3.42 \(\int (e x)^m (a+b x^n)^p (A+B x^n) (c+d x^n) \, dx\)

Optimal. Leaf size=271 \[ -\frac {(e x)^{m+1} \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac {m+1}{n},-p;\frac {m+n+1}{n};-\frac {b x^n}{a}\right ) (A b (m+n p+n+1) (a d (m+1)-b c (m+n (p+2)+1))-a (m+1) (a B d (m+n+1)-b (A d n+B c (m+n (p+2)+1))))}{b^2 e (m+1) (m+n p+n+1) (m+n (p+2)+1)}-\frac {(e x)^{m+1} \left (a+b x^n\right )^{p+1} (a B d (m+n+1)-b (A d n+B c (m+n (p+2)+1)))}{b^2 e (m+n p+n+1) (m+n (p+2)+1)}+\frac {d (e x)^{m+1} \left (A+B x^n\right ) \left (a+b x^n\right )^{p+1}}{b e (m+n (p+2)+1)} \]

[Out]

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

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Rubi [A]  time = 0.33, antiderivative size = 255, normalized size of antiderivative = 0.94, number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {596, 459, 365, 364} \[ \frac {(e x)^{m+1} \left (a+b x^n\right )^{p+1} (-a B d (m+n+1)+A b d n+b B c (m+n (p+2)+1))}{b^2 e (m+n p+n+1) (m+n (p+2)+1)}-\frac {(e x)^{m+1} \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac {m+1}{n},-p;\frac {m+n+1}{n};-\frac {b x^n}{a}\right ) \left (\frac {a (-a B d (m+n+1)+A b d n+b B c (m+n (p+2)+1))}{b (m+n p+n+1)}+a A d-\frac {A b c (m+n (p+2)+1)}{m+1}\right )}{b e (m+n (p+2)+1)}+\frac {d (e x)^{m+1} \left (A+B x^n\right ) \left (a+b x^n\right )^{p+1}}{b e (m+n (p+2)+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 365

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^FracPart[p])
/(1 + (b*x^n)/a)^FracPart[p], Int[(c*x)^m*(1 + (b*x^n)/a)^p, x], 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 596

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[(f*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q)/(b*g*(m + n*(p + q + 1) + 1)), x] + Dis
t[1/(b*(m + n*(p + q + 1) + 1)), Int[(g*x)^m*(a + b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*((b*e - a*f)*(m + 1) + b
*e*n*(p + q + 1)) + (d*(b*e - a*f)*(m + 1) + f*n*q*(b*c - a*d) + b*e*d*n*(p + q + 1))*x^n, x], x], x] /; FreeQ
[{a, b, c, d, e, f, g, m, n, p}, x] && GtQ[q, 0] &&  !(EqQ[q, 1] && SimplerQ[e + f*x^n, c + d*x^n])

Rubi steps

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

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Mathematica [A]  time = 0.29, size = 164, normalized size = 0.61 \[ x (e x)^m \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \left (x^n \left (\frac {(A d+B c) \, _2F_1\left (\frac {m+n+1}{n},-p;\frac {m+2 n+1}{n};-\frac {b x^n}{a}\right )}{m+n+1}+\frac {B d x^n \, _2F_1\left (\frac {m+2 n+1}{n},-p;\frac {m+3 n+1}{n};-\frac {b x^n}{a}\right )}{m+2 n+1}\right )+\frac {A c \, _2F_1\left (\frac {m+1}{n},-p;\frac {m+n+1}{n};-\frac {b x^n}{a}\right )}{m+1}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(e*x)^m*(a + b*x^n)^p*((A*c*Hypergeometric2F1[(1 + m)/n, -p, (1 + m + n)/n, -((b*x^n)/a)])/(1 + m) + x^n*((
(B*c + A*d)*Hypergeometric2F1[(1 + m + n)/n, -p, (1 + m + 2*n)/n, -((b*x^n)/a)])/(1 + m + n) + (B*d*x^n*Hyperg
eometric2F1[(1 + m + 2*n)/n, -p, (1 + m + 3*n)/n, -((b*x^n)/a)])/(1 + m + 2*n))))/(1 + (b*x^n)/a)^p

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fricas [F]  time = 0.84, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (B d x^{2 \, n} + A c + {\left (B c + A d\right )} x^{n}\right )} {\left (b x^{n} + a\right )}^{p} \left (e x\right )^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.68, size = 0, normalized size = 0.00 \[ \int \left (B \,x^{n}+A \right ) \left (d \,x^{n}+c \right ) \left (e x \right )^{m} \left (b \,x^{n}+a \right )^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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