3.1.27 \(\int \frac {(e x)^m (A+B x^n)}{(a+b x^n)^2 (c+d x^n)} \, dx\) [27]

Optimal. Leaf size=212 \[ \frac {(A b-a B) (e x)^{1+m}}{a (b c-a d) e n \left (a+b x^n\right )}+\frac {(A b (a d (1+m-2 n)-b c (1+m-n))+a B (b c (1+m)-a d (1+m-n))) (e x)^{1+m} \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b x^n}{a}\right )}{a^2 (b c-a d)^2 e (1+m) n}-\frac {d (B c-A d) (e x)^{1+m} \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {d x^n}{c}\right )}{c (b c-a d)^2 e (1+m)} \]

[Out]

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

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Rubi [A]
time = 0.35, antiderivative size = 212, normalized size of antiderivative = 1.00, 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 = {609, 611, 371} \begin {gather*} \frac {(e x)^{m+1} \, _2F_1\left (1,\frac {m+1}{n};\frac {m+n+1}{n};-\frac {b x^n}{a}\right ) (A b (a d (m-2 n+1)-b c (m-n+1))+a B (b c (m+1)-a d (m-n+1)))}{a^2 e (m+1) n (b c-a d)^2}-\frac {d (e x)^{m+1} (B c-A d) \, _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)^2}+\frac {(e x)^{m+1} (A b-a B)}{a e n (b c-a d) \left (a+b x^n\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 609

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 611

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]

Rubi steps

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

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Mathematica [A]
time = 0.36, size = 177, normalized size = 0.83 \begin {gather*} \frac {x (e x)^m \left (\frac {(-A b+a B) (-b c+a d)}{a n \left (a+b x^n\right )}-\frac {(A b (-a d (1+m-2 n)+b c (1+m-n))+a B (-b c (1+m)+a d (1+m-n))) \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b x^n}{a}\right )}{a^2 (1+m) n}+\frac {d (-B c+A d) \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {d x^n}{c}\right )}{c (1+m)}\right )}{(b c-a d)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.11, size = 0, normalized size = 0.00 \[\int \frac {\left (e x \right )^{m} \left (A +B \,x^{n}\right )}{\left (a +b \,x^{n}\right )^{2} \left (c +d \,x^{n}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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

[Out]

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

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: HeuristicGCDFailed} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: HeuristicGCDFailed >> no luck

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (e\,x\right )}^m\,\left (A+B\,x^n\right )}{{\left (a+b\,x^n\right )}^2\,\left (c+d\,x^n\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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