Optimal. Leaf size=339 \[ -\frac {2 c^2 \left (\frac {b}{\sqrt {b^2-4 a c}}+1\right ) \tan ^{-1}\left (\frac {x \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{a \sqrt {b-\sqrt {b^2-4 a c}} \left (2 c d-e \left (b-\sqrt {b^2-4 a c}\right )\right )^{3/2}}-\frac {2 c^2 \left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {x \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}{\sqrt {\sqrt {b^2-4 a c}+b} \sqrt {d+e x^2}}\right )}{a \sqrt {\sqrt {b^2-4 a c}+b} \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )^{3/2}}+\frac {e x (c d-b e)}{a d \sqrt {d+e x^2} \left (e (a e-b d)+c d^2\right )}+\frac {-d-2 e x^2}{a d^2 x \sqrt {d+e x^2}} \]
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Rubi [A] time = 2.84, antiderivative size = 462, normalized size of antiderivative = 1.36, number of steps used = 12, number of rules used = 8, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {1301, 271, 191, 6728, 264, 1692, 377, 205} \[ -\frac {c \left (\frac {2 a c e+b^2 (-e)+b c d}{\sqrt {b^2-4 a c}}-b e+c d\right ) \tan ^{-1}\left (\frac {x \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{a \sqrt {b-\sqrt {b^2-4 a c}} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )} \left (a e^2-b d e+c d^2\right )}-\frac {c \left (-\frac {2 a c e+b^2 (-e)+b c d}{\sqrt {b^2-4 a c}}-b e+c d\right ) \tan ^{-1}\left (\frac {x \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}{\sqrt {\sqrt {b^2-4 a c}+b} \sqrt {d+e x^2}}\right )}{a \sqrt {\sqrt {b^2-4 a c}+b} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )} \left (a e^2-b d e+c d^2\right )}-\frac {2 e^3 x}{d^2 \sqrt {d+e x^2} \left (a e^2-b d e+c d^2\right )}-\frac {e^2}{d x \sqrt {d+e x^2} \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {d+e x^2} (c d-b e)}{a d x \left (a e^2-b d e+c d^2\right )} \]
Antiderivative was successfully verified.
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Rule 191
Rule 205
Rule 264
Rule 271
Rule 377
Rule 1301
Rule 1692
Rule 6728
Rubi steps
\begin {align*} \int \frac {1}{x^2 \left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx &=\frac {\int \frac {c d-b e-c e x^2}{x^2 \sqrt {d+e x^2} \left (a+b x^2+c x^4\right )} \, dx}{c d^2-b d e+a e^2}+\frac {e^2 \int \frac {1}{x^2 \left (d+e x^2\right )^{3/2}} \, dx}{c d^2-b d e+a e^2}\\ &=-\frac {e^2}{d \left (c d^2-b d e+a e^2\right ) x \sqrt {d+e x^2}}+\frac {\int \left (\frac {c d-b e}{a x^2 \sqrt {d+e x^2}}+\frac {-b c d+b^2 e-a c e-c (c d-b e) x^2}{a \sqrt {d+e x^2} \left (a+b x^2+c x^4\right )}\right ) \, dx}{c d^2-b d e+a e^2}-\frac {\left (2 e^3\right ) \int \frac {1}{\left (d+e x^2\right )^{3/2}} \, dx}{d \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {e^2}{d \left (c d^2-b d e+a e^2\right ) x \sqrt {d+e x^2}}-\frac {2 e^3 x}{d^2 \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x^2}}+\frac {\int \frac {-b c d+b^2 e-a c e-c (c d-b e) x^2}{\sqrt {d+e x^2} \left (a+b x^2+c x^4\right )} \, dx}{a \left (c d^2-b d e+a e^2\right )}+\frac {(c d-b e) \int \frac {1}{x^2 \sqrt {d+e x^2}} \, dx}{a \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {e^2}{d \left (c d^2-b d e+a e^2\right ) x \sqrt {d+e x^2}}-\frac {2 e^3 x}{d^2 \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x^2}}-\frac {(c d-b e) \sqrt {d+e x^2}}{a d \left (c d^2-b d e+a e^2\right ) x}+\frac {\int \left (\frac {-c (c d-b e)+\frac {c \left (-b c d+b^2 e-2 a c e\right )}{\sqrt {b^2-4 a c}}}{\left (b-\sqrt {b^2-4 a c}+2 c x^2\right ) \sqrt {d+e x^2}}+\frac {-c (c d-b e)-\frac {c \left (-b c d+b^2 e-2 a c e\right )}{\sqrt {b^2-4 a c}}}{\left (b+\sqrt {b^2-4 a c}+2 c x^2\right ) \sqrt {d+e x^2}}\right ) \, dx}{a \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {e^2}{d \left (c d^2-b d e+a e^2\right ) x \sqrt {d+e x^2}}-\frac {2 e^3 x}{d^2 \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x^2}}-\frac {(c d-b e) \sqrt {d+e x^2}}{a d \left (c d^2-b d e+a e^2\right ) x}-\frac {\left (c \left (c d-b e-\frac {b c d-b^2 e+2 a c e}{\sqrt {b^2-4 a c}}\right )\right ) \int \frac {1}{\left (b+\sqrt {b^2-4 a c}+2 c x^2\right ) \sqrt {d+e x^2}} \, dx}{a \left (c d^2-b d e+a e^2\right )}-\frac {\left (c \left (c d-b e+\frac {b c d-b^2 e+2 a c e}{\sqrt {b^2-4 a c}}\right )\right ) \int \frac {1}{\left (b-\sqrt {b^2-4 a c}+2 c x^2\right ) \sqrt {d+e x^2}} \, dx}{a \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {e^2}{d \left (c d^2-b d e+a e^2\right ) x \sqrt {d+e x^2}}-\frac {2 e^3 x}{d^2 \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x^2}}-\frac {(c d-b e) \sqrt {d+e x^2}}{a d \left (c d^2-b d e+a e^2\right ) x}-\frac {\left (c \left (c d-b e-\frac {b c d-b^2 e+2 a c e}{\sqrt {b^2-4 a c}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{b+\sqrt {b^2-4 a c}-\left (-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e\right ) x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )}{a \left (c d^2-b d e+a e^2\right )}-\frac {\left (c \left (c d-b e+\frac {b c d-b^2 e+2 a c e}{\sqrt {b^2-4 a c}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{b-\sqrt {b^2-4 a c}-\left (-2 c d+\left (b-\sqrt {b^2-4 a c}\right ) e\right ) x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )}{a \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {e^2}{d \left (c d^2-b d e+a e^2\right ) x \sqrt {d+e x^2}}-\frac {2 e^3 x}{d^2 \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x^2}}-\frac {(c d-b e) \sqrt {d+e x^2}}{a d \left (c d^2-b d e+a e^2\right ) x}-\frac {c \left (c d-b e+\frac {b c d-b^2 e+2 a c e}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{a \sqrt {b-\sqrt {b^2-4 a c}} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )}-\frac {c \left (c d-b e-\frac {b c d-b^2 e+2 a c e}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b+\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{a \sqrt {b+\sqrt {b^2-4 a c}} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )}\\ \end {align*}
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Mathematica [C] time = 6.77, size = 2158, normalized size = 6.37 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.04, size = 387, normalized size = 1.14 \[ -\frac {8 b \,e^{\frac {3}{2}}}{\left (4 a \,e^{2}-4 d e b +4 c \,d^{2}\right ) \left (2 e \,x^{2}-2 \sqrt {e \,x^{2}+d}\, \sqrt {e}\, x +2 d \right ) a}+\frac {8 c d \sqrt {e}}{\left (4 a \,e^{2}-4 d e b +4 c \,d^{2}\right ) \left (2 e \,x^{2}-2 \sqrt {e \,x^{2}+d}\, \sqrt {e}\, x +2 d \right ) a}-\frac {2 \sqrt {e}\, \left (b c \,d^{2} e -c^{2} d^{3}+\left (b e -c d \right ) \RootOf \left (\textit {\_Z}^{4} c +c \,d^{4}+\left (4 b e -4 c d \right ) \textit {\_Z}^{3}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{2}+\left (4 b \,d^{2} e -4 c \,d^{3}\right ) \textit {\_Z} \right )^{2} c +2 \left (-2 a c \,e^{2}+2 b^{2} e^{2}-3 b c d e +c^{2} d^{2}\right ) \RootOf \left (\textit {\_Z}^{4} c +c \,d^{4}+\left (4 b e -4 c d \right ) \textit {\_Z}^{3}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{2}+\left (4 b \,d^{2} e -4 c \,d^{3}\right ) \textit {\_Z} \right )\right ) \ln \left (-\RootOf \left (\textit {\_Z}^{4} c +c \,d^{4}+\left (4 b e -4 c d \right ) \textit {\_Z}^{3}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{2}+\left (4 b \,d^{2} e -4 c \,d^{3}\right ) \textit {\_Z} \right )+\left (-\sqrt {e}\, x +\sqrt {e \,x^{2}+d}\right )^{2}\right )}{\left (4 a \,e^{2}-4 d e b +4 c \,d^{2}\right ) a \left (\RootOf \left (\textit {\_Z}^{4} c +c \,d^{4}+\left (4 b e -4 c d \right ) \textit {\_Z}^{3}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{2}+\left (4 b \,d^{2} e -4 c \,d^{3}\right ) \textit {\_Z} \right )^{3} c +3 \RootOf \left (\textit {\_Z}^{4} c +c \,d^{4}+\left (4 b e -4 c d \right ) \textit {\_Z}^{3}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{2}+\left (4 b \,d^{2} e -4 c \,d^{3}\right ) \textit {\_Z} \right )^{2} b e -3 \RootOf \left (\textit {\_Z}^{4} c +c \,d^{4}+\left (4 b e -4 c d \right ) \textit {\_Z}^{3}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{2}+\left (4 b \,d^{2} e -4 c \,d^{3}\right ) \textit {\_Z} \right )^{2} c d +8 \RootOf \left (\textit {\_Z}^{4} c +c \,d^{4}+\left (4 b e -4 c d \right ) \textit {\_Z}^{3}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{2}+\left (4 b \,d^{2} e -4 c \,d^{3}\right ) \textit {\_Z} \right ) a \,e^{2}-4 \RootOf \left (\textit {\_Z}^{4} c +c \,d^{4}+\left (4 b e -4 c d \right ) \textit {\_Z}^{3}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{2}+\left (4 b \,d^{2} e -4 c \,d^{3}\right ) \textit {\_Z} \right ) b d e +3 \RootOf \left (\textit {\_Z}^{4} c +c \,d^{4}+\left (4 b e -4 c d \right ) \textit {\_Z}^{3}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{2}+\left (4 b \,d^{2} e -4 c \,d^{3}\right ) \textit {\_Z} \right ) c \,d^{2}+b \,d^{2} e -c \,d^{3}\right )}-\frac {2 e x}{\sqrt {e \,x^{2}+d}\, a \,d^{2}}-\frac {1}{\sqrt {e \,x^{2}+d}\, a d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (c x^{4} + b x^{2} + a\right )} {\left (e x^{2} + d\right )}^{\frac {3}{2}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{x^2\,{\left (e\,x^2+d\right )}^{3/2}\,\left (c\,x^4+b\,x^2+a\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{2} \left (d + e x^{2}\right )^{\frac {3}{2}} \left (a + b x^{2} + c x^{4}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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