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

Optimal. Leaf size=112 \[ \frac{(e x)^{m+1} (B c (m+1)-A d (m-2 n+1)) \, _2F_1\left (2,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right )}{2 c^3 d e (m+1) n}-\frac{(e x)^{m+1} (B c-A d)}{2 c d e n \left (c+d x^n\right )^2} \]

[Out]

-((B*c - A*d)*(e*x)^(1 + m))/(2*c*d*e*n*(c + d*x^n)^2) + ((B*c*(1 + m) - A*d*(1 + m - 2*n))*(e*x)^(1 + m)*Hype
rgeometric2F1[2, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)])/(2*c^3*d*e*(1 + m)*n)

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Rubi [A]  time = 0.0556258, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {457, 364} \[ \frac{(e x)^{m+1} (B c (m+1)-A d (m-2 n+1)) \, _2F_1\left (2,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right )}{2 c^3 d e (m+1) n}-\frac{(e x)^{m+1} (B c-A d)}{2 c d e n \left (c+d x^n\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((B*c - A*d)*(e*x)^(1 + m))/(2*c*d*e*n*(c + d*x^n)^2) + ((B*c*(1 + m) - A*d*(1 + m - 2*n))*(e*x)^(1 + m)*Hype
rgeometric2F1[2, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)])/(2*c^3*d*e*(1 + m)*n)

Rule 457

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

Mathematica [A]  time = 0.0734161, size = 83, normalized size = 0.74 \[ \frac{x (e x)^m \left ((A d-B c) \, _2F_1\left (3,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right )+B c \, _2F_1\left (2,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right )\right )}{c^3 d (m+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(e*x)^m*(B*c*Hypergeometric2F1[2, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)] + (-(B*c) + A*d)*Hypergeometric2F
1[3, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)]))/(c^3*d*(1 + m))

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Maple [F]  time = 0.366, 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 ) ^{3}}}\, 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)^3,x)

[Out]

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

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (B x^{n} + A\right )} \left (e x\right )^{m}}{d^{3} x^{3 \, n} + 3 \, c d^{2} x^{2 \, n} + 3 \, c^{2} d x^{n} + c^{3}}, 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)^3,x, algorithm="fricas")

[Out]

integral((B*x^n + A)*(e*x)^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)]  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)**3,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 (e x\right )^{m}}{{\left (d x^{n} + c\right )}^{3}}\,{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)^3,x, algorithm="giac")

[Out]

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