\(\int x^{1-3 p} (a x^2+b x^3)^p \, dx\) [256]

Optimal result

Integrand size = 21, antiderivative size = 61 \[ \int x^{1-3 p} \left (a x^2+b x^3\right )^p \, dx=\frac {x^{2-3 p} \left (1+\frac {b x}{a}\right )^{-p} \left (a x^2+b x^3\right )^p \operatorname {Hypergeometric2F1}\left (2-p,-p,3-p,-\frac {b x}{a}\right )}{2-p} \] Output:

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.97 \[ \int x^{1-3 p} \left (a x^2+b x^3\right )^p \, dx=\frac {x^{2-3 p} \left (x^2 (a+b x)\right )^p \left (1+\frac {b x}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (2-p,-p,3-p,-\frac {b x}{a}\right )}{2-p} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {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.

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

\(\Big \downarrow \) 1938

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

\(\Big \downarrow \) 76

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

\(\Big \downarrow \) 74

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

Input:

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

Output:

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

((a*x**2 + b*x**3)**p*a*p*x + (a*x**2 + b*x**3)**p*b*x**2 + x**(3*p)*int(( 
a*x**2 + b*x**3)**p/(x**(3*p)*a + x**(3*p)*b*x),x)*a**2*p**2 - x**(3*p)*in 
t((a*x**2 + b*x**3)**p/(x**(3*p)*a + x**(3*p)*b*x),x)*a**2*p)/(2*x**(3*p)* 
b)