Optimal. Leaf size=173 \[ \frac{3 a e^{5/2} (8 b c-7 a d) \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{32 b^{11/4}}-\frac{3 a e^{5/2} (8 b c-7 a d) \tanh ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{32 b^{11/4}}+\frac{e (e x)^{3/2} \sqrt [4]{a+b x^2} (8 b c-7 a d)}{16 b^2}+\frac{d (e x)^{7/2} \sqrt [4]{a+b x^2}}{4 b e} \]
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Rubi [A] time = 0.360219, antiderivative size = 173, 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 \[ \frac{3 a e^{5/2} (8 b c-7 a d) \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{32 b^{11/4}}-\frac{3 a e^{5/2} (8 b c-7 a d) \tanh ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{32 b^{11/4}}+\frac{e (e x)^{3/2} \sqrt [4]{a+b x^2} (8 b c-7 a d)}{16 b^2}+\frac{d (e x)^{7/2} \sqrt [4]{a+b x^2}}{4 b e} \]
Antiderivative was successfully verified.
[In] Int[((e*x)^(5/2)*(c + d*x^2))/(a + b*x^2)^(3/4),x]
[Out]
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Rubi in Sympy [A] time = 35.4988, size = 165, normalized size = 0.95 \[ - \frac{3 a e^{\frac{5}{2}} \left (7 a d - 8 b c\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a + b x^{2}}} \right )}}{32 b^{\frac{11}{4}}} + \frac{3 a e^{\frac{5}{2}} \left (7 a d - 8 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a + b x^{2}}} \right )}}{32 b^{\frac{11}{4}}} + \frac{d \left (e x\right )^{\frac{7}{2}} \sqrt [4]{a + b x^{2}}}{4 b e} - \frac{e \left (e x\right )^{\frac{3}{2}} \sqrt [4]{a + b x^{2}} \left (7 a d - 8 b c\right )}{16 b^{2}} \]
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)**(3/4),x)
[Out]
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Mathematica [C] time = 0.13174, size = 97, normalized size = 0.56 \[ \frac{e (e x)^{3/2} \left (a \left (\frac{b x^2}{a}+1\right )^{3/4} (7 a d-8 b c) \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^2}{a}\right )-\left (a+b x^2\right ) \left (7 a d-4 b \left (2 c+d x^2\right )\right )\right )}{16 b^2 \left (a+b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
[In] Integrate[((e*x)^(5/2)*(c + d*x^2))/(a + b*x^2)^(3/4),x]
[Out]
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Maple [F] time = 0.065, size = 0, normalized size = 0. \[ \int{(d{x}^{2}+c) \left ( ex \right ) ^{{\frac{5}{2}}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{4}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x)^(5/2)*(d*x^2+c)/(b*x^2+a)^(3/4),x)
[Out]
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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)^(3/4),x, algorithm="maxima")
[Out]
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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)^(3/4),x, algorithm="fricas")
[Out]
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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)**(3/4),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x^{2} + c\right )} \left (e x\right )^{\frac{5}{2}}}{{\left (b x^{2} + a\right )}^{\frac{3}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)*(e*x)^(5/2)/(b*x^2 + a)^(3/4),x, algorithm="giac")
[Out]