Optimal. Leaf size=336 \[ \frac{54 a^2 x \sqrt{a+b x^3} (23 A b-8 a B)}{21505 b^2}-\frac{36\ 3^{3/4} \sqrt{2+\sqrt{3}} a^3 \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (23 A b-8 a B) F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{21505 b^{7/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{2 x^4 \left (a+b x^3\right )^{3/2} (23 A b-8 a B)}{391 b}+\frac{18 a x^4 \sqrt{a+b x^3} (23 A b-8 a B)}{4301 b}+\frac{2 B x^4 \left (a+b x^3\right )^{5/2}}{23 b} \]
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Rubi [A] time = 0.499769, antiderivative size = 336, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{54 a^2 x \sqrt{a+b x^3} (23 A b-8 a B)}{21505 b^2}-\frac{36\ 3^{3/4} \sqrt{2+\sqrt{3}} a^3 \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (23 A b-8 a B) F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{21505 b^{7/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{2 x^4 \left (a+b x^3\right )^{3/2} (23 A b-8 a B)}{391 b}+\frac{18 a x^4 \sqrt{a+b x^3} (23 A b-8 a B)}{4301 b}+\frac{2 B x^4 \left (a+b x^3\right )^{5/2}}{23 b} \]
Antiderivative was successfully verified.
[In] Int[x^3*(a + b*x^3)^(3/2)*(A + B*x^3),x]
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Rubi in Sympy [A] time = 34.3159, size = 309, normalized size = 0.92 \[ \frac{2 B x^{4} \left (a + b x^{3}\right )^{\frac{5}{2}}}{23 b} - \frac{36 \cdot 3^{\frac{3}{4}} a^{3} \sqrt{\frac{a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2}}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (\sqrt [3]{a} + \sqrt [3]{b} x\right ) \left (23 A b - 8 B a\right ) F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{b} x}{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x} \right )}\middle | -7 - 4 \sqrt{3}\right )}{21505 b^{\frac{7}{3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a} + \sqrt [3]{b} x\right )}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )^{2}}} \sqrt{a + b x^{3}}} + \frac{54 a^{2} x \sqrt{a + b x^{3}} \left (23 A b - 8 B a\right )}{21505 b^{2}} + \frac{18 a x^{4} \sqrt{a + b x^{3}} \left (23 A b - 8 B a\right )}{4301 b} + \frac{2 x^{4} \left (a + b x^{3}\right )^{\frac{3}{2}} \left (23 A b - 8 B a\right )}{391 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3*(b*x**3+a)**(3/2)*(B*x**3+A),x)
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Mathematica [C] time = 1.10423, size = 229, normalized size = 0.68 \[ \sqrt{a+b x^3} \left (-\frac{54 a^2 x (8 a B-23 A b)}{21505 b^2}+\frac{2}{391} x^7 (26 a B+23 A b)+\frac{2 a x^4 (27 a B+460 A b)}{4301 b}+\frac{2}{23} b B x^{10}\right )-\frac{36 i 3^{3/4} a^{10/3} \sqrt{(-1)^{5/6} \left (\frac{\sqrt [3]{-b} x}{\sqrt [3]{a}}-1\right )} \sqrt{\frac{(-b)^{2/3} x^2}{a^{2/3}}+\frac{\sqrt [3]{-b} x}{\sqrt [3]{a}}+1} (23 A b-8 a B) F\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{i \sqrt [3]{-b} x}{\sqrt [3]{a}}-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )}{21505 \sqrt [3]{-b} b^2 \sqrt{a+b x^3}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[x^3*(a + b*x^3)^(3/2)*(A + B*x^3),x]
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Maple [B] time = 0.01, size = 694, normalized size = 2.1 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3*(b*x^3+a)^(3/2)*(B*x^3+A),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (B x^{3} + A\right )}{\left (b x^{3} + a\right )}^{\frac{3}{2}} x^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*(b*x^3 + a)^(3/2)*x^3,x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (B b x^{9} +{\left (B a + A b\right )} x^{6} + A a x^{3}\right )} \sqrt{b x^{3} + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*(b*x^3 + a)^(3/2)*x^3,x, algorithm="fricas")
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Sympy [A] time = 11.866, size = 172, normalized size = 0.51 \[ \frac{A a^{\frac{3}{2}} x^{4} \Gamma \left (\frac{4}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{4}{3} \\ \frac{7}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac{7}{3}\right )} + \frac{A \sqrt{a} b x^{7} \Gamma \left (\frac{7}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{7}{3} \\ \frac{10}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac{10}{3}\right )} + \frac{B a^{\frac{3}{2}} x^{7} \Gamma \left (\frac{7}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{7}{3} \\ \frac{10}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac{10}{3}\right )} + \frac{B \sqrt{a} b x^{10} \Gamma \left (\frac{10}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{10}{3} \\ \frac{13}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac{13}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3*(b*x**3+a)**(3/2)*(B*x**3+A),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (B x^{3} + A\right )}{\left (b x^{3} + a\right )}^{\frac{3}{2}} x^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*(b*x^3 + a)^(3/2)*x^3,x, algorithm="giac")
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