3.76 \(\int \frac{(e x)^m (a c-b c x)}{a+b x} \, dx\)

Optimal. Leaf size=52 \[ \frac{2 c (e x)^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac{b x}{a}\right )}{e (m+1)}-\frac{c (e x)^{m+1}}{e (m+1)} \]

[Out]

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

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Rubi [A]  time = 0.014491, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {80, 64} \[ \frac{2 c (e x)^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac{b x}{a}\right )}{e (m+1)}-\frac{c (e x)^{m+1}}{e (m+1)} \]

Antiderivative was successfully verified.

[In]

Int[((e*x)^m*(a*c - b*c*x))/(a + b*x),x]

[Out]

-((c*(e*x)^(1 + m))/(e*(1 + m))) + (2*c*(e*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, -((b*x)/a)])/(e*(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{(e x)^m (a c-b c x)}{a+b x} \, dx &=-\frac{c (e x)^{1+m}}{e (1+m)}+(2 a c) \int \frac{(e x)^m}{a+b x} \, dx\\ &=-\frac{c (e x)^{1+m}}{e (1+m)}+\frac{2 c (e x)^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac{b x}{a}\right )}{e (1+m)}\\ \end{align*}

Mathematica [A]  time = 0.0077883, size = 32, normalized size = 0.62 \[ \frac{c x (e x)^m \left (2 \, _2F_1\left (1,m+1;m+2;-\frac{b x}{a}\right )-1\right )}{m+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-integrate((b*c*x - a*c)*(e*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 (b c x - a c\right )} \left (e x\right )^{m}}{b x + a}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(b*c*x - a*c)*(e*x)^m/(b*x + a), x)

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Sympy [C]  time = 2.96208, size = 150, normalized size = 2.88 \begin{align*} \frac{c e^{m} m x x^{m} \Phi \left (\frac{b x e^{i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{\Gamma \left (m + 2\right )} + \frac{c e^{m} x x^{m} \Phi \left (\frac{b x e^{i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{\Gamma \left (m + 2\right )} - \frac{b c e^{m} 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 b c e^{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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(-(b*c*x - a*c)*(e*x)^m/(b*x + a), x)