Optimal. Leaf size=614 \[ \frac{2 \sqrt [3]{2} 3^{3/4} \sqrt{2+\sqrt{3}} (b c-a d)^2 ((a+b x) (c+d x))^{2/3} \sqrt{(a d+b c+2 b d x)^2} \left (2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+(b c-a d)^{2/3}\right ) \sqrt{\frac{2 \sqrt [3]{2} b^{2/3} d^{2/3} ((a+b x) (c+d x))^{2/3}-2^{2/3} \sqrt [3]{b} \sqrt [3]{d} (b c-a d)^{2/3} \sqrt [3]{(a+b x) (c+d x)}+(b c-a d)^{4/3}}{\left (2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+\left (1+\sqrt{3}\right ) (b c-a d)^{2/3}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}{\left (1+\sqrt{3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}\right )|-7-4 \sqrt{3}\right )}{5 \sqrt [3]{b} d^{7/3} (a+b x)^{2/3} (c+d x)^{2/3} (a d+b c+2 b d x) \sqrt{\frac{(b c-a d)^{2/3} \left (2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+(b c-a d)^{2/3}\right )}{\left (2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+\left (1+\sqrt{3}\right ) (b c-a d)^{2/3}\right )^2}} \sqrt{(a d+b (c+2 d x))^2}}-\frac{6 \sqrt [3]{a+b x} \sqrt [3]{c+d x} (b c-a d)}{5 d^2}+\frac{3 (a+b x)^{4/3} \sqrt [3]{c+d x}}{5 d} \]
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Rubi [A] time = 1.80139, antiderivative size = 614, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{2 \sqrt [3]{2} 3^{3/4} \sqrt{2+\sqrt{3}} (b c-a d)^2 ((a+b x) (c+d x))^{2/3} \sqrt{(a d+b c+2 b d x)^2} \left (2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+(b c-a d)^{2/3}\right ) \sqrt{\frac{2 \sqrt [3]{2} b^{2/3} d^{2/3} ((a+b x) (c+d x))^{2/3}-2^{2/3} \sqrt [3]{b} \sqrt [3]{d} (b c-a d)^{2/3} \sqrt [3]{(a+b x) (c+d x)}+(b c-a d)^{4/3}}{\left (2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+\left (1+\sqrt{3}\right ) (b c-a d)^{2/3}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}{\left (1+\sqrt{3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}\right )|-7-4 \sqrt{3}\right )}{5 \sqrt [3]{b} d^{7/3} (a+b x)^{2/3} (c+d x)^{2/3} (a d+b c+2 b d x) \sqrt{\frac{(b c-a d)^{2/3} \left (2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+(b c-a d)^{2/3}\right )}{\left (2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+\left (1+\sqrt{3}\right ) (b c-a d)^{2/3}\right )^2}} \sqrt{(a d+b (c+2 d x))^2}}-\frac{6 \sqrt [3]{a+b x} \sqrt [3]{c+d x} (b c-a d)}{5 d^2}+\frac{3 (a+b x)^{4/3} \sqrt [3]{c+d x}}{5 d} \]
Warning: Unable to verify antiderivative.
[In] Int[(a + b*x)^(4/3)/(c + d*x)^(2/3),x]
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Rubi in Sympy [A] time = 70.1928, size = 648, normalized size = 1.06 \[ \frac{3 \left (a + b x\right )^{\frac{4}{3}} \sqrt [3]{c + d x}}{5 d} + \frac{6 \sqrt [3]{a + b x} \sqrt [3]{c + d x} \left (a d - b c\right )}{5 d^{2}} + \frac{2 \sqrt [3]{2} \cdot 3^{\frac{3}{4}} \sqrt{\frac{2 \sqrt [3]{2} b^{\frac{2}{3}} d^{\frac{2}{3}} \left (a c + b d x^{2} + x \left (a d + b c\right )\right )^{\frac{2}{3}} - 2^{\frac{2}{3}} \sqrt [3]{b} \sqrt [3]{d} \left (a d - b c\right )^{\frac{2}{3}} \sqrt [3]{a c + b d x^{2} + x \left (a d + b c\right )} + \left (a d - b c\right )^{\frac{4}{3}}}{\left (2^{\frac{2}{3}} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{a c + b d x^{2} + x \left (a d + b c\right )} + \left (1 + \sqrt{3}\right ) \left (a d - b c\right )^{\frac{2}{3}}\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (a d - b c\right )^{2} \left (2^{\frac{2}{3}} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{a c + b d x^{2} + x \left (a d + b c\right )} + \left (a d - b c\right )^{\frac{2}{3}}\right ) \left (a c + b d x^{2} + x \left (a d + b c\right )\right )^{\frac{2}{3}} \sqrt{\left (a d + b c + 2 b d x\right )^{2}} F\left (\operatorname{asin}{\left (\frac{2^{\frac{2}{3}} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{a c + b d x^{2} + x \left (a d + b c\right )} - \left (-1 + \sqrt{3}\right ) \left (a d - b c\right )^{\frac{2}{3}}}{2^{\frac{2}{3}} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{a c + b d x^{2} + x \left (a d + b c\right )} + \left (1 + \sqrt{3}\right ) \left (a d - b c\right )^{\frac{2}{3}}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{5 \sqrt [3]{b} d^{\frac{7}{3}} \sqrt{\frac{\left (a d - b c\right )^{\frac{2}{3}} \left (2^{\frac{2}{3}} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{a c + b d x^{2} + x \left (a d + b c\right )} + \left (a d - b c\right )^{\frac{2}{3}}\right )}{\left (2^{\frac{2}{3}} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{a c + b d x^{2} + x \left (a d + b c\right )} + \left (1 + \sqrt{3}\right ) \left (a d - b c\right )^{\frac{2}{3}}\right )^{2}}} \left (a + b x\right )^{\frac{2}{3}} \left (c + d x\right )^{\frac{2}{3}} \sqrt{b d \left (4 a c + 4 b d x^{2} + x \left (4 a d + 4 b c\right )\right ) + \left (a d - b c\right )^{2}} \left (a d + b c + 2 b d x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**(4/3)/(d*x+c)**(2/3),x)
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Mathematica [C] time = 0.188068, size = 106, normalized size = 0.17 \[ \frac{3 \sqrt [3]{c+d x} \left (2 (b c-a d)^2 \left (\frac{d (a+b x)}{a d-b c}\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};\frac{b (c+d x)}{b c-a d}\right )+d (a+b x) (3 a d-2 b c+b d x)\right )}{5 d^3 (a+b x)^{2/3}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)^(4/3)/(c + d*x)^(2/3),x]
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Maple [F] time = 0.036, size = 0, normalized size = 0. \[ \int{1 \left ( bx+a \right ) ^{{\frac{4}{3}}} \left ( dx+c \right ) ^{-{\frac{2}{3}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^(4/3)/(d*x+c)^(2/3),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{\frac{4}{3}}}{{\left (d x + c\right )}^{\frac{2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^(4/3)/(d*x + c)^(2/3),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x + a\right )}^{\frac{4}{3}}}{{\left (d x + c\right )}^{\frac{2}{3}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^(4/3)/(d*x + c)^(2/3),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x\right )^{\frac{4}{3}}}{\left (c + d x\right )^{\frac{2}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**(4/3)/(d*x+c)**(2/3),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{\frac{4}{3}}}{{\left (d x + c\right )}^{\frac{2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^(4/3)/(d*x + c)^(2/3),x, algorithm="giac")
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