Optimal. Leaf size=822 \[ -\frac {\sqrt {c x^4+b x^2+a} e^3}{\left (c d^4+b e^2 d^2+a e^4\right ) (d+e x)}-\frac {\sqrt [4]{a} \sqrt [4]{c} \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+b x^2+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) e^2}{\left (c d^4+b e^2 d^2+a e^4\right ) \sqrt {c x^4+b x^2+a}}+\frac {\sqrt {c} x \sqrt {c x^4+b x^2+a} e^2}{\left (c d^4+b e^2 d^2+a e^4\right ) \left (\sqrt {c} x^2+\sqrt {a}\right )}-\frac {d \left (2 c d^2+b e^2\right ) \tan ^{-1}\left (\frac {\sqrt {-c d^4-b e^2 d^2-a e^4} x}{d e \sqrt {c x^4+b x^2+a}}\right ) e}{2 \left (-c d^4-b e^2 d^2-a e^4\right )^{3/2}}-\frac {d \left (2 c d^2+b e^2\right ) \tanh ^{-1}\left (\frac {b d^2+2 a e^2+\left (2 c d^2+b e^2\right ) x^2}{2 \sqrt {c d^4+b e^2 d^2+a e^4} \sqrt {c x^4+b x^2+a}}\right ) e}{2 \left (c d^4+b e^2 d^2+a e^4\right )^{3/2}}+\frac {\sqrt [4]{c} \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+b x^2+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{a} \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \sqrt {c x^4+b x^2+a}}-\frac {\left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \left (2 c d^2+b e^2\right ) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+b x^2+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} \Pi \left (\frac {\left (\sqrt {c} d^2+\sqrt {a} e^2\right )^2}{4 \sqrt {a} \sqrt {c} d^2 e^2};2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{4 \sqrt [4]{a} \sqrt [4]{c} \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \left (c d^4+b e^2 d^2+a e^4\right ) \sqrt {c x^4+b x^2+a}} \]
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Rubi [A] time = 1.26, antiderivative size = 822, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {1726, 1741, 12, 1247, 724, 206, 1714, 1195, 1708, 1103, 1706} \[ -\frac {\sqrt {c x^4+b x^2+a} e^3}{\left (c d^4+b e^2 d^2+a e^4\right ) (d+e x)}-\frac {\sqrt [4]{a} \sqrt [4]{c} \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+b x^2+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) e^2}{\left (c d^4+b e^2 d^2+a e^4\right ) \sqrt {c x^4+b x^2+a}}+\frac {\sqrt {c} x \sqrt {c x^4+b x^2+a} e^2}{\left (c d^4+b e^2 d^2+a e^4\right ) \left (\sqrt {c} x^2+\sqrt {a}\right )}-\frac {d \left (2 c d^2+b e^2\right ) \tan ^{-1}\left (\frac {\sqrt {-c d^4-b e^2 d^2-a e^4} x}{d e \sqrt {c x^4+b x^2+a}}\right ) e}{2 \left (-c d^4-b e^2 d^2-a e^4\right )^{3/2}}-\frac {d \left (2 c d^2+b e^2\right ) \tanh ^{-1}\left (\frac {b d^2+2 a e^2+\left (2 c d^2+b e^2\right ) x^2}{2 \sqrt {c d^4+b e^2 d^2+a e^4} \sqrt {c x^4+b x^2+a}}\right ) e}{2 \left (c d^4+b e^2 d^2+a e^4\right )^{3/2}}+\frac {\sqrt [4]{c} \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+b x^2+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{a} \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \sqrt {c x^4+b x^2+a}}-\frac {\left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \left (2 c d^2+b e^2\right ) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+b x^2+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} \Pi \left (\frac {\left (\sqrt {c} d^2+\sqrt {a} e^2\right )^2}{4 \sqrt {a} \sqrt {c} d^2 e^2};2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{4 \sqrt [4]{a} \sqrt [4]{c} \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \left (c d^4+b e^2 d^2+a e^4\right ) \sqrt {c x^4+b x^2+a}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 206
Rule 724
Rule 1103
Rule 1195
Rule 1247
Rule 1706
Rule 1708
Rule 1714
Rule 1726
Rule 1741
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^2 \sqrt {a+b x^2+c x^4}} \, dx &=-\frac {e^3 \sqrt {a+b x^2+c x^4}}{\left (c d^4+b d^2 e^2+a e^4\right ) (d+e x)}-\frac {\int \frac {-d \left (c d^2+b e^2\right )+c d^2 e x-c d e^2 x^2-c e^3 x^3}{(d+e x) \sqrt {a+b x^2+c x^4}} \, dx}{c d^4+b d^2 e^2+a e^4}\\ &=-\frac {e^3 \sqrt {a+b x^2+c x^4}}{\left (c d^4+b d^2 e^2+a e^4\right ) (d+e x)}-\frac {\int \frac {\left (c d^3 e+d e \left (c d^2+b e^2\right )\right ) x}{\left (d^2-e^2 x^2\right ) \sqrt {a+b x^2+c x^4}} \, dx}{c d^4+b d^2 e^2+a e^4}-\frac {\int \frac {-d^2 \left (c d^2+b e^2\right )-2 c d^2 e^2 x^2+c e^4 x^4}{\left (d^2-e^2 x^2\right ) \sqrt {a+b x^2+c x^4}} \, dx}{c d^4+b d^2 e^2+a e^4}\\ &=-\frac {e^3 \sqrt {a+b x^2+c x^4}}{\left (c d^4+b d^2 e^2+a e^4\right ) (d+e x)}+\frac {\int \frac {\sqrt {a} c^{3/2} d^2 e^4+c d^2 e^2 \left (c d^2+b e^2\right )+\left (2 c^2 d^2 e^4-c e^4 \left (c d^2+\sqrt {a} \sqrt {c} e^2\right )\right ) x^2}{\left (d^2-e^2 x^2\right ) \sqrt {a+b x^2+c x^4}} \, dx}{c e^2 \left (c d^4+b d^2 e^2+a e^4\right )}-\frac {\left (\sqrt {a} \sqrt {c} e^2\right ) \int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}{\sqrt {a+b x^2+c x^4}} \, dx}{c d^4+b d^2 e^2+a e^4}-\frac {\left (d e \left (2 c d^2+b e^2\right )\right ) \int \frac {x}{\left (d^2-e^2 x^2\right ) \sqrt {a+b x^2+c x^4}} \, dx}{c d^4+b d^2 e^2+a e^4}\\ &=-\frac {e^3 \sqrt {a+b x^2+c x^4}}{\left (c d^4+b d^2 e^2+a e^4\right ) (d+e x)}+\frac {\sqrt {c} e^2 x \sqrt {a+b x^2+c x^4}}{\left (c d^4+b d^2 e^2+a e^4\right ) \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {\sqrt [4]{a} \sqrt [4]{c} e^2 \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\left (c d^4+b d^2 e^2+a e^4\right ) \sqrt {a+b x^2+c x^4}}+\frac {\sqrt {c} \int \frac {1}{\sqrt {a+b x^2+c x^4}} \, dx}{\sqrt {c} d^2+\sqrt {a} e^2}-\frac {\left (d e \left (2 c d^2+b e^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\left (d^2-e^2 x\right ) \sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{2 \left (c d^4+b d^2 e^2+a e^4\right )}+\frac {\left (\sqrt {a} d^2 e^2 \left (2 c d^2+b e^2\right )\right ) \int \frac {1+\frac {\sqrt {c} x^2}{\sqrt {a}}}{\left (d^2-e^2 x^2\right ) \sqrt {a+b x^2+c x^4}} \, dx}{\left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \left (c d^4+b d^2 e^2+a e^4\right )}\\ &=-\frac {e^3 \sqrt {a+b x^2+c x^4}}{\left (c d^4+b d^2 e^2+a e^4\right ) (d+e x)}+\frac {\sqrt {c} e^2 x \sqrt {a+b x^2+c x^4}}{\left (c d^4+b d^2 e^2+a e^4\right ) \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {d e \left (2 c d^2+b e^2\right ) \tan ^{-1}\left (\frac {\sqrt {-c d^4-b d^2 e^2-a e^4} x}{d e \sqrt {a+b x^2+c x^4}}\right )}{2 \left (-c d^4-b d^2 e^2-a e^4\right )^{3/2}}-\frac {\sqrt [4]{a} \sqrt [4]{c} e^2 \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\left (c d^4+b d^2 e^2+a e^4\right ) \sqrt {a+b x^2+c x^4}}+\frac {\sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{a} \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \sqrt {a+b x^2+c x^4}}-\frac {\left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \left (2 c d^2+b e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \Pi \left (\frac {\left (\sqrt {c} d^2+\sqrt {a} e^2\right )^2}{4 \sqrt {a} \sqrt {c} d^2 e^2};2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{4 \sqrt [4]{a} \sqrt [4]{c} \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \left (c d^4+b d^2 e^2+a e^4\right ) \sqrt {a+b x^2+c x^4}}+\frac {\left (d e \left (2 c d^2+b e^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4 c d^4+4 b d^2 e^2+4 a e^4-x^2} \, dx,x,\frac {-b d^2-2 a e^2-\left (2 c d^2+b e^2\right ) x^2}{\sqrt {a+b x^2+c x^4}}\right )}{c d^4+b d^2 e^2+a e^4}\\ &=-\frac {e^3 \sqrt {a+b x^2+c x^4}}{\left (c d^4+b d^2 e^2+a e^4\right ) (d+e x)}+\frac {\sqrt {c} e^2 x \sqrt {a+b x^2+c x^4}}{\left (c d^4+b d^2 e^2+a e^4\right ) \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {d e \left (2 c d^2+b e^2\right ) \tan ^{-1}\left (\frac {\sqrt {-c d^4-b d^2 e^2-a e^4} x}{d e \sqrt {a+b x^2+c x^4}}\right )}{2 \left (-c d^4-b d^2 e^2-a e^4\right )^{3/2}}-\frac {d e \left (2 c d^2+b e^2\right ) \tanh ^{-1}\left (\frac {b d^2+2 a e^2+\left (2 c d^2+b e^2\right ) x^2}{2 \sqrt {c d^4+b d^2 e^2+a e^4} \sqrt {a+b x^2+c x^4}}\right )}{2 \left (c d^4+b d^2 e^2+a e^4\right )^{3/2}}-\frac {\sqrt [4]{a} \sqrt [4]{c} e^2 \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\left (c d^4+b d^2 e^2+a e^4\right ) \sqrt {a+b x^2+c x^4}}+\frac {\sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{a} \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \sqrt {a+b x^2+c x^4}}-\frac {\left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \left (2 c d^2+b e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \Pi \left (\frac {\left (\sqrt {c} d^2+\sqrt {a} e^2\right )^2}{4 \sqrt {a} \sqrt {c} d^2 e^2};2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{4 \sqrt [4]{a} \sqrt [4]{c} \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \left (c d^4+b d^2 e^2+a e^4\right ) \sqrt {a+b x^2+c x^4}}\\ \end {align*}
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Mathematica [C] time = 7.77, size = 4019, normalized size = 4.89 \[ \text {Result too large to show} \]
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] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {c x^{4} + b x^{2} + a} {\left (e x + d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 771, normalized size = 0.94 \[ -\frac {\sqrt {2}\, \sqrt {-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}+4}\, \sqrt {\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}+4}\, \left (-\EllipticE \left (\frac {\sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, x}{2}, \frac {\sqrt {\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) b}{a c}-4}}{2}\right )+\EllipticF \left (\frac {\sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, x}{2}, \frac {\sqrt {\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) b}{a c}-4}}{2}\right )\right ) a c \,e^{2}}{2 \left (a \,e^{4}+b \,d^{2} e^{2}+c \,d^{4}\right ) \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}-\frac {\sqrt {2}\, \sqrt {-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}+4}\, \sqrt {\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}+4}\, c \,d^{2} \EllipticF \left (\frac {\sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, x}{2}, \frac {\sqrt {\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) b}{a c}-4}}{2}\right )}{4 \left (a \,e^{4}+b \,d^{2} e^{2}+c \,d^{4}\right ) \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, e^{3}}{\left (a \,e^{4}+b \,d^{2} e^{2}+c \,d^{4}\right ) \left (e x +d \right )}+\frac {\left (b \,e^{2}+2 c \,d^{2}\right ) \left (\frac {\sqrt {2}\, \sqrt {-\frac {\left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{2 a}+1}\, \sqrt {\frac {\left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{2 a}+1}\, e \EllipticPi \left (\frac {\sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, x}{2}, \frac {2 a \,e^{2}}{\left (-b +\sqrt {-4 a c +b^{2}}\right ) d^{2}}, \frac {\sqrt {-\frac {b +\sqrt {-4 a c +b^{2}}}{2 a}}\, \sqrt {2}}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}\right )}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, d}-\frac {\arctanh \left (\frac {b \,x^{2}+\frac {2 c \,d^{2} x^{2}}{e^{2}}+2 a +\frac {b \,d^{2}}{e^{2}}}{2 \sqrt {a +\frac {b \,d^{2}}{e^{2}}+\frac {c \,d^{4}}{e^{4}}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}\right )}{2 \sqrt {a +\frac {b \,d^{2}}{e^{2}}+\frac {c \,d^{4}}{e^{4}}}}\right ) d}{\left (a \,e^{4}+b \,d^{2} e^{2}+c \,d^{4}\right ) e} \]
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 \[ \int \frac {1}{\sqrt {c x^{4} + b x^{2} + a} {\left (e x + d\right )}^{2}}\,{d x} \]
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 \[ \int \frac {1}{{\left (d+e\,x\right )}^2\,\sqrt {c\,x^4+b\,x^2+a}} \,d x \]
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 \[ \int \frac {1}{\left (d + e x\right )^{2} \sqrt {a + b x^{2} + c x^{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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