3.254 \(\int \frac{A+B x^2}{x^{7/2} \sqrt{b x^2+c x^4}} \, dx\)

Optimal. Leaf size=167 \[ -\frac{c^{3/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (7 b B-5 A c) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{21 b^{9/4} \sqrt{b x^2+c x^4}}-\frac{2 \sqrt{b x^2+c x^4} (7 b B-5 A c)}{21 b^2 x^{5/2}}-\frac{2 A \sqrt{b x^2+c x^4}}{7 b x^{9/2}} \]

[Out]

(-2*A*Sqrt[b*x^2 + c*x^4])/(7*b*x^(9/2)) - (2*(7*b*B - 5*A*c)*Sqrt[b*x^2 + c*x^4
])/(21*b^2*x^(5/2)) - (c^(3/4)*(7*b*B - 5*A*c)*x*(Sqrt[b] + Sqrt[c]*x)*Sqrt[(b +
c*x^2)/(Sqrt[b] + Sqrt[c]*x)^2]*EllipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/b^(1/4)],
1/2])/(21*b^(9/4)*Sqrt[b*x^2 + c*x^4])

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Rubi [A]  time = 0.469348, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ -\frac{c^{3/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (7 b B-5 A c) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{21 b^{9/4} \sqrt{b x^2+c x^4}}-\frac{2 \sqrt{b x^2+c x^4} (7 b B-5 A c)}{21 b^2 x^{5/2}}-\frac{2 A \sqrt{b x^2+c x^4}}{7 b x^{9/2}} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x^2)/(x^(7/2)*Sqrt[b*x^2 + c*x^4]),x]

[Out]

(-2*A*Sqrt[b*x^2 + c*x^4])/(7*b*x^(9/2)) - (2*(7*b*B - 5*A*c)*Sqrt[b*x^2 + c*x^4
])/(21*b^2*x^(5/2)) - (c^(3/4)*(7*b*B - 5*A*c)*x*(Sqrt[b] + Sqrt[c]*x)*Sqrt[(b +
c*x^2)/(Sqrt[b] + Sqrt[c]*x)^2]*EllipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/b^(1/4)],
1/2])/(21*b^(9/4)*Sqrt[b*x^2 + c*x^4])

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Rubi in Sympy [A]  time = 36.6652, size = 162, normalized size = 0.97 \[ - \frac{2 A \sqrt{b x^{2} + c x^{4}}}{7 b x^{\frac{9}{2}}} + \frac{2 \left (5 A c - 7 B b\right ) \sqrt{b x^{2} + c x^{4}}}{21 b^{2} x^{\frac{5}{2}}} + \frac{c^{\frac{3}{4}} \sqrt{\frac{b + c x^{2}}{\left (\sqrt{b} + \sqrt{c} x\right )^{2}}} \left (\sqrt{b} + \sqrt{c} x\right ) \left (5 A c - 7 B b\right ) \sqrt{b x^{2} + c x^{4}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{21 b^{\frac{9}{4}} x \left (b + c x^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**2+A)/x**(7/2)/(c*x**4+b*x**2)**(1/2),x)

[Out]

-2*A*sqrt(b*x**2 + c*x**4)/(7*b*x**(9/2)) + 2*(5*A*c - 7*B*b)*sqrt(b*x**2 + c*x*
*4)/(21*b**2*x**(5/2)) + c**(3/4)*sqrt((b + c*x**2)/(sqrt(b) + sqrt(c)*x)**2)*(s
qrt(b) + sqrt(c)*x)*(5*A*c - 7*B*b)*sqrt(b*x**2 + c*x**4)*elliptic_f(2*atan(c**(
1/4)*sqrt(x)/b**(1/4)), 1/2)/(21*b**(9/4)*x*(b + c*x**2))

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Mathematica [C]  time = 0.275328, size = 156, normalized size = 0.93 \[ \frac{2 i c x^{9/2} \sqrt{\frac{b}{c x^2}+1} (5 A c-7 b B) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{b}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )-2 \sqrt{\frac{i \sqrt{b}}{\sqrt{c}}} \left (b+c x^2\right ) \left (3 A b-5 A c x^2+7 b B x^2\right )}{21 b^2 x^{5/2} \sqrt{\frac{i \sqrt{b}}{\sqrt{c}}} \sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x^2)/(x^(7/2)*Sqrt[b*x^2 + c*x^4]),x]

[Out]

(-2*Sqrt[(I*Sqrt[b])/Sqrt[c]]*(b + c*x^2)*(3*A*b + 7*b*B*x^2 - 5*A*c*x^2) + (2*I
)*c*(-7*b*B + 5*A*c)*Sqrt[1 + b/(c*x^2)]*x^(9/2)*EllipticF[I*ArcSinh[Sqrt[(I*Sqr
t[b])/Sqrt[c]]/Sqrt[x]], -1])/(21*b^2*Sqrt[(I*Sqrt[b])/Sqrt[c]]*x^(5/2)*Sqrt[x^2
*(b + c*x^2)])

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Maple [A]  time = 0.027, size = 247, normalized size = 1.5 \[{\frac{1}{21\,{b}^{2}} \left ( 5\,A\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ) \sqrt{-bc}{x}^{3}c-7\,B\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ) \sqrt{-bc}{x}^{3}b+10\,A{c}^{2}{x}^{4}-14\,B{x}^{4}bc+4\,Abc{x}^{2}-14\,B{b}^{2}{x}^{2}-6\,{b}^{2}A \right ){\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}}}}{x}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)/x^(7/2)/(c*x^4+b*x^2)^(1/2),x)

[Out]

1/21/(c*x^4+b*x^2)^(1/2)/x^(5/2)*(5*A*((c*x+(-b*c)^(1/2))/(-b*c)^(1/2))^(1/2)*2^
(1/2)*((-c*x+(-b*c)^(1/2))/(-b*c)^(1/2))^(1/2)*(-x*c/(-b*c)^(1/2))^(1/2)*Ellipti
cF(((c*x+(-b*c)^(1/2))/(-b*c)^(1/2))^(1/2),1/2*2^(1/2))*(-b*c)^(1/2)*x^3*c-7*B*(
(c*x+(-b*c)^(1/2))/(-b*c)^(1/2))^(1/2)*2^(1/2)*((-c*x+(-b*c)^(1/2))/(-b*c)^(1/2)
)^(1/2)*(-x*c/(-b*c)^(1/2))^(1/2)*EllipticF(((c*x+(-b*c)^(1/2))/(-b*c)^(1/2))^(1
/2),1/2*2^(1/2))*(-b*c)^(1/2)*x^3*b+10*A*c^2*x^4-14*B*x^4*b*c+4*A*b*c*x^2-14*B*b
^2*x^2-6*b^2*A)/b^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{B x^{2} + A}{\sqrt{c x^{4} + b x^{2}} x^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)/(sqrt(c*x^4 + b*x^2)*x^(7/2)),x, algorithm="maxima")

[Out]

integrate((B*x^2 + A)/(sqrt(c*x^4 + b*x^2)*x^(7/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{B x^{2} + A}{\sqrt{c x^{4} + b x^{2}} x^{\frac{7}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)/(sqrt(c*x^4 + b*x^2)*x^(7/2)),x, algorithm="fricas")

[Out]

integral((B*x^2 + A)/(sqrt(c*x^4 + b*x^2)*x^(7/2)), 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)/x**(7/2)/(c*x**4+b*x**2)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{B x^{2} + A}{\sqrt{c x^{4} + b x^{2}} x^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)/(sqrt(c*x^4 + b*x^2)*x^(7/2)),x, algorithm="giac")

[Out]

integrate((B*x^2 + A)/(sqrt(c*x^4 + b*x^2)*x^(7/2)), x)