3.6.94 \(\int (h x)^m (a+b x^n)^p (c+d x^n)^p (e+\frac {(b c+a d) e (1+m+n+n p) x^n}{a c (1+m)}+\frac {b d e (1+m+2 n+2 n p) x^{2 n}}{a c (1+m)}) \, dx\) [594]

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

[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 = 0.38, antiderivative size = 45, normalized size of antiderivative = 1.00, 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 = {1862} \begin {gather*} \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)} \end {gather*}

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 1862

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 = 0.98, size = 41, normalized size = 0.91 \begin {gather*} \frac {e x (h x)^m \left (a+b x^n\right )^{1+p} \left (c+d x^n\right )^{1+p}}{a c (1+m)} \end {gather*}

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] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.79, size = 136, normalized size = 3.02

method	result	size
risch	\(\frac {\left (a +b \,x^{n}\right )^{p} {\mathrm e}^{\frac {m \left (-i \pi \mathrm {csgn}\left (i h x \right )^{3}+i \pi \mathrm {csgn}\left (i h x \right )^{2} \mathrm {csgn}\left (i h \right )+i \pi \mathrm {csgn}\left (i h x \right )^{2} \mathrm {csgn}\left (i x \right )-i \pi \,\mathrm {csgn}\left (i h x \right ) \mathrm {csgn}\left (i h \right ) \mathrm {csgn}\left (i x \right )+2 \ln \left (h \right )+2 \ln \left (x \right )\right )}{2}} \left (b d \,x^{2 n}+a d \,x^{n}+b c \,x^{n}+a c \right ) e x \left (c +d \,x^{n}\right )^{p}}{a c \left (1+m \right )}\)	\(136\)

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,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (46) = 92\).
time = 0.40, size = 94, normalized size = 2.09 \begin {gather*} \frac {{\left (b d h^{m} x e^{\left (m \log \left (x\right ) + 2 \, n \log \left (x\right ) + 1\right )} + a c h^{m} x e^{\left (m \log \left (x\right ) + 1\right )} + {\left (b c h^{m} + a d h^{m}\right )} x e^{\left (m \log \left (x\right ) + n \log \left (x\right ) + 1\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 {gather*}

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]

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

________________________________________________________________________________________

Fricas [A]
time = 0.40, size = 88, normalized size = 1.96 \begin {gather*} \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 {gather*}

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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

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]

Exception raised: SystemError >> excessive stack use: stack is 3006 deep

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 155 vs. \(2 (46) = 92\).
time = 1.44, size = 155, normalized size = 3.44 \begin {gather*} \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 {gather*}

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)

________________________________________________________________________________________

Mupad [B]
time = 5.64, size = 106, normalized size = 2.36 \begin {gather*} {\left (c+d\,x^n\right )}^p\,\left (\frac {e\,x\,{\left (h\,x\right )}^m\,{\left (a+b\,x^n\right )}^p}{m+1}+\frac {e\,x\,x^n\,{\left (h\,x\right )}^m\,\left (a\,d+b\,c\right )\,{\left (a+b\,x^n\right )}^p}{a\,c\,\left (m+1\right )}+\frac {b\,d\,e\,x\,x^{2\,n}\,{\left (h\,x\right )}^m\,{\left (a+b\,x^n\right )}^p}{a\,c\,\left (m+1\right )}\right ) \end {gather*}

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

[Out]

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

________________________________________________________________________________________