3.580 \(\int (h x)^{-1-n-n p} \left (a+b x^n\right )^p \left (c+d x^n\right )^p \left (a c-b d x^{2 n}\right ) \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.241569, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.022 \[ -\frac{(h x)^{-n (p+1)} \left (a+b x^n\right )^{p+1} \left (c+d x^n\right )^{p+1}}{h n (p+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 38.0471, size = 36, normalized size = 0.8 \[ - \frac{\left (h x\right )^{- n \left (p + 1\right )} \left (a + b x^{n}\right )^{p + 1} \left (c + d x^{n}\right )^{p + 1}}{h n \left (p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.266415, size = 46, normalized size = 1.02 \[ -\frac{(h x)^{n (-p)-n} \left (a+b x^n\right )^{p+1} \left (c+d x^n\right )^{p+1}}{h n p+h n} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.677, size = 138, normalized size = 3.1 \[ -{\frac{ \left ( a+b{x}^{n} \right ) ^{p} \left ( bd \left ({x}^{n} \right ) ^{2}+ad{x}^{n}+bc{x}^{n}+ac \right ) x \left ( c+d{x}^{n} \right ) ^{p}}{n \left ( 1+p \right ) }{{\rm e}^{-{\frac{ \left ( np+n+1 \right ) \left ( -i \left ({\it csgn} \left ( ihx \right ) \right ) ^{3}\pi +i \left ({\it csgn} \left ( ihx \right ) \right ) ^{2}{\it csgn} \left ( ih \right ) \pi +i \left ({\it csgn} \left ( ihx \right ) \right ) ^{2}{\it csgn} \left ( ix \right ) \pi -i{\it csgn} \left ( ihx \right ){\it csgn} \left ( ih \right ){\it csgn} \left ( ix \right ) \pi +2\,\ln \left ( h \right ) +2\,\ln \left ( x \right ) \right ) }{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 2.20546, size = 104, normalized size = 2.31 \[ -\frac{{\left (b d x^{2 \, n} + a c +{\left (b c + a d\right )} x^{n}\right )} h^{-n p - n - 1} e^{\left (-n p \log \left (x\right ) + p \log \left (b x^{n} + a\right ) + p \log \left (d x^{n} + c\right ) - n \log \left (x\right )\right )}}{n{\left (p + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.246381, size = 161, normalized size = 3.58 \[ -\frac{{\left (b d x x^{2 \, n} e^{\left (-{\left (n p + n + 1\right )} \log \left (h\right ) -{\left (n p + n + 1\right )} \log \left (x\right )\right )} + a c x e^{\left (-{\left (n p + n + 1\right )} \log \left (h\right ) -{\left (n p + n + 1\right )} \log \left (x\right )\right )} +{\left (b c + a d\right )} x x^{n} e^{\left (-{\left (n p + n + 1\right )} \log \left (h\right ) -{\left (n p + n + 1\right )} \log \left (x\right )\right )}\right )}{\left (b x^{n} + a\right )}^{p}{\left (d x^{n} + c\right )}^{p}}{n p + n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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)**(-n*p-n-1)*(a+b*x**n)**p*(c+d*x**n)**p*(a*c-b*d*x**(2*n)),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.25732, size = 323, normalized size = 7.18 \[ -\frac{b d x e^{\left (-n p{\rm ln}\left (h\right ) - n p{\rm ln}\left (x\right ) + p{\rm ln}\left (b e^{\left (n{\rm ln}\left (x\right )\right )} + a\right ) + p{\rm ln}\left (d e^{\left (n{\rm ln}\left (x\right )\right )} + c\right ) - n{\rm ln}\left (h\right ) + n{\rm ln}\left (x\right ) -{\rm ln}\left (h\right ) -{\rm ln}\left (x\right )\right )} + a c x e^{\left (-n p{\rm ln}\left (h\right ) - n p{\rm ln}\left (x\right ) + p{\rm ln}\left (b e^{\left (n{\rm ln}\left (x\right )\right )} + a\right ) + p{\rm ln}\left (d e^{\left (n{\rm ln}\left (x\right )\right )} + c\right ) - n{\rm ln}\left (h\right ) - n{\rm ln}\left (x\right ) -{\rm ln}\left (h\right ) -{\rm ln}\left (x\right )\right )} + b c x e^{\left (-n p{\rm ln}\left (h\right ) - n p{\rm ln}\left (x\right ) + p{\rm ln}\left (b e^{\left (n{\rm ln}\left (x\right )\right )} + a\right ) + p{\rm ln}\left (d e^{\left (n{\rm ln}\left (x\right )\right )} + c\right ) - n{\rm ln}\left (h\right ) -{\rm ln}\left (h\right ) -{\rm ln}\left (x\right )\right )} + a d x e^{\left (-n p{\rm ln}\left (h\right ) - n p{\rm ln}\left (x\right ) + p{\rm ln}\left (b e^{\left (n{\rm ln}\left (x\right )\right )} + a\right ) + p{\rm ln}\left (d e^{\left (n{\rm ln}\left (x\right )\right )} + c\right ) - n{\rm ln}\left (h\right ) -{\rm ln}\left (h\right ) -{\rm ln}\left (x\right )\right )}}{n p + n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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