Optimal. Leaf size=192 \[ \frac{5 \sqrt{a} e^2 (e x)^{3/2} \left (\frac{a}{b x^2}+1\right )^{3/4} (2 b c-3 a d) F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{6 b^{5/2} \left (a+b x^2\right )^{3/4}}+\frac{5 e^3 \sqrt{e x} \sqrt [4]{a+b x^2} (2 b c-3 a d)}{6 b^3}-\frac{e (e x)^{5/2} \sqrt [4]{a+b x^2} (2 b c-3 a d)}{3 a b^2}+\frac{2 (e x)^{9/2} (b c-a d)}{3 a b e \left (a+b x^2\right )^{3/4}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.419147, antiderivative size = 192, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ \frac{5 \sqrt{a} e^2 (e x)^{3/2} \left (\frac{a}{b x^2}+1\right )^{3/4} (2 b c-3 a d) F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{6 b^{5/2} \left (a+b x^2\right )^{3/4}}+\frac{5 e^3 \sqrt{e x} \sqrt [4]{a+b x^2} (2 b c-3 a d)}{6 b^3}-\frac{e (e x)^{5/2} \sqrt [4]{a+b x^2} (2 b c-3 a d)}{3 a b^2}+\frac{2 (e x)^{9/2} (b c-a d)}{3 a b e \left (a+b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
[In] Int[((e*x)^(7/2)*(c + d*x^2))/(a + b*x^2)^(7/4),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 40.7811, size = 167, normalized size = 0.87 \[ - \frac{5 \sqrt{a} e^{2} \left (e x\right )^{\frac{3}{2}} \left (\frac{3 a d}{2} - b c\right ) \left (\frac{a}{b x^{2}} + 1\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{2}\middle | 2\right )}{3 b^{\frac{5}{2}} \left (a + b x^{2}\right )^{\frac{3}{4}}} + \frac{d \left (e x\right )^{\frac{9}{2}}}{3 b e \left (a + b x^{2}\right )^{\frac{3}{4}}} + \frac{e \left (e x\right )^{\frac{5}{2}} \left (3 a d - 2 b c\right )}{3 b^{2} \left (a + b x^{2}\right )^{\frac{3}{4}}} - \frac{5 e^{3} \sqrt{e x} \sqrt [4]{a + b x^{2}} \left (\frac{3 a d}{2} - b c\right )}{3 b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x)**(7/2)*(d*x**2+c)/(b*x**2+a)**(7/4),x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 0.145395, size = 110, normalized size = 0.57 \[ \frac{e^3 \sqrt{e x} \left (-15 a^2 d+5 a \left (\frac{b x^2}{a}+1\right )^{3/4} (3 a d-2 b c) \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};-\frac{b x^2}{a}\right )+a b \left (10 c-9 d x^2\right )+2 b^2 x^2 \left (3 c+d x^2\right )\right )}{6 b^3 \left (a+b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
[In] Integrate[((e*x)^(7/2)*(c + d*x^2))/(a + b*x^2)^(7/4),x]
[Out]
_______________________________________________________________________________________
Maple [F] time = 0.087, size = 0, normalized size = 0. \[ \int{(d{x}^{2}+c) \left ( ex \right ) ^{{\frac{7}{2}}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{4}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x)^(7/2)*(d*x^2+c)/(b*x^2+a)^(7/4),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x^{2} + c\right )} \left (e x\right )^{\frac{7}{2}}}{{\left (b x^{2} + a\right )}^{\frac{7}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)*(e*x)^(7/2)/(b*x^2 + a)^(7/4),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (d e^{3} x^{5} + c e^{3} x^{3}\right )} \sqrt{e x}}{{\left (b x^{2} + a\right )}^{\frac{7}{4}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)*(e*x)^(7/2)/(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)**(7/2)*(d*x**2+c)/(b*x**2+a)**(7/4),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x^{2} + c\right )} \left (e x\right )^{\frac{7}{2}}}{{\left (b x^{2} + a\right )}^{\frac{7}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)*(e*x)^(7/2)/(b*x^2 + a)^(7/4),x, algorithm="giac")
[Out]