Optimal. Leaf size=207 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (b e \left (-c^2 f^2-6 c d e f+15 d^2 e^2\right )-a f \left (3 c^2 f^2+2 c d e f+3 d^2 e^2\right )\right )}{8 e^{5/2} f^{7/2}}+\frac {d x (b e (15 d e-c f)-3 a f (c f+d e))}{8 e^2 f^3}-\frac {x \left (c+d x^2\right ) (b e (5 d e-c f)-a f (3 c f+d e))}{8 e^2 f^2 \left (e+f x^2\right )}-\frac {x \left (c+d x^2\right )^2 (b e-a f)}{4 e f \left (e+f x^2\right )^2} \]
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Rubi [A] time = 0.24, antiderivative size = 207, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 388
Rule 526
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*}
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Mathematica [A] time = 0.13, size = 183, normalized size = 0.88 \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b x^2\right ) \left (c+d x^2\right )^2}{\left (e+f x^2\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.43, size = 777, normalized size = 3.75 \begin {gather*} \left [\frac {16 \, b d^{2} e^{3} f^{3} x^{5} + 2 \, {\left (25 \, b d^{2} e^{4} f^{2} + 3 \, a c^{2} e f^{5} - 5 \, {\left (2 \, b c d + a d^{2}\right )} e^{3} f^{3} + {\left (b c^{2} + 2 \, a c d\right )} e^{2} f^{4}\right )} x^{3} + {\left (15 \, b d^{2} e^{5} - 3 \, a c^{2} e^{2} f^{3} - 3 \, {\left (2 \, b c d + a d^{2}\right )} e^{4} f - {\left (b c^{2} + 2 \, a c d\right )} e^{3} f^{2} + {\left (15 \, b d^{2} e^{3} f^{2} - 3 \, a c^{2} f^{5} - 3 \, {\left (2 \, b c d + a d^{2}\right )} e^{2} f^{3} - {\left (b c^{2} + 2 \, a c d\right )} e f^{4}\right )} x^{4} + 2 \, {\left (15 \, b d^{2} e^{4} f - 3 \, 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^{2}\right )} \sqrt {-e f} \log \left (\frac {f x^{2} - 2 \, \sqrt {-e f} x - e}{f x^{2} + e}\right ) + 2 \, {\left (15 \, b d^{2} e^{5} f + 5 \, a c^{2} e^{2} f^{4} - 3 \, {\left (2 \, b c d + a d^{2}\right )} e^{4} f^{2} - {\left (b c^{2} + 2 \, a c d\right )} e^{3} f^{3}\right )} x}{16 \, {\left (e^{3} f^{6} x^{4} + 2 \, e^{4} f^{5} x^{2} + e^{5} f^{4}\right )}}, \frac {8 \, b d^{2} e^{3} f^{3} x^{5} + {\left (25 \, b d^{2} e^{4} f^{2} + 3 \, a c^{2} e f^{5} - 5 \, {\left (2 \, b c d + a d^{2}\right )} e^{3} f^{3} + {\left (b c^{2} + 2 \, a c d\right )} e^{2} f^{4}\right )} x^{3} - {\left (15 \, b d^{2} e^{5} - 3 \, a c^{2} e^{2} f^{3} - 3 \, {\left (2 \, b c d + a d^{2}\right )} e^{4} f - {\left (b c^{2} + 2 \, a c d\right )} e^{3} f^{2} + {\left (15 \, b d^{2} e^{3} f^{2} - 3 \, a c^{2} f^{5} - 3 \, {\left (2 \, b c d + a d^{2}\right )} e^{2} f^{3} - {\left (b c^{2} + 2 \, a c d\right )} e f^{4}\right )} x^{4} + 2 \, {\left (15 \, b d^{2} e^{4} f - 3 \, 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^{2}\right )} \sqrt {e f} \arctan \left (\frac {\sqrt {e f} x}{e}\right ) + {\left (15 \, b d^{2} e^{5} f + 5 \, a c^{2} e^{2} f^{4} - 3 \, {\left (2 \, b c d + a d^{2}\right )} e^{4} f^{2} - {\left (b c^{2} + 2 \, a c d\right )} e^{3} f^{3}\right )} x}{8 \, {\left (e^{3} f^{6} x^{4} + 2 \, e^{4} f^{5} x^{2} + e^{5} f^{4}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.45, size = 238, normalized size = 1.15 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 397, normalized size = 1.92 \begin {gather*} \frac {3 a \,c^{2} f \,x^{3}}{8 \left (f \,x^{2}+e \right )^{2} e^{2}}+\frac {a c d \,x^{3}}{4 \left (f \,x^{2}+e \right )^{2} e}-\frac {5 a \,d^{2} x^{3}}{8 \left (f \,x^{2}+e \right )^{2} f}+\frac {b \,c^{2} x^{3}}{8 \left (f \,x^{2}+e \right )^{2} e}-\frac {5 b c d \,x^{3}}{4 \left (f \,x^{2}+e \right )^{2} f}+\frac {9 b \,d^{2} e \,x^{3}}{8 \left (f \,x^{2}+e \right )^{2} f^{2}}+\frac {5 a \,c^{2} x}{8 \left (f \,x^{2}+e \right )^{2} e}-\frac {a c d x}{4 \left (f \,x^{2}+e \right )^{2} f}-\frac {3 a \,d^{2} e x}{8 \left (f \,x^{2}+e \right )^{2} f^{2}}-\frac {b \,c^{2} x}{8 \left (f \,x^{2}+e \right )^{2} f}-\frac {3 b c d e x}{4 \left (f \,x^{2}+e \right )^{2} f^{2}}+\frac {7 b \,d^{2} e^{2} x}{8 \left (f \,x^{2}+e \right )^{2} f^{3}}+\frac {3 a \,c^{2} \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{8 \sqrt {e f}\, e^{2}}+\frac {a c d \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{4 \sqrt {e f}\, e f}+\frac {3 a \,d^{2} \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{8 \sqrt {e f}\, f^{2}}+\frac {b \,c^{2} \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{8 \sqrt {e f}\, e f}+\frac {3 b c d \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{4 \sqrt {e f}\, f^{2}}-\frac {15 b \,d^{2} e \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{8 \sqrt {e f}\, f^{3}}+\frac {b \,d^{2} x}{f^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.01, size = 236, normalized size = 1.14 \begin {gather*} \frac {b d^{2} x}{f^{3}} + \frac {{\left (9 \, b d^{2} e^{3} f + 3 \, a c^{2} f^{4} - 5 \, {\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^{3} + {\left (7 \, b d^{2} e^{4} + 5 \, 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}\right )} x}{8 \, {\left (e^{2} f^{5} x^{4} + 2 \, e^{3} f^{4} x^{2} + e^{4} f^{3}\right )}} - \frac {{\left (15 \, b d^{2} e^{3} - 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 )}{8 \, \sqrt {e f} e^{2} f^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.98, size = 243, normalized size = 1.17 \begin {gather*} \frac {\mathrm {atan}\left (\frac {\sqrt {f}\,x}{\sqrt {e}}\right )\,\left (b\,c^2\,e\,f^2+3\,a\,c^2\,f^3+6\,b\,c\,d\,e^2\,f+2\,a\,c\,d\,e\,f^2-15\,b\,d^2\,e^3+3\,a\,d^2\,e^2\,f\right )}{8\,e^{5/2}\,f^{7/2}}-\frac {\frac {x\,\left (b\,c^2\,e\,f^2-5\,a\,c^2\,f^3+6\,b\,c\,d\,e^2\,f+2\,a\,c\,d\,e\,f^2-7\,b\,d^2\,e^3+3\,a\,d^2\,e^2\,f\right )}{8\,e}-\frac {x^3\,\left (b\,c^2\,e\,f^3+3\,a\,c^2\,f^4-10\,b\,c\,d\,e^2\,f^2+2\,a\,c\,d\,e\,f^3+9\,b\,d^2\,e^3\,f-5\,a\,d^2\,e^2\,f^2\right )}{8\,e^2}}{e^2\,f^3+2\,e\,f^4\,x^2+f^5\,x^4}+\frac {b\,d^2\,x}{f^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 11.98, size = 400, normalized size = 1.93 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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