3.73 \(\int \frac{(e x)^m (a+b x)}{(a c-b c x)^2} \, dx\)
Optimal. Leaf size=66 \[ \frac{2 (e x)^{m+1}}{c^2 e (a-b x)}-\frac{(2 m+1) (e x)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{b x}{a}\right )}{a c^2 e (m+1)} \]
[Out]
(2*(e*x)^(1 + m))/(c^2*e*(a - b*x)) - ((1 + 2*m)*(e*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (b*x)/a])/(a
*c^2*e*(1 + m))
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Rubi [A] time = 0.0238615, antiderivative size = 66, 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 =
{78, 64} \[ \frac{2 (e x)^{m+1}}{c^2 e (a-b x)}-\frac{(2 m+1) (e x)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{b x}{a}\right )}{a c^2 e (m+1)} \]
Antiderivative was successfully verified.
[In]
Int[((e*x)^m*(a + b*x))/(a*c - b*c*x)^2,x]
[Out]
(2*(e*x)^(1 + m))/(c^2*e*(a - b*x)) - ((1 + 2*m)*(e*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (b*x)/a])/(a
*c^2*e*(1 + m))
Rule 78
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ
[p, n]))))
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+b x)}{(a c-b c x)^2} \, dx &=\frac{2 (e x)^{1+m}}{c^2 e (a-b x)}-\frac{(1+2 m) \int \frac{(e x)^m}{a c-b c x} \, dx}{c}\\ &=\frac{2 (e x)^{1+m}}{c^2 e (a-b x)}-\frac{(1+2 m) (e x)^{1+m} \, _2F_1\left (1,1+m;2+m;\frac{b x}{a}\right )}{a c^2 e (1+m)}\\ \end{align*}
Mathematica [A] time = 0.0224761, size = 62, normalized size = 0.94 \[ -\frac{x (e x)^m \left (2 a (m+1)-(2 m+1) (a-b x) \, _2F_1\left (1,m+1;m+2;\frac{b x}{a}\right )\right )}{a c^2 (m+1) (b x-a)} \]
Antiderivative was successfully verified.
[In]
Integrate[((e*x)^m*(a + b*x))/(a*c - b*c*x)^2,x]
[Out]
-((x*(e*x)^m*(2*a*(1 + m) - (1 + 2*m)*(a - b*x)*Hypergeometric2F1[1, 1 + m, 2 + m, (b*x)/a]))/(a*c^2*(1 + m)*(
-a + b*x)))
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Maple [F] time = 0.032, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex \right ) ^{m} \left ( bx+a \right ) }{ \left ( -bcx+ac \right ) ^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((e*x)^m*(b*x+a)/(-b*c*x+a*c)^2,x)
[Out]
int((e*x)^m*(b*x+a)/(-b*c*x+a*c)^2,x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )} \left (e x\right )^{m}}{{\left (b c x - a c\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x)^m*(b*x+a)/(-b*c*x+a*c)^2,x, algorithm="maxima")
[Out]
integrate((b*x + a)*(e*x)^m/(b*c*x - a*c)^2, x)
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b x + a\right )} \left (e x\right )^{m}}{b^{2} c^{2} x^{2} - 2 \, a b c^{2} x + a^{2} c^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x)^m*(b*x+a)/(-b*c*x+a*c)^2,x, algorithm="fricas")
[Out]
integral((b*x + a)*(e*x)^m/(b^2*c^2*x^2 - 2*a*b*c^2*x + a^2*c^2), x)
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Sympy [C] time = 5.74544, size = 799, normalized size = 12.11 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x)**m*(b*x+a)/(-b*c*x+a*c)**2,x)
[Out]
a*(a*e**m*m**2*x*x**m*lerchphi(b*x*exp_polar(2*I*pi)/a, 1, m + 1)*gamma(m + 1)/(-a**3*c**2*gamma(m + 2) + a**2
*b*c**2*x*gamma(m + 2)) + a*e**m*m*x*x**m*lerchphi(b*x*exp_polar(2*I*pi)/a, 1, m + 1)*gamma(m + 1)/(-a**3*c**2
*gamma(m + 2) + a**2*b*c**2*x*gamma(m + 2)) - a*e**m*m*x*x**m*gamma(m + 1)/(-a**3*c**2*gamma(m + 2) + a**2*b*c
**2*x*gamma(m + 2)) - a*e**m*x*x**m*gamma(m + 1)/(-a**3*c**2*gamma(m + 2) + a**2*b*c**2*x*gamma(m + 2)) - b*e*
*m*m**2*x**2*x**m*lerchphi(b*x*exp_polar(2*I*pi)/a, 1, m + 1)*gamma(m + 1)/(-a**3*c**2*gamma(m + 2) + a**2*b*c
**2*x*gamma(m + 2)) - b*e**m*m*x**2*x**m*lerchphi(b*x*exp_polar(2*I*pi)/a, 1, m + 1)*gamma(m + 1)/(-a**3*c**2*
gamma(m + 2) + a**2*b*c**2*x*gamma(m + 2))) + b*(a*e**m*m**2*x**2*x**m*lerchphi(b*x*exp_polar(2*I*pi)/a, 1, m
+ 2)*gamma(m + 2)/(-a**3*c**2*gamma(m + 3) + a**2*b*c**2*x*gamma(m + 3)) + 3*a*e**m*m*x**2*x**m*lerchphi(b*x*e
xp_polar(2*I*pi)/a, 1, m + 2)*gamma(m + 2)/(-a**3*c**2*gamma(m + 3) + a**2*b*c**2*x*gamma(m + 3)) - a*e**m*m*x
**2*x**m*gamma(m + 2)/(-a**3*c**2*gamma(m + 3) + a**2*b*c**2*x*gamma(m + 3)) + 2*a*e**m*x**2*x**m*lerchphi(b*x
*exp_polar(2*I*pi)/a, 1, m + 2)*gamma(m + 2)/(-a**3*c**2*gamma(m + 3) + a**2*b*c**2*x*gamma(m + 3)) - 2*a*e**m
*x**2*x**m*gamma(m + 2)/(-a**3*c**2*gamma(m + 3) + a**2*b*c**2*x*gamma(m + 3)) - b*e**m*m**2*x**3*x**m*lerchph
i(b*x*exp_polar(2*I*pi)/a, 1, m + 2)*gamma(m + 2)/(-a**3*c**2*gamma(m + 3) + a**2*b*c**2*x*gamma(m + 3)) - 3*b
*e**m*m*x**3*x**m*lerchphi(b*x*exp_polar(2*I*pi)/a, 1, m + 2)*gamma(m + 2)/(-a**3*c**2*gamma(m + 3) + a**2*b*c
**2*x*gamma(m + 3)) - 2*b*e**m*x**3*x**m*lerchphi(b*x*exp_polar(2*I*pi)/a, 1, m + 2)*gamma(m + 2)/(-a**3*c**2*
gamma(m + 3) + a**2*b*c**2*x*gamma(m + 3)))
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )} \left (e x\right )^{m}}{{\left (b c x - a c\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x)^m*(b*x+a)/(-b*c*x+a*c)^2,x, algorithm="giac")
[Out]
integrate((b*x + a)*(e*x)^m/(b*c*x - a*c)^2, x)