Optimal. Leaf size=221 \[ \frac{e^{7/2} (4 b c-9 a d) \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{4 b^{13/4}}+\frac{e^{7/2} (4 b c-9 a d) \tanh ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{4 b^{13/4}}-\frac{e^3 \sqrt{e x} (4 b c-9 a d)}{2 b^3 \sqrt [4]{a+b x^2}}-\frac{e (e x)^{5/2} (4 b c-9 a d)}{10 a b^2 \sqrt [4]{a+b x^2}}+\frac{2 (e x)^{9/2} (b c-a d)}{5 a b e \left (a+b x^2\right )^{5/4}} \]
[Out]
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Rubi [A] time = 0.391884, antiderivative size = 221, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{e^{7/2} (4 b c-9 a d) \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{4 b^{13/4}}+\frac{e^{7/2} (4 b c-9 a d) \tanh ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{4 b^{13/4}}-\frac{e^3 \sqrt{e x} (4 b c-9 a d)}{2 b^3 \sqrt [4]{a+b x^2}}-\frac{e (e x)^{5/2} (4 b c-9 a d)}{10 a b^2 \sqrt [4]{a+b x^2}}+\frac{2 (e x)^{9/2} (b c-a d)}{5 a b e \left (a+b x^2\right )^{5/4}} \]
Antiderivative was successfully verified.
[In] Int[((e*x)^(7/2)*(c + d*x^2))/(a + b*x^2)^(9/4),x]
[Out]
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Rubi in Sympy [A] time = 41.5549, size = 194, normalized size = 0.88 \[ \frac{d \left (e x\right )^{\frac{9}{2}}}{2 b e \left (a + b x^{2}\right )^{\frac{5}{4}}} + \frac{e \left (e x\right )^{\frac{5}{2}} \left (9 a d - 4 b c\right )}{10 b^{2} \left (a + b x^{2}\right )^{\frac{5}{4}}} + \frac{e^{3} \sqrt{e x} \left (9 a d - 4 b c\right )}{2 b^{3} \sqrt [4]{a + b x^{2}}} - \frac{e^{\frac{7}{2}} \left (9 a d - 4 b c\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a + b x^{2}}} \right )}}{4 b^{\frac{13}{4}}} - \frac{e^{\frac{7}{2}} \left (9 a d - 4 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a + b x^{2}}} \right )}}{4 b^{\frac{13}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x)**(7/2)*(d*x**2+c)/(b*x**2+a)**(9/4),x)
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Mathematica [C] time = 0.186947, size = 116, normalized size = 0.52 \[ \frac{e^3 \sqrt{e x} \left (45 a^2 d+5 \left (a+b x^2\right ) \sqrt [4]{\frac{b x^2}{a}+1} (4 b c-9 a d) \, _2F_1\left (\frac{1}{4},\frac{1}{4};\frac{5}{4};-\frac{b x^2}{a}\right )+a b \left (54 d x^2-20 c\right )+b^2 x^2 \left (5 d x^2-24 c\right )\right )}{10 b^3 \left (a+b x^2\right )^{5/4}} \]
Antiderivative was successfully verified.
[In] Integrate[((e*x)^(7/2)*(c + d*x^2))/(a + b*x^2)^(9/4),x]
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Maple [F] time = 0.103, size = 0, normalized size = 0. \[ \int{(d{x}^{2}+c) \left ( ex \right ) ^{{\frac{7}{2}}} \left ( b{x}^{2}+a \right ) ^{-{\frac{9}{4}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x)^(7/2)*(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. \[ \int \frac{{\left (d x^{2} + c\right )} \left (e x\right )^{\frac{7}{2}}}{{\left (b x^{2} + a\right )}^{\frac{9}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)*(e*x)^(7/2)/(b*x^2 + a)^(9/4),x, algorithm="maxima")
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Fricas [A] time = 0.284408, size = 1108, normalized size = 5.01 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)*(e*x)^(7/2)/(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)**(7/2)*(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. \[ \int \frac{{\left (d x^{2} + c\right )} \left (e x\right )^{\frac{7}{2}}}{{\left (b x^{2} + a\right )}^{\frac{9}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)*(e*x)^(7/2)/(b*x^2 + a)^(9/4),x, algorithm="giac")
[Out]