Optimal. Leaf size=349 \[ \frac {2 \left (c d^2-b d e+a e^2\right ) x}{15 d e^2 \left (d+e x^3\right )^{5/2}}-\frac {2 \left (17 c d^2-2 b d e-13 a e^2\right ) x}{135 d^2 e^2 \left (d+e x^3\right )^{3/2}}+\frac {2 \left (16 c d^2+14 b d e+91 a e^2\right ) x}{405 d^3 e^2 \sqrt {d+e x^3}}+\frac {2 \sqrt {2+\sqrt {3}} \left (16 c d^2+14 b d e+91 a e^2\right ) \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \sqrt {\frac {d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x}\right )|-7-4 \sqrt {3}\right )}{405 \sqrt [4]{3} d^3 e^{7/3} \sqrt {\frac {\sqrt [3]{d} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \sqrt {d+e x^3}} \]
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Rubi [A]
time = 0.22, antiderivative size = 349, 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 = {1423, 393, 205,
224} \begin {gather*} \frac {2 \sqrt {2+\sqrt {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 (91 a e^2+14 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 )}{405 \sqrt [4]{3} d^3 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 (-13 a e^2-2 b d e+17 c d^2\right )}{135 d^2 e^2 \left (d+e x^3\right )^{3/2}}+\frac {2 x \left (a e^2-b d e+c d^2\right )}{15 d e^2 \left (d+e x^3\right )^{5/2}}+\frac {2 x \left (91 a e^2+14 b d e+16 c d^2\right )}{405 d^3 e^2 \sqrt {d+e x^3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 224
Rule 393
Rule 1423
Rubi steps
\begin {align*} \int \frac {a+b x^3+c x^6}{\left (d+e x^3\right )^{7/2}} \, dx &=\frac {2 \left (c d^2-b d e+a e^2\right ) x}{15 d e^2 \left (d+e x^3\right )^{5/2}}-\frac {2 \int \frac {\frac {1}{2} \left (2 c d^2-e (2 b d+13 a e)\right )-\frac {15}{2} c d e x^3}{\left (d+e x^3\right )^{5/2}} \, dx}{15 d e^2}\\ &=\frac {2 \left (c d^2-b d e+a e^2\right ) x}{15 d e^2 \left (d+e x^3\right )^{5/2}}-\frac {2 \left (17 c d^2-2 b d e-13 a e^2\right ) x}{135 d^2 e^2 \left (d+e x^3\right )^{3/2}}+\frac {\left (16 c d^2+14 b d e+91 a e^2\right ) \int \frac {1}{\left (d+e x^3\right )^{3/2}} \, dx}{135 d^2 e^2}\\ &=\frac {2 \left (c d^2-b d e+a e^2\right ) x}{15 d e^2 \left (d+e x^3\right )^{5/2}}-\frac {2 \left (17 c d^2-2 b d e-13 a e^2\right ) x}{135 d^2 e^2 \left (d+e x^3\right )^{3/2}}+\frac {2 \left (16 c d^2+14 b d e+91 a e^2\right ) x}{405 d^3 e^2 \sqrt {d+e x^3}}+\frac {\left (16 c d^2+14 b d e+91 a e^2\right ) \int \frac {1}{\sqrt {d+e x^3}} \, dx}{405 d^3 e^2}\\ &=\frac {2 \left (c d^2-b d e+a e^2\right ) x}{15 d e^2 \left (d+e x^3\right )^{5/2}}-\frac {2 \left (17 c d^2-2 b d e-13 a e^2\right ) x}{135 d^2 e^2 \left (d+e x^3\right )^{3/2}}+\frac {2 \left (16 c d^2+14 b d e+91 a e^2\right ) x}{405 d^3 e^2 \sqrt {d+e x^3}}+\frac {2 \sqrt {2+\sqrt {3}} \left (16 c d^2+14 b d e+91 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 )}{405 \sqrt [4]{3} d^3 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 = 10.14, size = 166, normalized size = 0.48 \begin {gather*} \frac {2 x \left (c d^2 \left (-8 d^2-19 d e x^3+16 e^2 x^6\right )+e \left (b d \left (-7 d^2+34 d e x^3+14 e^2 x^6\right )+a e \left (157 d^2+221 d e x^3+91 e^2 x^6\right )\right )\right )+\left (16 c d^2+7 e (2 b d+13 a e)\right ) x \left (d+e x^3\right )^2 \sqrt {1+\frac {e x^3}{d}} \, _2F_1\left (\frac {1}{3},\frac {1}{2};\frac {4}{3};-\frac {e x^3}{d}\right )}{405 d^3 e^2 \left (d+e x^3\right )^{5/2}} \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. 1094 vs. \(2 (282 ) = 564\).
time = 0.22, size = 1095, normalized size = 3.14
method | result | size |
elliptic | \(\frac {2 x \left (a \,e^{2}-d e b +c \,d^{2}\right ) \sqrt {e \,x^{3}+d}}{15 d \,e^{5} \left (x^{3}+\frac {d}{e}\right )^{3}}+\frac {2 x \left (13 a \,e^{2}+2 d e b -17 c \,d^{2}\right ) \sqrt {e \,x^{3}+d}}{135 d^{2} e^{4} \left (x^{3}+\frac {d}{e}\right )^{2}}+\frac {2 x \left (91 a \,e^{2}+14 d e b +16 c \,d^{2}\right )}{405 e^{2} d^{3} \sqrt {\left (x^{3}+\frac {d}{e}\right ) e}}-\frac {2 i \left (91 a \,e^{2}+14 d e b +16 c \,d^{2}\right ) \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}-\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right ) \sqrt {3}\, e}{\left (-d \,e^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{e}}{-\frac {3 \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}+\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}+\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right ) \sqrt {3}\, e}{\left (-d \,e^{2}\right )^{\frac {1}{3}}}}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}-\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right ) \sqrt {3}\, e}{\left (-d \,e^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{e \left (-\frac {3 \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}+\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right )}}\right )}{1215 d^{3} e^{3} \sqrt {e \,x^{3}+d}}\) | \(437\) |
default | \(\text {Expression too large to display}\) | \(1095\) |
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 = 262, normalized size = 0.75 \begin {gather*} \frac {2 \, {\left ({\left (91 \, a x^{9} e^{5} + 16 \, c d^{5} + 7 \, {\left (2 \, b d x^{9} + 39 \, a d x^{6}\right )} e^{4} + {\left (16 \, c d^{2} x^{9} + 42 \, b d^{2} x^{6} + 273 \, a d^{2} x^{3}\right )} e^{3} + {\left (48 \, c d^{3} x^{6} + 42 \, b d^{3} x^{3} + 91 \, a d^{3}\right )} e^{2} + 2 \, {\left (24 \, c d^{4} x^{3} + 7 \, b d^{4}\right )} e\right )} e^{\frac {1}{2}} {\rm weierstrassPInverse}\left (0, -4 \, d e^{\left (-1\right )}, x\right ) + {\left (91 \, a x^{7} e^{5} - 8 \, c d^{4} x e + {\left (14 \, b d x^{7} + 221 \, a d x^{4}\right )} e^{4} + {\left (16 \, c d^{2} x^{7} + 34 \, b d^{2} x^{4} + 157 \, a d^{2} x\right )} e^{3} - {\left (19 \, c d^{3} x^{4} + 7 \, b d^{3} x\right )} e^{2}\right )} \sqrt {x^{3} e + d}\right )}}{405 \, {\left (d^{3} x^{9} e^{6} + 3 \, d^{4} x^{6} e^{5} + 3 \, d^{5} x^{3} e^{4} + d^{6} e^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 152.64, size = 119, normalized size = 0.34 \begin {gather*} \frac {a x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {7}{2} \\ \frac {4}{3} \end {matrix}\middle | {\frac {e x^{3} e^{i \pi }}{d}} \right )}}{3 d^{\frac {7}{2}} \Gamma \left (\frac {4}{3}\right )} + \frac {b x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {4}{3}, \frac {7}{2} \\ \frac {7}{3} \end {matrix}\middle | {\frac {e x^{3} e^{i \pi }}{d}} \right )}}{3 d^{\frac {7}{2}} \Gamma \left (\frac {7}{3}\right )} + \frac {c x^{7} \Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {7}{3}, \frac {7}{2} \\ \frac {10}{3} \end {matrix}\middle | {\frac {e x^{3} e^{i \pi }}{d}} \right )}}{3 d^{\frac {7}{2}} \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 \frac {c\,x^6+b\,x^3+a}{{\left (e\,x^3+d\right )}^{7/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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