Optimal. Leaf size=305 \[ \frac {(7 A b-16 a B) \sqrt {a+b x^3}}{80 a x^5}+\frac {3 b (7 A b-16 a B) \sqrt {a+b x^3}}{320 a^2 x^2}-\frac {A \left (a+b x^3\right )^{3/2}}{8 a x^8}+\frac {3^{3/4} \sqrt {2+\sqrt {3}} b^{5/3} (7 A b-16 a B) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{320 a^2 \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \]
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Rubi [A]
time = 0.10, antiderivative size = 305, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {464, 283, 331,
224} \begin {gather*} \frac {3 b \sqrt {a+b x^3} (7 A b-16 a B)}{320 a^2 x^2}+\frac {3^{3/4} \sqrt {2+\sqrt {3}} b^{5/3} \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}} (7 A b-16 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 )}{320 a^2 \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 {\sqrt {a+b x^3} (7 A b-16 a B)}{80 a x^5}-\frac {A \left (a+b x^3\right )^{3/2}}{8 a x^8} \end {gather*}
Antiderivative was successfully verified.
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Rule 224
Rule 283
Rule 331
Rule 464
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x^9} \, dx &=-\frac {A \left (a+b x^3\right )^{3/2}}{8 a x^8}-\frac {\left (\frac {7 A b}{2}-8 a B\right ) \int \frac {\sqrt {a+b x^3}}{x^6} \, dx}{8 a}\\ &=\frac {(7 A b-16 a B) \sqrt {a+b x^3}}{80 a x^5}-\frac {A \left (a+b x^3\right )^{3/2}}{8 a x^8}-\frac {(3 b (7 A b-16 a B)) \int \frac {1}{x^3 \sqrt {a+b x^3}} \, dx}{160 a}\\ &=\frac {(7 A b-16 a B) \sqrt {a+b x^3}}{80 a x^5}+\frac {3 b (7 A b-16 a B) \sqrt {a+b x^3}}{320 a^2 x^2}-\frac {A \left (a+b x^3\right )^{3/2}}{8 a x^8}+\frac {\left (3 b^2 (7 A b-16 a B)\right ) \int \frac {1}{\sqrt {a+b x^3}} \, dx}{640 a^2}\\ &=\frac {(7 A b-16 a B) \sqrt {a+b x^3}}{80 a x^5}+\frac {3 b (7 A b-16 a B) \sqrt {a+b x^3}}{320 a^2 x^2}-\frac {A \left (a+b x^3\right )^{3/2}}{8 a x^8}+\frac {3^{3/4} \sqrt {2+\sqrt {3}} b^{5/3} (7 A b-16 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 )}{320 a^2 \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 = 80, normalized size = 0.26 \begin {gather*} \frac {\sqrt {a+b x^3} \left (-5 A \left (a+b x^3\right )+\frac {\left (\frac {7 A b}{2}-8 a B\right ) x^3 \, _2F_1\left (-\frac {5}{3},-\frac {1}{2};-\frac {2}{3};-\frac {b x^3}{a}\right )}{\sqrt {1+\frac {b x^3}{a}}}\right )}{40 a x^8} \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. 659 vs. \(2 (238 ) = 476\).
time = 0.34, size = 660, normalized size = 2.16 Too large to display
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 = 0.32, size = 89, normalized size = 0.29 \begin {gather*} -\frac {3 \, {\left (16 \, B a b - 7 \, A b^{2}\right )} \sqrt {b} x^{8} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) + {\left (3 \, {\left (16 \, B a b - 7 \, A b^{2}\right )} x^{6} + 4 \, {\left (16 \, B a^{2} + 3 \, A a b\right )} x^{3} + 40 \, A a^{2}\right )} \sqrt {b x^{3} + a}}{320 \, a^{2} x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.67, size = 97, normalized size = 0.32 \begin {gather*} \frac {A \sqrt {a} \Gamma \left (- \frac {8}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {8}{3}, - \frac {1}{2} \\ - \frac {5}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{8} \Gamma \left (- \frac {5}{3}\right )} + \frac {B \sqrt {a} \Gamma \left (- \frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{3}, - \frac {1}{2} \\ - \frac {2}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{5} \Gamma \left (- \frac {2}{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 )\,\sqrt {b\,x^3+a}}{x^9} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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