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

Optimal. Leaf size=327 \[ \frac{\sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (\sqrt{a} B+3 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 a^{7/4} \sqrt [4]{c} e \sqrt{e x} \sqrt{a+c x^2}}-\frac{3 A \sqrt [4]{c} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{a^{7/4} e \sqrt{e x} \sqrt{a+c x^2}}-\frac{3 A \sqrt{a+c x^2}}{a^2 e \sqrt{e x}}+\frac{3 A \sqrt{c} x \sqrt{a+c x^2}}{a^2 e \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}+\frac{A+B x}{a e \sqrt{e x} \sqrt{a+c x^2}} \]

[Out]

(A + B*x)/(a*e*Sqrt[e*x]*Sqrt[a + c*x^2]) - (3*A*Sqrt[a + c*x^2])/(a^2*e*Sqrt[e*
x]) + (3*A*Sqrt[c]*x*Sqrt[a + c*x^2])/(a^2*e*Sqrt[e*x]*(Sqrt[a] + Sqrt[c]*x)) -
(3*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])/(a^(7/4)*e*Sqrt[e*x]*S
qrt[a + c*x^2]) + ((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])/(2*a^(7/4)*c^(1/4)*e*Sqrt[e*x]*Sqrt[a + c*x^2])

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Rubi [A]  time = 0.823885, antiderivative size = 327, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ \frac{\sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (\sqrt{a} B+3 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 a^{7/4} \sqrt [4]{c} e \sqrt{e x} \sqrt{a+c x^2}}-\frac{3 A \sqrt [4]{c} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{a^{7/4} e \sqrt{e x} \sqrt{a+c x^2}}-\frac{3 A \sqrt{a+c x^2}}{a^2 e \sqrt{e x}}+\frac{3 A \sqrt{c} x \sqrt{a+c x^2}}{a^2 e \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}+\frac{A+B x}{a e \sqrt{e x} \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(A + B*x)/(a*e*Sqrt[e*x]*Sqrt[a + c*x^2]) - (3*A*Sqrt[a + c*x^2])/(a^2*e*Sqrt[e*
x]) + (3*A*Sqrt[c]*x*Sqrt[a + c*x^2])/(a^2*e*Sqrt[e*x]*(Sqrt[a] + Sqrt[c]*x)) -
(3*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])/(a^(7/4)*e*Sqrt[e*x]*S
qrt[a + c*x^2]) + ((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])/(2*a^(7/4)*c^(1/4)*e*Sqrt[e*x]*Sqrt[a + c*x^2])

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Rubi in Sympy [A]  time = 112.311, size = 298, normalized size = 0.91 \[ \frac{3 A \sqrt{c} x \sqrt{a + c x^{2}}}{a^{2} e \sqrt{e x} \left (\sqrt{a} + \sqrt{c} x\right )} - \frac{3 A \sqrt{a + c x^{2}}}{a^{2} e \sqrt{e x}} - \frac{3 A \sqrt [4]{c} \sqrt{x} \sqrt{\frac{a + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{a^{\frac{7}{4}} e \sqrt{e x} \sqrt{a + c x^{2}}} + \frac{A + B x}{a e \sqrt{e x} \sqrt{a + c x^{2}}} + \frac{\sqrt{x} \sqrt{\frac{a + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) \left (3 A \sqrt{c} + B \sqrt{a}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{2 a^{\frac{7}{4}} \sqrt [4]{c} e \sqrt{e x} \sqrt{a + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*A*sqrt(c)*x*sqrt(a + c*x**2)/(a**2*e*sqrt(e*x)*(sqrt(a) + sqrt(c)*x)) - 3*A*sq
rt(a + c*x**2)/(a**2*e*sqrt(e*x)) - 3*A*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)/(a**(7/4)*e*sqrt(e*x)*sqrt(a + c*x**2)) + (A + B*x)/(a*e*sqrt(e*x)
*sqrt(a + c*x**2)) + 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)/(2*a**(7/4)*c**(1/4)*e*sqrt(e*x)*sqrt(a + c*x**2))

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Mathematica [C]  time = 0.310123, size = 201, normalized size = 0.61 \[ \frac{x \left (x^{3/2} \sqrt{\frac{a}{c x^2}+1} \left (3 A \sqrt{c}+i \sqrt{a} B\right ) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )+\sqrt{a} \sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} (A+B x)-3 A \sqrt{c} x^{3/2} \sqrt{\frac{a}{c x^2}+1} E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{a^{3/2} \sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} (e x)^{3/2} \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.03, size = 303, normalized size = 0.9 \[{\frac{1}{2\,ce{a}^{2}} \left ( 6\,A\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ) ac-3\,A\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ) ac+B\sqrt{{1 \left ( cx+\sqrt{-ac} \right ){\frac{1}{\sqrt{-ac}}}}}\sqrt{2}\sqrt{{1 \left ( -cx+\sqrt{-ac} \right ){\frac{1}{\sqrt{-ac}}}}}\sqrt{-{cx{\frac{1}{\sqrt{-ac}}}}}{\it EllipticF} \left ( \sqrt{{1 \left ( cx+\sqrt{-ac} \right ){\frac{1}{\sqrt{-ac}}}}},{\frac{\sqrt{2}}{2}} \right ) \sqrt{-ac}a-6\,A{c}^{2}{x}^{2}+2\,aBcx-4\,aAc \right ){\frac{1}{\sqrt{c{x}^{2}+a}}}{\frac{1}{\sqrt{ex}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(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-6*A*c^2*x^2+2*a*B*c*x-4*a*A*c)/(c*x^2+a)^(1/2)/c/e/(e*x)^(1/2)/a^2

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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