3.814 \(\int \frac{A+B x^2}{(e x)^{7/2} \left (a+b x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=379 \[ -\frac{3 \sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (7 A b-5 a B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{10 a^{11/4} e^{7/2} \sqrt{a+b x^2}}+\frac{3 \sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (7 A b-5 a B) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{5 a^{11/4} e^{7/2} \sqrt{a+b x^2}}-\frac{3 \sqrt{b} \sqrt{e x} \sqrt{a+b x^2} (7 A b-5 a B)}{5 a^3 e^4 \left (\sqrt{a}+\sqrt{b} x\right )}+\frac{3 \sqrt{a+b x^2} (7 A b-5 a B)}{5 a^3 e^3 \sqrt{e x}}-\frac{7 A b-5 a B}{5 a^2 e^3 \sqrt{e x} \sqrt{a+b x^2}}-\frac{2 A}{5 a e (e x)^{5/2} \sqrt{a+b x^2}} \]

[Out]

(-2*A)/(5*a*e*(e*x)^(5/2)*Sqrt[a + b*x^2]) - (7*A*b - 5*a*B)/(5*a^2*e^3*Sqrt[e*x
]*Sqrt[a + b*x^2]) + (3*(7*A*b - 5*a*B)*Sqrt[a + b*x^2])/(5*a^3*e^3*Sqrt[e*x]) -
(3*Sqrt[b]*(7*A*b - 5*a*B)*Sqrt[e*x]*Sqrt[a + b*x^2])/(5*a^3*e^4*(Sqrt[a] + Sqr
t[b]*x)) + (3*b^(1/4)*(7*A*b - 5*a*B)*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sq
rt[a] + Sqrt[b]*x)^2]*EllipticE[2*ArcTan[(b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])],
1/2])/(5*a^(11/4)*e^(7/2)*Sqrt[a + b*x^2]) - (3*b^(1/4)*(7*A*b - 5*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])/(10*a^(11/4)*e^(7/2)*Sqrt[a + b*x^2])

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Rubi [A]  time = 0.72855, antiderivative size = 379, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ -\frac{3 \sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (7 A b-5 a B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{10 a^{11/4} e^{7/2} \sqrt{a+b x^2}}+\frac{3 \sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (7 A b-5 a B) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{5 a^{11/4} e^{7/2} \sqrt{a+b x^2}}-\frac{3 \sqrt{b} \sqrt{e x} \sqrt{a+b x^2} (7 A b-5 a B)}{5 a^3 e^4 \left (\sqrt{a}+\sqrt{b} x\right )}+\frac{3 \sqrt{a+b x^2} (7 A b-5 a B)}{5 a^3 e^3 \sqrt{e x}}-\frac{7 A b-5 a B}{5 a^2 e^3 \sqrt{e x} \sqrt{a+b x^2}}-\frac{2 A}{5 a e (e x)^{5/2} \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x^2)/((e*x)^(7/2)*(a + b*x^2)^(3/2)),x]

[Out]

(-2*A)/(5*a*e*(e*x)^(5/2)*Sqrt[a + b*x^2]) - (7*A*b - 5*a*B)/(5*a^2*e^3*Sqrt[e*x
]*Sqrt[a + b*x^2]) + (3*(7*A*b - 5*a*B)*Sqrt[a + b*x^2])/(5*a^3*e^3*Sqrt[e*x]) -
(3*Sqrt[b]*(7*A*b - 5*a*B)*Sqrt[e*x]*Sqrt[a + b*x^2])/(5*a^3*e^4*(Sqrt[a] + Sqr
t[b]*x)) + (3*b^(1/4)*(7*A*b - 5*a*B)*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sq
rt[a] + Sqrt[b]*x)^2]*EllipticE[2*ArcTan[(b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])],
1/2])/(5*a^(11/4)*e^(7/2)*Sqrt[a + b*x^2]) - (3*b^(1/4)*(7*A*b - 5*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])/(10*a^(11/4)*e^(7/2)*Sqrt[a + b*x^2])

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Rubi in Sympy [A]  time = 79.6542, size = 359, normalized size = 0.95 \[ - \frac{2 A}{5 a e \left (e x\right )^{\frac{5}{2}} \sqrt{a + b x^{2}}} - \frac{7 A b - 5 B a}{5 a^{2} e^{3} \sqrt{e x} \sqrt{a + b x^{2}}} - \frac{3 \sqrt{b} \sqrt{e x} \sqrt{a + b x^{2}} \left (7 A b - 5 B a\right )}{5 a^{3} e^{4} \left (\sqrt{a} + \sqrt{b} x\right )} + \frac{3 \sqrt{a + b x^{2}} \left (7 A b - 5 B a\right )}{5 a^{3} e^{3} \sqrt{e x}} + \frac{3 \sqrt [4]{b} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \left (7 A b - 5 B a\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{5 a^{\frac{11}{4}} e^{\frac{7}{2}} \sqrt{a + b x^{2}}} - \frac{3 \sqrt [4]{b} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \left (7 A b - 5 B a\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{10 a^{\frac{11}{4}} e^{\frac{7}{2}} \sqrt{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**2+A)/(e*x)**(7/2)/(b*x**2+a)**(3/2),x)

[Out]

-2*A/(5*a*e*(e*x)**(5/2)*sqrt(a + b*x**2)) - (7*A*b - 5*B*a)/(5*a**2*e**3*sqrt(e
*x)*sqrt(a + b*x**2)) - 3*sqrt(b)*sqrt(e*x)*sqrt(a + b*x**2)*(7*A*b - 5*B*a)/(5*
a**3*e**4*(sqrt(a) + sqrt(b)*x)) + 3*sqrt(a + b*x**2)*(7*A*b - 5*B*a)/(5*a**3*e*
*3*sqrt(e*x)) + 3*b**(1/4)*sqrt((a + b*x**2)/(sqrt(a) + sqrt(b)*x)**2)*(sqrt(a)
+ sqrt(b)*x)*(7*A*b - 5*B*a)*elliptic_e(2*atan(b**(1/4)*sqrt(e*x)/(a**(1/4)*sqrt
(e))), 1/2)/(5*a**(11/4)*e**(7/2)*sqrt(a + b*x**2)) - 3*b**(1/4)*sqrt((a + b*x**
2)/(sqrt(a) + sqrt(b)*x)**2)*(sqrt(a) + sqrt(b)*x)*(7*A*b - 5*B*a)*elliptic_f(2*
atan(b**(1/4)*sqrt(e*x)/(a**(1/4)*sqrt(e))), 1/2)/(10*a**(11/4)*e**(7/2)*sqrt(a
+ b*x**2))

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Mathematica [C]  time = 0.626032, size = 233, normalized size = 0.61 \[ \frac{x \left (\sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}} \left (-2 a^2 \left (A+5 B x^2\right )+a b x^2 \left (14 A-15 B x^2\right )+21 A b^2 x^4\right )-3 \sqrt{a} \sqrt{b} x^3 \sqrt{\frac{b x^2}{a}+1} (5 a B-7 A b) F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}}\right )\right |-1\right )+3 \sqrt{a} \sqrt{b} x^3 \sqrt{\frac{b x^2}{a}+1} (5 a B-7 A b) E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}}\right )\right |-1\right )\right )}{5 a^3 (e x)^{7/2} \sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}} \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x^2)/((e*x)^(7/2)*(a + b*x^2)^(3/2)),x]

[Out]

(x*(Sqrt[(I*Sqrt[b]*x)/Sqrt[a]]*(21*A*b^2*x^4 + a*b*x^2*(14*A - 15*B*x^2) - 2*a^
2*(A + 5*B*x^2)) + 3*Sqrt[a]*Sqrt[b]*(-7*A*b + 5*a*B)*x^3*Sqrt[1 + (b*x^2)/a]*El
lipticE[I*ArcSinh[Sqrt[(I*Sqrt[b]*x)/Sqrt[a]]], -1] - 3*Sqrt[a]*Sqrt[b]*(-7*A*b
+ 5*a*B)*x^3*Sqrt[1 + (b*x^2)/a]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[b]*x)/Sqrt[a]]
], -1]))/(5*a^3*Sqrt[(I*Sqrt[b]*x)/Sqrt[a]]*(e*x)^(7/2)*Sqrt[a + b*x^2])

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Maple [A]  time = 0.033, size = 417, normalized size = 1.1 \[ -{\frac{1}{10\,{x}^{2}{e}^{3}{a}^{3}} \left ( 42\,A\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticE} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ){x}^{2}ab-21\,A\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ){x}^{2}ab-30\,B\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticE} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ){x}^{2}{a}^{2}+15\,B\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ){x}^{2}{a}^{2}-42\,A{b}^{2}{x}^{4}+30\,B{x}^{4}ab-28\,aAb{x}^{2}+20\,B{x}^{2}{a}^{2}+4\,A{a}^{2} \right ){\frac{1}{\sqrt{b{x}^{2}+a}}}{\frac{1}{\sqrt{ex}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)/(e*x)^(7/2)/(b*x^2+a)^(3/2),x)

[Out]

-1/10/x^2*(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-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))*x^2*a*b-30*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+15*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^2-42*A*b^2*x^4+30*B*x^4*a*b
-28*a*A*b*x^2+20*B*x^2*a^2+4*A*a^2)/(b*x^2+a)^(1/2)/e^3/(e*x)^(1/2)/a^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{B x^{2} + A}{{\left (b x^{2} + a\right )}^{\frac{3}{2}} \left (e x\right )^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)/((b*x^2 + a)^(3/2)*(e*x)^(7/2)),x, algorithm="maxima")

[Out]

integrate((B*x^2 + A)/((b*x^2 + a)^(3/2)*(e*x)^(7/2)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)/((b*x^2 + a)^(3/2)*(e*x)^(7/2)),x, algorithm="fricas")

[Out]

integral((B*x^2 + A)/((b*e^3*x^5 + a*e^3*x^3)*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)**(7/2)/(b*x**2+a)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{B x^{2} + A}{{\left (b x^{2} + a\right )}^{\frac{3}{2}} \left (e x\right )^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)/((b*x^2 + a)^(3/2)*(e*x)^(7/2)),x, algorithm="giac")

[Out]

integrate((B*x^2 + A)/((b*x^2 + a)^(3/2)*(e*x)^(7/2)), x)