3.1415 \(\int \frac{(b d+2 c d x)^m}{a+b x+c x^2} \, dx\)

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]

(-2*(d*(b + 2*c*x))^(1 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, (b + 2*c*
x)^2/(b^2 - 4*a*c)])/((b^2 - 4*a*c)*d*(1 + m))

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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]

(-2*(d*(b + 2*c*x))^(1 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, (b + 2*c*
x)^2/(b^2 - 4*a*c)])/((b^2 - 4*a*c)*d*(1 + m))

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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]

-2*(b*d + 2*c*d*x)**(m + 1)*hyper((1, m/2 + 1/2), (m/2 + 3/2,), -(b + 2*c*x)**2/
(4*a*c - b**2))/(d*(m + 1)*(-4*a*c + b**2))

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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]

((d*(b + 2*c*x))^m*(Hypergeometric2F1[-m, -m, 1 - m, Sqrt[b^2 - 4*a*c]/(-b + Sqr
t[b^2 - 4*a*c] - 2*c*x)]/((b + 2*c*x)/(b - Sqrt[b^2 - 4*a*c] + 2*c*x))^m - Hyper
geometric2F1[-m, -m, 1 - m, Sqrt[b^2 - 4*a*c]/(b + Sqrt[b^2 - 4*a*c] + 2*c*x)]/(
(b + 2*c*x)/(b + Sqrt[b^2 - 4*a*c] + 2*c*x))^m))/(Sqrt[b^2 - 4*a*c]*m)

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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]

int((2*c*d*x+b*d)^m/(c*x^2+b*x+a),x)

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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]

integrate((2*c*d*x + b*d)^m/(c*x^2 + b*x + a), x)

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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]

integral((2*c*d*x + b*d)^m/(c*x^2 + b*x + a), x)

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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]

Integral((d*(b + 2*c*x))**m/(a + b*x + c*x**2), x)

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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]

integrate((2*c*d*x + b*d)^m/(c*x^2 + b*x + a), x)