Optimal. Leaf size=173 \[ \frac {\sqrt {a+b x^2+c x^4}}{2 c e}-\frac {(2 c d+b e) \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{4 c^{3/2} e^2}+\frac {d^2 \tanh ^{-1}\left (\frac {b d-2 a e+(2 c d-b e) x^2}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x^2+c x^4}}\right )}{2 e^2 \sqrt {c d^2-b d e+a e^2}} \]
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Rubi [A]
time = 0.21, antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {1265, 1667,
857, 635, 212, 738} \begin {gather*} -\frac {(b e+2 c d) \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{4 c^{3/2} e^2}+\frac {d^2 \tanh ^{-1}\left (\frac {-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt {a+b x^2+c x^4} \sqrt {a e^2-b d e+c d^2}}\right )}{2 e^2 \sqrt {a e^2-b d e+c d^2}}+\frac {\sqrt {a+b x^2+c x^4}}{2 c e} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 635
Rule 738
Rule 857
Rule 1265
Rule 1667
Rubi steps
\begin {align*} \int \frac {x^5}{\left (d+e x^2\right ) \sqrt {a+b x^2+c x^4}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^2}{(d+e x) \sqrt {a+b x+c x^2}} \, dx,x,x^2\right )\\ &=\frac {\sqrt {a+b x^2+c x^4}}{2 c e}+\frac {\text {Subst}\left (\int \frac {-\frac {1}{2} b d e-\frac {1}{2} e (2 c d+b e) x}{(d+e x) \sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{2 c e^2}\\ &=\frac {\sqrt {a+b x^2+c x^4}}{2 c e}+\frac {d^2 \text {Subst}\left (\int \frac {1}{(d+e x) \sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{2 e^2}-\frac {(2 c d+b e) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{4 c e^2}\\ &=\frac {\sqrt {a+b x^2+c x^4}}{2 c e}-\frac {d^2 \text {Subst}\left (\int \frac {1}{4 c d^2-4 b d e+4 a e^2-x^2} \, dx,x,\frac {-b d+2 a e-(2 c d-b e) x^2}{\sqrt {a+b x^2+c x^4}}\right )}{e^2}-\frac {(2 c d+b e) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x^2}{\sqrt {a+b x^2+c x^4}}\right )}{2 c e^2}\\ &=\frac {\sqrt {a+b x^2+c x^4}}{2 c e}-\frac {(2 c d+b e) \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{4 c^{3/2} e^2}+\frac {d^2 \tanh ^{-1}\left (\frac {b d-2 a e+(2 c d-b e) x^2}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x^2+c x^4}}\right )}{2 e^2 \sqrt {c d^2-b d e+a e^2}}\\ \end {align*}
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Mathematica [A]
time = 0.66, size = 178, normalized size = 1.03 \begin {gather*} \frac {\frac {2 e \sqrt {a+b x^2+c x^4}}{c}+\frac {4 d^2 \sqrt {-c d^2+b d e-a e^2} \tan ^{-1}\left (\frac {\sqrt {c} \left (d+e x^2\right )-e \sqrt {a+b x^2+c x^4}}{\sqrt {-c d^2+e (b d-a e)}}\right )}{c d^2+e (-b d+a e)}+\frac {(2 c d+b e) \log \left (c \left (b+2 c x^2-2 \sqrt {c} \sqrt {a+b x^2+c x^4}\right )\right )}{c^{3/2}}}{4 e^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 266, normalized size = 1.54
method | result | size |
default | \(\frac {\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{2 c}-\frac {b \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{4 c^{\frac {3}{2}}}}{e}-\frac {d \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{2 e^{2} \sqrt {c}}-\frac {d^{2} \ln \left (\frac {\frac {2 a \,e^{2}-2 d e b +2 c \,d^{2}}{e^{2}}+\frac {\left (e b -2 c d \right ) \left (x^{2}+\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-d e b +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x^{2}+\frac {d}{e}\right )^{2}+\frac {\left (e b -2 c d \right ) \left (x^{2}+\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-d e b +c \,d^{2}}{e^{2}}}}{x^{2}+\frac {d}{e}}\right )}{2 e^{3} \sqrt {\frac {a \,e^{2}-d e b +c \,d^{2}}{e^{2}}}}\) | \(266\) |
risch | \(\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{2 c e}-\frac {\ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right ) b}{4 c^{\frac {3}{2}} e}-\frac {d \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{2 e^{2} \sqrt {c}}-\frac {d^{2} \ln \left (\frac {\frac {2 a \,e^{2}-2 d e b +2 c \,d^{2}}{e^{2}}+\frac {\left (e b -2 c d \right ) \left (x^{2}+\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-d e b +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x^{2}+\frac {d}{e}\right )^{2}+\frac {\left (e b -2 c d \right ) \left (x^{2}+\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-d e b +c \,d^{2}}{e^{2}}}}{x^{2}+\frac {d}{e}}\right )}{2 e^{3} \sqrt {\frac {a \,e^{2}-d e b +c \,d^{2}}{e^{2}}}}\) | \(267\) |
elliptic | \(\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{2 c e}-\frac {\ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right ) b}{4 c^{\frac {3}{2}} e}-\frac {d \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{2 e^{2} \sqrt {c}}-\frac {d^{2} \ln \left (\frac {\frac {2 a \,e^{2}-2 d e b +2 c \,d^{2}}{e^{2}}+\frac {\left (e b -2 c d \right ) \left (x^{2}+\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-d e b +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x^{2}+\frac {d}{e}\right )^{2}+\frac {\left (e b -2 c d \right ) \left (x^{2}+\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-d e b +c \,d^{2}}{e^{2}}}}{x^{2}+\frac {d}{e}}\right )}{2 e^{3} \sqrt {\frac {a \,e^{2}-d e b +c \,d^{2}}{e^{2}}}}\) | \(267\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 318 vs.
\(2 (149) = 298\).
time = 12.61, size = 1364, normalized size = 7.88 \begin {gather*} \left [\frac {2 \, \sqrt {c d^{2} - b d e + a e^{2}} c^{2} d^{2} \log \left (-\frac {8 \, c^{2} d^{2} x^{4} + 8 \, b c d^{2} x^{2} + {\left (b^{2} + 4 \, a c\right )} d^{2} + 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c d x^{2} + b d - {\left (b x^{2} + 2 \, a\right )} e\right )} \sqrt {c d^{2} - b d e + a e^{2}} + {\left ({\left (b^{2} + 4 \, a c\right )} x^{4} + 8 \, a b x^{2} + 8 \, a^{2}\right )} e^{2} - 2 \, {\left (4 \, b c d x^{4} + {\left (3 \, b^{2} + 4 \, a c\right )} d x^{2} + 4 \, a b d\right )} e}{x^{4} e^{2} + 2 \, d x^{2} e + d^{2}}\right ) + {\left (2 \, c^{2} d^{3} - b c d^{2} e + a b e^{3} - {\left (b^{2} - 2 \, a c\right )} d e^{2}\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{4} - 8 \, b c x^{2} - b^{2} + 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} + b\right )} \sqrt {c} - 4 \, a c\right ) + 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (c^{2} d^{2} e - b c d e^{2} + a c e^{3}\right )}}{8 \, {\left (c^{3} d^{2} e^{2} - b c^{2} d e^{3} + a c^{2} e^{4}\right )}}, \frac {4 \, \sqrt {-c d^{2} + b d e - a e^{2}} c^{2} d^{2} \arctan \left (-\frac {\sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c d x^{2} + b d - {\left (b x^{2} + 2 \, a\right )} e\right )} \sqrt {-c d^{2} + b d e - a e^{2}}}{2 \, {\left (c^{2} d^{2} x^{4} + b c d^{2} x^{2} + a c d^{2} + {\left (a c x^{4} + a b x^{2} + a^{2}\right )} e^{2} - {\left (b c d x^{4} + b^{2} d x^{2} + a b d\right )} e\right )}}\right ) + {\left (2 \, c^{2} d^{3} - b c d^{2} e + a b e^{3} - {\left (b^{2} - 2 \, a c\right )} d e^{2}\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{4} - 8 \, b c x^{2} - b^{2} + 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} + b\right )} \sqrt {c} - 4 \, a c\right ) + 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (c^{2} d^{2} e - b c d e^{2} + a c e^{3}\right )}}{8 \, {\left (c^{3} d^{2} e^{2} - b c^{2} d e^{3} + a c^{2} e^{4}\right )}}, \frac {\sqrt {c d^{2} - b d e + a e^{2}} c^{2} d^{2} \log \left (-\frac {8 \, c^{2} d^{2} x^{4} + 8 \, b c d^{2} x^{2} + {\left (b^{2} + 4 \, a c\right )} d^{2} + 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c d x^{2} + b d - {\left (b x^{2} + 2 \, a\right )} e\right )} \sqrt {c d^{2} - b d e + a e^{2}} + {\left ({\left (b^{2} + 4 \, a c\right )} x^{4} + 8 \, a b x^{2} + 8 \, a^{2}\right )} e^{2} - 2 \, {\left (4 \, b c d x^{4} + {\left (3 \, b^{2} + 4 \, a c\right )} d x^{2} + 4 \, a b d\right )} e}{x^{4} e^{2} + 2 \, d x^{2} e + d^{2}}\right ) + {\left (2 \, c^{2} d^{3} - b c d^{2} e + a b e^{3} - {\left (b^{2} - 2 \, a c\right )} d e^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{4} + b c x^{2} + a c\right )}}\right ) + 2 \, \sqrt {c x^{4} + b x^{2} + a} {\left (c^{2} d^{2} e - b c d e^{2} + a c e^{3}\right )}}{4 \, {\left (c^{3} d^{2} e^{2} - b c^{2} d e^{3} + a c^{2} e^{4}\right )}}, \frac {2 \, \sqrt {-c d^{2} + b d e - a e^{2}} c^{2} d^{2} \arctan \left (-\frac {\sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c d x^{2} + b d - {\left (b x^{2} + 2 \, a\right )} e\right )} \sqrt {-c d^{2} + b d e - a e^{2}}}{2 \, {\left (c^{2} d^{2} x^{4} + b c d^{2} x^{2} + a c d^{2} + {\left (a c x^{4} + a b x^{2} + a^{2}\right )} e^{2} - {\left (b c d x^{4} + b^{2} d x^{2} + a b d\right )} e\right )}}\right ) + {\left (2 \, c^{2} d^{3} - b c d^{2} e + a b e^{3} - {\left (b^{2} - 2 \, a c\right )} d e^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{4} + b c x^{2} + a c\right )}}\right ) + 2 \, \sqrt {c x^{4} + b x^{2} + a} {\left (c^{2} d^{2} e - b c d e^{2} + a c e^{3}\right )}}{4 \, {\left (c^{3} d^{2} e^{2} - b c^{2} d e^{3} + a c^{2} e^{4}\right )}}\right ] \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^{5}}{\left (d + e 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(-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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^5}{\left (e\,x^2+d\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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