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

Optimal. Leaf size=368 \[ \frac{3 \sqrt [4]{c} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (5 b B-7 A c) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{10 b^{11/4} \sqrt{b x^2+c x^4}}-\frac{3 \sqrt [4]{c} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (5 b B-7 A c) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 b^{11/4} \sqrt{b x^2+c x^4}}+\frac{3 \sqrt{c} x^{3/2} \left (b+c x^2\right ) (5 b B-7 A c)}{5 b^3 \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}-\frac{3 \sqrt{b x^2+c x^4} (5 b B-7 A c)}{5 b^3 x^{3/2}}+\frac{\sqrt{x} (5 b B-7 A c)}{5 b^2 \sqrt{b x^2+c x^4}}-\frac{2 A}{5 b x^{3/2} \sqrt{b x^2+c x^4}} \]

[Out]

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

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Rubi [A]  time = 0.850866, antiderivative size = 368, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ \frac{3 \sqrt [4]{c} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (5 b B-7 A c) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{10 b^{11/4} \sqrt{b x^2+c x^4}}-\frac{3 \sqrt [4]{c} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (5 b B-7 A c) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 b^{11/4} \sqrt{b x^2+c x^4}}+\frac{3 \sqrt{c} x^{3/2} \left (b+c x^2\right ) (5 b B-7 A c)}{5 b^3 \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}-\frac{3 \sqrt{b x^2+c x^4} (5 b B-7 A c)}{5 b^3 x^{3/2}}+\frac{\sqrt{x} (5 b B-7 A c)}{5 b^2 \sqrt{b x^2+c x^4}}-\frac{2 A}{5 b x^{3/2} \sqrt{b x^2+c x^4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 72.5374, size = 354, normalized size = 0.96 \[ - \frac{2 A}{5 b x^{\frac{3}{2}} \sqrt{b x^{2} + c x^{4}}} - \frac{\sqrt{x} \left (7 A c - 5 B b\right )}{5 b^{2} \sqrt{b x^{2} + c x^{4}}} - \frac{3 \sqrt{c} \left (7 A c - 5 B b\right ) \sqrt{b x^{2} + c x^{4}}}{5 b^{3} \sqrt{x} \left (\sqrt{b} + \sqrt{c} x\right )} + \frac{3 \left (7 A c - 5 B b\right ) \sqrt{b x^{2} + c x^{4}}}{5 b^{3} x^{\frac{3}{2}}} + \frac{3 \sqrt [4]{c} \sqrt{\frac{b + c x^{2}}{\left (\sqrt{b} + \sqrt{c} x\right )^{2}}} \left (\sqrt{b} + \sqrt{c} x\right ) \left (7 A c - 5 B b\right ) \sqrt{b x^{2} + c x^{4}} E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{5 b^{\frac{11}{4}} x \left (b + c x^{2}\right )} - \frac{3 \sqrt [4]{c} \sqrt{\frac{b + c x^{2}}{\left (\sqrt{b} + \sqrt{c} x\right )^{2}}} \left (\sqrt{b} + \sqrt{c} x\right ) \left (7 A c - 5 B b\right ) \sqrt{b x^{2} + c x^{4}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{10 b^{\frac{11}{4}} x \left (b + c x^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**2+A)/(c*x**4+b*x**2)**(3/2)/x**(1/2),x)

[Out]

-2*A/(5*b*x**(3/2)*sqrt(b*x**2 + c*x**4)) - sqrt(x)*(7*A*c - 5*B*b)/(5*b**2*sqrt
(b*x**2 + c*x**4)) - 3*sqrt(c)*(7*A*c - 5*B*b)*sqrt(b*x**2 + c*x**4)/(5*b**3*sqr
t(x)*(sqrt(b) + sqrt(c)*x)) + 3*(7*A*c - 5*B*b)*sqrt(b*x**2 + c*x**4)/(5*b**3*x*
*(3/2)) + 3*c**(1/4)*sqrt((b + c*x**2)/(sqrt(b) + sqrt(c)*x)**2)*(sqrt(b) + sqrt
(c)*x)*(7*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**(11/4)*x*(b + c*x**2)) - 3*c**(1/4)*sqrt((b + c*x**2)/(sqr
t(b) + sqrt(c)*x)**2)*(sqrt(b) + sqrt(c)*x)*(7*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)/(10*b**(11/4)*x*(b + c*x**2
))

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Mathematica [C]  time = 0.425833, size = 236, normalized size = 0.64 \[ \frac{\sqrt{\frac{i \sqrt{c} x}{\sqrt{b}}} \left (A \left (-2 b^2+14 b c x^2+21 c^2 x^4\right )-5 b B x^2 \left (2 b+3 c x^2\right )\right )-3 \sqrt{b} \sqrt{c} x^3 \sqrt{\frac{c x^2}{b}+1} (5 b B-7 A c) F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c} x}{\sqrt{b}}}\right )\right |-1\right )+3 \sqrt{b} \sqrt{c} x^3 \sqrt{\frac{c x^2}{b}+1} (5 b B-7 A c) E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c} x}{\sqrt{b}}}\right )\right |-1\right )}{5 b^3 x^{3/2} \sqrt{\frac{i \sqrt{c} x}{\sqrt{b}}} \sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[(I*Sqrt[c]*x)/Sqrt[b]]*(-5*b*B*x^2*(2*b + 3*c*x^2) + A*(-2*b^2 + 14*b*c*x^
2 + 21*c^2*x^4)) + 3*Sqrt[b]*Sqrt[c]*(5*b*B - 7*A*c)*x^3*Sqrt[1 + (c*x^2)/b]*Ell
ipticE[I*ArcSinh[Sqrt[(I*Sqrt[c]*x)/Sqrt[b]]], -1] - 3*Sqrt[b]*Sqrt[c]*(5*b*B -
7*A*c)*x^3*Sqrt[1 + (c*x^2)/b]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c]*x)/Sqrt[b]]],
-1])/(5*b^3*x^(3/2)*Sqrt[(I*Sqrt[c]*x)/Sqrt[b]]*Sqrt[x^2*(b + c*x^2)])

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Maple [A]  time = 0.032, size = 420, normalized size = 1.1 \[ -{\frac{c{x}^{2}+b}{10\,{b}^{3}}\sqrt{x} \left ( 42\,A\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ){x}^{2}bc-21\,A\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ){x}^{2}bc-30\,B\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ){x}^{2}{b}^{2}+15\,B\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ){x}^{2}{b}^{2}-42\,A{c}^{2}{x}^{4}+30\,B{x}^{4}bc-28\,Abc{x}^{2}+20\,B{b}^{2}{x}^{2}+4\,{b}^{2}A \right ) \left ( c{x}^{4}+b{x}^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)/(c*x^4+b*x^2)^(3/2)/x^(1/2),x)

[Out]

-1/10/(c*x^4+b*x^2)^(3/2)*x^(1/2)*(c*x^2+b)*(42*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-21*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-30*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+15*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^2-42*A*c^2*x^4+30*B*x^4*b*c-28*A*b*c*x^2+20*B*b^2*x^2+4*b^2*A)/b^3

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((B*x^2 + A)/((c*x^4 + b*x^2)^(3/2)*sqrt(x)), 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)**(3/2)/x**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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