3.376 \(\int \frac{x^m (A+B x)}{(a+b x)^2} \, dx\)
Optimal. Leaf size=73 \[ \frac{x^{m+1} (A b-a B)}{a b (a+b x)}-\frac{x^{m+1} (A b m-a B (m+1)) \, _2F_1\left (1,m+1;m+2;-\frac{b x}{a}\right )}{a^2 b (m+1)} \]
[Out]
((A*b - a*B)*x^(1 + m))/(a*b*(a + b*x)) - ((A*b*m - a*B*(1 + m))*x^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m,
-((b*x)/a)])/(a^2*b*(1 + m))
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Rubi [A] time = 0.0264038, antiderivative size = 73, 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 =
{78, 64} \[ \frac{x^{m+1} (A b-a B)}{a b (a+b x)}-\frac{x^{m+1} (A b m-a B (m+1)) \, _2F_1\left (1,m+1;m+2;-\frac{b x}{a}\right )}{a^2 b (m+1)} \]
Antiderivative was successfully verified.
[In]
Int[(x^m*(A + B*x))/(a + b*x)^2,x]
[Out]
((A*b - a*B)*x^(1 + m))/(a*b*(a + b*x)) - ((A*b*m - a*B*(1 + m))*x^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m,
-((b*x)/a)])/(a^2*b*(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{x^m (A+B x)}{(a+b x)^2} \, dx &=\frac{(A b-a B) x^{1+m}}{a b (a+b x)}-\frac{(A b m-a B (1+m)) \int \frac{x^m}{a+b x} \, dx}{a b}\\ &=\frac{(A b-a B) x^{1+m}}{a b (a+b x)}-\frac{(A b m-a B (1+m)) x^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac{b x}{a}\right )}{a^2 b (1+m)}\\ \end{align*}
Mathematica [A] time = 0.037713, size = 63, normalized size = 0.86 \[ \frac{x^{m+1} \left (\frac{(a B (m+1)-A b m) \, _2F_1\left (1,m+1;m+2;-\frac{b x}{a}\right )}{m+1}+\frac{a (A b-a B)}{a+b x}\right )}{a^2 b} \]
Antiderivative was successfully verified.
[In]
Integrate[(x^m*(A + B*x))/(a + b*x)^2,x]
[Out]
(x^(1 + m)*((a*(A*b - a*B))/(a + b*x) + ((-(A*b*m) + a*B*(1 + m))*Hypergeometric2F1[1, 1 + m, 2 + m, -((b*x)/a
)])/(1 + m)))/(a^2*b)
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Maple [F] time = 0.032, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( Bx+A \right ){x}^{m}}{ \left ( bx+a \right ) ^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^m*(B*x+A)/(b*x+a)^2,x)
[Out]
int(x^m*(B*x+A)/(b*x+a)^2,x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B x + A\right )} x^{m}}{{\left (b x + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^m*(B*x+A)/(b*x+a)^2,x, algorithm="maxima")
[Out]
integrate((B*x + A)*x^m/(b*x + a)^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 )} x^{m}}{b^{2} x^{2} + 2 \, a b x + a^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^m*(B*x+A)/(b*x+a)^2,x, algorithm="fricas")
[Out]
integral((B*x + A)*x^m/(b^2*x^2 + 2*a*b*x + a^2), x)
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Sympy [C] time = 6.32261, size = 639, normalized size = 8.75 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**m*(B*x+A)/(b*x+a)**2,x)
[Out]
A*(-a*m**2*x*x**m*lerchphi(b*x*exp_polar(I*pi)/a, 1, m + 1)*gamma(m + 1)/(a**3*gamma(m + 2) + a**2*b*x*gamma(m
+ 2)) - a*m*x*x**m*lerchphi(b*x*exp_polar(I*pi)/a, 1, m + 1)*gamma(m + 1)/(a**3*gamma(m + 2) + a**2*b*x*gamma
(m + 2)) + a*m*x*x**m*gamma(m + 1)/(a**3*gamma(m + 2) + a**2*b*x*gamma(m + 2)) + a*x*x**m*gamma(m + 1)/(a**3*g
amma(m + 2) + a**2*b*x*gamma(m + 2)) - b*m**2*x**2*x**m*lerchphi(b*x*exp_polar(I*pi)/a, 1, m + 1)*gamma(m + 1)
/(a**3*gamma(m + 2) + a**2*b*x*gamma(m + 2)) - b*m*x**2*x**m*lerchphi(b*x*exp_polar(I*pi)/a, 1, m + 1)*gamma(m
+ 1)/(a**3*gamma(m + 2) + a**2*b*x*gamma(m + 2))) + B*(-a*m**2*x**2*x**m*lerchphi(b*x*exp_polar(I*pi)/a, 1, m
+ 2)*gamma(m + 2)/(a**3*gamma(m + 3) + a**2*b*x*gamma(m + 3)) - 3*a*m*x**2*x**m*lerchphi(b*x*exp_polar(I*pi)/
a, 1, m + 2)*gamma(m + 2)/(a**3*gamma(m + 3) + a**2*b*x*gamma(m + 3)) + a*m*x**2*x**m*gamma(m + 2)/(a**3*gamma
(m + 3) + a**2*b*x*gamma(m + 3)) - 2*a*x**2*x**m*lerchphi(b*x*exp_polar(I*pi)/a, 1, m + 2)*gamma(m + 2)/(a**3*
gamma(m + 3) + a**2*b*x*gamma(m + 3)) + 2*a*x**2*x**m*gamma(m + 2)/(a**3*gamma(m + 3) + a**2*b*x*gamma(m + 3))
- b*m**2*x**3*x**m*lerchphi(b*x*exp_polar(I*pi)/a, 1, m + 2)*gamma(m + 2)/(a**3*gamma(m + 3) + a**2*b*x*gamma
(m + 3)) - 3*b*m*x**3*x**m*lerchphi(b*x*exp_polar(I*pi)/a, 1, m + 2)*gamma(m + 2)/(a**3*gamma(m + 3) + a**2*b*
x*gamma(m + 3)) - 2*b*x**3*x**m*lerchphi(b*x*exp_polar(I*pi)/a, 1, m + 2)*gamma(m + 2)/(a**3*gamma(m + 3) + a*
*2*b*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 )} x^{m}}{{\left (b x + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^m*(B*x+A)/(b*x+a)^2,x, algorithm="giac")
[Out]
integrate((B*x + A)*x^m/(b*x + a)^2, x)