3.77 \(\int \frac{(e x)^m}{(a+b x) (a c-b c x)} \, dx\)

Optimal. Leaf size=48 \[ \frac{(e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};\frac{b^2 x^2}{a^2}\right )}{a^2 c e (m+1)} \]

[Out]

((e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, (b^2*x^2)/a^2])/(a^2*c*e*(1 + m))

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Rubi [A]  time = 0.0121768, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {73, 364} \[ \frac{(e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};\frac{b^2 x^2}{a^2}\right )}{a^2 c e (m+1)} \]

Antiderivative was successfully verified.

[In]

Int[(e*x)^m/((a + b*x)*(a*c - b*c*x)),x]

[Out]

((e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, (b^2*x^2)/a^2])/(a^2*c*e*(1 + m))

Rule 73

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[(a*c + b*
d*x^2)^m*(e + f*x)^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[n, m] && Integer
Q[m]

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin{align*} \int \frac{(e x)^m}{(a+b x) (a c-b c x)} \, dx &=\int \frac{(e x)^m}{a^2 c-b^2 c x^2} \, dx\\ &=\frac{(e x)^{1+m} \, _2F_1\left (1,\frac{1+m}{2};\frac{3+m}{2};\frac{b^2 x^2}{a^2}\right )}{a^2 c e (1+m)}\\ \end{align*}

Mathematica [A]  time = 0.012466, size = 46, normalized size = 0.96 \[ \frac{x (e x)^m \, _2F_1\left (1,\frac{m+1}{2};\frac{m+1}{2}+1;\frac{b^2 x^2}{a^2}\right )}{a^2 c (m+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.043, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex \right ) ^{m}}{ \left ( bx+a \right ) \left ( -bcx+ac \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(e*x)^m/(b^2*c*x^2 - a^2*c), x)

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Sympy [C]  time = 1.60685, size = 73, normalized size = 1.52 \begin{align*} - \frac{e^{m} m x^{m} \Phi \left (\frac{a}{b x}, 1, m e^{i \pi }\right ) \Gamma \left (- m\right )}{2 a b c \Gamma \left (1 - m\right )} + \frac{e^{m} m x^{m} \Phi \left (\frac{b x e^{i \pi }}{a}, 1, m\right ) \Gamma \left (- m\right )}{2 a b c \Gamma \left (1 - m\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-e**m*m*x**m*lerchphi(a/(b*x), 1, m*exp_polar(I*pi))*gamma(-m)/(2*a*b*c*gamma(1 - m)) + e**m*m*x**m*lerchphi(b
*x*exp_polar(I*pi)/a, 1, m)*gamma(-m)/(2*a*b*c*gamma(1 - m))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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