3.1433 \(\int \frac{\left (a+b x+c x^2\right )^p}{(b d+2 c d x)^4} \, dx\)

Optimal. Leaf size=90 \[ \frac{2 \left (\frac{1}{4} \left (4 a-\frac{b^2}{c}\right )+\frac{(b+2 c x)^2}{4 c}\right )^{p+1} \, _2F_1\left (1,p-\frac{1}{2};-\frac{1}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{3 d^4 \left (b^2-4 a c\right ) (b+2 c x)^3} \]

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

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Rubi [A]  time = 0.159677, antiderivative size = 85, normalized size of antiderivative = 0.94, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{\left (a+b x+c x^2\right )^p \left (1-\frac{(b+2 c x)^2}{b^2-4 a c}\right )^{-p} \, _2F_1\left (-\frac{3}{2},-p;-\frac{1}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{6 c d^4 (b+2 c x)^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)^p/(b*d + 2*c*d*x)^4,x]

[Out]

-((a + b*x + c*x^2)^p*Hypergeometric2F1[-3/2, -p, -1/2, (b + 2*c*x)^2/(b^2 - 4*a
*c)])/(6*c*d^4*(b + 2*c*x)^3*(1 - (b + 2*c*x)^2/(b^2 - 4*a*c))^p)

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Rubi in Sympy [A]  time = 30.9453, size = 87, normalized size = 0.97 \[ - \frac{\left (\frac{\left (b + 2 c x\right )^{2}}{4 a c - b^{2}} + 1\right )^{- p} \left (a - \frac{b^{2}}{4 c} + \frac{\left (b + 2 c x\right )^{2}}{4 c}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, - \frac{3}{2} \\ - \frac{1}{2} \end{matrix}\middle |{- \frac{\left (b + 2 c x\right )^{2}}{4 a c - b^{2}}} \right )}}{6 c d^{4} \left (b + 2 c x\right )^{3}} \]

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)**4,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, -3/2), (-1/2,), -(b + 2*c*x)**2/(4*a*c - b**2))/(6*c*d**4*(b +
2*c*x)**3)

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Mathematica [A]  time = 0.0818152, size = 92, normalized size = 1.02 \[ -\frac{2^{-2 p-1} (a+x (b+c x))^p \left (\frac{c (a+x (b+c x))}{4 a c-b^2}\right )^{-p} \, _2F_1\left (-\frac{3}{2},-p;-\frac{1}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{3 c d^4 (b+2 c x)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)^p/(b*d + 2*c*d*x)^4,x]

[Out]

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

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Maple [F]  time = 0.21, size = 0, normalized size = 0. \[ \int{\frac{ \left ( c{x}^{2}+bx+a \right ) ^{p}}{ \left ( 2\,cdx+bd \right ) ^{4}}}\, dx \]

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)^4,x)

[Out]

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

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

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)^4,x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (c x^{2} + b x + a\right )}^{p}}{16 \, c^{4} d^{4} x^{4} + 32 \, b c^{3} d^{4} x^{3} + 24 \, b^{2} c^{2} d^{4} x^{2} + 8 \, b^{3} c d^{4} x + b^{4} d^{4}}, x\right ) \]

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)^4,x, algorithm="fricas")

[Out]

integral((c*x^2 + b*x + a)^p/(16*c^4*d^4*x^4 + 32*b*c^3*d^4*x^3 + 24*b^2*c^2*d^4
*x^2 + 8*b^3*c*d^4*x + b^4*d^4), 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)**4,x)

[Out]

Timed out

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

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)^4,x, algorithm="giac")

[Out]

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