3.1426 \(\int (b d+2 c d x)^4 \left (a+b x+c x^2\right )^p \, dx\)

Optimal. Leaf size=90 \[ -\frac{2 d^4 (b+2 c x)^5 \left (\frac{1}{4} \left (4 a-\frac{b^2}{c}\right )+\frac{(b+2 c x)^2}{4 c}\right )^{p+1} \, _2F_1\left (1,p+\frac{7}{2};\frac{7}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{5 \left (b^2-4 a c\right )} \]

[Out]

(-2*d^4*(b + 2*c*x)^5*((4*a - b^2/c)/4 + (b + 2*c*x)^2/(4*c))^(1 + p)*Hypergeome
tric2F1[1, 7/2 + p, 7/2, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(5*(b^2 - 4*a*c))

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Rubi [A]  time = 0.174678, 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 \[ \frac{d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^p \left (1-\frac{(b+2 c x)^2}{b^2-4 a c}\right )^{-p} \, _2F_1\left (\frac{5}{2},-p;\frac{7}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{10 c} \]

Antiderivative was successfully verified.

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

[Out]

(d^4*(b + 2*c*x)^5*(a + b*x + c*x^2)^p*Hypergeometric2F1[5/2, -p, 7/2, (b + 2*c*
x)^2/(b^2 - 4*a*c)])/(10*c*(1 - (b + 2*c*x)^2/(b^2 - 4*a*c))^p)

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Rubi in Sympy [A]  time = 32.2879, size = 82, normalized size = 0.91 \[ \frac{d^{4} \left (b + 2 c x\right )^{5} \left (\frac{\left (b + 2 c x\right )^{2}}{4 a c - b^{2}} + 1\right )^{- p} \left (a - \frac{b^{2}}{4 c} + \frac{\left (b + 2 c x\right )^{2}}{4 c}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{- \frac{\left (b + 2 c x\right )^{2}}{4 a c - b^{2}}} \right )}}{10 c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**4*(b + 2*c*x)**5*((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, 5/2), (7/2,), -(b + 2*c*x)**2/(4*a*c - b**2)
)/(10*c)

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Mathematica [A]  time = 0.0936216, size = 92, normalized size = 1.02 \[ \frac{d^4 2^{-2 p-1} (b+2 c x)^5 (a+x (b+c x))^p \left (\frac{c (a+x (b+c x))}{4 a c-b^2}\right )^{-p} \, _2F_1\left (\frac{5}{2},-p;\frac{7}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{5 c} \]

Antiderivative was successfully verified.

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

[Out]

(2^(-1 - 2*p)*d^4*(b + 2*c*x)^5*(a + x*(b + c*x))^p*Hypergeometric2F1[5/2, -p, 7
/2, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(5*c*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^p)

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Maple [F]  time = 0.166, size = 0, normalized size = 0. \[ \int \left ( 2\,cdx+bd \right ) ^{4} \left ( c{x}^{2}+bx+a \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^4*(c*x^2 + b*x + a)^p,x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^4*(c*x^2 + b*x + a)^p,x, algorithm="fricas")

[Out]

integral((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)*(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)**4*(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. \[ \int{\left (2 \, c d x + b d\right )}^{4}{\left (c x^{2} + b x + a\right )}^{p}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^4*(c*x^2 + b*x + a)^p,x, algorithm="giac")

[Out]

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