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

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))

________________________________________________________________________________________

Rubi [A]  time = 0.553113, 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, Rules used = {1848} \[ \frac{e (h x)^{m+1} \left (a+b x^n\right )^{p+1} \left (c+d x^n\right )^{p+1}}{a c h (m+1)} \]

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))

Rule 1848

Int[((h_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_.))^(p_.)*((c_) + (d_.)*(x_)^(n_.))^(p_.)*((e_) + (f_.)*(x_)^(n_.
) + (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, f, g, h, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[a*c*f*(m + 1) - e*(b*c + a*d)*
(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)^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 &=\frac{e (h x)^{1+m} \left (a+b x^n\right )^{1+p} \left (c+d x^n\right )^{1+p}}{a c h (1+m)}\\ \end{align*}

Mathematica [A]  time = 0.767357, 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))

________________________________________________________________________________________

Maple [C]  time = 0.429, size = 136, normalized size = 3. \begin{align*}{\frac{xe \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}}{ac \left ( 1+m \right ) }{{\rm e}^{-{\frac{m \left ( i \left ({\it csgn} \left ( ihx \right ) \right ) ^{3}\pi -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{\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}}}}} \end{align*}

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

________________________________________________________________________________________

Maxima [B]  time = 1.36721, size = 124, normalized size = 2.76 \begin{align*} \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 )}} \end{align*}

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, 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))

________________________________________________________________________________________

Fricas [A]  time = 1.7337, size = 223, normalized size = 4.96 \begin{align*} \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} \end{align*}

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, 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)

________________________________________________________________________________________

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)**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

________________________________________________________________________________________

Giac [B]  time = 1.18857, size = 209, normalized size = 4.64 \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 (m \log \left (h\right ) + m \log \left (x\right ) + 1\right )} +{\left (b x^{n} + a\right )}^{p}{\left (d x^{n} + c\right )}^{p} b c x x^{n} e^{\left (m \log \left (h\right ) + m \log \left (x\right ) + 1\right )} +{\left (b x^{n} + a\right )}^{p}{\left (d x^{n} + c\right )}^{p} a d x x^{n} e^{\left (m \log \left (h\right ) + m \log \left (x\right ) + 1\right )} +{\left (b x^{n} + a\right )}^{p}{\left (d x^{n} + c\right )}^{p} a c x e^{\left (m \log \left (h\right ) + m \log \left (x\right ) + 1\right )}}{a c m + a c} \end{align*}

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, algorithm="giac")

[Out]

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