Optimal. Leaf size=448 \[ \frac{3 \left (-\frac{c x (b+c x)}{b^2}\right )^{4/3} (b+2 c x) \left (b x+c x^2\right )^{4/3}}{22 c \left (-\frac{c \left (b x+c x^2\right )}{b^2}\right )^{4/3}}+\frac{3 \sqrt [3]{-\frac{c x (b+c x)}{b^2}} (b+2 c x) \left (b x+c x^2\right )^{4/3}}{55 c \left (-\frac{c \left (b x+c x^2\right )}{b^2}\right )^{4/3}}+\frac{\sqrt [3]{2} 3^{3/4} \sqrt{2-\sqrt{3}} b^2 \left (b x+c x^2\right )^{4/3} \left (1-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (-\frac{c x (b+c x)}{b^2}\right )^{2/3}+2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}+1}{\left (-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}+\sqrt{3}+1}{-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{55 c (b+2 c x) \left (-\frac{c \left (b x+c x^2\right )}{b^2}\right )^{4/3} \sqrt{-\frac{1-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}}{\left (-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1\right )^2}}} \]
[Out]
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Rubi [A] time = 1.33759, antiderivative size = 448, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385 \[ \frac{3 \left (-\frac{c x (b+c x)}{b^2}\right )^{4/3} (b+2 c x) \left (b x+c x^2\right )^{4/3}}{22 c \left (-\frac{c \left (b x+c x^2\right )}{b^2}\right )^{4/3}}+\frac{3 \sqrt [3]{-\frac{c x (b+c x)}{b^2}} (b+2 c x) \left (b x+c x^2\right )^{4/3}}{55 c \left (-\frac{c \left (b x+c x^2\right )}{b^2}\right )^{4/3}}+\frac{\sqrt [3]{2} 3^{3/4} \sqrt{2-\sqrt{3}} b^2 \left (b x+c x^2\right )^{4/3} \left (1-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (-\frac{c x (b+c x)}{b^2}\right )^{2/3}+2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}+1}{\left (-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}+\sqrt{3}+1}{-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{55 c (b+2 c x) \left (-\frac{c \left (b x+c x^2\right )}{b^2}\right )^{4/3} \sqrt{-\frac{1-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}}{\left (-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1\right )^2}}} \]
Antiderivative was successfully verified.
[In] Int[(b*x + c*x^2)^(4/3),x]
[Out]
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Rubi in Sympy [A] time = 41.3632, size = 398, normalized size = 0.89 \[ \frac{\sqrt [3]{2} \cdot 3^{\frac{3}{4}} b^{2} \sqrt{\frac{\left (1 - \frac{\left (- b - 2 c x\right )^{2}}{b^{2}}\right )^{\frac{2}{3}} + \sqrt [3]{1 - \frac{\left (- b - 2 c x\right )^{2}}{b^{2}}} + 1}{\left (- \sqrt [3]{1 - \frac{\left (- b - 2 c x\right )^{2}}{b^{2}}} - \sqrt{3} + 1\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (b x + c x^{2}\right )^{\frac{4}{3}} \left (- \sqrt [3]{1 - \frac{\left (- b - 2 c x\right )^{2}}{b^{2}}} + 1\right ) F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{1 - \frac{\left (- b - 2 c x\right )^{2}}{b^{2}}} + 1 + \sqrt{3}}{- \sqrt [3]{1 - \frac{\left (- b - 2 c x\right )^{2}}{b^{2}}} - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{55 c \sqrt{\frac{\sqrt [3]{1 - \frac{\left (- b - 2 c x\right )^{2}}{b^{2}}} - 1}{\left (- \sqrt [3]{1 - \frac{\left (- b - 2 c x\right )^{2}}{b^{2}}} - \sqrt{3} + 1\right )^{2}}} \left (\frac{c \left (- b x - c x^{2}\right )}{b^{2}}\right )^{\frac{4}{3}} \left (b + 2 c x\right )} + \frac{3 \sqrt [3]{2} \left (1 - \frac{\left (- b - 2 c x\right )^{2}}{b^{2}}\right )^{\frac{4}{3}} \left (b + 2 c x\right ) \left (b x + c x^{2}\right )^{\frac{4}{3}}}{176 c \left (\frac{c \left (- b x - c x^{2}\right )}{b^{2}}\right )^{\frac{4}{3}}} + \frac{3 \sqrt [3]{2} \sqrt [3]{1 - \frac{\left (- b - 2 c x\right )^{2}}{b^{2}}} \left (b + 2 c x\right ) \left (b x + c x^{2}\right )^{\frac{4}{3}}}{110 c \left (\frac{c \left (- b x - c x^{2}\right )}{b^{2}}\right )^{\frac{4}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+b*x)**(4/3),x)
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Mathematica [C] time = 0.0840112, size = 94, normalized size = 0.21 \[ \frac{3 x \left (2 b^4 \left (\frac{c x}{b}+1\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{c x}{b}\right )-2 b^4-b^3 c x+16 b^2 c^2 x^2+25 b c^3 x^3+10 c^4 x^4\right )}{110 c^2 (x (b+c x))^{2/3}} \]
Antiderivative was successfully verified.
[In] Integrate[(b*x + c*x^2)^(4/3),x]
[Out]
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Maple [F] time = 0.057, size = 0, normalized size = 0. \[ \int \left ( c{x}^{2}+bx \right ) ^{{\frac{4}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x)^(4/3),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + b x\right )}^{\frac{4}{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^(4/3),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (c x^{2} + b x\right )}^{\frac{4}{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^(4/3),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \left (b x + c x^{2}\right )^{\frac{4}{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+b*x)**(4/3),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + b x\right )}^{\frac{4}{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^(4/3),x, algorithm="giac")
[Out]