3.361 \(\int \frac{\sqrt{b x^2+c x^4}}{x^{11/2}} \, dx\)

Optimal. Leaf size=146 \[ -\frac{2 c^{7/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{21 b^{5/4} \sqrt{b x^2+c x^4}}-\frac{4 c \sqrt{b x^2+c x^4}}{21 b x^{5/2}}-\frac{2 \sqrt{b x^2+c x^4}}{7 x^{9/2}} \]

[Out]

(-2*Sqrt[b*x^2 + c*x^4])/(7*x^(9/2)) - (4*c*Sqrt[b*x^2 + c*x^4])/(21*b*x^(5/2))
- (2*c^(7/4)*x*(Sqrt[b] + Sqrt[c]*x)*Sqrt[(b + c*x^2)/(Sqrt[b] + Sqrt[c]*x)^2]*E
llipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/b^(1/4)], 1/2])/(21*b^(5/4)*Sqrt[b*x^2 + c*x
^4])

_______________________________________________________________________________________

Rubi [A]  time = 0.359456, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ -\frac{2 c^{7/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{21 b^{5/4} \sqrt{b x^2+c x^4}}-\frac{4 c \sqrt{b x^2+c x^4}}{21 b x^{5/2}}-\frac{2 \sqrt{b x^2+c x^4}}{7 x^{9/2}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[b*x^2 + c*x^4]/x^(11/2),x]

[Out]

(-2*Sqrt[b*x^2 + c*x^4])/(7*x^(9/2)) - (4*c*Sqrt[b*x^2 + c*x^4])/(21*b*x^(5/2))
- (2*c^(7/4)*x*(Sqrt[b] + Sqrt[c]*x)*Sqrt[(b + c*x^2)/(Sqrt[b] + Sqrt[c]*x)^2]*E
llipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/b^(1/4)], 1/2])/(21*b^(5/4)*Sqrt[b*x^2 + c*x
^4])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 32.5988, size = 141, normalized size = 0.97 \[ - \frac{2 \sqrt{b x^{2} + c x^{4}}}{7 x^{\frac{9}{2}}} - \frac{4 c \sqrt{b x^{2} + c x^{4}}}{21 b x^{\frac{5}{2}}} - \frac{2 c^{\frac{7}{4}} \sqrt{\frac{b + c x^{2}}{\left (\sqrt{b} + \sqrt{c} x\right )^{2}}} \left (\sqrt{b} + \sqrt{c} x\right ) \sqrt{b x^{2} + c x^{4}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{21 b^{\frac{5}{4}} x \left (b + c x^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**4+b*x**2)**(1/2)/x**(11/2),x)

[Out]

-2*sqrt(b*x**2 + c*x**4)/(7*x**(9/2)) - 4*c*sqrt(b*x**2 + c*x**4)/(21*b*x**(5/2)
) - 2*c**(7/4)*sqrt((b + c*x**2)/(sqrt(b) + sqrt(c)*x)**2)*(sqrt(b) + sqrt(c)*x)
*sqrt(b*x**2 + c*x**4)*elliptic_f(2*atan(c**(1/4)*sqrt(x)/b**(1/4)), 1/2)/(21*b*
*(5/4)*x*(b + c*x**2))

_______________________________________________________________________________________

Mathematica [C]  time = 0.429399, size = 122, normalized size = 0.84 \[ \frac{1}{21} \sqrt{x^2 \left (b+c x^2\right )} \left (-\frac{2 \left (3 b+2 c x^2\right )}{b x^{9/2}}-\frac{4 i c^2 \sqrt{\frac{b}{c x^2}+1} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{b}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )}{b \sqrt{\frac{i \sqrt{b}}{\sqrt{c}}} \left (b+c x^2\right )}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[b*x^2 + c*x^4]/x^(11/2),x]

[Out]

(Sqrt[x^2*(b + c*x^2)]*((-2*(3*b + 2*c*x^2))/(b*x^(9/2)) - ((4*I)*c^2*Sqrt[1 + b
/(c*x^2)]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[c]]/Sqrt[x]], -1])/(b*Sqrt[(
I*Sqrt[b])/Sqrt[c]]*(b + c*x^2))))/21

_______________________________________________________________________________________

Maple [A]  time = 0.039, size = 142, normalized size = 1. \[ -{\frac{2}{ \left ( 21\,c{x}^{2}+21\,b \right ) b}\sqrt{c{x}^{4}+b{x}^{2}} \left ( \sqrt{{1 \left ( cx+\sqrt{-bc} \right ){\frac{1}{\sqrt{-bc}}}}}\sqrt{{1 \left ( -cx+\sqrt{-bc} \right ){\frac{1}{\sqrt{-bc}}}}}\sqrt{-{cx{\frac{1}{\sqrt{-bc}}}}}{\it EllipticF} \left ( \sqrt{{1 \left ( cx+\sqrt{-bc} \right ){\frac{1}{\sqrt{-bc}}}}},{\frac{\sqrt{2}}{2}} \right ) \sqrt{-bc}\sqrt{2}{x}^{3}c+2\,{c}^{2}{x}^{4}+5\,bc{x}^{2}+3\,{b}^{2} \right ){x}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+b*x^2)^(1/2)/x^(11/2),x)

[Out]

-2/21*(c*x^4+b*x^2)^(1/2)/x^(9/2)/(c*x^2+b)*(((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)*Elliptic
F(((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^3*
c+2*c^2*x^4+5*b*c*x^2+3*b^2)/b

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x^{4} + b x^{2}}}{x^{\frac{11}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^4 + b*x^2)/x^(11/2),x, algorithm="maxima")

[Out]

integrate(sqrt(c*x^4 + b*x^2)/x^(11/2), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{c x^{4} + b x^{2}}}{x^{\frac{11}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^4 + b*x^2)/x^(11/2),x, algorithm="fricas")

[Out]

integral(sqrt(c*x^4 + b*x^2)/x^(11/2), 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**(11/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x^{4} + b x^{2}}}{x^{\frac{11}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^4 + b*x^2)/x^(11/2),x, algorithm="giac")

[Out]

integrate(sqrt(c*x^4 + b*x^2)/x^(11/2), x)