3.464 \(\int \frac{A+B x}{(e x)^{5/2} \sqrt{a+c x^2}} \, dx\)

Optimal. Leaf size=327 \[ \frac{\sqrt [4]{c} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (3 \sqrt{a} B-A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{3 a^{5/4} e^2 \sqrt{e x} \sqrt{a+c x^2}}-\frac{2 B \sqrt [4]{c} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{a^{3/4} e^2 \sqrt{e x} \sqrt{a+c x^2}}-\frac{2 A \sqrt{a+c x^2}}{3 a e (e x)^{3/2}}-\frac{2 B \sqrt{a+c x^2}}{a e^2 \sqrt{e x}}+\frac{2 B \sqrt{c} x \sqrt{a+c x^2}}{a e^2 \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )} \]

[Out]

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

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Rubi [A]  time = 0.791652, antiderivative size = 327, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{\sqrt [4]{c} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (3 \sqrt{a} B-A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{3 a^{5/4} e^2 \sqrt{e x} \sqrt{a+c x^2}}-\frac{2 B \sqrt [4]{c} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{a^{3/4} e^2 \sqrt{e x} \sqrt{a+c x^2}}-\frac{2 A \sqrt{a+c x^2}}{3 a e (e x)^{3/2}}-\frac{2 B \sqrt{a+c x^2}}{a e^2 \sqrt{e x}}+\frac{2 B \sqrt{c} x \sqrt{a+c x^2}}{a e^2 \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/((e*x)^(5/2)*Sqrt[a + c*x^2]),x]

[Out]

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

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Rubi in Sympy [A]  time = 97.7227, size = 301, normalized size = 0.92 \[ - \frac{2 A \sqrt{a + c x^{2}}}{3 a e \left (e x\right )^{\frac{3}{2}}} + \frac{2 B \sqrt{c} x \sqrt{a + c x^{2}}}{a e^{2} \sqrt{e x} \left (\sqrt{a} + \sqrt{c} x\right )} - \frac{2 B \sqrt{a + c x^{2}}}{a e^{2} \sqrt{e x}} - \frac{2 B \sqrt [4]{c} \sqrt{x} \sqrt{\frac{a + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{a^{\frac{3}{4}} e^{2} \sqrt{e x} \sqrt{a + c x^{2}}} - \frac{\sqrt [4]{c} \sqrt{x} \sqrt{\frac{a + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) \left (A \sqrt{c} - 3 B \sqrt{a}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{3 a^{\frac{5}{4}} e^{2} \sqrt{e x} \sqrt{a + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/(e*x)**(5/2)/(c*x**2+a)**(1/2),x)

[Out]

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

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Mathematica [C]  time = 0.716938, size = 212, normalized size = 0.65 \[ \frac{x \left (2 \sqrt{c} x^{5/2} \sqrt{\frac{a}{c x^2}+1} \left (3 \sqrt{a} B-i A \sqrt{c}\right ) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )-2 A \sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} \left (a+c x^2\right )-6 \sqrt{a} B \sqrt{c} x^{5/2} \sqrt{\frac{a}{c x^2}+1} E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{3 a \sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} (e x)^{5/2} \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/((e*x)^(5/2)*Sqrt[a + c*x^2]),x]

[Out]

(x*(-2*A*Sqrt[(I*Sqrt[a])/Sqrt[c]]*(a + c*x^2) - 6*Sqrt[a]*B*Sqrt[c]*Sqrt[1 + a/
(c*x^2)]*x^(5/2)*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[c]]/Sqrt[x]], -1] + 2
*(3*Sqrt[a]*B - I*A*Sqrt[c])*Sqrt[c]*Sqrt[1 + a/(c*x^2)]*x^(5/2)*EllipticF[I*Arc
Sinh[Sqrt[(I*Sqrt[a])/Sqrt[c]]/Sqrt[x]], -1]))/(3*a*Sqrt[(I*Sqrt[a])/Sqrt[c]]*(e
*x)^(5/2)*Sqrt[a + c*x^2])

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Maple [A]  time = 0.028, size = 306, normalized size = 0.9 \[ -{\frac{1}{3\,a{e}^{2}x} \left ( A\sqrt{{1 \left ( cx+\sqrt{-ac} \right ){\frac{1}{\sqrt{-ac}}}}}\sqrt{2}\sqrt{{1 \left ( -cx+\sqrt{-ac} \right ){\frac{1}{\sqrt{-ac}}}}}\sqrt{-{cx{\frac{1}{\sqrt{-ac}}}}}{\it EllipticF} \left ( \sqrt{{1 \left ( cx+\sqrt{-ac} \right ){\frac{1}{\sqrt{-ac}}}}},{\frac{\sqrt{2}}{2}} \right ) \sqrt{-ac}x+3\,B\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ) xa-6\,B\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ) xa+6\,Bc{x}^{3}+2\,Ac{x}^{2}+6\,aBx+2\,aA \right ){\frac{1}{\sqrt{c{x}^{2}+a}}}{\frac{1}{\sqrt{ex}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/(e*x)^(5/2)/(c*x^2+a)^(1/2),x)

[Out]

-1/3/x*(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)*x+3*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))*x*a-6*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))*x*a+6*B*c*x^3+2*A*c*x^2+6*a*B*x+2*a*A)/(c*x^2+a)^(1/2)/a/e^2/(e*x)^(1
/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{B x + A}{\sqrt{c x^{2} + a} \left (e x\right )^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/(sqrt(c*x^2 + a)*(e*x)^(5/2)),x, algorithm="maxima")

[Out]

integrate((B*x + A)/(sqrt(c*x^2 + a)*(e*x)^(5/2)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/(sqrt(c*x^2 + a)*(e*x)^(5/2)),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 85.83, size = 100, normalized size = 0.31 \[ \frac{A \Gamma \left (- \frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{1}{2} \\ \frac{1}{4} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt{a} e^{\frac{5}{2}} x^{\frac{3}{2}} \Gamma \left (\frac{1}{4}\right )} + \frac{B \Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{1}{2} \\ \frac{3}{4} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt{a} e^{\frac{5}{2}} \sqrt{x} \Gamma \left (\frac{3}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)/(e*x)**(5/2)/(c*x**2+a)**(1/2),x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{B x + A}{\sqrt{c x^{2} + a} \left (e x\right )^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/(sqrt(c*x^2 + a)*(e*x)^(5/2)),x, algorithm="giac")

[Out]

integrate((B*x + A)/(sqrt(c*x^2 + a)*(e*x)^(5/2)), x)