Optimal. Leaf size=57 \[ \frac{8 \left (a+b x^2\right )^{9/4}}{45 a^2 c (c x)^{9/2}}-\frac{2 \left (a+b x^2\right )^{5/4}}{5 a c (c x)^{9/2}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.056541, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{8 \left (a+b x^2\right )^{9/4}}{45 a^2 c (c x)^{9/2}}-\frac{2 \left (a+b x^2\right )^{5/4}}{5 a c (c x)^{9/2}} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^2)^(1/4)/(c*x)^(11/2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 6.69595, size = 48, normalized size = 0.84 \[ - \frac{2 \left (a + b x^{2}\right )^{\frac{5}{4}}}{5 a c \left (c x\right )^{\frac{9}{2}}} + \frac{8 \left (a + b x^{2}\right )^{\frac{9}{4}}}{45 a^{2} c \left (c x\right )^{\frac{9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)**(1/4)/(c*x)**(11/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0373974, size = 51, normalized size = 0.89 \[ -\frac{2 \sqrt{c x} \sqrt [4]{a+b x^2} \left (5 a^2+a b x^2-4 b^2 x^4\right )}{45 a^2 c^6 x^5} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^2)^(1/4)/(c*x)^(11/2),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.007, size = 31, normalized size = 0.5 \[ -{\frac{2\,x \left ( -4\,b{x}^{2}+5\,a \right ) }{45\,{a}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{4}}} \left ( cx \right ) ^{-{\frac{11}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)^(1/4)/(c*x)^(11/2),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 1.38785, size = 51, normalized size = 0.89 \[ \frac{2 \,{\left (\frac{9 \,{\left (b x^{2} + a\right )}^{\frac{5}{4}} b}{x^{\frac{5}{2}}} - \frac{5 \,{\left (b x^{2} + a\right )}^{\frac{9}{4}}}{x^{\frac{9}{2}}}\right )}}{45 \, a^{2} c^{\frac{11}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^(1/4)/(c*x)^(11/2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.256954, size = 62, normalized size = 1.09 \[ \frac{2 \,{\left (4 \, b^{2} x^{4} - a b x^{2} - 5 \, a^{2}\right )}{\left (b x^{2} + a\right )}^{\frac{1}{4}} \sqrt{c x}}{45 \, a^{2} c^{6} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^(1/4)/(c*x)^(11/2),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((b*x**2+a)**(1/4)/(c*x)**(11/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.228822, size = 144, normalized size = 2.53 \[ \frac{2 \,{\left (\frac{9 \,{\left (b c^{4} x^{2} + a c^{4}\right )}^{\frac{1}{4}}{\left (b c^{2} + \frac{a c^{2}}{x^{2}}\right )} b c^{2}}{\sqrt{c x}} - \frac{5 \,{\left (b^{2} c^{8} x^{4} + 2 \, a b c^{8} x^{2} + a^{2} c^{8}\right )}{\left (b c^{4} x^{2} + a c^{4}\right )}^{\frac{1}{4}}}{\sqrt{c x} c^{4} x^{4}}\right )}}{45 \, a^{2} c^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^(1/4)/(c*x)^(11/2),x, algorithm="giac")
[Out]