Optimal. Leaf size=762 \[ -\frac{6 \sqrt{a+b x}}{b^{2/3} \left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}-\frac{2 \sqrt{2} 3^{3/4} \sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+b^{2/3} (c+d x)^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}{\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}\right )|-7+4 \sqrt{3}\right )}{b^{2/3} d \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac{3 \sqrt [4]{3} \sqrt{2+\sqrt{3}} \sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+b^{2/3} (c+d x)^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}{\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}\right )|-7+4 \sqrt{3}\right )}{b^{2/3} d \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}} \]
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Rubi [A] time = 1.2798, antiderivative size = 762, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ -\frac{6 \sqrt{a+b x}}{b^{2/3} \left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}-\frac{2 \sqrt{2} 3^{3/4} \sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+b^{2/3} (c+d x)^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}{\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}\right )|-7+4 \sqrt{3}\right )}{b^{2/3} d \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac{3 \sqrt [4]{3} \sqrt{2+\sqrt{3}} \sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+b^{2/3} (c+d x)^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}{\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}\right )|-7+4 \sqrt{3}\right )}{b^{2/3} d \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}} \]
Antiderivative was successfully verified.
[In] Int[1/(Sqrt[a + b*x]*(c + d*x)^(1/3)),x]
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Rubi in Sympy [A] time = 64.1664, size = 654, normalized size = 0.86 \[ \frac{6 \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}}{b^{\frac{2}{3}} \left (\sqrt [3]{b} \sqrt [3]{c + d x} + \left (1 + \sqrt{3}\right ) \sqrt [3]{a d - b c}\right )} - \frac{3 \sqrt [4]{3} \sqrt{\frac{b^{\frac{2}{3}} \left (c + d x\right )^{\frac{2}{3}} - \sqrt [3]{b} \sqrt [3]{c + d x} \sqrt [3]{a d - b c} + \left (a d - b c\right )^{\frac{2}{3}}}{\left (\sqrt [3]{b} \sqrt [3]{c + d x} + \left (1 + \sqrt{3}\right ) \sqrt [3]{a d - b c}\right )^{2}}} \sqrt{- \sqrt{3} + 2} \sqrt [3]{a d - b c} \left (\sqrt [3]{b} \sqrt [3]{c + d x} + \sqrt [3]{a d - b c}\right ) E\left (\operatorname{asin}{\left (\frac{\sqrt [3]{b} \sqrt [3]{c + d x} - \left (-1 + \sqrt{3}\right ) \sqrt [3]{a d - b c}}{\sqrt [3]{b} \sqrt [3]{c + d x} + \left (1 + \sqrt{3}\right ) \sqrt [3]{a d - b c}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{b^{\frac{2}{3}} d \sqrt{\frac{\sqrt [3]{a d - b c} \left (\sqrt [3]{b} \sqrt [3]{c + d x} + \sqrt [3]{a d - b c}\right )}{\left (\sqrt [3]{b} \sqrt [3]{c + d x} + \left (1 + \sqrt{3}\right ) \sqrt [3]{a d - b c}\right )^{2}}} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}} + \frac{2 \sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt{\frac{b^{\frac{2}{3}} \left (c + d x\right )^{\frac{2}{3}} - \sqrt [3]{b} \sqrt [3]{c + d x} \sqrt [3]{a d - b c} + \left (a d - b c\right )^{\frac{2}{3}}}{\left (\sqrt [3]{b} \sqrt [3]{c + d x} + \left (1 + \sqrt{3}\right ) \sqrt [3]{a d - b c}\right )^{2}}} \sqrt [3]{a d - b c} \left (\sqrt [3]{b} \sqrt [3]{c + d x} + \sqrt [3]{a d - b c}\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt [3]{b} \sqrt [3]{c + d x} - \left (-1 + \sqrt{3}\right ) \sqrt [3]{a d - b c}}{\sqrt [3]{b} \sqrt [3]{c + d x} + \left (1 + \sqrt{3}\right ) \sqrt [3]{a d - b c}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{b^{\frac{2}{3}} d \sqrt{\frac{\sqrt [3]{a d - b c} \left (\sqrt [3]{b} \sqrt [3]{c + d x} + \sqrt [3]{a d - b c}\right )}{\left (\sqrt [3]{b} \sqrt [3]{c + d x} + \left (1 + \sqrt{3}\right ) \sqrt [3]{a d - b c}\right )^{2}}} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x+a)**(1/2)/(d*x+c)**(1/3),x)
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Mathematica [C] time = 0.066143, size = 73, normalized size = 0.1 \[ \frac{3 (c+d x)^{2/3} \sqrt{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{2},\frac{2}{3};\frac{5}{3};\frac{b (c+d x)}{b c-a d}\right )}{2 d \sqrt{a+b x}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(Sqrt[a + b*x]*(c + d*x)^(1/3)),x]
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Maple [F] time = 0.082, size = 0, normalized size = 0. \[ \int{1{\frac{1}{\sqrt{bx+a}}}{\frac{1}{\sqrt [3]{dx+c}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b*x+a)^(1/2)/(d*x+c)^(1/3),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b x + a}{\left (d x + c\right )}^{\frac{1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(b*x + a)*(d*x + c)^(1/3)),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{b x + a}{\left (d x + c\right )}^{\frac{1}{3}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(b*x + a)*(d*x + c)^(1/3)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a + b x} \sqrt [3]{c + d x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*x+a)**(1/2)/(d*x+c)**(1/3),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b x + a}{\left (d x + c\right )}^{\frac{1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(b*x + a)*(d*x + c)^(1/3)),x, algorithm="giac")
[Out]