3.377 \(\int \frac {x^m (A+B x)}{(a+b x)^3} \, dx\)

Optimal. Leaf size=81 \[ \frac {x^{m+1} (a B (m+1)+A b (1-m)) \, _2F_1\left (2,m+1;m+2;-\frac {b x}{a}\right )}{2 a^3 b (m+1)}+\frac {x^{m+1} (A b-a B)}{2 a b (a+b x)^2} \]

[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)

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Rubi [A]  time = 0.03, antiderivative size = 81, normalized size of antiderivative = 1.00, 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} \[ \frac {x^{m+1} (a B (m+1)+A b (1-m)) \, _2F_1\left (2,m+1;m+2;-\frac {b x}{a}\right )}{2 a^3 b (m+1)}+\frac {x^{m+1} (A b-a B)}{2 a b (a+b x)^2} \]

Antiderivative was successfully verified.

[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 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])))

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]))))

Rubi steps

\begin {align*} \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)) \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*}

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Mathematica [A]  time = 0.04, size = 71, normalized size = 0.88 \[ \frac {x^{m+1} \left (\frac {a^2 (A b-a B)}{(a+b x)^2}-\frac {(A b (m-1)-a B (m+1)) \, _2F_1\left (2,m+1;m+2;-\frac {b x}{a}\right )}{m+1}\right )}{2 a^3 b} \]

Antiderivative was successfully verified.

[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)

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fricas [F]  time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B x + A\right )} x^{m}}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B x + A\right )} x^{m}}{{\left (b x + a\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [F]  time = 0.11, size = 0, normalized size = 0.00 \[ \int \frac {\left (B x +A \right ) x^{m}}{\left (b x +a \right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*(B*x+A)/(b*x+a)^3,x)

[Out]

int(x^m*(B*x+A)/(b*x+a)^3,x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B x + A\right )} x^{m}}{{\left (b x + a\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^m\,\left (A+B\,x\right )}{{\left (a+b\,x\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^m*(A + B*x))/(a + b*x)^3,x)

[Out]

int((x^m*(A + B*x))/(a + b*x)^3, x)

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sympy [C]  time = 8.54, size = 1680, normalized size = 20.74 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m*(B*x+A)/(b*x+a)**3,x)

[Out]

A*(a**2*m**3*x*x**m*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*g
amma(m + 2) + 2*a**3*b**2*x**2*gamma(m + 2)) - a**2*m**2*x*x**m*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*x**m*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*x**m*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*x**m*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**2*x**m*l
erchphi(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**2*x**m*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**2*x**m*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**2*x**m*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**3*x**m*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**3*x**m*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**2*x**m*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**2*x**m*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*gam
ma(m + 3) + 2*a**3*b**2*x**2*gamma(m + 3)) - a**2*m**2*x**2*x**m*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**2*x**m*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**2*x*
*m*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**2*
x**m*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**3*x**m*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)) + 6*a*b*m**2*x**3*x**m*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**3*x**m*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**3*x**m*le
rchphi(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**3*x**m*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)) + b**2*m**3*x**4*x**m*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**4*x**m*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*ga
mma(m + 3)) + 2*b**2*m*x**4*x**m*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)))

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