3.1.37 \(\int \frac {(e x)^m (a+b x^n) (A+B x^n)}{(c+d x^n)^3} \, dx\) [37]

Optimal. Leaf size=228 \[ -\frac {(b c-a d) (e x)^{1+m} \left (A+B x^n\right )}{2 c d e n \left (c+d x^n\right )^2}-\frac {(a d (A d (1+m-2 n)-B c (1+m-n))-b c (A d (1+m)-B c (1+m+n))) (e x)^{1+m}}{2 c^2 d^2 e n^2 \left (c+d x^n\right )}-\frac {(A d (b c (1+m)-a d (1+m-2 n)) (1+m-n)+B c (1+m) (a d (1+m-n)-b c (1+m+n))) (e x)^{1+m} \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {d x^n}{c}\right )}{2 c^3 d^2 e (1+m) n^2} \]

[Out]

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

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Rubi [A]
time = 0.19, antiderivative size = 227, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {608, 468, 371} \begin {gather*} -\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 (m-n+1) (b c (m+1)-a d (m-2 n+1))+B c (m+1) (a d (m-n+1)-b c (m+n+1)))}{2 c^3 d^2 e (m+1) n^2}+\frac {(e x)^{m+1} (A d (b c (m+1)-a d (m-2 n+1))+B c (a d (m-n+1)-b c (m+n+1)))}{2 c^2 d^2 e n^2 \left (c+d x^n\right )}-\frac {(e x)^{m+1} (b c-a d) \left (A+B x^n\right )}{2 c d e n \left (c+d x^n\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*((b*c - a*d)*(e*x)^(1 + m)*(A + B*x^n))/(c*d*e*n*(c + d*x^n)^2) + ((A*d*(b*c*(1 + m) - a*d*(1 + m - 2*n))
+ B*c*(a*d*(1 + m - n) - b*c*(1 + m + n)))*(e*x)^(1 + m))/(2*c^2*d^2*e*n^2*(c + d*x^n)) - ((A*d*(b*c*(1 + m)
- a*d*(1 + m - 2*n))*(1 + m - n) + B*c*(1 + m)*(a*d*(1 + m - n) - b*c*(1 + m + n)))*(e*x)^(1 + m)*Hypergeometr
ic2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)])/(2*c^3*d^2*e*(1 + m)*n^2)

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 468

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*c - a*d
))*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*b*e*n*(p + 1))), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a
*b*n*(p + 1)), Int[(e*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0]
&& LtQ[p, -1] && (( !IntegerQ[p + 1/2] && NeQ[p, -5/4]) ||  !RationalQ[m] || (IGtQ[n, 0] && ILtQ[p + 1/2, 0]
&& LeQ[-1, m, (-n)*(p + 1)]))

Rule 608

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/(a*b*g*n*(p + 1))), x] + Dis
t[1/(a*b*n*(p + 1)), Int[(g*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1)*Simp[c*(b*e*n*(p + 1) + (b*e - a*f)*(
m + 1)) + d*(b*e*n*(p + 1) + (b*e - a*f)*(m + n*q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n},
x] && LtQ[p, -1] && GtQ[q, 0] &&  !(EqQ[q, 1] && SimplerQ[b*c - a*d, b*e - a*f])

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

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1153\) vs. \(2(228)=456\).
time = 0.59, size = 1153, normalized size = 5.06 \begin {gather*} \frac {x (e x)^m \left (b B c^4 (1+m) n-A b c^3 d (1+m) n-a B c^3 d (1+m) n+a A c^2 d^2 (1+m) n-b B c^3 (1+m) \left (c+d x^n\right )+A b c^2 d (1+m) \left (c+d x^n\right )+a B c^2 d (1+m) \left (c+d x^n\right )-a A c d^2 (1+m) \left (c+d x^n\right )-b B c^3 m (1+m) \left (c+d x^n\right )+A b c^2 d m (1+m) \left (c+d x^n\right )+a B c^2 d m (1+m) \left (c+d x^n\right )-a A c d^2 m (1+m) \left (c+d x^n\right )-2 b B c^3 (1+m) n \left (c+d x^n\right )+2 a A c d^2 (1+m) n \left (c+d x^n\right )+b B c^2 \left (c+d x^n\right )^2 \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {d x^n}{c}\right )-A b c d \left (c+d x^n\right )^2 \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {d x^n}{c}\right )-a B c d \left (c+d x^n\right )^2 \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {d x^n}{c}\right )+a A d^2 \left (c+d x^n\right )^2 \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {d x^n}{c}\right )+2 b B c^2 m \left (c+d x^n\right )^2 \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {d x^n}{c}\right )-2 A b c d m \left (c+d x^n\right )^2 \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {d x^n}{c}\right )-2 a B c d m \left (c+d x^n\right )^2 \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {d x^n}{c}\right )+2 a A d^2 m \left (c+d x^n\right )^2 \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {d x^n}{c}\right )+b B c^2 m^2 \left (c+d x^n\right )^2 \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {d x^n}{c}\right )-A b c d m^2 \left (c+d x^n\right )^2 \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {d x^n}{c}\right )-a B c d m^2 \left (c+d x^n\right )^2 \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {d x^n}{c}\right )+a A d^2 m^2 \left (c+d x^n\right )^2 \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {d x^n}{c}\right )+b B c^2 n \left (c+d x^n\right )^2 \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {d x^n}{c}\right )+A b c d n \left (c+d x^n\right )^2 \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {d x^n}{c}\right )+a B c d n \left (c+d x^n\right )^2 \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {d x^n}{c}\right )-3 a A d^2 n \left (c+d x^n\right )^2 \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {d x^n}{c}\right )+b B c^2 m n \left (c+d x^n\right )^2 \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {d x^n}{c}\right )+A b c d m n \left (c+d x^n\right )^2 \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {d x^n}{c}\right )+a B c d m n \left (c+d x^n\right )^2 \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {d x^n}{c}\right )-3 a A d^2 m n \left (c+d x^n\right )^2 \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {d x^n}{c}\right )+2 a A d^2 n^2 \left (c+d x^n\right )^2 \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {d x^n}{c}\right )\right )}{2 c^3 d^2 (1+m) n^2 \left (c+d x^n\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(e*x)^m*(b*B*c^4*(1 + m)*n - A*b*c^3*d*(1 + m)*n - a*B*c^3*d*(1 + m)*n + a*A*c^2*d^2*(1 + m)*n - b*B*c^3*(1
+ m)*(c + d*x^n) + A*b*c^2*d*(1 + m)*(c + d*x^n) + a*B*c^2*d*(1 + m)*(c + d*x^n) - a*A*c*d^2*(1 + m)*(c + d*x
^n) - b*B*c^3*m*(1 + m)*(c + d*x^n) + A*b*c^2*d*m*(1 + m)*(c + d*x^n) + a*B*c^2*d*m*(1 + m)*(c + d*x^n) - a*A*
c*d^2*m*(1 + m)*(c + d*x^n) - 2*b*B*c^3*(1 + m)*n*(c + d*x^n) + 2*a*A*c*d^2*(1 + m)*n*(c + d*x^n) + b*B*c^2*(c
+ d*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)] - A*b*c*d*(c + d*x^n)^2*Hypergeometri
c2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)] - a*B*c*d*(c + d*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m
+ n)/n, -((d*x^n)/c)] + a*A*d^2*(c + d*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)] +
2*b*B*c^2*m*(c + d*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)] - 2*A*b*c*d*m*(c + d*x^
n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)] - 2*a*B*c*d*m*(c + d*x^n)^2*Hypergeometric2F
1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)] + 2*a*A*d^2*m*(c + d*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 +
m + n)/n, -((d*x^n)/c)] + b*B*c^2*m^2*(c + d*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c
)] - A*b*c*d*m^2*(c + d*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)] - a*B*c*d*m^2*(c +
d*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)] + a*A*d^2*m^2*(c + d*x^n)^2*Hypergeomet
ric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)] + b*B*c^2*n*(c + d*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1
+ m + n)/n, -((d*x^n)/c)] + A*b*c*d*n*(c + d*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/
c)] + a*B*c*d*n*(c + d*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)] - 3*a*A*d^2*n*(c +
d*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)] + b*B*c^2*m*n*(c + d*x^n)^2*Hypergeometr
ic2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)] + A*b*c*d*m*n*(c + d*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (
1 + m + n)/n, -((d*x^n)/c)] + a*B*c*d*m*n*(c + d*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^
n)/c)] - 3*a*A*d^2*m*n*(c + d*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)] + 2*a*A*d^2*
n^2*(c + d*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)]))/(2*c^3*d^2*(1 + m)*n^2*(c + d
*x^n)^2)

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Maple [F]
time = 0.07, size = 0, normalized size = 0.00 \[\int \frac {\left (e x \right )^{m} \left (a +b \,x^{n}\right ) \left (A +B \,x^{n}\right )}{\left (c +d \,x^{n}\right )^{3}}\, dx\]

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)^3,x)

[Out]

int((e*x)^m*(a+b*x^n)*(A+B*x^n)/(c+d*x^n)^3,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)/(c+d*x^n)^3,x, algorithm="maxima")

[Out]

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

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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)/(c+d*x^n)^3,x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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)/(c+d*x**n)**3,x)

[Out]

Timed out

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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)/(c+d*x^n)^3,x, algorithm="giac")

[Out]

integrate((B*x^n + A)*(b*x^n + a)*(x*e)^m/(d*x^n + c)^3, 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 )}{{\left (c+d\,x^n\right )}^3} \,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))/(c + d*x^n)^3,x)

[Out]

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

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