Optimal. Leaf size=608 \[ \frac {9 b (A b-4 a B) \sqrt {a+b x^3}}{224 a x^4}+\frac {27 b^2 (A b-4 a B) \sqrt {a+b x^3}}{448 a^2 x}-\frac {27 b^{7/3} (A b-4 a B) \sqrt {a+b x^3}}{448 a^2 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {(A b-4 a B) \left (a+b x^3\right )^{3/2}}{28 a x^7}-\frac {A \left (a+b x^3\right )^{5/2}}{10 a x^{10}}+\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b^{7/3} (A b-4 a B) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{896 a^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {9\ 3^{3/4} b^{7/3} (A b-4 a B) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{224 \sqrt {2} a^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \]
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Rubi [A]
time = 0.26, antiderivative size = 608, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {464, 283, 331,
309, 224, 1891} \begin {gather*} -\frac {9\ 3^{3/4} b^{7/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (A b-4 a B) F\left (\text {ArcSin}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{224 \sqrt {2} a^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b^{7/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (A b-4 a B) E\left (\text {ArcSin}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{896 a^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {27 b^{7/3} \sqrt {a+b x^3} (A b-4 a B)}{448 a^2 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {27 b^2 \sqrt {a+b x^3} (A b-4 a B)}{448 a^2 x}+\frac {\left (a+b x^3\right )^{3/2} (A b-4 a B)}{28 a x^7}+\frac {9 b \sqrt {a+b x^3} (A b-4 a B)}{224 a x^4}-\frac {A \left (a+b x^3\right )^{5/2}}{10 a x^{10}} \end {gather*}
Antiderivative was successfully verified.
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Rule 224
Rule 283
Rule 309
Rule 331
Rule 464
Rule 1891
Rubi steps
\begin {align*} \int \frac {\left (a+b x^3\right )^{3/2} \left (A+B x^3\right )}{x^{11}} \, dx &=-\frac {A \left (a+b x^3\right )^{5/2}}{10 a x^{10}}-\frac {\left (\frac {5 A b}{2}-10 a B\right ) \int \frac {\left (a+b x^3\right )^{3/2}}{x^8} \, dx}{10 a}\\ &=\frac {(A b-4 a B) \left (a+b x^3\right )^{3/2}}{28 a x^7}-\frac {A \left (a+b x^3\right )^{5/2}}{10 a x^{10}}-\frac {(9 b (A b-4 a B)) \int \frac {\sqrt {a+b x^3}}{x^5} \, dx}{56 a}\\ &=\frac {9 b (A b-4 a B) \sqrt {a+b x^3}}{224 a x^4}+\frac {(A b-4 a B) \left (a+b x^3\right )^{3/2}}{28 a x^7}-\frac {A \left (a+b x^3\right )^{5/2}}{10 a x^{10}}-\frac {\left (27 b^2 (A b-4 a B)\right ) \int \frac {1}{x^2 \sqrt {a+b x^3}} \, dx}{448 a}\\ &=\frac {9 b (A b-4 a B) \sqrt {a+b x^3}}{224 a x^4}+\frac {27 b^2 (A b-4 a B) \sqrt {a+b x^3}}{448 a^2 x}+\frac {(A b-4 a B) \left (a+b x^3\right )^{3/2}}{28 a x^7}-\frac {A \left (a+b x^3\right )^{5/2}}{10 a x^{10}}-\frac {\left (27 b^3 (A b-4 a B)\right ) \int \frac {x}{\sqrt {a+b x^3}} \, dx}{896 a^2}\\ &=\frac {9 b (A b-4 a B) \sqrt {a+b x^3}}{224 a x^4}+\frac {27 b^2 (A b-4 a B) \sqrt {a+b x^3}}{448 a^2 x}+\frac {(A b-4 a B) \left (a+b x^3\right )^{3/2}}{28 a x^7}-\frac {A \left (a+b x^3\right )^{5/2}}{10 a x^{10}}-\frac {\left (27 b^{8/3} (A b-4 a B)\right ) \int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt {a+b x^3}} \, dx}{896 a^2}-\frac {\left (27 \sqrt {\frac {1}{2} \left (2-\sqrt {3}\right )} b^{8/3} (A b-4 a B)\right ) \int \frac {1}{\sqrt {a+b x^3}} \, dx}{448 a^{5/3}}\\ &=\frac {9 b (A b-4 a B) \sqrt {a+b x^3}}{224 a x^4}+\frac {27 b^2 (A b-4 a B) \sqrt {a+b x^3}}{448 a^2 x}-\frac {27 b^{7/3} (A b-4 a B) \sqrt {a+b x^3}}{448 a^2 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {(A b-4 a B) \left (a+b x^3\right )^{3/2}}{28 a x^7}-\frac {A \left (a+b x^3\right )^{5/2}}{10 a x^{10}}+\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b^{7/3} (A b-4 a B) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{896 a^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {9\ 3^{3/4} b^{7/3} (A b-4 a B) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{224 \sqrt {2} a^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b 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 = 10.06, size = 82, normalized size = 0.13 \begin {gather*} \frac {\sqrt {a+b x^3} \left (-\frac {7 A \left (a+b x^3\right )^2}{a}+\frac {5 (A b-4 a B) x^3 \, _2F_1\left (-\frac {7}{3},-\frac {3}{2};-\frac {4}{3};-\frac {b x^3}{a}\right )}{2 \sqrt {1+\frac {b x^3}{a}}}\right )}{70 x^{10}} \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. 1001 vs. \(2 (462 ) = 924\).
time = 0.34, size = 1002, normalized size = 1.65
method | result | size |
risch | \(-\frac {\sqrt {b \,x^{3}+a}\, \left (-135 x^{9} A \,b^{3}+540 x^{9} B a \,b^{2}+54 x^{6} A a \,b^{2}+680 x^{6} B \,a^{2} b +368 x^{3} A \,a^{2} b +320 a^{3} B \,x^{3}+224 A \,a^{3}\right )}{2240 x^{10} a^{2}}+\frac {9 i b^{2} \left (A b -4 B a \right ) \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}}{-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \left (\left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \EllipticE \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )+\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}} \EllipticF \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{b}\right )}{448 a^{2} \sqrt {b \,x^{3}+a}}\) | \(529\) |
elliptic | \(-\frac {A a \sqrt {b \,x^{3}+a}}{10 x^{10}}-\frac {\left (\frac {23 A b}{20}+B a \right ) \sqrt {b \,x^{3}+a}}{7 x^{7}}-\frac {b \left (27 A b +340 B a \right ) \sqrt {b \,x^{3}+a}}{1120 a \,x^{4}}+\frac {27 b^{2} \left (A b -4 B a \right ) \sqrt {b \,x^{3}+a}}{448 a^{2} x}+\frac {9 i b^{2} \left (A b -4 B a \right ) \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}}{-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \left (\left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \EllipticE \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )+\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}} \EllipticF \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{b}\right )}{448 a^{2} \sqrt {b \,x^{3}+a}}\) | \(540\) |
default | \(\text {Expression too large to display}\) | \(1002\) |
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 = 0.52, size = 123, normalized size = 0.20 \begin {gather*} -\frac {135 \, {\left (4 \, B a b^{2} - A b^{3}\right )} \sqrt {b} x^{10} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) + {\left (135 \, {\left (4 \, B a b^{2} - A b^{3}\right )} x^{9} + 2 \, {\left (340 \, B a^{2} b + 27 \, A a b^{2}\right )} x^{6} + 224 \, A a^{3} + 16 \, {\left (20 \, B a^{3} + 23 \, A a^{2} b\right )} x^{3}\right )} \sqrt {b x^{3} + a}}{2240 \, a^{2} x^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 3.06, size = 199, normalized size = 0.33 \begin {gather*} \frac {A a^{\frac {3}{2}} \Gamma \left (- \frac {10}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {10}{3}, - \frac {1}{2} \\ - \frac {7}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{10} \Gamma \left (- \frac {7}{3}\right )} + \frac {A \sqrt {a} b \Gamma \left (- \frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{3}, - \frac {1}{2} \\ - \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{7} \Gamma \left (- \frac {4}{3}\right )} + \frac {B a^{\frac {3}{2}} \Gamma \left (- \frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{3}, - \frac {1}{2} \\ - \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{7} \Gamma \left (- \frac {4}{3}\right )} + \frac {B \sqrt {a} b \Gamma \left (- \frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {4}{3}, - \frac {1}{2} \\ - \frac {1}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{4} \Gamma \left (- \frac {1}{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 {\left (B\,x^3+A\right )\,{\left (b\,x^3+a\right )}^{3/2}}{x^{11}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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