Optimal. Leaf size=67 \[ -\frac{2 (d (b+2 c x))^{m+1} \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{d (m+1) \left (b^2-4 a c\right )} \]
[Out]
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Rubi [A] time = 0.108881, antiderivative size = 67, 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 \[ -\frac{2 (d (b+2 c x))^{m+1} \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{d (m+1) \left (b^2-4 a c\right )} \]
Antiderivative was successfully verified.
[In] Int[(b*d + 2*c*d*x)^m/(a + b*x + c*x^2),x]
[Out]
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Rubi in Sympy [A] time = 25.8711, size = 60, normalized size = 0.9 \[ - \frac{2 \left (b d + 2 c d x\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{- \frac{\left (b + 2 c x\right )^{2}}{4 a c - b^{2}}} \right )}}{d \left (m + 1\right ) \left (- 4 a c + b^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2*c*d*x+b*d)**m/(c*x**2+b*x+a),x)
[Out]
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Mathematica [B] time = 0.385535, size = 186, normalized size = 2.78 \[ \frac{(d (b+2 c x))^m \left (\left (\frac{b+2 c x}{-\sqrt{b^2-4 a c}+b+2 c x}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{\sqrt{b^2-4 a c}}{-b-2 c x+\sqrt{b^2-4 a c}}\right )-\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}+b+2 c x}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{\sqrt{b^2-4 a c}}{b+2 c x+\sqrt{b^2-4 a c}}\right )\right )}{m \sqrt{b^2-4 a c}} \]
Antiderivative was successfully verified.
[In] Integrate[(b*d + 2*c*d*x)^m/(a + b*x + c*x^2),x]
[Out]
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Maple [F] time = 0.149, size = 0, normalized size = 0. \[ \int{\frac{ \left ( 2\,cdx+bd \right ) ^{m}}{c{x}^{2}+bx+a}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2*c*d*x+b*d)^m/(c*x^2+b*x+a),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (2 \, c d x + b d\right )}^{m}}{c x^{2} + b x + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x + b*d)^m/(c*x^2 + b*x + a),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (2 \, c d x + b d\right )}^{m}}{c x^{2} + b x + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x + b*d)^m/(c*x^2 + b*x + a),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d \left (b + 2 c x\right )\right )^{m}}{a + b x + c x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x+b*d)**m/(c*x**2+b*x+a),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (2 \, c d x + b d\right )}^{m}}{c x^{2} + b x + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x + b*d)^m/(c*x^2 + b*x + a),x, algorithm="giac")
[Out]