3.4.82 \(\int \frac {x^m}{(a+b x) (c+d x)} \, dx\) [382]

Optimal. Leaf size=82 \[ \frac {b x^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac {b x}{a}\right )}{a (b c-a d) (1+m)}-\frac {d x^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac {d x}{c}\right )}{c (b c-a d) (1+m)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.02, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {88, 66} \begin {gather*} \frac {b x^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac {b x}{a}\right )}{a (m+1) (b c-a d)}-\frac {d x^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac {d x}{c}\right )}{c (m+1) (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^m/((a + b*x)*(c + d*x)),x]

[Out]

(b*x^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, -((b*x)/a)])/(a*(b*c - a*d)*(1 + m)) - (d*x^(1 + m)*Hypergeome
tric2F1[1, 1 + m, 2 + m, -((d*x)/c)])/(c*(b*c - a*d)*(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 88

Int[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), In
t[(e + f*x)^p/(a + b*x), x], x] - Dist[d/(b*c - a*d), Int[(e + f*x)^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d,
e, f, p}, x] &&  !IntegerQ[p]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 0.07, size = 65, normalized size = 0.79 \begin {gather*} \frac {x^{1+m} \left (-b c \, _2F_1\left (1,1+m;2+m;-\frac {b x}{a}\right )+a d \, _2F_1\left (1,1+m;2+m;-\frac {d x}{c}\right )\right )}{a c (-b c+a d) (1+m)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {x^{m}}{\left (b x +a \right ) \left (d x +c \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x^m/(b*d*x^2 + a*c + (b*c + a*d)*x), x)

________________________________________________________________________________________

Sympy [C] Result contains complex when optimal does not.
time = 0.90, size = 102, normalized size = 1.24 \begin {gather*} - \frac {b^{m} m x^{m} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m\right ) \Gamma \left (- m\right )}{a b^{m} d \Gamma \left (1 - m\right ) - b b^{m} c \Gamma \left (1 - m\right )} + \frac {b^{m} m x^{m} \Phi \left (\frac {c e^{i \pi }}{d x}, 1, m e^{i \pi }\right ) \Gamma \left (- m\right )}{a b^{m} d \Gamma \left (1 - m\right ) - b b^{m} c \Gamma \left (1 - m\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b**m*m*x**m*lerchphi(b*x*exp_polar(I*pi)/a, 1, m)*gamma(-m)/(a*b**m*d*gamma(1 - m) - b*b**m*c*gamma(1 - m)) +
b**m*m*x**m*lerchphi(c*exp_polar(I*pi)/(d*x), 1, m*exp_polar(I*pi))*gamma(-m)/(a*b**m*d*gamma(1 - m) - b*b**m
*c*gamma(1 - m))

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^m}{\left (a+b\,x\right )\,\left (c+d\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m/((a + b*x)*(c + d*x)),x)

[Out]

int(x^m/((a + b*x)*(c + d*x)), x)

________________________________________________________________________________________