Optimal. Leaf size=611 \[ \frac{55 b^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (17 A b-14 a B) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt{3}\right )}{168 \sqrt{2} \sqrt [4]{3} a^{11/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{55 b^{4/3} \sqrt{a+b x^3} (17 A b-14 a B)}{336 a^4 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac{55 \sqrt{2-\sqrt{3}} b^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (17 A b-14 a B) E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{224\ 3^{3/4} a^{11/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{55 b \sqrt{a+b x^3} (17 A b-14 a B)}{336 a^4 x}+\frac{11 \sqrt{a+b x^3} (17 A b-14 a B)}{168 a^3 x^4}-\frac{17 A b-14 a B}{21 a^2 x^4 \sqrt{a+b x^3}}-\frac{A}{7 a x^7 \sqrt{a+b x^3}} \]
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Rubi [A] time = 0.361824, antiderivative size = 611, normalized size of antiderivative = 1., 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 = {453, 290, 325, 303, 218, 1877} \[ \frac{55 b^{4/3} \sqrt{a+b x^3} (17 A b-14 a B)}{336 a^4 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac{55 b^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (17 A b-14 a B) F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{168 \sqrt{2} \sqrt [4]{3} a^{11/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{55 \sqrt{2-\sqrt{3}} b^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (17 A b-14 a B) E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{224\ 3^{3/4} a^{11/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{55 b \sqrt{a+b x^3} (17 A b-14 a B)}{336 a^4 x}+\frac{11 \sqrt{a+b x^3} (17 A b-14 a B)}{168 a^3 x^4}-\frac{17 A b-14 a B}{21 a^2 x^4 \sqrt{a+b x^3}}-\frac{A}{7 a x^7 \sqrt{a+b x^3}} \]
Antiderivative was successfully verified.
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Rule 453
Rule 290
Rule 325
Rule 303
Rule 218
Rule 1877
Rubi steps
\begin{align*} \int \frac{A+B x^3}{x^8 \left (a+b x^3\right )^{3/2}} \, dx &=-\frac{A}{7 a x^7 \sqrt{a+b x^3}}-\frac{\left (\frac{17 A b}{2}-7 a B\right ) \int \frac{1}{x^5 \left (a+b x^3\right )^{3/2}} \, dx}{7 a}\\ &=-\frac{A}{7 a x^7 \sqrt{a+b x^3}}-\frac{17 A b-14 a B}{21 a^2 x^4 \sqrt{a+b x^3}}-\frac{(11 (17 A b-14 a B)) \int \frac{1}{x^5 \sqrt{a+b x^3}} \, dx}{42 a^2}\\ &=-\frac{A}{7 a x^7 \sqrt{a+b x^3}}-\frac{17 A b-14 a B}{21 a^2 x^4 \sqrt{a+b x^3}}+\frac{11 (17 A b-14 a B) \sqrt{a+b x^3}}{168 a^3 x^4}+\frac{(55 b (17 A b-14 a B)) \int \frac{1}{x^2 \sqrt{a+b x^3}} \, dx}{336 a^3}\\ &=-\frac{A}{7 a x^7 \sqrt{a+b x^3}}-\frac{17 A b-14 a B}{21 a^2 x^4 \sqrt{a+b x^3}}+\frac{11 (17 A b-14 a B) \sqrt{a+b x^3}}{168 a^3 x^4}-\frac{55 b (17 A b-14 a B) \sqrt{a+b x^3}}{336 a^4 x}+\frac{\left (55 b^2 (17 A b-14 a B)\right ) \int \frac{x}{\sqrt{a+b x^3}} \, dx}{672 a^4}\\ &=-\frac{A}{7 a x^7 \sqrt{a+b x^3}}-\frac{17 A b-14 a B}{21 a^2 x^4 \sqrt{a+b x^3}}+\frac{11 (17 A b-14 a B) \sqrt{a+b x^3}}{168 a^3 x^4}-\frac{55 b (17 A b-14 a B) \sqrt{a+b x^3}}{336 a^4 x}+\frac{\left (55 b^{5/3} (17 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}{672 a^4}+\frac{\left (55 \sqrt{\frac{1}{2} \left (2-\sqrt{3}\right )} b^{5/3} (17 A b-14 a B)\right ) \int \frac{1}{\sqrt{a+b x^3}} \, dx}{336 a^{11/3}}\\ &=-\frac{A}{7 a x^7 \sqrt{a+b x^3}}-\frac{17 A b-14 a B}{21 a^2 x^4 \sqrt{a+b x^3}}+\frac{11 (17 A b-14 a B) \sqrt{a+b x^3}}{168 a^3 x^4}-\frac{55 b (17 A b-14 a B) \sqrt{a+b x^3}}{336 a^4 x}+\frac{55 b^{4/3} (17 A b-14 a B) \sqrt{a+b x^3}}{336 a^4 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac{55 \sqrt{2-\sqrt{3}} b^{4/3} (17 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\ 3^{3/4} a^{11/3} \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{55 b^{4/3} (17 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 )}{168 \sqrt{2} \sqrt [4]{3} a^{11/3} \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*}
Mathematica [C] time = 0.0340703, size = 72, normalized size = 0.12 \[ \frac{x^3 \sqrt{\frac{b x^3}{a}+1} (17 A b-14 a B) \, _2F_1\left (-\frac{4}{3},\frac{3}{2};-\frac{1}{3};-\frac{b x^3}{a}\right )-8 a A}{56 a^2 x^7 \sqrt{a+b x^3}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.025, size = 1018, normalized size = 1.7 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{B x^{3} + A}{{\left (b x^{3} + a\right )}^{\frac{3}{2}} x^{8}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (B x^{3} + A\right )} \sqrt{b x^{3} + a}}{b^{2} x^{14} + 2 \, a b x^{11} + a^{2} x^{8}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{B x^{3} + A}{{\left (b x^{3} + a\right )}^{\frac{3}{2}} x^{8}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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