Optimal. Leaf size=403 \[ -\frac {(4 b B-5 A c) x \sqrt {a+b x^2+c x^4}}{15 c^2}+\frac {B x^3 \sqrt {a+b x^2+c x^4}}{5 c}+\frac {\left (8 b^2 B-10 A b c-9 a B c\right ) x \sqrt {a+b x^2+c x^4}}{15 c^{5/2} \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {\sqrt [4]{a} \left (8 b^2 B-10 A b c-9 a B c\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{15 c^{11/4} \sqrt {a+b x^2+c x^4}}+\frac {\sqrt [4]{a} \left (8 b^2 B-10 A b c-9 a B c+\sqrt {a} \sqrt {c} (4 b B-5 A c)\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{30 c^{11/4} \sqrt {a+b x^2+c x^4}} \]
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Rubi [A]
time = 0.18, antiderivative size = 403, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {1293, 1211,
1117, 1209} \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 (\sqrt {a} \sqrt {c} (4 b B-5 A c)-9 a B c-10 A b c+8 b^2 B\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^{11/4} \sqrt {a+b x^2+c x^4}}-\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 (-9 a B c-10 A b c+8 b^2 B\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^{11/4} \sqrt {a+b x^2+c x^4}}+\frac {x \sqrt {a+b x^2+c x^4} \left (-9 a B c-10 A b c+8 b^2 B\right )}{15 c^{5/2} \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {x \sqrt {a+b x^2+c x^4} (4 b B-5 A c)}{15 c^2}+\frac {B x^3 \sqrt {a+b x^2+c x^4}}{5 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 1117
Rule 1209
Rule 1211
Rule 1293
Rubi steps
\begin {align*} \int \frac {x^4 \left (A+B x^2\right )}{\sqrt {a+b x^2+c x^4}} \, dx &=\frac {B x^3 \sqrt {a+b x^2+c x^4}}{5 c}-\frac {\int \frac {x^2 \left (3 a B+(4 b B-5 A c) x^2\right )}{\sqrt {a+b x^2+c x^4}} \, dx}{5 c}\\ &=-\frac {(4 b B-5 A c) x \sqrt {a+b x^2+c x^4}}{15 c^2}+\frac {B x^3 \sqrt {a+b x^2+c x^4}}{5 c}+\frac {\int \frac {a (4 b B-5 A c)+\left (8 b^2 B-10 A b c-9 a B c\right ) x^2}{\sqrt {a+b x^2+c x^4}} \, dx}{15 c^2}\\ &=-\frac {(4 b B-5 A c) x \sqrt {a+b x^2+c x^4}}{15 c^2}+\frac {B x^3 \sqrt {a+b x^2+c x^4}}{5 c}-\frac {\left (\sqrt {a} \left (8 b^2 B-10 A b c-9 a B c\right )\right ) \int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}{\sqrt {a+b x^2+c x^4}} \, dx}{15 c^{5/2}}+\frac {\left (\sqrt {a} \left (8 b^2 B-10 A b c-9 a B c+\sqrt {a} \sqrt {c} (4 b B-5 A c)\right )\right ) \int \frac {1}{\sqrt {a+b x^2+c x^4}} \, dx}{15 c^{5/2}}\\ &=-\frac {(4 b B-5 A c) x \sqrt {a+b x^2+c x^4}}{15 c^2}+\frac {B x^3 \sqrt {a+b x^2+c x^4}}{5 c}+\frac {\left (8 b^2 B-10 A b c-9 a B c\right ) x \sqrt {a+b x^2+c x^4}}{15 c^{5/2} \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {\sqrt [4]{a} \left (8 b^2 B-10 A b c-9 a B 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 )}{15 c^{11/4} \sqrt {a+b x^2+c x^4}}+\frac {\sqrt [4]{a} \left (8 b^2 B-10 A b c-9 a B c+\sqrt {a} \sqrt {c} (4 b B-5 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}} 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^{11/4} \sqrt {a+b x^2+c x^4}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 11.36, size = 532, normalized size = 1.32 \begin {gather*} \frac {4 c \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x \left (-4 b B+5 A c+3 B c x^2\right ) \left (a+b x^2+c x^4\right )+i \left (8 b^2 B-10 A b c-9 a B c\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^3 B+b c \left (17 a B-10 A \sqrt {b^2-4 a c}\right )+2 b^2 \left (5 A c+4 B \sqrt {b^2-4 a c}\right )-a c \left (10 A c+9 B \sqrt {b^2-4 a c}\right )\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 )}{60 c^3 \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 [B] Leaf count of result is larger than twice the leaf count of optimal. \(814\) vs.
\(2(391)=782\).
time = 0.04, size = 815, normalized size = 2.02 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 \frac {x^{4} \left (A + B x^{2}\right )}{\sqrt {a + b x^{2} + c x^{4}}}\, 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 \frac {x^4\,\left (B\,x^2+A\right )}{\sqrt {c\,x^4+b\,x^2+a}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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