Optimal. Leaf size=160 \[ \frac{(d x)^{m+1} \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1} F_1\left (\frac{m+1}{n};\frac{1}{2},\frac{1}{2};\frac{m+n+1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{d (m+1) \sqrt{a+b x^n+c x^{2 n}}} \]
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Rubi [A] time = 0.469492, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{(d x)^{m+1} \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1} F_1\left (\frac{m+1}{n};\frac{1}{2},\frac{1}{2};\frac{m+n+1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{d (m+1) \sqrt{a+b x^n+c x^{2 n}}} \]
Antiderivative was successfully verified.
[In] Int[(d*x)^m/Sqrt[a + b*x^n + c*x^(2*n)],x]
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Rubi in Sympy [A] time = 36.6309, size = 136, normalized size = 0.85 \[ \frac{\left (d x\right )^{m + 1} \sqrt{a + b x^{n} + c x^{2 n}} \operatorname{appellf_{1}}{\left (\frac{m + 1}{n},\frac{1}{2},\frac{1}{2},\frac{m + n + 1}{n},- \frac{2 c x^{n}}{b - \sqrt{- 4 a c + b^{2}}},- \frac{2 c x^{n}}{b + \sqrt{- 4 a c + b^{2}}} \right )}}{a d \left (m + 1\right ) \sqrt{\frac{2 c x^{n}}{b - \sqrt{- 4 a c + b^{2}}} + 1} \sqrt{\frac{2 c x^{n}}{b + \sqrt{- 4 a c + b^{2}}} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x)**m/(a+b*x**n+c*x**(2*n))**(1/2),x)
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Mathematica [B] time = 0.502606, size = 440, normalized size = 2.75 \[ \frac{4 a^2 x (m+n+1) (d x)^m \left (-\sqrt{b^2-4 a c}+b+2 c x^n\right ) \left (\sqrt{b^2-4 a c}+b+2 c x^n\right ) F_1\left (\frac{m+1}{n};\frac{1}{2},\frac{1}{2};\frac{m+n+1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )}{(m+1) \left (b-\sqrt{b^2-4 a c}\right ) \left (\sqrt{b^2-4 a c}+b\right ) \left (a+x^n \left (b+c x^n\right )\right )^{3/2} \left (4 a (m+n+1) F_1\left (\frac{m+1}{n};\frac{1}{2},\frac{1}{2};\frac{m+n+1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )-n x^n \left (\left (\sqrt{b^2-4 a c}+b\right ) F_1\left (\frac{m+n+1}{n};\frac{1}{2},\frac{3}{2};\frac{m+2 n+1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )+\left (b-\sqrt{b^2-4 a c}\right ) F_1\left (\frac{m+n+1}{n};\frac{3}{2},\frac{1}{2};\frac{m+2 n+1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )\right )\right )} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(d*x)^m/Sqrt[a + b*x^n + c*x^(2*n)],x]
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Maple [F] time = 0.047, size = 0, normalized size = 0. \[ \int{ \left ( dx \right ) ^{m}{\frac{1}{\sqrt{a+b{x}^{n}+c{x}^{2\,n}}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x)^m/(a+b*x^n+c*x^(2*n))^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d x\right )^{m}}{\sqrt{c x^{2 \, n} + b x^{n} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x)^m/sqrt(c*x^(2*n) + b*x^n + a),x, algorithm="maxima")
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x)^m/sqrt(c*x^(2*n) + b*x^n + a),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d x\right )^{m}}{\sqrt{a + b x^{n} + c x^{2 n}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x)**m/(a+b*x**n+c*x**(2*n))**(1/2),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d x\right )^{m}}{\sqrt{c x^{2 \, n} + b x^{n} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x)^m/sqrt(c*x^(2*n) + b*x^n + a),x, algorithm="giac")
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