Optimal. Leaf size=207 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \left (b e \left (-c^2 f^2-6 c d e f+15 d^2 e^2\right )-a f \left (3 c^2 f^2+2 c d e f+3 d^2 e^2\right )\right )}{8 e^{5/2} f^{7/2}}-\frac{x \left (c+d x^2\right ) (b e (5 d e-c f)-a f (3 c f+d e))}{8 e^2 f^2 \left (e+f x^2\right )}+\frac{d x (b e (15 d e-c f)-3 a f (c f+d e))}{8 e^2 f^3}-\frac{x \left (c+d x^2\right )^2 (b e-a f)}{4 e f \left (e+f x^2\right )^2} \]
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Rubi [A] time = 0.238958, antiderivative size = 207, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {526, 388, 205} \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \left (b e \left (-c^2 f^2-6 c d e f+15 d^2 e^2\right )-a f \left (3 c^2 f^2+2 c d e f+3 d^2 e^2\right )\right )}{8 e^{5/2} f^{7/2}}-\frac{x \left (c+d x^2\right ) (b e (5 d e-c f)-a f (3 c f+d e))}{8 e^2 f^2 \left (e+f x^2\right )}+\frac{d x (b e (15 d e-c f)-3 a f (c f+d e))}{8 e^2 f^3}-\frac{x \left (c+d x^2\right )^2 (b e-a f)}{4 e f \left (e+f x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 526
Rule 388
Rule 205
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right ) \left (c+d x^2\right )^2}{\left (e+f x^2\right )^3} \, dx &=-\frac{(b e-a f) x \left (c+d x^2\right )^2}{4 e f \left (e+f x^2\right )^2}-\frac{\int \frac{\left (c+d x^2\right ) \left (-c (b e+3 a f)-d (5 b e-a f) x^2\right )}{\left (e+f x^2\right )^2} \, dx}{4 e f}\\ &=-\frac{(b e-a f) x \left (c+d x^2\right )^2}{4 e f \left (e+f x^2\right )^2}-\frac{(b e (5 d e-c f)-a f (d e+3 c f)) x \left (c+d x^2\right )}{8 e^2 f^2 \left (e+f x^2\right )}+\frac{\int \frac{-c (a f (d e-3 c f)-b e (5 d e+c f))+d (b e (15 d e-c f)-3 a f (d e+c f)) x^2}{e+f x^2} \, dx}{8 e^2 f^2}\\ &=\frac{d (b e (15 d e-c f)-3 a f (d e+c f)) x}{8 e^2 f^3}-\frac{(b e-a f) x \left (c+d x^2\right )^2}{4 e f \left (e+f x^2\right )^2}-\frac{(b e (5 d e-c f)-a f (d e+3 c f)) x \left (c+d x^2\right )}{8 e^2 f^2 \left (e+f x^2\right )}-\frac{\left (b e \left (15 d^2 e^2-6 c d e f-c^2 f^2\right )-a f \left (3 d^2 e^2+2 c d e f+3 c^2 f^2\right )\right ) \int \frac{1}{e+f x^2} \, dx}{8 e^2 f^3}\\ &=\frac{d (b e (15 d e-c f)-3 a f (d e+c f)) x}{8 e^2 f^3}-\frac{(b e-a f) x \left (c+d x^2\right )^2}{4 e f \left (e+f x^2\right )^2}-\frac{(b e (5 d e-c f)-a f (d e+3 c f)) x \left (c+d x^2\right )}{8 e^2 f^2 \left (e+f x^2\right )}-\frac{\left (b e \left (15 d^2 e^2-6 c d e f-c^2 f^2\right )-a f \left (3 d^2 e^2+2 c d e f+3 c^2 f^2\right )\right ) \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{8 e^{5/2} f^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.127148, size = 183, normalized size = 0.88 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \left (b e \left (-c^2 f^2-6 c d e f+15 d^2 e^2\right )-a f \left (3 c^2 f^2+2 c d e f+3 d^2 e^2\right )\right )}{8 e^{5/2} f^{7/2}}+\frac{x (d e-c f) (b e (9 d e-c f)-a f (3 c f+5 d e))}{8 e^2 f^3 \left (e+f x^2\right )}-\frac{x (b e-a f) (d e-c f)^2}{4 e f^3 \left (e+f x^2\right )^2}+\frac{b d^2 x}{f^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.011, size = 397, normalized size = 1.9 \begin{align*}{\frac{b{d}^{2}x}{{f}^{3}}}+{\frac{3\,f{x}^{3}a{c}^{2}}{8\, \left ( f{x}^{2}+e \right ) ^{2}{e}^{2}}}+{\frac{{x}^{3}acd}{4\, \left ( f{x}^{2}+e \right ) ^{2}e}}-{\frac{5\,{x}^{3}a{d}^{2}}{8\,f \left ( f{x}^{2}+e \right ) ^{2}}}+{\frac{{x}^{3}b{c}^{2}}{8\, \left ( f{x}^{2}+e \right ) ^{2}e}}-{\frac{5\,{x}^{3}bcd}{4\,f \left ( f{x}^{2}+e \right ) ^{2}}}+{\frac{9\,{x}^{3}b{d}^{2}e}{8\,{f}^{2} \left ( f{x}^{2}+e \right ) ^{2}}}+{\frac{5\,ax{c}^{2}}{8\, \left ( f{x}^{2}+e \right ) ^{2}e}}-{\frac{acdx}{4\,f \left ( f{x}^{2}+e \right ) ^{2}}}-{\frac{3\,a{d}^{2}ex}{8\,{f}^{2} \left ( f{x}^{2}+e \right ) ^{2}}}-{\frac{b{c}^{2}x}{8\,f \left ( f{x}^{2}+e \right ) ^{2}}}-{\frac{3\,bcdex}{4\,{f}^{2} \left ( f{x}^{2}+e \right ) ^{2}}}+{\frac{7\,b{d}^{2}{e}^{2}x}{8\,{f}^{3} \left ( f{x}^{2}+e \right ) ^{2}}}+{\frac{3\,a{c}^{2}}{8\,{e}^{2}}\arctan \left ({fx{\frac{1}{\sqrt{ef}}}} \right ){\frac{1}{\sqrt{ef}}}}+{\frac{acd}{4\,ef}\arctan \left ({fx{\frac{1}{\sqrt{ef}}}} \right ){\frac{1}{\sqrt{ef}}}}+{\frac{3\,a{d}^{2}}{8\,{f}^{2}}\arctan \left ({fx{\frac{1}{\sqrt{ef}}}} \right ){\frac{1}{\sqrt{ef}}}}+{\frac{b{c}^{2}}{8\,ef}\arctan \left ({fx{\frac{1}{\sqrt{ef}}}} \right ){\frac{1}{\sqrt{ef}}}}+{\frac{3\,bcd}{4\,{f}^{2}}\arctan \left ({fx{\frac{1}{\sqrt{ef}}}} \right ){\frac{1}{\sqrt{ef}}}}-{\frac{15\,b{d}^{2}e}{8\,{f}^{3}}\arctan \left ({fx{\frac{1}{\sqrt{ef}}}} \right ){\frac{1}{\sqrt{ef}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.5857, size = 1613, normalized size = 7.79 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 18.9233, size = 400, normalized size = 1.93 \begin{align*} \frac{b d^{2} x}{f^{3}} - \frac{\sqrt{- \frac{1}{e^{5} f^{7}}} \left (3 a c^{2} f^{3} + 2 a c d e f^{2} + 3 a d^{2} e^{2} f + b c^{2} e f^{2} + 6 b c d e^{2} f - 15 b d^{2} e^{3}\right ) \log{\left (- e^{3} f^{3} \sqrt{- \frac{1}{e^{5} f^{7}}} + x \right )}}{16} + \frac{\sqrt{- \frac{1}{e^{5} f^{7}}} \left (3 a c^{2} f^{3} + 2 a c d e f^{2} + 3 a d^{2} e^{2} f + b c^{2} e f^{2} + 6 b c d e^{2} f - 15 b d^{2} e^{3}\right ) \log{\left (e^{3} f^{3} \sqrt{- \frac{1}{e^{5} f^{7}}} + x \right )}}{16} + \frac{x^{3} \left (3 a c^{2} f^{4} + 2 a c d e f^{3} - 5 a d^{2} e^{2} f^{2} + b c^{2} e f^{3} - 10 b c d e^{2} f^{2} + 9 b d^{2} e^{3} f\right ) + x \left (5 a c^{2} e f^{3} - 2 a c d e^{2} f^{2} - 3 a d^{2} e^{3} f - b c^{2} e^{2} f^{2} - 6 b c d e^{3} f + 7 b d^{2} e^{4}\right )}{8 e^{4} f^{3} + 16 e^{3} f^{4} x^{2} + 8 e^{2} f^{5} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19987, size = 321, normalized size = 1.55 \begin{align*} \frac{b d^{2} x}{f^{3}} + \frac{{\left (3 \, a c^{2} f^{3} + b c^{2} f^{2} e + 2 \, a c d f^{2} e + 6 \, b c d f e^{2} + 3 \, a d^{2} f e^{2} - 15 \, b d^{2} e^{3}\right )} \arctan \left (\sqrt{f} x e^{\left (-\frac{1}{2}\right )}\right ) e^{\left (-\frac{5}{2}\right )}}{8 \, f^{\frac{7}{2}}} + \frac{{\left (3 \, a c^{2} f^{4} x^{3} + b c^{2} f^{3} x^{3} e + 2 \, a c d f^{3} x^{3} e - 10 \, b c d f^{2} x^{3} e^{2} - 5 \, a d^{2} f^{2} x^{3} e^{2} + 9 \, b d^{2} f x^{3} e^{3} + 5 \, a c^{2} f^{3} x e - b c^{2} f^{2} x e^{2} - 2 \, a c d f^{2} x e^{2} - 6 \, b c d f x e^{3} - 3 \, a d^{2} f x e^{3} + 7 \, b d^{2} x e^{4}\right )} e^{\left (-2\right )}}{8 \,{\left (f x^{2} + e\right )}^{2} f^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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