3.2555 \(\int (d+e x) \left (a+b x+c x^2\right )^p \, dx\)

Optimal. Leaf size=160 \[ \frac{e \left (a+b x+c x^2\right )^{p+1}}{2 c (p+1)}-\frac{2^p (2 c d-b e) \left (-\frac{-\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}\right )^{-p-1} \left (a+b x+c x^2\right )^{p+1} \, _2F_1\left (-p,p+1;p+2;\frac{b+2 c x+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right )}{c (p+1) \sqrt{b^2-4 a c}} \]

[Out]

(e*(a + b*x + c*x^2)^(1 + p))/(2*c*(1 + p)) - (2^p*(2*c*d - b*e)*(-((b - Sqrt[b^
2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]))^(-1 - p)*(a + b*x + c*x^2)^(1 + p)*Hyper
geometric2F1[-p, 1 + p, 2 + p, (b + Sqrt[b^2 - 4*a*c] + 2*c*x)/(2*Sqrt[b^2 - 4*a
*c])])/(c*Sqrt[b^2 - 4*a*c]*(1 + p))

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Rubi [A]  time = 0.141904, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{e \left (a+b x+c x^2\right )^{p+1}}{2 c (p+1)}-\frac{2^p (2 c d-b e) \left (-\frac{-\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}\right )^{-p-1} \left (a+b x+c x^2\right )^{p+1} \, _2F_1\left (-p,p+1;p+2;\frac{b+2 c x+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right )}{c (p+1) \sqrt{b^2-4 a c}} \]

Antiderivative was successfully verified.

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

[Out]

(e*(a + b*x + c*x^2)^(1 + p))/(2*c*(1 + p)) - (2^p*(2*c*d - b*e)*(-((b - Sqrt[b^
2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]))^(-1 - p)*(a + b*x + c*x^2)^(1 + p)*Hyper
geometric2F1[-p, 1 + p, 2 + p, (b + Sqrt[b^2 - 4*a*c] + 2*c*x)/(2*Sqrt[b^2 - 4*a
*c])])/(c*Sqrt[b^2 - 4*a*c]*(1 + p))

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Rubi in Sympy [A]  time = 14.5307, size = 138, normalized size = 0.86 \[ \frac{e \left (a + b x + c x^{2}\right )^{p + 1}}{2 c \left (p + 1\right )} + \frac{\left (\frac{- \frac{b}{2} - c x + \frac{\sqrt{- 4 a c + b^{2}}}{2}}{\sqrt{- 4 a c + b^{2}}}\right )^{- p - 1} \left (b e - 2 c d\right ) \left (a + b x + c x^{2}\right )^{p + 1}{{}_{2}F_{1}\left (\begin{matrix} - p, p + 1 \\ p + 2 \end{matrix}\middle |{\frac{\frac{b}{2} + c x + \frac{\sqrt{- 4 a c + b^{2}}}{2}}{\sqrt{- 4 a c + b^{2}}}} \right )}}{2 c \left (p + 1\right ) \sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e*(a + b*x + c*x**2)**(p + 1)/(2*c*(p + 1)) + ((-b/2 - c*x + sqrt(-4*a*c + b**2)
/2)/sqrt(-4*a*c + b**2))**(-p - 1)*(b*e - 2*c*d)*(a + b*x + c*x**2)**(p + 1)*hyp
er((-p, p + 1), (p + 2,), (b/2 + c*x + sqrt(-4*a*c + b**2)/2)/sqrt(-4*a*c + b**2
))/(2*c*(p + 1)*sqrt(-4*a*c + b**2))

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Mathematica [C]  time = 1.86782, size = 476, normalized size = 2.98 \[ \frac{1}{4} \left (-\sqrt{b^2-4 a c}+b+2 c x\right ) (a+x (b+c x))^p \left (\frac{3 e x^2 \left (\sqrt{b^2-4 a c}+b\right ) \left (x \left (b-\sqrt{b^2-4 a c}\right )+2 a\right )^2 F_1\left (2;-p,-p;3;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right )}{\left (\sqrt{b^2-4 a c}-b\right ) \left (\sqrt{b^2-4 a c}+b+2 c x\right ) (a+x (b+c x)) \left (p x \left (\left (\sqrt{b^2-4 a c}-b\right ) F_1\left (3;1-p,-p;4;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right )-\left (\sqrt{b^2-4 a c}+b\right ) F_1\left (3;-p,1-p;4;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right )\right )-6 a F_1\left (2;-p,-p;3;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right )\right )}+\frac{d 2^{p+1} \left (\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}\right )^{-p} \, _2F_1\left (-p,p+1;p+2;-\frac{b}{2 \sqrt{b^2-4 a c}}-\frac{c x}{\sqrt{b^2-4 a c}}+\frac{1}{2}\right )}{c p+c}\right ) \]

Warning: Unable to verify antiderivative.

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

[Out]

((b - Sqrt[b^2 - 4*a*c] + 2*c*x)*(a + x*(b + c*x))^p*((3*(b + Sqrt[b^2 - 4*a*c])
*e*x^2*(2*a + (b - Sqrt[b^2 - 4*a*c])*x)^2*AppellF1[2, -p, -p, 3, (-2*c*x)/(b +
Sqrt[b^2 - 4*a*c]), (2*c*x)/(-b + Sqrt[b^2 - 4*a*c])])/((-b + Sqrt[b^2 - 4*a*c])
*(b + Sqrt[b^2 - 4*a*c] + 2*c*x)*(a + x*(b + c*x))*(-6*a*AppellF1[2, -p, -p, 3,
(-2*c*x)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x)/(-b + Sqrt[b^2 - 4*a*c])] + p*x*((-b +
Sqrt[b^2 - 4*a*c])*AppellF1[3, 1 - p, -p, 4, (-2*c*x)/(b + Sqrt[b^2 - 4*a*c]),
(2*c*x)/(-b + Sqrt[b^2 - 4*a*c])] - (b + Sqrt[b^2 - 4*a*c])*AppellF1[3, -p, 1 -
p, 4, (-2*c*x)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x)/(-b + Sqrt[b^2 - 4*a*c])]))) + (
2^(1 + p)*d*Hypergeometric2F1[-p, 1 + p, 2 + p, 1/2 - b/(2*Sqrt[b^2 - 4*a*c]) -
(c*x)/Sqrt[b^2 - 4*a*c]])/((c + c*p)*((b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 -
4*a*c])^p)))/4

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Maple [F]  time = 0.126, size = 0, normalized size = 0. \[ \int \left ( ex+d \right ) \left ( c{x}^{2}+bx+a \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)*(c*x^2+b*x+a)^p,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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