3.10 \(\int (a+\frac{c e x}{f}+c x^2)^p (\frac{a f}{c}+e x+f x^2)^q \, dx\)

Optimal. Leaf size=200 \[ -\frac{\sqrt{c} 2^{p+q+1} \left (a+\frac{c e x}{f}+c x^2\right )^p \left (\frac{a f}{c}+e x+f x^2\right )^{q+1} \left (-\frac{\sqrt{c} \left (-\frac{\sqrt{c e^2-4 a f^2}}{\sqrt{c}}+e+2 f x\right )}{\sqrt{c e^2-4 a f^2}}\right )^{-p-q-1} \, _2F_1\left (-p-q,p+q+1;p+q+2;\frac{\sqrt{c} \left (e+2 f x+\frac{\sqrt{c e^2-4 a f^2}}{\sqrt{c}}\right )}{2 \sqrt{c e^2-4 a f^2}}\right )}{(p+q+1) \sqrt{c e^2-4 a f^2}} \]

[Out]

-((2^(1 + p + q)*Sqrt[c]*(-((Sqrt[c]*(e - Sqrt[c*e^2 - 4*a*f^2]/Sqrt[c] + 2*f*x))/Sqrt[c*e^2 - 4*a*f^2]))^(-1
- p - q)*(a + (c*e*x)/f + c*x^2)^p*((a*f)/c + e*x + f*x^2)^(1 + q)*Hypergeometric2F1[-p - q, 1 + p + q, 2 + p
+ q, (Sqrt[c]*(e + Sqrt[c*e^2 - 4*a*f^2]/Sqrt[c] + 2*f*x))/(2*Sqrt[c*e^2 - 4*a*f^2])])/(Sqrt[c*e^2 - 4*a*f^2]*
(1 + p + q)))

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Rubi [A]  time = 0.134029, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {968, 624} \[ -\frac{\sqrt{c} 2^{p+q+1} \left (a+\frac{c e x}{f}+c x^2\right )^p \left (\frac{a f}{c}+e x+f x^2\right )^{q+1} \left (-\frac{\sqrt{c} \left (-\frac{\sqrt{c e^2-4 a f^2}}{\sqrt{c}}+e+2 f x\right )}{\sqrt{c e^2-4 a f^2}}\right )^{-p-q-1} \, _2F_1\left (-p-q,p+q+1;p+q+2;\frac{\sqrt{c} \left (e+2 f x+\frac{\sqrt{c e^2-4 a f^2}}{\sqrt{c}}\right )}{2 \sqrt{c e^2-4 a f^2}}\right )}{(p+q+1) \sqrt{c e^2-4 a f^2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + (c*e*x)/f + c*x^2)^p*((a*f)/c + e*x + f*x^2)^q,x]

[Out]

-((2^(1 + p + q)*Sqrt[c]*(-((Sqrt[c]*(e - Sqrt[c*e^2 - 4*a*f^2]/Sqrt[c] + 2*f*x))/Sqrt[c*e^2 - 4*a*f^2]))^(-1
- p - q)*(a + (c*e*x)/f + c*x^2)^p*((a*f)/c + e*x + f*x^2)^(1 + q)*Hypergeometric2F1[-p - q, 1 + p + q, 2 + p
+ q, (Sqrt[c]*(e + Sqrt[c*e^2 - 4*a*f^2]/Sqrt[c] + 2*f*x))/(2*Sqrt[c*e^2 - 4*a*f^2])])/(Sqrt[c*e^2 - 4*a*f^2]*
(1 + p + q)))

Rule 968

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (e_.)*(x_) + (f_.)*(x_)^2)^(q_.), x_Symbol] :> Dist[(a^Int
Part[p]*(a + b*x + c*x^2)^FracPart[p])/(d^IntPart[p]*(d + e*x + f*x^2)^FracPart[p]), Int[(d + e*x + f*x^2)^(p
+ q), x], x] /; FreeQ[{a, b, c, d, e, f, p, q}, x] && EqQ[c*d - a*f, 0] && EqQ[b*d - a*e, 0] &&  !IntegerQ[p]
&&  !IntegerQ[q] &&  !GtQ[c/f, 0]

Rule 624

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, -Simp[((a + b*x + c*
x^2)^(p + 1)*Hypergeometric2F1[-p, p + 1, p + 2, (b + q + 2*c*x)/(2*q)])/(q*(p + 1)*((q - b - 2*c*x)/(2*q))^(p
+ 1)), x]] /; FreeQ[{a, b, c, p}, x] && NeQ[b^2 - 4*a*c, 0] &&  !IntegerQ[4*p]

Rubi steps

\begin{align*} \int \left (a+\frac{c e x}{f}+c x^2\right )^p \left (\frac{a f}{c}+e x+f x^2\right )^q \, dx &=\left (\left (a+\frac{c e x}{f}+c x^2\right )^p \left (\frac{a f}{c}+e x+f x^2\right )^{-p}\right ) \int \left (\frac{a f}{c}+e x+f x^2\right )^{p+q} \, dx\\ &=-\frac{2^{1+p+q} \sqrt{c} \left (-\frac{\sqrt{c} \left (e-\frac{\sqrt{c e^2-4 a f^2}}{\sqrt{c}}+2 f x\right )}{\sqrt{c e^2-4 a f^2}}\right )^{-1-p-q} \left (a+\frac{c e x}{f}+c x^2\right )^p \left (\frac{a f}{c}+e x+f x^2\right )^{1+q} \, _2F_1\left (-p-q,1+p+q;2+p+q;\frac{\sqrt{c} \left (e+\frac{\sqrt{c e^2-4 a f^2}}{\sqrt{c}}+2 f x\right )}{2 \sqrt{c e^2-4 a f^2}}\right )}{\sqrt{c e^2-4 a f^2} (1+p+q)}\\ \end{align*}

Mathematica [A]  time = 0.238339, size = 172, normalized size = 0.86 \[ \frac{2^{p+q-1} \left (\sqrt{c} (e+2 f x)-\sqrt{c e^2-4 a f^2}\right ) \left (a+\frac{c x (e+f x)}{f}\right )^p \left (\frac{a f}{c}+x (e+f x)\right )^q \left (\frac{\sqrt{c} (e+2 f x)}{\sqrt{c e^2-4 a f^2}}+1\right )^{-p-q} \, _2F_1\left (-p-q,p+q+1;p+q+2;\frac{1}{2}-\frac{\sqrt{c} (e+2 f x)}{2 \sqrt{c e^2-4 a f^2}}\right )}{\sqrt{c} f (p+q+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + (c*e*x)/f + c*x^2)^p*((a*f)/c + e*x + f*x^2)^q,x]

[Out]

(2^(-1 + p + q)*((a*f)/c + x*(e + f*x))^q*(a + (c*x*(e + f*x))/f)^p*(-Sqrt[c*e^2 - 4*a*f^2] + Sqrt[c]*(e + 2*f
*x))*(1 + (Sqrt[c]*(e + 2*f*x))/Sqrt[c*e^2 - 4*a*f^2])^(-p - q)*Hypergeometric2F1[-p - q, 1 + p + q, 2 + p + q
, 1/2 - (Sqrt[c]*(e + 2*f*x))/(2*Sqrt[c*e^2 - 4*a*f^2])])/(Sqrt[c]*f*(1 + p + q))

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Maple [F]  time = 3.482, size = 0, normalized size = 0. \begin{align*} \int \left ( a+{\frac{cex}{f}}+c{x}^{2} \right ) ^{p} \left ({\frac{af}{c}}+ex+f{x}^{2} \right ) ^{q}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+c*e*x/f+c*x^2)^p*(a*f/c+e*x+f*x^2)^q,x)

[Out]

int((a+c*e*x/f+c*x^2)^p*(a*f/c+e*x+f*x^2)^q,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c x^{2} + \frac{c e x}{f} + a\right )}^{p}{\left (f x^{2} + e x + \frac{a f}{c}\right )}^{q}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+c*e*x/f+c*x^2)^p*(a*f/c+e*x+f*x^2)^q,x, algorithm="maxima")

[Out]

integrate((c*x^2 + c*e*x/f + a)^p*(f*x^2 + e*x + a*f/c)^q, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (\frac{c f x^{2} + c e x + a f}{c}\right )^{q} \left (\frac{c f x^{2} + c e x + a f}{f}\right )^{p}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+c*e*x/f+c*x^2)^p*(a*f/c+e*x+f*x^2)^q,x, algorithm="fricas")

[Out]

integral(((c*f*x^2 + c*e*x + a*f)/c)^q*((c*f*x^2 + c*e*x + a*f)/f)^p, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+c*e*x/f+c*x**2)**p*(a*f/c+e*x+f*x**2)**q,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c x^{2} + \frac{c e x}{f} + a\right )}^{p}{\left (f x^{2} + e x + \frac{a f}{c}\right )}^{q}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+c*e*x/f+c*x^2)^p*(a*f/c+e*x+f*x^2)^q,x, algorithm="giac")

[Out]

integrate((c*x^2 + c*e*x/f + a)^p*(f*x^2 + e*x + a*f/c)^q, x)