx \right ){\frac{1}{\sqrt{c{x}^{2}+a}}}{\frac{1}{\sqrt{ex}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/(e*x)^(1/2)/(c*x^2+a)^(3/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{B x + A}{{\left (c x^{2} + a\right )}^{\frac{3}{2}} \sqrt{e x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((c*x^2 + a)^(3/2)*sqrt(e*x)),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((c*x^2 + a)^(3/2)*sqrt(e*x)),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 68.1008, size = 94, normalized size = 0.32 \[ \frac{A \sqrt{x} \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{3}{2} \\ \frac{5}{4} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac{3}{2}} \sqrt{e} \Gamma \left (\frac{5}{4}\right )} + \frac{B x^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{3}{2} \\ \frac{7}{4} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac{3}{2}} \sqrt{e} \Gamma \left (\frac{7}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)/(e*x)**(1/2)/(c*x**2+a)**(3/2),x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{B x + A}{{\left (c x^{2} + a\right )}^{\frac{3}{2}} \sqrt{e x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((c*x^2 + a)^(3/2)*sqrt(e*x)),x, algorithm="giac")

[Out]

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