Integrand size = 29, antiderivative size = 265 \[ \int \frac {e+f x}{(c+d x) \sqrt {c^3+4 d^3 x^3}} \, dx=\frac {2 (d e-c f) \arctan \left (\frac {\sqrt {3} \sqrt {c} (c+2 d x)}{\sqrt {c^3+4 d^3 x^3}}\right )}{3 \sqrt {3} c^{3/2} d^2}+\frac {\sqrt [3]{2} \sqrt {2+\sqrt {3}} (2 d e+c f) \left (c+2^{2/3} d x\right ) \sqrt {\frac {c^2-2^{2/3} c d x+2 \sqrt [3]{2} d^2 x^2}{\left (\left (1+\sqrt {3}\right ) c+2^{2/3} d x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) c+2^{2/3} d x}{\left (1+\sqrt {3}\right ) c+2^{2/3} d x}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} c d^2 \sqrt {\frac {c \left (c+2^{2/3} d x\right )}{\left (\left (1+\sqrt {3}\right ) c+2^{2/3} d x\right )^2}} \sqrt {c^3+4 d^3 x^3}} \]
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Time = 0.23 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2164, 224, 2162, 209} \[ \int \frac {e+f x}{(c+d x) \sqrt {c^3+4 d^3 x^3}} \, dx=\frac {\sqrt [3]{2} \sqrt {2+\sqrt {3}} \left (c+2^{2/3} d x\right ) \sqrt {\frac {c^2-2^{2/3} c d x+2 \sqrt [3]{2} d^2 x^2}{\left (\left (1+\sqrt {3}\right ) c+2^{2/3} d x\right )^2}} (c f+2 d e) \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) c+2^{2/3} d x}{\left (1+\sqrt {3}\right ) c+2^{2/3} d x}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} c d^2 \sqrt {\frac {c \left (c+2^{2/3} d x\right )}{\left (\left (1+\sqrt {3}\right ) c+2^{2/3} d x\right )^2}} \sqrt {c^3+4 d^3 x^3}}+\frac {2 \arctan \left (\frac {\sqrt {3} \sqrt {c} (c+2 d x)}{\sqrt {c^3+4 d^3 x^3}}\right ) (d e-c f)}{3 \sqrt {3} c^{3/2} d^2} \]
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Rule 209
Rule 224
Rule 2162
Rule 2164
Rubi steps \begin{align*} \text {integral}& = \frac {(d e-c f) \int \frac {c-2 d x}{(c+d x) \sqrt {c^3+4 d^3 x^3}} \, dx}{3 c d}+\frac {(2 d e+c f) \int \frac {1}{\sqrt {c^3+4 d^3 x^3}} \, dx}{3 c d} \\ & = \frac {\sqrt [3]{2} \sqrt {2+\sqrt {3}} (2 d e+c f) \left (c+2^{2/3} d x\right ) \sqrt {\frac {c^2-2^{2/3} c d x+2 \sqrt [3]{2} d^2 x^2}{\left (\left (1+\sqrt {3}\right ) c+2^{2/3} d x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) c+2^{2/3} d x}{\left (1+\sqrt {3}\right ) c+2^{2/3} d x}\right )|-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} c d^2 \sqrt {\frac {c \left (c+2^{2/3} d x\right )}{\left (\left (1+\sqrt {3}\right ) c+2^{2/3} d x\right )^2}} \sqrt {c^3+4 d^3 x^3}}+\frac {(2 (d e-c f)) \text {Subst}\left (\int \frac {1}{1+3 c^3 x^2} \, dx,x,\frac {1+\frac {2 d x}{c}}{\sqrt {c^3+4 d^3 x^3}}\right )}{3 d^2} \\ & = \frac {2 (d e-c f) \tan ^{-1}\left (\frac {\sqrt {3} \sqrt {c} (c+2 d x)}{\sqrt {c^3+4 d^3 x^3}}\right )}{3 \sqrt {3} c^{3/2} d^2}+\frac {\sqrt [3]{2} \sqrt {2+\sqrt {3}} (2 d e+c f) \left (c+2^{2/3} d x\right ) \sqrt {\frac {c^2-2^{2/3} c d x+2 \sqrt [3]{2} d^2 x^2}{\left (\left (1+\sqrt {3}\right ) c+2^{2/3} d x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) c+2^{2/3} d x}{\left (1+\sqrt {3}\right ) c+2^{2/3} d x}\right )|-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} c d^2 \sqrt {\frac {c \left (c+2^{2/3} d x\right )}{\left (\left (1+\sqrt {3}\right ) c+2^{2/3} d x\right )^2}} \sqrt {c^3+4 d^3 x^3}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 11.07 (sec) , antiderivative size = 380, normalized size of antiderivative = 1.43 \[ \int \frac {e+f x}{(c+d x) \sqrt {c^3+4 d^3 x^3}} \, dx=\frac {\sqrt [6]{2} \sqrt {\frac {\sqrt [3]{2} c+2 d x}{\left (1+\sqrt [3]{-1}\right ) c}} \left (-f \sqrt {\frac {\sqrt [3]{-2} c-2 (-1)^{2/3} d x}{\left (1+\sqrt [3]{-1}\right ) c}} \left (\sqrt [3]{-1} \left (2+\sqrt [3]{-2}\right ) c-2 \left (\sqrt [3]{-1}+2^{2/3}\right ) d x\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {\sqrt [3]{2} c+2 (-1)^{2/3} d x}{\left (1+\sqrt [3]{-1}\right ) c}}}{\sqrt [6]{2}}\right ),\sqrt [3]{-1}\right )+\frac {\sqrt [3]{-1} 2^{2/3} \left (1+\sqrt [3]{-1}\right ) (-d e+c f) \sqrt {\frac {\sqrt [3]{2} c+2 (-1)^{2/3} d x}{\left (1+\sqrt [3]{-1}\right ) c}} \sqrt {2^{2/3}-\frac {2 \sqrt [3]{2} d x}{c}+\frac {4 d^2 x^2}{c^2}} \operatorname {EllipticPi}\left (\frac {i \sqrt [3]{2} \sqrt {3}}{2+\sqrt [3]{-2}},\arcsin \left (\frac {\sqrt {\frac {\sqrt [3]{2} c+2 (-1)^{2/3} d x}{\left (1+\sqrt [3]{-1}\right ) c}}}{\sqrt [6]{2}}\right ),\sqrt [3]{-1}\right )}{\sqrt {3}}\right )}{\left (2+\sqrt [3]{-2}\right ) d^2 \sqrt {\frac {\sqrt [3]{2} c+2 (-1)^{2/3} d x}{\left (1+\sqrt [3]{-1}\right ) c}} \sqrt {c^3+4 d^3 x^3}} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 899 vs. \(2 (216 ) = 432\).
Time = 0.96 (sec) , antiderivative size = 900, normalized size of antiderivative = 3.40
method | result | size |
default | \(\text {Expression too large to display}\) | \(900\) |
elliptic | \(\text {Expression too large to display}\) | \(900\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.32 (sec) , antiderivative size = 390, normalized size of antiderivative = 1.47 \[ \int \frac {e+f x}{(c+d x) \sqrt {c^3+4 d^3 x^3}} \, dx=\left [\frac {\sqrt {3} {\left (d^{3} e - c d^{2} f\right )} \sqrt {-c} \log \left (\frac {2 \, d^{6} x^{6} - 36 \, c d^{5} x^{5} - 18 \, c^{2} d^{4} x^{4} + 28 \, c^{3} d^{3} x^{3} + 18 \, c^{4} d^{2} x^{2} - c^{6} - \sqrt {3} {\left (4 \, d^{4} x^{4} - 10 \, c d^{3} x^{3} - 18 \, c^{2} d^{2} x^{2} - 8 \, c^{3} d x - c^{4}\right )} \sqrt {4 \, d^{3} x^{3} + c^{3}} \sqrt {-c}}{d^{6} x^{6} + 6 \, c d^{5} x^{5} + 15 \, c^{2} d^{4} x^{4} + 20 \, c^{3} d^{3} x^{3} + 15 \, c^{4} d^{2} x^{2} + 6 \, c^{5} d x + c^{6}}\right ) + 6 \, \sqrt {d^{3}} {\left (2 \, c d e + c^{2} f\right )} {\rm weierstrassPInverse}\left (0, -\frac {c^{3}}{d^{3}}, x\right )}{18 \, c^{2} d^{4}}, -\frac {\sqrt {3} {\left (d^{3} e - c d^{2} f\right )} \sqrt {c} \arctan \left (\frac {\sqrt {3} \sqrt {4 \, d^{3} x^{3} + c^{3}} {\left (2 \, d^{3} x^{3} - 6 \, c d^{2} x^{2} - 6 \, c^{2} d x - c^{3}\right )} \sqrt {c}}{3 \, {\left (8 \, c d^{4} x^{4} + 4 \, c^{2} d^{3} x^{3} + 2 \, c^{4} d x + c^{5}\right )}}\right ) - 3 \, \sqrt {d^{3}} {\left (2 \, c d e + c^{2} f\right )} {\rm weierstrassPInverse}\left (0, -\frac {c^{3}}{d^{3}}, x\right )}{9 \, c^{2} d^{4}}\right ] \]
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\[ \int \frac {e+f x}{(c+d x) \sqrt {c^3+4 d^3 x^3}} \, dx=\int \frac {e + f x}{\left (c + d x\right ) \sqrt {c^{3} + 4 d^{3} x^{3}}}\, dx \]
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\[ \int \frac {e+f x}{(c+d x) \sqrt {c^3+4 d^3 x^3}} \, dx=\int { \frac {f x + e}{\sqrt {4 \, d^{3} x^{3} + c^{3}} {\left (d x + c\right )}} \,d x } \]
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\[ \int \frac {e+f x}{(c+d x) \sqrt {c^3+4 d^3 x^3}} \, dx=\int { \frac {f x + e}{\sqrt {4 \, d^{3} x^{3} + c^{3}} {\left (d x + c\right )}} \,d x } \]
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Timed out. \[ \int \frac {e+f x}{(c+d x) \sqrt {c^3+4 d^3 x^3}} \, dx=\int \frac {e+f\,x}{\sqrt {c^3+4\,d^3\,x^3}\,\left (c+d\,x\right )} \,d x \]
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