Integrand size = 19, antiderivative size = 1221 \[ \int \frac {\sqrt {a+c x^4}}{(d+e x)^2} \, dx=\frac {2 \sqrt {c} x \sqrt {a+c x^4}}{e^2 \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {d \sqrt {a+c x^4}}{e \left (d^2-e^2 x^2\right )}+\frac {x \sqrt {a+c x^4}}{d^2-e^2 x^2}+\frac {\sqrt {-c d^4-a e^4} \arctan \left (\frac {\sqrt {-c d^4-a e^4} x}{d e \sqrt {a+c x^4}}\right )}{2 d e^3}-\frac {\left (c d^4-a e^4\right ) \arctan \left (\frac {\sqrt {-c d^4-a e^4} x}{d e \sqrt {a+c x^4}}\right )}{2 d e^3 \sqrt {-c d^4-a e^4}}-\frac {\sqrt {c} d \text {arctanh}\left (\frac {\sqrt {c} x^2}{\sqrt {a+c x^4}}\right )}{e^3}+\frac {c d^3 \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 )}{e^3 \sqrt {c d^4+a e^4}}-\frac {2 \sqrt [4]{a} \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}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{e^2 \sqrt {a+c x^4}}+\frac {3 \sqrt [4]{a} \sqrt [4]{c} \left (\frac {\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 {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 e^4 \sqrt {a+c x^4}}-\frac {\sqrt [4]{c} \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 {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} e^4 \sqrt {a+c x^4}}+\frac {\sqrt [4]{c} \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 {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} e^4 \sqrt {a+c x^4}}-\frac {\sqrt [4]{c} \left (c d^4+a e^4\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 {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} e^4 \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \sqrt {a+c x^4}}+\frac {\left (\sqrt {c} d^2-\sqrt {a} e^2\right )^2 \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 )}{4 \sqrt [4]{a} \sqrt [4]{c} d^2 e^4 \sqrt {a+c x^4}}+\frac {\left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \left (c d^4+a e^4\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 )}{4 \sqrt [4]{a} \sqrt [4]{c} d^2 e^4 \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \sqrt {a+c x^4}} \]
[Out]
Time = 1.33 (sec) , antiderivative size = 1221, normalized size of antiderivative = 1.00, number of steps used = 32, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.789, Rules used = {2184, 1241, 1212, 226, 1210, 1231, 1721, 1262, 747, 858, 223, 212, 739, 1350, 1223} \[ \int \frac {\sqrt {a+c x^4}}{(d+e x)^2} \, dx=\frac {c \text {arctanh}\left (\frac {a e^2+c d^2 x^2}{\sqrt {c d^4+a e^4} \sqrt {c x^4+a}}\right ) d^3}{e^3 \sqrt {c d^4+a e^4}}-\frac {\sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} x^2}{\sqrt {c x^4+a}}\right ) d}{e^3}-\frac {\sqrt {c x^4+a} d}{e \left (d^2-e^2 x^2\right )}-\frac {2 \sqrt [4]{a} \sqrt [4]{c} \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{e^2 \sqrt {c x^4+a}}+\frac {3 \sqrt [4]{a} \sqrt [4]{c} \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 e^4 \sqrt {c x^4+a}}-\frac {\sqrt [4]{c} \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+a}{\left (\sqrt {c} x^2+\sqrt {a}\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} e^4 \sqrt {c x^4+a}}+\frac {\sqrt [4]{c} \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} e^4 \sqrt {c x^4+a}}-\frac {\sqrt [4]{c} \left (c d^4+a e^4\right ) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+a}{\left (\sqrt {c} x^2+\sqrt {a}\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} e^4 \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \sqrt {c x^4+a}}+\frac {2 \sqrt {c} x \sqrt {c x^4+a}}{e^2 \left (\sqrt {c} x^2+\sqrt {a}\right )}+\frac {x \sqrt {c x^4+a}}{d^2-e^2 x^2}-\frac {\left (c d^4-a e^4\right ) \arctan \left (\frac {\sqrt {-c d^4-a e^4} x}{d e \sqrt {c x^4+a}}\right )}{2 e^3 \sqrt {-c d^4-a e^4} d}+\frac {\sqrt {-c d^4-a e^4} \arctan \left (\frac {\sqrt {-c d^4-a e^4} x}{d e \sqrt {c x^4+a}}\right )}{2 e^3 d}+\frac {\left (\sqrt {c} d^2-\sqrt {a} e^2\right )^2 \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+a}{\left (\sqrt {c} x^2+\sqrt {a}\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 )}{4 \sqrt [4]{a} \sqrt [4]{c} e^4 \sqrt {c x^4+a} d^2}+\frac {\left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \left (c d^4+a e^4\right ) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+a}{\left (\sqrt {c} x^2+\sqrt {a}\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 )}{4 \sqrt [4]{a} \sqrt [4]{c} e^4 \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \sqrt {c x^4+a} d^2} \]
[In]
[Out]
Rule 212
Rule 223
Rule 226
Rule 739
Rule 747
Rule 858
Rule 1210
Rule 1212
Rule 1223
Rule 1231
Rule 1241
Rule 1262
Rule 1350
Rule 1721
Rule 2184
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {d^2 \sqrt {a+c x^4}}{\left (d^2-e^2 x^2\right )^2}-\frac {2 d e x \sqrt {a+c x^4}}{\left (d^2-e^2 x^2\right )^2}+\frac {e^2 x^2 \sqrt {a+c x^4}}{\left (-d^2+e^2 x^2\right )^2}\right ) \, dx \\ & = d^2 \int \frac {\sqrt {a+c x^4}}{\left (d^2-e^2 x^2\right )^2} \, dx-(2 d e) \int \frac {x \sqrt {a+c x^4}}{\left (d^2-e^2 x^2\right )^2} \, dx+e^2 \int \frac {x^2 \sqrt {a+c x^4}}{\left (-d^2+e^2 x^2\right )^2} \, dx \\ & = \frac {x \sqrt {a+c x^4}}{2 \left (d^2-e^2 x^2\right )}+\frac {1}{2} \left (a-\frac {c d^4}{e^4}\right ) \int \frac {1}{\left (d^2-e^2 x^2\right ) \sqrt {a+c x^4}} \, dx+\frac {c \int \frac {d^2+e^2 x^2}{\sqrt {a+c x^4}} \, dx}{2 e^4}-(d e) \text {Subst}\left (\int \frac {\sqrt {a+c x^2}}{\left (d^2-e^2 x\right )^2} \, dx,x,x^2\right )+e^2 \int \left (\frac {d^2 \sqrt {a+c x^4}}{e^2 \left (-d^2+e^2 x^2\right )^2}+\frac {\sqrt {a+c x^4}}{e^2 \left (-d^2+e^2 x^2\right )}\right ) \, dx \\ & = -\frac {d \sqrt {a+c x^4}}{e \left (d^2-e^2 x^2\right )}+\frac {x \sqrt {a+c x^4}}{2 \left (d^2-e^2 x^2\right )}+d^2 \int \frac {\sqrt {a+c x^4}}{\left (-d^2+e^2 x^2\right )^2} \, dx+\frac {1}{2} \left (\sqrt {a} \left (\sqrt {a}-\frac {\sqrt {c} d^2}{e^2}\right )\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-\frac {\left (\sqrt {a} \sqrt {c}\right ) \int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}{\sqrt {a+c x^4}} \, dx}{2 e^2}+\frac {(c d) \text {Subst}\left (\int \frac {x}{\left (d^2-e^2 x\right ) \sqrt {a+c x^2}} \, dx,x,x^2\right )}{e}-\frac {\left (\sqrt {c} \left (\sqrt {c} d^2-\sqrt {a} e^2\right )\right ) \int \frac {1}{\sqrt {a+c x^4}} \, dx}{2 e^4}+\frac {\left (c \left (d^2+\frac {\sqrt {a} e^2}{\sqrt {c}}\right )\right ) \int \frac {1}{\sqrt {a+c x^4}} \, dx}{2 e^4}+\int \frac {\sqrt {a+c x^4}}{-d^2+e^2 x^2} \, dx \\ & = \frac {\sqrt {c} x \sqrt {a+c x^4}}{2 e^2 \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {d \sqrt {a+c x^4}}{e \left (d^2-e^2 x^2\right )}+\frac {x \sqrt {a+c x^4}}{d^2-e^2 x^2}-\frac {\left (c d^4-a e^4\right ) \tan ^{-1}\left (\frac {\sqrt {-c d^4-a e^4} x}{d e \sqrt {a+c x^4}}\right )}{4 d e^3 \sqrt {-c d^4-a e^4}}-\frac {\sqrt [4]{a} \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}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 e^2 \sqrt {a+c x^4}}-\frac {\sqrt [4]{c} \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}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{a} e^4 \sqrt {a+c x^4}}+\frac {\sqrt [4]{c} \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}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{a} e^4 \sqrt {a+c x^4}}+\frac {\left (\sqrt {c} d^2-\sqrt {a} e^2\right )^2 \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 )}{8 \sqrt [4]{a} \sqrt [4]{c} d^2 e^4 \sqrt {a+c x^4}}+\frac {1}{2} \left (-a+\frac {c d^4}{e^4}\right ) \int \frac {1}{\left (-d^2+e^2 x^2\right ) \sqrt {a+c x^4}} \, dx+\left (a+\frac {c d^4}{e^4}\right ) \int \frac {1}{\left (-d^2+e^2 x^2\right ) \sqrt {a+c x^4}} \, dx-\frac {\int \frac {-c d^2-c e^2 x^2}{\sqrt {a+c x^4}} \, dx}{e^4}-\frac {c \int \frac {-d^2-e^2 x^2}{\sqrt {a+c x^4}} \, dx}{2 e^4}-\frac {(c d) \text {Subst}\left (\int \frac {1}{\sqrt {a+c x^2}} \, dx,x,x^2\right )}{e^3}+\frac {\left (c d^3\right ) \text {Subst}\left (\int \frac {1}{\left (d^2-e^2 x\right ) \sqrt {a+c x^2}} \, dx,x,x^2\right )}{e^3} \\ & = \frac {\sqrt {c} x \sqrt {a+c x^4}}{2 e^2 \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {d \sqrt {a+c x^4}}{e \left (d^2-e^2 x^2\right )}+\frac {x \sqrt {a+c x^4}}{d^2-e^2 x^2}-\frac {\left (c d^4-a e^4\right ) \tan ^{-1}\left (\frac {\sqrt {-c d^4-a e^4} x}{d e \sqrt {a+c x^4}}\right )}{4 d e^3 \sqrt {-c d^4-a e^4}}-\frac {\sqrt [4]{a} \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}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 e^2 \sqrt {a+c x^4}}-\frac {\sqrt [4]{c} \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}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{a} e^4 \sqrt {a+c x^4}}+\frac {\sqrt [4]{c} \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}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{a} e^4 \sqrt {a+c x^4}}+\frac {\left (\sqrt {c} d^2-\sqrt {a} e^2\right )^2 \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 )}{8 \sqrt [4]{a} \sqrt [4]{c} d^2 e^4 \sqrt {a+c x^4}}-\frac {1}{2} \left (\sqrt {a} \left (\sqrt {a}-\frac {\sqrt {c} d^2}{e^2}\right )\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-\frac {(c d) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x^2}{\sqrt {a+c x^4}}\right )}{e^3}-\frac {\left (c d^3\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 )}{e^3}-\frac {\left (\sqrt {a} \sqrt {c}\right ) \int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}{\sqrt {a+c x^4}} \, dx}{2 e^2}-\frac {\left (\sqrt {a} \sqrt {c}\right ) \int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}{\sqrt {a+c x^4}} \, dx}{e^2}-\frac {\left (\sqrt {c} \left (\sqrt {c} d^2-\sqrt {a} e^2\right )\right ) \int \frac {1}{\sqrt {a+c x^4}} \, dx}{2 e^4}-\frac {\left (\sqrt {c} \left (a+\frac {c d^4}{e^4}\right )\right ) \int \frac {1}{\sqrt {a+c x^4}} \, dx}{\sqrt {c} d^2+\sqrt {a} e^2}+\frac {\left (\sqrt {a} \left (a+\frac {c d^4}{e^4}\right ) 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}{\sqrt {c} d^2+\sqrt {a} e^2}+\frac {\left (\sqrt {c} \left (\sqrt {c} d^2+\sqrt {a} e^2\right )\right ) \int \frac {1}{\sqrt {a+c x^4}} \, dx}{2 e^4}+\frac {\left (\sqrt {c} \left (\sqrt {c} d^2+\sqrt {a} e^2\right )\right ) \int \frac {1}{\sqrt {a+c x^4}} \, dx}{e^4} \\ & = \frac {2 \sqrt {c} x \sqrt {a+c x^4}}{e^2 \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {d \sqrt {a+c x^4}}{e \left (d^2-e^2 x^2\right )}+\frac {x \sqrt {a+c x^4}}{d^2-e^2 x^2}+\frac {\sqrt {-c d^4-a e^4} \tan ^{-1}\left (\frac {\sqrt {-c d^4-a e^4} x}{d e \sqrt {a+c x^4}}\right )}{2 d e^3}-\frac {\left (c d^4-a e^4\right ) \tan ^{-1}\left (\frac {\sqrt {-c d^4-a e^4} x}{d e \sqrt {a+c x^4}}\right )}{2 d e^3 \sqrt {-c d^4-a e^4}}-\frac {\sqrt {c} d \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a+c x^4}}\right )}{e^3}+\frac {c d^3 \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 )}{e^3 \sqrt {c d^4+a e^4}}-\frac {2 \sqrt [4]{a} \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}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{e^2 \sqrt {a+c x^4}}-\frac {\sqrt [4]{c} \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}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{a} e^4 \sqrt {a+c x^4}}-\frac {\sqrt [4]{c} \left (a+\frac {c d^4}{e^4}\right ) \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 {\sqrt [4]{c} \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}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{a} e^4 \sqrt {a+c x^4}}+\frac {\left (\sqrt {c} d^2-\sqrt {a} e^2\right )^2 \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 )}{4 \sqrt [4]{a} \sqrt [4]{c} d^2 e^4 \sqrt {a+c x^4}}+\frac {\left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \left (c d^4+a e^4\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 )}{4 \sqrt [4]{a} \sqrt [4]{c} d^2 e^4 \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \sqrt {a+c x^4}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 11.46 (sec) , antiderivative size = 394, normalized size of antiderivative = 0.32 \[ \int \frac {\sqrt {a+c x^4}}{(d+e x)^2} \, dx=\frac {-\frac {e^3 \left (a+c x^4\right )}{d+e x}-\frac {2 c d^3 e \sqrt {a+c x^4} \arctan \left (\frac {\sqrt {c} \left (d^2-e^2 x^2\right )+e^2 \sqrt {a+c x^4}}{\sqrt {-c d^4-a e^4}}\right )}{\sqrt {-c d^4-a e^4}}-2 i a \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} e^2 \sqrt {1+\frac {c x^4}{a}} E\left (\left .i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )-\frac {2 \sqrt {c} \left (i \sqrt {c} d^2+\sqrt {a} e^2\right ) \sqrt {1+\frac {c x^4}{a}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right ),-1\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}+2 \sqrt [4]{-1} \sqrt [4]{a} c^{3/4} d^2 \sqrt {1+\frac {c x^4}{a}} \operatorname {EllipticPi}\left (\frac {i \sqrt {a} e^2}{\sqrt {c} d^2},\arcsin \left (\frac {(-1)^{3/4} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )+\sqrt {c} d e \sqrt {a+c x^4} \log \left (-\sqrt {c} x^2+\sqrt {a+c x^4}\right )}{e^4 \sqrt {a+c x^4}} \]
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Result contains complex when optimal does not.
Time = 1.02 (sec) , antiderivative size = 402, normalized size of antiderivative = 0.33
method | result | size |
default | \(-\frac {\sqrt {c \,x^{4}+a}}{e \left (e x +d \right )}+\frac {2 c \,d^{2} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{e^{4} \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}-\frac {d \sqrt {c}\, \ln \left (2 x^{2} \sqrt {c}+2 \sqrt {c \,x^{4}+a}\right )}{e^{3}}+\frac {2 i \sqrt {c}\, \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{e^{2} \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}-\frac {2 d^{3} c \left (-\frac {\operatorname {arctanh}\left (\frac {\frac {2 c \,x^{2} d^{2}}{e^{2}}+2 a}{2 \sqrt {\frac {c \,d^{4}}{e^{4}}+a}\, \sqrt {c \,x^{4}+a}}\right )}{2 \sqrt {\frac {c \,d^{4}}{e^{4}}+a}}+\frac {e \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \Pi \left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, -\frac {i \sqrt {a}\, e^{2}}{\sqrt {c}\, d^{2}}, \frac {\sqrt {-\frac {i \sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, d \sqrt {c \,x^{4}+a}}\right )}{e^{5}}\) | \(402\) |
elliptic | \(-\frac {\sqrt {c \,x^{4}+a}}{e \left (e x +d \right )}+\frac {2 c \,d^{2} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{e^{4} \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}-\frac {d \sqrt {c}\, \ln \left (2 x^{2} \sqrt {c}+2 \sqrt {c \,x^{4}+a}\right )}{e^{3}}+\frac {2 i \sqrt {c}\, \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{e^{2} \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}-\frac {2 d^{3} c \left (-\frac {\operatorname {arctanh}\left (\frac {\frac {2 c \,x^{2} d^{2}}{e^{2}}+2 a}{2 \sqrt {\frac {c \,d^{4}}{e^{4}}+a}\, \sqrt {c \,x^{4}+a}}\right )}{2 \sqrt {\frac {c \,d^{4}}{e^{4}}+a}}+\frac {e \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \Pi \left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, -\frac {i \sqrt {a}\, e^{2}}{\sqrt {c}\, d^{2}}, \frac {\sqrt {-\frac {i \sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, d \sqrt {c \,x^{4}+a}}\right )}{e^{5}}\) | \(402\) |
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Timed out. \[ \int \frac {\sqrt {a+c x^4}}{(d+e x)^2} \, dx=\text {Timed out} \]
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\[ \int \frac {\sqrt {a+c x^4}}{(d+e x)^2} \, dx=\int \frac {\sqrt {a + c x^{4}}}{\left (d + e x\right )^{2}}\, dx \]
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\[ \int \frac {\sqrt {a+c x^4}}{(d+e x)^2} \, dx=\int { \frac {\sqrt {c x^{4} + a}}{{\left (e x + d\right )}^{2}} \,d x } \]
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\[ \int \frac {\sqrt {a+c x^4}}{(d+e x)^2} \, dx=\int { \frac {\sqrt {c x^{4} + a}}{{\left (e x + d\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {a+c x^4}}{(d+e x)^2} \, dx=\int \frac {\sqrt {c\,x^4+a}}{{\left (d+e\,x\right )}^2} \,d x \]
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