Optimal. Leaf size=227 \[ \frac {\sqrt [4]{4 a d^2+c^3} \sqrt {\frac {d^2 \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )}{\left (4 a d^2+c^3\right ) \left (\frac {d^2 \left (\frac {c}{d}+x\right )^2}{\sqrt {4 a d^2+c^3}}+\sqrt {c}\right )^2}} \left (\frac {d^2 \left (\frac {c}{d}+x\right )^2}{\sqrt {4 a d^2+c^3}}+\sqrt {c}\right ) F\left (2 \tan ^{-1}\left (\frac {c+d x}{\sqrt [4]{c} \sqrt [4]{c^3+4 a d^2}}\right )|\frac {1}{2} \left (\frac {c^{3/2}}{\sqrt {c^3+4 a d^2}}+1\right )\right )}{2 \sqrt [4]{c} d \sqrt {4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}} \]
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Rubi [A] time = 0.17, antiderivative size = 227, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {1106, 1103} \[ \frac {\sqrt [4]{4 a d^2+c^3} \sqrt {\frac {d^2 \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )}{\left (4 a d^2+c^3\right ) \left (\frac {d^2 \left (\frac {c}{d}+x\right )^2}{\sqrt {4 a d^2+c^3}}+\sqrt {c}\right )^2}} \left (\frac {d^2 \left (\frac {c}{d}+x\right )^2}{\sqrt {4 a d^2+c^3}}+\sqrt {c}\right ) F\left (2 \tan ^{-1}\left (\frac {c+d x}{\sqrt [4]{c} \sqrt [4]{c^3+4 a d^2}}\right )|\frac {1}{2} \left (\frac {c^{3/2}}{\sqrt {c^3+4 a d^2}}+1\right )\right )}{2 \sqrt [4]{c} d \sqrt {4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}} \]
Antiderivative was successfully verified.
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Rule 1103
Rule 1106
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{\sqrt {c \left (4 a+\frac {c^3}{d^2}\right )-2 c^2 x^2+d^2 x^4}} \, dx,x,\frac {c}{d}+x\right )\\ &=\frac {\sqrt [4]{c^3+4 a d^2} \sqrt {\frac {d^2 \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )}{\left (c^3+4 a d^2\right ) \left (\sqrt {c}+\frac {d^2 \left (\frac {c}{d}+x\right )^2}{\sqrt {c^3+4 a d^2}}\right )^2}} \left (\sqrt {c}+\frac {d^2 \left (\frac {c}{d}+x\right )^2}{\sqrt {c^3+4 a d^2}}\right ) F\left (2 \tan ^{-1}\left (\frac {c+d x}{\sqrt [4]{c} \sqrt [4]{c^3+4 a d^2}}\right )|\frac {1}{2} \left (1+\frac {c^{3/2}}{\sqrt {c^3+4 a d^2}}\right )\right )}{2 \sqrt [4]{c} d \sqrt {4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}}\\ \end {align*}
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Mathematica [C] time = 2.25, size = 822, normalized size = 3.62 \[ \frac {2 \left (-c-d x+\sqrt {c^2-2 i \sqrt {a} \sqrt {c} d}\right ) \left (c+d x+\sqrt {c^2-2 i \sqrt {a} \sqrt {c} d}\right ) \sqrt {-\frac {\sqrt {c^2-2 i \sqrt {a} \sqrt {c} d} \left (c+d x-\sqrt {c^2+2 i \sqrt {a} d \sqrt {c}}\right )}{\left (\sqrt {c^2-2 i \sqrt {a} \sqrt {c} d}+\sqrt {c^2+2 i \sqrt {a} d \sqrt {c}}\right ) \left (-c-d x+\sqrt {c^2-2 i \sqrt {a} \sqrt {c} d}\right )}} \sqrt {-\frac {\sqrt {c^2-2 i \sqrt {a} \sqrt {c} d} \left (c+d x+\sqrt {c^2+2 i \sqrt {a} d \sqrt {c}}\right )}{\left (\sqrt {c^2-2 i \sqrt {a} \sqrt {c} d}-\sqrt {c^2+2 i \sqrt {a} d \sqrt {c}}\right ) \left (-c-d x+\sqrt {c^2-2 i \sqrt {a} \sqrt {c} d}\right )}} F\left (\sin ^{-1}\left (\sqrt {\frac {\left (\sqrt {c^2-2 i \sqrt {a} \sqrt {c} d}-\sqrt {c^2+2 i \sqrt {a} d \sqrt {c}}\right ) \left (c+d x+\sqrt {c^2-2 i \sqrt {a} \sqrt {c} d}\right )}{\left (\sqrt {c^2-2 i \sqrt {a} \sqrt {c} d}+\sqrt {c^2+2 i \sqrt {a} d \sqrt {c}}\right ) \left (-c-d x+\sqrt {c^2-2 i \sqrt {a} \sqrt {c} d}\right )}}\right )|\frac {\left (\sqrt {c^2-2 i \sqrt {a} \sqrt {c} d}+\sqrt {c^2+2 i \sqrt {a} d \sqrt {c}}\right )^2}{\left (\sqrt {c^2-2 i \sqrt {a} \sqrt {c} d}-\sqrt {c^2+2 i \sqrt {a} d \sqrt {c}}\right )^2}\right )}{d \sqrt {c^2-2 i \sqrt {a} \sqrt {c} d} \sqrt {\frac {\left (\sqrt {c^2-2 i \sqrt {a} \sqrt {c} d}-\sqrt {c^2+2 i \sqrt {a} d \sqrt {c}}\right ) \left (c+d x+\sqrt {c^2-2 i \sqrt {a} \sqrt {c} d}\right )}{\left (\sqrt {c^2-2 i \sqrt {a} \sqrt {c} d}+\sqrt {c^2+2 i \sqrt {a} d \sqrt {c}}\right ) \left (-c-d x+\sqrt {c^2-2 i \sqrt {a} \sqrt {c} d}\right )}} \sqrt {x^2 (2 c+d x)^2+4 a c}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\sqrt {d^{2} x^{4} + 4 \, c d x^{3} + 4 \, c^{2} x^{2} + 4 \, a c}}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {d^{2} x^{4} + 4 \, c d x^{3} + 4 \, c^{2} x^{2} + 4 \, a c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 1056, normalized size = 4.65 \[ \frac {2 \left (\frac {-c +\sqrt {c^{2}+2 \sqrt {-a c}\, d}}{d}+\frac {c +\sqrt {c^{2}-2 \sqrt {-a c}\, d}}{d}\right ) \sqrt {\frac {\left (-\frac {c +\sqrt {c^{2}-2 \sqrt {-a c}\, d}}{d}+\frac {c +\sqrt {c^{2}+2 \sqrt {-a c}\, d}}{d}\right ) \left (x -\frac {-c +\sqrt {c^{2}+2 \sqrt {-a c}\, d}}{d}\right )}{\left (-\frac {c +\sqrt {c^{2}-2 \sqrt {-a c}\, d}}{d}-\frac {-c +\sqrt {c^{2}+2 \sqrt {-a c}\, d}}{d}\right ) \left (x +\frac {c +\sqrt {c^{2}+2 \sqrt {-a c}\, d}}{d}\right )}}\, \left (x +\frac {c +\sqrt {c^{2}+2 \sqrt {-a c}\, d}}{d}\right )^{2} \sqrt {\frac {\left (-\frac {c +\sqrt {c^{2}+2 \sqrt {-a c}\, d}}{d}-\frac {-c +\sqrt {c^{2}+2 \sqrt {-a c}\, d}}{d}\right ) \left (x -\frac {-c +\sqrt {c^{2}-2 \sqrt {-a c}\, d}}{d}\right )}{\left (\frac {-c +\sqrt {c^{2}-2 \sqrt {-a c}\, d}}{d}-\frac {-c +\sqrt {c^{2}+2 \sqrt {-a c}\, d}}{d}\right ) \left (x +\frac {c +\sqrt {c^{2}+2 \sqrt {-a c}\, d}}{d}\right )}}\, \sqrt {\frac {\left (-\frac {c +\sqrt {c^{2}+2 \sqrt {-a c}\, d}}{d}-\frac {-c +\sqrt {c^{2}+2 \sqrt {-a c}\, d}}{d}\right ) \left (x +\frac {c +\sqrt {c^{2}-2 \sqrt {-a c}\, d}}{d}\right )}{\left (-\frac {c +\sqrt {c^{2}-2 \sqrt {-a c}\, d}}{d}-\frac {-c +\sqrt {c^{2}+2 \sqrt {-a c}\, d}}{d}\right ) \left (x +\frac {c +\sqrt {c^{2}+2 \sqrt {-a c}\, d}}{d}\right )}}\, \EllipticF \left (\sqrt {\frac {\left (-\frac {c +\sqrt {c^{2}-2 \sqrt {-a c}\, d}}{d}+\frac {c +\sqrt {c^{2}+2 \sqrt {-a c}\, d}}{d}\right ) \left (x -\frac {-c +\sqrt {c^{2}+2 \sqrt {-a c}\, d}}{d}\right )}{\left (-\frac {c +\sqrt {c^{2}-2 \sqrt {-a c}\, d}}{d}-\frac {-c +\sqrt {c^{2}+2 \sqrt {-a c}\, d}}{d}\right ) \left (x +\frac {c +\sqrt {c^{2}+2 \sqrt {-a c}\, d}}{d}\right )}}, \sqrt {\frac {\left (-\frac {c +\sqrt {c^{2}+2 \sqrt {-a c}\, d}}{d}-\frac {-c +\sqrt {c^{2}-2 \sqrt {-a c}\, d}}{d}\right ) \left (\frac {-c +\sqrt {c^{2}+2 \sqrt {-a c}\, d}}{d}+\frac {c +\sqrt {c^{2}-2 \sqrt {-a c}\, d}}{d}\right )}{\left (\frac {-c +\sqrt {c^{2}+2 \sqrt {-a c}\, d}}{d}-\frac {-c +\sqrt {c^{2}-2 \sqrt {-a c}\, d}}{d}\right ) \left (-\frac {c +\sqrt {c^{2}+2 \sqrt {-a c}\, d}}{d}+\frac {c +\sqrt {c^{2}-2 \sqrt {-a c}\, d}}{d}\right )}}\right )}{\left (-\frac {c +\sqrt {c^{2}-2 \sqrt {-a c}\, d}}{d}+\frac {c +\sqrt {c^{2}+2 \sqrt {-a c}\, d}}{d}\right ) \left (-\frac {c +\sqrt {c^{2}+2 \sqrt {-a c}\, d}}{d}-\frac {-c +\sqrt {c^{2}+2 \sqrt {-a c}\, d}}{d}\right ) \sqrt {\left (x -\frac {-c +\sqrt {c^{2}+2 \sqrt {-a c}\, d}}{d}\right ) \left (x +\frac {c +\sqrt {c^{2}+2 \sqrt {-a c}\, d}}{d}\right ) \left (x -\frac {-c +\sqrt {c^{2}-2 \sqrt {-a c}\, d}}{d}\right ) \left (x +\frac {c +\sqrt {c^{2}-2 \sqrt {-a c}\, d}}{d}\right ) d^{2}}} \]
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 \[ \int \frac {1}{\sqrt {d^{2} x^{4} + 4 \, c d x^{3} + 4 \, c^{2} x^{2} + 4 \, a c}}\,{d x} \]
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 \[ \int \frac {1}{\sqrt {4\,c^2\,x^2+4\,c\,d\,x^3+4\,a\,c+d^2\,x^4}} \,d x \]
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 \[ \int \frac {1}{\sqrt {4 a c + 4 c^{2} x^{2} + 4 c d x^{3} + d^{2} x^{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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