\(\int \frac {x^m (A+B x)}{(a+b x)^3} \, dx\) [377]
Optimal result
Integrand size = 16, antiderivative size = 81 \[
\int \frac {x^m (A+B x)}{(a+b x)^3} \, dx=\frac {(A b-a B) x^{1+m}}{2 a b (a+b x)^2}+\frac {(A b (1-m)+a B (1+m)) x^{1+m} \operatorname {Hypergeometric2F1}\left (2,1+m,2+m,-\frac {b x}{a}\right )}{2 a^3 b (1+m)}
\]
[Out]
1/2*(A*b-B*a)*x^(1+m)/a/b/(b*x+a)^2+1/2*(A*b*(1-m)+a*B*(1+m))*x^(1+m)*hypergeom([2, 1+m],[2+m],-b*x/a)/a^3/b/(
1+m)
Rubi [A] (verified)
Time = 0.02 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00,
number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {79, 66}
\[
\int \frac {x^m (A+B x)}{(a+b x)^3} \, dx=\frac {x^{m+1} (a B (m+1)+A b (1-m)) \operatorname {Hypergeometric2F1}\left (2,m+1,m+2,-\frac {b x}{a}\right )}{2 a^3 b (m+1)}+\frac {x^{m+1} (A b-a B)}{2 a b (a+b x)^2}
\]
[In]
Int[(x^m*(A + B*x))/(a + b*x)^3,x]
[Out]
((A*b - a*B)*x^(1 + m))/(2*a*b*(a + b*x)^2) + ((A*b*(1 - m) + a*B*(1 + m))*x^(1 + m)*Hypergeometric2F1[2, 1 +
m, 2 + m, -((b*x)/a)])/(2*a^3*b*(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 79
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] || L
tQ[p, n]))))
Rubi steps \begin{align*}
\text {integral}& = \frac {(A b-a B) x^{1+m}}{2 a b (a+b x)^2}-\frac {(A b (-1+m)-a B (1+m)) \int \frac {x^m}{(a+b x)^2} \, dx}{2 a b} \\ & = \frac {(A b-a B) x^{1+m}}{2 a b (a+b x)^2}+\frac {(A b (1-m)+a B (1+m)) x^{1+m} \, _2F_1\left (2,1+m;2+m;-\frac {b x}{a}\right )}{2 a^3 b (1+m)} \\
\end{align*}
Mathematica [A] (verified)
Time = 0.05 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.88
\[
\int \frac {x^m (A+B x)}{(a+b x)^3} \, dx=\frac {x^{1+m} \left (\frac {a^2 (A b-a B)}{(a+b x)^2}-\frac {(A b (-1+m)-a B (1+m)) \operatorname {Hypergeometric2F1}\left (2,1+m,2+m,-\frac {b x}{a}\right )}{1+m}\right )}{2 a^3 b}
\]
[In]
Integrate[(x^m*(A + B*x))/(a + b*x)^3,x]
[Out]
(x^(1 + m)*((a^2*(A*b - a*B))/(a + b*x)^2 - ((A*b*(-1 + m) - a*B*(1 + m))*Hypergeometric2F1[2, 1 + m, 2 + m, -
((b*x)/a)])/(1 + m)))/(2*a^3*b)
Maple [F]
\[\int \frac {x^{m} \left (B x +A \right )}{\left (b x +a \right )^{3}}d x\]
[In]
int(x^m*(B*x+A)/(b*x+a)^3,x)
[Out]
int(x^m*(B*x+A)/(b*x+a)^3,x)
Fricas [F]
\[
\int \frac {x^m (A+B x)}{(a+b x)^3} \, dx=\int { \frac {{\left (B x + A\right )} x^{m}}{{\left (b x + a\right )}^{3}} \,d x }
\]
[In]
integrate(x^m*(B*x+A)/(b*x+a)^3,x, algorithm="fricas")
[Out]
integral((B*x + A)*x^m/(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a^3), x)
Sympy [C] (verification not implemented)
Result contains complex when optimal does not.
Time = 3.64 (sec) , antiderivative size = 1678, normalized size of antiderivative = 20.72
\[
\int \frac {x^m (A+B x)}{(a+b x)^3} \, dx=\text {Too large to display}
\]
[In]
integrate(x**m*(B*x+A)/(b*x+a)**3,x)
[Out]
A*(a**2*m**3*x**(m + 1)*lerchphi(b*x*exp_polar(I*pi)/a, 1, m + 1)*gamma(m + 1)/(2*a**5*gamma(m + 2) + 4*a**4*b
*x*gamma(m + 2) + 2*a**3*b**2*x**2*gamma(m + 2)) - a**2*m**2*x**(m + 1)*gamma(m + 1)/(2*a**5*gamma(m + 2) + 4*
a**4*b*x*gamma(m + 2) + 2*a**3*b**2*x**2*gamma(m + 2)) - a**2*m*x**(m + 1)*lerchphi(b*x*exp_polar(I*pi)/a, 1,
m + 1)*gamma(m + 1)/(2*a**5*gamma(m + 2) + 4*a**4*b*x*gamma(m + 2) + 2*a**3*b**2*x**2*gamma(m + 2)) + a**2*m*x
**(m + 1)*gamma(m + 1)/(2*a**5*gamma(m + 2) + 4*a**4*b*x*gamma(m + 2) + 2*a**3*b**2*x**2*gamma(m + 2)) + 2*a**
2*x**(m + 1)*gamma(m + 1)/(2*a**5*gamma(m + 2) + 4*a**4*b*x*gamma(m + 2) + 2*a**3*b**2*x**2*gamma(m + 2)) + 2*
a*b*m**3*x*x**(m + 1)*lerchphi(b*x*exp_polar(I*pi)/a, 1, m + 1)*gamma(m + 1)/(2*a**5*gamma(m + 2) + 4*a**4*b*x
*gamma(m + 2) + 2*a**3*b**2*x**2*gamma(m + 2)) - a*b*m**2*x*x**(m + 1)*gamma(m + 1)/(2*a**5*gamma(m + 2) + 4*a
**4*b*x*gamma(m + 2) + 2*a**3*b**2*x**2*gamma(m + 2)) - 2*a*b*m*x*x**(m + 1)*lerchphi(b*x*exp_polar(I*pi)/a, 1
, m + 1)*gamma(m + 1)/(2*a**5*gamma(m + 2) + 4*a**4*b*x*gamma(m + 2) + 2*a**3*b**2*x**2*gamma(m + 2)) + a*b*x*
x**(m + 1)*gamma(m + 1)/(2*a**5*gamma(m + 2) + 4*a**4*b*x*gamma(m + 2) + 2*a**3*b**2*x**2*gamma(m + 2)) + b**2
*m**3*x**2*x**(m + 1)*lerchphi(b*x*exp_polar(I*pi)/a, 1, m + 1)*gamma(m + 1)/(2*a**5*gamma(m + 2) + 4*a**4*b*x
*gamma(m + 2) + 2*a**3*b**2*x**2*gamma(m + 2)) - b**2*m*x**2*x**(m + 1)*lerchphi(b*x*exp_polar(I*pi)/a, 1, m +
1)*gamma(m + 1)/(2*a**5*gamma(m + 2) + 4*a**4*b*x*gamma(m + 2) + 2*a**3*b**2*x**2*gamma(m + 2))) + B*(a**2*m*
*3*x**(m + 2)*lerchphi(b*x*exp_polar(I*pi)/a, 1, m + 2)*gamma(m + 2)/(2*a**5*gamma(m + 3) + 4*a**4*b*x*gamma(m
+ 3) + 2*a**3*b**2*x**2*gamma(m + 3)) + 3*a**2*m**2*x**(m + 2)*lerchphi(b*x*exp_polar(I*pi)/a, 1, m + 2)*gamm
a(m + 2)/(2*a**5*gamma(m + 3) + 4*a**4*b*x*gamma(m + 3) + 2*a**3*b**2*x**2*gamma(m + 3)) - a**2*m**2*x**(m + 2
)*gamma(m + 2)/(2*a**5*gamma(m + 3) + 4*a**4*b*x*gamma(m + 3) + 2*a**3*b**2*x**2*gamma(m + 3)) + 2*a**2*m*x**(
m + 2)*lerchphi(b*x*exp_polar(I*pi)/a, 1, m + 2)*gamma(m + 2)/(2*a**5*gamma(m + 3) + 4*a**4*b*x*gamma(m + 3) +
2*a**3*b**2*x**2*gamma(m + 3)) - a**2*m*x**(m + 2)*gamma(m + 2)/(2*a**5*gamma(m + 3) + 4*a**4*b*x*gamma(m + 3
) + 2*a**3*b**2*x**2*gamma(m + 3)) + 2*a**2*x**(m + 2)*gamma(m + 2)/(2*a**5*gamma(m + 3) + 4*a**4*b*x*gamma(m
+ 3) + 2*a**3*b**2*x**2*gamma(m + 3)) + 2*a*b*m**3*x*x**(m + 2)*lerchphi(b*x*exp_polar(I*pi)/a, 1, m + 2)*gamm
a(m + 2)/(2*a**5*gamma(m + 3) + 4*a**4*b*x*gamma(m + 3) + 2*a**3*b**2*x**2*gamma(m + 3)) + 6*a*b*m**2*x*x**(m
+ 2)*lerchphi(b*x*exp_polar(I*pi)/a, 1, m + 2)*gamma(m + 2)/(2*a**5*gamma(m + 3) + 4*a**4*b*x*gamma(m + 3) + 2
*a**3*b**2*x**2*gamma(m + 3)) - a*b*m**2*x*x**(m + 2)*gamma(m + 2)/(2*a**5*gamma(m + 3) + 4*a**4*b*x*gamma(m +
3) + 2*a**3*b**2*x**2*gamma(m + 3)) + 4*a*b*m*x*x**(m + 2)*lerchphi(b*x*exp_polar(I*pi)/a, 1, m + 2)*gamma(m
+ 2)/(2*a**5*gamma(m + 3) + 4*a**4*b*x*gamma(m + 3) + 2*a**3*b**2*x**2*gamma(m + 3)) - 2*a*b*m*x*x**(m + 2)*ga
mma(m + 2)/(2*a**5*gamma(m + 3) + 4*a**4*b*x*gamma(m + 3) + 2*a**3*b**2*x**2*gamma(m + 3)) + b**2*m**3*x**2*x*
*(m + 2)*lerchphi(b*x*exp_polar(I*pi)/a, 1, m + 2)*gamma(m + 2)/(2*a**5*gamma(m + 3) + 4*a**4*b*x*gamma(m + 3)
+ 2*a**3*b**2*x**2*gamma(m + 3)) + 3*b**2*m**2*x**2*x**(m + 2)*lerchphi(b*x*exp_polar(I*pi)/a, 1, m + 2)*gamm
a(m + 2)/(2*a**5*gamma(m + 3) + 4*a**4*b*x*gamma(m + 3) + 2*a**3*b**2*x**2*gamma(m + 3)) + 2*b**2*m*x**2*x**(m
+ 2)*lerchphi(b*x*exp_polar(I*pi)/a, 1, m + 2)*gamma(m + 2)/(2*a**5*gamma(m + 3) + 4*a**4*b*x*gamma(m + 3) +
2*a**3*b**2*x**2*gamma(m + 3)))
Maxima [F]
\[
\int \frac {x^m (A+B x)}{(a+b x)^3} \, dx=\int { \frac {{\left (B x + A\right )} x^{m}}{{\left (b x + a\right )}^{3}} \,d x }
\]
[In]
integrate(x^m*(B*x+A)/(b*x+a)^3,x, algorithm="maxima")
[Out]
integrate((B*x + A)*x^m/(b*x + a)^3, x)
Giac [F]
\[
\int \frac {x^m (A+B x)}{(a+b x)^3} \, dx=\int { \frac {{\left (B x + A\right )} x^{m}}{{\left (b x + a\right )}^{3}} \,d x }
\]
[In]
integrate(x^m*(B*x+A)/(b*x+a)^3,x, algorithm="giac")
[Out]
integrate((B*x + A)*x^m/(b*x + a)^3, x)
Mupad [F(-1)]
Timed out. \[
\int \frac {x^m (A+B x)}{(a+b x)^3} \, dx=\int \frac {x^m\,\left (A+B\,x\right )}{{\left (a+b\,x\right )}^3} \,d x
\]
[In]
int((x^m*(A + B*x))/(a + b*x)^3,x)
[Out]
int((x^m*(A + B*x))/(a + b*x)^3, x)