Integrand size = 19, antiderivative size = 610 \[ \int \frac {1}{(d+e x)^2 \sqrt {a+c x^4}} \, dx=-\frac {e^3 \sqrt {a+c x^4}}{\left (c d^4+a e^4\right ) (d+e x)}+\frac {\sqrt {c} e^2 x \sqrt {a+c x^4}}{\left (c d^4+a e^4\right ) \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {c d^3 e \arctan \left (\frac {\sqrt {-c d^4-a e^4} x}{d e \sqrt {a+c x^4}}\right )}{\left (-c d^4-a e^4\right )^{3/2}}-\frac {c d^3 e \text {arctanh}\left (\frac {a e^2+c d^2 x^2}{\sqrt {c d^4+a e^4} \sqrt {a+c x^4}}\right )}{\left (c d^4+a e^4\right )^{3/2}}-\frac {\sqrt [4]{a} \sqrt [4]{c} e^2 \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\left (c d^4+a e^4\right ) \sqrt {a+c x^4}}+\frac {\sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \sqrt {a+c x^4}}-\frac {c^{3/4} d^2 \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {c} d^2+\sqrt {a} e^2\right )^2}{4 \sqrt {a} \sqrt {c} d^2 e^2},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \left (c d^4+a e^4\right ) \sqrt {a+c x^4}} \]
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Time = 0.56 (sec) , antiderivative size = 610, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.579, Rules used = {1741, 1756, 12, 1262, 739, 212, 1729, 1210, 1723, 226, 1721} \[ \int \frac {1}{(d+e x)^2 \sqrt {a+c x^4}} \, dx=-\frac {c^{3/4} d^2 \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \operatorname {EllipticPi}\left (\frac {\left (\sqrt {c} d^2+\sqrt {a} e^2\right )^2}{4 \sqrt {a} \sqrt {c} d^2 e^2},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt {a+c x^4} \left (\sqrt {a} e^2+\sqrt {c} d^2\right ) \left (a e^4+c d^4\right )}-\frac {\sqrt [4]{a} \sqrt [4]{c} e^2 \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt {a+c x^4} \left (a e^4+c d^4\right )}+\frac {\sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt {a+c x^4} \left (\sqrt {a} e^2+\sqrt {c} d^2\right )}-\frac {c d^3 e \arctan \left (\frac {x \sqrt {-a e^4-c d^4}}{d e \sqrt {a+c x^4}}\right )}{\left (-a e^4-c d^4\right )^{3/2}}-\frac {c d^3 e \text {arctanh}\left (\frac {a e^2+c d^2 x^2}{\sqrt {a+c x^4} \sqrt {a e^4+c d^4}}\right )}{\left (a e^4+c d^4\right )^{3/2}}-\frac {e^3 \sqrt {a+c x^4}}{(d+e x) \left (a e^4+c d^4\right )}+\frac {\sqrt {c} e^2 x \sqrt {a+c x^4}}{\left (\sqrt {a}+\sqrt {c} x^2\right ) \left (a e^4+c d^4\right )} \]
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Rule 12
Rule 212
Rule 226
Rule 739
Rule 1210
Rule 1262
Rule 1721
Rule 1723
Rule 1729
Rule 1741
Rule 1756
Rubi steps \begin{align*} \text {integral}& = -\frac {e^3 \sqrt {a+c x^4}}{\left (c d^4+a e^4\right ) (d+e x)}-\frac {c \int \frac {-d^3+d^2 e x-d e^2 x^2-e^3 x^3}{(d+e x) \sqrt {a+c x^4}} \, dx}{c d^4+a e^4} \\ & = -\frac {e^3 \sqrt {a+c x^4}}{\left (c d^4+a e^4\right ) (d+e x)}-\frac {c \int \frac {2 d^3 e x}{\left (d^2-e^2 x^2\right ) \sqrt {a+c x^4}} \, dx}{c d^4+a e^4}-\frac {c \int \frac {-d^4-2 d^2 e^2 x^2+e^4 x^4}{\left (d^2-e^2 x^2\right ) \sqrt {a+c x^4}} \, dx}{c d^4+a e^4} \\ & = -\frac {e^3 \sqrt {a+c x^4}}{\left (c d^4+a e^4\right ) (d+e x)}+\frac {\int \frac {c d^4 e^2+\sqrt {a} \sqrt {c} d^2 e^4+\left (2 c d^2 e^4-e^4 \left (c d^2+\sqrt {a} \sqrt {c} e^2\right )\right ) x^2}{\left (d^2-e^2 x^2\right ) \sqrt {a+c x^4}} \, dx}{e^2 \left (c d^4+a e^4\right )}-\frac {\left (2 c d^3 e\right ) \int \frac {x}{\left (d^2-e^2 x^2\right ) \sqrt {a+c x^4}} \, dx}{c d^4+a e^4}-\frac {\left (\sqrt {a} \sqrt {c} e^2\right ) \int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}{\sqrt {a+c x^4}} \, dx}{c d^4+a e^4} \\ & = -\frac {e^3 \sqrt {a+c x^4}}{\left (c d^4+a e^4\right ) (d+e x)}+\frac {\sqrt {c} e^2 x \sqrt {a+c x^4}}{\left (c d^4+a e^4\right ) \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {\sqrt [4]{a} \sqrt [4]{c} e^2 \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\left (c d^4+a e^4\right ) \sqrt {a+c x^4}}+\frac {\sqrt {c} \int \frac {1}{\sqrt {a+c x^4}} \, dx}{\sqrt {c} d^2+\sqrt {a} e^2}-\frac {\left (c d^3 e\right ) \text {Subst}\left (\int \frac {1}{\left (d^2-e^2 x\right ) \sqrt {a+c x^2}} \, dx,x,x^2\right )}{c d^4+a e^4}+\frac {\left (2 \sqrt {a} c d^4 e^2\right ) \int \frac {1+\frac {\sqrt {c} x^2}{\sqrt {a}}}{\left (d^2-e^2 x^2\right ) \sqrt {a+c x^4}} \, dx}{\left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \left (c d^4+a e^4\right )} \\ & = -\frac {e^3 \sqrt {a+c x^4}}{\left (c d^4+a e^4\right ) (d+e x)}+\frac {\sqrt {c} e^2 x \sqrt {a+c x^4}}{\left (c d^4+a e^4\right ) \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {c d^3 e \tan ^{-1}\left (\frac {\sqrt {-c d^4-a e^4} x}{d e \sqrt {a+c x^4}}\right )}{\left (-c d^4-a e^4\right )^{3/2}}-\frac {\sqrt [4]{a} \sqrt [4]{c} e^2 \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\left (c d^4+a e^4\right ) \sqrt {a+c x^4}}+\frac {\sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+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}{2}\right )}{2 \sqrt [4]{a} \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \sqrt {a+c x^4}}-\frac {c^{3/4} d^2 \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+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}{2}\right )}{2 \sqrt [4]{a} \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \left (c d^4+a e^4\right ) \sqrt {a+c x^4}}+\frac {\left (c d^3 e\right ) \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 )}{c d^4+a e^4} \\ & = -\frac {e^3 \sqrt {a+c x^4}}{\left (c d^4+a e^4\right ) (d+e x)}+\frac {\sqrt {c} e^2 x \sqrt {a+c x^4}}{\left (c d^4+a e^4\right ) \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {c d^3 e \tan ^{-1}\left (\frac {\sqrt {-c d^4-a e^4} x}{d e \sqrt {a+c x^4}}\right )}{\left (-c d^4-a e^4\right )^{3/2}}-\frac {c d^3 e \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 )}{\left (c d^4+a e^4\right )^{3/2}}-\frac {\sqrt [4]{a} \sqrt [4]{c} e^2 \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\left (c d^4+a e^4\right ) \sqrt {a+c x^4}}+\frac {\sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+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}{2}\right )}{2 \sqrt [4]{a} \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \sqrt {a+c x^4}}-\frac {c^{3/4} d^2 \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+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}{2}\right )}{2 \sqrt [4]{a} \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \left (c d^4+a e^4\right ) \sqrt {a+c x^4}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 10.89 (sec) , antiderivative size = 448, normalized size of antiderivative = 0.73 \[ \int \frac {1}{(d+e x)^2 \sqrt {a+c x^4}} \, dx=-\frac {\sqrt {a} \sqrt {c} e^2 \sqrt {-c d^4-a e^4} (d+e x) \sqrt {1+\frac {c x^4}{a}} E\left (\left .i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )+i \sqrt {c} \left (\sqrt {c} d^2+i \sqrt {a} e^2\right ) \sqrt {-c d^4-a e^4} (d+e x) \sqrt {1+\frac {c x^4}{a}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right ),-1\right )-\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} \left (e^3 \sqrt {-c d^4-a e^4} \left (a+c x^4\right )-2 c d^3 e (d+e x) \sqrt {a+c x^4} \arctan \left (\frac {\sqrt {c} \left (d^2-e^2 x^2\right )+e^2 \sqrt {a+c x^4}}{\sqrt {-c d^4-a e^4}}\right )+2 \sqrt [4]{-1} \sqrt [4]{a} c^{3/4} d^2 \sqrt {-c d^4-a e^4} (d+e x) \sqrt {1+\frac {c x^4}{a}} \operatorname {EllipticPi}\left (\frac {i \sqrt {a} e^2}{\sqrt {c} d^2},\arcsin \left (\frac {(-1)^{3/4} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} \left (-c d^4-a e^4\right )^{3/2} (d+e x) \sqrt {a+c x^4}} \]
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Result contains complex when optimal does not.
Time = 0.55 (sec) , antiderivative size = 421, normalized size of antiderivative = 0.69
method | result | size |
default | \(-\frac {e^{3} \sqrt {c \,x^{4}+a}}{\left (e^{4} a +d^{4} c \right ) \left (e x +d \right )}-\frac {c \,d^{2} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\left (e^{4} a +d^{4} c \right ) \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {i e^{2} \sqrt {c}\, \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\left (e^{4} a +d^{4} c \right ) \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {2 d^{3} c \left (-\frac {\operatorname {arctanh}\left (\frac {\frac {2 c \,x^{2} d^{2}}{e^{2}}+2 a}{2 \sqrt {\frac {c \,d^{4}}{e^{4}}+a}\, \sqrt {c \,x^{4}+a}}\right )}{2 \sqrt {\frac {c \,d^{4}}{e^{4}}+a}}+\frac {e \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \Pi \left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, -\frac {i \sqrt {a}\, e^{2}}{\sqrt {c}\, d^{2}}, \frac {\sqrt {-\frac {i \sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, d \sqrt {c \,x^{4}+a}}\right )}{\left (e^{4} a +d^{4} c \right ) e}\) | \(421\) |
elliptic | \(-\frac {e^{3} \sqrt {c \,x^{4}+a}}{\left (e^{4} a +d^{4} c \right ) \left (e x +d \right )}-\frac {c \,d^{2} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\left (e^{4} a +d^{4} c \right ) \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {i e^{2} \sqrt {c}\, \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\left (e^{4} a +d^{4} c \right ) \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {2 d^{3} c \left (-\frac {\operatorname {arctanh}\left (\frac {\frac {2 c \,x^{2} d^{2}}{e^{2}}+2 a}{2 \sqrt {\frac {c \,d^{4}}{e^{4}}+a}\, \sqrt {c \,x^{4}+a}}\right )}{2 \sqrt {\frac {c \,d^{4}}{e^{4}}+a}}+\frac {e \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \Pi \left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, -\frac {i \sqrt {a}\, e^{2}}{\sqrt {c}\, d^{2}}, \frac {\sqrt {-\frac {i \sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, d \sqrt {c \,x^{4}+a}}\right )}{\left (e^{4} a +d^{4} c \right ) e}\) | \(421\) |
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Timed out. \[ \int \frac {1}{(d+e x)^2 \sqrt {a+c x^4}} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{(d+e x)^2 \sqrt {a+c x^4}} \, dx=\int \frac {1}{\sqrt {a + c x^{4}} \left (d + e x\right )^{2}}\, dx \]
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\[ \int \frac {1}{(d+e x)^2 \sqrt {a+c x^4}} \, dx=\int { \frac {1}{\sqrt {c x^{4} + a} {\left (e x + d\right )}^{2}} \,d x } \]
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\[ \int \frac {1}{(d+e x)^2 \sqrt {a+c x^4}} \, dx=\int { \frac {1}{\sqrt {c x^{4} + a} {\left (e x + d\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {1}{(d+e x)^2 \sqrt {a+c x^4}} \, dx=\int \frac {1}{\sqrt {c\,x^4+a}\,{\left (d+e\,x\right )}^2} \,d x \]
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