3.229 \(\int \frac{\left (A+B x^2\right ) \sqrt{b x^2+c x^4}}{x^{13/2}} \, dx\)

Optimal. Leaf size=369 \[ \frac{2 c^{5/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (3 b B-A c) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{15 b^{7/4} \sqrt{b x^2+c x^4}}-\frac{4 c^{5/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (3 b B-A c) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{15 b^{7/4} \sqrt{b x^2+c x^4}}+\frac{4 c^{3/2} x^{3/2} \left (b+c x^2\right ) (3 b B-A c)}{15 b^2 \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}-\frac{4 c \sqrt{b x^2+c x^4} (3 b B-A c)}{15 b^2 x^{3/2}}-\frac{2 \sqrt{b x^2+c x^4} (3 b B-A c)}{15 b x^{7/2}}-\frac{2 A \left (b x^2+c x^4\right )^{3/2}}{9 b x^{15/2}} \]

[Out]

(4*c^(3/2)*(3*b*B - A*c)*x^(3/2)*(b + c*x^2))/(15*b^2*(Sqrt[b] + Sqrt[c]*x)*Sqrt
[b*x^2 + c*x^4]) - (2*(3*b*B - A*c)*Sqrt[b*x^2 + c*x^4])/(15*b*x^(7/2)) - (4*c*(
3*b*B - A*c)*Sqrt[b*x^2 + c*x^4])/(15*b^2*x^(3/2)) - (2*A*(b*x^2 + c*x^4)^(3/2))
/(9*b*x^(15/2)) - (4*c^(5/4)*(3*b*B - A*c)*x*(Sqrt[b] + Sqrt[c]*x)*Sqrt[(b + c*x
^2)/(Sqrt[b] + Sqrt[c]*x)^2]*EllipticE[2*ArcTan[(c^(1/4)*Sqrt[x])/b^(1/4)], 1/2]
)/(15*b^(7/4)*Sqrt[b*x^2 + c*x^4]) + (2*c^(5/4)*(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])/(15*b^(7/4)*Sqrt[b*x^2 + c*x^4])

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Rubi [A]  time = 0.83408, antiderivative size = 369, 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 \[ \frac{2 c^{5/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (3 b B-A c) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{15 b^{7/4} \sqrt{b x^2+c x^4}}-\frac{4 c^{5/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (3 b B-A c) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{15 b^{7/4} \sqrt{b x^2+c x^4}}+\frac{4 c^{3/2} x^{3/2} \left (b+c x^2\right ) (3 b B-A c)}{15 b^2 \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}-\frac{4 c \sqrt{b x^2+c x^4} (3 b B-A c)}{15 b^2 x^{3/2}}-\frac{2 \sqrt{b x^2+c x^4} (3 b B-A c)}{15 b x^{7/2}}-\frac{2 A \left (b x^2+c x^4\right )^{3/2}}{9 b x^{15/2}} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x^2)*Sqrt[b*x^2 + c*x^4])/x^(13/2),x]

[Out]

(4*c^(3/2)*(3*b*B - A*c)*x^(3/2)*(b + c*x^2))/(15*b^2*(Sqrt[b] + Sqrt[c]*x)*Sqrt
[b*x^2 + c*x^4]) - (2*(3*b*B - A*c)*Sqrt[b*x^2 + c*x^4])/(15*b*x^(7/2)) - (4*c*(
3*b*B - A*c)*Sqrt[b*x^2 + c*x^4])/(15*b^2*x^(3/2)) - (2*A*(b*x^2 + c*x^4)^(3/2))
/(9*b*x^(15/2)) - (4*c^(5/4)*(3*b*B - A*c)*x*(Sqrt[b] + Sqrt[c]*x)*Sqrt[(b + c*x
^2)/(Sqrt[b] + Sqrt[c]*x)^2]*EllipticE[2*ArcTan[(c^(1/4)*Sqrt[x])/b^(1/4)], 1/2]
)/(15*b^(7/4)*Sqrt[b*x^2 + c*x^4]) + (2*c^(5/4)*(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])/(15*b^(7/4)*Sqrt[b*x^2 + c*x^4])

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Rubi in Sympy [A]  time = 71.9903, size = 347, normalized size = 0.94 \[ - \frac{2 A \left (b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{9 b x^{\frac{15}{2}}} + \frac{2 \left (A c - 3 B b\right ) \sqrt{b x^{2} + c x^{4}}}{15 b x^{\frac{7}{2}}} - \frac{4 c^{\frac{3}{2}} \left (A c - 3 B b\right ) \sqrt{b x^{2} + c x^{4}}}{15 b^{2} \sqrt{x} \left (\sqrt{b} + \sqrt{c} x\right )} + \frac{4 c \left (A c - 3 B b\right ) \sqrt{b x^{2} + c x^{4}}}{15 b^{2} x^{\frac{3}{2}}} + \frac{4 c^{\frac{5}{4}} \sqrt{\frac{b + c x^{2}}{\left (\sqrt{b} + \sqrt{c} x\right )^{2}}} \left (\sqrt{b} + \sqrt{c} x\right ) \left (A c - 3 B b\right ) \sqrt{b x^{2} + c x^{4}} E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{15 b^{\frac{7}{4}} x \left (b + c x^{2}\right )} - \frac{2 c^{\frac{5}{4}} \sqrt{\frac{b + c x^{2}}{\left (\sqrt{b} + \sqrt{c} x\right )^{2}}} \left (\sqrt{b} + \sqrt{c} x\right ) \left (A c - 3 B b\right ) \sqrt{b x^{2} + c x^{4}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{15 b^{\frac{7}{4}} x \left (b + c x^{2}\right )} \]

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**(13/2),x)

[Out]

-2*A*(b*x**2 + c*x**4)**(3/2)/(9*b*x**(15/2)) + 2*(A*c - 3*B*b)*sqrt(b*x**2 + c*
x**4)/(15*b*x**(7/2)) - 4*c**(3/2)*(A*c - 3*B*b)*sqrt(b*x**2 + c*x**4)/(15*b**2*
sqrt(x)*(sqrt(b) + sqrt(c)*x)) + 4*c*(A*c - 3*B*b)*sqrt(b*x**2 + c*x**4)/(15*b**
2*x**(3/2)) + 4*c**(5/4)*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_e(2*atan(c**(1/4)*sqrt(x
)/b**(1/4)), 1/2)/(15*b**(7/4)*x*(b + c*x**2)) - 2*c**(5/4)*sqrt((b + c*x**2)/(s
qrt(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)/(15*b**(7/4)*x*(b + c*x**2)
)

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Mathematica [C]  time = 0.779728, size = 241, normalized size = 0.65 \[ -\frac{2 \left (\sqrt{\frac{i \sqrt{c} x}{\sqrt{b}}} \left (b+c x^2\right ) \left (A \left (5 b^2+2 b c x^2-6 c^2 x^4\right )+9 b B x^2 \left (b+2 c x^2\right )\right )+6 \sqrt{b} c^{3/2} x^5 \sqrt{\frac{c x^2}{b}+1} (3 b B-A c) F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c} x}{\sqrt{b}}}\right )\right |-1\right )-6 \sqrt{b} c^{3/2} x^5 \sqrt{\frac{c x^2}{b}+1} (3 b B-A c) E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c} x}{\sqrt{b}}}\right )\right |-1\right )\right )}{45 b^2 x^{7/2} \sqrt{\frac{i \sqrt{c} x}{\sqrt{b}}} \sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x^2)*Sqrt[b*x^2 + c*x^4])/x^(13/2),x]

[Out]

(-2*(Sqrt[(I*Sqrt[c]*x)/Sqrt[b]]*(b + c*x^2)*(9*b*B*x^2*(b + 2*c*x^2) + A*(5*b^2
+ 2*b*c*x^2 - 6*c^2*x^4)) - 6*Sqrt[b]*c^(3/2)*(3*b*B - A*c)*x^5*Sqrt[1 + (c*x^2
)/b]*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[c]*x)/Sqrt[b]]], -1] + 6*Sqrt[b]*c^(3/2)*(
3*b*B - A*c)*x^5*Sqrt[1 + (c*x^2)/b]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c]*x)/Sqrt
[b]]], -1]))/(45*b^2*x^(7/2)*Sqrt[(I*Sqrt[c]*x)/Sqrt[b]]*Sqrt[x^2*(b + c*x^2)])

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Maple [A]  time = 0.052, size = 452, normalized size = 1.2 \[ -{\frac{2}{ \left ( 45\,c{x}^{2}+45\,b \right ){b}^{2}}\sqrt{c{x}^{4}+b{x}^{2}} \left ( 6\,A\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ){x}^{4}b{c}^{2}-3\,A\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ){x}^{4}b{c}^{2}-18\,B\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ){x}^{4}{b}^{2}c+9\,B\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ){x}^{4}{b}^{2}c-6\,A{c}^{3}{x}^{6}+18\,B{x}^{6}b{c}^{2}-4\,Ab{c}^{2}{x}^{4}+27\,B{x}^{4}{b}^{2}c+7\,A{b}^{2}c{x}^{2}+9\,B{x}^{2}{b}^{3}+5\,A{b}^{3} \right ){x}^{-{\frac{11}{2}}}} \]

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^(13/2),x)

[Out]

-2/45*(c*x^4+b*x^2)^(1/2)/x^(11/2)/(c*x^2+b)*(6*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^4*b*c^2-3*
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^4*b*c^2-18*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^4*b^2*c+9*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^4*b^2*c-6*A*c^3*x^6+18*B*x^6*b*c^2-4*A*b*c^2*x^4+27*B*x^4*b^2*c+7*A*b^2*c
*x^2+9*B*x^2*b^3+5*A*b^3)/b^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x^{4} + b x^{2}}{\left (B x^{2} + A\right )}}{x^{\frac{13}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^4 + b*x^2)*(B*x^2 + A)/x^(13/2),x, algorithm="maxima")

[Out]

integrate(sqrt(c*x^4 + b*x^2)*(B*x^2 + A)/x^(13/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{c x^{4} + b x^{2}}{\left (B x^{2} + A\right )}}{x^{\frac{13}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^4 + b*x^2)*(B*x^2 + A)/x^(13/2),x, algorithm="fricas")

[Out]

integral(sqrt(c*x^4 + b*x^2)*(B*x^2 + A)/x^(13/2), 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**(13/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x^{4} + b x^{2}}{\left (B x^{2} + A\right )}}{x^{\frac{13}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^4 + b*x^2)*(B*x^2 + A)/x^(13/2),x, algorithm="giac")

[Out]

integrate(sqrt(c*x^4 + b*x^2)*(B*x^2 + A)/x^(13/2), x)