\(\int (a+b \sqrt [3]{x}+c x^{2/3})^p (d x)^m \, dx\) [93]

Optimal result

Integrand size = 24, antiderivative size = 163 \[ \int \left (a+b \sqrt [3]{x}+c x^{2/3}\right )^p (d x)^m \, dx=\frac {\left (1+\frac {2 c \sqrt [3]{x}}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (1+\frac {2 c \sqrt [3]{x}}{b+\sqrt {b^2-4 a c}}\right )^{-p} \left (a+b \sqrt [3]{x}+c x^{2/3}\right )^p (d x)^{1+m} \operatorname {AppellF1}\left (3 (1+m),-p,-p,4+3 m,-\frac {2 c \sqrt [3]{x}}{b-\sqrt {b^2-4 a c}},-\frac {2 c \sqrt [3]{x}}{b+\sqrt {b^2-4 a c}}\right )}{d (1+m)} \] Output:

(a+b*x^(1/3)+c*x^(2/3))^p*(d*x)^(1+m)*AppellF1(3+3*m,-p,-p,4+3*m,-2*c*x^(1 
/3)/(b-(-4*a*c+b^2)^(1/2)),-2*c*x^(1/3)/(b+(-4*a*c+b^2)^(1/2)))/d/(1+m)/(( 
1+2*c*x^(1/3)/(b-(-4*a*c+b^2)^(1/2)))^p)/((1+2*c*x^(1/3)/(b+(-4*a*c+b^2)^( 
1/2)))^p)
 

Mathematica [A] (warning: unable to verify)

Time = 0.73 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.15 \[ \int \left (a+b \sqrt [3]{x}+c x^{2/3}\right )^p (d x)^m \, dx=\frac {\left (\frac {b-\sqrt {b^2-4 a c}+2 c \sqrt [3]{x}}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (\frac {b+\sqrt {b^2-4 a c}+2 c \sqrt [3]{x}}{b+\sqrt {b^2-4 a c}}\right )^{-p} \left (a+b \sqrt [3]{x}+c x^{2/3}\right )^p x (d x)^m \operatorname {AppellF1}\left (3 (1+m),-p,-p,4+3 m,-\frac {2 c \sqrt [3]{x}}{b+\sqrt {b^2-4 a c}},\frac {2 c \sqrt [3]{x}}{-b+\sqrt {b^2-4 a c}}\right )}{1+m} \] Input:

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

Output:

((a + b*x^(1/3) + c*x^(2/3))^p*x*(d*x)^m*AppellF1[3*(1 + m), -p, -p, 4 + 3 
*m, (-2*c*x^(1/3))/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^(1/3))/(-b + Sqrt[b^2 - 
 4*a*c])])/((1 + m)*((b - Sqrt[b^2 - 4*a*c] + 2*c*x^(1/3))/(b - Sqrt[b^2 - 
 4*a*c]))^p*((b + Sqrt[b^2 - 4*a*c] + 2*c*x^(1/3))/(b + Sqrt[b^2 - 4*a*c]) 
)^p)
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.98, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1716, 1715, 1179, 150}

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.

\(\displaystyle \int (d x)^m \left (a+b \sqrt [3]{x}+c x^{2/3}\right )^p \, dx\)

\(\Big \downarrow \) 1716

\(\displaystyle x^{-m} (d x)^m \int \left (a+c x^{2/3}+b \sqrt [3]{x}\right )^p x^mdx\)

\(\Big \downarrow \) 1715

\(\displaystyle 3 x^{-m} (d x)^m \int \left (a+c x^{2/3}+b \sqrt [3]{x}\right )^p x^{\frac {1}{3} (3 m+2)}d\sqrt [3]{x}\)

\(\Big \downarrow \) 1179

\(\displaystyle 3 x^{-m} (d x)^m \left (\frac {2 c \sqrt [3]{x}}{b-\sqrt {b^2-4 a c}}+1\right )^{-p} \left (\frac {2 c \sqrt [3]{x}}{\sqrt {b^2-4 a c}+b}+1\right )^{-p} \left (a+b \sqrt [3]{x}+c x^{2/3}\right )^p \int \left (\frac {2 \sqrt [3]{x} c}{b-\sqrt {b^2-4 a c}}+1\right )^p \left (\frac {2 \sqrt [3]{x} c}{b+\sqrt {b^2-4 a c}}+1\right )^p x^{\frac {1}{3} (3 m+2)}d\sqrt [3]{x}\)

\(\Big \downarrow \) 150

\(\displaystyle \frac {x (d x)^m \left (\frac {2 c \sqrt [3]{x}}{b-\sqrt {b^2-4 a c}}+1\right )^{-p} \left (\frac {2 c \sqrt [3]{x}}{\sqrt {b^2-4 a c}+b}+1\right )^{-p} \left (a+b \sqrt [3]{x}+c x^{2/3}\right )^p \operatorname {AppellF1}\left (3 (m+1),-p,-p,3 m+4,-\frac {2 c \sqrt [3]{x}}{b-\sqrt {b^2-4 a c}},-\frac {2 c \sqrt [3]{x}}{b+\sqrt {b^2-4 a c}}\right )}{m+1}\)

Input:

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

Output:

((a + b*x^(1/3) + c*x^(2/3))^p*x*(d*x)^m*AppellF1[3*(1 + m), -p, -p, 4 + 3 
*m, (-2*c*x^(1/3))/(b - Sqrt[b^2 - 4*a*c]), (-2*c*x^(1/3))/(b + Sqrt[b^2 - 
 4*a*c])])/((1 + m)*(1 + (2*c*x^(1/3))/(b - Sqrt[b^2 - 4*a*c]))^p*(1 + (2* 
c*x^(1/3))/(b + Sqrt[b^2 - 4*a*c]))^p)
 

Defintions of rubi rules used

rule 150

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ 
] :> Simp[c^n*e^p*((b*x)^(m + 1)/(b*(m + 1)))*AppellF1[m + 1, -n, -p, m + 2 
, (-d)*(x/c), (-f)*(x/e)], x] /; FreeQ[{b, c, d, e, f, m, n, p}, x] &&  !In 
tegerQ[m] &&  !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])
 

rule 1179

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(a + b*x + c*x^2)^p/(e*(1 - ( 
d + e*x)/(d - e*((b - q)/(2*c))))^p*(1 - (d + e*x)/(d - e*((b + q)/(2*c)))) 
^p)   Subst[Int[x^m*Simp[1 - x/(d - e*((b - q)/(2*c))), x]^p*Simp[1 - x/(d 
- e*((b + q)/(2*c))), x]^p, x], x, d + e*x], x]] /; FreeQ[{a, b, c, d, e, m 
, p}, x]
 

rule 1715

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] 
 :> With[{k = Denominator[n]}, Simp[k   Subst[Int[x^(k*(m + 1) - 1)*(a + b* 
x^(k*n) + c*x^(2*k*n))^p, x], x, x^(1/k)], x]] /; FreeQ[{a, b, c, m, p}, x] 
 && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && FractionQ[n]
 

rule 1716

Int[((d_)*(x_))^(m_)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_S 
ymbol] :> Simp[d^IntPart[m]*((d*x)^FracPart[m]/x^FracPart[m])   Int[x^m*(a 
+ b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, m, p}, x] && EqQ[n2, 2 
*n] && NeQ[b^2 - 4*a*c, 0] && FractionQ[n]
 

Maple [F]

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

Input:

int((a+b*x^(1/3)+c*x^(2/3))^p*(d*x)^m,x)
 

Output:

int((a+b*x^(1/3)+c*x^(2/3))^p*(d*x)^m,x)
 

Fricas [F(-2)]

Exception generated. \[ \int \left (a+b \sqrt [3]{x}+c x^{2/3}\right )^p (d x)^m \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >>  Error detected within library code:   algl 
ogextint: unimplemented
 

Sympy [F(-1)]

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

integrate((a+b*x**(1/3)+c*x**(2/3))**p*(d*x)**m,x)
 

Output:

Timed out
 

Maxima [F]

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

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

Output:

integrate((d*x)^m*(c*x^(2/3) + b*x^(1/3) + a)^p, x)
 

Giac [F]

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

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

Output:

integrate((d*x)^m*(c*x^(2/3) + b*x^(1/3) + a)^p, x)
 

Mupad [F(-1)]

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

int((d*x)^m*(a + b*x^(1/3) + c*x^(2/3))^p,x)
 

Output:

int((d*x)^m*(a + b*x^(1/3) + c*x^(2/3))^p, x)
 

Reduce [F]

\[ \int \left (a+b \sqrt [3]{x}+c x^{2/3}\right )^p (d x)^m \, dx=\text {too large to display} \] Input:

int((a+b*x^(1/3)+c*x^(2/3))^p*(d*x)^m,x)
 

Output:

(d**m*(27*x**((3*m + 2)/3)*(x**(2/3)*c + x**(1/3)*b + a)**p*b**2*c*m**2*p 
+ 27*x**((3*m + 2)/3)*(x**(2/3)*c + x**(1/3)*b + a)**p*b**2*c*m*p**2 + 9*x 
**((3*m + 2)/3)*(x**(2/3)*c + x**(1/3)*b + a)**p*b**2*c*m*p + 6*x**((3*m + 
 2)/3)*(x**(2/3)*c + x**(1/3)*b + a)**p*b**2*c*p**3 + 3*x**((3*m + 2)/3)*( 
x**(2/3)*c + x**(1/3)*b + a)**p*b**2*c*p**2 + 54*x**((3*m + 1)/3)*(x**(2/3 
)*c + x**(1/3)*b + a)**p*a*b*c*m**2*p + 54*x**((3*m + 1)/3)*(x**(2/3)*c + 
x**(1/3)*b + a)**p*a*b*c*m*p**2 + 36*x**((3*m + 1)/3)*(x**(2/3)*c + x**(1/ 
3)*b + a)**p*a*b*c*m*p + 12*x**((3*m + 1)/3)*(x**(2/3)*c + x**(1/3)*b + a) 
**p*a*b*c*p**3 + 12*x**((3*m + 1)/3)*(x**(2/3)*c + x**(1/3)*b + a)**p*a*b* 
c*p**2 - 27*x**((3*m + 1)/3)*(x**(2/3)*c + x**(1/3)*b + a)**p*b**3*m**2*p 
- 18*x**((3*m + 1)/3)*(x**(2/3)*c + x**(1/3)*b + a)**p*b**3*m*p**2 - 18*x* 
*((3*m + 1)/3)*(x**(2/3)*c + x**(1/3)*b + a)**p*b**3*m*p - 3*x**((3*m + 1) 
/3)*(x**(2/3)*c + x**(1/3)*b + a)**p*b**3*p**3 - 6*x**((3*m + 1)/3)*(x**(2 
/3)*c + x**(1/3)*b + a)**p*b**3*p**2 - 54*x**m*(x**(2/3)*c + x**(1/3)*b + 
a)**p*a**2*c*m**2*p - 36*x**m*(x**(2/3)*c + x**(1/3)*b + a)**p*a**2*c*m*p* 
*2 - 54*x**m*(x**(2/3)*c + x**(1/3)*b + a)**p*a**2*c*m*p - 12*x**m*(x**(2/ 
3)*c + x**(1/3)*b + a)**p*a**2*c*p**2 - 12*x**m*(x**(2/3)*c + x**(1/3)*b + 
 a)**p*a**2*c*p + 27*x**m*(x**(2/3)*c + x**(1/3)*b + a)**p*a*b**2*m**2*p + 
 9*x**m*(x**(2/3)*c + x**(1/3)*b + a)**p*a*b**2*m*p**2 + 27*x**m*(x**(2/3) 
*c + x**(1/3)*b + a)**p*a*b**2*m*p + 3*x**m*(x**(2/3)*c + x**(1/3)*b + ...