Optimal. Leaf size=610 \[ -\frac {c^{3/4} d^2 \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \Pi \left (\frac {\left (\sqrt {c} d^2+\sqrt {a} e^2\right )^2}{4 \sqrt {a} \sqrt {c} d^2 e^2};2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt {a+c x^4} \left (\sqrt {a} e^2+\sqrt {c} d^2\right ) \left (a e^4+c d^4\right )}-\frac {e^3 \sqrt {a+c x^4}}{(d+e x) \left (a e^4+c d^4\right )}+\frac {\sqrt {c} e^2 x \sqrt {a+c x^4}}{\left (\sqrt {a}+\sqrt {c} x^2\right ) \left (a e^4+c d^4\right )}-\frac {\sqrt [4]{a} \sqrt [4]{c} e^2 \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt {a+c x^4} \left (a e^4+c d^4\right )}+\frac {\sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt {a+c x^4} \left (\sqrt {a} e^2+\sqrt {c} d^2\right )}-\frac {c d^3 e \tan ^{-1}\left (\frac {x \sqrt {-a e^4-c d^4}}{d e \sqrt {a+c x^4}}\right )}{\left (-a e^4-c d^4\right )^{3/2}}-\frac {c d^3 e \tanh ^{-1}\left (\frac {a e^2+c d^2 x^2}{\sqrt {a+c x^4} \sqrt {a e^4+c d^4}}\right )}{\left (a e^4+c d^4\right )^{3/2}} \]
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Rubi [A] time = 0.76, antiderivative size = 610, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.579, Rules used = {1727, 1742, 12, 1248, 725, 206, 1715, 1196, 1709, 220, 1707} \[ -\frac {c^{3/4} d^2 \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \Pi \left (\frac {\left (\sqrt {c} d^2+\sqrt {a} e^2\right )^2}{4 \sqrt {a} \sqrt {c} d^2 e^2};2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt {a+c x^4} \left (\sqrt {a} e^2+\sqrt {c} d^2\right ) \left (a e^4+c d^4\right )}-\frac {e^3 \sqrt {a+c x^4}}{(d+e x) \left (a e^4+c d^4\right )}+\frac {\sqrt {c} e^2 x \sqrt {a+c x^4}}{\left (\sqrt {a}+\sqrt {c} x^2\right ) \left (a e^4+c d^4\right )}-\frac {c d^3 e \tan ^{-1}\left (\frac {x \sqrt {-a e^4-c d^4}}{d e \sqrt {a+c x^4}}\right )}{\left (-a e^4-c d^4\right )^{3/2}}-\frac {c d^3 e \tanh ^{-1}\left (\frac {a e^2+c d^2 x^2}{\sqrt {a+c x^4} \sqrt {a e^4+c d^4}}\right )}{\left (a e^4+c d^4\right )^{3/2}}+\frac {\sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt {a+c x^4} \left (\sqrt {a} e^2+\sqrt {c} d^2\right )}-\frac {\sqrt [4]{a} \sqrt [4]{c} e^2 \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt {a+c x^4} \left (a e^4+c d^4\right )} \]
Antiderivative was successfully verified.
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Rule 12
Rule 206
Rule 220
Rule 725
Rule 1196
Rule 1248
Rule 1707
Rule 1709
Rule 1715
Rule 1727
Rule 1742
Rubi steps
\begin {align*} \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 {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 ) \operatorname {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 ) \operatorname {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*}
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Mathematica [C] time = 1.11, size = 425, normalized size = 0.70 \[ \frac {-\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} \left (2 \sqrt [4]{-1} \sqrt [4]{a} c^{3/4} d^2 \sqrt {\frac {c x^4}{a}+1} (d+e x) \sqrt {a e^4+c d^4} \Pi \left (\frac {i \sqrt {a} e^2}{\sqrt {c} d^2};\left .\sin ^{-1}\left (\frac {(-1)^{3/4} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )+e^3 \left (a+c x^4\right ) \sqrt {a e^4+c d^4}+c d^3 e \sqrt {a+c x^4} (d+e x) \tanh ^{-1}\left (\frac {a e^2+c d^2 x^2}{\sqrt {a+c x^4} \sqrt {a e^4+c d^4}}\right )\right )+\sqrt {a} \sqrt {c} e^2 \sqrt {\frac {c x^4}{a}+1} (d+e x) \sqrt {a e^4+c d^4} E\left (\left .i \sinh ^{-1}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )+i \sqrt {c} \sqrt {\frac {c x^4}{a}+1} (d+e x) \left (\sqrt {c} d^2+i \sqrt {a} e^2\right ) \sqrt {a e^4+c d^4} F\left (\left .i \sinh ^{-1}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} \sqrt {a+c x^4} (d+e x) \left (a e^4+c d^4\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {1}{\sqrt {c x^{4} + a} {\left (e x + d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.03, size = 421, normalized size = 0.69 \[ -\frac {\sqrt {-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}+1}\, c \,d^{2} \EllipticF \left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, x , i\right )}{\left (a \,e^{4}+c \,d^{4}\right ) \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {i \sqrt {-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}+1}\, \left (-\EllipticE \left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, x , i\right )+\EllipticF \left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, x , i\right )\right ) \sqrt {a}\, \sqrt {c}\, e^{2}}{\left (a \,e^{4}+c \,d^{4}\right ) \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {2 \left (\frac {\sqrt {-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}+1}\, e \EllipticPi \left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, x , -\frac {i \sqrt {a}\, e^{2}}{\sqrt {c}\, d^{2}}, \frac {\sqrt {-\frac {i \sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, d}-\frac {\arctanh \left (\frac {\frac {2 c \,d^{2} x^{2}}{e^{2}}+2 a}{2 \sqrt {a +\frac {c \,d^{4}}{e^{4}}}\, \sqrt {c \,x^{4}+a}}\right )}{2 \sqrt {a +\frac {c \,d^{4}}{e^{4}}}}\right ) c \,d^{3}}{\left (a \,e^{4}+c \,d^{4}\right ) e}-\frac {\sqrt {c \,x^{4}+a}\, e^{3}}{\left (a \,e^{4}+c \,d^{4}\right ) \left (e x +d \right )} \]
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 \[ \int \frac {1}{\sqrt {c x^{4} + a} {\left (e x + d\right )}^{2}}\,{d x} \]
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 \[ \int \frac {1}{\sqrt {c\,x^4+a}\,{\left (d+e\,x\right )}^2} \,d x \]
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 \[ \int \frac {1}{\sqrt {a + c x^{4}} \left (d + e x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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