3.33 \(\int \frac{(e x)^m (A+B x^n)}{(a+b x^n) (c+d x^n)^2} \, dx\)

Optimal. Leaf size=211 \[ \frac{(e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right ) (a d (B c (m+1)-A d (m-n+1))+b c (A d (m-2 n+1)-B c (m-n+1)))}{c^2 e (m+1) n (b c-a d)^2}+\frac{b (e x)^{m+1} (A b-a B) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{a e (m+1) (b c-a d)^2}+\frac{(e x)^{m+1} (B c-A d)}{c e n (b c-a d) \left (c+d x^n\right )} \]

[Out]

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

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Rubi [A]  time = 0.520973, antiderivative size = 211, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {595, 597, 364} \[ \frac{(e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right ) (a d (B c (m+1)-A d (m-n+1))+b c (A d (m-2 n+1)-B c (m-n+1)))}{c^2 e (m+1) n (b c-a d)^2}+\frac{b (e x)^{m+1} (A b-a B) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{a e (m+1) (b c-a d)^2}+\frac{(e x)^{m+1} (B c-A d)}{c e n (b c-a d) \left (c+d x^n\right )} \]

Antiderivative was successfully verified.

[In]

Int[((e*x)^m*(A + B*x^n))/((a + b*x^n)*(c + d*x^n)^2),x]

[Out]

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

Rule 595

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_)*((e_) + (f_.)*(x_)^(n_)), x
_Symbol] :> -Simp[((b*e - a*f)*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*g*n*(b*c - a*d)*(p +
1)), x] + Dist[1/(a*n*(b*c - a*d)*(p + 1)), Int[(g*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f)*(
m + 1) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d,
e, f, g, m, n, q}, x] && LtQ[p, -1]

Rule 597

Int[(((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n_)))/((c_) + (d_.)*(x_)^(n_)), x_Sy
mbol] :> Int[ExpandIntegrand[((g*x)^m*(a + b*x^n)^p*(e + f*x^n))/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e,
f, g, m, n, p}, x]

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

Mathematica [A]  time = 0.208247, size = 150, normalized size = 0.71 \[ \frac{x (e x)^m \left (b c^2 (A b-a B) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )+a c d (a B-A b) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right )+a (b c-a d) (B c-A d) \, _2F_1\left (2,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right )\right )}{a c^2 (m+1) (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[((e*x)^m*(A + B*x^n))/((a + b*x^n)*(c + d*x^n)^2),x]

[Out]

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

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Maple [F]  time = 0.667, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex \right ) ^{m} \left ( A+B{x}^{n} \right ) }{ \left ( a+b{x}^{n} \right ) \left ( c+d{x}^{n} \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (B c e^{m} - A d e^{m}\right )} x x^{m}}{b c^{3} n - a c^{2} d n +{\left (b c^{2} d n - a c d^{2} n\right )} x^{n}} -{\left ({\left (a d^{2} e^{m}{\left (m - n + 1\right )} - b c d e^{m}{\left (m - 2 \, n + 1\right )}\right )} A +{\left (b c^{2} e^{m}{\left (m - n + 1\right )} - a c d e^{m}{\left (m + 1\right )}\right )} B\right )} \int \frac{x^{m}}{b^{2} c^{4} n - 2 \, a b c^{3} d n + a^{2} c^{2} d^{2} n +{\left (b^{2} c^{3} d n - 2 \, a b c^{2} d^{2} n + a^{2} c d^{3} n\right )} x^{n}}\,{d x} -{\left (B a b e^{m} - A b^{2} e^{m}\right )} \int \frac{x^{m}}{a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2} +{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{n}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*(A+B*x^n)/(a+b*x^n)/(c+d*x^n)^2,x, algorithm="maxima")

[Out]

(B*c*e^m - A*d*e^m)*x*x^m/(b*c^3*n - a*c^2*d*n + (b*c^2*d*n - a*c*d^2*n)*x^n) - ((a*d^2*e^m*(m - n + 1) - b*c*
d*e^m*(m - 2*n + 1))*A + (b*c^2*e^m*(m - n + 1) - a*c*d*e^m*(m + 1))*B)*integrate(x^m/(b^2*c^4*n - 2*a*b*c^3*d
*n + a^2*c^2*d^2*n + (b^2*c^3*d*n - 2*a*b*c^2*d^2*n + a^2*c*d^3*n)*x^n), x) - (B*a*b*e^m - A*b^2*e^m)*integrat
e(x^m/(a*b^2*c^2 - 2*a^2*b*c*d + a^3*d^2 + (b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)*x^n), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (B x^{n} + A\right )} \left (e x\right )^{m}}{b d^{2} x^{3 \, n} + a c^{2} +{\left (2 \, b c d + a d^{2}\right )} x^{2 \, n} +{\left (b c^{2} + 2 \, a c 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)/(a+b*x^n)/(c+d*x^n)^2,x, algorithm="fricas")

[Out]

integral((B*x^n + A)*(e*x)^m/(b*d^2*x^(3*n) + a*c^2 + (2*b*c*d + a*d^2)*x^(2*n) + (b*c^2 + 2*a*c*d)*x^n), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)**m*(A+B*x**n)/(a+b*x**n)/(c+d*x**n)**2,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*(A+B*x^n)/(a+b*x^n)/(c+d*x^n)^2,x, algorithm="giac")

[Out]

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