Optimal. Leaf size=447 \[ \frac {\sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} (b d-2 a f) E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{a^{3/4} \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (\sqrt {c} d-\sqrt {a} f\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 a^{3/4} \sqrt [4]{c} \left (b-2 \sqrt {a} \sqrt {c}\right ) \sqrt {a+b x^2+c x^4}}-\frac {\sqrt {c} x \sqrt {a+b x^2+c x^4} (b d-2 a f)}{a \left (b^2-4 a c\right ) \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {x \left (c x^2 (b d-2 a f)-a b f-2 a c d+b^2 d\right )}{a \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {-2 a g+x^2 (2 c e-b g)+b e}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}} \]
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Rubi [A] time = 0.27, antiderivative size = 447, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.219, Rules used = {1673, 1178, 1197, 1103, 1195, 1247, 636} \[ \frac {\sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} (b d-2 a f) E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{a^{3/4} \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (\sqrt {c} d-\sqrt {a} f\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 a^{3/4} \sqrt [4]{c} \left (b-2 \sqrt {a} \sqrt {c}\right ) \sqrt {a+b x^2+c x^4}}-\frac {\sqrt {c} x \sqrt {a+b x^2+c x^4} (b d-2 a f)}{a \left (b^2-4 a c\right ) \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {x \left (c x^2 (b d-2 a f)-a b f-2 a c d+b^2 d\right )}{a \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {-2 a g+x^2 (2 c e-b g)+b e}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}} \]
Antiderivative was successfully verified.
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Rule 636
Rule 1103
Rule 1178
Rule 1195
Rule 1197
Rule 1247
Rule 1673
Rubi steps
\begin {align*} \int \frac {d+e x+f x^2+g x^3}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx &=\int \frac {d+f x^2}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx+\int \frac {x \left (e+g x^2\right )}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx\\ &=\frac {x \left (b^2 d-2 a c d-a b f+c (b d-2 a f) x^2\right )}{a \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {e+g x}{\left (a+b x+c x^2\right )^{3/2}} \, dx,x,x^2\right )-\frac {\int \frac {a (2 c d-b f)+c (b d-2 a f) x^2}{\sqrt {a+b x^2+c x^4}} \, dx}{a \left (b^2-4 a c\right )}\\ &=\frac {x \left (b^2 d-2 a c d-a b f+c (b d-2 a f) x^2\right )}{a \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {b e-2 a g+(2 c e-b g) x^2}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}+\frac {\left (\sqrt {c} (b d-2 a f)\right ) \int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}{\sqrt {a+b x^2+c x^4}} \, dx}{\sqrt {a} \left (b^2-4 a c\right )}-\frac {\left (\sqrt {c} (b d-2 a f)+\sqrt {a} (2 c d-b f)\right ) \int \frac {1}{\sqrt {a+b x^2+c x^4}} \, dx}{\sqrt {a} \left (b^2-4 a c\right )}\\ &=\frac {x \left (b^2 d-2 a c d-a b f+c (b d-2 a f) x^2\right )}{a \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {b e-2 a g+(2 c e-b g) x^2}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {\sqrt {c} (b d-2 a f) x \sqrt {a+b x^2+c x^4}}{a \left (b^2-4 a c\right ) \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {\sqrt [4]{c} (b d-2 a f) \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 )}{a^{3/4} \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {\left (\sqrt {c} d-\sqrt {a} f\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}} 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 a^{3/4} \left (b-2 \sqrt {a} \sqrt {c}\right ) \sqrt [4]{c} \sqrt {a+b x^2+c x^4}}\\ \end {align*}
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Mathematica [F] time = 0.00, size = 0, normalized size = 0.00 \[ \text {\$Aborted} \]
Verification is Not applicable to the result.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{4} + b x^{2} + a} {\left (g x^{3} + f x^{2} + e x + d\right )}}{c^{2} x^{8} + 2 \, b c x^{6} + {\left (b^{2} + 2 \, a c\right )} x^{4} + 2 \, a b x^{2} + a^{2}}, x\right ) \]
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 \[ \int \frac {g x^{3} + f x^{2} + e x + d}{{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 1005, normalized size = 2.25 \[ \left (-\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 ) b c}{2 \left (4 a c -b^{2}\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 {2 \left (\frac {b \,x^{3}}{2 \left (4 a c -b^{2}\right ) a}-\frac {\left (2 a c -b^{2}\right ) x}{2 \left (4 a c -b^{2}\right ) a c}\right ) c}{\sqrt {\left (x^{4}+\frac {b \,x^{2}}{c}+\frac {a}{c}\right ) c}}+\frac {\left (\frac {1}{a}-\frac {2 a c -b^{2}}{\left (4 a c -b^{2}\right ) a}\right ) \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}\, \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 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}\right ) d +\frac {\left (2 c \,x^{2}+b \right ) e}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}+\left (\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}{\left (4 a c -b^{2}\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}\, b \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 (4 a c -b^{2}\right ) \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {2 \left (-\frac {x^{3}}{4 a c -b^{2}}-\frac {b x}{2 \left (4 a c -b^{2}\right ) c}\right ) c}{\sqrt {\left (x^{4}+\frac {b \,x^{2}}{c}+\frac {a}{c}\right ) c}}\right ) f -\frac {\left (b \,x^{2}+2 a \right ) g}{\sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (4 a c -b^{2}\right )} \]
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 \[ \int \frac {g x^{3} + f x^{2} + e x + d}{{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {g\,x^3+f\,x^2+e\,x+d}{{\left (c\,x^4+b\,x^2+a\right )}^{3/2}} \,d x \]
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 \[ \int \frac {d + e x + f x^{2} + g x^{3}}{\left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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