Optimal. Leaf size=142 \[ \frac{\sqrt{a} e^2 \sqrt{e x} \sqrt [4]{\frac{a}{b x^2}+1} (6 b c-7 a d) E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{2 b^{5/2} \sqrt [4]{a+b x^2}}+\frac{e (e x)^{3/2} (6 b c-7 a d)}{6 b^2 \sqrt [4]{a+b x^2}}+\frac{d (e x)^{7/2}}{3 b e \sqrt [4]{a+b x^2}} \]
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Rubi [A] time = 0.243145, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{\sqrt{a} e^2 \sqrt{e x} \sqrt [4]{\frac{a}{b x^2}+1} (6 b c-7 a d) E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{2 b^{5/2} \sqrt [4]{a+b x^2}}+\frac{e (e x)^{3/2} (6 b c-7 a d)}{6 b^2 \sqrt [4]{a+b x^2}}+\frac{d (e x)^{7/2}}{3 b e \sqrt [4]{a+b x^2}} \]
Antiderivative was successfully verified.
[In] Int[((e*x)^(5/2)*(c + d*x^2))/(a + b*x^2)^(5/4),x]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{a e^{2} \sqrt{e x} \left (7 a d - 6 b c\right ) \sqrt [4]{\frac{a}{b x^{2}} + 1} \int ^{\frac{1}{x}} \frac{1}{\sqrt [4]{\frac{a x^{2}}{b} + 1}}\, dx}{4 b^{3} \sqrt [4]{a + b x^{2}}} - \frac{a e^{2} \sqrt{e x} \left (7 a d - 6 b c\right )}{2 b^{3} x \sqrt [4]{a + b x^{2}}} + \frac{d \left (e x\right )^{\frac{7}{2}}}{3 b e \sqrt [4]{a + b x^{2}}} - \frac{e \left (e x\right )^{\frac{3}{2}} \left (7 a d - 6 b c\right )}{6 b^{2} \sqrt [4]{a + b x^{2}}} \]
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)**(5/4),x)
[Out]
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Mathematica [C] time = 0.125334, size = 84, normalized size = 0.59 \[ \frac{e (e x)^{3/2} \left (\sqrt [4]{\frac{b x^2}{a}+1} (6 b c-7 a d) \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^2}{a}\right )+7 a d-6 b c+b d x^2\right )}{3 b^2 \sqrt [4]{a+b x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[((e*x)^(5/2)*(c + d*x^2))/(a + b*x^2)^(5/4),x]
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Maple [F] time = 0.088, size = 0, normalized size = 0. \[ \int{(d{x}^{2}+c) \left ( ex \right ) ^{{\frac{5}{2}}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{4}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x)^(5/2)*(d*x^2+c)/(b*x^2+a)^(5/4),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x^{2} + c\right )} \left (e x\right )^{\frac{5}{2}}}{{\left (b x^{2} + a\right )}^{\frac{5}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)*(e*x)^(5/2)/(b*x^2 + a)^(5/4),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (d e^{2} x^{4} + c e^{2} x^{2}\right )} \sqrt{e x}}{{\left (b x^{2} + a\right )}^{\frac{5}{4}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)*(e*x)^(5/2)/(b*x^2 + a)^(5/4),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)**(5/4),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x^{2} + c\right )} \left (e x\right )^{\frac{5}{2}}}{{\left (b x^{2} + a\right )}^{\frac{5}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)*(e*x)^(5/2)/(b*x^2 + a)^(5/4),x, algorithm="giac")
[Out]