Integrand size = 24, antiderivative size = 357 \[ \int (e x)^m \left (a+b x^n\right )^p \left (c+d x^n\right )^3 \, dx=\frac {d \left (3 b^2 c^2+\frac {a d (1+m+n) (a d (1+m+2 n)-3 b c (1+m+n (3+p)))}{(1+m+n (2+p)) (1+m+n (3+p))}\right ) (e x)^{1+m} \left (a+b x^n\right )^{1+p}}{b^3 e (1+m+n+n p)}-\frac {d^2 (a d (1+m+2 n)-3 b c (1+m+n (3+p))) x^n (e x)^{1+m} \left (a+b x^n\right )^{1+p}}{b^2 e (1+m+n (2+p)) (1+m+n (3+p))}+\frac {d^3 x^{2 n} (e x)^{1+m} \left (a+b x^n\right )^{1+p}}{b e (1+m+n (3+p))}+\frac {\left (\frac {c^3}{1+m}-\frac {a d \left (3 b^2 c^2+\frac {a d (1+m+n) (a d (1+m+2 n)-3 b c (1+m+n (3+p)))}{(1+m+n (2+p)) (1+m+n (3+p))}\right )}{b^3 (1+m+n+n p)}\right ) (e x)^{1+m} \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{n},-p,\frac {1+m+n}{n},-\frac {b x^n}{a}\right )}{e} \] Output:
d*(3*b^2*c^2+a*d*(1+m+n)*(a*d*(1+m+2*n)-3*b*c*(1+m+n*(3+p)))/(1+m+n*(2+p)) /(1+m+n*(3+p)))*(e*x)^(1+m)*(a+b*x^n)^(p+1)/b^3/e/(n*p+m+n+1)-d^2*(a*d*(1+ m+2*n)-3*b*c*(1+m+n*(3+p)))*x^n*(e*x)^(1+m)*(a+b*x^n)^(p+1)/b^2/e/(1+m+n*( 2+p))/(1+m+n*(3+p))+d^3*x^(2*n)*(e*x)^(1+m)*(a+b*x^n)^(p+1)/b/e/(1+m+n*(3+ p))+(c^3/(1+m)-a*d*(3*b^2*c^2+a*d*(1+m+n)*(a*d*(1+m+2*n)-3*b*c*(1+m+n*(3+p )))/(1+m+n*(2+p))/(1+m+n*(3+p)))/b^3/(n*p+m+n+1))*(e*x)^(1+m)*(a+b*x^n)^p* hypergeom([-p, (1+m)/n],[(1+m+n)/n],-b*x^n/a)/e/((1+b*x^n/a)^p)
Time = 6.69 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.59 \[ \int (e x)^m \left (a+b x^n\right )^p \left (c+d x^n\right )^3 \, dx=x (e x)^m \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \left (\frac {c^3 \operatorname {Hypergeometric2F1}\left (\frac {1+m}{n},-p,\frac {1+m+n}{n},-\frac {b x^n}{a}\right )}{1+m}+d x^n \left (\frac {3 c^2 \operatorname {Hypergeometric2F1}\left (\frac {1+m+n}{n},-p,\frac {1+m+2 n}{n},-\frac {b x^n}{a}\right )}{1+m+n}+d x^n \left (\frac {3 c \operatorname {Hypergeometric2F1}\left (\frac {1+m+2 n}{n},-p,\frac {1+m+3 n}{n},-\frac {b x^n}{a}\right )}{1+m+2 n}+\frac {d x^n \operatorname {Hypergeometric2F1}\left (\frac {1+m+3 n}{n},-p,\frac {1+m+4 n}{n},-\frac {b x^n}{a}\right )}{1+m+3 n}\right )\right )\right ) \] Input:
Integrate[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^3,x]
Output:
(x*(e*x)^m*(a + b*x^n)^p*((c^3*Hypergeometric2F1[(1 + m)/n, -p, (1 + m + n )/n, -((b*x^n)/a)])/(1 + m) + d*x^n*((3*c^2*Hypergeometric2F1[(1 + m + n)/ n, -p, (1 + m + 2*n)/n, -((b*x^n)/a)])/(1 + m + n) + d*x^n*((3*c*Hypergeom etric2F1[(1 + m + 2*n)/n, -p, (1 + m + 3*n)/n, -((b*x^n)/a)])/(1 + m + 2*n ) + (d*x^n*Hypergeometric2F1[(1 + m + 3*n)/n, -p, (1 + m + 4*n)/n, -((b*x^ n)/a)])/(1 + m + 3*n)))))/(1 + (b*x^n)/a)^p
Time = 1.84 (sec) , antiderivative size = 513, normalized size of antiderivative = 1.44, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1008, 25, 1066, 959, 889, 888}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int (e x)^m \left (c+d x^n\right )^3 \left (a+b x^n\right )^p \, dx\) |
\(\Big \downarrow \) 1008 |
\(\displaystyle \frac {\int -(e x)^m \left (b x^n+a\right )^p \left (d x^n+c\right ) \left (d (a d (m+2 n+1)-b c (m+n (p+5)+1)) x^n+c (a d (m+1)-b c (m+n (p+3)+1))\right )dx}{b (m+n (p+3)+1)}+\frac {d (e x)^{m+1} \left (c+d x^n\right )^2 \left (a+b x^n\right )^{p+1}}{b e (m+n (p+3)+1)}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {d (e x)^{m+1} \left (c+d x^n\right )^2 \left (a+b x^n\right )^{p+1}}{b e (m+n (p+3)+1)}-\frac {\int (e x)^m \left (b x^n+a\right )^p \left (d x^n+c\right ) \left (d (a d (m+2 n+1)-b c (m+n (p+5)+1)) x^n+c (a d (m+1)-b c (m+n (p+3)+1))\right )dx}{b (m+n (p+3)+1)}\) |
\(\Big \downarrow \) 1066 |
\(\displaystyle \frac {d (e x)^{m+1} \left (c+d x^n\right )^2 \left (a+b x^n\right )^{p+1}}{b e (m+n (p+3)+1)}-\frac {\frac {\int (e x)^m \left (b x^n+a\right )^p \left (d (b c n (p+2) (a d (m+1)-b c (m+n (p+3)+1))+(b c-a d) (m+1) (a d (m+2 n+1)-b c (m+n (p+3)+1))+(b c-a d) n (a d (m+2 n+1)-b c (m+n (p+5)+1))) x^n+c (b c n (p+2) (a d (m+1)-b c (m+n (p+3)+1))+(b c-a d) (m+1) (a d (m+2 n+1)-b c (m+n (p+3)+1)))\right )dx}{b (m+n (p+2)+1)}+\frac {d (e x)^{m+1} \left (c+d x^n\right ) \left (a+b x^n\right )^{p+1} (a d (m+2 n+1)-b c (m+n (p+5)+1))}{b e (m+n (p+2)+1)}}{b (m+n (p+3)+1)}\) |
\(\Big \downarrow \) 959 |
\(\displaystyle \frac {d (e x)^{m+1} \left (c+d x^n\right )^2 \left (a+b x^n\right )^{p+1}}{b e (m+n (p+3)+1)}-\frac {\frac {\left (c (b c n (p+2) (a d (m+1)-b c (m+n (p+3)+1))+(m+1) (b c-a d) (a d (m+2 n+1)-b c (m+n (p+3)+1)))-\frac {a d (m+1) (b c n (p+2) (a d (m+1)-b c (m+n (p+3)+1))+(m+1) (b c-a d) (a d (m+2 n+1)-b c (m+n (p+3)+1))+n (b c-a d) (a d (m+2 n+1)-b c (m+n (p+5)+1)))}{b (m+n p+n+1)}\right ) \int (e x)^m \left (b x^n+a\right )^pdx+\frac {d (e x)^{m+1} \left (a+b x^n\right )^{p+1} (b c n (p+2) (a d (m+1)-b c (m+n (p+3)+1))+(m+1) (b c-a d) (a d (m+2 n+1)-b c (m+n (p+3)+1))+n (b c-a d) (a d (m+2 n+1)-b c (m+n (p+5)+1)))}{b e (m+n p+n+1)}}{b (m+n (p+2)+1)}+\frac {d (e x)^{m+1} \left (c+d x^n\right ) \left (a+b x^n\right )^{p+1} (a d (m+2 n+1)-b c (m+n (p+5)+1))}{b e (m+n (p+2)+1)}}{b (m+n (p+3)+1)}\) |
\(\Big \downarrow \) 889 |
\(\displaystyle \frac {d (e x)^{m+1} \left (c+d x^n\right )^2 \left (a+b x^n\right )^{p+1}}{b e (m+n (p+3)+1)}-\frac {\frac {\left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \left (c (b c n (p+2) (a d (m+1)-b c (m+n (p+3)+1))+(m+1) (b c-a d) (a d (m+2 n+1)-b c (m+n (p+3)+1)))-\frac {a d (m+1) (b c n (p+2) (a d (m+1)-b c (m+n (p+3)+1))+(m+1) (b c-a d) (a d (m+2 n+1)-b c (m+n (p+3)+1))+n (b c-a d) (a d (m+2 n+1)-b c (m+n (p+5)+1)))}{b (m+n p+n+1)}\right ) \int (e x)^m \left (\frac {b x^n}{a}+1\right )^pdx+\frac {d (e x)^{m+1} \left (a+b x^n\right )^{p+1} (b c n (p+2) (a d (m+1)-b c (m+n (p+3)+1))+(m+1) (b c-a d) (a d (m+2 n+1)-b c (m+n (p+3)+1))+n (b c-a d) (a d (m+2 n+1)-b c (m+n (p+5)+1)))}{b e (m+n p+n+1)}}{b (m+n (p+2)+1)}+\frac {d (e x)^{m+1} \left (c+d x^n\right ) \left (a+b x^n\right )^{p+1} (a d (m+2 n+1)-b c (m+n (p+5)+1))}{b e (m+n (p+2)+1)}}{b (m+n (p+3)+1)}\) |
\(\Big \downarrow \) 888 |
\(\displaystyle \frac {d (e x)^{m+1} \left (c+d x^n\right )^2 \left (a+b x^n\right )^{p+1}}{b e (m+n (p+3)+1)}-\frac {\frac {\frac {(e x)^{m+1} \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \left (c (b c n (p+2) (a d (m+1)-b c (m+n (p+3)+1))+(m+1) (b c-a d) (a d (m+2 n+1)-b c (m+n (p+3)+1)))-\frac {a d (m+1) (b c n (p+2) (a d (m+1)-b c (m+n (p+3)+1))+(m+1) (b c-a d) (a d (m+2 n+1)-b c (m+n (p+3)+1))+n (b c-a d) (a d (m+2 n+1)-b c (m+n (p+5)+1)))}{b (m+n p+n+1)}\right ) \operatorname {Hypergeometric2F1}\left (\frac {m+1}{n},-p,\frac {m+n+1}{n},-\frac {b x^n}{a}\right )}{e (m+1)}+\frac {d (e x)^{m+1} \left (a+b x^n\right )^{p+1} (b c n (p+2) (a d (m+1)-b c (m+n (p+3)+1))+(m+1) (b c-a d) (a d (m+2 n+1)-b c (m+n (p+3)+1))+n (b c-a d) (a d (m+2 n+1)-b c (m+n (p+5)+1)))}{b e (m+n p+n+1)}}{b (m+n (p+2)+1)}+\frac {d (e x)^{m+1} \left (c+d x^n\right ) \left (a+b x^n\right )^{p+1} (a d (m+2 n+1)-b c (m+n (p+5)+1))}{b e (m+n (p+2)+1)}}{b (m+n (p+3)+1)}\) |
Input:
Int[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^3,x]
Output:
(d*(e*x)^(1 + m)*(a + b*x^n)^(1 + p)*(c + d*x^n)^2)/(b*e*(1 + m + n*(3 + p ))) - ((d*(a*d*(1 + m + 2*n) - b*c*(1 + m + n*(5 + p)))*(e*x)^(1 + m)*(a + b*x^n)^(1 + p)*(c + d*x^n))/(b*e*(1 + m + n*(2 + p))) + ((d*(b*c*n*(2 + p )*(a*d*(1 + m) - b*c*(1 + m + n*(3 + p))) + (b*c - a*d)*(1 + m)*(a*d*(1 + m + 2*n) - b*c*(1 + m + n*(3 + p))) + (b*c - a*d)*n*(a*d*(1 + m + 2*n) - b *c*(1 + m + n*(5 + p))))*(e*x)^(1 + m)*(a + b*x^n)^(1 + p))/(b*e*(1 + m + n + n*p)) + ((c*(b*c*n*(2 + p)*(a*d*(1 + m) - b*c*(1 + m + n*(3 + p))) + ( b*c - a*d)*(1 + m)*(a*d*(1 + m + 2*n) - b*c*(1 + m + n*(3 + p)))) - (a*d*( 1 + m)*(b*c*n*(2 + p)*(a*d*(1 + m) - b*c*(1 + m + n*(3 + p))) + (b*c - a*d )*(1 + m)*(a*d*(1 + m + 2*n) - b*c*(1 + m + n*(3 + p))) + (b*c - a*d)*n*(a *d*(1 + m + 2*n) - b*c*(1 + m + n*(5 + p)))))/(b*(1 + m + n + n*p)))*(e*x) ^(1 + m)*(a + b*x^n)^p*Hypergeometric2F1[(1 + m)/n, -p, (1 + m + n)/n, -(( b*x^n)/a)])/(e*(1 + m)*(1 + (b*x^n)/a)^p))/(b*(1 + m + n*(2 + p))))/(b*(1 + m + n*(3 + p)))
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p *((c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/n, (m + 1)/n + 1 , (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] && !IGtQ[p, 0] && (ILt Q[p, 0] || GtQ[a, 0])
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^I ntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p]) Int[(c*x) ^m*(1 + b*(x^n/a))^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] && !IGtQ[p, 0 ] && !(ILtQ[p, 0] || GtQ[a, 0])
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _)), x_Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(b*e*(m + n*(p + 1) + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m + n*(p + 1) + 1)) Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0] && NeQ[m + n*(p + 1) + 1, 0]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[d*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n) ^(q - 1)/(b*e*(m + n*(p + q) + 1))), x] + Simp[1/(b*(m + n*(p + q) + 1)) Int[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^(q - 2)*Simp[c*((c*b - a*d)*(m + 1) + c*b*n*(p + q)) + (d*(c*b - a*d)*(m + 1) + d*n*(q - 1)*(b*c - a*d) + c*b*d* n*(p + q))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[f*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/(b*g*(m + n*(p + q + 1) + 1))), x] + Simp[1/( b*(m + n*(p + q + 1) + 1)) Int[(g*x)^m*(a + b*x^n)^p*(c + d*x^n)^(q - 1)* Simp[c*((b*e - a*f)*(m + 1) + b*e*n*(p + q + 1)) + (d*(b*e - a*f)*(m + 1) + f*n*q*(b*c - a*d) + b*e*d*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c , d, e, f, g, m, n, p}, x] && GtQ[q, 0] && !(EqQ[q, 1] && SimplerQ[e + f*x ^n, c + d*x^n])
\[\int \left (e x \right )^{m} \left (a +b \,x^{n}\right )^{p} \left (c +d \,x^{n}\right )^{3}d x\]
Input:
int((e*x)^m*(a+b*x^n)^p*(c+d*x^n)^3,x)
Output:
int((e*x)^m*(a+b*x^n)^p*(c+d*x^n)^3,x)
\[ \int (e x)^m \left (a+b x^n\right )^p \left (c+d x^n\right )^3 \, dx=\int { {\left (d x^{n} + c\right )}^{3} {\left (b x^{n} + a\right )}^{p} \left (e x\right )^{m} \,d x } \] Input:
integrate((e*x)^m*(a+b*x^n)^p*(c+d*x^n)^3,x, algorithm="fricas")
Output:
integral((d^3*x^(3*n) + 3*c*d^2*x^(2*n) + 3*c^2*d*x^n + c^3)*(b*x^n + a)^p *(e*x)^m, x)
Timed out. \[ \int (e x)^m \left (a+b x^n\right )^p \left (c+d x^n\right )^3 \, dx=\text {Timed out} \] Input:
integrate((e*x)**m*(a+b*x**n)**p*(c+d*x**n)**3,x)
Output:
Timed out
\[ \int (e x)^m \left (a+b x^n\right )^p \left (c+d x^n\right )^3 \, dx=\int { {\left (d x^{n} + c\right )}^{3} {\left (b x^{n} + a\right )}^{p} \left (e x\right )^{m} \,d x } \] Input:
integrate((e*x)^m*(a+b*x^n)^p*(c+d*x^n)^3,x, algorithm="maxima")
Output:
integrate((d*x^n + c)^3*(b*x^n + a)^p*(e*x)^m, x)
Exception generated. \[ \int (e x)^m \left (a+b x^n\right )^p \left (c+d x^n\right )^3 \, dx=\text {Exception raised: TypeError} \] Input:
integrate((e*x)^m*(a+b*x^n)^p*(c+d*x^n)^3,x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro unding error%%%{1,[2,0,6,4,0,2,4,4,3,0]%%%}+%%%{4,[2,0,6,4,0,2,3,4,3,0]%%% }+%%%{6,[
Timed out. \[ \int (e x)^m \left (a+b x^n\right )^p \left (c+d x^n\right )^3 \, dx=\int {\left (e\,x\right )}^m\,{\left (a+b\,x^n\right )}^p\,{\left (c+d\,x^n\right )}^3 \,d x \] Input:
int((e*x)^m*(a + b*x^n)^p*(c + d*x^n)^3,x)
Output:
int((e*x)^m*(a + b*x^n)^p*(c + d*x^n)^3, x)
\[ \int (e x)^m \left (a+b x^n\right )^p \left (c+d x^n\right )^3 \, dx=\text {too large to display} \] Input:
int((e*x)^m*(a+b*x^n)^p*(c+d*x^n)^3,x)
Output:
(e**m*(x**(m + 3*n)*(x**n*b + a)**p*b**3*d**3*m**3*x + 3*x**(m + 3*n)*(x** n*b + a)**p*b**3*d**3*m**2*n*p*x + 3*x**(m + 3*n)*(x**n*b + a)**p*b**3*d** 3*m**2*n*x + 3*x**(m + 3*n)*(x**n*b + a)**p*b**3*d**3*m**2*x + 3*x**(m + 3 *n)*(x**n*b + a)**p*b**3*d**3*m*n**2*p**2*x + 6*x**(m + 3*n)*(x**n*b + a)* *p*b**3*d**3*m*n**2*p*x + 2*x**(m + 3*n)*(x**n*b + a)**p*b**3*d**3*m*n**2* x + 6*x**(m + 3*n)*(x**n*b + a)**p*b**3*d**3*m*n*p*x + 6*x**(m + 3*n)*(x** n*b + a)**p*b**3*d**3*m*n*x + 3*x**(m + 3*n)*(x**n*b + a)**p*b**3*d**3*m*x + x**(m + 3*n)*(x**n*b + a)**p*b**3*d**3*n**3*p**3*x + 3*x**(m + 3*n)*(x* *n*b + a)**p*b**3*d**3*n**3*p**2*x + 2*x**(m + 3*n)*(x**n*b + a)**p*b**3*d **3*n**3*p*x + 3*x**(m + 3*n)*(x**n*b + a)**p*b**3*d**3*n**2*p**2*x + 6*x* *(m + 3*n)*(x**n*b + a)**p*b**3*d**3*n**2*p*x + 2*x**(m + 3*n)*(x**n*b + a )**p*b**3*d**3*n**2*x + 3*x**(m + 3*n)*(x**n*b + a)**p*b**3*d**3*n*p*x + 3 *x**(m + 3*n)*(x**n*b + a)**p*b**3*d**3*n*x + x**(m + 3*n)*(x**n*b + a)**p *b**3*d**3*x + x**(m + 2*n)*(x**n*b + a)**p*a*b**2*d**3*m**2*n*p*x + 2*x** (m + 2*n)*(x**n*b + a)**p*a*b**2*d**3*m*n**2*p**2*x + x**(m + 2*n)*(x**n*b + a)**p*a*b**2*d**3*m*n**2*p*x + 2*x**(m + 2*n)*(x**n*b + a)**p*a*b**2*d* *3*m*n*p*x + x**(m + 2*n)*(x**n*b + a)**p*a*b**2*d**3*n**3*p**3*x + x**(m + 2*n)*(x**n*b + a)**p*a*b**2*d**3*n**3*p**2*x + 2*x**(m + 2*n)*(x**n*b + a)**p*a*b**2*d**3*n**2*p**2*x + x**(m + 2*n)*(x**n*b + a)**p*a*b**2*d**3*n **2*p*x + x**(m + 2*n)*(x**n*b + a)**p*a*b**2*d**3*n*p*x + 3*x**(m + 2*...