Optimal. Leaf size=621 \[ \frac{2 \sqrt [3]{2} 3^{3/4} \sqrt{2+\sqrt{3}} d^{5/3} ((a+b x) (c+d x))^{2/3} \sqrt{(a d+b c+2 b d x)^2} \left (2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+(b c-a d)^{2/3}\right ) \sqrt{\frac{2 \sqrt [3]{2} b^{2/3} d^{2/3} ((a+b x) (c+d x))^{2/3}-2^{2/3} \sqrt [3]{b} \sqrt [3]{d} (b c-a d)^{2/3} \sqrt [3]{(a+b x) (c+d x)}+(b c-a d)^{4/3}}{\left (2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+\left (1+\sqrt{3}\right ) (b c-a d)^{2/3}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}{\left (1+\sqrt{3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}\right )|-7-4 \sqrt{3}\right )}{5 \sqrt [3]{b} (a+b x)^{2/3} (c+d x)^{2/3} (b c-a d)^2 (a d+b c+2 b d x) \sqrt{\frac{(b c-a d)^{2/3} \left (2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+(b c-a d)^{2/3}\right )}{\left (2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+\left (1+\sqrt{3}\right ) (b c-a d)^{2/3}\right )^2}} \sqrt{(a d+b (c+2 d x))^2}}+\frac{6 d \sqrt [3]{c+d x}}{5 (a+b x)^{2/3} (b c-a d)^2}-\frac{3 \sqrt [3]{c+d x}}{5 (a+b x)^{5/3} (b c-a d)} \]
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Rubi [A] time = 1.66537, antiderivative size = 621, 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 \[ \frac{2 \sqrt [3]{2} 3^{3/4} \sqrt{2+\sqrt{3}} d^{5/3} ((a+b x) (c+d x))^{2/3} \sqrt{(a d+b c+2 b d x)^2} \left (2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+(b c-a d)^{2/3}\right ) \sqrt{\frac{2 \sqrt [3]{2} b^{2/3} d^{2/3} ((a+b x) (c+d x))^{2/3}-2^{2/3} \sqrt [3]{b} \sqrt [3]{d} (b c-a d)^{2/3} \sqrt [3]{(a+b x) (c+d x)}+(b c-a d)^{4/3}}{\left (2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+\left (1+\sqrt{3}\right ) (b c-a d)^{2/3}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}{\left (1+\sqrt{3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}\right )|-7-4 \sqrt{3}\right )}{5 \sqrt [3]{b} (a+b x)^{2/3} (c+d x)^{2/3} (b c-a d)^2 (a d+b c+2 b d x) \sqrt{\frac{(b c-a d)^{2/3} \left (2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+(b c-a d)^{2/3}\right )}{\left (2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+\left (1+\sqrt{3}\right ) (b c-a d)^{2/3}\right )^2}} \sqrt{(a d+b (c+2 d x))^2}}+\frac{6 d \sqrt [3]{c+d x}}{5 (a+b x)^{2/3} (b c-a d)^2}-\frac{3 \sqrt [3]{c+d x}}{5 (a+b x)^{5/3} (b c-a d)} \]
Warning: Unable to verify antiderivative.
[In] Int[1/((a + b*x)^(8/3)*(c + d*x)^(2/3)),x]
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Rubi in Sympy [A] time = 74.1483, size = 653, normalized size = 1.05 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x+a)**(8/3)/(d*x+c)**(2/3),x)
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Mathematica [C] time = 0.233637, size = 102, normalized size = 0.16 \[ \frac{3 \sqrt [3]{c+d x} \left (2 d (a+b x) \left (\frac{d (a+b x)}{a d-b c}\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};\frac{b (c+d x)}{b c-a d}\right )+3 a d-b c+2 b d x\right )}{5 (a+b x)^{5/3} (b c-a d)^2} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b*x)^(8/3)*(c + d*x)^(2/3)),x]
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Maple [F] time = 0.066, size = 0, normalized size = 0. \[ \int{1 \left ( bx+a \right ) ^{-{\frac{8}{3}}} \left ( dx+c \right ) ^{-{\frac{2}{3}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b*x+a)^(8/3)/(d*x+c)^(2/3),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{8}{3}}{\left (d x + c\right )}^{\frac{2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)^(8/3)*(d*x + c)^(2/3)),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}{\left (b x + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{2}{3}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)^(8/3)*(d*x + c)^(2/3)),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(1/(b*x+a)**(8/3)/(d*x+c)**(2/3),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{8}{3}}{\left (d x + c\right )}^{\frac{2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)^(8/3)*(d*x + c)^(2/3)),x, algorithm="giac")
[Out]