3.581 \(\int \left (a+b x^n\right )^p \left (c+d x^n\right )^p \left (e+\frac{(b c+a d) e (1+n+n p) x^n}{a c}+\frac{b d e (1+2 n+2 n p) x^{2 n}}{a c}\right ) \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.313953, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 69, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.014 \[ \frac{e x \left (a+b x^n\right )^{p+1} \left (c+d x^n\right )^{p+1}}{a c} \]

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)

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Rubi in Sympy [A]  time = 75.3723, size = 26, normalized size = 0.84 \[ \frac{e x \left (a + b x^{n}\right )^{p + 1} \left (c + d x^{n}\right )^{p + 1}}{a c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_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]

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

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Mathematica [A]  time = 0.292104, size = 31, normalized size = 1. \[ \frac{e x \left (a+b x^n\right )^{p+1} \left (c+d x^n\right )^{p+1}}{a c} \]

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 = 0.185, size = 52, normalized size = 1.7 \[{\frac{ \left ( a+b{x}^{n} \right ) ^{p} \left ( bd \left ({x}^{n} \right ) ^{2}+ad{x}^{n}+bc{x}^{n}+ac \right ) ex \left ( c+d{x}^{n} \right ) ^{p}}{ac}} \]

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)

[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 = 1.93932, size = 80, normalized size = 2.58 \[ \frac{{\left (b d e x x^{2 \, n} + a c e x +{\left (b c e + a d e\right )} x x^{n}\right )} e^{\left (p \log \left (b x^{n} + a\right ) + p \log \left (d x^{n} + c\right )\right )}}{a c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((2*n*p + 2*n + 1)*b*d*e*x^(2*n)/(a*c) + (b*c + a*d)*(n*p + n + 1)*e*x^n/(a*c) + e)*(b*x^n + a)^p*(d*x^n + c)^p,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.248956, size = 73, normalized size = 2.35 \[ \frac{{\left (b d e x x^{2 \, n} + a c e x +{\left (b c + a d\right )} e x x^{n}\right )}{\left (b x^{n} + a\right )}^{p}{\left (d x^{n} + c\right )}^{p}}{a c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((2*n*p + 2*n + 1)*b*d*e*x^(2*n)/(a*c) + (b*c + a*d)*(n*p + n + 1)*e*x^n/(a*c) + e)*(b*x^n + a)^p*(d*x^n + c)^p,x, algorithm="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(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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]

Timed out

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GIAC/XCAS [A]  time = 0.264626, size = 196, normalized size = 6.32 \[ \frac{b d x e^{\left (p{\rm ln}\left (b e^{\left (n{\rm ln}\left (x\right )\right )} + a\right ) + p{\rm ln}\left (d e^{\left (n{\rm ln}\left (x\right )\right )} + c\right ) + 2 \, n{\rm ln}\left (x\right ) + 1\right )} + b c x e^{\left (p{\rm ln}\left (b e^{\left (n{\rm ln}\left (x\right )\right )} + a\right ) + p{\rm ln}\left (d e^{\left (n{\rm ln}\left (x\right )\right )} + c\right ) + n{\rm ln}\left (x\right ) + 1\right )} + a d x e^{\left (p{\rm ln}\left (b e^{\left (n{\rm ln}\left (x\right )\right )} + a\right ) + p{\rm ln}\left (d e^{\left (n{\rm ln}\left (x\right )\right )} + c\right ) + n{\rm ln}\left (x\right ) + 1\right )} + a c x e^{\left (p{\rm ln}\left (b e^{\left (n{\rm ln}\left (x\right )\right )} + a\right ) + p{\rm ln}\left (d e^{\left (n{\rm ln}\left (x\right )\right )} + c\right ) + 1\right )}}{a c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((2*n*p + 2*n + 1)*b*d*e*x^(2*n)/(a*c) + (b*c + a*d)*(n*p + n + 1)*e*x^n/(a*c) + e)*(b*x^n + a)^p*(d*x^n + c)^p,x, algorithm="giac")

[Out]

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