Optimal. Leaf size=409 \[ -\frac {x \sqrt {a+b x^2-c x^4}}{3 c}-\frac {b \left (b-\sqrt {b^2+4 a c}\right ) \sqrt {b+\sqrt {b^2+4 a c}} \sqrt {1-\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}} \sqrt {1-\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}} E\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2+4 a c}}}\right )|\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{3 \sqrt {2} c^{5/2} \sqrt {a+b x^2-c x^4}}+\frac {\sqrt {b+\sqrt {b^2+4 a c}} \left (b^2+a c-b \sqrt {b^2+4 a c}\right ) \sqrt {1-\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}} \sqrt {1-\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2+4 a c}}}\right )|\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{3 \sqrt {2} c^{5/2} \sqrt {a+b x^2-c x^4}} \]
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Rubi [A]
time = 0.41, antiderivative size = 409, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {1136, 1216,
538, 435, 430} \begin {gather*} \frac {\sqrt {\sqrt {4 a c+b^2}+b} \left (-b \sqrt {4 a c+b^2}+a c+b^2\right ) \sqrt {1-\frac {2 c x^2}{b-\sqrt {4 a c+b^2}}} \sqrt {1-\frac {2 c x^2}{\sqrt {4 a c+b^2}+b}} F\left (\text {ArcSin}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2+4 a c}}}\right )|\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{3 \sqrt {2} c^{5/2} \sqrt {a+b x^2-c x^4}}-\frac {b \left (b-\sqrt {4 a c+b^2}\right ) \sqrt {\sqrt {4 a c+b^2}+b} \sqrt {1-\frac {2 c x^2}{b-\sqrt {4 a c+b^2}}} \sqrt {1-\frac {2 c x^2}{\sqrt {4 a c+b^2}+b}} E\left (\text {ArcSin}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2+4 a c}}}\right )|\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{3 \sqrt {2} c^{5/2} \sqrt {a+b x^2-c x^4}}-\frac {x \sqrt {a+b x^2-c x^4}}{3 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 430
Rule 435
Rule 538
Rule 1136
Rule 1216
Rubi steps
\begin {align*} \int \frac {x^4}{\sqrt {a+b x^2-c x^4}} \, dx &=-\frac {x \sqrt {a+b x^2-c x^4}}{3 c}+\frac {\int \frac {a+2 b x^2}{\sqrt {a+b x^2-c x^4}} \, dx}{3 c}\\ &=-\frac {x \sqrt {a+b x^2-c x^4}}{3 c}+\frac {\left (\sqrt {1-\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}} \sqrt {1-\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}}\right ) \int \frac {a+2 b x^2}{\sqrt {1-\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}} \sqrt {1-\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}}} \, dx}{3 c \sqrt {a+b x^2-c x^4}}\\ &=-\frac {x \sqrt {a+b x^2-c x^4}}{3 c}-\frac {\left (b \left (b-\sqrt {b^2+4 a c}\right ) \sqrt {1-\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}} \sqrt {1-\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}}\right ) \int \frac {\sqrt {1-\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}}}{\sqrt {1-\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}}} \, dx}{3 c^2 \sqrt {a+b x^2-c x^4}}+\frac {\left (\left (b^2+a c-b \sqrt {b^2+4 a c}\right ) \sqrt {1-\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}} \sqrt {1-\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}}\right ) \int \frac {1}{\sqrt {1-\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}} \sqrt {1-\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}}} \, dx}{3 c^2 \sqrt {a+b x^2-c x^4}}\\ &=-\frac {x \sqrt {a+b x^2-c x^4}}{3 c}-\frac {b \left (b-\sqrt {b^2+4 a c}\right ) \sqrt {b+\sqrt {b^2+4 a c}} \sqrt {1-\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}} \sqrt {1-\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}} E\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2+4 a c}}}\right )|\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{3 \sqrt {2} c^{5/2} \sqrt {a+b x^2-c x^4}}+\frac {\sqrt {b+\sqrt {b^2+4 a c}} \left (b^2+a c-b \sqrt {b^2+4 a c}\right ) \sqrt {1-\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}} \sqrt {1-\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2+4 a c}}}\right )|\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{3 \sqrt {2} c^{5/2} \sqrt {a+b x^2-c x^4}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 10.52, size = 459, normalized size = 1.12 \begin {gather*} \frac {2 c \sqrt {-\frac {c}{b+\sqrt {b^2+4 a c}}} x \left (-a-b x^2+c x^4\right )-i \sqrt {2} b \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 {-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 )+i \sqrt {2} \left (-b^2-a c+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 {-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 )}{6 c^2 \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 = 0.04, size = 391, normalized size = 0.96
method | result | size |
default | \(-\frac {x \sqrt {-c \,x^{4}+b \,x^{2}+a}}{3 c}+\frac {a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \EllipticF \left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )}{12 c \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}\, \sqrt {-c \,x^{4}+b \,x^{2}+a}}-\frac {b a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\EllipticF \left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )-\EllipticE \left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{3 c \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}\, \sqrt {-c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {4 a c +b^{2}}\right )}\) | \(391\) |
risch | \(-\frac {x \sqrt {-c \,x^{4}+b \,x^{2}+a}}{3 c}+\frac {-\frac {b a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\EllipticF \left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )-\EllipticE \left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{\sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}\, \sqrt {-c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {4 a c +b^{2}}\right )}+\frac {a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \EllipticF \left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}\, \sqrt {-c \,x^{4}+b \,x^{2}+a}}}{3 c}\) | \(391\) |
elliptic | \(-\frac {x \sqrt {-c \,x^{4}+b \,x^{2}+a}}{3 c}+\frac {a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \EllipticF \left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )}{12 c \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}\, \sqrt {-c \,x^{4}+b \,x^{2}+a}}-\frac {b a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\EllipticF \left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )-\EllipticE \left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{3 c \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}\, \sqrt {-c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {4 a c +b^{2}}\right )}\) | \(391\) |
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}}{\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}{\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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