3.382 \(\int \frac{x^m}{(a+b x) (c+d x)} \, dx\)

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

[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))

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Rubi [A]  time = 0.0285017, antiderivative size = 82, normalized size of antiderivative = 1., 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 = {86, 64} \[ \frac{b x^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac{b x}{a}\right )}{a (m+1) (b c-a d)}-\frac{d x^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac{d x}{c}\right )}{c (m+1) (b c-a d)} \]

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 86

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]

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) (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 = 0.0161422, size = 65, normalized size = 0.79 \[ \frac{x^{m+1} \left (a d \, _2F_1\left (1,m+1;m+2;-\frac{d x}{c}\right )-b c \, _2F_1\left (1,m+1;m+2;-\frac{b x}{a}\right )\right )}{a c (m+1) (a d-b c)} \]

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))

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Maple [F]  time = 0.045, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{m}}{ \left ( bx+a \right ) \left ( dx+c \right ) }}\, dx \end{align*}

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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{m}}{{\left (b x + a\right )}{\left (d x + c\right )}}\,{d x} \end{align*}

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)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{m}}{b d x^{2} + a c +{\left (b c + a d\right )} x}, x\right ) \end{align*}

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)

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Sympy [C]  time = 2.01332, size = 102, normalized size = 1.24 \begin{align*} - \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{align*}

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))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{m}}{{\left (b x + a\right )}{\left (d x + c\right )}}\,{d x} \end{align*}

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)