Optimal. Leaf size=208 \[ \frac {\left (-4 c e (b d-a e)-b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{16 c^{3/2} e^3}-\frac {d \sqrt {a e^2-b d e+c d^2} \tanh ^{-1}\left (\frac {-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt {a+b x^2+c x^4} \sqrt {a e^2-b d e+c d^2}}\right )}{2 e^3}-\frac {\sqrt {a+b x^2+c x^4} \left (-b e+4 c d-2 c e x^2\right )}{8 c e^2} \]
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Rubi [A] time = 0.31, 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 = {1251, 814, 843, 621, 206, 724} \[ \frac {\left (-4 c e (b d-a e)-b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{16 c^{3/2} e^3}-\frac {d \sqrt {a e^2-b d e+c d^2} \tanh ^{-1}\left (\frac {-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt {a+b x^2+c x^4} \sqrt {a e^2-b d e+c d^2}}\right )}{2 e^3}-\frac {\sqrt {a+b x^2+c x^4} \left (-b e+4 c d-2 c e x^2\right )}{8 c e^2} \]
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 724
Rule 814
Rule 843
Rule 1251
Rubi steps
\begin {align*} \int \frac {x^3 \sqrt {a+b x^2+c x^4}}{d+e x^2} \, dx &=\frac {1}{2} \operatorname {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 {\operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 = 0.26, size = 205, normalized size = 0.99 \[ \frac {\left (4 c e (a e-b d)-b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )+2 \sqrt {c} \left (4 c d \sqrt {a e^2-b d e+c d^2} \tanh ^{-1}\left (\frac {2 a e-b d+b e x^2-2 c d x^2}{2 \sqrt {a+b x^2+c x^4} \sqrt {a e^2-b d e+c d^2}}\right )+e \sqrt {a+b x^2+c x^4} \left (b e-4 c d+2 c e x^2\right )\right )}{16 c^{3/2} e^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 46.63, size = 1231, normalized size = 5.92 \[ \text {result too large to display} \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 887, normalized size = 4.26 \[ \frac {a d \ln \left (\frac {\frac {\left (b e -2 c d \right ) \left (x^{2}+\frac {d}{e}\right )}{e}+\frac {2 a \,e^{2}-2 d e b +2 c \,d^{2}}{e^{2}}+2 \sqrt {\frac {a \,e^{2}-d e b +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x^{2}+\frac {d}{e}\right )^{2} c +\frac {\left (b e -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 \sqrt {\frac {a \,e^{2}-d e b +c \,d^{2}}{e^{2}}}\, e^{2}}-\frac {b \,d^{2} \ln \left (\frac {\frac {\left (b e -2 c d \right ) \left (x^{2}+\frac {d}{e}\right )}{e}+\frac {2 a \,e^{2}-2 d e b +2 c \,d^{2}}{e^{2}}+2 \sqrt {\frac {a \,e^{2}-d e b +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x^{2}+\frac {d}{e}\right )^{2} c +\frac {\left (b e -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 \sqrt {\frac {a \,e^{2}-d e b +c \,d^{2}}{e^{2}}}\, e^{3}}+\frac {c \,d^{3} \ln \left (\frac {\frac {\left (b e -2 c d \right ) \left (x^{2}+\frac {d}{e}\right )}{e}+\frac {2 a \,e^{2}-2 d e b +2 c \,d^{2}}{e^{2}}+2 \sqrt {\frac {a \,e^{2}-d e b +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x^{2}+\frac {d}{e}\right )^{2} c +\frac {\left (b e -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 \sqrt {\frac {a \,e^{2}-d e b +c \,d^{2}}{e^{2}}}\, e^{4}}+\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{4 e}+\frac {a \ln \left (\frac {c \,x^{2}+\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{4 \sqrt {c}\, e}-\frac {b^{2} \ln \left (\frac {c \,x^{2}+\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{16 c^{\frac {3}{2}} e}-\frac {b d \ln \left (\frac {\left (x^{2}+\frac {d}{e}\right ) c +\frac {b e -2 c d}{2 e}}{\sqrt {c}}+\sqrt {\left (x^{2}+\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x^{2}+\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-d e b +c \,d^{2}}{e^{2}}}\right )}{4 \sqrt {c}\, e^{2}}+\frac {\sqrt {c}\, d^{2} \ln \left (\frac {\left (x^{2}+\frac {d}{e}\right ) c +\frac {b e -2 c d}{2 e}}{\sqrt {c}}+\sqrt {\left (x^{2}+\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x^{2}+\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-d e b +c \,d^{2}}{e^{2}}}\right )}{2 e^{3}}+\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, b}{8 c e}-\frac {\sqrt {\left (x^{2}+\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x^{2}+\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-d e b +c \,d^{2}}{e^{2}}}\, d}{2 e^{2}} \]
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 \[ \text {Exception raised: ValueError} \]
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 \[ \int \frac {x^3\,\sqrt {c\,x^4+b\,x^2+a}}{e\,x^2+d} \,d x \]
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 \[ \int \frac {x^{3} \sqrt {a + b x^{2} + c x^{4}}}{d + e x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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