Optimal. Leaf size=581 \[ -\frac {3^{3/4} b^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (5 A b-14 a B) F\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{56 \sqrt {2} a^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (5 A b-14 a B) E\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{224 a^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {3 b^{4/3} \sqrt {a+b x^3} (5 A b-14 a B)}{112 a^2 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {3 b \sqrt {a+b x^3} (5 A b-14 a B)}{112 a^2 x}+\frac {\sqrt {a+b x^3} (5 A b-14 a B)}{56 a x^4}-\frac {A \left (a+b x^3\right )^{3/2}}{7 a x^7} \]
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Rubi [A] time = 0.32, antiderivative size = 581, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {453, 277, 325, 303, 218, 1877} \[ -\frac {3 b^{4/3} \sqrt {a+b x^3} (5 A b-14 a B)}{112 a^2 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {3^{3/4} b^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (5 A b-14 a B) F\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{56 \sqrt {2} a^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (5 A b-14 a B) E\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{224 a^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {3 b \sqrt {a+b x^3} (5 A b-14 a B)}{112 a^2 x}+\frac {\sqrt {a+b x^3} (5 A b-14 a B)}{56 a x^4}-\frac {A \left (a+b x^3\right )^{3/2}}{7 a x^7} \]
Antiderivative was successfully verified.
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Rule 218
Rule 277
Rule 303
Rule 325
Rule 453
Rule 1877
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x^8} \, dx &=-\frac {A \left (a+b x^3\right )^{3/2}}{7 a x^7}-\frac {\left (\frac {5 A b}{2}-7 a B\right ) \int \frac {\sqrt {a+b x^3}}{x^5} \, dx}{7 a}\\ &=\frac {(5 A b-14 a B) \sqrt {a+b x^3}}{56 a x^4}-\frac {A \left (a+b x^3\right )^{3/2}}{7 a x^7}-\frac {(3 b (5 A b-14 a B)) \int \frac {1}{x^2 \sqrt {a+b x^3}} \, dx}{112 a}\\ &=\frac {(5 A b-14 a B) \sqrt {a+b x^3}}{56 a x^4}+\frac {3 b (5 A b-14 a B) \sqrt {a+b x^3}}{112 a^2 x}-\frac {A \left (a+b x^3\right )^{3/2}}{7 a x^7}-\frac {\left (3 b^2 (5 A b-14 a B)\right ) \int \frac {x}{\sqrt {a+b x^3}} \, dx}{224 a^2}\\ &=\frac {(5 A b-14 a B) \sqrt {a+b x^3}}{56 a x^4}+\frac {3 b (5 A b-14 a B) \sqrt {a+b x^3}}{112 a^2 x}-\frac {A \left (a+b x^3\right )^{3/2}}{7 a x^7}-\frac {\left (3 b^{5/3} (5 A b-14 a B)\right ) \int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt {a+b x^3}} \, dx}{224 a^2}-\frac {\left (3 \sqrt {\frac {1}{2} \left (2-\sqrt {3}\right )} b^{5/3} (5 A b-14 a B)\right ) \int \frac {1}{\sqrt {a+b x^3}} \, dx}{112 a^{5/3}}\\ &=\frac {(5 A b-14 a B) \sqrt {a+b x^3}}{56 a x^4}+\frac {3 b (5 A b-14 a B) \sqrt {a+b x^3}}{112 a^2 x}-\frac {3 b^{4/3} (5 A b-14 a B) \sqrt {a+b x^3}}{112 a^2 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {A \left (a+b x^3\right )^{3/2}}{7 a x^7}+\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b^{4/3} (5 A b-14 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 )}{224 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 {3^{3/4} b^{4/3} (5 A b-14 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 )}{56 \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] time = 0.10, size = 80, normalized size = 0.14 \[ \frac {\sqrt {a+b x^3} \left (\frac {x^3 \left (\frac {5 A b}{2}-7 a B\right ) \, _2F_1\left (-\frac {4}{3},-\frac {1}{2};-\frac {1}{3};-\frac {b x^3}{a}\right )}{\sqrt {\frac {b x^3}{a}+1}}-4 A \left (a+b x^3\right )\right )}{28 a x^7} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.88, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B x^{3} + A\right )} \sqrt {b x^{3} + a}}{x^{8}}, x\right ) \]
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 \[ \int \frac {{\left (B x^{3} + A\right )} \sqrt {b x^{3} + a}}{x^{8}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.09, size = 964, normalized size = 1.66 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (B x^{3} + A\right )} \sqrt {b x^{3} + a}}{x^{8}}\,{d x} \]
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 \[ \int \frac {\left (B\,x^3+A\right )\,\sqrt {b\,x^3+a}}{x^8} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.78, size = 97, normalized size = 0.17 \[ \frac {A \sqrt {a} \Gamma \left (- \frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{3}, - \frac {1}{2} \\ - \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{7} \Gamma \left (- \frac {4}{3}\right )} + \frac {B \sqrt {a} \Gamma \left (- \frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {4}{3}, - \frac {1}{2} \\ - \frac {1}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{4} \Gamma \left (- \frac {1}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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