3.256 \(\int \frac{A+B x^2}{x^{11/2} \sqrt{b x^2+c x^4}} \, dx\)

Optimal. Leaf size=204 \[ \frac{5 c^{7/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (11 b B-9 A c) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{231 b^{13/4} \sqrt{b x^2+c x^4}}+\frac{10 c \sqrt{b x^2+c x^4} (11 b B-9 A c)}{231 b^3 x^{5/2}}-\frac{2 \sqrt{b x^2+c x^4} (11 b B-9 A c)}{77 b^2 x^{9/2}}-\frac{2 A \sqrt{b x^2+c x^4}}{11 b x^{13/2}} \]

[Out]

(-2*A*Sqrt[b*x^2 + c*x^4])/(11*b*x^(13/2)) - (2*(11*b*B - 9*A*c)*Sqrt[b*x^2 + c*
x^4])/(77*b^2*x^(9/2)) + (10*c*(11*b*B - 9*A*c)*Sqrt[b*x^2 + c*x^4])/(231*b^3*x^
(5/2)) + (5*c^(7/4)*(11*b*B - 9*A*c)*x*(Sqrt[b] + Sqrt[c]*x)*Sqrt[(b + c*x^2)/(S
qrt[b] + Sqrt[c]*x)^2]*EllipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/b^(1/4)], 1/2])/(231
*b^(13/4)*Sqrt[b*x^2 + c*x^4])

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Rubi [A]  time = 0.572306, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{5 c^{7/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (11 b B-9 A c) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{231 b^{13/4} \sqrt{b x^2+c x^4}}+\frac{10 c \sqrt{b x^2+c x^4} (11 b B-9 A c)}{231 b^3 x^{5/2}}-\frac{2 \sqrt{b x^2+c x^4} (11 b B-9 A c)}{77 b^2 x^{9/2}}-\frac{2 A \sqrt{b x^2+c x^4}}{11 b x^{13/2}} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x^2)/(x^(11/2)*Sqrt[b*x^2 + c*x^4]),x]

[Out]

(-2*A*Sqrt[b*x^2 + c*x^4])/(11*b*x^(13/2)) - (2*(11*b*B - 9*A*c)*Sqrt[b*x^2 + c*
x^4])/(77*b^2*x^(9/2)) + (10*c*(11*b*B - 9*A*c)*Sqrt[b*x^2 + c*x^4])/(231*b^3*x^
(5/2)) + (5*c^(7/4)*(11*b*B - 9*A*c)*x*(Sqrt[b] + Sqrt[c]*x)*Sqrt[(b + c*x^2)/(S
qrt[b] + Sqrt[c]*x)^2]*EllipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/b^(1/4)], 1/2])/(231
*b^(13/4)*Sqrt[b*x^2 + c*x^4])

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Rubi in Sympy [A]  time = 44.6411, size = 201, normalized size = 0.99 \[ - \frac{2 A \sqrt{b x^{2} + c x^{4}}}{11 b x^{\frac{13}{2}}} + \frac{2 \left (9 A c - 11 B b\right ) \sqrt{b x^{2} + c x^{4}}}{77 b^{2} x^{\frac{9}{2}}} - \frac{10 c \left (9 A c - 11 B b\right ) \sqrt{b x^{2} + c x^{4}}}{231 b^{3} x^{\frac{5}{2}}} - \frac{5 c^{\frac{7}{4}} \sqrt{\frac{b + c x^{2}}{\left (\sqrt{b} + \sqrt{c} x\right )^{2}}} \left (\sqrt{b} + \sqrt{c} x\right ) \left (9 A c - 11 B b\right ) \sqrt{b x^{2} + c x^{4}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{231 b^{\frac{13}{4}} x \left (b + c x^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**2+A)/x**(11/2)/(c*x**4+b*x**2)**(1/2),x)

[Out]

-2*A*sqrt(b*x**2 + c*x**4)/(11*b*x**(13/2)) + 2*(9*A*c - 11*B*b)*sqrt(b*x**2 + c
*x**4)/(77*b**2*x**(9/2)) - 10*c*(9*A*c - 11*B*b)*sqrt(b*x**2 + c*x**4)/(231*b**
3*x**(5/2)) - 5*c**(7/4)*sqrt((b + c*x**2)/(sqrt(b) + sqrt(c)*x)**2)*(sqrt(b) +
sqrt(c)*x)*(9*A*c - 11*B*b)*sqrt(b*x**2 + c*x**4)*elliptic_f(2*atan(c**(1/4)*sqr
t(x)/b**(1/4)), 1/2)/(231*b**(13/4)*x*(b + c*x**2))

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Mathematica [C]  time = 0.350868, size = 181, normalized size = 0.89 \[ \frac{-2 \sqrt{\frac{i \sqrt{b}}{\sqrt{c}}} \left (b+c x^2\right ) \left (3 A \left (7 b^2-9 b c x^2+15 c^2 x^4\right )+11 b B x^2 \left (3 b-5 c x^2\right )\right )-10 i c^2 x^{13/2} \sqrt{\frac{b}{c x^2}+1} (9 A c-11 b B) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{b}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )}{231 b^3 x^{9/2} \sqrt{\frac{i \sqrt{b}}{\sqrt{c}}} \sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x^2)/(x^(11/2)*Sqrt[b*x^2 + c*x^4]),x]

[Out]

(-2*Sqrt[(I*Sqrt[b])/Sqrt[c]]*(b + c*x^2)*(11*b*B*x^2*(3*b - 5*c*x^2) + 3*A*(7*b
^2 - 9*b*c*x^2 + 15*c^2*x^4)) - (10*I)*c^2*(-11*b*B + 9*A*c)*Sqrt[1 + b/(c*x^2)]
*x^(13/2)*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[c]]/Sqrt[x]], -1])/(231*b^3*
Sqrt[(I*Sqrt[b])/Sqrt[c]]*x^(9/2)*Sqrt[x^2*(b + c*x^2)])

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Maple [A]  time = 0.026, size = 274, normalized size = 1.3 \[ -{\frac{1}{231\,{b}^{3}} \left ( 45\,A\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ) \sqrt{-bc}{x}^{5}{c}^{2}-55\,B\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ) \sqrt{-bc}{x}^{5}bc+90\,A{c}^{3}{x}^{6}-110\,B{x}^{6}b{c}^{2}+36\,Ab{c}^{2}{x}^{4}-44\,B{x}^{4}{b}^{2}c-12\,A{b}^{2}c{x}^{2}+66\,B{x}^{2}{b}^{3}+42\,A{b}^{3} \right ){\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}}}}{x}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)/x^(11/2)/(c*x^4+b*x^2)^(1/2),x)

[Out]

-1/231/(c*x^4+b*x^2)^(1/2)/x^(9/2)*(45*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)*Elli
pticF(((c*x+(-b*c)^(1/2))/(-b*c)^(1/2))^(1/2),1/2*2^(1/2))*(-b*c)^(1/2)*x^5*c^2-
55*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^5*b*c+90*A*c^3*x^6-110*B*x^6*b*c^2+36*A*b*
c^2*x^4-44*B*x^4*b^2*c-12*A*b^2*c*x^2+66*B*x^2*b^3+42*A*b^3)/b^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{B x^{2} + A}{\sqrt{c x^{4} + b x^{2}} x^{\frac{11}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)/(sqrt(c*x^4 + b*x^2)*x^(11/2)),x, algorithm="maxima")

[Out]

integrate((B*x^2 + A)/(sqrt(c*x^4 + b*x^2)*x^(11/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{B x^{2} + A}{\sqrt{c x^{4} + b x^{2}} x^{\frac{11}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)/(sqrt(c*x^4 + b*x^2)*x^(11/2)),x, algorithm="fricas")

[Out]

integral((B*x^2 + A)/(sqrt(c*x^4 + b*x^2)*x^(11/2)), 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**(11/2)/(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. \[ \int \frac{B x^{2} + A}{\sqrt{c x^{4} + b x^{2}} x^{\frac{11}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)/(sqrt(c*x^4 + b*x^2)*x^(11/2)),x, algorithm="giac")

[Out]

integrate((B*x^2 + A)/(sqrt(c*x^4 + b*x^2)*x^(11/2)), x)