Optimal. Leaf size=560 \[ \frac {(e f-d g) \tan ^{-1}\left (\frac {\sqrt {-c d^4-b d^2 e^2-a e^4} x}{d e \sqrt {a+b x^2+c x^4}}\right )}{2 \sqrt {-c d^4-e^2 \left (b d^2+a e^2\right )}}-\frac {(e f-d g) \tanh ^{-1}\left (\frac {b d^2+2 a e^2+\left (2 c d^2+b e^2\right ) x^2}{2 \sqrt {c d^4+b d^2 e^2+a e^4} \sqrt {a+b x^2+c x^4}}\right )}{2 \sqrt {c d^4+b d^2 e^2+a e^4}}+\frac {\left (\sqrt {c} d f+\sqrt {a} e g\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \sqrt {a+b x^2+c x^4}}-\frac {\left (\sqrt {c} d^2-\sqrt {a} e^2\right ) (e f-d g) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \Pi \left (\frac {\left (\sqrt {c} d^2+\sqrt {a} e^2\right )^2}{4 \sqrt {a} \sqrt {c} d^2 e^2};2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{4 \sqrt [4]{a} \sqrt [4]{c} d e \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \sqrt {a+b x^2+c x^4}} \]
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Rubi [A]
time = 0.40, antiderivative size = 560, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {1755, 12, 1261,
738, 212, 1722, 1117, 1720} \begin {gather*} \frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) \left (\sqrt {a} e g+\sqrt {c} d f\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+b x^2+c x^4} \left (\sqrt {a} e^2+\sqrt {c} d^2\right )}-\frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) (e f-d g) \Pi \left (\frac {\left (\sqrt {c} d^2+\sqrt {a} e^2\right )^2}{4 \sqrt {a} \sqrt {c} d^2 e^2};2 \text {ArcTan}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{4 \sqrt [4]{a} \sqrt [4]{c} d e \sqrt {a+b x^2+c x^4} \left (\sqrt {a} e^2+\sqrt {c} d^2\right )}+\frac {(e f-d g) \text {ArcTan}\left (\frac {x \sqrt {-a e^4-b d^2 e^2-c d^4}}{d e \sqrt {a+b x^2+c x^4}}\right )}{2 \sqrt {-e^2 \left (a e^2+b d^2\right )-c d^4}}-\frac {(e f-d g) \tanh ^{-1}\left (\frac {2 a e^2+x^2 \left (b e^2+2 c d^2\right )+b d^2}{2 \sqrt {a+b x^2+c x^4} \sqrt {a e^4+b d^2 e^2+c d^4}}\right )}{2 \sqrt {a e^4+b d^2 e^2+c d^4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 212
Rule 738
Rule 1117
Rule 1261
Rule 1720
Rule 1722
Rule 1755
Rubi steps
\begin {align*} \int \frac {f+g x}{(d+e x) \sqrt {a+b x^2+c x^4}} \, dx &=\int \frac {(-e f+d g) x}{\left (d^2-e^2 x^2\right ) \sqrt {a+b x^2+c x^4}} \, dx+\int \frac {d f-e g x^2}{\left (d^2-e^2 x^2\right ) \sqrt {a+b x^2+c x^4}} \, dx\\ &=\frac {\left (\sqrt {a} d e (e f-d g)\right ) \int \frac {1+\frac {\sqrt {c} x^2}{\sqrt {a}}}{\left (d^2-e^2 x^2\right ) \sqrt {a+b x^2+c x^4}} \, dx}{\sqrt {c} d^2+\sqrt {a} e^2}+(-e f+d g) \int \frac {x}{\left (d^2-e^2 x^2\right ) \sqrt {a+b x^2+c x^4}} \, dx+\frac {\left (\sqrt {c} d f+\sqrt {a} e g\right ) \int \frac {1}{\sqrt {a+b x^2+c x^4}} \, dx}{\sqrt {c} d^2+\sqrt {a} e^2}\\ &=\frac {(e f-d g) \tan ^{-1}\left (\frac {\sqrt {-c d^4-b d^2 e^2-a e^4} x}{d e \sqrt {a+b x^2+c x^4}}\right )}{2 \sqrt {-c d^4-e^2 \left (b d^2+a e^2\right )}}+\frac {\left (\sqrt {c} d f+\sqrt {a} e g\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \sqrt {a+b x^2+c x^4}}-\frac {\left (\sqrt {c} d^2-\sqrt {a} e^2\right ) (e f-d g) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \Pi \left (\frac {\left (\sqrt {c} d^2+\sqrt {a} e^2\right )^2}{4 \sqrt {a} \sqrt {c} d^2 e^2};2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{4 \sqrt [4]{a} \sqrt [4]{c} d e \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \sqrt {a+b x^2+c x^4}}+\frac {1}{2} (-e f+d g) \text {Subst}\left (\int \frac {1}{\left (d^2-e^2 x\right ) \sqrt {a+b x+c x^2}} \, dx,x,x^2\right )\\ &=\frac {(e f-d g) \tan ^{-1}\left (\frac {\sqrt {-c d^4-b d^2 e^2-a e^4} x}{d e \sqrt {a+b x^2+c x^4}}\right )}{2 \sqrt {-c d^4-e^2 \left (b d^2+a e^2\right )}}+\frac {\left (\sqrt {c} d f+\sqrt {a} e g\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \sqrt {a+b x^2+c x^4}}-\frac {\left (\sqrt {c} d^2-\sqrt {a} e^2\right ) (e f-d g) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \Pi \left (\frac {\left (\sqrt {c} d^2+\sqrt {a} e^2\right )^2}{4 \sqrt {a} \sqrt {c} d^2 e^2};2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{4 \sqrt [4]{a} \sqrt [4]{c} d e \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \sqrt {a+b x^2+c x^4}}+(e f-d g) \text {Subst}\left (\int \frac {1}{4 c d^4+4 b d^2 e^2+4 a e^4-x^2} \, dx,x,\frac {-b d^2-2 a e^2-\left (2 c d^2+b e^2\right ) x^2}{\sqrt {a+b x^2+c x^4}}\right )\\ &=\frac {(e f-d g) \tan ^{-1}\left (\frac {\sqrt {-c d^4-b d^2 e^2-a e^4} x}{d e \sqrt {a+b x^2+c x^4}}\right )}{2 \sqrt {-c d^4-e^2 \left (b d^2+a e^2\right )}}-\frac {(e f-d g) \tanh ^{-1}\left (\frac {b d^2+2 a e^2+\left (2 c d^2+b e^2\right ) x^2}{2 \sqrt {c d^4+b d^2 e^2+a e^4} \sqrt {a+b x^2+c x^4}}\right )}{2 \sqrt {c d^4+b d^2 e^2+a e^4}}+\frac {\left (\sqrt {c} d f+\sqrt {a} e g\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \sqrt {a+b x^2+c x^4}}-\frac {\left (\sqrt {c} d^2-\sqrt {a} e^2\right ) (e f-d g) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \Pi \left (\frac {\left (\sqrt {c} d^2+\sqrt {a} e^2\right )^2}{4 \sqrt {a} \sqrt {c} d^2 e^2};2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{4 \sqrt [4]{a} \sqrt [4]{c} d e \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \sqrt {a+b x^2+c x^4}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 17.34, size = 3652, normalized size = 6.52 \begin {gather*} \text {Result too large to show} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 437, normalized size = 0.78
method | result | size |
default | \(\frac {g \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \EllipticF \left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 e \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {\left (-d g +e f \right ) \left (-\frac {\arctanh \left (\frac {\frac {2 c \,x^{2} d^{2}}{e^{2}}+\frac {b \,d^{2}}{e^{2}}+b \,x^{2}+2 a}{2 \sqrt {\frac {c \,d^{4}}{e^{4}}+\frac {b \,d^{2}}{e^{2}}+a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}\right )}{2 \sqrt {\frac {c \,d^{4}}{e^{4}}+\frac {b \,d^{2}}{e^{2}}+a}}+\frac {\sqrt {2}\, e \sqrt {1-\frac {\left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{2 a}}\, \sqrt {1+\frac {\left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{2 a}}\, \EllipticPi \left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {2 a \,e^{2}}{\left (-b +\sqrt {-4 a c +b^{2}}\right ) d^{2}}, \frac {\sqrt {-\frac {b +\sqrt {-4 a c +b^{2}}}{2 a}}\, \sqrt {2}}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}\right )}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, d \sqrt {c \,x^{4}+b \,x^{2}+a}}\right )}{e^{2}}\) | \(437\) |
elliptic | \(\frac {g \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \EllipticF \left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 e \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {\left (d g -e f \right ) \left (-\frac {\arctanh \left (\frac {\frac {2 c \,x^{2} d^{2}}{e^{2}}+\frac {b \,d^{2}}{e^{2}}+b \,x^{2}+2 a}{2 \sqrt {\frac {c \,d^{4}}{e^{4}}+\frac {b \,d^{2}}{e^{2}}+a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}\right )}{2 \sqrt {\frac {c \,d^{4}}{e^{4}}+\frac {b \,d^{2}}{e^{2}}+a}}+\frac {\sqrt {2}\, e \sqrt {1-\frac {\left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{2 a}}\, \sqrt {1+\frac {\left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{2 a}}\, \EllipticPi \left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {2 a \,e^{2}}{\left (-b +\sqrt {-4 a c +b^{2}}\right ) d^{2}}, \frac {\sqrt {-\frac {b +\sqrt {-4 a c +b^{2}}}{2 a}}\, \sqrt {2}}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}\right )}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, d \sqrt {c \,x^{4}+b \,x^{2}+a}}\right )}{e^{2}}\) | \(438\) |
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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 {f + g x}{\left (d + e x\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]
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 {f+g\,x}{\left (d+e\,x\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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