3.1431 \(\int \frac{\left (a+b x+c x^2\right )^p}{(b d+2 c d x)^2} \, dx\)

Optimal. Leaf size=88 \[ \frac{2 \left (\frac{1}{4} \left (4 a-\frac{b^2}{c}\right )+\frac{(b+2 c x)^2}{4 c}\right )^{p+1} \, _2F_1\left (1,p+\frac{1}{2};\frac{1}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{d^2 \left (b^2-4 a c\right ) (b+2 c x)} \]

[Out]

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

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Rubi [A]  time = 0.162256, antiderivative size = 85, normalized size of antiderivative = 0.97, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{\left (a+b x+c x^2\right )^p \left (1-\frac{(b+2 c x)^2}{b^2-4 a c}\right )^{-p} \, _2F_1\left (-\frac{1}{2},-p;\frac{1}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{2 c d^2 (b+2 c x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 30.9914, size = 83, normalized size = 0.94 \[ - \frac{\left (\frac{\left (b + 2 c x\right )^{2}}{4 a c - b^{2}} + 1\right )^{- p} \left (a - \frac{b^{2}}{4 c} + \frac{\left (b + 2 c x\right )^{2}}{4 c}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, - \frac{1}{2} \\ \frac{1}{2} \end{matrix}\middle |{- \frac{\left (b + 2 c x\right )^{2}}{4 a c - b^{2}}} \right )}}{2 c d^{2} \left (b + 2 c x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x+a)**p/(2*c*d*x+b*d)**2,x)

[Out]

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

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Mathematica [A]  time = 0.0661984, size = 90, normalized size = 1.02 \[ -\frac{2^{-2 p-1} (a+x (b+c x))^p \left (\frac{c (a+x (b+c x))}{4 a c-b^2}\right )^{-p} \, _2F_1\left (-\frac{1}{2},-p;\frac{1}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{c d^2 (b+2 c x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.152, size = 0, normalized size = 0. \[ \int{\frac{ \left ( c{x}^{2}+bx+a \right ) ^{p}}{ \left ( 2\,cdx+bd \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^p/(2*c*d*x + b*d)^2,x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^p/(2*c*d*x + b*d)^2,x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^p/(2*c*d*x + b*d)^2,x, algorithm="giac")

[Out]

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