Optimal. Leaf size=184 \[ -\frac{e^{5/2} (4 b c-7 a d) \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{4 b^{11/4}}+\frac{e^{5/2} (4 b c-7 a d) \tanh ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{4 b^{11/4}}-\frac{e (e x)^{3/2} \sqrt [4]{a+b x^2} (4 b c-7 a d)}{6 a b^2}+\frac{2 (e x)^{7/2} (b c-a d)}{3 a b e \left (a+b x^2\right )^{3/4}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.372869, antiderivative size = 184, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ -\frac{e^{5/2} (4 b c-7 a d) \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{4 b^{11/4}}+\frac{e^{5/2} (4 b c-7 a d) \tanh ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{4 b^{11/4}}-\frac{e (e x)^{3/2} \sqrt [4]{a+b x^2} (4 b c-7 a d)}{6 a b^2}+\frac{2 (e x)^{7/2} (b c-a d)}{3 a b e \left (a+b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
[In] Int[((e*x)^(5/2)*(c + d*x^2))/(a + b*x^2)^(7/4),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 36.135, size = 158, normalized size = 0.86 \[ \frac{d \left (e x\right )^{\frac{7}{2}}}{2 b e \left (a + b x^{2}\right )^{\frac{3}{4}}} + \frac{e \left (e x\right )^{\frac{3}{2}} \left (7 a d - 4 b c\right )}{6 b^{2} \left (a + b x^{2}\right )^{\frac{3}{4}}} + \frac{e^{\frac{5}{2}} \left (7 a d - 4 b c\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a + b x^{2}}} \right )}}{4 b^{\frac{11}{4}}} - \frac{e^{\frac{5}{2}} \left (7 a d - 4 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a + b x^{2}}} \right )}}{4 b^{\frac{11}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x)**(5/2)*(d*x**2+c)/(b*x**2+a)**(7/4),x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 0.143481, size = 85, normalized size = 0.46 \[ \frac{e (e x)^{3/2} \left (\left (\frac{b x^2}{a}+1\right )^{3/4} (4 b c-7 a d) \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^2}{a}\right )+7 a d-4 b c+3 b d x^2\right )}{6 b^2 \left (a+b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
[In] Integrate[((e*x)^(5/2)*(c + d*x^2))/(a + b*x^2)^(7/4),x]
[Out]
_______________________________________________________________________________________
Maple [F] time = 0.084, size = 0, normalized size = 0. \[ \int{(d{x}^{2}+c) \left ( ex \right ) ^{{\frac{5}{2}}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{4}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x)^(5/2)*(d*x^2+c)/(b*x^2+a)^(7/4),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)*(e*x)^(5/2)/(b*x^2 + a)^(7/4),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [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)^(5/2)/(b*x^2 + a)^(7/4),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((e*x)**(5/2)*(d*x**2+c)/(b*x**2+a)**(7/4),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x^{2} + c\right )} \left (e x\right )^{\frac{5}{2}}}{{\left (b x^{2} + a\right )}^{\frac{7}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)*(e*x)^(5/2)/(b*x^2 + a)^(7/4),x, algorithm="giac")
[Out]