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)))

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Rubi [A]  time = 0.16, antiderivative size = 45, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.41, 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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fricas [B]  time = 1.09, size = 119, normalized size = 2.64 \[ -\frac {{\left (b d x x^{2 \, n} e^{\left (-{\left (n p + n + 1\right )} \log \relax (h) - {\left (n p + n + 1\right )} \log \relax (x)\right )} + a c x e^{\left (-{\left (n p + n + 1\right )} \log \relax (h) - {\left (n p + n + 1\right )} \log \relax (x)\right )} + {\left (b c + a d\right )} x x^{n} e^{\left (-{\left (n p + n + 1\right )} \log \relax (h) - {\left (n p + n + 1\right )} \log \relax (x)\right )}\right )} {\left (b x^{n} + a\right )}^{p} {\left (d x^{n} + c\right )}^{p}}{n p + n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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)

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giac [B]  time = 0.43, size = 237, normalized size = 5.27 \[ -\frac {{\left (b x^{n} + a\right )}^{p} {\left (d x^{n} + c\right )}^{p} b d x x^{2 \, n} e^{\left (-n p \log \relax (h) - n p \log \relax (x) - n \log \relax (h) - n \log \relax (x) - \log \relax (h) - \log \relax (x)\right )} + {\left (b x^{n} + a\right )}^{p} {\left (d x^{n} + c\right )}^{p} b c x x^{n} e^{\left (-n p \log \relax (h) - n p \log \relax (x) - n \log \relax (h) - n \log \relax (x) - \log \relax (h) - \log \relax (x)\right )} + {\left (b x^{n} + a\right )}^{p} {\left (d x^{n} + c\right )}^{p} a d x x^{n} e^{\left (-n p \log \relax (h) - n p \log \relax (x) - n \log \relax (h) - n \log \relax (x) - \log \relax (h) - \log \relax (x)\right )} + {\left (b x^{n} + a\right )}^{p} {\left (d x^{n} + c\right )}^{p} a c x e^{\left (-n p \log \relax (h) - n p \log \relax (x) - n \log \relax (h) - n \log \relax (x) - \log \relax (h) - \log \relax (x)\right )}}{n p + n} \]

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)

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maple [C]  time = 0.58, size = 138, normalized size = 3.07 \[ -\frac {\left (a d \,x^{n}+b c \,x^{n}+b d \,x^{2 n}+a c \right ) x \left (b \,x^{n}+a \right )^{p} \left (d \,x^{n}+c \right )^{p} {\mathrm e}^{-\frac {\left (n p +n +1\right ) \left (-i \pi \,\mathrm {csgn}\left (i h \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i h x \right )+i \pi \,\mathrm {csgn}\left (i h \right ) \mathrm {csgn}\left (i h x \right )^{2}+i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i h x \right )^{2}-i \pi \mathrm {csgn}\left (i h x \right )^{3}+2 \ln \relax (h )+2 \ln \relax (x )\right )}{2}}}{\left (p +1\right ) n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 3.04, size = 77, normalized size = 1.71 \[ -\frac {{\left (b d x^{2 \, n} + a c + {\left (b c + a d\right )} x^{n}\right )} h^{-n p - n - 1} e^{\left (-n p \log \relax (x) + p \log \left (b x^{n} + a\right ) + p \log \left (d x^{n} + c\right ) - n \log \relax (x)\right )}}{n {\left (p + 1\right )}} \]

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))

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mupad [B]  time = 5.37, size = 124, normalized size = 2.76 \[ -{\left (c+d\,x^n\right )}^p\,\left (\frac {a\,c\,x\,{\left (a+b\,x^n\right )}^p}{n\,{\left (h\,x\right )}^{n+n\,p+1}\,\left (p+1\right )}+\frac {x\,x^n\,\left (a\,d+b\,c\right )\,{\left (a+b\,x^n\right )}^p}{n\,{\left (h\,x\right )}^{n+n\,p+1}\,\left (p+1\right )}+\frac {b\,d\,x\,x^{2\,n}\,{\left (a+b\,x^n\right )}^p}{n\,{\left (h\,x\right )}^{n+n\,p+1}\,\left (p+1\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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