3.921 \(\int \frac{(a+b x)^n (c+d x)}{x} \, dx\)

Optimal. Leaf size=56 \[ \frac{d (a+b x)^{n+1}}{b (n+1)}-\frac{c (a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )}{a (n+1)} \]

[Out]

(d*(a + b*x)^(1 + n))/(b*(1 + n)) - (c*(a + b*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, 1 + (b*x)/a])/(a*(
1 + n))

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Rubi [A]  time = 0.0155928, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {80, 65} \[ \frac{d (a+b x)^{n+1}}{b (n+1)}-\frac{c (a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )}{a (n+1)} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^n*(c + d*x))/x,x]

[Out]

(d*(a + b*x)^(1 + n))/(b*(1 + n)) - (c*(a + b*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, 1 + (b*x)/a])/(a*(
1 + n))

Rule 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 65

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x)^(n + 1)*Hypergeometric2F1[-m, n +
1, n + 2, 1 + (d*x)/c])/(d*(n + 1)*(-(d/(b*c)))^m), x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[n] && (Inte
gerQ[m] || GtQ[-(d/(b*c)), 0])

Rubi steps

\begin{align*} \int \frac{(a+b x)^n (c+d x)}{x} \, dx &=\frac{d (a+b x)^{1+n}}{b (1+n)}+c \int \frac{(a+b x)^n}{x} \, dx\\ &=\frac{d (a+b x)^{1+n}}{b (1+n)}-\frac{c (a+b x)^{1+n} \, _2F_1\left (1,1+n;2+n;1+\frac{b x}{a}\right )}{a (1+n)}\\ \end{align*}

Mathematica [A]  time = 0.0131629, size = 45, normalized size = 0.8 \[ \frac{(a+b x)^{n+1} \left (a d-b c \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )\right )}{a b (n+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^n*(c + d*x))/x,x]

[Out]

((a + b*x)^(1 + n)*(a*d - b*c*Hypergeometric2F1[1, 1 + n, 2 + n, 1 + (b*x)/a]))/(a*b*(1 + n))

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Maple [F]  time = 0.021, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( bx+a \right ) ^{n} \left ( dx+c \right ) }{x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^n*(d*x+c)/x,x)

[Out]

int((b*x+a)^n*(d*x+c)/x,x)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^n*(d*x+c)/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^n*(d*x+c)/x,x, algorithm="fricas")

[Out]

integral((d*x + c)*(b*x + a)^n/x, x)

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Sympy [A]  time = 7.37829, size = 170, normalized size = 3.04 \begin{align*} - \frac{b^{n} c n \left (\frac{a}{b} + x\right )^{n} \Phi \left (1 + \frac{b x}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{\Gamma \left (n + 2\right )} - \frac{b^{n} c \left (\frac{a}{b} + x\right )^{n} \Phi \left (1 + \frac{b x}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{\Gamma \left (n + 2\right )} + d \left (\begin{cases} a^{n} x & \text{for}\: b = 0 \\\frac{\begin{cases} \frac{\left (a + b x\right )^{n + 1}}{n + 1} & \text{for}\: n \neq -1 \\\log{\left (a + b x \right )} & \text{otherwise} \end{cases}}{b} & \text{otherwise} \end{cases}\right ) - \frac{b b^{n} c n x \left (\frac{a}{b} + x\right )^{n} \Phi \left (1 + \frac{b x}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{a \Gamma \left (n + 2\right )} - \frac{b b^{n} c x \left (\frac{a}{b} + x\right )^{n} \Phi \left (1 + \frac{b x}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{a \Gamma \left (n + 2\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**n*(d*x+c)/x,x)

[Out]

-b**n*c*n*(a/b + x)**n*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/gamma(n + 2) - b**n*c*(a/b + x)**n*lerchphi(
1 + b*x/a, 1, n + 1)*gamma(n + 1)/gamma(n + 2) + d*Piecewise((a**n*x, Eq(b, 0)), (Piecewise(((a + b*x)**(n + 1
)/(n + 1), Ne(n, -1)), (log(a + b*x), True))/b, True)) - b*b**n*c*n*x*(a/b + x)**n*lerchphi(1 + b*x/a, 1, n +
1)*gamma(n + 1)/(a*gamma(n + 2)) - b*b**n*c*x*(a/b + x)**n*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(a*gamma
(n + 2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^n*(d*x+c)/x,x, algorithm="giac")

[Out]

integrate((d*x + c)*(b*x + a)^n/x, x)