### 3.2571 $$\int \sqrt{d+e x} (a+b x+c x^2)^p \, dx$$

Optimal. Leaf size=185 $\frac{2 (d+e x)^{3/2} \left (a+b x+c x^2\right )^p \left (1-\frac{2 c (d+e x)}{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}\right )^{-p} \left (1-\frac{2 c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}\right )^{-p} F_1\left (\frac{3}{2};-p,-p;\frac{5}{2};\frac{2 c (d+e x)}{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e},\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{3 e}$

[Out]

(2*(d + e*x)^(3/2)*(a + b*x + c*x^2)^p*AppellF1[3/2, -p, -p, 5/2, (2*c*(d + e*x))/(2*c*d - (b - Sqrt[b^2 - 4*a
*c])*e), (2*c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(3*e*(1 - (2*c*(d + e*x))/(2*c*d - (b - Sqrt[b^
2 - 4*a*c])*e))^p*(1 - (2*c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e))^p)

________________________________________________________________________________________

Rubi [A]  time = 0.10627, antiderivative size = 185, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.091, Rules used = {759, 133} $\frac{2 (d+e x)^{3/2} \left (a+b x+c x^2\right )^p \left (1-\frac{2 c (d+e x)}{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}\right )^{-p} \left (1-\frac{2 c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}\right )^{-p} F_1\left (\frac{3}{2};-p,-p;\frac{5}{2};\frac{2 c (d+e x)}{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e},\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{3 e}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*(d + e*x)^(3/2)*(a + b*x + c*x^2)^p*AppellF1[3/2, -p, -p, 5/2, (2*c*(d + e*x))/(2*c*d - (b - Sqrt[b^2 - 4*a
*c])*e), (2*c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(3*e*(1 - (2*c*(d + e*x))/(2*c*d - (b - Sqrt[b^
2 - 4*a*c])*e))^p*(1 - (2*c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e))^p)

Rule 759

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

Rule 133

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[(c^n*e^p*(b*x)^(m +
1)*AppellF1[m + 1, -n, -p, m + 2, -((d*x)/c), -((f*x)/e)])/(b*(m + 1)), x] /; FreeQ[{b, c, d, e, f, m, n, p},
x] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])

Rubi steps

\begin{align*} \int \sqrt{d+e x} \left (a+b x+c x^2\right )^p \, dx &=\frac{\left (\left (a+b x+c x^2\right )^p \left (1-\frac{d+e x}{d-\frac{\left (b-\sqrt{b^2-4 a c}\right ) e}{2 c}}\right )^{-p} \left (1-\frac{d+e x}{d-\frac{\left (b+\sqrt{b^2-4 a c}\right ) e}{2 c}}\right )^{-p}\right ) \operatorname{Subst}\left (\int \sqrt{x} \left (1-\frac{2 c x}{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e}\right )^p \left (1-\frac{2 c x}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )^p \, dx,x,d+e x\right )}{e}\\ &=\frac{2 (d+e x)^{3/2} \left (a+b x+c x^2\right )^p \left (1-\frac{2 c (d+e x)}{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e}\right )^{-p} \left (1-\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )^{-p} F_1\left (\frac{3}{2};-p,-p;\frac{5}{2};\frac{2 c (d+e x)}{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e},\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{3 e}\\ \end{align*}

Mathematica [A]  time = 0.482945, size = 211, normalized size = 1.14 $\frac{2^{1-2 p} (d+e x)^{3/2} (a+x (b+c x))^p \left (\frac{e \left (\sqrt{b^2-4 a c}-b-2 c x\right )}{4 e \left (\sqrt{b^2-4 a c}-b\right )+8 c d}\right )^{-p} \left (\frac{e \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{e \left (\sqrt{b^2-4 a c}+b\right )-2 c d}\right )^{-p} F_1\left (\frac{3}{2};-p,-p;\frac{5}{2};\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e},\frac{2 c (d+e x)}{2 c d+\left (\sqrt{b^2-4 a c}-b\right ) e}\right )}{3 e}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2^(1 - 2*p)*(d + e*x)^(3/2)*(a + x*(b + c*x))^p*AppellF1[3/2, -p, -p, 5/2, (2*c*(d + e*x))/(2*c*d - (b + Sqrt
[b^2 - 4*a*c])*e), (2*c*(d + e*x))/(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)])/(3*e*((e*(-b + Sqrt[b^2 - 4*a*c] - 2
*c*x))/(8*c*d + 4*(-b + Sqrt[b^2 - 4*a*c])*e))^p*((e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(-2*c*d + (b + Sqrt[b^2
- 4*a*c])*e))^p)

________________________________________________________________________________________

Maple [F]  time = 1.284, size = 0, normalized size = 0. \begin{align*} \int \sqrt{ex+d} \left ( c{x}^{2}+bx+a \right ) ^{p}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(1/2)*(c*x^2+b*x+a)^p,x)

[Out]

int((e*x+d)^(1/2)*(c*x^2+b*x+a)^p,x)

________________________________________________________________________________________

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(1/2)*(c*x^2+b*x+a)^p,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(1/2)*(c*x^2+b*x+a)^p,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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((e*x+d)**(1/2)*(c*x**2+b*x+a)**p,x)

[Out]

Timed out

________________________________________________________________________________________

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(1/2)*(c*x^2+b*x+a)^p,x, algorithm="giac")

[Out]

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