3.1424 \(\int (b d+2 c d x)^m \left (a+b x+c x^2\right )^p \, dx\)

Optimal. Leaf size=107 \[ -\frac{2 \left (\frac{1}{4} \left (4 a-\frac{b^2}{c}\right )+\frac{(b+2 c x)^2}{4 c}\right )^{p+1} (b d+2 c d x)^{m+1} \, _2F_1\left (1,\frac{1}{2} (m+2 p+3);\frac{m+3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{d (m+1) \left (b^2-4 a c\right )} \]

[Out]

(-2*(b*d + 2*c*d*x)^(1 + m)*((4*a - b^2/c)/4 + (b + 2*c*x)^2/(4*c))^(1 + p)*Hype
rgeometric2F1[1, (3 + m + 2*p)/2, (3 + m)/2, (b + 2*c*x)^2/(b^2 - 4*a*c)])/((b^2
- 4*a*c)*d*(1 + m))

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Rubi [A]  time = 0.175715, antiderivative size = 102, normalized size of antiderivative = 0.95, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{\left (a+b x+c x^2\right )^p \left (1-\frac{(b+2 c x)^2}{b^2-4 a c}\right )^{-p} (d (b+2 c x))^{m+1} \, _2F_1\left (\frac{m+1}{2},-p;\frac{m+3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{2 c d (m+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 32.7979, size = 95, normalized size = 0.89 \[ \frac{\left (b d + 2 c d x\right )^{m + 1} \left (\frac{\left (b + 2 c x\right )^{2}}{4 a c - b^{2}} + 1\right )^{- p} \left (a - \frac{b^{2}}{4 c} + \frac{\left (b + 2 c x\right )^{2}}{4 c}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{- \frac{\left (b + 2 c x\right )^{2}}{4 a c - b^{2}}} \right )}}{2 c d \left (m + 1\right )} \]

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)**p,x)

[Out]

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

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Mathematica [A]  time = 0.375695, size = 113, normalized size = 1.06 \[ \frac{2^{-2 p-1} (b+2 c x) (a+x (b+c x))^p \left (\frac{c (a+x (b+c x))}{4 a c-b^2}\right )^{-p} (d (b+2 c x))^m \, _2F_1\left (\frac{m}{2}+\frac{1}{2},-p;\frac{m}{2}+\frac{3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{c (m+1)} \]

Antiderivative was successfully verified.

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

[Out]

(2^(-1 - 2*p)*(b + 2*c*x)*(d*(b + 2*c*x))^m*(a + x*(b + c*x))^p*HypergeometricPF
Q[{1/2 + m/2, -p}, {3/2 + m/2}, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(c*(1 + m)*((c*(a
+ x*(b + c*x)))/(-b^2 + 4*a*c))^p)

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Maple [F]  time = 0.232, size = 0, normalized size = 0. \[ \int \left ( 2\,cdx+bd \right ) ^{m} \left ( c{x}^{2}+bx+a \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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