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

Optimal. Leaf size=45 \[ \frac{e (h x)^{m+1} \left (a+b x^n\right )^{p+1} \left (c+d x^n\right )^{p+1}}{a c h (m+1)} \]

[Out]

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

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Rubi [A]  time = 0.839767, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 86, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.012 \[ \frac{e (h x)^{m+1} \left (a+b x^n\right )^{p+1} \left (c+d x^n\right )^{p+1}}{a c h (m+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((h*x)**m*(a+b*x**n)**p*(c+d*x**n)**p*(e+(a*d+b*c)*e*(n*p+m+n+1)*x**n/a/c/(1+m)+b*d*e*(2*n*p+m+2*n+1)*x**(2*n)/a/c/(1+m)),x)

[Out]

Timed out

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Mathematica [A]  time = 0.423018, size = 41, normalized size = 0.91 \[ \frac{e x (h x)^m \left (a+b x^n\right )^{p+1} \left (c+d x^n\right )^{p+1}}{a c (m+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.605, size = 136, normalized size = 3. \[{\frac{ \left ( a+b{x}^{n} \right ) ^{p} \left ( bd \left ({x}^{n} \right ) ^{2}+ad{x}^{n}+bc{x}^{n}+ac \right ) ex \left ( c+d{x}^{n} \right ) ^{p}}{ac \left ( 1+m \right ) }{{\rm e}^{{\frac{m \left ( -i \left ({\it csgn} \left ( ihx \right ) \right ) ^{3}\pi +i \left ({\it csgn} \left ( ihx \right ) \right ) ^{2}{\it csgn} \left ( ih \right ) \pi +i \left ({\it csgn} \left ( ihx \right ) \right ) ^{2}{\it csgn} \left ( ix \right ) \pi -i{\it csgn} \left ( ihx \right ){\it csgn} \left ( ih \right ){\it csgn} \left ( ix \right ) \pi +2\,\ln \left ( h \right ) +2\,\ln \left ( x \right ) \right ) }{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((h*x)^m*(a+b*x^n)^p*(c+d*x^n)^p*(e+(a*d+b*c)*e*(n*p+m+n+1)*x^n/a/c/(1+m)+b*d*e*(2*n*p+m+2*n+1)*x^(2*n)/a/c/(1+m)),x)

[Out]

(a+b*x^n)^p*exp(1/2*m*(-I*csgn(I*h*x)^3*Pi+I*csgn(I*h*x)^2*csgn(I*h)*Pi+I*csgn(I
*h*x)^2*csgn(I*x)*Pi-I*csgn(I*h*x)*csgn(I*h)*csgn(I*x)*Pi+2*ln(h)+2*ln(x)))*(b*d
*(x^n)^2+a*d*x^n+b*c*x^n+a*c)*e*x/a/c/(1+m)*(c+d*x^n)^p

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Maxima [A]  time = 2.11185, size = 124, normalized size = 2.76 \[ \frac{{\left (a c e h^{m} x x^{m} + b d e h^{m} x e^{\left (m \log \left (x\right ) + 2 \, n \log \left (x\right )\right )} +{\left (b c e h^{m} + a d e h^{m}\right )} x e^{\left (m \log \left (x\right ) + n \log \left (x\right )\right )}\right )} e^{\left (p \log \left (b x^{n} + a\right ) + p \log \left (d x^{n} + c\right )\right )}}{a c{\left (m + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.254125, size = 119, normalized size = 2.64 \[ \frac{{\left (b d e x x^{2 \, n} e^{\left (m \log \left (h\right ) + m \log \left (x\right )\right )} + a c e x e^{\left (m \log \left (h\right ) + m \log \left (x\right )\right )} +{\left (b c + a d\right )} e x x^{n} e^{\left (m \log \left (h\right ) + m \log \left (x\right )\right )}\right )}{\left (b x^{n} + a\right )}^{p}{\left (d x^{n} + c\right )}^{p}}{a c m + a c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.274291, size = 244, normalized size = 5.42 \[ \frac{b d x e^{\left (p{\rm ln}\left (b e^{\left (n{\rm ln}\left (x\right )\right )} + a\right ) + p{\rm ln}\left (d e^{\left (n{\rm ln}\left (x\right )\right )} + c\right ) + m{\rm ln}\left (h\right ) + m{\rm ln}\left (x\right ) + 2 \, n{\rm ln}\left (x\right ) + 1\right )} + b c x e^{\left (p{\rm ln}\left (b e^{\left (n{\rm ln}\left (x\right )\right )} + a\right ) + p{\rm ln}\left (d e^{\left (n{\rm ln}\left (x\right )\right )} + c\right ) + m{\rm ln}\left (h\right ) + m{\rm ln}\left (x\right ) + n{\rm ln}\left (x\right ) + 1\right )} + a d x e^{\left (p{\rm ln}\left (b e^{\left (n{\rm ln}\left (x\right )\right )} + a\right ) + p{\rm ln}\left (d e^{\left (n{\rm ln}\left (x\right )\right )} + c\right ) + m{\rm ln}\left (h\right ) + m{\rm ln}\left (x\right ) + n{\rm ln}\left (x\right ) + 1\right )} + a c x e^{\left (p{\rm ln}\left (b e^{\left (n{\rm ln}\left (x\right )\right )} + a\right ) + p{\rm ln}\left (d e^{\left (n{\rm ln}\left (x\right )\right )} + c\right ) + m{\rm ln}\left (h\right ) + m{\rm ln}\left (x\right ) + 1\right )}}{a c m + a c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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