3.1428 \(\int (b d+2 c d x)^2 \left (a+b x+c x^2\right )^p \, dx\)

Optimal. Leaf size=90 \[ -\frac{2 d^2 (b+2 c x)^3 \left (\frac{1}{4} \left (4 a-\frac{b^2}{c}\right )+\frac{(b+2 c x)^2}{4 c}\right )^{p+1} \, _2F_1\left (1,p+\frac{5}{2};\frac{5}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{3 \left (b^2-4 a c\right )} \]

[Out]

(-2*d^2*(b + 2*c*x)^3*((4*a - b^2/c)/4 + (b + 2*c*x)^2/(4*c))^(1 + p)*Hypergeome
tric2F1[1, 5/2 + p, 5/2, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(3*(b^2 - 4*a*c))

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Rubi [A]  time = 0.165893, 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 \[ \frac{d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^p \left (1-\frac{(b+2 c x)^2}{b^2-4 a c}\right )^{-p} \, _2F_1\left (\frac{3}{2},-p;\frac{5}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{6 c} \]

Antiderivative was successfully verified.

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

[Out]

(d^2*(b + 2*c*x)^3*(a + b*x + c*x^2)^p*Hypergeometric2F1[3/2, -p, 5/2, (b + 2*c*
x)^2/(b^2 - 4*a*c)])/(6*c*(1 - (b + 2*c*x)^2/(b^2 - 4*a*c))^p)

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Rubi in Sympy [A]  time = 33.0523, size = 82, normalized size = 0.91 \[ \frac{d^{2} \left (b + 2 c x\right )^{3} \left (\frac{\left (b + 2 c x\right )^{2}}{4 a c - b^{2}} + 1\right )^{- p} \left (a - \frac{b^{2}}{4 c} + \frac{\left (b + 2 c x\right )^{2}}{4 c}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{- \frac{\left (b + 2 c x\right )^{2}}{4 a c - b^{2}}} \right )}}{6 c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**2*(b + 2*c*x)**3*((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), (5/2,), -(b + 2*c*x)**2/(4*a*c - b**2)
)/(6*c)

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Mathematica [A]  time = 0.0728451, size = 92, normalized size = 1.02 \[ \frac{d^2 2^{-2 p-1} (b+2 c x)^3 (a+x (b+c x))^p \left (\frac{c (a+x (b+c x))}{4 a c-b^2}\right )^{-p} \, _2F_1\left (\frac{3}{2},-p;\frac{5}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{3 c} \]

Antiderivative was successfully verified.

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

[Out]

(2^(-1 - 2*p)*d^2*(b + 2*c*x)^3*(a + x*(b + c*x))^p*Hypergeometric2F1[3/2, -p, 5
/2, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(3*c*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^p)

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Maple [F]  time = 0.145, size = 0, normalized size = 0. \[ \int \left ( 2\,cdx+bd \right ) ^{2} \left ( c{x}^{2}+bx+a \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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