Optimal. Leaf size=114 \[ \frac{2 (e x)^{3/2} (b c-a d)}{5 a b e \left (a+b x^2\right )^{5/4}}-\frac{2 \sqrt{e x} \sqrt [4]{\frac{a}{b x^2}+1} (3 a d+2 b c) E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{5 a^{3/2} b^{3/2} \sqrt [4]{a+b x^2}} \]
[Out]
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Rubi [A] time = 0.185584, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{2 (e x)^{3/2} (b c-a d)}{5 a b e \left (a+b x^2\right )^{5/4}}-\frac{2 \sqrt{e x} \sqrt [4]{\frac{a}{b x^2}+1} (3 a d+2 b c) E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{5 a^{3/2} b^{3/2} \sqrt [4]{a+b x^2}} \]
Antiderivative was successfully verified.
[In] Int[(Sqrt[e*x]*(c + d*x^2))/(a + b*x^2)^(9/4),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{d \left (e x\right )^{\frac{3}{2}}}{b e \left (a + b x^{2}\right )^{\frac{5}{4}}} + \frac{2 \left (e x\right )^{\frac{3}{2}} \left (\frac{3 a d}{2} + b c\right )}{5 a b e \left (a + b x^{2}\right )^{\frac{5}{4}}} + \frac{2 \sqrt{e x} \left (\frac{3 a d}{2} + b c\right ) \sqrt [4]{\frac{a}{b x^{2}} + 1} \int ^{\frac{1}{x}} \frac{1}{\sqrt [4]{\frac{a x^{2}}{b} + 1}}\, dx}{5 a b^{2} \sqrt [4]{a + b x^{2}}} - \frac{4 \sqrt{e x} \left (\frac{3 a d}{2} + b c\right )}{5 a b^{2} x \sqrt [4]{a + b x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x)**(1/2)*(d*x**2+c)/(b*x**2+a)**(9/4),x)
[Out]
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Mathematica [C] time = 0.157829, size = 111, normalized size = 0.97 \[ -\frac{2 \sqrt{e x} \left (2 x \left (a+b x^2\right ) \sqrt [4]{\frac{b x^2}{a}+1} (3 a d+2 b c) \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^2}{a}\right )-3 x \left (2 a^2 d+3 a b \left (c+d x^2\right )+2 b^2 c x^2\right )\right )}{15 a^2 b \left (a+b x^2\right )^{5/4}} \]
Antiderivative was successfully verified.
[In] Integrate[(Sqrt[e*x]*(c + d*x^2))/(a + b*x^2)^(9/4),x]
[Out]
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Maple [F] time = 0.05, size = 0, normalized size = 0. \[ \int{(d{x}^{2}+c)\sqrt{ex} \left ( b{x}^{2}+a \right ) ^{-{\frac{9}{4}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x)^(1/2)*(d*x^2+c)/(b*x^2+a)^(9/4),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x^{2} + c\right )} \sqrt{e x}}{{\left (b x^{2} + a\right )}^{\frac{9}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)*sqrt(e*x)/(b*x^2 + a)^(9/4),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (d x^{2} + c\right )} \sqrt{e x}}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}{\left (b x^{2} + a\right )}^{\frac{1}{4}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)*sqrt(e*x)/(b*x^2 + a)^(9/4),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)**(1/2)*(d*x**2+c)/(b*x**2+a)**(9/4),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x^{2} + c\right )} \sqrt{e x}}{{\left (b x^{2} + a\right )}^{\frac{9}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)*sqrt(e*x)/(b*x^2 + a)^(9/4),x, algorithm="giac")
[Out]