3.246 \(\int \frac{x^{9/2} \left (A+B x^2\right )}{\sqrt{b x^2+c x^4}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.556507, 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 \[ -\frac{5 b^{7/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (9 b B-11 A c) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{231 c^{13/4} \sqrt{b x^2+c x^4}}+\frac{10 b \sqrt{b x^2+c x^4} (9 b B-11 A c)}{231 c^3 \sqrt{x}}-\frac{2 x^{3/2} \sqrt{b x^2+c x^4} (9 b B-11 A c)}{77 c^2}+\frac{2 B x^{7/2} \sqrt{b x^2+c x^4}}{11 c} \]

Antiderivative was successfully verified.

[In]  Int[(x^(9/2)*(A + B*x^2))/Sqrt[b*x^2 + c*x^4],x]

[Out]

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

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Rubi in Sympy [A]  time = 44.1369, size = 201, normalized size = 0.99 \[ \frac{2 B x^{\frac{7}{2}} \sqrt{b x^{2} + c x^{4}}}{11 c} + \frac{5 b^{\frac{7}{4}} \sqrt{\frac{b + c x^{2}}{\left (\sqrt{b} + \sqrt{c} x\right )^{2}}} \left (\sqrt{b} + \sqrt{c} x\right ) \left (11 A c - 9 B b\right ) \sqrt{b x^{2} + c x^{4}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{231 c^{\frac{13}{4}} x \left (b + c x^{2}\right )} - \frac{10 b \left (11 A c - 9 B b\right ) \sqrt{b x^{2} + c x^{4}}}{231 c^{3} \sqrt{x}} + \frac{2 x^{\frac{3}{2}} \left (11 A c - 9 B b\right ) \sqrt{b x^{2} + c x^{4}}}{77 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(9/2)*(B*x**2+A)/(c*x**4+b*x**2)**(1/2),x)

[Out]

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

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Mathematica [C]  time = 0.305409, size = 176, normalized size = 0.86 \[ \frac{2 x^{3/2} \sqrt{\frac{i \sqrt{b}}{\sqrt{c}}} \left (b+c x^2\right ) \left (-b c \left (55 A+27 B x^2\right )+3 c^2 x^2 \left (11 A+7 B x^2\right )+45 b^2 B\right )+10 i b^2 x^2 \sqrt{\frac{b}{c x^2}+1} (11 A c-9 b B) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{b}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )}{231 c^3 \sqrt{\frac{i \sqrt{b}}{\sqrt{c}}} \sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^(9/2)*(A + B*x^2))/Sqrt[b*x^2 + c*x^4],x]

[Out]

(2*Sqrt[(I*Sqrt[b])/Sqrt[c]]*x^(3/2)*(b + c*x^2)*(45*b^2*B + 3*c^2*x^2*(11*A + 7
*B*x^2) - b*c*(55*A + 27*B*x^2)) + (10*I)*b^2*(-9*b*B + 11*A*c)*Sqrt[1 + b/(c*x^
2)]*x^2*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[c]]/Sqrt[x]], -1])/(231*Sqrt[(
I*Sqrt[b])/Sqrt[c]]*c^3*Sqrt[x^2*(b + c*x^2)])

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Maple [A]  time = 0.025, size = 274, normalized size = 1.3 \[{\frac{1}{231\,{c}^{4}}\sqrt{x} \left ( 42\,B{x}^{7}{c}^{4}+55\,A\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ) \sqrt{-bc}{b}^{2}c+66\,A{x}^{5}{c}^{4}-45\,B\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ) \sqrt{-bc}{b}^{3}-12\,B{x}^{5}b{c}^{3}-44\,A{x}^{3}b{c}^{3}+36\,B{x}^{3}{b}^{2}{c}^{2}-110\,Ax{b}^{2}{c}^{2}+90\,Bx{b}^{3}c \right ){\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(9/2)*(B*x^2+A)/(c*x^4+b*x^2)^(1/2),x)

[Out]

1/231/(c*x^4+b*x^2)^(1/2)*x^(1/2)*(42*B*x^7*c^4+55*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)*EllipticF(((c*x+(-b*c)^(1/2))/(-b*c)^(1/2))^(1/2),1/2*2^(1/2))*(-b*c)^(1
/2)*b^2*c+66*A*x^5*c^4-45*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)*b^3-12*B*x^5*b*c^3-44
*A*x^3*b*c^3+36*B*x^3*b^2*c^2-110*A*x*b^2*c^2+90*B*x*b^3*c)/c^4

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x^{2} + A\right )} x^{\frac{9}{2}}}{\sqrt{c x^{4} + b x^{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^(9/2)/sqrt(c*x^4 + b*x^2),x, algorithm="maxima")

[Out]

integrate((B*x^2 + A)*x^(9/2)/sqrt(c*x^4 + b*x^2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (B x^{6} + A x^{4}\right )} \sqrt{x}}{\sqrt{c x^{4} + b x^{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^(9/2)/sqrt(c*x^4 + b*x^2),x, algorithm="fricas")

[Out]

integral((B*x^6 + A*x^4)*sqrt(x)/sqrt(c*x^4 + b*x^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(x**(9/2)*(B*x**2+A)/(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. \[ \int \frac{{\left (B x^{2} + A\right )} x^{\frac{9}{2}}}{\sqrt{c x^{4} + b x^{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^(9/2)/sqrt(c*x^4 + b*x^2),x, algorithm="giac")

[Out]

integrate((B*x^2 + A)*x^(9/2)/sqrt(c*x^4 + b*x^2), x)