3.1.34 \(\int \frac {(e x)^m (A+B x^n)}{(a+b x^n)^2 (c+d x^n)^2} \, dx\) [34]

Optimal. Leaf size=315 \[ \frac {d (A b c-2 a B c+a A d) (e x)^{1+m}}{a c (b c-a d)^2 e n \left (c+d x^n\right )}+\frac {(A b-a B) (e x)^{1+m}}{a (b c-a d) e n \left (a+b x^n\right ) \left (c+d x^n\right )}+\frac {b (a B (b c (1+m)-a d (1+m-2 n))+A b (a d (1+m-3 n)-b c (1+m-n))) (e x)^{1+m} \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b x^n}{a}\right )}{a^2 (b c-a d)^3 e (1+m) n}-\frac {d (b c (A d (1+m-3 n)-B c (1+m-2 n))+a d (B c (1+m)-A d (1+m-n))) (e x)^{1+m} \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {d x^n}{c}\right )}{c^2 (b c-a d)^3 e (1+m) n} \]

[Out]

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

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Rubi [A]
time = 0.72, antiderivative size = 315, normalized size of antiderivative = 1.00, number of steps used = 6, 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 {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 b (a d (m-3 n+1)-b c (m-n+1))+a B (b c (m+1)-a d (m-2 n+1)))}{a^2 e (m+1) n (b c-a d)^3}-\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 ) (a d (B c (m+1)-A d (m-n+1))+b c (A d (m-3 n+1)-B c (m-2 n+1)))}{c^2 e (m+1) n (b c-a d)^3}+\frac {d (e x)^{m+1} (a A d-2 a B c+A b c)}{a c e n (b c-a d)^2 \left (c+d x^n\right )}+\frac {(e x)^{m+1} (A b-a B)}{a e n (b c-a d) \left (a+b x^n\right ) \left (c+d 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)^2),x]

[Out]

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

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

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Mathematica [A]
time = 0.74, size = 242, normalized size = 0.77 \begin {gather*} \frac {x (e x)^m \left (\frac {b (-A b+a B) (-b c+a d)}{a \left (a+b x^n\right )}+\frac {d (-b c+a d) (B c-A d)}{c \left (c+d x^n\right )}-\frac {b (a B (-b c (1+m)+a d (1+m-2 n))+A b (-a d (1+m-3 n)+b c (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)}+\frac {d (b c (-A d (1+m-3 n)+B c (1+m-2 n))+a d (-B c (1+m)+A d (1+m-n))) \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {d x^n}{c}\right )}{c^2 (1+m)}\right )}{(b c-a d)^3 n} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(e*x)^m*((b*(-(A*b) + a*B)*(-(b*c) + a*d))/(a*(a + b*x^n)) + (d*(-(b*c) + a*d)*(B*c - A*d))/(c*(c + d*x^n))
- (b*(a*B*(-(b*c*(1 + m)) + a*d*(1 + m - 2*n)) + A*b*(-(a*d*(1 + m - 3*n)) + b*c*(1 + m - n)))*Hypergeometric
2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)])/(a^2*(1 + m)) + (d*(b*c*(-(A*d*(1 + m - 3*n)) + B*c*(1 + m - 2
*n)) + a*d*(-(B*c*(1 + m)) + A*d*(1 + m - n)))*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)])/(
c^2*(1 + m))))/((b*c - a*d)^3*n)

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Maple [F]
time = 0.15, size = 0, normalized size = 0.00 \[\int \frac {\left (e x \right )^{m} \left (A +B \,x^{n}\right )}{\left (a +b \,x^{n}\right )^{2} \left (c +d \,x^{n}\right )^{2}}\, dx\]

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)^2,x)

[Out]

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

[Out]

(((m*e^m - (n - 1)*e^m)*b^3*c - (m*e^m - (3*n - 1)*e^m)*a*b^2*d)*A - ((m*e^m + e^m)*a*b^2*c - (m*e^m - (2*n -
1)*e^m)*a^2*b*d)*B)*integrate(-x^m/(a^2*b^3*c^3*n - 3*a^3*b^2*c^2*d*n + 3*a^4*b*c*d^2*n - a^5*d^3*n + (a*b^4*c
^3*n - 3*a^2*b^3*c^2*d*n + 3*a^3*b^2*c*d^2*n - a^4*b*d^3*n)*x^n), x) + (((m*e^m - (3*n - 1)*e^m)*b*c*d^2 - (m*
e^m - (n - 1)*e^m)*a*d^3)*A - ((m*e^m - (2*n - 1)*e^m)*b*c^2*d - (m*e^m + e^m)*a*c*d^2)*B)*integrate(-x^m/(b^3
*c^5*n - 3*a*b^2*c^4*d*n + 3*a^2*b*c^3*d^2*n - a^3*c^2*d^3*n + (b^3*c^4*d*n - 3*a*b^2*c^3*d^2*n + 3*a^2*b*c^2*
d^3*n - a^3*c*d^4*n)*x^n), x) + (((b^2*c^2*e^m + a^2*d^2*e^m)*A - (a*b*c^2*e^m + a^2*c*d*e^m)*B)*x*x^m - (2*B*
a*b*c*d*e^m - (b^2*c*d*e^m + a*b*d^2*e^m)*A)*x*e^(m*log(x) + n*log(x)))/(a^2*b^2*c^4*n - 2*a^3*b*c^3*d*n + a^4
*c^2*d^2*n + (a*b^3*c^3*d*n - 2*a^2*b^2*c^2*d^2*n + a^3*b*c*d^3*n)*x^(2*n) + (a*b^3*c^4*n - a^2*b^2*c^3*d*n -
a^3*b*c^2*d^2*n + a^4*c*d^3*n)*x^n)

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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)^2,x, algorithm="fricas")

[Out]

integral((B*x^n + A)*(x*e)^m/(b^2*d^2*x^(4*n) + a^2*c^2 + 2*(b^2*c*d + a*b*d^2)*x^(3*n) + (b^2*c^2 + 4*a*b*c*d
+ a^2*d^2)*x^(2*n) + 2*(a*b*c^2 + a^2*c*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)**2,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)^2,x, algorithm="giac")

[Out]

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

[Out]

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

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