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

Optimal. Leaf size=268 \[ -\frac{(e x)^{m+1} (b c-a d) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right ) (A b (b c (m-n+1)-a d (m+n+1))-a B (b c (m+1)-a d (m+2 n+1)))}{a^2 b^3 e (m+1) n}-\frac{d (e x)^{m+1} (A b (2 b c (m+1)-a d (m+n+1))-a B (2 b c (m+n+1)-a d (m+2 n+1)))}{a b^3 e (m+1) n}-\frac{d^2 x^{n+1} (e x)^m (A b (m+n+1)-a B (m+2 n+1))}{a b^2 n (m+n+1)}+\frac{(e x)^{m+1} (A b-a B) \left (c+d x^n\right )^2}{a b e n \left (a+b x^n\right )} \]

[Out]

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

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Rubi [A]  time = 0.668707, antiderivative size = 268, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161, Rules used = {594, 570, 20, 30, 364} \[ -\frac{(e x)^{m+1} (b c-a d) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right ) (A b (b c (m-n+1)-a d (m+n+1))-a B (b c (m+1)-a d (m+2 n+1)))}{a^2 b^3 e (m+1) n}-\frac{d (e x)^{m+1} (A b (2 b c (m+1)-a d (m+n+1))-a B (2 b c (m+n+1)-a d (m+2 n+1)))}{a b^3 e (m+1) n}-\frac{d^2 x^{n+1} (e x)^m (A b (m+n+1)-a B (m+2 n+1))}{a b^2 n (m+n+1)}+\frac{(e x)^{m+1} (A b-a B) \left (c+d x^n\right )^2}{a b e n \left (a+b x^n\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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])

Rule 570

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_))^
(r_.), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^r, x], x] /; FreeQ[{a,
b, c, d, e, f, g, m, n}, x] && IGtQ[p, -2] && IGtQ[q, 0] && IGtQ[r, 0]

Rule 20

Int[(u_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Dist[(b^IntPart[n]*(b*v)^FracPart[n])/(a^IntPart[n
]*(a*v)^FracPart[n]), Int[u*(a*v)^(m + n), x], x] /; FreeQ[{a, b, m, n}, x] &&  !IntegerQ[m] &&  !IntegerQ[n]
&&  !IntegerQ[m + n]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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

Mathematica [A]  time = 0.245594, size = 159, normalized size = 0.59 \[ \frac{x (e x)^m \left (\frac{(A b-a B) (b c-a d)^2 \, _2F_1\left (2,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{a^2 (m+1)}+\frac{(b c-a d) (-3 a B d+2 A b d+b B c) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{a (m+1)}+\frac{d (-2 a B d+A b d+2 b B c)}{m+1}+\frac{b B d^2 x^n}{m+n+1}\right )}{b^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.5, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex \right ) ^{m} \left ( A+B{x}^{n} \right ) \left ( c+d{x}^{n} \right ) ^{2}}{ \left ( a+b{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)*(c+d*x^n)^2/(a+b*x^n)^2,x)

[Out]

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

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

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)^2,x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\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^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}}, x\right ) \end{align*}

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)^2,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^2*x^(2*n) + 2*
a*b*x^n + a^2), 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)*(c+d*x**n)**2/(a+b*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 (d x^{n} + c\right )}^{2} \left (e x\right )^{m}}{{\left (b x^{n} + a\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)*(c+d*x^n)^2/(a+b*x^n)^2,x, algorithm="giac")

[Out]

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