Optimal. Leaf size=249 \[ \frac {2 \tan ^{-1}\left (\frac {\sqrt {3} \sqrt {c} (c+2 d x)}{\sqrt {c^3+4 d^3 x^3}}\right )}{3 \sqrt {3} c^{3/2} d}+\frac {2 \sqrt [3]{2} \sqrt {2+\sqrt {3}} \left (c+2^{2/3} d x\right ) \sqrt {\frac {c^2-2^{2/3} c d x+2 \sqrt [3]{2} d^2 x^2}{\left (\left (1+\sqrt {3}\right ) c+2^{2/3} d x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) c+2^{2/3} d x}{\left (1+\sqrt {3}\right ) c+2^{2/3} d x}\right )|-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} c d \sqrt {\frac {c \left (c+2^{2/3} d x\right )}{\left (\left (1+\sqrt {3}\right ) c+2^{2/3} d x\right )^2}} \sqrt {c^3+4 d^3 x^3}} \]
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Rubi [A]
time = 0.18, antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2159, 224,
2162, 209} \begin {gather*} \frac {2 \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}} F\left (\text {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 \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 \text {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} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 224
Rule 2159
Rule 2162
Rubi steps
\begin {align*} \int \frac {1}{(c+d x) \sqrt {c^3+4 d^3 x^3}} \, dx &=\frac {\int \frac {c-2 d x}{(c+d x) \sqrt {c^3+4 d^3 x^3}} \, dx}{3 c}+\frac {2 \int \frac {1}{\sqrt {c^3+4 d^3 x^3}} \, dx}{3 c}\\ &=\frac {2 \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}} 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 \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 \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}\\ &=\frac {2 \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}+\frac {2 \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}} 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 \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*}
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Mathematica [C] Result contains complex when optimal does not.
time = 10.16, size = 169, normalized size = 0.68 \begin {gather*} -\frac {i 2^{5/6} \sqrt {\frac {\sqrt [3]{2} c+2 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}} \Pi \left (\frac {i \sqrt [3]{2} \sqrt {3}}{2+\sqrt [3]{-2}};\sin ^{-1}\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 )}{\left (2+\sqrt [3]{-2}\right ) d \sqrt {c^3+4 d^3 x^3}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 494 vs. \(2 (200 ) = 400\).
time = 0.40, size = 495, normalized size = 1.99
method | result | size |
default | \(\frac {2 \left (\frac {\left (\frac {2^{\frac {1}{3}}}{4}-\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}-\frac {\left (\frac {2^{\frac {1}{3}}}{4}+\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}\right ) \sqrt {\frac {x -\frac {\left (\frac {2^{\frac {1}{3}}}{4}+\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}}{\frac {\left (\frac {2^{\frac {1}{3}}}{4}-\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}-\frac {\left (\frac {2^{\frac {1}{3}}}{4}+\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}}}\, \sqrt {\frac {x +\frac {2^{\frac {1}{3}} c}{2 d}}{\frac {\left (\frac {2^{\frac {1}{3}}}{4}+\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}+\frac {2^{\frac {1}{3}} c}{2 d}}}\, \sqrt {\frac {x -\frac {\left (\frac {2^{\frac {1}{3}}}{4}-\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}}{\frac {\left (\frac {2^{\frac {1}{3}}}{4}+\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}-\frac {\left (\frac {2^{\frac {1}{3}}}{4}-\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}}}\, \EllipticPi \left (\sqrt {\frac {x -\frac {\left (\frac {2^{\frac {1}{3}}}{4}+\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}}{\frac {\left (\frac {2^{\frac {1}{3}}}{4}-\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}-\frac {\left (\frac {2^{\frac {1}{3}}}{4}+\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}}}, \frac {\frac {\left (\frac {2^{\frac {1}{3}}}{4}+\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}-\frac {\left (\frac {2^{\frac {1}{3}}}{4}-\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}}{\frac {\left (\frac {2^{\frac {1}{3}}}{4}+\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}+\frac {c}{d}}, \sqrt {\frac {\frac {\left (\frac {2^{\frac {1}{3}}}{4}+\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}-\frac {\left (\frac {2^{\frac {1}{3}}}{4}-\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}}{\frac {\left (\frac {2^{\frac {1}{3}}}{4}+\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}+\frac {2^{\frac {1}{3}} c}{2 d}}}\right )}{d \sqrt {4 d^{3} x^{3}+c^{3}}\, \left (\frac {\left (\frac {2^{\frac {1}{3}}}{4}+\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}+\frac {c}{d}\right )}\) | \(495\) |
elliptic | \(\frac {2 \left (\frac {\left (\frac {2^{\frac {1}{3}}}{4}-\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}-\frac {\left (\frac {2^{\frac {1}{3}}}{4}+\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}\right ) \sqrt {\frac {x -\frac {\left (\frac {2^{\frac {1}{3}}}{4}+\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}}{\frac {\left (\frac {2^{\frac {1}{3}}}{4}-\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}-\frac {\left (\frac {2^{\frac {1}{3}}}{4}+\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}}}\, \sqrt {\frac {x +\frac {2^{\frac {1}{3}} c}{2 d}}{\frac {\left (\frac {2^{\frac {1}{3}}}{4}+\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}+\frac {2^{\frac {1}{3}} c}{2 d}}}\, \sqrt {\frac {x -\frac {\left (\frac {2^{\frac {1}{3}}}{4}-\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}}{\frac {\left (\frac {2^{\frac {1}{3}}}{4}+\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}-\frac {\left (\frac {2^{\frac {1}{3}}}{4}-\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}}}\, \EllipticPi \left (\sqrt {\frac {x -\frac {\left (\frac {2^{\frac {1}{3}}}{4}+\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}}{\frac {\left (\frac {2^{\frac {1}{3}}}{4}-\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}-\frac {\left (\frac {2^{\frac {1}{3}}}{4}+\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}}}, \frac {\frac {\left (\frac {2^{\frac {1}{3}}}{4}+\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}-\frac {\left (\frac {2^{\frac {1}{3}}}{4}-\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}}{\frac {\left (\frac {2^{\frac {1}{3}}}{4}+\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}+\frac {c}{d}}, \sqrt {\frac {\frac {\left (\frac {2^{\frac {1}{3}}}{4}+\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}-\frac {\left (\frac {2^{\frac {1}{3}}}{4}-\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}}{\frac {\left (\frac {2^{\frac {1}{3}}}{4}+\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}+\frac {2^{\frac {1}{3}} c}{2 d}}}\right )}{d \sqrt {4 d^{3} x^{3}+c^{3}}\, \left (\frac {\left (\frac {2^{\frac {1}{3}}}{4}+\frac {i \sqrt {3}\, 2^{\frac {1}{3}}}{4}\right ) c}{d}+\frac {c}{d}\right )}\) | \(495\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.23, size = 349, normalized size = 1.40 \begin {gather*} \left [-\frac {\sqrt {3} \sqrt {-c} d^{2} \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 ) - 12 \, c \sqrt {d^{3}} {\rm weierstrassPInverse}\left (0, -\frac {c^{3}}{d^{3}}, x\right )}{18 \, c^{2} d^{3}}, -\frac {\sqrt {3} \sqrt {c} d^{2} \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 ) - 6 \, c \sqrt {d^{3}} {\rm weierstrassPInverse}\left (0, -\frac {c^{3}}{d^{3}}, x\right )}{9 \, c^{2} d^{3}}\right ] \end {gather*}
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 \begin {gather*} \int \frac {1}{\left (c + d x\right ) \sqrt {c^{3} + 4 d^{3} x^{3}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt {c^3+4\,d^3\,x^3}\,\left (c+d\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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