3.14 \(\int \frac {(e x)^m (A+B x^n) (c+d x^n)^2}{(a+b x^n)^3} \, dx\)

Optimal. Leaf size=322 \[ \frac {(e x)^{m+1} \, _2F_1\left (1,\frac {m+1}{n};\frac {m+n+1}{n};-\frac {b x^n}{a}\right ) (b c (a B (m+1)-A b (m-2 n+1)) (a d (m+1)-b c (m-n+1))-a d (A b (m+1)-a B (m+2 n+1)) (b c (m+1)-a d (m+n+1)))}{2 a^3 b^3 e (m+1) n^2}+\frac {d (e x)^{m+1} (A b (m+1)-a B (m+2 n+1)) (b c (m+1)-a d (m+n+1))}{2 a^2 b^3 e (m+1) n^2}+\frac {(e x)^{m+1} (b c-a d) \left (c (a B (m+1)-A b (m-2 n+1))-d x^n (A b (m+1)-a B (m+2 n+1))\right )}{2 a^2 b^2 e n^2 \left (a+b x^n\right )}+\frac {(e x)^{m+1} (A b-a B) \left (c+d x^n\right )^2}{2 a b e n \left (a+b x^n\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.54, antiderivative size = 322, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {594, 459, 364} \[ \frac {(e x)^{m+1} \, _2F_1\left (1,\frac {m+1}{n};\frac {m+n+1}{n};-\frac {b x^n}{a}\right ) (b c (a B (m+1)-A b (m-2 n+1)) (a d (m+1)-b c (m-n+1))-a d (A b (m+1)-a B (m+2 n+1)) (b c (m+1)-a d (m+n+1)))}{2 a^3 b^3 e (m+1) n^2}+\frac {(e x)^{m+1} (b c-a d) \left (c (a B (m+1)-A b (m-2 n+1))-d x^n (A b (m+1)-a B (m+2 n+1))\right )}{2 a^2 b^2 e n^2 \left (a+b x^n\right )}+\frac {d (e x)^{m+1} (A b (m+1)-a B (m+2 n+1)) (b c (m+1)-a d (m+n+1))}{2 a^2 b^3 e (m+1) n^2}+\frac {(e x)^{m+1} (A b-a B) \left (c+d x^n\right )^2}{2 a b e n \left (a+b x^n\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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])

Rule 459

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(d*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(b*e*(m + n*(p + 1) + 1)), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m +
n*(p + 1) + 1)), Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0]
&& NeQ[m + n*(p + 1) + 1, 0]

Rule 594

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] + Dist[
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 (c+d x^n\right )^2}{\left (a+b x^n\right )^3} \, dx &=\frac {(A b-a B) (e x)^{1+m} \left (c+d x^n\right )^2}{2 a b e n \left (a+b x^n\right )^2}-\frac {\int \frac {(e x)^m \left (c+d x^n\right ) \left (-c (a B (1+m)-A b (1+m-2 n))+d (A b (1+m)-a B (1+m+2 n)) x^n\right )}{\left (a+b x^n\right )^2} \, dx}{2 a b n}\\ &=\frac {(A b-a B) (e x)^{1+m} \left (c+d x^n\right )^2}{2 a b e n \left (a+b x^n\right )^2}+\frac {(b c-a d) (e x)^{1+m} \left (c (a B (1+m)-A b (1+m-2 n))-d (A b (1+m)-a B (1+m+2 n)) x^n\right )}{2 a^2 b^2 e n^2 \left (a+b x^n\right )}+\frac {\int \frac {(e x)^m \left (c (a B (1+m)-A b (1+m-2 n)) (a d (1+m)-b c (1+m-n))+d (b c (1+m)-a d (1+m+n)) (A b (1+m)-a B (1+m+2 n)) x^n\right )}{a+b x^n} \, dx}{2 a^2 b^2 n^2}\\ &=\frac {d (b c (1+m)-a d (1+m+n)) (A b (1+m)-a B (1+m+2 n)) (e x)^{1+m}}{2 a^2 b^3 e (1+m) n^2}+\frac {(A b-a B) (e x)^{1+m} \left (c+d x^n\right )^2}{2 a b e n \left (a+b x^n\right )^2}+\frac {(b c-a d) (e x)^{1+m} \left (c (a B (1+m)-A b (1+m-2 n))-d (A b (1+m)-a B (1+m+2 n)) x^n\right )}{2 a^2 b^2 e n^2 \left (a+b x^n\right )}+\frac {\left (c (a B (1+m)-A b (1+m-2 n)) (a d (1+m)-b c (1+m-n))-\frac {a d (b c (1+m)-a d (1+m+n)) (A b (1+m)-a B (1+m+2 n))}{b}\right ) \int \frac {(e x)^m}{a+b x^n} \, dx}{2 a^2 b^2 n^2}\\ &=\frac {d (b c (1+m)-a d (1+m+n)) (A b (1+m)-a B (1+m+2 n)) (e x)^{1+m}}{2 a^2 b^3 e (1+m) n^2}+\frac {(A b-a B) (e x)^{1+m} \left (c+d x^n\right )^2}{2 a b e n \left (a+b x^n\right )^2}+\frac {(b c-a d) (e x)^{1+m} \left (c (a B (1+m)-A b (1+m-2 n))-d (A b (1+m)-a B (1+m+2 n)) x^n\right )}{2 a^2 b^2 e n^2 \left (a+b x^n\right )}+\frac {\left (c (a B (1+m)-A b (1+m-2 n)) (a d (1+m)-b c (1+m-n))-\frac {a d (b c (1+m)-a d (1+m+n)) (A b (1+m)-a B (1+m+2 n))}{b}\right ) (e x)^{1+m} \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b x^n}{a}\right )}{2 a^3 b^2 e (1+m) n^2}\\ \end {align*}

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Mathematica [A]  time = 0.31, size = 168, normalized size = 0.52 \[ \frac {x (e x)^m \left (\frac {(A b-a B) (b c-a d)^2 \, _2F_1\left (3,\frac {m+1}{n};\frac {m+n+1}{n};-\frac {b x^n}{a}\right )}{a^3}+\frac {(b c-a d) (-3 a B d+2 A b d+b B c) \, _2F_1\left (2,\frac {m+1}{n};\frac {m+n+1}{n};-\frac {b x^n}{a}\right )}{a^2}+\frac {d (-3 a B d+A b d+2 b B c) \, _2F_1\left (1,\frac {m+1}{n};\frac {m+n+1}{n};-\frac {b x^n}{a}\right )}{a}+B d^2\right )}{b^3 (m+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B d^{2} x^{3 \, n} + A c^{2} + {\left (2 \, B c d + A d^{2}\right )} x^{2 \, n} + {\left (B c^{2} + 2 \, A c d\right )} x^{n}\right )} \left (e x\right )^{m}}{b^{3} x^{3 \, n} + 3 \, a b^{2} x^{2 \, n} + 3 \, a^{2} b x^{n} + a^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B x^{n} + A\right )} {\left (d x^{n} + c\right )}^{2} \left (e x\right )^{m}}{{\left (b x^{n} + a\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.81, size = 0, normalized size = 0.00 \[ \int \frac {\left (B \,x^{n}+A \right ) \left (d \,x^{n}+c \right )^{2} \left (e x \right )^{m}}{\left (b \,x^{n}+a \right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ {\left ({\left ({\left (m^{2} - m {\left (3 \, n - 2\right )} + 2 \, n^{2} - 3 \, n + 1\right )} b^{3} c^{2} e^{m} - 2 \, {\left (m^{2} - m {\left (n - 2\right )} - n + 1\right )} a b^{2} c d e^{m} + {\left (m^{2} + m {\left (n + 2\right )} + n + 1\right )} a^{2} b d^{2} e^{m}\right )} A - {\left ({\left (m^{2} - m {\left (n - 2\right )} - n + 1\right )} a b^{2} c^{2} e^{m} - 2 \, {\left (m^{2} + m {\left (n + 2\right )} + n + 1\right )} a^{2} b c d e^{m} + {\left (m^{2} + m {\left (3 \, n + 2\right )} + 2 \, n^{2} + 3 \, n + 1\right )} a^{3} d^{2} e^{m}\right )} B\right )} \int \frac {x^{m}}{2 \, {\left (a^{2} b^{4} n^{2} x^{n} + a^{3} b^{3} n^{2}\right )}}\,{d x} + \frac {2 \, B a^{2} b^{2} d^{2} e^{m} n^{2} x e^{\left (m \log \relax (x) + 2 \, n \log \relax (x)\right )} - {\left ({\left ({\left (m^{2} - m {\left (3 \, n - 2\right )} - 3 \, n + 1\right )} a b^{3} c^{2} e^{m} - 2 \, {\left (m^{2} - m {\left (n - 2\right )} - n + 1\right )} a^{2} b^{2} c d e^{m} + {\left (m^{2} + m {\left (n + 2\right )} + n + 1\right )} a^{3} b d^{2} e^{m}\right )} A - {\left ({\left (m^{2} - m {\left (n - 2\right )} - n + 1\right )} a^{2} b^{2} c^{2} e^{m} - 2 \, {\left (m^{2} + m {\left (n + 2\right )} + n + 1\right )} a^{3} b c d e^{m} + {\left (m^{2} + m {\left (3 \, n + 2\right )} + 2 \, n^{2} + 3 \, n + 1\right )} a^{4} d^{2} e^{m}\right )} B\right )} x x^{m} - {\left ({\left ({\left (m^{2} - 2 \, m {\left (n - 1\right )} - 2 \, n + 1\right )} b^{4} c^{2} e^{m} - 2 \, {\left (m^{2} + 2 \, m + 1\right )} a b^{3} c d e^{m} + {\left (m^{2} + 2 \, m {\left (n + 1\right )} + 2 \, n + 1\right )} a^{2} b^{2} d^{2} e^{m}\right )} A - {\left ({\left (m^{2} + 2 \, m + 1\right )} a b^{3} c^{2} e^{m} - 2 \, {\left (m^{2} + 2 \, m {\left (n + 1\right )} + 2 \, n + 1\right )} a^{2} b^{2} c d e^{m} + {\left (m^{2} + 2 \, m {\left (2 \, n + 1\right )} + 4 \, n^{2} + 4 \, n + 1\right )} a^{3} b d^{2} e^{m}\right )} B\right )} x e^{\left (m \log \relax (x) + n \log \relax (x)\right )}}{2 \, {\left ({\left (m n^{2} + n^{2}\right )} a^{2} b^{5} x^{2 \, n} + 2 \, {\left (m n^{2} + n^{2}\right )} a^{3} b^{4} x^{n} + {\left (m n^{2} + n^{2}\right )} a^{4} b^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (e\,x\right )}^m\,\left (A+B\,x^n\right )\,{\left (c+d\,x^n\right )}^2}{{\left (a+b\,x^n\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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