\(\int (e x)^{1+p} (2 b+3 c x) (a x^2+b x^3)^p \, dx\) [363]

Optimal result

Integrand size = 29, antiderivative size = 106 \[ \int (e x)^{1+p} (2 b+3 c x) \left (a x^2+b x^3\right )^p \, dx=\frac {3 c e (e x)^p \left (a x^2+b x^3\right )^{1+p}}{b (3+4 p)}-\frac {e \left (\frac {3 c}{3+4 p}-\frac {2 b^2}{2 a+3 a p}\right ) (e x)^p \left (a x^2+b x^3\right )^{1+p} \operatorname {Hypergeometric2F1}\left (1,3+4 p,3 (1+p),-\frac {b x}{a}\right )}{b} \] Output:

3*c*e*(e*x)^p*(b*x^3+a*x^2)^(p+1)/b/(3+4*p)-e*(3*c/(3+4*p)-2*b^2/(3*a*p+2* 
a))*(e*x)^p*(b*x^3+a*x^2)^(p+1)*hypergeom([1, 3+4*p],[3*p+3],-b*x/a)/b
 

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.08 \[ \int (e x)^{1+p} (2 b+3 c x) \left (a x^2+b x^3\right )^p \, dx=\frac {e x^2 (e x)^p \left (x^2 (a+b x)\right )^p \left (1+\frac {b x}{a}\right )^{-p} \left (3 c (2+3 p) (a+b x) \left (1+\frac {b x}{a}\right )^p+\left (-3 a c (2+3 p)+b^2 (6+8 p)\right ) \operatorname {Hypergeometric2F1}\left (-p,2+3 p,3+3 p,-\frac {b x}{a}\right )\right )}{b (2+3 p) (3+4 p)} \] Input:

Integrate[(e*x)^(1 + p)*(2*b + 3*c*x)*(a*x^2 + b*x^3)^p,x]
 

Output:

(e*x^2*(e*x)^p*(x^2*(a + b*x))^p*(3*c*(2 + 3*p)*(a + b*x)*(1 + (b*x)/a)^p 
+ (-3*a*c*(2 + 3*p) + b^2*(6 + 8*p))*Hypergeometric2F1[-p, 2 + 3*p, 3 + 3* 
p, -((b*x)/a)]))/(b*(2 + 3*p)*(3 + 4*p)*(1 + (b*x)/a)^p)
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.14, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {1945, 1938, 76, 74}

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.

\(\displaystyle \int (2 b+3 c x) (e x)^{p+1} \left (a x^2+b x^3\right )^p \, dx\)

\(\Big \downarrow \) 1945

\(\displaystyle \left (2 b-\frac {3 a c (3 p+2)}{b (4 p+3)}\right ) \int (e x)^{p+1} \left (b x^3+a x^2\right )^pdx+\frac {3 c e (e x)^p \left (a x^2+b x^3\right )^{p+1}}{b (4 p+3)}\)

\(\Big \downarrow \) 1938

\(\displaystyle e x^{-3 p} (e x)^p \left (2 b-\frac {3 a c (3 p+2)}{b (4 p+3)}\right ) (a+b x)^{-p} \left (a x^2+b x^3\right )^p \int x^{3 p+1} (a+b x)^pdx+\frac {3 c e (e x)^p \left (a x^2+b x^3\right )^{p+1}}{b (4 p+3)}\)

\(\Big \downarrow \) 76

\(\displaystyle e x^{-3 p} (e x)^p \left (2 b-\frac {3 a c (3 p+2)}{b (4 p+3)}\right ) \left (\frac {b x}{a}+1\right )^{-p} \left (a x^2+b x^3\right )^p \int x^{3 p+1} \left (\frac {b x}{a}+1\right )^pdx+\frac {3 c e (e x)^p \left (a x^2+b x^3\right )^{p+1}}{b (4 p+3)}\)

\(\Big \downarrow \) 74

\(\displaystyle \frac {e x^2 (e x)^p \left (2 b-\frac {3 a c (3 p+2)}{b (4 p+3)}\right ) \left (a x^2+b x^3\right )^p \left (\frac {b x}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,3 p+2,3 (p+1),-\frac {b x}{a}\right )}{3 p+2}+\frac {3 c e (e x)^p \left (a x^2+b x^3\right )^{p+1}}{b (4 p+3)}\)

Input:

Int[(e*x)^(1 + p)*(2*b + 3*c*x)*(a*x^2 + b*x^3)^p,x]
 

Output:

(3*c*e*(e*x)^p*(a*x^2 + b*x^3)^(1 + p))/(b*(3 + 4*p)) + (e*(2*b - (3*a*c*( 
2 + 3*p))/(b*(3 + 4*p)))*x^2*(e*x)^p*(a*x^2 + b*x^3)^p*Hypergeometric2F1[- 
p, 2 + 3*p, 3*(1 + p), -((b*x)/a)])/((2 + 3*p)*(1 + (b*x)/a)^p)
 

Defintions of rubi rules used

rule 74

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^n*((b*x 
)^(m + 1)/(b*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*(x/c)], x] 
/; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] && (IntegerQ[n] || (GtQ[c, 0] 
 &&  !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-d/(b*c), 0])))
 

rule 76

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^IntPart 
[n]*((c + d*x)^FracPart[n]/(1 + d*(x/c))^FracPart[n])   Int[(b*x)^m*(1 + d* 
(x/c))^n, x], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] &&  !Integer 
Q[n] &&  !GtQ[c, 0] &&  !GtQ[-d/(b*c), 0] && ((RationalQ[m] &&  !(EqQ[n, -2 
^(-1)] && EqQ[c^2 - d^2, 0])) ||  !RationalQ[n])
 

rule 1938

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol 
] :> Simp[c^IntPart[m]*(c*x)^FracPart[m]*((a*x^j + b*x^n)^FracPart[p]/(x^(F 
racPart[m] + j*FracPart[p])*(a + b*x^(n - j))^FracPart[p]))   Int[x^(m + j* 
p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !Inte 
gerQ[p] && NeQ[n, j] && PosQ[n - j]
 

rule 1945

Int[((e_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(jn_.))^(p_)*((c_) + 
(d_.)*(x_)^(n_.)), x_Symbol] :> Simp[d*e^(j - 1)*(e*x)^(m - j + 1)*((a*x^j 
+ b*x^(j + n))^(p + 1)/(b*(m + n + p*(j + n) + 1))), x] - Simp[(a*d*(m + j* 
p + 1) - b*c*(m + n + p*(j + n) + 1))/(b*(m + n + p*(j + n) + 1))   Int[(e* 
x)^m*(a*x^j + b*x^(j + n))^p, x], x] /; FreeQ[{a, b, c, d, e, j, m, n, p}, 
x] && EqQ[jn, j + n] &&  !IntegerQ[p] && NeQ[b*c - a*d, 0] && NeQ[m + n + p 
*(j + n) + 1, 0] && (GtQ[e, 0] || IntegerQ[j])
 

Maple [F]

\[\int \left (e x \right )^{p +1} \left (3 c x +2 b \right ) \left (b \,x^{3}+a \,x^{2}\right )^{p}d x\]

Input:

int((e*x)^(p+1)*(3*c*x+2*b)*(b*x^3+a*x^2)^p,x)
 

Output:

int((e*x)^(p+1)*(3*c*x+2*b)*(b*x^3+a*x^2)^p,x)
 

Fricas [F]

\[ \int (e x)^{1+p} (2 b+3 c x) \left (a x^2+b x^3\right )^p \, dx=\int { {\left (3 \, c x + 2 \, b\right )} {\left (b x^{3} + a x^{2}\right )}^{p} \left (e x\right )^{p + 1} \,d x } \] Input:

integrate((e*x)^(p+1)*(3*c*x+2*b)*(b*x^3+a*x^2)^p,x, algorithm="fricas")
 

Output:

integral((3*c*x + 2*b)*(b*x^3 + a*x^2)^p*(e*x)^(p + 1), x)
 

Sympy [F]

\[ \int (e x)^{1+p} (2 b+3 c x) \left (a x^2+b x^3\right )^p \, dx=\int \left (e x\right )^{p + 1} \left (x^{2} \left (a + b x\right )\right )^{p} \left (2 b + 3 c x\right )\, dx \] Input:

integrate((e*x)**(p+1)*(3*c*x+2*b)*(b*x**3+a*x**2)**p,x)
 

Output:

Integral((e*x)**(p + 1)*(x**2*(a + b*x))**p*(2*b + 3*c*x), x)
 

Maxima [F]

\[ \int (e x)^{1+p} (2 b+3 c x) \left (a x^2+b x^3\right )^p \, dx=\int { {\left (3 \, c x + 2 \, b\right )} {\left (b x^{3} + a x^{2}\right )}^{p} \left (e x\right )^{p + 1} \,d x } \] Input:

integrate((e*x)^(p+1)*(3*c*x+2*b)*(b*x^3+a*x^2)^p,x, algorithm="maxima")
 

Output:

integrate((3*c*x + 2*b)*(b*x^3 + a*x^2)^p*(e*x)^(p + 1), x)
 

Giac [F]

\[ \int (e x)^{1+p} (2 b+3 c x) \left (a x^2+b x^3\right )^p \, dx=\int { {\left (3 \, c x + 2 \, b\right )} {\left (b x^{3} + a x^{2}\right )}^{p} \left (e x\right )^{p + 1} \,d x } \] Input:

integrate((e*x)^(p+1)*(3*c*x+2*b)*(b*x^3+a*x^2)^p,x, algorithm="giac")
 

Output:

integrate((3*c*x + 2*b)*(b*x^3 + a*x^2)^p*(e*x)^(p + 1), x)
 

Mupad [F(-1)]

Timed out. \[ \int (e x)^{1+p} (2 b+3 c x) \left (a x^2+b x^3\right )^p \, dx=\int {\left (e\,x\right )}^{p+1}\,\left (2\,b+3\,c\,x\right )\,{\left (b\,x^3+a\,x^2\right )}^p \,d x \] Input:

int((e*x)^(p + 1)*(2*b + 3*c*x)*(a*x^2 + b*x^3)^p,x)
 

Output:

int((e*x)^(p + 1)*(2*b + 3*c*x)*(a*x^2 + b*x^3)^p, x)
 

Reduce [F]

\[ \int (e x)^{1+p} (2 b+3 c x) \left (a x^2+b x^3\right )^p \, dx =\text {Too large to display} \] Input:

int((e*x)^(p+1)*(3*c*x+2*b)*(b*x^3+a*x^2)^p,x)
 

Output:

(e**p*e*(27*x**p*(a*x**2 + b*x**3)**p*a**3*c*p**2 + 27*x**p*(a*x**2 + b*x* 
*3)**p*a**3*c*p + 6*x**p*(a*x**2 + b*x**3)**p*a**3*c - 24*x**p*(a*x**2 + b 
*x**3)**p*a**2*b**2*p**2 - 26*x**p*(a*x**2 + b*x**3)**p*a**2*b**2*p - 6*x* 
*p*(a*x**2 + b*x**3)**p*a**2*b**2 - 36*x**p*(a*x**2 + b*x**3)**p*a**2*b*c* 
p**2*x - 24*x**p*(a*x**2 + b*x**3)**p*a**2*b*c*p*x + 32*x**p*(a*x**2 + b*x 
**3)**p*a*b**3*p**2*x + 24*x**p*(a*x**2 + b*x**3)**p*a*b**3*p*x + 48*x**p* 
(a*x**2 + b*x**3)**p*a*b**2*c*p**2*x**2 + 12*x**p*(a*x**2 + b*x**3)**p*a*b 
**2*c*p*x**2 + 128*x**p*(a*x**2 + b*x**3)**p*b**4*p**2*x**2 + 128*x**p*(a* 
x**2 + b*x**3)**p*b**4*p*x**2 + 24*x**p*(a*x**2 + b*x**3)**p*b**4*x**2 + 1 
92*x**p*(a*x**2 + b*x**3)**p*b**3*c*p**2*x**3 + 144*x**p*(a*x**2 + b*x**3) 
**p*b**3*c*p*x**3 + 24*x**p*(a*x**2 + b*x**3)**p*b**3*c*x**3 - 2592*int((x 
**p*(a*x**2 + b*x**3)**p)/(32*a*p**3*x + 48*a*p**2*x + 22*a*p*x + 3*a*x + 
32*b*p**3*x**2 + 48*b*p**2*x**2 + 22*b*p*x**2 + 3*b*x**2),x)*a**4*c*p**6 - 
 6480*int((x**p*(a*x**2 + b*x**3)**p)/(32*a*p**3*x + 48*a*p**2*x + 22*a*p* 
x + 3*a*x + 32*b*p**3*x**2 + 48*b*p**2*x**2 + 22*b*p*x**2 + 3*b*x**2),x)*a 
**4*c*p**5 - 6246*int((x**p*(a*x**2 + b*x**3)**p)/(32*a*p**3*x + 48*a*p**2 
*x + 22*a*p*x + 3*a*x + 32*b*p**3*x**2 + 48*b*p**2*x**2 + 22*b*p*x**2 + 3* 
b*x**2),x)*a**4*c*p**4 - 2889*int((x**p*(a*x**2 + b*x**3)**p)/(32*a*p**3*x 
 + 48*a*p**2*x + 22*a*p*x + 3*a*x + 32*b*p**3*x**2 + 48*b*p**2*x**2 + 22*b 
*p*x**2 + 3*b*x**2),x)*a**4*c*p**3 - 639*int((x**p*(a*x**2 + b*x**3)**p...