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

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))))

________________________________________________________________________________________

Rubi [A]  time = 0.156862, 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, Rules used = {1849} \[ -\frac{(h x)^{-n (p+1)} \left (a+b x^n\right )^{p+1} \left (c+d x^n\right )^{p+1}}{h n (p+1)} \]

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))))

Rule 1849

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

Rubi steps

\begin{align*} \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 &=-\frac{(h x)^{-n (1+p)} \left (a+b x^n\right )^{1+p} \left (c+d x^n\right )^{1+p}}{h n (1+p)}\\ \end{align*}

Mathematica [A]  time = 0.354939, 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))

________________________________________________________________________________________

Maple [C]  time = 0.51, size = 138, normalized size = 3.1 \begin{align*} -{\frac{x \left ( bd \left ({x}^{n} \right ) ^{2}+ad{x}^{n}+bc{x}^{n}+ac \right ) \left ( a+b{x}^{n} \right ) ^{p} \left ( c+d{x}^{n} \right ) ^{p}}{n \left ( 1+p \right ) }{{\rm e}^{-{\frac{ \left ( np+n+1 \right ) \left ( -i\pi \, \left ({\it csgn} \left ( ihx \right ) \right ) ^{3}+i\pi \, \left ({\it csgn} \left ( ihx \right ) \right ) ^{2}{\it csgn} \left ( ih \right ) +i\pi \, \left ({\it csgn} \left ( ihx \right ) \right ) ^{2}{\it csgn} \left ( ix \right ) -i\pi \,{\it csgn} \left ( ihx \right ){\it csgn} \left ( ih \right ){\it csgn} \left ( ix \right ) +2\,\ln \left ( h \right ) +2\,\ln \left ( x \right ) \right ) }{2}}}}} \end{align*}

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*Pi*csgn(I*h*x)^3+I*Pi*csgn(I*h*x)^2*csgn(I*h)+I*Pi*csgn(I*h*x)^2*csgn(I*x)
-I*Pi*csgn(I*h*x)*csgn(I*h)*csgn(I*x)+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

________________________________________________________________________________________

Maxima [A]  time = 1.36532, size = 104, normalized size = 2.31 \begin{align*} -\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 )}} \end{align*}

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, 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))

________________________________________________________________________________________

Fricas [B]  time = 2.0071, size = 312, normalized size = 6.93 \begin{align*} -\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} \end{align*}

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, 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)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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

________________________________________________________________________________________

Giac [B]  time = 1.1456, size = 320, normalized size = 7.11 \begin{align*} -\frac{{\left (b x^{n} + a\right )}^{p}{\left (d x^{n} + c\right )}^{p} b d x x^{2 \, n} e^{\left (-n p \log \left (h\right ) - n p \log \left (x\right ) - n \log \left (h\right ) - n \log \left (x\right ) - \log \left (h\right ) - \log \left (x\right )\right )} +{\left (b x^{n} + a\right )}^{p}{\left (d x^{n} + c\right )}^{p} b c x x^{n} e^{\left (-n p \log \left (h\right ) - n p \log \left (x\right ) - n \log \left (h\right ) - n \log \left (x\right ) - \log \left (h\right ) - \log \left (x\right )\right )} +{\left (b x^{n} + a\right )}^{p}{\left (d x^{n} + c\right )}^{p} a d x x^{n} e^{\left (-n p \log \left (h\right ) - n p \log \left (x\right ) - n \log \left (h\right ) - n \log \left (x\right ) - \log \left (h\right ) - \log \left (x\right )\right )} +{\left (b x^{n} + a\right )}^{p}{\left (d x^{n} + c\right )}^{p} a c x e^{\left (-n p \log \left (h\right ) - n p \log \left (x\right ) - n \log \left (h\right ) - n \log \left (x\right ) - \log \left (h\right ) - \log \left (x\right )\right )}}{n p + n} \end{align*}

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, algorithm="giac")

[Out]

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