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

Optimal. Leaf size=187 \[ \frac{\left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (a B+5 A b) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{12 a^{9/4} b^{5/4} \sqrt{e} \sqrt{a+b x^2}}+\frac{\sqrt{e x} (a B+5 A b)}{6 a^2 b e \sqrt{a+b x^2}}+\frac{\sqrt{e x} (A b-a B)}{3 a b e \left (a+b x^2\right )^{3/2}} \]

[Out]

((A*b - a*B)*Sqrt[e*x])/(3*a*b*e*(a + b*x^2)^(3/2)) + ((5*A*b + a*B)*Sqrt[e*x])/
(6*a^2*b*e*Sqrt[a + b*x^2]) + ((5*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)*Sqr
t[e])], 1/2])/(12*a^(9/4)*b^(5/4)*Sqrt[e]*Sqrt[a + b*x^2])

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Rubi [A]  time = 0.312689, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{\left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (a B+5 A b) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{12 a^{9/4} b^{5/4} \sqrt{e} \sqrt{a+b x^2}}+\frac{\sqrt{e x} (a B+5 A b)}{6 a^2 b e \sqrt{a+b x^2}}+\frac{\sqrt{e x} (A b-a B)}{3 a b e \left (a+b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

((A*b - a*B)*Sqrt[e*x])/(3*a*b*e*(a + b*x^2)^(3/2)) + ((5*A*b + a*B)*Sqrt[e*x])/
(6*a^2*b*e*Sqrt[a + b*x^2]) + ((5*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)*Sqr
t[e])], 1/2])/(12*a^(9/4)*b^(5/4)*Sqrt[e]*Sqrt[a + b*x^2])

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Rubi in Sympy [A]  time = 31.2919, size = 163, normalized size = 0.87 \[ \frac{\sqrt{e x} \left (A b - B a\right )}{3 a b e \left (a + b x^{2}\right )^{\frac{3}{2}}} + \frac{\sqrt{e x} \left (5 A b + B a\right )}{6 a^{2} b e \sqrt{a + b x^{2}}} + \frac{\sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \left (5 A b + B a\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{12 a^{\frac{9}{4}} b^{\frac{5}{4}} \sqrt{e} \sqrt{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(e*x)*(A*b - B*a)/(3*a*b*e*(a + b*x**2)**(3/2)) + sqrt(e*x)*(5*A*b + B*a)/(6
*a**2*b*e*sqrt(a + b*x**2)) + sqrt((a + b*x**2)/(sqrt(a) + sqrt(b)*x)**2)*(sqrt(
a) + sqrt(b)*x)*(5*A*b + B*a)*elliptic_f(2*atan(b**(1/4)*sqrt(e*x)/(a**(1/4)*sqr
t(e))), 1/2)/(12*a**(9/4)*b**(5/4)*sqrt(e)*sqrt(a + b*x**2))

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Mathematica [C]  time = 0.254897, size = 164, normalized size = 0.88 \[ \frac{x \sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \left (a^2 (-B)+a b \left (7 A+B x^2\right )+5 A b^2 x^2\right )+i x^{3/2} \sqrt{\frac{a}{b x^2}+1} \left (a+b x^2\right ) (a B+5 A b) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}{\sqrt{x}}\right )\right |-1\right )}{6 a^2 b \sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \sqrt{e x} \left (a+b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[(I*Sqrt[a])/Sqrt[b]]*x*(-(a^2*B) + 5*A*b^2*x^2 + a*b*(7*A + B*x^2)) + I*(5
*A*b + a*B)*Sqrt[1 + a/(b*x^2)]*x^(3/2)*(a + b*x^2)*EllipticF[I*ArcSinh[Sqrt[(I*
Sqrt[a])/Sqrt[b]]/Sqrt[x]], -1])/(6*a^2*Sqrt[(I*Sqrt[a])/Sqrt[b]]*b*Sqrt[e*x]*(a
+ b*x^2)^(3/2))

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Maple [B]  time = 0.032, size = 425, normalized size = 2.3 \[{\frac{1}{12\,{a}^{2}{b}^{2}} \left ( 5\,A\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) \sqrt{-ab}{x}^{2}{b}^{2}+B\sqrt{{1 \left ( bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{2}\sqrt{{1 \left ( -bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{-{bx{\frac{1}{\sqrt{-ab}}}}}{\it EllipticF} \left ( \sqrt{{1 \left ( bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}},{\frac{\sqrt{2}}{2}} \right ) \sqrt{-ab}{x}^{2}ab+5\,A\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) \sqrt{-ab}ab+B\sqrt{{1 \left ( bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{2}\sqrt{{1 \left ( -bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{-{bx{\frac{1}{\sqrt{-ab}}}}}{\it EllipticF} \left ( \sqrt{{1 \left ( bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}},{\frac{\sqrt{2}}{2}} \right ) \sqrt{-ab}{a}^{2}+10\,A{x}^{3}{b}^{3}+2\,B{x}^{3}a{b}^{2}+14\,Axa{b}^{2}-2\,Bx{a}^{2}b \right ){\frac{1}{\sqrt{ex}}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*(5*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*b)^(1/2)*x^2*b^2+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*b)^(1/2
)*x^2*a*b+5*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*b)^(1/2)*a*b+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*b)^(1/2
)*a^2+10*A*x^3*b^3+2*B*x^3*a*b^2+14*A*x*a*b^2-2*B*x*a^2*b)/(e*x)^(1/2)/a^2/b^2/(
b*x^2+a)^(3/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((B*x^2 + A)/((b^2*x^4 + 2*a*b*x^2 + a^2)*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)/(b*x**2+a)**(5/2)/(e*x)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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