3.6.93 \(\int (a+b x^n)^p (c+d x^n)^p (e+\frac {(b c+a d) e (1+n+n p) x^n}{a c}+\frac {b d e (1+2 n+2 n p) x^{2 n}}{a c}) \, dx\) [593]

Optimal. Leaf size=31 \[ \frac {e x \left (a+b x^n\right )^{1+p} \left (c+d x^n\right )^{1+p}}{a c} \]

[Out]

e*x*(a+b*x^n)^(1+p)*(c+d*x^n)^(1+p)/a/c

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Rubi [A]
time = 0.14, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 69, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.014, Rules used = {1911} \begin {gather*} \frac {e x \left (a+b x^n\right )^{p+1} \left (c+d x^n\right )^{p+1}}{a c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^n)^p*(c + d*x^n)^p*(e + ((b*c + a*d)*e*(1 + n + n*p)*x^n)/(a*c) + (b*d*e*(1 + 2*n + 2*n*p)*x^(2*n
))/(a*c)),x]

[Out]

(e*x*(a + b*x^n)^(1 + p)*(c + d*x^n)^(1 + p))/(a*c)

Rule 1911

Int[((a_) + (b_.)*(x_)^(n_.))^(p_.)*((c_) + (d_.)*(x_)^(n_.))^(p_.)*((e_) + (f_.)*(x_)^(n_.) + (g_.)*(x_)^(n2_
.)), x_Symbol] :> Simp[e*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(p + 1)/(a*c)), x] /; FreeQ[{a, b, c, d, e, f, g,
n, p}, x] && EqQ[n2, 2*n] && EqQ[a*c*f - e*(b*c + a*d)*(n*(p + 1) + 1), 0] && EqQ[a*c*g - b*d*e*(2*n*(p + 1) +
1), 0]

Rubi steps

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

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Mathematica [A]
time = 0.62, size = 31, normalized size = 1.00 \begin {gather*} \frac {e x \left (a+b x^n\right )^{1+p} \left (c+d x^n\right )^{1+p}}{a c} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^n)^p*(c + d*x^n)^p*(e + ((b*c + a*d)*e*(1 + n + n*p)*x^n)/(a*c) + (b*d*e*(1 + 2*n + 2*n*p)*
x^(2*n))/(a*c)),x]

[Out]

(e*x*(a + b*x^n)^(1 + p)*(c + d*x^n)^(1 + p))/(a*c)

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Maple [A]
time = 0.51, size = 52, normalized size = 1.68

method	result	size
risch	\(\frac {\left (a +b \,x^{n}\right )^{p} \left (b d \,x^{2 n}+a d \,x^{n}+b c \,x^{n}+a c \right ) e x \left (c +d \,x^{n}\right )^{p}}{a c}\)	\(52\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*x^n)^p*(c+d*x^n)^p*(e+(a*d+b*c)*e*(n*p+n+1)*x^n/a/c+b*d*e*(2*n*p+2*n+1)*x^(2*n)/a/c),x,method=_RETURN
VERBOSE)

[Out]

(a+b*x^n)^p*(b*d*(x^n)^2+a*d*x^n+b*c*x^n+a*c)*e*x/a/c*(c+d*x^n)^p

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Maxima [A]
time = 0.37, size = 64, normalized size = 2.06 \begin {gather*} \frac {{\left (a c x e + b d x e^{\left (2 \, n \log \left (x\right ) + 1\right )} + {\left (b c + a d\right )} x e^{\left (n \log \left (x\right ) + 1\right )}\right )} e^{\left (p \log \left (b x^{n} + a\right ) + p \log \left (d x^{n} + c\right )\right )}}{a c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^n)^p*(c+d*x^n)^p*(e+(a*d+b*c)*e*(n*p+n+1)*x^n/a/c+b*d*e*(2*n*p+2*n+1)*x^(2*n)/a/c),x, algorit
hm="maxima")

[Out]

(a*c*x*e + b*d*x*e^(2*n*log(x) + 1) + (b*c + a*d)*x*e^(n*log(x) + 1))*e^(p*log(b*x^n + a) + p*log(d*x^n + c))/
(a*c)

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Fricas [A]
time = 0.40, size = 54, normalized size = 1.74 \begin {gather*} \frac {{\left (b d e x x^{2 \, n} + a c e x + {\left (b c + a d\right )} e x x^{n}\right )} {\left (b x^{n} + a\right )}^{p} {\left (d x^{n} + c\right )}^{p}}{a c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^n)^p*(c+d*x^n)^p*(e+(a*d+b*c)*e*(n*p+n+1)*x^n/a/c+b*d*e*(2*n*p+2*n+1)*x^(2*n)/a/c),x, algorit
hm="fricas")

[Out]

(b*d*e*x*x^(2*n) + a*c*e*x + (b*c + a*d)*e*x*x^n)*(b*x^n + a)^p*(d*x^n + c)^p/(a*c)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: HeuristicGCDFailed} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**n)**p*(c+d*x**n)**p*(e+(a*d+b*c)*e*(n*p+n+1)*x**n/a/c+b*d*e*(2*n*p+2*n+1)*x**(2*n)/a/c),x)

[Out]

Exception raised: HeuristicGCDFailed >> no luck

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 115 vs. \(2 (32) = 64\).
time = 0.87, size = 115, normalized size = 3.71 \begin {gather*} \frac {{\left (b x^{n} + a\right )}^{p} {\left (d x^{n} + c\right )}^{p} b d x x^{2 \, n} e + {\left (b x^{n} + a\right )}^{p} {\left (d x^{n} + c\right )}^{p} b c x x^{n} e + {\left (b x^{n} + a\right )}^{p} {\left (d x^{n} + c\right )}^{p} a d x x^{n} e + {\left (b x^{n} + a\right )}^{p} {\left (d x^{n} + c\right )}^{p} a c x e}{a c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^n)^p*(c+d*x^n)^p*(e+(a*d+b*c)*e*(n*p+n+1)*x^n/a/c+b*d*e*(2*n*p+2*n+1)*x^(2*n)/a/c),x, algorit
hm="giac")

[Out]

((b*x^n + a)^p*(d*x^n + c)^p*b*d*x*x^(2*n)*e + (b*x^n + a)^p*(d*x^n + c)^p*b*c*x*x^n*e + (b*x^n + a)^p*(d*x^n
+ c)^p*a*d*x*x^n*e + (b*x^n + a)^p*(d*x^n + c)^p*a*c*x*e)/(a*c)

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Mupad [B]
time = 5.30, size = 76, normalized size = 2.45 \begin {gather*} {\left (c+d\,x^n\right )}^p\,\left (e\,x\,{\left (a+b\,x^n\right )}^p+\frac {e\,x\,x^n\,\left (a\,d+b\,c\right )\,{\left (a+b\,x^n\right )}^p}{a\,c}+\frac {b\,d\,e\,x\,x^{2\,n}\,{\left (a+b\,x^n\right )}^p}{a\,c}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^n)^p*(c + d*x^n)^p*(e + (e*x^n*(a*d + b*c)*(n + n*p + 1))/(a*c) + (b*d*e*x^(2*n)*(2*n + 2*n*p + 1
))/(a*c)),x)

[Out]

(c + d*x^n)^p*(e*x*(a + b*x^n)^p + (e*x*x^n*(a*d + b*c)*(a + b*x^n)^p)/(a*c) + (b*d*e*x*x^(2*n)*(a + b*x^n)^p)
/(a*c))

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