Optimal. Leaf size=384 \[ \frac{3 (b+2 c x) \left (-\frac{c \left (b x+c x^2\right )}{b^2}\right )^{5/3}}{2 c \left (-\frac{c x (b+c x)}{b^2}\right )^{2/3} \left (b x+c x^2\right )^{5/3}}+\frac{\sqrt [3]{2} 3^{3/4} \sqrt{2-\sqrt{3}} b^2 \left (-\frac{c \left (b x+c x^2\right )}{b^2}\right )^{5/3} \left (1-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (-\frac{c x (b+c x)}{b^2}\right )^{2/3}+2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}+1}{\left (-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}+\sqrt{3}+1}{-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{c (b+2 c x) \left (b x+c x^2\right )^{5/3} \sqrt{-\frac{1-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}}{\left (-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1\right )^2}}} \]
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Rubi [A] time = 0.921619, antiderivative size = 384, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385 \[ \frac{3 (b+2 c x) \left (-\frac{c \left (b x+c x^2\right )}{b^2}\right )^{5/3}}{2 c \left (-\frac{c x (b+c x)}{b^2}\right )^{2/3} \left (b x+c x^2\right )^{5/3}}+\frac{\sqrt [3]{2} 3^{3/4} \sqrt{2-\sqrt{3}} b^2 \left (-\frac{c \left (b x+c x^2\right )}{b^2}\right )^{5/3} \left (1-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (-\frac{c x (b+c x)}{b^2}\right )^{2/3}+2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}+1}{\left (-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}+\sqrt{3}+1}{-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{c (b+2 c x) \left (b x+c x^2\right )^{5/3} \sqrt{-\frac{1-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}}{\left (-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1\right )^2}}} \]
Antiderivative was successfully verified.
[In] Int[(b*x + c*x^2)^(-5/3),x]
[Out]
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Rubi in Sympy [A] time = 32.4619, size = 328, normalized size = 0.85 \[ \frac{\sqrt [3]{2} \cdot 3^{\frac{3}{4}} b^{2} \sqrt{\frac{\left (1 - \frac{\left (- b - 2 c x\right )^{2}}{b^{2}}\right )^{\frac{2}{3}} + \sqrt [3]{1 - \frac{\left (- b - 2 c x\right )^{2}}{b^{2}}} + 1}{\left (- \sqrt [3]{1 - \frac{\left (- b - 2 c x\right )^{2}}{b^{2}}} - \sqrt{3} + 1\right )^{2}}} \left (\frac{c \left (- b x - c x^{2}\right )}{b^{2}}\right )^{\frac{5}{3}} \sqrt{- \sqrt{3} + 2} \left (- \sqrt [3]{1 - \frac{\left (- b - 2 c x\right )^{2}}{b^{2}}} + 1\right ) F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{1 - \frac{\left (- b - 2 c x\right )^{2}}{b^{2}}} + 1 + \sqrt{3}}{- \sqrt [3]{1 - \frac{\left (- b - 2 c x\right )^{2}}{b^{2}}} - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{c \sqrt{\frac{\sqrt [3]{1 - \frac{\left (- b - 2 c x\right )^{2}}{b^{2}}} - 1}{\left (- \sqrt [3]{1 - \frac{\left (- b - 2 c x\right )^{2}}{b^{2}}} - \sqrt{3} + 1\right )^{2}}} \left (b + 2 c x\right ) \left (b x + c x^{2}\right )^{\frac{5}{3}}} + \frac{3 \sqrt [3]{2} \left (\frac{c \left (- b x - c x^{2}\right )}{b^{2}}\right )^{\frac{5}{3}} \left (b + 2 c x\right )}{c \left (1 - \frac{\left (- b - 2 c x\right )^{2}}{b^{2}}\right )^{\frac{2}{3}} \left (b x + c x^{2}\right )^{\frac{5}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(c*x**2+b*x)**(5/3),x)
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Mathematica [C] time = 0.048871, size = 57, normalized size = 0.15 \[ -\frac{3 \left (2 c x \left (\frac{c x}{b}+1\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{c x}{b}\right )+b+2 c x\right )}{2 b^2 (x (b+c x))^{2/3}} \]
Antiderivative was successfully verified.
[In] Integrate[(b*x + c*x^2)^(-5/3),x]
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Maple [F] time = 0.084, size = 0, normalized size = 0. \[ \int \left ( c{x}^{2}+bx \right ) ^{-{\frac{5}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(c*x^2+b*x)^(5/3),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{2} + b x\right )}^{\frac{5}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^(-5/3),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (c x^{2} + b x\right )}^{\frac{5}{3}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^(-5/3),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (b x + c x^{2}\right )^{\frac{5}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(c*x**2+b*x)**(5/3),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{2} + b x\right )}^{\frac{5}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^(-5/3),x, algorithm="giac")
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