3.262 \(\int \frac{x^{7/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=299 \[ \frac{x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (3 b B-A c) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{2 b^{3/4} c^{7/4} \sqrt{b x^2+c x^4}}-\frac{x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (3 b B-A c) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{b^{3/4} c^{7/4} \sqrt{b x^2+c x^4}}+\frac{x^{3/2} \left (b+c x^2\right ) (3 b B-A c)}{b c^{3/2} \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}-\frac{x^{5/2} (b B-A c)}{b c \sqrt{b x^2+c x^4}} \]

[Out]

-(((b*B - A*c)*x^(5/2))/(b*c*Sqrt[b*x^2 + c*x^4])) + ((3*b*B - A*c)*x^(3/2)*(b +
c*x^2))/(b*c^(3/2)*(Sqrt[b] + Sqrt[c]*x)*Sqrt[b*x^2 + c*x^4]) - ((3*b*B - A*c)*
x*(Sqrt[b] + Sqrt[c]*x)*Sqrt[(b + c*x^2)/(Sqrt[b] + Sqrt[c]*x)^2]*EllipticE[2*Ar
cTan[(c^(1/4)*Sqrt[x])/b^(1/4)], 1/2])/(b^(3/4)*c^(7/4)*Sqrt[b*x^2 + c*x^4]) + (
(3*b*B - 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])/(2*b^(3/4)*c^(7/4)*Sqrt[b*x
^2 + c*x^4])

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Rubi [A]  time = 0.63033, antiderivative size = 299, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (3 b B-A c) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{2 b^{3/4} c^{7/4} \sqrt{b x^2+c x^4}}-\frac{x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (3 b B-A c) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{b^{3/4} c^{7/4} \sqrt{b x^2+c x^4}}+\frac{x^{3/2} \left (b+c x^2\right ) (3 b B-A c)}{b c^{3/2} \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}-\frac{x^{5/2} (b B-A c)}{b c \sqrt{b x^2+c x^4}} \]

Antiderivative was successfully verified.

[In]  Int[(x^(7/2)*(A + B*x^2))/(b*x^2 + c*x^4)^(3/2),x]

[Out]

-(((b*B - A*c)*x^(5/2))/(b*c*Sqrt[b*x^2 + c*x^4])) + ((3*b*B - A*c)*x^(3/2)*(b +
c*x^2))/(b*c^(3/2)*(Sqrt[b] + Sqrt[c]*x)*Sqrt[b*x^2 + c*x^4]) - ((3*b*B - A*c)*
x*(Sqrt[b] + Sqrt[c]*x)*Sqrt[(b + c*x^2)/(Sqrt[b] + Sqrt[c]*x)^2]*EllipticE[2*Ar
cTan[(c^(1/4)*Sqrt[x])/b^(1/4)], 1/2])/(b^(3/4)*c^(7/4)*Sqrt[b*x^2 + c*x^4]) + (
(3*b*B - 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])/(2*b^(3/4)*c^(7/4)*Sqrt[b*x
^2 + c*x^4])

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Rubi in Sympy [A]  time = 53.2912, size = 272, normalized size = 0.91 \[ \frac{x^{\frac{5}{2}} \left (A c - B b\right )}{b c \sqrt{b x^{2} + c x^{4}}} - \frac{\left (A c - 3 B b\right ) \sqrt{b x^{2} + c x^{4}}}{b c^{\frac{3}{2}} \sqrt{x} \left (\sqrt{b} + \sqrt{c} x\right )} + \frac{\sqrt{\frac{b + c x^{2}}{\left (\sqrt{b} + \sqrt{c} x\right )^{2}}} \left (\sqrt{b} + \sqrt{c} x\right ) \left (A c - 3 B b\right ) \sqrt{b x^{2} + c x^{4}} E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{b^{\frac{3}{4}} c^{\frac{7}{4}} x \left (b + c x^{2}\right )} - \frac{\sqrt{\frac{b + c x^{2}}{\left (\sqrt{b} + \sqrt{c} x\right )^{2}}} \left (\sqrt{b} + \sqrt{c} x\right ) \left (A c - 3 B b\right ) \sqrt{b x^{2} + c x^{4}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{2 b^{\frac{3}{4}} c^{\frac{7}{4}} x \left (b + c x^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(7/2)*(B*x**2+A)/(c*x**4+b*x**2)**(3/2),x)

[Out]

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

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Mathematica [C]  time = 0.750669, size = 213, normalized size = 0.71 \[ \frac{i \left (\sqrt{b} \sqrt{x} \sqrt{\frac{i \sqrt{b}}{\sqrt{c}}} \left (-A c+3 b B+2 B c x^2\right )-\sqrt{c} x^2 \sqrt{\frac{b}{c x^2}+1} (A c-3 b B) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{b}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )+\sqrt{c} x^2 \sqrt{\frac{b}{c x^2}+1} (A c-3 b B) E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{b}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{c^{5/2} \left (\frac{i \sqrt{b}}{\sqrt{c}}\right )^{3/2} \sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^(7/2)*(A + B*x^2))/(b*x^2 + c*x^4)^(3/2),x]

[Out]

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

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Maple [A]  time = 0.027, size = 388, normalized size = 1.3 \[ -{\frac{c{x}^{2}+b}{2\,b{c}^{2}}{x}^{{\frac{5}{2}}} \left ( 2\,A\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ) bc-A\sqrt{{1 \left ( cx+\sqrt{-bc} \right ){\frac{1}{\sqrt{-bc}}}}}\sqrt{2}\sqrt{{1 \left ( -cx+\sqrt{-bc} \right ){\frac{1}{\sqrt{-bc}}}}}\sqrt{-{cx{\frac{1}{\sqrt{-bc}}}}}{\it EllipticF} \left ( \sqrt{{1 \left ( cx+\sqrt{-bc} \right ){\frac{1}{\sqrt{-bc}}}}},{\frac{\sqrt{2}}{2}} \right ) bc-6\,B\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ){b}^{2}+3\,B\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ){b}^{2}-2\,A{x}^{2}{c}^{2}+2\,B{x}^{2}bc \right ) \left ( c{x}^{4}+b{x}^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(7/2)*(B*x^2+A)/(c*x^4+b*x^2)^(3/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x^{2} + A\right )} x^{\frac{7}{2}}}{{\left (c x^{4} + b x^{2}\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^(7/2)/(c*x^4 + b*x^2)^(3/2),x, algorithm="maxima")

[Out]

integrate((B*x^2 + A)*x^(7/2)/(c*x^4 + b*x^2)^(3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (B x^{3} + A x\right )} \sqrt{x}}{\sqrt{c x^{4} + b x^{2}}{\left (c x^{2} + b\right )}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^(7/2)/(c*x^4 + b*x^2)^(3/2),x, algorithm="fricas")

[Out]

integral((B*x^3 + A*x)*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**(7/2)*(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. \[ \int \frac{{\left (B x^{2} + A\right )} x^{\frac{7}{2}}}{{\left (c x^{4} + b x^{2}\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^(7/2)/(c*x^4 + b*x^2)^(3/2),x, algorithm="giac")

[Out]

integrate((B*x^2 + A)*x^(7/2)/(c*x^4 + b*x^2)^(3/2), x)