3.1097 \(\int x (A+B x) \left (a+b x+c x^2\right )^p \, dx\)

Optimal. Leaf size=211 \[ \frac{2^p \left (-\frac{-\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}\right )^{-p-1} \left (a+b x+c x^2\right )^{p+1} \left (2 a B c+A b c (2 p+3)+b^2 (-B) (p+2)\right ) \, _2F_1\left (-p,p+1;p+2;\frac{b+2 c x+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right )}{c^2 (p+1) (2 p+3) \sqrt{b^2-4 a c}}-\frac{\left (a+b x+c x^2\right )^{p+1} (-A c (2 p+3)+b B (p+2)-2 B c (p+1) x)}{2 c^2 (p+1) (2 p+3)} \]

[Out]

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

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Rubi [A]  time = 0.275378, antiderivative size = 211, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{2^p \left (-\frac{-\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}\right )^{-p-1} \left (a+b x+c x^2\right )^{p+1} \left (2 a B c+A b c (2 p+3)+b^2 (-B) (p+2)\right ) \, _2F_1\left (-p,p+1;p+2;\frac{b+2 c x+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right )}{c^2 (p+1) (2 p+3) \sqrt{b^2-4 a c}}-\frac{\left (a+b x+c x^2\right )^{p+1} (-A c (2 p+3)+b B (p+2)-2 B c (p+1) x)}{2 c^2 (p+1) (2 p+3)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 25.4114, size = 192, normalized size = 0.91 \[ \frac{\left (\frac{- \frac{b}{2} - c x + \frac{\sqrt{- 4 a c + b^{2}}}{2}}{\sqrt{- 4 a c + b^{2}}}\right )^{- p - 1} \left (a + b x + c x^{2}\right )^{p + 1} \left (A b c \left (2 p + 3\right ) + 2 B a c - B b^{2} \left (p + 2\right )\right ){{}_{2}F_{1}\left (\begin{matrix} - p, p + 1 \\ p + 2 \end{matrix}\middle |{\frac{\frac{b}{2} + c x + \frac{\sqrt{- 4 a c + b^{2}}}{2}}{\sqrt{- 4 a c + b^{2}}}} \right )}}{2 c^{2} \left (p + 1\right ) \left (2 p + 3\right ) \sqrt{- 4 a c + b^{2}}} + \frac{\left (a + b x + c x^{2}\right )^{p + 1} \left (A c \left (2 p + 3\right ) - B b \left (p + 2\right ) + 2 B c x \left (p + 1\right )\right )}{2 c^{2} \left (p + 1\right ) \left (2 p + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

((-b/2 - c*x + sqrt(-4*a*c + b**2)/2)/sqrt(-4*a*c + b**2))**(-p - 1)*(a + b*x +
c*x**2)**(p + 1)*(A*b*c*(2*p + 3) + 2*B*a*c - B*b**2*(p + 2))*hyper((-p, p + 1),
(p + 2,), (b/2 + c*x + sqrt(-4*a*c + b**2)/2)/sqrt(-4*a*c + b**2))/(2*c**2*(p +
1)*(2*p + 3)*sqrt(-4*a*c + b**2)) + (a + b*x + c*x**2)**(p + 1)*(A*c*(2*p + 3)
- B*b*(p + 2) + 2*B*c*x*(p + 1))/(2*c**2*(p + 1)*(2*p + 3))

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Mathematica [C]  time = 1.40687, size = 588, normalized size = 2.79 \[ -\frac{x^2 \left (\sqrt{b^2-4 a c}+b\right ) \left (-\sqrt{b^2-4 a c}+b+2 c x\right ) \left (x \left (b-\sqrt{b^2-4 a c}\right )+2 a\right ) (a+x (b+c x))^{p-1} \left (\frac{8 B x F_1\left (3;-p,-p;4;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right )}{p x \left (\left (\sqrt{b^2-4 a c}-b\right ) F_1\left (4;1-p,-p;5;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right )-\left (\sqrt{b^2-4 a c}+b\right ) F_1\left (4;-p,1-p;5;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right )\right )-8 a F_1\left (3;-p,-p;4;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right )}-\frac{9 A F_1\left (2;-p,-p;3;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right )}{6 a F_1\left (2;-p,-p;3;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right )+p x \left (\left (b-\sqrt{b^2-4 a c}\right ) F_1\left (3;1-p,-p;4;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right )+\left (\sqrt{b^2-4 a c}+b\right ) F_1\left (3;-p,1-p;4;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right )\right )}\right )}{24 c} \]

Warning: Unable to verify antiderivative.

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

[Out]

-((b + Sqrt[b^2 - 4*a*c])*x^2*(b - Sqrt[b^2 - 4*a*c] + 2*c*x)*(2*a + (b - Sqrt[b
^2 - 4*a*c])*x)*(a + x*(b + c*x))^(-1 + p)*((-9*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])])/(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]
)])) + (8*B*x*AppellF1[3, -p, -p, 4, (-2*c*x)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x)/(
-b + Sqrt[b^2 - 4*a*c])])/(-8*a*AppellF1[3, -p, -p, 4, (-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])*Appel
lF1[4, 1 - p, -p, 5, (-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[4, -p, 1 - p, 5, (-2*c*x)/(b + Sqrt[
b^2 - 4*a*c]), (2*c*x)/(-b + Sqrt[b^2 - 4*a*c])]))))/(24*c)

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Maple [F]  time = 0.15, size = 0, normalized size = 0. \[ \int x \left ( Bx+A \right ) \left ( c{x}^{2}+bx+a \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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