Optimal. Leaf size=495 \[ \frac {\left (8 b^4-51 a b^2 c+60 a^2 c^2\right ) x \sqrt {a+b x^2+c x^4}}{1155 c^3}-\frac {8 b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right ) x \sqrt {a+b x^2+c x^4}}{1155 c^{7/2} \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {x^3 \left (b \left (2 b^2+a c\right )+10 c \left (b^2-3 a c\right ) x^2\right ) \sqrt {a+b x^2+c x^4}}{385 c^2}+\frac {x^3 \left (b+3 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{33 c}+\frac {8 \sqrt [4]{a} b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{1155 c^{15/4} \sqrt {a+b x^2+c x^4}}-\frac {\sqrt [4]{a} \left (8 b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right )+\sqrt {a} \sqrt {c} \left (8 b^4-51 a b^2 c+60 a^2 c^2\right )\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2310 c^{15/4} \sqrt {a+b x^2+c x^4}} \]
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Rubi [A]
time = 0.28, 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 = {1130, 1287,
1293, 1211, 1117, 1209} \begin {gather*} -\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 \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 )}{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 \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 \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 )}{1155 c^{15/4} \sqrt {a+b x^2+c x^4}}-\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 {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} \end {gather*}
Antiderivative was successfully verified.
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Rule 1117
Rule 1130
Rule 1209
Rule 1211
Rule 1287
Rule 1293
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] Result contains complex when optimal does not.
time = 11.46, size = 657, normalized size = 1.33 \begin {gather*} \frac {2 c \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x \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 )-4 i b \left (2 b^4-15 a b^2 c+27 a^2 c^2\right ) \left (-b+\sqrt {b^2-4 a c}\right ) \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+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 )+i \left (-8 b^6+68 a b^4 c-159 a^2 b^2 c^2+60 a^3 c^3+8 b^5 \sqrt {b^2-4 a c}-60 a b^3 c \sqrt {b^2-4 a c}+108 a^2 b c^2 \sqrt {b^2-4 a c}\right ) \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+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}{b+\sqrt {b^2-4 a c}}} \sqrt {a+b x^2+c x^4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 674, normalized size = 1.36 Too large to display
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 [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 x^{4} \left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}\, 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 x^4\,{\left (c\,x^4+b\,x^2+a\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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