Optimal. Leaf size=299 \[ \frac{32 \sqrt{2+\sqrt{3}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (5 A b-14 a B) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt{3}\right )}{135 \sqrt [4]{3} b^{10/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{2 x^4 (5 A b-14 a B)}{45 b^2 \left (a+b x^3\right )^{3/2}}-\frac{16 x (5 A b-14 a B)}{135 b^3 \sqrt{a+b x^3}}+\frac{2 B x^7}{5 b \left (a+b x^3\right )^{3/2}} \]
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Rubi [A] time = 0.131619, antiderivative size = 299, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {459, 288, 218} \[ \frac{32 \sqrt{2+\sqrt{3}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (5 A b-14 a B) F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{135 \sqrt [4]{3} b^{10/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{2 x^4 (5 A b-14 a B)}{45 b^2 \left (a+b x^3\right )^{3/2}}-\frac{16 x (5 A b-14 a B)}{135 b^3 \sqrt{a+b x^3}}+\frac{2 B x^7}{5 b \left (a+b x^3\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 459
Rule 288
Rule 218
Rubi steps
\begin{align*} \int \frac{x^6 \left (A+B x^3\right )}{\left (a+b x^3\right )^{5/2}} \, dx &=\frac{2 B x^7}{5 b \left (a+b x^3\right )^{3/2}}-\frac{\left (2 \left (-\frac{5 A b}{2}+7 a B\right )\right ) \int \frac{x^6}{\left (a+b x^3\right )^{5/2}} \, dx}{5 b}\\ &=-\frac{2 (5 A b-14 a B) x^4}{45 b^2 \left (a+b x^3\right )^{3/2}}+\frac{2 B x^7}{5 b \left (a+b x^3\right )^{3/2}}+\frac{(8 (5 A b-14 a B)) \int \frac{x^3}{\left (a+b x^3\right )^{3/2}} \, dx}{45 b^2}\\ &=-\frac{2 (5 A b-14 a B) x^4}{45 b^2 \left (a+b x^3\right )^{3/2}}+\frac{2 B x^7}{5 b \left (a+b x^3\right )^{3/2}}-\frac{16 (5 A b-14 a B) x}{135 b^3 \sqrt{a+b x^3}}+\frac{(16 (5 A b-14 a B)) \int \frac{1}{\sqrt{a+b x^3}} \, dx}{135 b^3}\\ &=-\frac{2 (5 A b-14 a B) x^4}{45 b^2 \left (a+b x^3\right )^{3/2}}+\frac{2 B x^7}{5 b \left (a+b x^3\right )^{3/2}}-\frac{16 (5 A b-14 a B) x}{135 b^3 \sqrt{a+b x^3}}+\frac{32 \sqrt{2+\sqrt{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 )}{135 \sqrt [4]{3} b^{10/3} \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 = 0.128289, size = 108, normalized size = 0.36 \[ \frac{2 x \left (112 a^2 B+8 \left (a+b x^3\right ) \sqrt{\frac{b x^3}{a}+1} (5 A b-14 a B) \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{4}{3};-\frac{b x^3}{a}\right )+a \left (154 b B x^3-40 A b\right )+b^2 x^3 \left (27 B x^3-55 A\right )\right )}{135 b^3 \left (a+b x^3\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.035, size = 683, normalized size = 2.3 \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{{\left (B x^{3} + A\right )} x^{6}}{{\left (b x^{3} + a\right )}^{\frac{5}{2}}}\,{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^{9} + A x^{6}\right )} \sqrt{b x^{3} + a}}{b^{3} x^{9} + 3 \, a b^{2} x^{6} + 3 \, a^{2} b x^{3} + a^{3}}, 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{{\left (B x^{3} + A\right )} x^{6}}{{\left (b x^{3} + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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