Optimal. Leaf size=505 \[ \frac {\left (5 b c d-2 b^2 f+6 a c f\right ) x \sqrt {a+b x^2+c x^4}}{15 c^{3/2} \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {(2 c e-b g) \left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{16 c^2}+\frac {x \left (5 c d+b f+3 c f x^2\right ) \sqrt {a+b x^2+c x^4}}{15 c}+\frac {g \left (a+b x^2+c x^4\right )^{3/2}}{6 c}-\frac {\left (b^2-4 a c\right ) (2 c e-b g) \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{32 c^{5/2}}-\frac {\sqrt [4]{a} \left (5 b c d-2 b^2 f+6 a c f\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{15 c^{7/4} \sqrt {a+b x^2+c x^4}}+\frac {\sqrt [4]{a} \left (b+2 \sqrt {a} \sqrt {c}\right ) \left (5 c d-2 b f+3 \sqrt {a} \sqrt {c} f\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{30 c^{7/4} \sqrt {a+b x^2+c x^4}} \]
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Rubi [A]
time = 0.18, antiderivative size = 505, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 10, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {1687, 1190,
1211, 1117, 1209, 1261, 654, 626, 635, 212} \begin {gather*} -\frac {\sqrt [4]{a} \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 (6 a c f-2 b^2 f+5 b c d\right ) E\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{15 c^{7/4} \sqrt {a+b x^2+c x^4}}+\frac {\sqrt [4]{a} \left (2 \sqrt {a} \sqrt {c}+b\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}} \left (3 \sqrt {a} \sqrt {c} f-2 b f+5 c d\right ) F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{30 c^{7/4} \sqrt {a+b x^2+c x^4}}+\frac {x \sqrt {a+b x^2+c x^4} \left (6 a c f-2 b^2 f+5 b c d\right )}{15 c^{3/2} \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {\left (b^2-4 a c\right ) (2 c e-b g) \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{32 c^{5/2}}+\frac {\left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4} (2 c e-b g)}{16 c^2}+\frac {x \sqrt {a+b x^2+c x^4} \left (b f+5 c d+3 c f x^2\right )}{15 c}+\frac {g \left (a+b x^2+c x^4\right )^{3/2}}{6 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 626
Rule 635
Rule 654
Rule 1117
Rule 1190
Rule 1209
Rule 1211
Rule 1261
Rule 1687
Rubi steps
\begin {align*} \int \left (d+e x+f x^2+g x^3\right ) \sqrt {a+b x^2+c x^4} \, dx &=\int \left (d+f x^2\right ) \sqrt {a+b x^2+c x^4} \, dx+\int x \left (e+g x^2\right ) \sqrt {a+b x^2+c x^4} \, dx\\ &=\frac {x \left (5 c d+b f+3 c f x^2\right ) \sqrt {a+b x^2+c x^4}}{15 c}+\frac {1}{2} \text {Subst}\left (\int (e+g x) \sqrt {a+b x+c x^2} \, dx,x,x^2\right )+\frac {\int \frac {a (10 c d-b f)+\left (5 b c d-2 b^2 f+6 a c f\right ) x^2}{\sqrt {a+b x^2+c x^4}} \, dx}{15 c}\\ &=\frac {x \left (5 c d+b f+3 c f x^2\right ) \sqrt {a+b x^2+c x^4}}{15 c}+\frac {g \left (a+b x^2+c x^4\right )^{3/2}}{6 c}+\frac {\left (\sqrt {a} \left (b+2 \sqrt {a} \sqrt {c}\right ) \left (5 c d-2 b f+3 \sqrt {a} \sqrt {c} f\right )\right ) \int \frac {1}{\sqrt {a+b x^2+c x^4}} \, dx}{15 c^{3/2}}-\frac {\left (\sqrt {a} \left (5 b c d-2 b^2 f+6 a c f\right )\right ) \int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}{\sqrt {a+b x^2+c x^4}} \, dx}{15 c^{3/2}}+\frac {(2 c e-b g) \text {Subst}\left (\int \sqrt {a+b x+c x^2} \, dx,x,x^2\right )}{4 c}\\ &=\frac {\left (5 b c d-2 b^2 f+6 a c f\right ) x \sqrt {a+b x^2+c x^4}}{15 c^{3/2} \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {(2 c e-b g) \left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{16 c^2}+\frac {x \left (5 c d+b f+3 c f x^2\right ) \sqrt {a+b x^2+c x^4}}{15 c}+\frac {g \left (a+b x^2+c x^4\right )^{3/2}}{6 c}-\frac {\sqrt [4]{a} \left (5 b c d-2 b^2 f+6 a c 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}} 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 )}{15 c^{7/4} \sqrt {a+b x^2+c x^4}}+\frac {\sqrt [4]{a} \left (b+2 \sqrt {a} \sqrt {c}\right ) \left (5 c d-2 b f+3 \sqrt {a} \sqrt {c} 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 )}{30 c^{7/4} \sqrt {a+b x^2+c x^4}}-\frac {\left (\left (b^2-4 a c\right ) (2 c e-b g)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{32 c^2}\\ &=\frac {\left (5 b c d-2 b^2 f+6 a c f\right ) x \sqrt {a+b x^2+c x^4}}{15 c^{3/2} \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {(2 c e-b g) \left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{16 c^2}+\frac {x \left (5 c d+b f+3 c f x^2\right ) \sqrt {a+b x^2+c x^4}}{15 c}+\frac {g \left (a+b x^2+c x^4\right )^{3/2}}{6 c}-\frac {\sqrt [4]{a} \left (5 b c d-2 b^2 f+6 a c 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}} 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 )}{15 c^{7/4} \sqrt {a+b x^2+c x^4}}+\frac {\sqrt [4]{a} \left (b+2 \sqrt {a} \sqrt {c}\right ) \left (5 c d-2 b f+3 \sqrt {a} \sqrt {c} 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 )}{30 c^{7/4} \sqrt {a+b x^2+c x^4}}-\frac {\left (\left (b^2-4 a c\right ) (2 c e-b g)\right ) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x^2}{\sqrt {a+b x^2+c x^4}}\right )}{16 c^2}\\ &=\frac {\left (5 b c d-2 b^2 f+6 a c f\right ) x \sqrt {a+b x^2+c x^4}}{15 c^{3/2} \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {(2 c e-b g) \left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{16 c^2}+\frac {x \left (5 c d+b f+3 c f x^2\right ) \sqrt {a+b x^2+c x^4}}{15 c}+\frac {g \left (a+b x^2+c x^4\right )^{3/2}}{6 c}-\frac {\left (b^2-4 a c\right ) (2 c e-b g) \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{32 c^{5/2}}-\frac {\sqrt [4]{a} \left (5 b c d-2 b^2 f+6 a c 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}} 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 )}{15 c^{7/4} \sqrt {a+b x^2+c x^4}}+\frac {\sqrt [4]{a} \left (b+2 \sqrt {a} \sqrt {c}\right ) \left (5 c d-2 b f+3 \sqrt {a} \sqrt {c} 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 )}{30 c^{7/4} \sqrt {a+b x^2+c x^4}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 12.55, size = 661, normalized size = 1.31 \begin {gather*} \frac {2 \sqrt {c} \left (a+b x^2+c x^4\right ) \left (-15 b^2 g+2 b c (15 e+x (8 f+5 g x))+4 c (10 a g+c x (20 d+x (15 e+2 x (6 f+5 g x))))\right )+\frac {-8 i \sqrt {2} \sqrt {c} \left (-b+\sqrt {b^2-4 a c}\right ) \left (-5 b c d+2 b^2 f-6 a c f\right ) \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} E\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )+8 i \sqrt {2} \sqrt {c} \left (-2 b^3 f+b c \left (-5 \sqrt {b^2-4 a c} d+8 a f\right )+b^2 \left (5 c d+2 \sqrt {b^2-4 a c} f\right )-2 a c \left (10 c d+3 \sqrt {b^2-4 a c} f\right )\right ) \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} F\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )-15 \left (b^2-4 a c\right ) \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} (-2 c e+b g) \sqrt {a+b x^2+c x^4} \log \left (b+2 c x^2-2 \sqrt {c} \sqrt {a+b x^2+c x^4}\right )}{\sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}}}}{480 c^{5/2} \sqrt {a+b x^2+c x^4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1041\) vs.
\(2(475)=950\).
time = 0.07, size = 1042, normalized size = 2.06
method | result | size |
elliptic | \(\frac {g \,x^{4} \sqrt {c \,x^{4}+b \,x^{2}+a}}{6}+\frac {f \,x^{3} \sqrt {c \,x^{4}+b \,x^{2}+a}}{5}+\frac {\left (\frac {b g}{6}+c e \right ) x^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{4 c}+\frac {\left (\frac {b f}{5}+c d \right ) x \sqrt {c \,x^{4}+b \,x^{2}+a}}{3 c}+\frac {\left (\frac {a g}{3}+e b -\frac {3 b \left (\frac {b g}{6}+c e \right )}{4 c}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{2 c}+\frac {\left (a d -\frac {a \left (\frac {b f}{5}+c d \right )}{3 c}\right ) \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \EllipticF \left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {\left (a e -\frac {a \left (\frac {b g}{6}+c e \right )}{2 c}-\frac {b \left (\frac {a g}{3}+e b -\frac {3 b \left (\frac {b g}{6}+c e \right )}{4 c}\right )}{2 c}\right ) \ln \left (\frac {2 c \,x^{2}+b}{\sqrt {c}}+2 \sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{2 \sqrt {c}}-\frac {\left (\frac {2 f a}{5}+b d -\frac {2 b \left (\frac {b f}{5}+c d \right )}{3 c}\right ) a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\EllipticF \left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-\EllipticE \left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{2 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}\) | \(620\) |
default | \(g \left (\frac {\left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{6 c}-\frac {b \,x^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{8 c}-\frac {b^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{16 c^{2}}-\frac {b \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right ) a}{8 c^{\frac {3}{2}}}+\frac {b^{3} \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{32 c^{\frac {5}{2}}}\right )+f \left (\frac {x^{3} \sqrt {c \,x^{4}+b \,x^{2}+a}}{5}+\frac {b x \sqrt {c \,x^{4}+b \,x^{2}+a}}{15 c}-\frac {b a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \EllipticF \left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{60 c \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {\left (\frac {2 a}{5}-\frac {2 b^{2}}{15 c}\right ) a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\EllipticF \left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-\EllipticE \left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{2 \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 )}\right )+e \left (\frac {\left (2 c \,x^{2}+b \right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{8 c}+\frac {\ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right ) a}{4 \sqrt {c}}-\frac {\ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right ) b^{2}}{16 c^{\frac {3}{2}}}\right )+d \left (\frac {x \sqrt {c \,x^{4}+b \,x^{2}+a}}{3}+\frac {a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \EllipticF \left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{6 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {b a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\EllipticF \left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-\EllipticE \left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{6 \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 )}\right )\) | \(1042\) |
risch | \(\text {Expression too large to display}\) | \(1501\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.26, size = 574, normalized size = 1.14 \begin {gather*} \frac {32 \, \sqrt {\frac {1}{2}} {\left ({\left (5 \, b c^{2} d - 2 \, {\left (b^{2} c - 3 \, a c^{2}\right )} f\right )} x \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - {\left (5 \, b^{2} c d - 2 \, {\left (b^{3} - 3 \, a b c\right )} f\right )} x\right )} \sqrt {c} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}} E(\arcsin \left (\frac {\sqrt {\frac {1}{2}} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}}}{x}\right )\,|\,\frac {b c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} + b^{2} - 2 \, a c}{2 \, a c}) - 32 \, \sqrt {\frac {1}{2}} {\left ({\left (5 \, {\left (b c^{2} - 2 \, c^{3}\right )} d - {\left (2 \, b^{2} c - {\left (6 \, a + b\right )} c^{2}\right )} f\right )} x \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - {\left (5 \, {\left (b^{2} c + 2 \, b c^{2}\right )} d - {\left (2 \, b^{3} - {\left (6 \, a b - b^{2}\right )} c\right )} f\right )} x\right )} \sqrt {c} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}} F(\arcsin \left (\frac {\sqrt {\frac {1}{2}} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}}}{x}\right )\,|\,\frac {b c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} + b^{2} - 2 \, a c}{2 \, a c}) + 15 \, {\left (2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} e - {\left (b^{3} - 4 \, a b c\right )} g\right )} \sqrt {c} x \log \left (8 \, c^{2} x^{4} + 8 \, b c x^{2} + b^{2} - 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} + b\right )} \sqrt {c} + 4 \, a c\right ) + 4 \, {\left (40 \, c^{3} g x^{5} + 48 \, c^{3} f x^{4} + 80 \, b c^{2} d + 10 \, {\left (6 \, c^{3} e + b c^{2} g\right )} x^{3} + 16 \, {\left (5 \, c^{3} d + b c^{2} f\right )} x^{2} - 32 \, {\left (b^{2} c - 3 \, a c^{2}\right )} f + 5 \, {\left (6 \, b c^{2} e - {\left (3 \, b^{2} c - 8 \, a c^{2}\right )} g\right )} x\right )} \sqrt {c x^{4} + b x^{2} + a}}{960 \, c^{3} x} \end {gather*}
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 \begin {gather*} \int \sqrt {a + b x^{2} + c x^{4}} \left (d + e x + f x^{2} + g x^{3}\right )\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \sqrt {c\,x^4+b\,x^2+a}\,\left (g\,x^3+f\,x^2+e\,x+d\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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