3.1025 \(\int \frac{(e x)^m}{(a+b x^n) (c+d x^n)} \, dx\)

Optimal. Leaf size=114 \[ \frac{b (e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{a e (m+1) (b c-a d)}-\frac{d (e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right )}{c e (m+1) (b c-a d)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0624926, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {508, 364} \[ \frac{b (e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{a e (m+1) (b c-a d)}-\frac{d (e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right )}{c e (m+1) (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 508

Int[((e_.)*(x_))^(m_.)/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[b/(b*c - a*d), I
nt[(e*x)^m/(a + b*x^n), x], x] - Dist[d/(b*c - a*d), Int[(e*x)^m/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e,
n, m}, x] && NeQ[b*c - a*d, 0]

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

Mathematica [A]  time = 0.0689063, size = 88, normalized size = 0.77 \[ \frac{x (e x)^m \left (a d \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right )-b c \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )\right )}{a c (m+1) (a d-b c)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.07, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex \right ) ^{m}}{ \left ( a+b{x}^{n} \right ) \left ( c+d{x}^{n} \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((e*x)**m/((a + b*x**n)*(c + d*x**n)), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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