3.261 \(\int \frac{x^{9/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=178 \[ -\frac{x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (5 b B-3 A c) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{6 \sqrt [4]{b} c^{9/4} \sqrt{b x^2+c x^4}}+\frac{\sqrt{b x^2+c x^4} (5 b B-3 A c)}{3 b c^2 \sqrt{x}}-\frac{x^{7/2} (b B-A c)}{b c \sqrt{b x^2+c x^4}} \]

[Out]

-(((b*B - A*c)*x^(7/2))/(b*c*Sqrt[b*x^2 + c*x^4])) + ((5*b*B - 3*A*c)*Sqrt[b*x^2
+ c*x^4])/(3*b*c^2*Sqrt[x]) - ((5*b*B - 3*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])/(6*b^(1/4)*c^(9/4)*Sqrt[b*x^2 + c*x^4])

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Rubi [A]  time = 0.485518, antiderivative size = 178, 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 \[ -\frac{x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (5 b B-3 A c) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{6 \sqrt [4]{b} c^{9/4} \sqrt{b x^2+c x^4}}+\frac{\sqrt{b x^2+c x^4} (5 b B-3 A c)}{3 b c^2 \sqrt{x}}-\frac{x^{7/2} (b B-A c)}{b c \sqrt{b x^2+c x^4}} \]

Antiderivative was successfully verified.

[In]  Int[(x^(9/2)*(A + B*x^2))/(b*x^2 + c*x^4)^(3/2),x]

[Out]

-(((b*B - A*c)*x^(7/2))/(b*c*Sqrt[b*x^2 + c*x^4])) + ((5*b*B - 3*A*c)*Sqrt[b*x^2
+ c*x^4])/(3*b*c^2*Sqrt[x]) - ((5*b*B - 3*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])/(6*b^(1/4)*c^(9/4)*Sqrt[b*x^2 + c*x^4])

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Rubi in Sympy [A]  time = 39.1247, size = 165, normalized size = 0.93 \[ \frac{x^{\frac{7}{2}} \left (A c - B b\right )}{b c \sqrt{b x^{2} + c x^{4}}} - \frac{\left (3 A c - 5 B b\right ) \sqrt{b x^{2} + c x^{4}}}{3 b c^{2} \sqrt{x}} + \frac{\sqrt{\frac{b + c x^{2}}{\left (\sqrt{b} + \sqrt{c} x\right )^{2}}} \left (\sqrt{b} + \sqrt{c} x\right ) \left (3 A c - 5 B b\right ) \sqrt{b x^{2} + c x^{4}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{6 \sqrt [4]{b} c^{\frac{9}{4}} x \left (b + c x^{2}\right )} \]

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)**(3/2),x)

[Out]

x**(7/2)*(A*c - B*b)/(b*c*sqrt(b*x**2 + c*x**4)) - (3*A*c - 5*B*b)*sqrt(b*x**2 +
c*x**4)/(3*b*c**2*sqrt(x)) + sqrt((b + c*x**2)/(sqrt(b) + sqrt(c)*x)**2)*(sqrt(
b) + sqrt(c)*x)*(3*A*c - 5*B*b)*sqrt(b*x**2 + c*x**4)*elliptic_f(2*atan(c**(1/4)
*sqrt(x)/b**(1/4)), 1/2)/(6*b**(1/4)*c**(9/4)*x*(b + c*x**2))

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Mathematica [C]  time = 0.221854, size = 142, normalized size = 0.8 \[ \frac{i x^2 \sqrt{\frac{b}{c x^2}+1} (3 A c-5 b B) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{b}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )+x^{3/2} \sqrt{\frac{i \sqrt{b}}{\sqrt{c}}} \left (-3 A c+5 b B+2 B c x^2\right )}{3 c^2 \sqrt{\frac{i \sqrt{b}}{\sqrt{c}}} \sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^(9/2)*(A + B*x^2))/(b*x^2 + c*x^4)^(3/2),x]

[Out]

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

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Maple [A]  time = 0.028, size = 230, normalized size = 1.3 \[{\frac{c{x}^{2}+b}{6\,{c}^{3}}{x}^{{\frac{5}{2}}} \left ( 3\,A\sqrt{-bc}\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ) c-5\,B\sqrt{-bc}\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ) b+4\,B{c}^{2}{x}^{3}-6\,Ax{c}^{2}+10\,Bxbc \right ) \left ( c{x}^{4}+b{x}^{2} \right ) ^{-{\frac{3}{2}}}} \]

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)^(3/2),x)

[Out]

1/6/(c*x^4+b*x^2)^(3/2)*x^(5/2)*(c*x^2+b)*(3*A*(-b*c)^(1/2)*((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))*c-
5*B*(-b*c)^(1/2)*((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+4*B*c^2*x^3-6*A*x*c^2+10*B*x*b*c)/c^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x^{2} + A\right )} x^{\frac{9}{2}}}{{\left (c x^{4} + b x^{2}\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^(9/2)/(c*x^4 + b*x^2)^(3/2),x, algorithm="maxima")

[Out]

integrate((B*x^2 + A)*x^(9/2)/(c*x^4 + b*x^2)^(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^(9/2)/(c*x^4 + b*x^2)^(3/2),x, algorithm="fricas")

[Out]

integral((B*x^4 + A*x^2)*sqrt(x)/(sqrt(c*x^4 + b*x^2)*(c*x^2 + b)), 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)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x^{2} + A\right )} x^{\frac{9}{2}}}{{\left (c x^{4} + b x^{2}\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^(9/2)/(c*x^4 + b*x^2)^(3/2),x, algorithm="giac")

[Out]

integrate((B*x^2 + A)*x^(9/2)/(c*x^4 + b*x^2)^(3/2), x)