Optimal. Leaf size=251 \[ \frac{2 \sqrt{2+\sqrt{3}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (2 a B+A b) F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{3 \sqrt [4]{3} a b^{4/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{2 x (A b-a B)}{3 a b \sqrt{a+b x^3}} \]
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Rubi [A] time = 0.20803, antiderivative size = 251, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{2 \sqrt{2+\sqrt{3}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (2 a B+A b) F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{3 \sqrt [4]{3} a b^{4/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{2 x (A b-a B)}{3 a b \sqrt{a+b x^3}} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x^3)/(a + b*x^3)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 13.6957, size = 219, normalized size = 0.87 \[ \frac{2 x \left (A b - B a\right )}{3 a b \sqrt{a + b x^{3}}} + \frac{4 \cdot 3^{\frac{3}{4}} \sqrt{\frac{a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2}}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (\sqrt [3]{a} + \sqrt [3]{b} x\right ) \left (\frac{A b}{2} + B a\right ) F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{b} x}{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x} \right )}\middle | -7 - 4 \sqrt{3}\right )}{9 a b^{\frac{4}{3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a} + \sqrt [3]{b} x\right )}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )^{2}}} \sqrt{a + b x^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x**3+A)/(b*x**3+a)**(3/2),x)
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Mathematica [C] time = 0.905129, size = 176, normalized size = 0.7 \[ -\frac{6 \sqrt [3]{-b} x (A b-a B)+2 i 3^{3/4} \sqrt [3]{a} \sqrt{\frac{(-1)^{5/6} \left (\sqrt [3]{-b} x-\sqrt [3]{a}\right )}{\sqrt [3]{a}}} \sqrt{\frac{(-b)^{2/3} x^2}{a^{2/3}}+\frac{\sqrt [3]{-b} x}{\sqrt [3]{a}}+1} (2 a B+A b) F\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{i \sqrt [3]{-b} x}{\sqrt [3]{a}}-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )}{9 a (-b)^{4/3} \sqrt{a+b x^3}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(A + B*x^3)/(a + b*x^3)^(3/2),x]
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Maple [B] time = 0.007, size = 613, normalized size = 2.4 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x^3+A)/(b*x^3+a)^(3/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{B x^{3} + A}{{\left (b x^{3} + a\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)/(b*x^3 + a)^(3/2),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{B x^{3} + A}{{\left (b x^{3} + a\right )}^{\frac{3}{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)/(b*x^3 + a)^(3/2),x, algorithm="fricas")
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Sympy [A] time = 34.4522, size = 78, normalized size = 0.31 \[ \frac{A x \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{3}{2} \\ \frac{4}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac{3}{2}} \Gamma \left (\frac{4}{3}\right )} + \frac{B x^{4} \Gamma \left (\frac{4}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{4}{3}, \frac{3}{2} \\ \frac{7}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac{3}{2}} \Gamma \left (\frac{7}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x**3+A)/(b*x**3+a)**(3/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{B x^{3} + A}{{\left (b x^{3} + a\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)/(b*x^3 + a)^(3/2),x, algorithm="giac")
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