3.472 \(\int \frac{A+B x}{(e x)^{5/2} \left (a+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=357 \[ \frac{\sqrt [4]{c} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (9 \sqrt{a} B-5 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{6 a^{9/4} e^2 \sqrt{e x} \sqrt{a+c x^2}}-\frac{3 B \sqrt [4]{c} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{a^{7/4} e^2 \sqrt{e x} \sqrt{a+c x^2}}-\frac{5 A \sqrt{a+c x^2}}{3 a^2 e (e x)^{3/2}}-\frac{3 B \sqrt{a+c x^2}}{a^2 e^2 \sqrt{e x}}+\frac{3 B \sqrt{c} x \sqrt{a+c x^2}}{a^2 e^2 \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}+\frac{A+B x}{a e (e x)^{3/2} \sqrt{a+c x^2}} \]

[Out]

(A + B*x)/(a*e*(e*x)^(3/2)*Sqrt[a + c*x^2]) - (5*A*Sqrt[a + c*x^2])/(3*a^2*e*(e*
x)^(3/2)) - (3*B*Sqrt[a + c*x^2])/(a^2*e^2*Sqrt[e*x]) + (3*B*Sqrt[c]*x*Sqrt[a +
c*x^2])/(a^2*e^2*Sqrt[e*x]*(Sqrt[a] + Sqrt[c]*x)) - (3*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^(7/4)*e^2*Sqrt[e*x]*Sqrt[a + c*x^2]) + ((9*Sqrt
[a]*B - 5*A*Sqrt[c])*c^(1/4)*Sqrt[x]*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + c*x^2)/(Sqr
t[a] + Sqrt[c]*x)^2]*EllipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(6*a^(
9/4)*e^2*Sqrt[e*x]*Sqrt[a + c*x^2])

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Rubi [A]  time = 0.999568, antiderivative size = 357, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ \frac{\sqrt [4]{c} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (9 \sqrt{a} B-5 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{6 a^{9/4} e^2 \sqrt{e x} \sqrt{a+c x^2}}-\frac{3 B \sqrt [4]{c} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{a^{7/4} e^2 \sqrt{e x} \sqrt{a+c x^2}}-\frac{5 A \sqrt{a+c x^2}}{3 a^2 e (e x)^{3/2}}-\frac{3 B \sqrt{a+c x^2}}{a^2 e^2 \sqrt{e x}}+\frac{3 B \sqrt{c} x \sqrt{a+c x^2}}{a^2 e^2 \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}+\frac{A+B x}{a e (e x)^{3/2} \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/((e*x)^(5/2)*(a + c*x^2)^(3/2)),x]

[Out]

(A + B*x)/(a*e*(e*x)^(3/2)*Sqrt[a + c*x^2]) - (5*A*Sqrt[a + c*x^2])/(3*a^2*e*(e*
x)^(3/2)) - (3*B*Sqrt[a + c*x^2])/(a^2*e^2*Sqrt[e*x]) + (3*B*Sqrt[c]*x*Sqrt[a +
c*x^2])/(a^2*e^2*Sqrt[e*x]*(Sqrt[a] + Sqrt[c]*x)) - (3*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^(7/4)*e^2*Sqrt[e*x]*Sqrt[a + c*x^2]) + ((9*Sqrt
[a]*B - 5*A*Sqrt[c])*c^(1/4)*Sqrt[x]*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + c*x^2)/(Sqr
t[a] + Sqrt[c]*x)^2]*EllipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(6*a^(
9/4)*e^2*Sqrt[e*x]*Sqrt[a + c*x^2])

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Rubi in Sympy [A]  time = 141.251, size = 333, normalized size = 0.93 \[ - \frac{5 A \sqrt{a + c x^{2}}}{3 a^{2} e \left (e x\right )^{\frac{3}{2}}} + \frac{3 B \sqrt{c} x \sqrt{a + c x^{2}}}{a^{2} e^{2} \sqrt{e x} \left (\sqrt{a} + \sqrt{c} x\right )} - \frac{3 B \sqrt{a + c x^{2}}}{a^{2} e^{2} \sqrt{e x}} - \frac{3 B \sqrt [4]{c} \sqrt{x} \sqrt{\frac{a + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{a^{\frac{7}{4}} e^{2} \sqrt{e x} \sqrt{a + c x^{2}}} + \frac{A + B x}{a e \left (e x\right )^{\frac{3}{2}} \sqrt{a + c x^{2}}} - \frac{\sqrt [4]{c} \sqrt{x} \sqrt{\frac{a + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) \left (5 A \sqrt{c} - 9 B \sqrt{a}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{6 a^{\frac{9}{4}} e^{2} \sqrt{e x} \sqrt{a + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/(e*x)**(5/2)/(c*x**2+a)**(3/2),x)

[Out]

-5*A*sqrt(a + c*x**2)/(3*a**2*e*(e*x)**(3/2)) + 3*B*sqrt(c)*x*sqrt(a + c*x**2)/(
a**2*e**2*sqrt(e*x)*(sqrt(a) + sqrt(c)*x)) - 3*B*sqrt(a + c*x**2)/(a**2*e**2*sqr
t(e*x)) - 3*B*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*
*2*sqrt(e*x)*sqrt(a + c*x**2)) + (A + B*x)/(a*e*(e*x)**(3/2)*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)*(5*A*sqrt(c) - 9*B*sqrt(a))*elliptic_f(2*atan(c**(1/4)*sqrt(x)/a**(1/4)), 1/
2)/(6*a**(9/4)*e**2*sqrt(e*x)*sqrt(a + c*x**2))

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Mathematica [C]  time = 0.936249, size = 219, normalized size = 0.61 \[ \frac{x \left (\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} \left (-2 a A+3 a B x-5 A c x^2\right )+\sqrt{c} x^{5/2} \sqrt{\frac{a}{c x^2}+1} \left (9 \sqrt{a} B-5 i A \sqrt{c}\right ) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )-9 \sqrt{a} B \sqrt{c} x^{5/2} \sqrt{\frac{a}{c x^2}+1} E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{3 a^2 \sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} (e x)^{5/2} \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/((e*x)^(5/2)*(a + c*x^2)^(3/2)),x]

[Out]

(x*(Sqrt[(I*Sqrt[a])/Sqrt[c]]*(-2*a*A + 3*a*B*x - 5*A*c*x^2) - 9*Sqrt[a]*B*Sqrt[
c]*Sqrt[1 + a/(c*x^2)]*x^(5/2)*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[c]]/Sqr
t[x]], -1] + (9*Sqrt[a]*B - (5*I)*A*Sqrt[c])*Sqrt[c]*Sqrt[1 + a/(c*x^2)]*x^(5/2)
*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[c]]/Sqrt[x]], -1]))/(3*a^2*Sqrt[(I*Sq
rt[a])/Sqrt[c]]*(e*x)^(5/2)*Sqrt[a + c*x^2])

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Maple [A]  time = 0.051, size = 307, normalized size = 0.9 \[ -{\frac{1}{6\,{a}^{2}x{e}^{2}} \left ( 5\,A\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ) \sqrt{-ac}x+9\,B\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ) xa-18\,B\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ) xa+18\,Bc{x}^{3}+10\,Ac{x}^{2}+12\,aBx+4\,aA \right ){\frac{1}{\sqrt{c{x}^{2}+a}}}{\frac{1}{\sqrt{ex}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/(e*x)^(5/2)/(c*x^2+a)^(3/2),x)

[Out]

-1/6/x*(5*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+9*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-18*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+18*B*c*x^3+10*A*c*x^2+12*a*B*x+4*a*A)/(c*x^2+a)^(1/2)/a^2/e^2/
(e*x)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{B x + A}{{\left (c x^{2} + a\right )}^{\frac{3}{2}} \left (e x\right )^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((c*x^2 + a)^(3/2)*(e*x)^(5/2)),x, algorithm="maxima")

[Out]

integrate((B*x + A)/((c*x^2 + a)^(3/2)*(e*x)^(5/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{B x + A}{{\left (c e^{2} x^{4} + a e^{2} x^{2}\right )} \sqrt{c x^{2} + a} \sqrt{e x}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((c*x^2 + a)^(3/2)*(e*x)^(5/2)),x, algorithm="fricas")

[Out]

integral((B*x + A)/((c*e^2*x^4 + a*e^2*x^2)*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)**(5/2)/(c*x**2+a)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{B x + A}{{\left (c x^{2} + a\right )}^{\frac{3}{2}} \left (e x\right )^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((c*x^2 + a)^(3/2)*(e*x)^(5/2)),x, algorithm="giac")

[Out]

integrate((B*x + A)/((c*x^2 + a)^(3/2)*(e*x)^(5/2)), x)