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

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

[Out]

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

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Rubi [A]  time = 0.0238552, 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, 64} \[ \frac{x^{m+1} (b c-a d) \, _2F_1\left (1,m+1;m+2;-\frac{b x}{a}\right )}{a b (m+1)}+\frac{d x^{m+1}}{b (m+1)} \]

Antiderivative was successfully verified.

[In]

Int[(x^m*(c + d*x))/(a + b*x),x]

[Out]

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

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 64

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

Rubi steps

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

Mathematica [A]  time = 0.0170473, size = 45, normalized size = 0.8 \[ \frac{x^{m+1} \left ((b c-a d) \, _2F_1\left (1,m+1;m+2;-\frac{b x}{a}\right )+a d\right )}{a b (m+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.027, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( dx+c \right ){x}^{m}}{bx+a}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((d*x + c)*x^m/(b*x + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((d*x + c)*x^m/(b*x + a), x)

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Sympy [C]  time = 3.00578, size = 136, normalized size = 2.43 \begin{align*} \frac{c m x x^{m} \Phi \left (\frac{b x e^{i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{a \Gamma \left (m + 2\right )} + \frac{c x x^{m} \Phi \left (\frac{b x e^{i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{a \Gamma \left (m + 2\right )} + \frac{d m x^{2} x^{m} \Phi \left (\frac{b x e^{i \pi }}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{a \Gamma \left (m + 3\right )} + \frac{2 d x^{2} x^{m} \Phi \left (\frac{b x e^{i \pi }}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{a \Gamma \left (m + 3\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c*m*x*x**m*lerchphi(b*x*exp_polar(I*pi)/a, 1, m + 1)*gamma(m + 1)/(a*gamma(m + 2)) + c*x*x**m*lerchphi(b*x*exp
_polar(I*pi)/a, 1, m + 1)*gamma(m + 1)/(a*gamma(m + 2)) + d*m*x**2*x**m*lerchphi(b*x*exp_polar(I*pi)/a, 1, m +
2)*gamma(m + 2)/(a*gamma(m + 3)) + 2*d*x**2*x**m*lerchphi(b*x*exp_polar(I*pi)/a, 1, m + 2)*gamma(m + 2)/(a*ga
mma(m + 3))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((d*x + c)*x^m/(b*x + a), x)