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

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

[Out]

(-2*(3*A - B*x)*Sqrt[a + c*x^2])/(3*e*Sqrt[e*x]) + (4*A*Sqrt[c]*x*Sqrt[a + c*x^2
])/(e*Sqrt[e*x]*(Sqrt[a] + Sqrt[c]*x)) - (4*a^(1/4)*A*c^(1/4)*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])/(e*Sqrt[e*x]*Sqrt[a + c*x^2]) + (2*a^(1/4)*(Sqrt[a]*B
+ 3*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])/(3*c^(1/4)*e*Sqrt[e
*x]*Sqrt[a + c*x^2])

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Rubi [A]  time = 0.607861, antiderivative size = 300, 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 \[ -\frac{2 \sqrt{a+c x^2} (3 A-B x)}{3 e \sqrt{e x}}+\frac{2 \sqrt [4]{a} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (\sqrt{a} B+3 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{3 \sqrt [4]{c} e \sqrt{e x} \sqrt{a+c x^2}}+\frac{4 A \sqrt{c} x \sqrt{a+c x^2}}{e \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}-\frac{4 \sqrt [4]{a} A \sqrt [4]{c} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{e \sqrt{e x} \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(3*A - B*x)*Sqrt[a + c*x^2])/(3*e*Sqrt[e*x]) + (4*A*Sqrt[c]*x*Sqrt[a + c*x^2
])/(e*Sqrt[e*x]*(Sqrt[a] + Sqrt[c]*x)) - (4*a^(1/4)*A*c^(1/4)*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])/(e*Sqrt[e*x]*Sqrt[a + c*x^2]) + (2*a^(1/4)*(Sqrt[a]*B
+ 3*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])/(3*c^(1/4)*e*Sqrt[e
*x]*Sqrt[a + c*x^2])

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Rubi in Sympy [A]  time = 58.1271, size = 277, normalized size = 0.92 \[ - \frac{4 A \sqrt [4]{a} \sqrt [4]{c} \sqrt{x} \sqrt{\frac{a + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{e \sqrt{e x} \sqrt{a + c x^{2}}} + \frac{4 A \sqrt{c} x \sqrt{a + c x^{2}}}{e \sqrt{e x} \left (\sqrt{a} + \sqrt{c} x\right )} + \frac{2 \sqrt [4]{a} \sqrt{x} \sqrt{\frac{a + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) \left (3 A \sqrt{c} + B \sqrt{a}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{3 \sqrt [4]{c} e \sqrt{e x} \sqrt{a + c x^{2}}} - \frac{4 \left (\frac{3 A}{2} - \frac{B x}{2}\right ) \sqrt{a + c x^{2}}}{3 e \sqrt{e x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4*A*a**(1/4)*c**(1/4)*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)/(e*sqrt(e*x)
*sqrt(a + c*x**2)) + 4*A*sqrt(c)*x*sqrt(a + c*x**2)/(e*sqrt(e*x)*(sqrt(a) + sqrt
(c)*x)) + 2*a**(1/4)*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**(1/4)*e*sqrt(e*x)*sqrt(a + c*x**2)) - 4*(3*A/2 - B*x/2)*sqrt(a
+ c*x**2)/(3*e*sqrt(e*x))

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Mathematica [C]  time = 0.763077, size = 215, normalized size = 0.72 \[ \frac{x \left (2 \sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} \left (a+c x^2\right ) (3 A+B x)+4 \sqrt{a} x^{3/2} \sqrt{\frac{a}{c x^2}+1} \left (3 A \sqrt{c}+i \sqrt{a} B\right ) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )-12 \sqrt{a} A \sqrt{c} x^{3/2} \sqrt{\frac{a}{c x^2}+1} E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{3 \sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} (e x)^{3/2} \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.05, size = 307, normalized size = 1. \[{\frac{2}{3\,ce} \left ( 6\,A\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ) ac-3\,A\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ) ac+B\sqrt{{1 \left ( cx+\sqrt{-ac} \right ){\frac{1}{\sqrt{-ac}}}}}\sqrt{2}\sqrt{{1 \left ( -cx+\sqrt{-ac} \right ){\frac{1}{\sqrt{-ac}}}}}\sqrt{-{cx{\frac{1}{\sqrt{-ac}}}}}{\it EllipticF} \left ( \sqrt{{1 \left ( cx+\sqrt{-ac} \right ){\frac{1}{\sqrt{-ac}}}}},{\frac{\sqrt{2}}{2}} \right ) \sqrt{-ac}a+B{c}^{2}{x}^{3}-3\,A{c}^{2}{x}^{2}+aBcx-3\,aAc \right ){\frac{1}{\sqrt{c{x}^{2}+a}}}{\frac{1}{\sqrt{ex}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(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-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)*EllipticF(
((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+B*c^2*x^3-3*A*c^2*x^2+a*B*c*x-3*a*A*c)/(c*x^2+a)^(1/2)/e/(e*x)^(1/2)
/c

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 11.8341, size = 100, normalized size = 0.33 \[ \frac{A \sqrt{a} \Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, - \frac{1}{4} \\ \frac{3}{4} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{2 e^{\frac{3}{2}} \sqrt{x} \Gamma \left (\frac{3}{4}\right )} + \frac{B \sqrt{a} \sqrt{x} \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{2 e^{\frac{3}{2}} \Gamma \left (\frac{5}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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