### 3.2572 $$\int \frac{(a+b x+c x^2)^p}{\sqrt{d+e x}} \, dx$$

Optimal. Leaf size=183 $\frac{2 \sqrt{d+e x} \left (a+b x+c x^2\right )^p \left (1-\frac{2 c (d+e x)}{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}\right )^{-p} \left (1-\frac{2 c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}\right )^{-p} F_1\left (\frac{1}{2};-p,-p;\frac{3}{2};\frac{2 c (d+e x)}{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e},\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{e}$

[Out]

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

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Rubi [A]  time = 0.1046, antiderivative size = 183, 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} $\frac{2 \sqrt{d+e x} \left (a+b x+c x^2\right )^p \left (1-\frac{2 c (d+e x)}{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}\right )^{-p} \left (1-\frac{2 c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}\right )^{-p} F_1\left (\frac{1}{2};-p,-p;\frac{3}{2};\frac{2 c (d+e x)}{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e},\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{e}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*Sqrt[d + e*x]*(a + b*x + c*x^2)^p*AppellF1[1/2, -p, -p, 3/2, (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)])/(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 \frac{\left (a+b x+c x^2\right )^p}{\sqrt{d+e x}} \, 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 \frac{\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}{\sqrt{x}} \, dx,x,d+e x\right )}{e}\\ &=\frac{2 \sqrt{d+e x} \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{1}{2};-p,-p;\frac{3}{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 )}{e}\\ \end{align*}

Mathematica [A]  time = 0.370982, size = 208, normalized size = 1.14 $\frac{2^{1-2 p} \sqrt{d+e x} (a+x (b+c x))^p \left (\frac{e \left (\sqrt{b^2-4 a c}-b-2 c x\right )}{4 e \left (\sqrt{b^2-4 a c}-b\right )+8 c d}\right )^{-p} \left (\frac{e \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{e \left (\sqrt{b^2-4 a c}+b\right )-2 c d}\right )^{-p} F_1\left (\frac{1}{2};-p,-p;\frac{3}{2};\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e},\frac{2 c (d+e x)}{2 c d+\left (\sqrt{b^2-4 a c}-b\right ) e}\right )}{e}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [F]  time = 1.238, size = 0, normalized size = 0. \begin{align*} \int{ \left ( c{x}^{2}+bx+a \right ) ^{p}{\frac{1}{\sqrt{ex+d}}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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