3.1416 \(\int \frac{(b d+2 c d x)^m}{\left (a+b x+c x^2\right )^2} \, dx\)

Optimal. Leaf size=68 \[ \frac{8 c (d (b+2 c x))^{m+1} \, _2F_1\left (2,\frac{m+1}{2};\frac{m+3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{d (m+1) \left (b^2-4 a c\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.116313, antiderivative size = 68, 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 \[ \frac{8 c (d (b+2 c x))^{m+1} \, _2F_1\left (2,\frac{m+1}{2};\frac{m+3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{d (m+1) \left (b^2-4 a c\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(b*d + 2*c*d*x)^m/(a + b*x + c*x^2)^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 25.3134, size = 61, normalized size = 0.9 \[ \frac{8 c \left (b d + 2 c d x\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 2, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{- \frac{\left (b + 2 c x\right )^{2}}{4 a c - b^{2}}} \right )}}{d \left (m + 1\right ) \left (- 4 a c + b^{2}\right )^{2}} \]

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)**2,x)

[Out]

8*c*(b*d + 2*c*d*x)**(m + 1)*hyper((2, 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)**2)

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Mathematica [A]  time = 0.0840909, size = 0, normalized size = 0. \[ \int \frac{(b d+2 c d x)^m}{\left (a+b x+c x^2\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[(b*d + 2*c*d*x)^m/(a + b*x + c*x^2)^2,x]

[Out]

Integrate[(b*d + 2*c*d*x)^m/(a + b*x + c*x^2)^2, x]

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Maple [F]  time = 0.168, size = 0, normalized size = 0. \[ \int{\frac{ \left ( 2\,cdx+bd \right ) ^{m}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}\, dx \]

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)^2,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (2 \, c d x + b d\right )}^{m}}{{\left (c x^{2} + b x + a\right )}^{2}}\,{d x} \]

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)^2,x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (2 \, c d x + b d\right )}^{m}}{c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}}, x\right ) \]

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)^2,x, algorithm="fricas")

[Out]

integral((2*c*d*x + b*d)^m/(c^2*x^4 + 2*b*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 +
a^2), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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)**2,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (2 \, c d x + b d\right )}^{m}}{{\left (c x^{2} + b x + a\right )}^{2}}\,{d x} \]

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)^2,x, algorithm="giac")

[Out]

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