Optimal. Leaf size=65 \[ -\frac{2 (e x)^{3/2} (4 b c-a d)}{3 a^2 e^3 \left (a+b x^2\right )^{3/4}}-\frac{2 c}{a e \sqrt{e x} \left (a+b x^2\right )^{3/4}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.117032, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{2 (e x)^{3/2} (4 b c-a d)}{3 a^2 e^3 \left (a+b x^2\right )^{3/4}}-\frac{2 c}{a e \sqrt{e x} \left (a+b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x^2)/((e*x)^(3/2)*(a + b*x^2)^(7/4)),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 11.8793, size = 58, normalized size = 0.89 \[ - \frac{2 c}{a e \sqrt{e x} \left (a + b x^{2}\right )^{\frac{3}{4}}} + \frac{2 \left (e x\right )^{\frac{3}{2}} \left (a d - 4 b c\right )}{3 a^{2} e^{3} \left (a + b x^{2}\right )^{\frac{3}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x**2+c)/(e*x)**(3/2)/(b*x**2+a)**(7/4),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0570357, size = 44, normalized size = 0.68 \[ \frac{2 x \left (-3 a c+a d x^2-4 b c x^2\right )}{3 a^2 (e x)^{3/2} \left (a+b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x^2)/((e*x)^(3/2)*(a + b*x^2)^(7/4)),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.009, size = 40, normalized size = 0.6 \[ -{\frac{2\,x \left ( -ad{x}^{2}+4\,c{x}^{2}b+3\,ac \right ) }{3\,{a}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{4}}} \left ( ex \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x^2+c)/(e*x)^(3/2)/(b*x^2+a)^(7/4),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac{7}{4}} \left (e x\right )^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)/((b*x^2 + a)^(7/4)*(e*x)^(3/2)),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.242057, size = 76, normalized size = 1.17 \[ -\frac{2 \,{\left ({\left (4 \, b c - a d\right )} x^{2} + 3 \, a c\right )}{\left (b x^{2} + a\right )}^{\frac{1}{4}} \sqrt{e x}}{3 \,{\left (a^{2} b e^{2} x^{3} + a^{3} e^{2} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)/((b*x^2 + a)^(7/4)*(e*x)^(3/2)),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((d*x**2+c)/(e*x)**(3/2)/(b*x**2+a)**(7/4),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac{7}{4}} \left (e x\right )^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)/((b*x^2 + a)^(7/4)*(e*x)^(3/2)),x, algorithm="giac")
[Out]