Optimal. Leaf size=164 \[ \frac {(d e-c f) \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) (b e (5 d e-c f)-a f (c f+3 d e))}{2 e^{3/2} f^{7/2}}-\frac {d x (b e (15 d e-13 c f)-3 a f (3 d e-c f))}{6 e f^3}+\frac {d x \left (c+d x^2\right ) (5 b e-3 a f)}{6 e f^2}-\frac {x \left (c+d x^2\right )^2 (b e-a f)}{2 e f \left (e+f x^2\right )} \]
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Rubi [A] time = 0.23, antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {526, 528, 388, 205} \[ \frac {(d e-c f) \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) (b e (5 d e-c f)-a f (c f+3 d e))}{2 e^{3/2} f^{7/2}}+\frac {d x \left (c+d x^2\right ) (5 b e-3 a f)}{6 e f^2}-\frac {d x (b e (15 d e-13 c f)-3 a f (3 d e-c f))}{6 e f^3}-\frac {x \left (c+d x^2\right )^2 (b e-a f)}{2 e f \left (e+f x^2\right )} \]
Antiderivative was successfully verified.
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Rule 205
Rule 388
Rule 526
Rule 528
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 )^2} \, dx &=-\frac {(b e-a f) x \left (c+d x^2\right )^2}{2 e f \left (e+f x^2\right )}-\frac {\int \frac {\left (c+d x^2\right ) \left (-c (b e+a f)-d (5 b e-3 a f) x^2\right )}{e+f x^2} \, dx}{2 e f}\\ &=\frac {d (5 b e-3 a f) x \left (c+d x^2\right )}{6 e f^2}-\frac {(b e-a f) x \left (c+d x^2\right )^2}{2 e f \left (e+f x^2\right )}-\frac {\int \frac {c (b e (5 d e-3 c f)-3 a f (d e+c f))+d (b e (15 d e-13 c f)-3 a f (3 d e-c f)) x^2}{e+f x^2} \, dx}{6 e f^2}\\ &=-\frac {d (b e (15 d e-13 c f)-3 a f (3 d e-c f)) x}{6 e f^3}+\frac {d (5 b e-3 a f) x \left (c+d x^2\right )}{6 e f^2}-\frac {(b e-a f) x \left (c+d x^2\right )^2}{2 e f \left (e+f x^2\right )}+\frac {((d e-c f) (b e (5 d e-c f)-a f (3 d e+c f))) \int \frac {1}{e+f x^2} \, dx}{2 e f^3}\\ &=-\frac {d (b e (15 d e-13 c f)-3 a f (3 d e-c f)) x}{6 e f^3}+\frac {d (5 b e-3 a f) x \left (c+d x^2\right )}{6 e f^2}-\frac {(b e-a f) x \left (c+d x^2\right )^2}{2 e f \left (e+f x^2\right )}+\frac {(d e-c f) (b e (5 d e-c f)-a f (3 d e+c f)) \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{2 e^{3/2} f^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 134, normalized size = 0.82 \[ \frac {(d e-c f) \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) (b e (5 d e-c f)-a f (c f+3 d e))}{2 e^{3/2} f^{7/2}}-\frac {x (b e-a f) (d e-c f)^2}{2 e f^3 \left (e+f x^2\right )}+\frac {d x (a d f+2 b c f-2 b d e)}{f^3}+\frac {b d^2 x^3}{3 f^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 552, normalized size = 3.37 \[ \left [\frac {4 \, b d^{2} e^{2} f^{3} x^{5} - 4 \, {\left (5 \, b d^{2} e^{3} f^{2} - 3 \, {\left (2 \, b c d + a d^{2}\right )} e^{2} f^{3}\right )} x^{3} - 3 \, {\left (5 \, b d^{2} e^{4} + a c^{2} e f^{3} - 3 \, {\left (2 \, b c d + a d^{2}\right )} e^{3} f + {\left (b c^{2} + 2 \, a c d\right )} e^{2} f^{2} + {\left (5 \, b d^{2} e^{3} f + a c^{2} f^{4} - 3 \, {\left (2 \, b c d + a d^{2}\right )} e^{2} f^{2} + {\left (b c^{2} + 2 \, a c d\right )} e f^{3}\right )} x^{2}\right )} \sqrt {-e f} \log \left (\frac {f x^{2} - 2 \, \sqrt {-e f} x - e}{f x^{2} + e}\right ) - 6 \, {\left (5 \, b d^{2} e^{4} f - a c^{2} e f^{4} - 3 \, {\left (2 \, b c d + a d^{2}\right )} e^{3} f^{2} + {\left (b c^{2} + 2 \, a c d\right )} e^{2} f^{3}\right )} x}{12 \, {\left (e^{2} f^{5} x^{2} + e^{3} f^{4}\right )}}, \frac {2 \, b d^{2} e^{2} f^{3} x^{5} - 2 \, {\left (5 \, b d^{2} e^{3} f^{2} - 3 \, {\left (2 \, b c d + a d^{2}\right )} e^{2} f^{3}\right )} x^{3} + 3 \, {\left (5 \, b d^{2} e^{4} + a c^{2} e f^{3} - 3 \, {\left (2 \, b c d + a d^{2}\right )} e^{3} f + {\left (b c^{2} + 2 \, a c d\right )} e^{2} f^{2} + {\left (5 \, b d^{2} e^{3} f + a c^{2} f^{4} - 3 \, {\left (2 \, b c d + a d^{2}\right )} e^{2} f^{2} + {\left (b c^{2} + 2 \, a c d\right )} e f^{3}\right )} x^{2}\right )} \sqrt {e f} \arctan \left (\frac {\sqrt {e f} x}{e}\right ) - 3 \, {\left (5 \, b d^{2} e^{4} f - a c^{2} e f^{4} - 3 \, {\left (2 \, b c d + a d^{2}\right )} e^{3} f^{2} + {\left (b c^{2} + 2 \, a c d\right )} e^{2} f^{3}\right )} x}{6 \, {\left (e^{2} f^{5} x^{2} + e^{3} f^{4}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 195, normalized size = 1.19 \[ \frac {{\left (a c^{2} f^{3} + b c^{2} f^{2} e + 2 \, a c d f^{2} e - 6 \, b c d f e^{2} - 3 \, a d^{2} f e^{2} + 5 \, b d^{2} e^{3}\right )} \arctan \left (\sqrt {f} x e^{\left (-\frac {1}{2}\right )}\right ) e^{\left (-\frac {3}{2}\right )}}{2 \, f^{\frac {7}{2}}} + \frac {{\left (a c^{2} f^{3} x - b c^{2} f^{2} x e - 2 \, a c d f^{2} x e + 2 \, b c d f x e^{2} + a d^{2} f x e^{2} - b d^{2} x e^{3}\right )} e^{\left (-1\right )}}{2 \, {\left (f x^{2} + e\right )} f^{3}} + \frac {b d^{2} f^{4} x^{3} + 6 \, b c d f^{4} x + 3 \, a d^{2} f^{4} x - 6 \, b d^{2} f^{3} x e}{3 \, f^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 299, normalized size = 1.82 \[ \frac {b \,d^{2} x^{3}}{3 f^{2}}+\frac {a \,c^{2} x}{2 \left (f \,x^{2}+e \right ) e}+\frac {a \,c^{2} \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{2 \sqrt {e f}\, e}-\frac {a c d x}{\left (f \,x^{2}+e \right ) f}+\frac {a c d \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{\sqrt {e f}\, f}+\frac {a \,d^{2} e x}{2 \left (f \,x^{2}+e \right ) f^{2}}-\frac {3 a \,d^{2} e \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{2 \sqrt {e f}\, f^{2}}-\frac {b \,c^{2} x}{2 \left (f \,x^{2}+e \right ) f}+\frac {b \,c^{2} \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{2 \sqrt {e f}\, f}+\frac {b c d e x}{\left (f \,x^{2}+e \right ) f^{2}}-\frac {3 b c d e \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{\sqrt {e f}\, f^{2}}-\frac {b \,d^{2} e^{2} x}{2 \left (f \,x^{2}+e \right ) f^{3}}+\frac {5 b \,d^{2} e^{2} \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{2 \sqrt {e f}\, f^{3}}+\frac {a \,d^{2} x}{f^{2}}+\frac {2 b c d x}{f^{2}}-\frac {2 b \,d^{2} e x}{f^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.94, size = 186, normalized size = 1.13 \[ -\frac {{\left (b d^{2} e^{3} - a c^{2} f^{3} - {\left (2 \, b c d + a d^{2}\right )} e^{2} f + {\left (b c^{2} + 2 \, a c d\right )} e f^{2}\right )} x}{2 \, {\left (e f^{4} x^{2} + e^{2} f^{3}\right )}} + \frac {b d^{2} f x^{3} - 3 \, {\left (2 \, b d^{2} e - {\left (2 \, b c d + a d^{2}\right )} f\right )} x}{3 \, f^{3}} + \frac {{\left (5 \, b d^{2} e^{3} + a c^{2} f^{3} - 3 \, {\left (2 \, b c d + a d^{2}\right )} e^{2} f + {\left (b c^{2} + 2 \, a c d\right )} e f^{2}\right )} \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{2 \, \sqrt {e f} e f^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.17, size = 257, normalized size = 1.57 \[ x\,\left (\frac {a\,d^2+2\,b\,c\,d}{f^2}-\frac {2\,b\,d^2\,e}{f^3}\right )+\frac {b\,d^2\,x^3}{3\,f^2}+\frac {x\,\left (-b\,c^2\,e\,f^2+a\,c^2\,f^3+2\,b\,c\,d\,e^2\,f-2\,a\,c\,d\,e\,f^2-b\,d^2\,e^3+a\,d^2\,e^2\,f\right )}{2\,e\,\left (f^4\,x^2+e\,f^3\right )}+\frac {\mathrm {atan}\left (\frac {\sqrt {f}\,x\,\left (c\,f-d\,e\right )\,\left (a\,c\,f^2-5\,b\,d\,e^2+3\,a\,d\,e\,f+b\,c\,e\,f\right )}{\sqrt {e}\,\left (b\,c^2\,e\,f^2+a\,c^2\,f^3-6\,b\,c\,d\,e^2\,f+2\,a\,c\,d\,e\,f^2+5\,b\,d^2\,e^3-3\,a\,d^2\,e^2\,f\right )}\right )\,\left (c\,f-d\,e\right )\,\left (a\,c\,f^2-5\,b\,d\,e^2+3\,a\,d\,e\,f+b\,c\,e\,f\right )}{2\,e^{3/2}\,f^{7/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.83, size = 483, normalized size = 2.95 \[ \frac {b d^{2} x^{3}}{3 f^{2}} + x \left (\frac {a d^{2}}{f^{2}} + \frac {2 b c d}{f^{2}} - \frac {2 b d^{2} e}{f^{3}}\right ) + \frac {x \left (a c^{2} f^{3} - 2 a c d e f^{2} + a d^{2} e^{2} f - b c^{2} e f^{2} + 2 b c d e^{2} f - b d^{2} e^{3}\right )}{2 e^{2} f^{3} + 2 e f^{4} x^{2}} - \frac {\sqrt {- \frac {1}{e^{3} f^{7}}} \left (c f - d e\right ) \left (a c f^{2} + 3 a d e f + b c e f - 5 b d e^{2}\right ) \log {\left (- \frac {e^{2} f^{3} \sqrt {- \frac {1}{e^{3} f^{7}}} \left (c f - d e\right ) \left (a c f^{2} + 3 a d e f + b c e f - 5 b d e^{2}\right )}{a c^{2} f^{3} + 2 a c d e f^{2} - 3 a d^{2} e^{2} f + b c^{2} e f^{2} - 6 b c d e^{2} f + 5 b d^{2} e^{3}} + x \right )}}{4} + \frac {\sqrt {- \frac {1}{e^{3} f^{7}}} \left (c f - d e\right ) \left (a c f^{2} + 3 a d e f + b c e f - 5 b d e^{2}\right ) \log {\left (\frac {e^{2} f^{3} \sqrt {- \frac {1}{e^{3} f^{7}}} \left (c f - d e\right ) \left (a c f^{2} + 3 a d e f + b c e f - 5 b d e^{2}\right )}{a c^{2} f^{3} + 2 a c d e f^{2} - 3 a d^{2} e^{2} f + b c^{2} e f^{2} - 6 b c d e^{2} f + 5 b d^{2} e^{3}} + x \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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