Optimal. Leaf size=396 \[ \frac {54 d^2 \left (16 c d^2-58 b d e+667 a e^2\right ) x \sqrt {d+e x^3}}{124729 e^2}+\frac {30 d \left (16 c d^2-58 b d e+667 a e^2\right ) x \left (d+e x^3\right )^{3/2}}{124729 e^2}+\frac {2 \left (16 c d^2-58 b d e+667 a e^2\right ) x \left (d+e x^3\right )^{5/2}}{11339 e^2}-\frac {2 (8 c d-29 b e) x \left (d+e x^3\right )^{7/2}}{667 e^2}+\frac {2 c x^4 \left (d+e x^3\right )^{7/2}}{29 e}+\frac {54\ 3^{3/4} \sqrt {2+\sqrt {3}} d^3 \left (16 c d^2-58 b d e+667 a e^2\right ) \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \sqrt {\frac {d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x}\right )|-7-4 \sqrt {3}\right )}{124729 e^{7/3} \sqrt {\frac {\sqrt [3]{d} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \sqrt {d+e x^3}} \]
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Rubi [A]
time = 0.27, antiderivative size = 396, normalized size of antiderivative = 1.00, number of steps
used = 6, 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 {54\ 3^{3/4} \sqrt {2+\sqrt {3}} d^3 \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 (667 a e^2-58 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 )}{124729 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 \left (d+e x^3\right )^{5/2} \left (667 a e^2-58 b d e+16 c d^2\right )}{11339 e^2}+\frac {30 d x \left (d+e x^3\right )^{3/2} \left (667 a e^2-58 b d e+16 c d^2\right )}{124729 e^2}+\frac {54 d^2 x \sqrt {d+e x^3} \left (667 a e^2-58 b d e+16 c d^2\right )}{124729 e^2}-\frac {2 x \left (d+e x^3\right )^{7/2} (8 c d-29 b e)}{667 e^2}+\frac {2 c x^4 \left (d+e x^3\right )^{7/2}}{29 e} \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 224
Rule 396
Rule 1425
Rubi steps
\begin {align*} \int \left (d+e x^3\right )^{5/2} \left (a+b x^3+c x^6\right ) \, dx &=\frac {2 c x^4 \left (d+e x^3\right )^{7/2}}{29 e}+\frac {2 \int \left (d+e x^3\right )^{5/2} \left (\frac {29 a e}{2}-\left (4 c d-\frac {29 b e}{2}\right ) x^3\right ) \, dx}{29 e}\\ &=-\frac {2 (8 c d-29 b e) x \left (d+e x^3\right )^{7/2}}{667 e^2}+\frac {2 c x^4 \left (d+e x^3\right )^{7/2}}{29 e}-\frac {1}{667} \left (-667 a-\frac {2 d (8 c d-29 b e)}{e^2}\right ) \int \left (d+e x^3\right )^{5/2} \, dx\\ &=\frac {2 \left (667 a+\frac {2 d (8 c d-29 b e)}{e^2}\right ) x \left (d+e x^3\right )^{5/2}}{11339}-\frac {2 (8 c d-29 b e) x \left (d+e x^3\right )^{7/2}}{667 e^2}+\frac {2 c x^4 \left (d+e x^3\right )^{7/2}}{29 e}+\frac {\left (15 d \left (667 a+\frac {2 d (8 c d-29 b e)}{e^2}\right )\right ) \int \left (d+e x^3\right )^{3/2} \, dx}{11339}\\ &=\frac {30 d \left (667 a+\frac {2 d (8 c d-29 b e)}{e^2}\right ) x \left (d+e x^3\right )^{3/2}}{124729}+\frac {2 \left (667 a+\frac {2 d (8 c d-29 b e)}{e^2}\right ) x \left (d+e x^3\right )^{5/2}}{11339}-\frac {2 (8 c d-29 b e) x \left (d+e x^3\right )^{7/2}}{667 e^2}+\frac {2 c x^4 \left (d+e x^3\right )^{7/2}}{29 e}+\frac {\left (135 d^2 \left (667 a+\frac {2 d (8 c d-29 b e)}{e^2}\right )\right ) \int \sqrt {d+e x^3} \, dx}{124729}\\ &=\frac {54 d^2 \left (667 a+\frac {2 d (8 c d-29 b e)}{e^2}\right ) x \sqrt {d+e x^3}}{124729}+\frac {30 d \left (667 a+\frac {2 d (8 c d-29 b e)}{e^2}\right ) x \left (d+e x^3\right )^{3/2}}{124729}+\frac {2 \left (667 a+\frac {2 d (8 c d-29 b e)}{e^2}\right ) x \left (d+e x^3\right )^{5/2}}{11339}-\frac {2 (8 c d-29 b e) x \left (d+e x^3\right )^{7/2}}{667 e^2}+\frac {2 c x^4 \left (d+e x^3\right )^{7/2}}{29 e}+\frac {\left (81 d^3 \left (667 a+\frac {2 d (8 c d-29 b e)}{e^2}\right )\right ) \int \frac {1}{\sqrt {d+e x^3}} \, dx}{124729}\\ &=\frac {54 d^2 \left (667 a+\frac {2 d (8 c d-29 b e)}{e^2}\right ) x \sqrt {d+e x^3}}{124729}+\frac {30 d \left (667 a+\frac {2 d (8 c d-29 b e)}{e^2}\right ) x \left (d+e x^3\right )^{3/2}}{124729}+\frac {2 \left (667 a+\frac {2 d (8 c d-29 b e)}{e^2}\right ) x \left (d+e x^3\right )^{5/2}}{11339}-\frac {2 (8 c d-29 b e) x \left (d+e x^3\right )^{7/2}}{667 e^2}+\frac {2 c x^4 \left (d+e x^3\right )^{7/2}}{29 e}+\frac {54\ 3^{3/4} \sqrt {2+\sqrt {3}} d^3 \left (16 c d^2-58 b d e+667 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 )}{124729 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 = 8.11, size = 103, normalized size = 0.26 \begin {gather*} \frac {x \sqrt {d+e x^3} \left (-2 \left (d+e x^3\right )^3 \left (8 c d-29 b e-23 c e x^3\right )+\frac {\left (16 c d^4+29 d^2 e (-2 b d+23 a e)\right ) \, _2F_1\left (-\frac {5}{2},\frac {1}{3};\frac {4}{3};-\frac {e x^3}{d}\right )}{\sqrt {1+\frac {e x^3}{d}}}\right )}{667 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. 1069 vs. \(2 (321 ) = 642\).
time = 0.61, size = 1070, normalized size = 2.70
method | result | size |
risch | \(\frac {2 x \left (4301 e^{4} c \,x^{12}+5423 e^{4} b \,x^{9}+11407 d \,e^{3} c \,x^{9}+7337 a \,e^{4} x^{6}+15631 b d \,e^{3} x^{6}+8591 c \,d^{2} e^{2} x^{6}+24679 d \,e^{3} a \,x^{3}+14123 d^{2} e^{2} b \,x^{3}+405 d^{3} e c \,x^{3}+35351 d^{2} e^{2} a +2349 d^{3} e b -648 d^{4} c \right ) \sqrt {e \,x^{3}+d}}{124729 e^{2}}-\frac {54 i d^{3} \left (667 a \,e^{2}-58 d e b +16 c \,d^{2}\right ) \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}-\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right ) \sqrt {3}\, e}{\left (-d \,e^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{e}}{-\frac {3 \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}+\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}+\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right ) \sqrt {3}\, e}{\left (-d \,e^{2}\right )^{\frac {1}{3}}}}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}-\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right ) \sqrt {3}\, e}{\left (-d \,e^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{e \left (-\frac {3 \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}+\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right )}}\right )}{124729 e^{3} \sqrt {e \,x^{3}+d}}\) | \(434\) |
elliptic | \(\frac {2 c \,e^{2} x^{13} \sqrt {e \,x^{3}+d}}{29}+\frac {2 \left (e^{3} b +\frac {61}{29} d \,e^{2} c \right ) x^{10} \sqrt {e \,x^{3}+d}}{23 e}+\frac {2 \left (a \,e^{3}+3 d \,e^{2} b +3 c \,d^{2} e -\frac {20 d \left (e^{3} b +\frac {61}{29} d \,e^{2} c \right )}{23 e}\right ) x^{7} \sqrt {e \,x^{3}+d}}{17 e}+\frac {2 \left (3 d \,e^{2} a +3 d^{2} e b +c \,d^{3}-\frac {14 d \left (a \,e^{3}+3 d \,e^{2} b +3 c \,d^{2} e -\frac {20 d \left (e^{3} b +\frac {61}{29} d \,e^{2} c \right )}{23 e}\right )}{17 e}\right ) x^{4} \sqrt {e \,x^{3}+d}}{11 e}+\frac {2 \left (3 a \,d^{2} e +d^{3} b -\frac {8 d \left (3 d \,e^{2} a +3 d^{2} e b +c \,d^{3}-\frac {14 d \left (a \,e^{3}+3 d \,e^{2} b +3 c \,d^{2} e -\frac {20 d \left (e^{3} b +\frac {61}{29} d \,e^{2} c \right )}{23 e}\right )}{17 e}\right )}{11 e}\right ) x \sqrt {e \,x^{3}+d}}{5 e}-\frac {2 i \left (d^{3} a -\frac {2 d \left (3 a \,d^{2} e +d^{3} b -\frac {8 d \left (3 d \,e^{2} a +3 d^{2} e b +c \,d^{3}-\frac {14 d \left (a \,e^{3}+3 d \,e^{2} b +3 c \,d^{2} e -\frac {20 d \left (e^{3} b +\frac {61}{29} d \,e^{2} c \right )}{23 e}\right )}{17 e}\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}}\) | \(665\) |
default | \(\text {Expression too large to display}\) | \(1070\) |
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 = 0.10, size = 162, normalized size = 0.41 \begin {gather*} \frac {2}{124729} \, {\left (81 \, {\left (16 \, c d^{5} - 58 \, b d^{4} e + 667 \, a d^{3} e^{2}\right )} e^{\frac {1}{2}} {\rm weierstrassPInverse}\left (0, -4 \, d e^{\left (-1\right )}, x\right ) - {\left (648 \, c d^{4} x e - 11 \, {\left (391 \, c x^{13} + 493 \, b x^{10} + 667 \, a x^{7}\right )} e^{5} - {\left (11407 \, c d x^{10} + 15631 \, b d x^{7} + 24679 \, a d x^{4}\right )} e^{4} - {\left (8591 \, c d^{2} x^{7} + 14123 \, b d^{2} x^{4} + 35351 \, a d^{2} x\right )} e^{3} - 81 \, {\left (5 \, c d^{3} x^{4} + 29 \, b d^{3} 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 = 4.30, size = 400, normalized size = 1.01 \begin {gather*} \frac {a d^{\frac {5}{2}} 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 {2 a d^{\frac {3}{2}} e 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 {a \sqrt {d} e^{2} 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 )} + \frac {b d^{\frac {5}{2}} 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 {2 b d^{\frac {3}{2}} e 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 )} + \frac {b \sqrt {d} e^{2} x^{10} \Gamma \left (\frac {10}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {10}{3} \\ \frac {13}{3} \end {matrix}\middle | {\frac {e x^{3} e^{i \pi }}{d}} \right )}}{3 \Gamma \left (\frac {13}{3}\right )} + \frac {c d^{\frac {5}{2}} 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 )} + \frac {2 c d^{\frac {3}{2}} e x^{10} \Gamma \left (\frac {10}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {10}{3} \\ \frac {13}{3} \end {matrix}\middle | {\frac {e x^{3} e^{i \pi }}{d}} \right )}}{3 \Gamma \left (\frac {13}{3}\right )} + \frac {c \sqrt {d} e^{2} x^{13} \Gamma \left (\frac {13}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {13}{3} \\ \frac {16}{3} \end {matrix}\middle | {\frac {e x^{3} e^{i \pi }}{d}} \right )}}{3 \Gamma \left (\frac {16}{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 {\left (e\,x^3+d\right )}^{5/2}\,\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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