Optimal. Leaf size=149 \[ \frac{d e^{3/2} \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{b^{9/4}}+\frac{d e^{3/2} \tanh ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{b^{9/4}}-\frac{2 d e \sqrt{e x}}{b^2 \sqrt [4]{a+b x^2}}+\frac{2 (e x)^{5/2} (b c-a d)}{5 a b e \left (a+b x^2\right )^{5/4}} \]
[Out]
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Rubi [A] time = 0.247068, antiderivative size = 149, 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 \[ \frac{d e^{3/2} \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{b^{9/4}}+\frac{d e^{3/2} \tanh ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{b^{9/4}}-\frac{2 d e \sqrt{e x}}{b^2 \sqrt [4]{a+b x^2}}+\frac{2 (e x)^{5/2} (b c-a d)}{5 a b e \left (a+b x^2\right )^{5/4}} \]
Antiderivative was successfully verified.
[In] Int[((e*x)^(3/2)*(c + d*x^2))/(a + b*x^2)^(9/4),x]
[Out]
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Rubi in Sympy [A] time = 33.3222, size = 138, normalized size = 0.93 \[ - \frac{2 d e \sqrt{e x}}{b^{2} \sqrt [4]{a + b x^{2}}} + \frac{d e^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a + b x^{2}}} \right )}}{b^{\frac{9}{4}}} + \frac{d e^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a + b x^{2}}} \right )}}{b^{\frac{9}{4}}} - \frac{2 \left (e x\right )^{\frac{5}{2}} \left (a d - b c\right )}{5 a b e \left (a + b x^{2}\right )^{\frac{5}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x)**(3/2)*(d*x**2+c)/(b*x**2+a)**(9/4),x)
[Out]
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Mathematica [C] time = 0.127835, size = 96, normalized size = 0.64 \[ \frac{2 e \sqrt{e x} \left (-5 a^2 d+5 a d \left (a+b x^2\right ) \sqrt [4]{\frac{b x^2}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{4};\frac{5}{4};-\frac{b x^2}{a}\right )-6 a b d x^2+b^2 c x^2\right )}{5 a b^2 \left (a+b x^2\right )^{5/4}} \]
Antiderivative was successfully verified.
[In] Integrate[((e*x)^(3/2)*(c + d*x^2))/(a + b*x^2)^(9/4),x]
[Out]
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Maple [F] time = 0.053, size = 0, normalized size = 0. \[ \int{(d{x}^{2}+c) \left ( ex \right ) ^{{\frac{3}{2}}} \left ( b{x}^{2}+a \right ) ^{-{\frac{9}{4}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x)^(3/2)*(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. \[ \int \frac{{\left (d x^{2} + c\right )} \left (e x\right )^{\frac{3}{2}}}{{\left (b x^{2} + a\right )}^{\frac{9}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)*(e*x)^(3/2)/(b*x^2 + a)^(9/4),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.271478, size = 564, normalized size = 3.79 \[ -\frac{4 \,{\left (5 \, a^{2} d e -{\left (b^{2} c - 6 \, a b d\right )} e x^{2}\right )}{\left (b x^{2} + a\right )}^{\frac{3}{4}} \sqrt{e x} + 20 \,{\left (a b^{4} x^{4} + 2 \, a^{2} b^{3} x^{2} + a^{3} b^{2}\right )} \left (\frac{d^{4} e^{6}}{b^{9}}\right )^{\frac{1}{4}} \arctan \left (\frac{{\left (b^{3} x^{2} + a b^{2}\right )} \left (\frac{d^{4} e^{6}}{b^{9}}\right )^{\frac{1}{4}}}{{\left (b x^{2} + a\right )}^{\frac{3}{4}} \sqrt{e x} d e +{\left (b x^{2} + a\right )} \sqrt{\frac{\sqrt{b x^{2} + a} d^{2} e^{3} x +{\left (b^{5} x^{2} + a b^{4}\right )} \sqrt{\frac{d^{4} e^{6}}{b^{9}}}}{b x^{2} + a}}}\right ) - 5 \,{\left (a b^{4} x^{4} + 2 \, a^{2} b^{3} x^{2} + a^{3} b^{2}\right )} \left (\frac{d^{4} e^{6}}{b^{9}}\right )^{\frac{1}{4}} \log \left (\frac{{\left (b x^{2} + a\right )}^{\frac{3}{4}} \sqrt{e x} d e +{\left (b^{3} x^{2} + a b^{2}\right )} \left (\frac{d^{4} e^{6}}{b^{9}}\right )^{\frac{1}{4}}}{b x^{2} + a}\right ) + 5 \,{\left (a b^{4} x^{4} + 2 \, a^{2} b^{3} x^{2} + a^{3} b^{2}\right )} \left (\frac{d^{4} e^{6}}{b^{9}}\right )^{\frac{1}{4}} \log \left (\frac{{\left (b x^{2} + a\right )}^{\frac{3}{4}} \sqrt{e x} d e -{\left (b^{3} x^{2} + a b^{2}\right )} \left (\frac{d^{4} e^{6}}{b^{9}}\right )^{\frac{1}{4}}}{b x^{2} + a}\right )}{10 \,{\left (a b^{4} x^{4} + 2 \, a^{2} b^{3} x^{2} + a^{3} b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)*(e*x)^(3/2)/(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)**(3/2)*(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. \[ \int \frac{{\left (d x^{2} + c\right )} \left (e x\right )^{\frac{3}{2}}}{{\left (b x^{2} + a\right )}^{\frac{9}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)*(e*x)^(3/2)/(b*x^2 + a)^(9/4),x, algorithm="giac")
[Out]