Optimal. Leaf size=577 \[ \frac{80 \sqrt{2} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (7 A b-16 a B) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt{3}\right )}{189 \sqrt [4]{3} b^{11/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{40 \sqrt{2-\sqrt{3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (7 A b-16 a B) E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{63\ 3^{3/4} b^{11/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{2 x^5 (7 A b-16 a B)}{63 b^2 \left (a+b x^3\right )^{3/2}}-\frac{20 x^2 (7 A b-16 a B)}{189 b^3 \sqrt{a+b x^3}}+\frac{80 \sqrt{a+b x^3} (7 A b-16 a B)}{189 b^{11/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac{2 B x^8}{7 b \left (a+b x^3\right )^{3/2}} \]
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Rubi [A] time = 0.317913, antiderivative size = 577, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {459, 288, 303, 218, 1877} \[ \frac{80 \sqrt{2} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (7 A b-16 a B) F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{189 \sqrt [4]{3} b^{11/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{40 \sqrt{2-\sqrt{3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (7 A b-16 a B) E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{63\ 3^{3/4} b^{11/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{2 x^5 (7 A b-16 a B)}{63 b^2 \left (a+b x^3\right )^{3/2}}-\frac{20 x^2 (7 A b-16 a B)}{189 b^3 \sqrt{a+b x^3}}+\frac{80 \sqrt{a+b x^3} (7 A b-16 a B)}{189 b^{11/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac{2 B x^8}{7 b \left (a+b x^3\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 459
Rule 288
Rule 303
Rule 218
Rule 1877
Rubi steps
\begin{align*} \int \frac{x^7 \left (A+B x^3\right )}{\left (a+b x^3\right )^{5/2}} \, dx &=\frac{2 B x^8}{7 b \left (a+b x^3\right )^{3/2}}-\frac{\left (2 \left (-\frac{7 A b}{2}+8 a B\right )\right ) \int \frac{x^7}{\left (a+b x^3\right )^{5/2}} \, dx}{7 b}\\ &=-\frac{2 (7 A b-16 a B) x^5}{63 b^2 \left (a+b x^3\right )^{3/2}}+\frac{2 B x^8}{7 b \left (a+b x^3\right )^{3/2}}+\frac{(10 (7 A b-16 a B)) \int \frac{x^4}{\left (a+b x^3\right )^{3/2}} \, dx}{63 b^2}\\ &=-\frac{2 (7 A b-16 a B) x^5}{63 b^2 \left (a+b x^3\right )^{3/2}}+\frac{2 B x^8}{7 b \left (a+b x^3\right )^{3/2}}-\frac{20 (7 A b-16 a B) x^2}{189 b^3 \sqrt{a+b x^3}}+\frac{(40 (7 A b-16 a B)) \int \frac{x}{\sqrt{a+b x^3}} \, dx}{189 b^3}\\ &=-\frac{2 (7 A b-16 a B) x^5}{63 b^2 \left (a+b x^3\right )^{3/2}}+\frac{2 B x^8}{7 b \left (a+b x^3\right )^{3/2}}-\frac{20 (7 A b-16 a B) x^2}{189 b^3 \sqrt{a+b x^3}}+\frac{(40 (7 A b-16 a B)) \int \frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt{a+b x^3}} \, dx}{189 b^{10/3}}+\frac{\left (40 \sqrt{2 \left (2-\sqrt{3}\right )} \sqrt [3]{a} (7 A b-16 a B)\right ) \int \frac{1}{\sqrt{a+b x^3}} \, dx}{189 b^{10/3}}\\ &=-\frac{2 (7 A b-16 a B) x^5}{63 b^2 \left (a+b x^3\right )^{3/2}}+\frac{2 B x^8}{7 b \left (a+b x^3\right )^{3/2}}-\frac{20 (7 A b-16 a B) x^2}{189 b^3 \sqrt{a+b x^3}}+\frac{80 (7 A b-16 a B) \sqrt{a+b x^3}}{189 b^{11/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac{40 \sqrt{2-\sqrt{3}} \sqrt [3]{a} (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}} 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 )}{63\ 3^{3/4} b^{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{80 \sqrt{2} \sqrt [3]{a} (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 )}{189 \sqrt [4]{3} b^{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 = 0.104084, size = 109, normalized size = 0.19 \[ \frac{2 x^2 \left (-32 a^2 B+2 \left (a+b x^3\right ) \sqrt{\frac{b x^3}{a}+1} (16 a B-7 A b) \, _2F_1\left (\frac{2}{3},\frac{5}{2};\frac{5}{3};-\frac{b x^3}{a}\right )+2 a b \left (7 A-8 B x^3\right )+b^2 x^3 \left (7 A+B x^3\right )\right )}{7 b^3 \left (a+b x^3\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.036, size = 997, 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{{\left (B x^{3} + A\right )} x^{7}}{{\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^{10} + A x^{7}\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^{7}}{{\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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