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

Optimal. Leaf size=328 \[ \frac{2 \sqrt [4]{c} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (A c+5 b B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 b^{3/4} \sqrt{b x^2+c x^4}}-\frac{4 \sqrt [4]{c} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (A c+5 b B) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 b^{3/4} \sqrt{b x^2+c x^4}}+\frac{4 \sqrt{c} x^{3/2} \left (b+c x^2\right ) (A c+5 b B)}{5 b \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}-\frac{2 \sqrt{b x^2+c x^4} (A c+5 b B)}{5 b x^{3/2}}-\frac{2 A \left (b x^2+c x^4\right )^{3/2}}{5 b x^{11/2}} \]

[Out]

(4*Sqrt[c]*(5*b*B + A*c)*x^(3/2)*(b + c*x^2))/(5*b*(Sqrt[b] + Sqrt[c]*x)*Sqrt[b*
x^2 + c*x^4]) - (2*(5*b*B + A*c)*Sqrt[b*x^2 + c*x^4])/(5*b*x^(3/2)) - (2*A*(b*x^
2 + c*x^4)^(3/2))/(5*b*x^(11/2)) - (4*c^(1/4)*(5*b*B + A*c)*x*(Sqrt[b] + Sqrt[c]
*x)*Sqrt[(b + c*x^2)/(Sqrt[b] + Sqrt[c]*x)^2]*EllipticE[2*ArcTan[(c^(1/4)*Sqrt[x
])/b^(1/4)], 1/2])/(5*b^(3/4)*Sqrt[b*x^2 + c*x^4]) + (2*c^(1/4)*(5*b*B + A*c)*x*
(Sqrt[b] + Sqrt[c]*x)*Sqrt[(b + c*x^2)/(Sqrt[b] + Sqrt[c]*x)^2]*EllipticF[2*ArcT
an[(c^(1/4)*Sqrt[x])/b^(1/4)], 1/2])/(5*b^(3/4)*Sqrt[b*x^2 + c*x^4])

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Rubi [A]  time = 0.701611, antiderivative size = 328, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{2 \sqrt [4]{c} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (A c+5 b B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 b^{3/4} \sqrt{b x^2+c x^4}}-\frac{4 \sqrt [4]{c} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (A c+5 b B) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 b^{3/4} \sqrt{b x^2+c x^4}}+\frac{4 \sqrt{c} x^{3/2} \left (b+c x^2\right ) (A c+5 b B)}{5 b \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}-\frac{2 \sqrt{b x^2+c x^4} (A c+5 b B)}{5 b x^{3/2}}-\frac{2 A \left (b x^2+c x^4\right )^{3/2}}{5 b x^{11/2}} \]

Antiderivative was successfully verified.

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

[Out]

(4*Sqrt[c]*(5*b*B + A*c)*x^(3/2)*(b + c*x^2))/(5*b*(Sqrt[b] + Sqrt[c]*x)*Sqrt[b*
x^2 + c*x^4]) - (2*(5*b*B + A*c)*Sqrt[b*x^2 + c*x^4])/(5*b*x^(3/2)) - (2*A*(b*x^
2 + c*x^4)^(3/2))/(5*b*x^(11/2)) - (4*c^(1/4)*(5*b*B + A*c)*x*(Sqrt[b] + Sqrt[c]
*x)*Sqrt[(b + c*x^2)/(Sqrt[b] + Sqrt[c]*x)^2]*EllipticE[2*ArcTan[(c^(1/4)*Sqrt[x
])/b^(1/4)], 1/2])/(5*b^(3/4)*Sqrt[b*x^2 + c*x^4]) + (2*c^(1/4)*(5*b*B + A*c)*x*
(Sqrt[b] + Sqrt[c]*x)*Sqrt[(b + c*x^2)/(Sqrt[b] + Sqrt[c]*x)^2]*EllipticF[2*ArcT
an[(c^(1/4)*Sqrt[x])/b^(1/4)], 1/2])/(5*b^(3/4)*Sqrt[b*x^2 + c*x^4])

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Rubi in Sympy [A]  time = 59.9583, size = 309, normalized size = 0.94 \[ - \frac{2 A \left (b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{5 b x^{\frac{11}{2}}} + \frac{4 \sqrt{c} \left (A c + 5 B b\right ) \sqrt{b x^{2} + c x^{4}}}{5 b \sqrt{x} \left (\sqrt{b} + \sqrt{c} x\right )} - \frac{2 \left (A c + 5 B b\right ) \sqrt{b x^{2} + c x^{4}}}{5 b x^{\frac{3}{2}}} - \frac{4 \sqrt [4]{c} \sqrt{\frac{b + c x^{2}}{\left (\sqrt{b} + \sqrt{c} x\right )^{2}}} \left (\sqrt{b} + \sqrt{c} x\right ) \left (A c + 5 B b\right ) \sqrt{b x^{2} + c x^{4}} E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{5 b^{\frac{3}{4}} x \left (b + c x^{2}\right )} + \frac{2 \sqrt [4]{c} \sqrt{\frac{b + c x^{2}}{\left (\sqrt{b} + \sqrt{c} x\right )^{2}}} \left (\sqrt{b} + \sqrt{c} x\right ) \left (A c + 5 B b\right ) \sqrt{b x^{2} + c x^{4}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{5 b^{\frac{3}{4}} x \left (b + c x^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A*(b*x**2 + c*x**4)**(3/2)/(5*b*x**(11/2)) + 4*sqrt(c)*(A*c + 5*B*b)*sqrt(b*x
**2 + c*x**4)/(5*b*sqrt(x)*(sqrt(b) + sqrt(c)*x)) - 2*(A*c + 5*B*b)*sqrt(b*x**2
+ c*x**4)/(5*b*x**(3/2)) - 4*c**(1/4)*sqrt((b + c*x**2)/(sqrt(b) + sqrt(c)*x)**2
)*(sqrt(b) + sqrt(c)*x)*(A*c + 5*B*b)*sqrt(b*x**2 + c*x**4)*elliptic_e(2*atan(c*
*(1/4)*sqrt(x)/b**(1/4)), 1/2)/(5*b**(3/4)*x*(b + c*x**2)) + 2*c**(1/4)*sqrt((b
+ c*x**2)/(sqrt(b) + sqrt(c)*x)**2)*(sqrt(b) + sqrt(c)*x)*(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)/(5*b**(3/4)*x*(
b + c*x**2))

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Mathematica [C]  time = 1.03788, size = 219, normalized size = 0.67 \[ \frac{2 \left (\sqrt{b} \sqrt{\frac{i \sqrt{b}}{\sqrt{c}}} \left (5 B x^2-A\right ) \left (b+c x^2\right )+2 \sqrt{c} x^{7/2} \sqrt{\frac{b}{c x^2}+1} (A c+5 b B) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{b}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )-2 \sqrt{c} x^{7/2} \sqrt{\frac{b}{c x^2}+1} (A c+5 b B) E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{b}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{5 \sqrt{b} x^{3/2} \sqrt{\frac{i \sqrt{b}}{\sqrt{c}}} \sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(Sqrt[b]*Sqrt[(I*Sqrt[b])/Sqrt[c]]*(-A + 5*B*x^2)*(b + c*x^2) - 2*Sqrt[c]*(5*
b*B + A*c)*Sqrt[1 + b/(c*x^2)]*x^(7/2)*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt
[c]]/Sqrt[x]], -1] + 2*Sqrt[c]*(5*b*B + A*c)*Sqrt[1 + b/(c*x^2)]*x^(7/2)*Ellipti
cF[I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[c]]/Sqrt[x]], -1]))/(5*Sqrt[b]*Sqrt[(I*Sqrt[b
])/Sqrt[c]]*x^(3/2)*Sqrt[x^2*(b + c*x^2)])

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Maple [A]  time = 0.05, size = 422, normalized size = 1.3 \[{\frac{2}{ \left ( 5\,c{x}^{2}+5\,b \right ) b}\sqrt{c{x}^{4}+b{x}^{2}} \left ( 2\,A\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ){x}^{2}bc-A\sqrt{{1 \left ( cx+\sqrt{-bc} \right ){\frac{1}{\sqrt{-bc}}}}}\sqrt{2}\sqrt{{1 \left ( -cx+\sqrt{-bc} \right ){\frac{1}{\sqrt{-bc}}}}}\sqrt{-{cx{\frac{1}{\sqrt{-bc}}}}}{\it EllipticF} \left ( \sqrt{{1 \left ( cx+\sqrt{-bc} \right ){\frac{1}{\sqrt{-bc}}}}},{\frac{\sqrt{2}}{2}} \right ){x}^{2}bc+10\,B\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ){x}^{2}{b}^{2}-5\,B\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ){x}^{2}{b}^{2}-2\,A{c}^{2}{x}^{4}-5\,B{x}^{4}bc-3\,Abc{x}^{2}-5\,B{b}^{2}{x}^{2}-{b}^{2}A \right ){x}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5*(c*x^4+b*x^2)^(1/2)/x^(7/2)/(c*x^2+b)*(2*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
)*EllipticE(((c*x+(-b*c)^(1/2))/(-b*c)^(1/2))^(1/2),1/2*2^(1/2))*x^2*b*c-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))*x^2*b*c+10*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)*EllipticE(((c*x+(-b
*c)^(1/2))/(-b*c)^(1/2))^(1/2),1/2*2^(1/2))*x^2*b^2-5*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))*x^2*b^
2-2*A*c^2*x^4-5*B*x^4*b*c-3*A*b*c*x^2-5*B*b^2*x^2-b^2*A)/b

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(sqrt(c*x^4 + b*x^2)*(B*x^2 + A)/x^(9/2), 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)*(c*x**4+b*x**2)**(1/2)/x**(9/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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