Optimal. Leaf size=340 \[ \frac{(d+e x)^2 \sqrt{\frac{2 c (d+e x)^3}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c (d+e x)^3}{\sqrt{b^2-4 a c}+b}+1} F_1\left (\frac{2}{3};\frac{1}{2},\frac{1}{2};\frac{5}{3};-\frac{2 c (d+e x)^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c (d+e x)^3}{b+\sqrt{b^2-4 a c}}\right )}{2 e^2 \sqrt{a+b (d+e x)^3+c (d+e x)^6}}-\frac{d (d+e x) \sqrt{\frac{2 c (d+e x)^3}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c (d+e x)^3}{\sqrt{b^2-4 a c}+b}+1} F_1\left (\frac{1}{3};\frac{1}{2},\frac{1}{2};\frac{4}{3};-\frac{2 c (d+e x)^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c (d+e x)^3}{b+\sqrt{b^2-4 a c}}\right )}{e^2 \sqrt{a+b (d+e x)^3+c (d+e x)^6}} \]
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Rubi [A] time = 1.5102, antiderivative size = 340, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{(d+e x)^2 \sqrt{\frac{2 c (d+e x)^3}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c (d+e x)^3}{\sqrt{b^2-4 a c}+b}+1} F_1\left (\frac{2}{3};\frac{1}{2},\frac{1}{2};\frac{5}{3};-\frac{2 c (d+e x)^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c (d+e x)^3}{b+\sqrt{b^2-4 a c}}\right )}{2 e^2 \sqrt{a+b (d+e x)^3+c (d+e x)^6}}-\frac{d (d+e x) \sqrt{\frac{2 c (d+e x)^3}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c (d+e x)^3}{\sqrt{b^2-4 a c}+b}+1} F_1\left (\frac{1}{3};\frac{1}{2},\frac{1}{2};\frac{4}{3};-\frac{2 c (d+e x)^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c (d+e x)^3}{b+\sqrt{b^2-4 a c}}\right )}{e^2 \sqrt{a+b (d+e x)^3+c (d+e x)^6}} \]
Antiderivative was successfully verified.
[In] Int[x/Sqrt[a + b*(d + e*x)^3 + c*(d + e*x)^6],x]
[Out]
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Rubi in Sympy [A] time = 85.5295, size = 299, normalized size = 0.88 \[ - \frac{d \left (d + e x\right ) \sqrt{a + b \left (d + e x\right )^{3} + c \left (d + e x\right )^{6}} \operatorname{appellf_{1}}{\left (\frac{1}{3},\frac{1}{2},\frac{1}{2},\frac{4}{3},- \frac{2 c \left (d + e x\right )^{3}}{b - \sqrt{- 4 a c + b^{2}}},- \frac{2 c \left (d + e x\right )^{3}}{b + \sqrt{- 4 a c + b^{2}}} \right )}}{a e^{2} \sqrt{\frac{2 c \left (d + e x\right )^{3}}{b - \sqrt{- 4 a c + b^{2}}} + 1} \sqrt{\frac{2 c \left (d + e x\right )^{3}}{b + \sqrt{- 4 a c + b^{2}}} + 1}} + \frac{\left (d + e x\right )^{2} \sqrt{a + b \left (d + e x\right )^{3} + c \left (d + e x\right )^{6}} \operatorname{appellf_{1}}{\left (\frac{2}{3},\frac{1}{2},\frac{1}{2},\frac{5}{3},- \frac{2 c \left (d + e x\right )^{3}}{b - \sqrt{- 4 a c + b^{2}}},- \frac{2 c \left (d + e x\right )^{3}}{b + \sqrt{- 4 a c + b^{2}}} \right )}}{2 a e^{2} \sqrt{\frac{2 c \left (d + e x\right )^{3}}{b - \sqrt{- 4 a c + b^{2}}} + 1} \sqrt{\frac{2 c \left (d + e x\right )^{3}}{b + \sqrt{- 4 a c + b^{2}}} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/(a+b*(e*x+d)**3+c*(e*x+d)**6)**(1/2),x)
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Mathematica [A] time = 20.6174, size = 0, normalized size = 0. \[ \int \frac{x}{\sqrt{a+b (d+e x)^3+c (d+e x)^6}} \, dx \]
Verification is Not applicable to the result.
[In] Integrate[x/Sqrt[a + b*(d + e*x)^3 + c*(d + e*x)^6],x]
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Maple [F] time = 0.037, size = 0, normalized size = 0. \[ \int{x{\frac{1}{\sqrt{a+b \left ( ex+d \right ) ^{3}+c \left ( ex+d \right ) ^{6}}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/(a+b*(e*x+d)^3+c*(e*x+d)^6)^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\sqrt{{\left (e x + d\right )}^{6} c +{\left (e x + d\right )}^{3} b + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/sqrt((e*x + d)^6*c + (e*x + d)^3*b + a),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x}{\sqrt{c e^{6} x^{6} + 6 \, c d e^{5} x^{5} + 15 \, c d^{2} e^{4} x^{4} + c d^{6} +{\left (20 \, c d^{3} + b\right )} e^{3} x^{3} + 3 \,{\left (5 \, c d^{4} + b d\right )} e^{2} x^{2} + b d^{3} + 3 \,{\left (2 \, c d^{5} + b d^{2}\right )} e x + a}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/sqrt((e*x + d)^6*c + (e*x + d)^3*b + a),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\sqrt{a + b d^{3} + 3 b d^{2} e x + 3 b d e^{2} x^{2} + b e^{3} x^{3} + c d^{6} + 6 c d^{5} e x + 15 c d^{4} e^{2} x^{2} + 20 c d^{3} e^{3} x^{3} + 15 c d^{2} e^{4} x^{4} + 6 c d e^{5} x^{5} + c e^{6} x^{6}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(a+b*(e*x+d)**3+c*(e*x+d)**6)**(1/2),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\sqrt{{\left (e x + d\right )}^{6} c +{\left (e x + d\right )}^{3} b + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/sqrt((e*x + d)^6*c + (e*x + d)^3*b + a),x, algorithm="giac")
[Out]