\(\int \frac {(e x)^m (a+b x)}{a c-b c x} \, dx\) [72]

Optimal result

Integrand size = 22, antiderivative size = 55 \[ \int \frac {(e x)^m (a+b x)}{a c-b c x} \, dx=-\frac {(e x)^{1+m}}{c e (1+m)}+\frac {2 (e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {b x}{a}\right )}{c e (1+m)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {81, 66} \[ \int \frac {(e x)^m (a+b x)}{a c-b c x} \, dx=\frac {2 (e x)^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,\frac {b x}{a}\right )}{c e (m+1)}-\frac {(e x)^{m+1}}{c e (m+1)} \]

[In]

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

[Out]

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

Rule 66

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^n*((b*x)^(m + 1)/(b*(m + 1)))*Hypergeometr
ic2F1[-n, m + 1, m + 2, (-d)*(x/c)], 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])))

Rule 81

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]

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

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.60 \[ \int \frac {(e x)^m (a+b x)}{a c-b c x} \, dx=\frac {x (e x)^m \left (-1+2 \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {b x}{a}\right )\right )}{c (1+m)} \]

[In]

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

[Out]

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

Maple [F]

\[\int \frac {\left (e x \right )^{m} \left (b x +a \right )}{-b c x +a c}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {(e x)^m (a+b x)}{a c-b c x} \, dx=\int { -\frac {{\left (b x + a\right )} \left (e x\right )^{m}}{b c x - a c} \,d x } \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 126 vs. \(2 (37) = 74\).

Time = 1.56 (sec) , antiderivative size = 126, normalized size of antiderivative = 2.29 \[ \int \frac {(e x)^m (a+b x)}{a c-b c x} \, dx=\frac {e^{m} m x^{m + 1} \Phi \left (\frac {b x}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{c \Gamma \left (m + 2\right )} + \frac {e^{m} x^{m + 1} \Phi \left (\frac {b x}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{c \Gamma \left (m + 2\right )} + \frac {b e^{m} m x^{m + 2} \Phi \left (\frac {b x}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{a c \Gamma \left (m + 3\right )} + \frac {2 b e^{m} x^{m + 2} \Phi \left (\frac {b x}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{a c \Gamma \left (m + 3\right )} \]

[In]

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

[Out]

e**m*m*x**(m + 1)*lerchphi(b*x/a, 1, m + 1)*gamma(m + 1)/(c*gamma(m + 2)) + e**m*x**(m + 1)*lerchphi(b*x/a, 1,
m + 1)*gamma(m + 1)/(c*gamma(m + 2)) + b*e**m*m*x**(m + 2)*lerchphi(b*x/a, 1, m + 2)*gamma(m + 2)/(a*c*gamma(
m + 3)) + 2*b*e**m*x**(m + 2)*lerchphi(b*x/a, 1, m + 2)*gamma(m + 2)/(a*c*gamma(m + 3))

Maxima [F]

\[ \int \frac {(e x)^m (a+b x)}{a c-b c x} \, dx=\int { -\frac {{\left (b x + a\right )} \left (e x\right )^{m}}{b c x - a c} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {(e x)^m (a+b x)}{a c-b c x} \, dx=\int { -\frac {{\left (b x + a\right )} \left (e x\right )^{m}}{b c x - a c} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(e x)^m (a+b x)}{a c-b c x} \, dx=\int \frac {{\left (e\,x\right )}^m\,\left (a+b\,x\right )}{a\,c-b\,c\,x} \,d x \]

[In]

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

[Out]

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