Optimal. Leaf size=85 \[ -\frac{64 \left (a+b x^2\right )^{11/4}}{231 a^3 c (c x)^{11/2}}+\frac{16 \left (a+b x^2\right )^{7/4}}{21 a^2 c (c x)^{11/2}}-\frac{2 \left (a+b x^2\right )^{3/4}}{3 a c (c x)^{11/2}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0874795, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ -\frac{64 \left (a+b x^2\right )^{11/4}}{231 a^3 c (c x)^{11/2}}+\frac{16 \left (a+b x^2\right )^{7/4}}{21 a^2 c (c x)^{11/2}}-\frac{2 \left (a+b x^2\right )^{3/4}}{3 a c (c x)^{11/2}} \]
Antiderivative was successfully verified.
[In] Int[1/((c*x)^(13/2)*(a + b*x^2)^(1/4)),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 10.7774, size = 73, normalized size = 0.86 \[ - \frac{2 \left (a + b x^{2}\right )^{\frac{3}{4}}}{3 a c \left (c x\right )^{\frac{11}{2}}} + \frac{16 \left (a + b x^{2}\right )^{\frac{7}{4}}}{21 a^{2} c \left (c x\right )^{\frac{11}{2}}} - \frac{64 \left (a + b x^{2}\right )^{\frac{11}{4}}}{231 a^{3} c \left (c x\right )^{\frac{11}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(c*x)**(13/2)/(b*x**2+a)**(1/4),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0436354, size = 52, normalized size = 0.61 \[ -\frac{2 \sqrt{c x} \left (a+b x^2\right )^{3/4} \left (21 a^2-24 a b x^2+32 b^2 x^4\right )}{231 a^3 c^7 x^6} \]
Antiderivative was successfully verified.
[In] Integrate[1/((c*x)^(13/2)*(a + b*x^2)^(1/4)),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.008, size = 42, normalized size = 0.5 \[ -{\frac{2\,x \left ( 32\,{b}^{2}{x}^{4}-24\,ab{x}^{2}+21\,{a}^{2} \right ) }{231\,{a}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{4}}} \left ( cx \right ) ^{-{\frac{13}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(c*x)^(13/2)/(b*x^2+a)^(1/4),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 1.40343, size = 74, normalized size = 0.87 \[ -\frac{2 \,{\left (\frac{77 \,{\left (b x^{2} + a\right )}^{\frac{3}{4}} b^{2}}{x^{\frac{3}{2}}} - \frac{66 \,{\left (b x^{2} + a\right )}^{\frac{7}{4}} b}{x^{\frac{7}{2}}} + \frac{21 \,{\left (b x^{2} + a\right )}^{\frac{11}{4}}}{x^{\frac{11}{2}}}\right )}}{231 \, a^{3} c^{\frac{13}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)^(1/4)*(c*x)^(13/2)),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.213683, size = 77, normalized size = 0.91 \[ -\frac{2 \,{\left (32 \, b^{3} x^{6} + 8 \, a b^{2} x^{4} - 3 \, a^{2} b x^{2} + 21 \, a^{3}\right )}}{231 \,{\left (b x^{2} + a\right )}^{\frac{1}{4}} \sqrt{c x} a^{3} c^{6} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)^(1/4)*(c*x)^(13/2)),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
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(1/(c*x)**(13/2)/(b*x**2+a)**(1/4),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{1}{4}} \left (c x\right )^{\frac{13}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)^(1/4)*(c*x)^(13/2)),x, algorithm="giac")
[Out]