Optimal. Leaf size=218 \[ \frac {(e f-d g) \tanh ^{-1}\left (\frac {a e^2-c d^2 x^2}{\sqrt {c d^4-a e^4} \sqrt {-a+c x^4}}\right )}{2 \sqrt {c d^4-a e^4}}+\frac {\sqrt [4]{a} g \sqrt {1-\frac {c x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{c} e \sqrt {-a+c x^4}}+\frac {\sqrt [4]{a} (e f-d g) \sqrt {1-\frac {c x^4}{a}} \Pi \left (\frac {\sqrt {a} e^2}{\sqrt {c} d^2};\left .\sin ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{c} d e \sqrt {-a+c x^4}} \]
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Rubi [A]
time = 0.17, antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 10, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {1756, 12,
1262, 739, 212, 1725, 230, 227, 1233, 1232} \begin {gather*} \frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} (e f-d g) \Pi \left (\frac {\sqrt {a} e^2}{\sqrt {c} d^2};\left .\text {ArcSin}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{c} d e \sqrt {c x^4-a}}+\frac {\sqrt [4]{a} g \sqrt {1-\frac {c x^4}{a}} F\left (\left .\text {ArcSin}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{c} e \sqrt {c x^4-a}}+\frac {(e f-d g) \tanh ^{-1}\left (\frac {a e^2-c d^2 x^2}{\sqrt {c x^4-a} \sqrt {c d^4-a e^4}}\right )}{2 \sqrt {c d^4-a e^4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 212
Rule 227
Rule 230
Rule 739
Rule 1232
Rule 1233
Rule 1262
Rule 1725
Rule 1756
Rubi steps
\begin {align*} \int \frac {f+g x}{(d+e x) \sqrt {-a+c x^4}} \, dx &=\int \frac {(-e f+d g) x}{\left (d^2-e^2 x^2\right ) \sqrt {-a+c x^4}} \, dx+\int \frac {d f-e g x^2}{\left (d^2-e^2 x^2\right ) \sqrt {-a+c x^4}} \, dx\\ &=\frac {g \int \frac {1}{\sqrt {-a+c x^4}} \, dx}{e}+\frac {(d (e f-d g)) \int \frac {1}{\left (d^2-e^2 x^2\right ) \sqrt {-a+c x^4}} \, dx}{e}+(-e f+d g) \int \frac {x}{\left (d^2-e^2 x^2\right ) \sqrt {-a+c x^4}} \, dx\\ &=\frac {1}{2} (-e f+d g) \text {Subst}\left (\int \frac {1}{\left (d^2-e^2 x\right ) \sqrt {-a+c x^2}} \, dx,x,x^2\right )+\frac {\left (g \sqrt {1-\frac {c x^4}{a}}\right ) \int \frac {1}{\sqrt {1-\frac {c x^4}{a}}} \, dx}{e \sqrt {-a+c x^4}}+\frac {\left (d (e f-d g) \sqrt {1-\frac {c x^4}{a}}\right ) \int \frac {1}{\left (d^2-e^2 x^2\right ) \sqrt {1-\frac {c x^4}{a}}} \, dx}{e \sqrt {-a+c x^4}}\\ &=\frac {\sqrt [4]{a} g \sqrt {1-\frac {c x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{c} e \sqrt {-a+c x^4}}+\frac {\sqrt [4]{a} (e f-d g) \sqrt {1-\frac {c x^4}{a}} \Pi \left (\frac {\sqrt {a} e^2}{\sqrt {c} d^2};\left .\sin ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{c} d e \sqrt {-a+c x^4}}+\frac {1}{2} (e f-d g) \text {Subst}\left (\int \frac {1}{c d^4-a e^4-x^2} \, dx,x,\frac {a e^2-c d^2 x^2}{\sqrt {-a+c x^4}}\right )\\ &=\frac {(e f-d g) \tanh ^{-1}\left (\frac {a e^2-c d^2 x^2}{\sqrt {c d^4-a e^4} \sqrt {-a+c x^4}}\right )}{2 \sqrt {c d^4-a e^4}}+\frac {\sqrt [4]{a} g \sqrt {1-\frac {c x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{c} e \sqrt {-a+c x^4}}+\frac {\sqrt [4]{a} (e f-d g) \sqrt {1-\frac {c x^4}{a}} \Pi \left (\frac {\sqrt {a} e^2}{\sqrt {c} d^2};\left .\sin ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{c} d e \sqrt {-a+c x^4}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 11.00, size = 719, normalized size = 3.30 \begin {gather*} \frac {-\frac {i g \sqrt {1-\frac {c x^4}{a}} F\left (\left .i \sinh ^{-1}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )}{\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} e}+\frac {i f \left (\sqrt [4]{a}-i \sqrt [4]{c} x\right )^2 \sqrt {-\frac {(1-i) \left (\sqrt [4]{a}-\sqrt [4]{c} x\right )}{i \sqrt [4]{a}+\sqrt [4]{c} x}} \sqrt {\frac {(1+i) \left (\sqrt [4]{a}+i \sqrt [4]{c} x\right ) \left (\sqrt [4]{a}+\sqrt [4]{c} x\right )}{\left (\sqrt [4]{a}-i \sqrt [4]{c} x\right )^2}} \left (\left (-\sqrt [4]{c} d+\sqrt [4]{a} e\right ) F\left (\left .\sin ^{-1}\left (\sqrt {\frac {(1+i) \left (\sqrt [4]{a}+\sqrt [4]{c} x\right )}{2 i \sqrt [4]{a}+2 \sqrt [4]{c} x}}\right )\right |2\right )-(1-i) \sqrt [4]{a} e \Pi \left (\frac {(1-i) \left (\sqrt [4]{c} d-i \sqrt [4]{a} e\right )}{\sqrt [4]{c} d-\sqrt [4]{a} e};\left .\sin ^{-1}\left (\sqrt {\frac {(1+i) \left (\sqrt [4]{a}+\sqrt [4]{c} x\right )}{2 i \sqrt [4]{a}+2 \sqrt [4]{c} x}}\right )\right |2\right )\right )}{\sqrt [4]{a} \left (-\sqrt [4]{c} d+\sqrt [4]{a} e\right ) \left (i \sqrt [4]{c} d+\sqrt [4]{a} e\right )}+\frac {d g \left (\sqrt [4]{a}-i \sqrt [4]{c} x\right )^2 \sqrt {-\frac {(1-i) \left (\sqrt [4]{a}-\sqrt [4]{c} x\right )}{i \sqrt [4]{a}+\sqrt [4]{c} x}} \sqrt {\frac {(1+i) \left (\sqrt [4]{a}+i \sqrt [4]{c} x\right ) \left (\sqrt [4]{a}+\sqrt [4]{c} x\right )}{\left (\sqrt [4]{a}-i \sqrt [4]{c} x\right )^2}} \left (i \left (\sqrt [4]{c} d-\sqrt [4]{a} e\right ) F\left (\left .\sin ^{-1}\left (\sqrt {\frac {(1+i) \left (\sqrt [4]{a}+\sqrt [4]{c} x\right )}{2 i \sqrt [4]{a}+2 \sqrt [4]{c} x}}\right )\right |2\right )+(1+i) \sqrt [4]{a} e \Pi \left (\frac {(1-i) \left (\sqrt [4]{c} d-i \sqrt [4]{a} e\right )}{\sqrt [4]{c} d-\sqrt [4]{a} e};\left .\sin ^{-1}\left (\sqrt {\frac {(1+i) \left (\sqrt [4]{a}+\sqrt [4]{c} x\right )}{2 i \sqrt [4]{a}+2 \sqrt [4]{c} x}}\right )\right |2\right )\right )}{\sqrt [4]{a} e \left (-\sqrt [4]{c} d+\sqrt [4]{a} e\right ) \left (i \sqrt [4]{c} d+\sqrt [4]{a} e\right )}}{\sqrt {-a+c x^4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 247, normalized size = 1.13
method | result | size |
default | \(\frac {g \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{e \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}-a}}+\frac {\left (-d g +e f \right ) \left (-\frac {\arctanh \left (\frac {\frac {2 c \,x^{2} d^{2}}{e^{2}}-2 a}{2 \sqrt {\frac {c \,d^{4}}{e^{4}}-a}\, \sqrt {c \,x^{4}-a}}\right )}{2 \sqrt {\frac {c \,d^{4}}{e^{4}}-a}}+\frac {e \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \EllipticPi \left (x \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {e^{2} \sqrt {a}}{d^{2} \sqrt {c}}, \frac {\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}\right )}{\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}\, d \sqrt {c \,x^{4}-a}}\right )}{e^{2}}\) | \(247\) |
elliptic | \(\frac {g \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{e \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}-a}}-\frac {\left (d g -e f \right ) \left (-\frac {\arctanh \left (\frac {\frac {2 c \,x^{2} d^{2}}{e^{2}}-2 a}{2 \sqrt {\frac {c \,d^{4}}{e^{4}}-a}\, \sqrt {c \,x^{4}-a}}\right )}{2 \sqrt {\frac {c \,d^{4}}{e^{4}}-a}}+\frac {e \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \EllipticPi \left (x \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {e^{2} \sqrt {a}}{d^{2} \sqrt {c}}, \frac {\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}\right )}{\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}\, d \sqrt {c \,x^{4}-a}}\right )}{e^{2}}\) | \(248\) |
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]
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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {f + g x}{\sqrt {- a + c x^{4}} \left (d + e x\right )}\, 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}{\sqrt {c\,x^4-a}\,\left (d+e\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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