3.363 \(\int \frac{\sqrt{b x^2+c x^4}}{x^{15/2}} \, dx\)

Optimal. Leaf size=176 \[ \frac{10 c^{11/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{231 b^{9/4} \sqrt{b x^2+c x^4}}+\frac{20 c^2 \sqrt{b x^2+c x^4}}{231 b^2 x^{5/2}}-\frac{4 c \sqrt{b x^2+c x^4}}{77 b x^{9/2}}-\frac{2 \sqrt{b x^2+c x^4}}{11 x^{13/2}} \]

[Out]

(-2*Sqrt[b*x^2 + c*x^4])/(11*x^(13/2)) - (4*c*Sqrt[b*x^2 + c*x^4])/(77*b*x^(9/2)
) + (20*c^2*Sqrt[b*x^2 + c*x^4])/(231*b^2*x^(5/2)) + (10*c^(11/4)*x*(Sqrt[b] + S
qrt[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])/(231*b^(9/4)*Sqrt[b*x^2 + c*x^4])

_______________________________________________________________________________________

Rubi [A]  time = 0.453419, antiderivative size = 176, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ \frac{10 c^{11/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{231 b^{9/4} \sqrt{b x^2+c x^4}}+\frac{20 c^2 \sqrt{b x^2+c x^4}}{231 b^2 x^{5/2}}-\frac{4 c \sqrt{b x^2+c x^4}}{77 b x^{9/2}}-\frac{2 \sqrt{b x^2+c x^4}}{11 x^{13/2}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[b*x^2 + c*x^4]/x^(15/2),x]

[Out]

(-2*Sqrt[b*x^2 + c*x^4])/(11*x^(13/2)) - (4*c*Sqrt[b*x^2 + c*x^4])/(77*b*x^(9/2)
) + (20*c^2*Sqrt[b*x^2 + c*x^4])/(231*b^2*x^(5/2)) + (10*c^(11/4)*x*(Sqrt[b] + S
qrt[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])/(231*b^(9/4)*Sqrt[b*x^2 + c*x^4])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 41.2058, size = 168, normalized size = 0.95 \[ - \frac{2 \sqrt{b x^{2} + c x^{4}}}{11 x^{\frac{13}{2}}} - \frac{4 c \sqrt{b x^{2} + c x^{4}}}{77 b x^{\frac{9}{2}}} + \frac{20 c^{2} \sqrt{b x^{2} + c x^{4}}}{231 b^{2} x^{\frac{5}{2}}} + \frac{10 c^{\frac{11}{4}} \sqrt{\frac{b + c x^{2}}{\left (\sqrt{b} + \sqrt{c} x\right )^{2}}} \left (\sqrt{b} + \sqrt{c} x\right ) \sqrt{b x^{2} + c x^{4}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{231 b^{\frac{9}{4}} x \left (b + c x^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**4+b*x**2)**(1/2)/x**(15/2),x)

[Out]

-2*sqrt(b*x**2 + c*x**4)/(11*x**(13/2)) - 4*c*sqrt(b*x**2 + c*x**4)/(77*b*x**(9/
2)) + 20*c**2*sqrt(b*x**2 + c*x**4)/(231*b**2*x**(5/2)) + 10*c**(11/4)*sqrt((b +
c*x**2)/(sqrt(b) + sqrt(c)*x)**2)*(sqrt(b) + sqrt(c)*x)*sqrt(b*x**2 + c*x**4)*e
lliptic_f(2*atan(c**(1/4)*sqrt(x)/b**(1/4)), 1/2)/(231*b**(9/4)*x*(b + c*x**2))

_______________________________________________________________________________________

Mathematica [C]  time = 0.515758, size = 133, normalized size = 0.76 \[ \frac{1}{231} \sqrt{x^2 \left (b+c x^2\right )} \left (\frac{2 \left (-21 b^2-6 b c x^2+10 c^2 x^4\right )}{b^2 x^{13/2}}+\frac{20 i c^3 \sqrt{\frac{b}{c x^2}+1} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{b}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )}{b^2 \sqrt{\frac{i \sqrt{b}}{\sqrt{c}}} \left (b+c x^2\right )}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[b*x^2 + c*x^4]/x^(15/2),x]

[Out]

(Sqrt[x^2*(b + c*x^2)]*((2*(-21*b^2 - 6*b*c*x^2 + 10*c^2*x^4))/(b^2*x^(13/2)) +
((20*I)*c^3*Sqrt[1 + b/(c*x^2)]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[c]]/Sq
rt[x]], -1])/(b^2*Sqrt[(I*Sqrt[b])/Sqrt[c]]*(b + c*x^2))))/231

_______________________________________________________________________________________

Maple [A]  time = 0.04, size = 156, normalized size = 0.9 \[{\frac{2}{ \left ( 231\,c{x}^{2}+231\,b \right ){b}^{2}}\sqrt{c{x}^{4}+b{x}^{2}} \left ( 5\,\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ) \sqrt{-bc}\sqrt{2}{x}^{5}{c}^{2}+10\,{c}^{3}{x}^{6}+4\,b{c}^{2}{x}^{4}-27\,{b}^{2}c{x}^{2}-21\,{b}^{3} \right ){x}^{-{\frac{13}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+b*x^2)^(1/2)/x^(15/2),x)

[Out]

2/231*(c*x^4+b*x^2)^(1/2)/x^(13/2)/(c*x^2+b)*(5*((c*x+(-b*c)^(1/2))/(-b*c)^(1/2)
)^(1/2)*((-c*x+(-b*c)^(1/2))/(-b*c)^(1/2))^(1/2)*(-x*c/(-b*c)^(1/2))^(1/2)*Ellip
ticF(((c*x+(-b*c)^(1/2))/(-b*c)^(1/2))^(1/2),1/2*2^(1/2))*(-b*c)^(1/2)*2^(1/2)*x
^5*c^2+10*c^3*x^6+4*b*c^2*x^4-27*b^2*c*x^2-21*b^3)/b^2

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x^{4} + b x^{2}}}{x^{\frac{15}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^4 + b*x^2)/x^(15/2),x, algorithm="maxima")

[Out]

integrate(sqrt(c*x^4 + b*x^2)/x^(15/2), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{c x^{4} + b x^{2}}}{x^{\frac{15}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^4 + b*x^2)/x^(15/2),x, algorithm="fricas")

[Out]

integral(sqrt(c*x^4 + b*x^2)/x^(15/2), x)

_______________________________________________________________________________________

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((c*x**4+b*x**2)**(1/2)/x**(15/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x^{4} + b x^{2}}}{x^{\frac{15}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^4 + b*x^2)/x^(15/2),x, algorithm="giac")

[Out]

integrate(sqrt(c*x^4 + b*x^2)/x^(15/2), x)