### 3.448 $$\int \frac{(d+e x)^m}{\sqrt{b x+c x^2}} \, dx$$

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

[Out]

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

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Rubi [A]  time = 0.0510385, 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} $\frac{\sqrt{-\frac{e x}{d}} (d+e x)^{m+1} \sqrt{1-\frac{c (d+e x)}{c d-b e}} F_1\left (m+1;\frac{1}{2},\frac{1}{2};m+2;\frac{d+e x}{d},\frac{c (d+e x)}{c d-b e}\right )}{e (m+1) \sqrt{b x+c x^2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^m/Sqrt[b*x + c*x^2],x]

[Out]

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

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

Mathematica [A]  time = 0.0402688, size = 74, normalized size = 0.7 $\frac{2 x \sqrt{\frac{b+c x}{b}} (d+e x)^m \left (\frac{d+e x}{d}\right )^{-m} F_1\left (\frac{1}{2};\frac{1}{2},-m;\frac{3}{2};-\frac{c x}{b},-\frac{e x}{d}\right )}{\sqrt{x (b+c x)}}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [F]  time = 0.613, size = 0, normalized size = 0. \begin{align*} \int{ \left ( ex+d \right ) ^{m}{\frac{1}{\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 \frac{{\left (e x + d\right )}^{m}}{\sqrt{c x^{2} + b x}}\,{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((e*x + d)^m/sqrt(c*x^2 + b*x), x)

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

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

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