### 3.447 $$\int (d+e x)^m \sqrt{b x+c x^2} \, dx$$

Optimal. Leaf size=105 $\frac{\sqrt{b x+c x^2} (d+e x)^{m+1} F_1\left (m+1;-\frac{1}{2},-\frac{1}{2};m+2;\frac{d+e x}{d},\frac{c (d+e x)}{c d-b e}\right )}{e (m+1) \sqrt{-\frac{e x}{d}} \sqrt{1-\frac{c (d+e x)}{c d-b e}}}$

[Out]

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

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Rubi [A]  time = 0.0493167, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.095, Rules used = {759, 133} $\frac{\sqrt{b x+c x^2} (d+e x)^{m+1} F_1\left (m+1;-\frac{1}{2},-\frac{1}{2};m+2;\frac{d+e x}{d},\frac{c (d+e x)}{c d-b e}\right )}{e (m+1) \sqrt{-\frac{e x}{d}} \sqrt{1-\frac{c (d+e x)}{c d-b e}}}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^m*Sqrt[b*x + c*x^2],x]

[Out]

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

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 (d+e x)^m \sqrt{b x+c x^2} \, dx &=\frac{\sqrt{b x+c x^2} \operatorname{Subst}\left (\int x^m \sqrt{1-\frac{x}{d}} \sqrt{1-\frac{c x}{c d-b e}} \, dx,x,d+e x\right )}{e \sqrt{1-\frac{d+e x}{d}} \sqrt{1-\frac{d+e x}{d-\frac{b e}{c}}}}\\ &=\frac{(d+e x)^{1+m} \sqrt{b x+c x^2} F_1\left (1+m;-\frac{1}{2},-\frac{1}{2};2+m;\frac{d+e x}{d},\frac{c (d+e x)}{c d-b e}\right )}{e (1+m) \sqrt{-\frac{e x}{d}} \sqrt{1-\frac{c (d+e x)}{c d-b e}}}\\ \end{align*}

Mathematica [A]  time = 0.0352619, size = 76, normalized size = 0.72 $\frac{2 x \sqrt{x (b+c x)} (d+e x)^m \left (\frac{d+e x}{d}\right )^{-m} F_1\left (\frac{3}{2};-\frac{1}{2},-m;\frac{5}{2};-\frac{c x}{b},-\frac{e x}{d}\right )}{3 \sqrt{\frac{b+c x}{b}}}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [F]  time = 0.614, size = 0, normalized size = 0. \begin{align*} \int \left ( ex+d \right ) ^{m}\sqrt{c{x}^{2}+bx}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(x*(b + c*x))*(d + e*x)**m, x)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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