3.2551 \(\int (d x)^m \left (a+b x+c x^2\right )^p \, dx\)

Optimal. Leaf size=137 \[ \frac{(d x)^{m+1} \left (\frac{2 c x}{b-\sqrt{b^2-4 a c}}+1\right )^{-p} \left (\frac{2 c x}{\sqrt{b^2-4 a c}+b}+1\right )^{-p} \left (a+b x+c x^2\right )^p F_1\left (m+1;-p,-p;m+2;-\frac{2 c x}{b-\sqrt{b^2-4 a c}},-\frac{2 c x}{b+\sqrt{b^2-4 a c}}\right )}{d (m+1)} \]

[Out]

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

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Rubi [A]  time = 0.60048, antiderivative size = 137, 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 \[ \frac{(d x)^{m+1} \left (\frac{2 c x}{b-\sqrt{b^2-4 a c}}+1\right )^{-p} \left (\frac{2 c x}{\sqrt{b^2-4 a c}+b}+1\right )^{-p} \left (a+b x+c x^2\right )^p F_1\left (m+1;-p,-p;m+2;-\frac{2 c x}{b-\sqrt{b^2-4 a c}},-\frac{2 c x}{b+\sqrt{b^2-4 a c}}\right )}{d (m+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 40.2935, size = 114, normalized size = 0.83 \[ \frac{\left (d x\right )^{m + 1} \left (\frac{2 c x}{b - \sqrt{- 4 a c + b^{2}}} + 1\right )^{- p} \left (\frac{2 c x}{b + \sqrt{- 4 a c + b^{2}}} + 1\right )^{- p} \left (a + b x + c x^{2}\right )^{p} \operatorname{appellf_{1}}{\left (m + 1,- p,- p,m + 2,- \frac{2 c x}{b - \sqrt{- 4 a c + b^{2}}},- \frac{2 c x}{b + \sqrt{- 4 a c + b^{2}}} \right )}}{d \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(d*x)**(m + 1)*(2*c*x/(b - sqrt(-4*a*c + b**2)) + 1)**(-p)*(2*c*x/(b + sqrt(-4*a
*c + b**2)) + 1)**(-p)*(a + b*x + c*x**2)**p*appellf1(m + 1, -p, -p, m + 2, -2*c
*x/(b - sqrt(-4*a*c + b**2)), -2*c*x/(b + sqrt(-4*a*c + b**2)))/(d*(m + 1))

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Mathematica [B]  time = 4.59026, size = 439, normalized size = 3.2 \[ \frac{c (m+2) 2^{-p-1} x \left (\sqrt{b^2-4 a c}+b\right ) (d x)^m \left (x \left (b-\sqrt{b^2-4 a c}\right )+2 a\right )^2 \left (\frac{b-\sqrt{b^2-4 a c}}{2 c}+x\right )^{-p} \left (\frac{-\sqrt{b^2-4 a c}+b+2 c x}{c}\right )^{p+1} (a+x (b+c x))^{p-1} F_1\left (m+1;-p,-p;m+2;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right )}{(m+1) \left (\sqrt{b^2-4 a c}-b\right ) \left (\sqrt{b^2-4 a c}+b+2 c x\right ) \left (p x \left (\left (\sqrt{b^2-4 a c}-b\right ) F_1\left (m+2;1-p,-p;m+3;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right )-\left (\sqrt{b^2-4 a c}+b\right ) F_1\left (m+2;-p,1-p;m+3;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right )\right )-2 a (m+2) F_1\left (m+1;-p,-p;m+2;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [F]  time = 0.184, size = 0, normalized size = 0. \[ \int \left ( dx \right ) ^{m} \left ( c{x}^{2}+bx+a \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x)^m*(c*x^2+b*x+a)^p,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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