Optimal. Leaf size=495 \[ -\frac {\sqrt [4]{a} \left (\sqrt {a} \sqrt {c} \left (60 a^2 c^2-51 a b^2 c+8 b^4\right )+8 b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right )\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2310 c^{15/4} \sqrt {a+b x^2+c x^4}}+\frac {x \left (60 a^2 c^2-51 a b^2 c+8 b^4\right ) \sqrt {a+b x^2+c x^4}}{1155 c^3}-\frac {8 b x \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right ) \sqrt {a+b x^2+c x^4}}{1155 c^{7/2} \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {8 \sqrt [4]{a} b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{1155 c^{15/4} \sqrt {a+b x^2+c x^4}}-\frac {x^3 \left (10 c x^2 \left (b^2-3 a c\right )+b \left (a c+2 b^2\right )\right ) \sqrt {a+b x^2+c x^4}}{385 c^2}+\frac {x^3 \left (b+3 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{33 c} \]
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Rubi [A] time = 0.44, antiderivative size = 495, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1116, 1273, 1279, 1197, 1103, 1195} \[ \frac {x \left (60 a^2 c^2-51 a b^2 c+8 b^4\right ) \sqrt {a+b x^2+c x^4}}{1155 c^3}-\frac {\sqrt [4]{a} \left (\sqrt {a} \sqrt {c} \left (60 a^2 c^2-51 a b^2 c+8 b^4\right )+8 b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right )\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2310 c^{15/4} \sqrt {a+b x^2+c x^4}}-\frac {x^3 \left (10 c x^2 \left (b^2-3 a c\right )+b \left (a c+2 b^2\right )\right ) \sqrt {a+b x^2+c x^4}}{385 c^2}-\frac {8 b x \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right ) \sqrt {a+b x^2+c x^4}}{1155 c^{7/2} \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {8 \sqrt [4]{a} b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{1155 c^{15/4} \sqrt {a+b x^2+c x^4}}+\frac {x^3 \left (b+3 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{33 c} \]
Antiderivative was successfully verified.
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Rule 1103
Rule 1116
Rule 1195
Rule 1197
Rule 1273
Rule 1279
Rubi steps
\begin {align*} \int x^4 \left (a+b x^2+c x^4\right )^{3/2} \, dx &=\frac {x^3 \left (b+3 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{33 c}-\frac {\int x^2 \left (3 a b+6 \left (b^2-3 a c\right ) x^2\right ) \sqrt {a+b x^2+c x^4} \, dx}{33 c}\\ &=-\frac {x^3 \left (b \left (2 b^2+a c\right )+10 c \left (b^2-3 a c\right ) x^2\right ) \sqrt {a+b x^2+c x^4}}{385 c^2}+\frac {x^3 \left (b+3 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{33 c}-\frac {\int \frac {x^2 \left (-6 a b \left (3 b^2-16 a c\right )-3 \left (8 b^4-51 a b^2 c+60 a^2 c^2\right ) x^2\right )}{\sqrt {a+b x^2+c x^4}} \, dx}{1155 c^2}\\ &=\frac {\left (8 b^4-51 a b^2 c+60 a^2 c^2\right ) x \sqrt {a+b x^2+c x^4}}{1155 c^3}-\frac {x^3 \left (b \left (2 b^2+a c\right )+10 c \left (b^2-3 a c\right ) x^2\right ) \sqrt {a+b x^2+c x^4}}{385 c^2}+\frac {x^3 \left (b+3 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{33 c}+\frac {\int \frac {-3 a \left (8 b^4-51 a b^2 c+60 a^2 c^2\right )-24 b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right ) x^2}{\sqrt {a+b x^2+c x^4}} \, dx}{3465 c^3}\\ &=\frac {\left (8 b^4-51 a b^2 c+60 a^2 c^2\right ) x \sqrt {a+b x^2+c x^4}}{1155 c^3}-\frac {x^3 \left (b \left (2 b^2+a c\right )+10 c \left (b^2-3 a c\right ) x^2\right ) \sqrt {a+b x^2+c x^4}}{385 c^2}+\frac {x^3 \left (b+3 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{33 c}+\frac {\left (8 \sqrt {a} b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right )\right ) \int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}{\sqrt {a+b x^2+c x^4}} \, dx}{1155 c^{7/2}}-\frac {\left (\sqrt {a} \left (8 b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right )+\sqrt {a} \sqrt {c} \left (8 b^4-51 a b^2 c+60 a^2 c^2\right )\right )\right ) \int \frac {1}{\sqrt {a+b x^2+c x^4}} \, dx}{1155 c^{7/2}}\\ &=\frac {\left (8 b^4-51 a b^2 c+60 a^2 c^2\right ) x \sqrt {a+b x^2+c x^4}}{1155 c^3}-\frac {8 b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right ) x \sqrt {a+b x^2+c x^4}}{1155 c^{7/2} \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {x^3 \left (b \left (2 b^2+a c\right )+10 c \left (b^2-3 a c\right ) x^2\right ) \sqrt {a+b x^2+c x^4}}{385 c^2}+\frac {x^3 \left (b+3 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{33 c}+\frac {8 \sqrt [4]{a} b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\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 )}{1155 c^{15/4} \sqrt {a+b x^2+c x^4}}-\frac {\sqrt [4]{a} \left (8 b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right )+\sqrt {a} \sqrt {c} \left (8 b^4-51 a b^2 c+60 a^2 c^2\right )\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 )}{2310 c^{15/4} \sqrt {a+b x^2+c x^4}}\\ \end {align*}
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Mathematica [C] time = 2.21, size = 657, normalized size = 1.33 \[ \frac {-4 i b \left (27 a^2 c^2-15 a b^2 c+2 b^4\right ) \left (\sqrt {b^2-4 a c}-b\right ) \sqrt {\frac {\sqrt {b^2-4 a c}+b+2 c x^2}{\sqrt {b^2-4 a c}+b}} \sqrt {\frac {-2 \sqrt {b^2-4 a c}+2 b+4 c x^2}{b-\sqrt {b^2-4 a c}}} E\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )+2 c x \sqrt {\frac {c}{\sqrt {b^2-4 a c}+b}} \left (60 a^3 c^2+a^2 c \left (-51 b^2+92 b c x^2+255 c^2 x^4\right )+a \left (8 b^4-57 b^3 c x^2-14 b^2 c^2 x^4+367 b c^3 x^6+300 c^4 x^8\right )+x^2 \left (8 b^5+2 b^4 c x^2-b^3 c^2 x^4+145 b^2 c^3 x^6+245 b c^4 x^8+105 c^5 x^{10}\right )\right )+i \left (60 a^3 c^3-159 a^2 b^2 c^2+108 a^2 b c^2 \sqrt {b^2-4 a c}+68 a b^4 c+8 b^5 \sqrt {b^2-4 a c}-60 a b^3 c \sqrt {b^2-4 a c}-8 b^6\right ) \sqrt {\frac {\sqrt {b^2-4 a c}+b+2 c x^2}{\sqrt {b^2-4 a c}+b}} \sqrt {\frac {-2 \sqrt {b^2-4 a c}+2 b+4 c x^2}{b-\sqrt {b^2-4 a c}}} F\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{2310 c^4 \sqrt {\frac {c}{\sqrt {b^2-4 a c}+b}} \sqrt {a+b x^2+c x^4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.89, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (c x^{8} + b x^{6} + a x^{4}\right )} \sqrt {c x^{4} + b x^{2} + a}, x\right ) \]
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 \[ \int {\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}} x^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 674, normalized size = 1.36 \[ \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, c \,x^{9}}{11}+\frac {4 \sqrt {c \,x^{4}+b \,x^{2}+a}\, b \,x^{7}}{33}+\frac {\left (\frac {13 a c}{11}+\frac {b^{2}}{33}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{5}}{7 c}+\frac {\left (\frac {38 a b}{33}-\frac {6 \left (\frac {13 a c}{11}+\frac {b^{2}}{33}\right ) b}{7 c}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{3}}{5 c}-\frac {\left (a^{2}-\frac {5 \left (\frac {13 a c}{11}+\frac {b^{2}}{33}\right ) a}{7 c}-\frac {4 \left (\frac {38 a b}{33}-\frac {6 \left (\frac {13 a c}{11}+\frac {b^{2}}{33}\right ) b}{7 c}\right ) b}{5 c}\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}\, a \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 )}{12 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, c}-\frac {\left (-\frac {3 \left (\frac {38 a b}{33}-\frac {6 \left (\frac {13 a c}{11}+\frac {b^{2}}{33}\right ) b}{7 c}\right ) a}{5 c}-\frac {2 \left (a^{2}-\frac {5 \left (\frac {13 a c}{11}+\frac {b^{2}}{33}\right ) a}{7 c}-\frac {4 \left (\frac {38 a b}{33}-\frac {6 \left (\frac {13 a c}{11}+\frac {b^{2}}{33}\right ) b}{7 c}\right ) b}{5 c}\right ) b}{3 c}\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}\, \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}{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 )}+\frac {\left (a^{2}-\frac {5 \left (\frac {13 a c}{11}+\frac {b^{2}}{33}\right ) a}{7 c}-\frac {4 \left (\frac {38 a b}{33}-\frac {6 \left (\frac {13 a c}{11}+\frac {b^{2}}{33}\right ) b}{7 c}\right ) b}{5 c}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}\, x}{3 c} \]
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 \[ \int {\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}} x^{4}\,{d x} \]
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 \[ \int x^4\,{\left (c\,x^4+b\,x^2+a\right )}^{3/2} \,d x \]
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 \[ \int x^{4} \left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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