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

Optimal. Leaf size=298 \[ \frac{e \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (\sqrt{a} B-A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 a^{3/4} c^{5/4} \sqrt{e x} \sqrt{a+c x^2}}+\frac{A e \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{a^{3/4} c^{3/4} \sqrt{e x} \sqrt{a+c x^2}}-\frac{\sqrt{e x} (a B-A c x)}{a c \sqrt{a+c x^2}}-\frac{A e x \sqrt{a+c x^2}}{a \sqrt{c} \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )} \]

[Out]

-((Sqrt[e*x]*(a*B - A*c*x))/(a*c*Sqrt[a + c*x^2])) - (A*e*x*Sqrt[a + c*x^2])/(a*
Sqrt[c]*Sqrt[e*x]*(Sqrt[a] + Sqrt[c]*x)) + (A*e*Sqrt[x]*(Sqrt[a] + Sqrt[c]*x)*Sq
rt[(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])*e*Sqrt[x]*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*El
lipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(2*a^(3/4)*c^(5/4)*Sqrt[e*x]*
Sqrt[a + c*x^2])

_______________________________________________________________________________________

Rubi [A]  time = 0.601901, antiderivative size = 298, 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 \[ \frac{e \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (\sqrt{a} B-A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 a^{3/4} c^{5/4} \sqrt{e x} \sqrt{a+c x^2}}+\frac{A e \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{a^{3/4} c^{3/4} \sqrt{e x} \sqrt{a+c x^2}}-\frac{\sqrt{e x} (a B-A c x)}{a c \sqrt{a+c x^2}}-\frac{A e x \sqrt{a+c x^2}}{a \sqrt{c} \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )} \]

Antiderivative was successfully verified.

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

[Out]

-((Sqrt[e*x]*(a*B - A*c*x))/(a*c*Sqrt[a + c*x^2])) - (A*e*x*Sqrt[a + c*x^2])/(a*
Sqrt[c]*Sqrt[e*x]*(Sqrt[a] + Sqrt[c]*x)) + (A*e*Sqrt[x]*(Sqrt[a] + Sqrt[c]*x)*Sq
rt[(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])*e*Sqrt[x]*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*El
lipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(2*a^(3/4)*c^(5/4)*Sqrt[e*x]*
Sqrt[a + c*x^2])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 59.6342, size = 269, normalized size = 0.9 \[ - \frac{A e x \sqrt{a + c x^{2}}}{a \sqrt{c} \sqrt{e x} \left (\sqrt{a} + \sqrt{c} x\right )} + \frac{A e \sqrt{x} \sqrt{\frac{a + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{a^{\frac{3}{4}} c^{\frac{3}{4}} \sqrt{e x} \sqrt{a + c x^{2}}} - \frac{\sqrt{e x} \left (- A c x + B a\right )}{a c \sqrt{a + c x^{2}}} - \frac{e \sqrt{x} \sqrt{\frac{a + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) \left (A \sqrt{c} - B \sqrt{a}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{2 a^{\frac{3}{4}} c^{\frac{5}{4}} \sqrt{e x} \sqrt{a + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A*e*x*sqrt(a + c*x**2)/(a*sqrt(c)*sqrt(e*x)*(sqrt(a) + sqrt(c)*x)) + A*e*sqrt(x
)*sqrt((a + 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)/(a**(3/4)*c**(3/4)*sqrt(e*x)*sqrt(a + c*x
**2)) - sqrt(e*x)*(-A*c*x + B*a)/(a*c*sqrt(a + c*x**2)) - e*sqrt(x)*sqrt((a + c*
x**2)/(sqrt(a) + sqrt(c)*x)**2)*(sqrt(a) + sqrt(c)*x)*(A*sqrt(c) - B*sqrt(a))*el
liptic_f(2*atan(c**(1/4)*sqrt(x)/a**(1/4)), 1/2)/(2*a**(3/4)*c**(5/4)*sqrt(e*x)*
sqrt(a + c*x**2))

_______________________________________________________________________________________

Mathematica [C]  time = 1.0937, size = 204, normalized size = 0.68 \[ \frac{i e \left (-x^{3/2} \sqrt{\frac{a}{c x^2}+1} \left (A \sqrt{c}-i \sqrt{a} B\right ) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )-\sqrt{a} \sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} (A+B x)+A \sqrt{c} x^{3/2} \sqrt{\frac{a}{c x^2}+1} E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{c^{3/2} \left (\frac{i \sqrt{a}}{\sqrt{c}}\right )^{3/2} \sqrt{e x} \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.02, size = 297, normalized size = 1. \[{\frac{1}{2\,ax{c}^{2}}\sqrt{ex} \left ( A\sqrt{{1 \left ( cx+\sqrt{-ac} \right ){\frac{1}{\sqrt{-ac}}}}}\sqrt{2}\sqrt{{1 \left ( -cx+\sqrt{-ac} \right ){\frac{1}{\sqrt{-ac}}}}}\sqrt{-{cx{\frac{1}{\sqrt{-ac}}}}}{\it EllipticF} \left ( \sqrt{{1 \left ( cx+\sqrt{-ac} \right ){\frac{1}{\sqrt{-ac}}}}},{\frac{\sqrt{2}}{2}} \right ) ac-2\,A\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ) ac+B\sqrt{{1 \left ( cx+\sqrt{-ac} \right ){\frac{1}{\sqrt{-ac}}}}}\sqrt{2}\sqrt{{1 \left ( -cx+\sqrt{-ac} \right ){\frac{1}{\sqrt{-ac}}}}}\sqrt{-{cx{\frac{1}{\sqrt{-ac}}}}}{\it EllipticF} \left ( \sqrt{{1 \left ( cx+\sqrt{-ac} \right ){\frac{1}{\sqrt{-ac}}}}},{\frac{\sqrt{2}}{2}} \right ) \sqrt{-ac}a+2\,A{c}^{2}{x}^{2}-2\,aBcx \right ){\frac{1}{\sqrt{c{x}^{2}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(e*x)^(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-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)*
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*A*c^2*x^2-2*a*B*c*x)/(c*x^2+a)^(1/2)/x/a/c^2

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 41.6011, size = 94, normalized size = 0.32 \[ \frac{A \sqrt{e} x^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{3}{2} \\ \frac{7}{4} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac{3}{2}} \Gamma \left (\frac{7}{4}\right )} + \frac{B \sqrt{e} x^{\frac{5}{2}} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{5}{4}, \frac{3}{2} \\ \frac{9}{4} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac{3}{2}} \Gamma \left (\frac{9}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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