Optimal. Leaf size=168 \[ \frac {\sqrt {a e^2-b d e+c d^2} \tanh ^{-1}\left (\frac {-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt {a+b x^2+c x^4} \sqrt {a e^2-b d e+c d^2}}\right )}{2 e^2}-\frac {(2 c d-b e) \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{4 \sqrt {c} e^2}+\frac {\sqrt {a+b x^2+c x^4}}{2 e} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.22, antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1247, 734, 843, 621, 206, 724} \[ \frac {\sqrt {a e^2-b d e+c d^2} \tanh ^{-1}\left (\frac {-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt {a+b x^2+c x^4} \sqrt {a e^2-b d e+c d^2}}\right )}{2 e^2}-\frac {(2 c d-b e) \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{4 \sqrt {c} e^2}+\frac {\sqrt {a+b x^2+c x^4}}{2 e} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 621
Rule 724
Rule 734
Rule 843
Rule 1247
Rubi steps
\begin {align*} \int \frac {x \sqrt {a+b x^2+c x^4}}{d+e x^2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\sqrt {a+b x+c x^2}}{d+e x} \, dx,x,x^2\right )\\ &=\frac {\sqrt {a+b x^2+c x^4}}{2 e}-\frac {\operatorname {Subst}\left (\int \frac {b d-2 a e+(2 c d-b e) x}{(d+e x) \sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{4 e}\\ &=\frac {\sqrt {a+b x^2+c x^4}}{2 e}-\frac {(2 c d-b e) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{4 e^2}+\frac {\left (c d^2-b d e+a e^2\right ) \operatorname {Subst}\left (\int \frac {1}{(d+e x) \sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{2 e^2}\\ &=\frac {\sqrt {a+b x^2+c x^4}}{2 e}-\frac {(2 c d-b e) \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 )}{2 e^2}-\frac {\left (c d^2-b d e+a e^2\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^2}\\ &=\frac {\sqrt {a+b x^2+c x^4}}{2 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 \sqrt {c} e^2}+\frac {\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^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.13, size = 167, normalized size = 0.99 \[ \frac {2 \sqrt {c} \left (e \sqrt {a+b x^2+c x^4}-\sqrt {a e^2-b d e+c d^2} \tanh ^{-1}\left (\frac {2 a e-b d+b e x^2-2 c d x^2}{2 \sqrt {a+b x^2+c x^4} \sqrt {a e^2-b d e+c d^2}}\right )\right )+(b e-2 c d) \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{4 \sqrt {c} e^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 3.69, size = 1050, normalized size = 6.25 \[ \left [\frac {4 \, \sqrt {c x^{4} + b x^{2} + a} c e - {\left (2 \, c d - b e\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 ) + 2 \, \sqrt {c d^{2} - b d e + a e^{2}} c \log \left (-\frac {{\left (8 \, c^{2} d^{2} - 8 \, b c d e + {\left (b^{2} + 4 \, a c\right )} e^{2}\right )} x^{4} - 8 \, a b d e + 8 \, a^{2} e^{2} + {\left (b^{2} + 4 \, a c\right )} d^{2} + 2 \, {\left (4 \, b c d^{2} + 4 \, a b e^{2} - {\left (3 \, b^{2} + 4 \, a c\right )} d e\right )} x^{2} + 4 \, \sqrt {c x^{4} + b x^{2} + a} \sqrt {c d^{2} - b d e + a e^{2}} {\left ({\left (2 \, c d - b e\right )} x^{2} + b d - 2 \, a e\right )}}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}\right )}{8 \, c e^{2}}, \frac {2 \, \sqrt {c x^{4} + b x^{2} + a} c e + {\left (2 \, c d - b e\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 ) + \sqrt {c d^{2} - b d e + a e^{2}} c \log \left (-\frac {{\left (8 \, c^{2} d^{2} - 8 \, b c d e + {\left (b^{2} + 4 \, a c\right )} e^{2}\right )} x^{4} - 8 \, a b d e + 8 \, a^{2} e^{2} + {\left (b^{2} + 4 \, a c\right )} d^{2} + 2 \, {\left (4 \, b c d^{2} + 4 \, a b e^{2} - {\left (3 \, b^{2} + 4 \, a c\right )} d e\right )} x^{2} + 4 \, \sqrt {c x^{4} + b x^{2} + a} \sqrt {c d^{2} - b d e + a e^{2}} {\left ({\left (2 \, c d - b e\right )} x^{2} + b d - 2 \, a e\right )}}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}\right )}{4 \, c e^{2}}, \frac {4 \, \sqrt {c x^{4} + b x^{2} + a} c e + 4 \, \sqrt {-c d^{2} + b d e - a e^{2}} c \arctan \left (-\frac {\sqrt {c x^{4} + b x^{2} + a} \sqrt {-c d^{2} + b d e - a e^{2}} {\left ({\left (2 \, c d - b e\right )} x^{2} + b d - 2 \, a e\right )}}{2 \, {\left ({\left (c^{2} d^{2} - b c d e + a c e^{2}\right )} x^{4} + a c d^{2} - a b d e + a^{2} e^{2} + {\left (b c d^{2} - b^{2} d e + a b e^{2}\right )} x^{2}\right )}}\right ) - {\left (2 \, c d - b e\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 )}{8 \, c e^{2}}, \frac {2 \, \sqrt {c x^{4} + b x^{2} + a} c e + 2 \, \sqrt {-c d^{2} + b d e - a e^{2}} c \arctan \left (-\frac {\sqrt {c x^{4} + b x^{2} + a} \sqrt {-c d^{2} + b d e - a e^{2}} {\left ({\left (2 \, c d - b e\right )} x^{2} + b d - 2 \, a e\right )}}{2 \, {\left ({\left (c^{2} d^{2} - b c d e + a c e^{2}\right )} x^{4} + a c d^{2} - a b d e + a^{2} e^{2} + {\left (b c d^{2} - b^{2} d e + a b e^{2}\right )} x^{2}\right )}}\right ) + {\left (2 \, c d - b e\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 )}{4 \, c e^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.00, size = 757, normalized size = 4.51 \[ -\frac {a \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}+\frac {b 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 {c \,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 {b \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}-\frac {\sqrt {c}\, 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 )}{2 e^{2}}+\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}}}}{2 e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x\,\sqrt {c\,x^4+b\,x^2+a}}{e\,x^2+d} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \sqrt {a + b x^{2} + c x^{4}}}{d + e x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________