Optimal. Leaf size=208 \[ -\frac {\left (4 c d-b e-2 c e x^2\right ) \sqrt {a+b x^2+c x^4}}{8 c e^2}+\frac {\left (8 c^2 d^2-b^2 e^2-4 c e (b d-a e)\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{16 c^{3/2} e^3}-\frac {d \sqrt {c d^2-b d e+a e^2} \tanh ^{-1}\left (\frac {b d-2 a e+(2 c d-b e) x^2}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x^2+c x^4}}\right )}{2 e^3} \]
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Rubi [A]
time = 0.21, antiderivative size = 208, 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, 828, 857,
635, 212, 738} \begin {gather*} \frac {\left (-4 c e (b d-a e)-b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{16 c^{3/2} e^3}-\frac {d \sqrt {a e^2-b d e+c 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^3}-\frac {\sqrt {a+b x^2+c x^4} \left (-b e+4 c d-2 c e x^2\right )}{8 c e^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 635
Rule 738
Rule 828
Rule 857
Rule 1265
Rubi steps
\begin {align*} \int \frac {x^3 \sqrt {a+b x^2+c x^4}}{d+e x^2} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x \sqrt {a+b x+c x^2}}{d+e x} \, dx,x,x^2\right )\\ &=-\frac {\left (4 c d-b e-2 c e x^2\right ) \sqrt {a+b x^2+c x^4}}{8 c e^2}-\frac {\text {Subst}\left (\int \frac {-\frac {1}{2} d \left (4 b c d-b^2 e-4 a c e\right )-\frac {1}{2} \left (8 c^2 d^2-b^2 e^2-4 c e (b d-a e)\right ) x}{(d+e x) \sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{8 c e^2}\\ &=-\frac {\left (4 c d-b e-2 c e x^2\right ) \sqrt {a+b x^2+c x^4}}{8 c e^2}-\frac {\left (d \left (c d^2-b d e+a e^2\right )\right ) \text {Subst}\left (\int \frac {1}{(d+e x) \sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{2 e^3}+\frac {\left (8 c^2 d^2-b^2 e^2-4 c e (b d-a e)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{16 c e^3}\\ &=-\frac {\left (4 c d-b e-2 c e x^2\right ) \sqrt {a+b x^2+c x^4}}{8 c e^2}+\frac {\left (d \left (c d^2-b d e+a e^2\right )\right ) \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^3}+\frac {\left (8 c^2 d^2-b^2 e^2-4 c e (b d-a e)\right ) \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 )}{8 c e^3}\\ &=-\frac {\left (4 c d-b e-2 c e x^2\right ) \sqrt {a+b x^2+c x^4}}{8 c e^2}+\frac {\left (8 c^2 d^2-b^2 e^2-4 c e (b d-a e)\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{16 c^{3/2} e^3}-\frac {d \sqrt {c d^2-b d e+a e^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^3}\\ \end {align*}
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Mathematica [A]
time = 0.60, size = 200, normalized size = 0.96 \begin {gather*} \frac {2 \sqrt {c} e \left (-4 c d+b e+2 c e x^2\right ) \sqrt {a+b x^2+c x^4}-16 c^{3/2} d \sqrt {-c d^2+e (b d-a e)} \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 )+\left (-8 c^2 d^2+b^2 e^2+4 c e (b d-a e)\right ) \log \left (c \left (b+2 c x^2-2 \sqrt {c} \sqrt {a+b x^2+c x^4}\right )\right )}{16 c^{3/2} e^3} \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. \(867\) vs.
\(2(184)=368\).
time = 0.16, size = 868, normalized size = 4.17 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 11.53, size = 1247, normalized size = 6.00 \begin {gather*} \left [\frac {{\left (8 \, \sqrt {c d^{2} - b d e + a e^{2}} c^{2} d \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 (8 \, c^{2} d^{2} - 4 \, b c d e - {\left (b^{2} - 4 \, a c\right )} 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 (4 \, c^{2} d e - {\left (2 \, c^{2} x^{2} + b c\right )} e^{2}\right )}\right )} e^{\left (-3\right )}}{32 \, c^{2}}, -\frac {{\left (16 \, \sqrt {-c d^{2} + b d e - a e^{2}} c^{2} d \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 (8 \, c^{2} d^{2} - 4 \, b c d e - {\left (b^{2} - 4 \, a c\right )} 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 (4 \, c^{2} d e - {\left (2 \, c^{2} x^{2} + b c\right )} e^{2}\right )}\right )} e^{\left (-3\right )}}{32 \, c^{2}}, \frac {{\left (4 \, \sqrt {c d^{2} - b d e + a e^{2}} c^{2} d \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 (8 \, c^{2} d^{2} - 4 \, b c d e - {\left (b^{2} - 4 \, a c\right )} 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 (4 \, c^{2} d e - {\left (2 \, c^{2} x^{2} + b c\right )} e^{2}\right )}\right )} e^{\left (-3\right )}}{16 \, c^{2}}, -\frac {{\left (8 \, \sqrt {-c d^{2} + b d e - a e^{2}} c^{2} d \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 (8 \, c^{2} d^{2} - 4 \, b c d e - {\left (b^{2} - 4 \, a c\right )} 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 (4 \, c^{2} d e - {\left (2 \, c^{2} x^{2} + b c\right )} e^{2}\right )}\right )} e^{\left (-3\right )}}{16 \, c^{2}}\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^{3} \sqrt {a + b x^{2} + c x^{4}}}{d + e x^{2}}\, 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 = -0.00 \begin {gather*} \int \frac {x^3\,\sqrt {c\,x^4+b\,x^2+a}}{e\,x^2+d} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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