Optimal. Leaf size=99 \[ \frac{3 \left (b^2-4 a c\right ) \sqrt [3]{a+b x+c x^2} \, _2F_1\left (-\frac{4}{3},-\frac{7}{6};-\frac{1}{6};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{56 c^2 d \sqrt [3]{1-\frac{(b+2 c x)^2}{b^2-4 a c}} (d (b+2 c x))^{7/3}} \]
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Rubi [A] time = 0.29168, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107 \[ \frac{3 \left (b^2-4 a c\right ) \sqrt [3]{a+b x+c x^2} \, _2F_1\left (-\frac{4}{3},-\frac{7}{6};-\frac{1}{6};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{56 c^2 d \sqrt [3]{1-\frac{(b+2 c x)^2}{b^2-4 a c}} (d (b+2 c x))^{7/3}} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x + c*x^2)^(4/3)/(b*d + 2*c*d*x)^(10/3),x]
[Out]
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Rubi in Sympy [A] time = 31.1135, size = 105, normalized size = 1.06 \[ \frac{3 \left (- 4 a c + b^{2}\right ) \sqrt [3]{a - \frac{b^{2}}{4 c} + \frac{\left (b + 2 c x\right )^{2}}{4 c}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{4}{3}, - \frac{7}{6} \\ - \frac{1}{6} \end{matrix}\middle |{- \frac{\left (b + 2 c x\right )^{2}}{4 a c - b^{2}}} \right )}}{56 c^{2} d \left (b d + 2 c d x\right )^{\frac{7}{3}} \sqrt [3]{\frac{\left (b + 2 c x\right )^{2}}{4 a c - b^{2}} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+b*x+a)**(4/3)/(2*c*d*x+b*d)**(10/3),x)
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Mathematica [A] time = 0.603119, size = 170, normalized size = 1.72 \[ \frac{6 \sqrt [3]{2} (b+2 c x)^4 \left (\frac{c (a+x (b+c x))}{4 a c-b^2}\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{5}{6};\frac{11}{6};\frac{(b+2 c x)^2}{b^2-4 a c}\right )-15 c \left (a^2 c+2 a \left (b^2+5 b c x+5 c^2 x^2\right )+x \left (2 b^3+11 b^2 c x+18 b c^2 x^2+9 c^3 x^3\right )\right )}{70 c^3 d (a+x (b+c x))^{2/3} (d (b+2 c x))^{7/3}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x + c*x^2)^(4/3)/(b*d + 2*c*d*x)^(10/3),x]
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Maple [F] time = 0.155, size = 0, normalized size = 0. \[ \int{1 \left ( c{x}^{2}+bx+a \right ) ^{{\frac{4}{3}}} \left ( 2\,cdx+bd \right ) ^{-{\frac{10}{3}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x+a)^(4/3)/(2*c*d*x+b*d)^(10/3),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + b x + a\right )}^{\frac{4}{3}}}{{\left (2 \, c d x + b d\right )}^{\frac{10}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^(4/3)/(2*c*d*x + b*d)^(10/3),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (c x^{2} + b x + a\right )}^{\frac{4}{3}}}{{\left (8 \, c^{3} d^{3} x^{3} + 12 \, b c^{2} d^{3} x^{2} + 6 \, b^{2} c d^{3} x + b^{3} d^{3}\right )}{\left (2 \, c d x + b d\right )}^{\frac{1}{3}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^(4/3)/(2*c*d*x + b*d)^(10/3),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+b*x+a)**(4/3)/(2*c*d*x+b*d)**(10/3),x)
[Out]
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GIAC/XCAS [A] time = 1.44285, size = 1, normalized size = 0.01 \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^(4/3)/(2*c*d*x + b*d)^(10/3),x, algorithm="giac")
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