Optimal. Leaf size=316 \[ \frac {2 \left (16 c d^2-34 b d e+187 a e^2\right ) x \sqrt {d+e x^3}}{935 e^2}-\frac {2 (8 c d-17 b e) x \left (d+e x^3\right )^{3/2}}{187 e^2}+\frac {2 c x^4 \left (d+e x^3\right )^{3/2}}{17 e}+\frac {2\ 3^{3/4} \sqrt {2+\sqrt {3}} d \left (16 c d^2-34 b d e+187 a e^2\right ) \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \sqrt {\frac {d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x}\right )|-7-4 \sqrt {3}\right )}{935 e^{7/3} \sqrt {\frac {\sqrt [3]{d} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \sqrt {d+e x^3}} \]
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Rubi [A]
time = 0.16, antiderivative size = 316, 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 = {1425, 396, 201,
224} \begin {gather*} \frac {2\ 3^{3/4} \sqrt {2+\sqrt {3}} d \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \sqrt {\frac {d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \left (187 a e^2-34 b d e+16 c d^2\right ) F\left (\text {ArcSin}\left (\frac {\sqrt [3]{e} x+\left (1-\sqrt {3}\right ) \sqrt [3]{d}}{\sqrt [3]{e} x+\left (1+\sqrt {3}\right ) \sqrt [3]{d}}\right )|-7-4 \sqrt {3}\right )}{935 e^{7/3} \sqrt {\frac {\sqrt [3]{d} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \sqrt {d+e x^3}}+\frac {2 x \sqrt {d+e x^3} \left (187 a e^2-34 b d e+16 c d^2\right )}{935 e^2}-\frac {2 x \left (d+e x^3\right )^{3/2} (8 c d-17 b e)}{187 e^2}+\frac {2 c x^4 \left (d+e x^3\right )^{3/2}}{17 e} \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 224
Rule 396
Rule 1425
Rubi steps
\begin {align*} \int \sqrt {d+e x^3} \left (a+b x^3+c x^6\right ) \, dx &=\frac {2 c x^4 \left (d+e x^3\right )^{3/2}}{17 e}+\frac {2 \int \sqrt {d+e x^3} \left (\frac {17 a e}{2}-\left (4 c d-\frac {17 b e}{2}\right ) x^3\right ) \, dx}{17 e}\\ &=-\frac {2 (8 c d-17 b e) x \left (d+e x^3\right )^{3/2}}{187 e^2}+\frac {2 c x^4 \left (d+e x^3\right )^{3/2}}{17 e}-\frac {1}{187} \left (-187 a-\frac {2 d (8 c d-17 b e)}{e^2}\right ) \int \sqrt {d+e x^3} \, dx\\ &=\frac {2}{935} \left (187 a+\frac {2 d (8 c d-17 b e)}{e^2}\right ) x \sqrt {d+e x^3}-\frac {2 (8 c d-17 b e) x \left (d+e x^3\right )^{3/2}}{187 e^2}+\frac {2 c x^4 \left (d+e x^3\right )^{3/2}}{17 e}+\frac {1}{935} \left (3 d \left (187 a+\frac {2 d (8 c d-17 b e)}{e^2}\right )\right ) \int \frac {1}{\sqrt {d+e x^3}} \, dx\\ &=\frac {2}{935} \left (187 a+\frac {2 d (8 c d-17 b e)}{e^2}\right ) x \sqrt {d+e x^3}-\frac {2 (8 c d-17 b e) x \left (d+e x^3\right )^{3/2}}{187 e^2}+\frac {2 c x^4 \left (d+e x^3\right )^{3/2}}{17 e}+\frac {2\ 3^{3/4} \sqrt {2+\sqrt {3}} d \left (16 c d^2-34 b d e+187 a e^2\right ) \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \sqrt {\frac {d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x}\right )|-7-4 \sqrt {3}\right )}{935 e^{7/3} \sqrt {\frac {\sqrt [3]{d} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \sqrt {d+e x^3}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 5.22, size = 98, normalized size = 0.31 \begin {gather*} \frac {x \sqrt {d+e x^3} \left (-2 \left (d+e x^3\right ) \left (8 c d-17 b e-11 c e x^3\right )+\frac {\left (16 c d^2+17 e (-2 b d+11 a e)\right ) \, _2F_1\left (-\frac {1}{2},\frac {1}{3};\frac {4}{3};-\frac {e x^3}{d}\right )}{\sqrt {1+\frac {e x^3}{d}}}\right )}{187 e^2} \end {gather*}
Antiderivative was successfully verified.
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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. 955 vs. \(2 (249 ) = 498\).
time = 0.21, size = 956, normalized size = 3.03
method | result | size |
risch | \(\frac {2 x \left (55 c \,e^{2} x^{6}+85 b \,e^{2} x^{3}+15 c d e \,x^{3}+187 a \,e^{2}+51 d e b -24 c \,d^{2}\right ) \sqrt {e \,x^{3}+d}}{935 e^{2}}-\frac {2 i d \left (187 a \,e^{2}-34 d e b +16 c \,d^{2}\right ) \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}-\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right ) \sqrt {3}\, e}{\left (-d \,e^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{e}}{-\frac {3 \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}+\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}+\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right ) \sqrt {3}\, e}{\left (-d \,e^{2}\right )^{\frac {1}{3}}}}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}-\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right ) \sqrt {3}\, e}{\left (-d \,e^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{e \left (-\frac {3 \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}+\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right )}}\right )}{935 e^{3} \sqrt {e \,x^{3}+d}}\) | \(362\) |
elliptic | \(\frac {2 c \,x^{7} \sqrt {e \,x^{3}+d}}{17}+\frac {2 \left (e b +\frac {3 c d}{17}\right ) x^{4} \sqrt {e \,x^{3}+d}}{11 e}+\frac {2 \left (a e +b d -\frac {8 d \left (e b +\frac {3 c d}{17}\right )}{11 e}\right ) x \sqrt {e \,x^{3}+d}}{5 e}-\frac {2 i \left (a d -\frac {2 d \left (a e +b d -\frac {8 d \left (e b +\frac {3 c d}{17}\right )}{11 e}\right )}{5 e}\right ) \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}-\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right ) \sqrt {3}\, e}{\left (-d \,e^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{e}}{-\frac {3 \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}+\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}+\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right ) \sqrt {3}\, e}{\left (-d \,e^{2}\right )^{\frac {1}{3}}}}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}-\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right ) \sqrt {3}\, e}{\left (-d \,e^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{e \left (-\frac {3 \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}+\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right )}}\right )}{3 e \sqrt {e \,x^{3}+d}}\) | \(391\) |
default | \(\text {Expression too large to display}\) | \(956\) |
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 = 0.08, size = 98, normalized size = 0.31 \begin {gather*} \frac {2}{935} \, {\left (3 \, {\left (16 \, c d^{3} - 34 \, b d^{2} e + 187 \, a d e^{2}\right )} e^{\frac {1}{2}} {\rm weierstrassPInverse}\left (0, -4 \, d e^{\left (-1\right )}, x\right ) - {\left (24 \, c d^{2} x e - {\left (55 \, c x^{7} + 85 \, b x^{4} + 187 \, a x\right )} e^{3} - 3 \, {\left (5 \, c d x^{4} + 17 \, b d x\right )} e^{2}\right )} \sqrt {x^{3} e + d}\right )} e^{\left (-3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.64, size = 124, normalized size = 0.39 \begin {gather*} \frac {a \sqrt {d} x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {e x^{3} e^{i \pi }}{d}} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} + \frac {b \sqrt {d} x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {e x^{3} e^{i \pi }}{d}} \right )}}{3 \Gamma \left (\frac {7}{3}\right )} + \frac {c \sqrt {d} x^{7} \Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{3} \\ \frac {10}{3} \end {matrix}\middle | {\frac {e x^{3} e^{i \pi }}{d}} \right )}}{3 \Gamma \left (\frac {10}{3}\right )} \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 \sqrt {e\,x^3+d}\,\left (c\,x^6+b\,x^3+a\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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