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

Optimal. Leaf size=377 \[ \frac{\left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (7 A b-a B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{4 a^{11/4} b^{3/4} e^{3/2} \sqrt{a+b x^2}}-\frac{\left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (7 A b-a B) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{2 a^{11/4} b^{3/4} e^{3/2} \sqrt{a+b x^2}}-\frac{(e x)^{3/2} (7 A b-a B)}{2 a^3 e^3 \sqrt{a+b x^2}}+\frac{\sqrt{e x} \sqrt{a+b x^2} (7 A b-a B)}{2 a^3 \sqrt{b} e^2 \left (\sqrt{a}+\sqrt{b} x\right )}-\frac{(e x)^{3/2} (7 A b-a B)}{3 a^2 e^3 \left (a+b x^2\right )^{3/2}}-\frac{2 A}{a e \sqrt{e x} \left (a+b x^2\right )^{3/2}} \]

[Out]

(-2*A)/(a*e*Sqrt[e*x]*(a + b*x^2)^(3/2)) - ((7*A*b - a*B)*(e*x)^(3/2))/(3*a^2*e^
3*(a + b*x^2)^(3/2)) - ((7*A*b - a*B)*(e*x)^(3/2))/(2*a^3*e^3*Sqrt[a + b*x^2]) +
((7*A*b - a*B)*Sqrt[e*x]*Sqrt[a + b*x^2])/(2*a^3*Sqrt[b]*e^2*(Sqrt[a] + Sqrt[b]
*x)) - ((7*A*b - a*B)*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*
x)^2]*EllipticE[2*ArcTan[(b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])], 1/2])/(2*a^(11/
4)*b^(3/4)*e^(3/2)*Sqrt[a + b*x^2]) + ((7*A*b - a*B)*(Sqrt[a] + Sqrt[b]*x)*Sqrt[
(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*EllipticF[2*ArcTan[(b^(1/4)*Sqrt[e*x])/(a^(
1/4)*Sqrt[e])], 1/2])/(4*a^(11/4)*b^(3/4)*e^(3/2)*Sqrt[a + b*x^2])

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Rubi [A]  time = 0.718684, antiderivative size = 377, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{\left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (7 A b-a B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{4 a^{11/4} b^{3/4} e^{3/2} \sqrt{a+b x^2}}-\frac{\left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (7 A b-a B) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{2 a^{11/4} b^{3/4} e^{3/2} \sqrt{a+b x^2}}-\frac{(e x)^{3/2} (7 A b-a B)}{2 a^3 e^3 \sqrt{a+b x^2}}+\frac{\sqrt{e x} \sqrt{a+b x^2} (7 A b-a B)}{2 a^3 \sqrt{b} e^2 \left (\sqrt{a}+\sqrt{b} x\right )}-\frac{(e x)^{3/2} (7 A b-a B)}{3 a^2 e^3 \left (a+b x^2\right )^{3/2}}-\frac{2 A}{a e \sqrt{e x} \left (a+b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*A)/(a*e*Sqrt[e*x]*(a + b*x^2)^(3/2)) - ((7*A*b - a*B)*(e*x)^(3/2))/(3*a^2*e^
3*(a + b*x^2)^(3/2)) - ((7*A*b - a*B)*(e*x)^(3/2))/(2*a^3*e^3*Sqrt[a + b*x^2]) +
((7*A*b - a*B)*Sqrt[e*x]*Sqrt[a + b*x^2])/(2*a^3*Sqrt[b]*e^2*(Sqrt[a] + Sqrt[b]
*x)) - ((7*A*b - a*B)*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*
x)^2]*EllipticE[2*ArcTan[(b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])], 1/2])/(2*a^(11/
4)*b^(3/4)*e^(3/2)*Sqrt[a + b*x^2]) + ((7*A*b - a*B)*(Sqrt[a] + Sqrt[b]*x)*Sqrt[
(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*EllipticF[2*ArcTan[(b^(1/4)*Sqrt[e*x])/(a^(
1/4)*Sqrt[e])], 1/2])/(4*a^(11/4)*b^(3/4)*e^(3/2)*Sqrt[a + b*x^2])

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Rubi in Sympy [A]  time = 78.1897, size = 342, normalized size = 0.91 \[ - \frac{2 A}{a e \sqrt{e x} \left (a + b x^{2}\right )^{\frac{3}{2}}} - \frac{\left (e x\right )^{\frac{3}{2}} \left (7 A b - B a\right )}{3 a^{2} e^{3} \left (a + b x^{2}\right )^{\frac{3}{2}}} - \frac{\left (e x\right )^{\frac{3}{2}} \left (7 A b - B a\right )}{2 a^{3} e^{3} \sqrt{a + b x^{2}}} + \frac{\sqrt{e x} \sqrt{a + b x^{2}} \left (7 A b - B a\right )}{2 a^{3} \sqrt{b} e^{2} \left (\sqrt{a} + \sqrt{b} x\right )} - \frac{\sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \left (7 A b - B a\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{2 a^{\frac{11}{4}} b^{\frac{3}{4}} e^{\frac{3}{2}} \sqrt{a + b x^{2}}} + \frac{\sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \left (7 A b - B a\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{4 a^{\frac{11}{4}} b^{\frac{3}{4}} e^{\frac{3}{2}} \sqrt{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A/(a*e*sqrt(e*x)*(a + b*x**2)**(3/2)) - (e*x)**(3/2)*(7*A*b - B*a)/(3*a**2*e*
*3*(a + b*x**2)**(3/2)) - (e*x)**(3/2)*(7*A*b - B*a)/(2*a**3*e**3*sqrt(a + b*x**
2)) + sqrt(e*x)*sqrt(a + b*x**2)*(7*A*b - B*a)/(2*a**3*sqrt(b)*e**2*(sqrt(a) + s
qrt(b)*x)) - sqrt((a + b*x**2)/(sqrt(a) + sqrt(b)*x)**2)*(sqrt(a) + sqrt(b)*x)*(
7*A*b - B*a)*elliptic_e(2*atan(b**(1/4)*sqrt(e*x)/(a**(1/4)*sqrt(e))), 1/2)/(2*a
**(11/4)*b**(3/4)*e**(3/2)*sqrt(a + b*x**2)) + sqrt((a + b*x**2)/(sqrt(a) + sqrt
(b)*x)**2)*(sqrt(a) + sqrt(b)*x)*(7*A*b - B*a)*elliptic_f(2*atan(b**(1/4)*sqrt(e
*x)/(a**(1/4)*sqrt(e))), 1/2)/(4*a**(11/4)*b**(3/4)*e**(3/2)*sqrt(a + b*x**2))

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Mathematica [C]  time = 0.539367, size = 182, normalized size = 0.48 \[ \frac{x \left (\frac{a^2 \left (5 B x^2-12 A\right )+a \left (3 b B x^4-35 A b x^2\right )-21 A b^2 x^4}{a+b x^2}+\frac{3 i a \sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}} \sqrt{\frac{b x^2}{a}+1} (a B-7 A b) \left (E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}}\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}}\right )\right |-1\right )\right )}{b}\right )}{6 a^3 (e x)^{3/2} \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(x*((-21*A*b^2*x^4 + a^2*(-12*A + 5*B*x^2) + a*(-35*A*b*x^2 + 3*b*B*x^4))/(a + b
*x^2) + ((3*I)*a*(-7*A*b + a*B)*Sqrt[(I*Sqrt[b]*x)/Sqrt[a]]*Sqrt[1 + (b*x^2)/a]*
(EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[b]*x)/Sqrt[a]]], -1] - EllipticF[I*ArcSinh[Sqr
t[(I*Sqrt[b]*x)/Sqrt[a]]], -1]))/b))/(6*a^3*(e*x)^(3/2)*Sqrt[a + b*x^2])

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Maple [A]  time = 0.034, size = 771, normalized size = 2.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*(42*A*((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*2^(1/2)*((-b*x+(-a*b)^(1/2))/
(-a*b)^(1/2))^(1/2)*(-x*b/(-a*b)^(1/2))^(1/2)*EllipticE(((b*x+(-a*b)^(1/2))/(-a*
b)^(1/2))^(1/2),1/2*2^(1/2))*x^2*a*b^2-21*A*((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1
/2)*2^(1/2)*((-b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*(-x*b/(-a*b)^(1/2))^(1/2)*E
llipticF(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))*x^2*a*b^2-6*B*((b*
x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*2^(1/2)*((-b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(
1/2)*(-x*b/(-a*b)^(1/2))^(1/2)*EllipticE(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)
,1/2*2^(1/2))*x^2*a^2*b+3*B*((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*2^(1/2)*((-b
*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*(-x*b/(-a*b)^(1/2))^(1/2)*EllipticF(((b*x+(
-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))*x^2*a^2*b+42*A*((b*x+(-a*b)^(1/2))
/(-a*b)^(1/2))^(1/2)*2^(1/2)*((-b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*(-x*b/(-a*
b)^(1/2))^(1/2)*EllipticE(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))*a
^2*b-21*A*((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*2^(1/2)*((-b*x+(-a*b)^(1/2))/(
-a*b)^(1/2))^(1/2)*(-x*b/(-a*b)^(1/2))^(1/2)*EllipticF(((b*x+(-a*b)^(1/2))/(-a*b
)^(1/2))^(1/2),1/2*2^(1/2))*a^2*b-6*B*((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*2^
(1/2)*((-b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*(-x*b/(-a*b)^(1/2))^(1/2)*Ellipti
cE(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))*a^3+3*B*((b*x+(-a*b)^(1/
2))/(-a*b)^(1/2))^(1/2)*2^(1/2)*((-b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*(-x*b/(
-a*b)^(1/2))^(1/2)*EllipticF(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2)
)*a^3-42*A*x^4*b^3+6*B*x^4*a*b^2-70*A*x^2*a*b^2+10*B*x^2*a^2*b-24*A*a^2*b)/b/a^3
/e/(e*x)^(1/2)/(b*x^2+a)^(3/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{B x^{2} + A}{{\left (b^{2} e x^{5} + 2 \, a b e x^{3} + a^{2} e x\right )} \sqrt{b x^{2} + a} \sqrt{e x}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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