Optimal. Leaf size=443 \[ \frac {x \left (84 a^2 c^2-57 a b^2 c+8 b^4\right ) \sqrt {a+b x^2+c x^4}}{315 c^{5/2} \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {\sqrt [4]{a} \left (84 a^2 c^2-57 a b^2 c+4 \sqrt {a} b \sqrt {c} \left (b^2-6 a c\right )+8 b^4\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{630 c^{11/4} \sqrt {a+b x^2+c x^4}}-\frac {\sqrt [4]{a} \left (84 a^2 c^2-57 a b^2 c+8 b^4\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{315 c^{11/4} \sqrt {a+b x^2+c x^4}}-\frac {x \left (6 c x^2 \left (2 b^2-7 a c\right )+b \left (4 b^2-9 a c\right )\right ) \sqrt {a+b x^2+c x^4}}{315 c^2}+\frac {x \left (3 b+7 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{63 c} \]
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Rubi [A] time = 0.28, antiderivative size = 443, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1116, 1176, 1197, 1103, 1195} \[ \frac {x \left (84 a^2 c^2-57 a b^2 c+8 b^4\right ) \sqrt {a+b x^2+c x^4}}{315 c^{5/2} \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {\sqrt [4]{a} \left (84 a^2 c^2-57 a b^2 c+4 \sqrt {a} b \sqrt {c} \left (b^2-6 a c\right )+8 b^4\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{630 c^{11/4} \sqrt {a+b x^2+c x^4}}-\frac {\sqrt [4]{a} \left (84 a^2 c^2-57 a b^2 c+8 b^4\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{315 c^{11/4} \sqrt {a+b x^2+c x^4}}-\frac {x \left (6 c x^2 \left (2 b^2-7 a c\right )+b \left (4 b^2-9 a c\right )\right ) \sqrt {a+b x^2+c x^4}}{315 c^2}+\frac {x \left (3 b+7 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{63 c} \]
Antiderivative was successfully verified.
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Rule 1103
Rule 1116
Rule 1176
Rule 1195
Rule 1197
Rubi steps
\begin {align*} \int x^2 \left (a+b x^2+c x^4\right )^{3/2} \, dx &=\frac {x \left (3 b+7 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{63 c}-\frac {\int \left (a b+2 \left (2 b^2-7 a c\right ) x^2\right ) \sqrt {a+b x^2+c x^4} \, dx}{21 c}\\ &=-\frac {x \left (b \left (4 b^2-9 a c\right )+6 c \left (2 b^2-7 a c\right ) x^2\right ) \sqrt {a+b x^2+c x^4}}{315 c^2}+\frac {x \left (3 b+7 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{63 c}-\frac {\int \frac {-4 a b \left (b^2-6 a c\right )+\left (-8 b^4+57 a b^2 c-84 a^2 c^2\right ) x^2}{\sqrt {a+b x^2+c x^4}} \, dx}{315 c^2}\\ &=-\frac {x \left (b \left (4 b^2-9 a c\right )+6 c \left (2 b^2-7 a c\right ) x^2\right ) \sqrt {a+b x^2+c x^4}}{315 c^2}+\frac {x \left (3 b+7 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{63 c}-\frac {\left (\sqrt {a} \left (8 b^4-57 a b^2 c+84 a^2 c^2\right )\right ) \int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}{\sqrt {a+b x^2+c x^4}} \, dx}{315 c^{5/2}}+\frac {\left (\sqrt {a} \left (8 b^4-57 a b^2 c+84 a^2 c^2+4 \sqrt {a} b \sqrt {c} \left (b^2-6 a c\right )\right )\right ) \int \frac {1}{\sqrt {a+b x^2+c x^4}} \, dx}{315 c^{5/2}}\\ &=\frac {\left (8 b^4-57 a b^2 c+84 a^2 c^2\right ) x \sqrt {a+b x^2+c x^4}}{315 c^{5/2} \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {x \left (b \left (4 b^2-9 a c\right )+6 c \left (2 b^2-7 a c\right ) x^2\right ) \sqrt {a+b x^2+c x^4}}{315 c^2}+\frac {x \left (3 b+7 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{63 c}-\frac {\sqrt [4]{a} \left (8 b^4-57 a b^2 c+84 a^2 c^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}} 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 )}{315 c^{11/4} \sqrt {a+b x^2+c x^4}}+\frac {\sqrt [4]{a} \left (8 b^4-57 a b^2 c+84 a^2 c^2+4 \sqrt {a} b \sqrt {c} \left (b^2-6 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 )}{630 c^{11/4} \sqrt {a+b x^2+c x^4}}\\ \end {align*}
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Mathematica [C] time = 1.89, size = 602, normalized size = 1.36 \[ \frac {i \left (84 a^2 c^2-57 a b^2 c+8 b^4\right ) \left (\sqrt {b^2-4 a c}-b\right ) \sqrt {\frac {\sqrt {b^2-4 a c}+b+2 c x^2}{\sqrt {b^2-4 a c}+b}} \sqrt {\frac {-2 \sqrt {b^2-4 a c}+2 b+4 c x^2}{b-\sqrt {b^2-4 a c}}} E\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )+4 c x \sqrt {\frac {c}{\sqrt {b^2-4 a c}+b}} \left (a^2 c \left (24 b+77 c x^2\right )+a \left (-4 b^3+27 b^2 c x^2+151 b c^2 x^4+112 c^3 x^6\right )-4 b^4 x^2-b^3 c x^4+53 b^2 c^2 x^6+85 b c^3 x^8+35 c^4 x^{10}\right )-i \left (84 a^2 c^2 \sqrt {b^2-4 a c}-132 a^2 b c^2+65 a b^3 c-57 a b^2 c \sqrt {b^2-4 a c}+8 b^4 \sqrt {b^2-4 a c}-8 b^5\right ) \sqrt {\frac {\sqrt {b^2-4 a c}+b+2 c x^2}{\sqrt {b^2-4 a c}+b}} \sqrt {\frac {-2 \sqrt {b^2-4 a c}+2 b+4 c x^2}{b-\sqrt {b^2-4 a c}}} F\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{1260 c^3 \sqrt {\frac {c}{\sqrt {b^2-4 a c}+b}} \sqrt {a+b x^2+c x^4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.73, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (c x^{6} + b x^{4} + a x^{2}\right )} \sqrt {c x^{4} + b x^{2} + a}, x\right ) \]
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 \[ \int {\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}} x^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 545, normalized size = 1.23 \[ \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, c \,x^{7}}{9}+\frac {10 \sqrt {c \,x^{4}+b \,x^{2}+a}\, b \,x^{5}}{63}+\frac {\left (\frac {11 a c}{9}+\frac {b^{2}}{21}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{3}}{5 c}-\frac {\left (\frac {76 a b}{63}-\frac {4 \left (\frac {11 a c}{9}+\frac {b^{2}}{21}\right ) b}{5 c}\right ) \sqrt {2}\, \sqrt {-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}+4}\, \sqrt {\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}+4}\, a \EllipticF \left (\frac {\sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, x}{2}, \frac {\sqrt {\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) b}{a c}-4}}{2}\right )}{12 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, c}-\frac {\left (a^{2}-\frac {3 \left (\frac {11 a c}{9}+\frac {b^{2}}{21}\right ) a}{5 c}-\frac {2 \left (\frac {76 a b}{63}-\frac {4 \left (\frac {11 a c}{9}+\frac {b^{2}}{21}\right ) b}{5 c}\right ) b}{3 c}\right ) \sqrt {2}\, \sqrt {-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}+4}\, \sqrt {\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}+4}\, \left (-\EllipticE \left (\frac {\sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, x}{2}, \frac {\sqrt {\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) b}{a c}-4}}{2}\right )+\EllipticF \left (\frac {\sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, x}{2}, \frac {\sqrt {\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) b}{a c}-4}}{2}\right )\right ) a}{2 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}+\frac {\left (\frac {76 a b}{63}-\frac {4 \left (\frac {11 a c}{9}+\frac {b^{2}}{21}\right ) b}{5 c}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}\, x}{3 c} \]
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 \[ \int {\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}} x^{2}\,{d x} \]
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 \[ \int x^2\,{\left (c\,x^4+b\,x^2+a\right )}^{3/2} \,d x \]
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 \[ \int x^{2} \left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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