Optimal. Leaf size=230 \[ -\frac{77 a^{3/2} e^6 \sqrt{e x} \sqrt [4]{\frac{a}{b x^2}+1} (2 b c-3 a d) E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{20 b^{9/2} \sqrt [4]{a+b x^2}}-\frac{77 a e^5 (e x)^{3/2} (2 b c-3 a d)}{60 b^4 \sqrt [4]{a+b x^2}}+\frac{11 e^3 (e x)^{7/2} (2 b c-3 a d)}{30 b^3 \sqrt [4]{a+b x^2}}-\frac{e (e x)^{11/2} (2 b c-3 a d)}{5 a b^2 \sqrt [4]{a+b x^2}}+\frac{2 (e x)^{15/2} (b c-a d)}{5 a b e \left (a+b x^2\right )^{5/4}} \]
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Rubi [A] time = 0.404226, antiderivative size = 230, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ -\frac{77 a^{3/2} e^6 \sqrt{e x} \sqrt [4]{\frac{a}{b x^2}+1} (2 b c-3 a d) E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{20 b^{9/2} \sqrt [4]{a+b x^2}}-\frac{77 a e^5 (e x)^{3/2} (2 b c-3 a d)}{60 b^4 \sqrt [4]{a+b x^2}}+\frac{11 e^3 (e x)^{7/2} (2 b c-3 a d)}{30 b^3 \sqrt [4]{a+b x^2}}-\frac{e (e x)^{11/2} (2 b c-3 a d)}{5 a b^2 \sqrt [4]{a+b x^2}}+\frac{2 (e x)^{15/2} (b c-a d)}{5 a b e \left (a+b x^2\right )^{5/4}} \]
Antiderivative was successfully verified.
[In] Int[((e*x)^(13/2)*(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. \[ - \frac{77 a^{2} e^{6} \sqrt{e x} \left (\frac{3 a d}{2} - b c\right ) \sqrt [4]{\frac{a}{b x^{2}} + 1} \int ^{\frac{1}{x}} \frac{1}{\sqrt [4]{\frac{a x^{2}}{b} + 1}}\, dx}{20 b^{5} \sqrt [4]{a + b x^{2}}} + \frac{77 a^{2} e^{6} \sqrt{e x} \left (\frac{3 a d}{2} - b c\right )}{10 b^{5} x \sqrt [4]{a + b x^{2}}} + \frac{77 a e^{5} \left (e x\right )^{\frac{3}{2}} \left (3 a d - 2 b c\right )}{60 b^{4} \sqrt [4]{a + b x^{2}}} + \frac{d \left (e x\right )^{\frac{15}{2}}}{5 b e \left (a + b x^{2}\right )^{\frac{5}{4}}} + \frac{e \left (e x\right )^{\frac{11}{2}} \left (3 a d - 2 b c\right )}{5 b^{2} \left (a + b x^{2}\right )^{\frac{5}{4}}} - \frac{11 e^{3} \left (e x\right )^{\frac{7}{2}} \left (\frac{3 a d}{2} - b c\right )}{15 b^{3} \sqrt [4]{a + b x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x)**(13/2)*(d*x**2+c)/(b*x**2+a)**(9/4),x)
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Mathematica [C] time = 0.232949, size = 139, normalized size = 0.6 \[ \frac{e^5 (e x)^{3/2} \left (-231 a^3 d+22 a^2 b \left (7 c-12 d x^2\right )+a b^2 x^2 \left (176 c-15 d x^2\right )+77 a \left (a+b x^2\right ) \sqrt [4]{\frac{b x^2}{a}+1} (3 a d-2 b c) \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^2}{a}\right )+2 b^3 x^4 \left (5 c+3 d x^2\right )\right )}{30 b^4 \left (a+b x^2\right )^{5/4}} \]
Antiderivative was successfully verified.
[In] Integrate[((e*x)^(13/2)*(c + d*x^2))/(a + b*x^2)^(9/4),x]
[Out]
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Maple [F] time = 0.104, size = 0, normalized size = 0. \[ \int{(d{x}^{2}+c) \left ( ex \right ) ^{{\frac{13}{2}}} \left ( b{x}^{2}+a \right ) ^{-{\frac{9}{4}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x)^(13/2)*(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. \[ \int \frac{{\left (d x^{2} + c\right )} \left (e x\right )^{\frac{13}{2}}}{{\left (b x^{2} + a\right )}^{\frac{9}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)*(e*x)^(13/2)/(b*x^2 + a)^(9/4),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (d e^{6} x^{8} + c e^{6} x^{6}\right )} \sqrt{e x}}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}{\left (b x^{2} + a\right )}^{\frac{1}{4}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)*(e*x)^(13/2)/(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)**(13/2)*(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. \[ \int \frac{{\left (d x^{2} + c\right )} \left (e x\right )^{\frac{13}{2}}}{{\left (b x^{2} + a\right )}^{\frac{9}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)*(e*x)^(13/2)/(b*x^2 + a)^(9/4),x, algorithm="giac")
[Out]