3.965 \(\int \frac{(c+d x)^n (e+f x)^p}{\sqrt{b x}} \, dx\)

Optimal. Leaf size=77 \[ \frac{2 \sqrt{b x} (c+d x)^n \left (\frac{d x}{c}+1\right )^{-n} (e+f x)^p \left (\frac{f x}{e}+1\right )^{-p} F_1\left (\frac{1}{2};-n,-p;\frac{3}{2};-\frac{d x}{c},-\frac{f x}{e}\right )}{b} \]

[Out]

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

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Rubi [A]  time = 0.0468082, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {135, 133} \[ \frac{2 \sqrt{b x} (c+d x)^n \left (\frac{d x}{c}+1\right )^{-n} (e+f x)^p \left (\frac{f x}{e}+1\right )^{-p} F_1\left (\frac{1}{2};-n,-p;\frac{3}{2};-\frac{d x}{c},-\frac{f x}{e}\right )}{b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 135

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

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{(c+d x)^n (e+f x)^p}{\sqrt{b x}} \, dx &=\left ((c+d x)^n \left (1+\frac{d x}{c}\right )^{-n}\right ) \int \frac{\left (1+\frac{d x}{c}\right )^n (e+f x)^p}{\sqrt{b x}} \, dx\\ &=\left ((c+d x)^n \left (1+\frac{d x}{c}\right )^{-n} (e+f x)^p \left (1+\frac{f x}{e}\right )^{-p}\right ) \int \frac{\left (1+\frac{d x}{c}\right )^n \left (1+\frac{f x}{e}\right )^p}{\sqrt{b x}} \, dx\\ &=\frac{2 \sqrt{b x} (c+d x)^n \left (1+\frac{d x}{c}\right )^{-n} (e+f x)^p \left (1+\frac{f x}{e}\right )^{-p} F_1\left (\frac{1}{2};-n,-p;\frac{3}{2};-\frac{d x}{c},-\frac{f x}{e}\right )}{b}\\ \end{align*}

Mathematica [A]  time = 0.0499169, size = 77, normalized size = 1. \[ \frac{2 x (c+d x)^n \left (\frac{c+d x}{c}\right )^{-n} (e+f x)^p \left (\frac{e+f x}{e}\right )^{-p} F_1\left (\frac{1}{2};-n,-p;\frac{3}{2};-\frac{d x}{c},-\frac{f x}{e}\right )}{\sqrt{b x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.042, size = 0, normalized size = 0. \begin{align*} \int{ \left ( dx+c \right ) ^{n} \left ( fx+e \right ) ^{p}{\frac{1}{\sqrt{bx}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((d*x + c)^n*(f*x + e)^p/sqrt(b*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(b*x)*(d*x + c)^n*(f*x + e)^p/(b*x), 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((d*x+c)**n*(f*x+e)**p/(b*x)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((d*x + c)^n*(f*x + e)^p/sqrt(b*x), x)