\(\int \frac {x^2 (c+d x)^{1+n}}{a+b x^3} \, dx\) [47]

Optimal result

Integrand size = 22, antiderivative size = 253 \[ \int \frac {x^2 (c+d x)^{1+n}}{a+b x^3} \, dx=-\frac {(c+d x)^{2+n} \operatorname {Hypergeometric2F1}\left (1,2+n,3+n,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 b^{2/3} \left (\sqrt [3]{b} c-\sqrt [3]{a} d\right ) (2+n)}-\frac {(c+d x)^{2+n} \operatorname {Hypergeometric2F1}\left (1,2+n,3+n,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d}\right )}{3 b^{2/3} \left (\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d\right ) (2+n)}-\frac {(c+d x)^{2+n} \operatorname {Hypergeometric2F1}\left (1,2+n,3+n,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right )}{3 b^{2/3} \left (\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d\right ) (2+n)} \] Output:

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

Mathematica [A] (verified)

Time = 0.55 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.84 \[ \int \frac {x^2 (c+d x)^{1+n}}{a+b x^3} \, dx=\frac {(c+d x)^{2+n} \left (-\frac {\operatorname {Hypergeometric2F1}\left (1,2+n,3+n,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{\sqrt [3]{b} c-\sqrt [3]{a} d}-\frac {\operatorname {Hypergeometric2F1}\left (1,2+n,3+n,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d}\right )}{\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d}-\frac {\operatorname {Hypergeometric2F1}\left (1,2+n,3+n,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right )}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right )}{3 b^{2/3} (2+n)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.17 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {7276, 2009}

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.

\(\displaystyle \int \frac {x^2 (c+d x)^{n+1}}{a+b x^3} \, dx\)

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {(c+d x)^{n+2} \operatorname {Hypergeometric2F1}\left (1,n+2,n+3,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 b^{2/3} (n+2) \left (\sqrt [3]{b} c-\sqrt [3]{a} d\right )}-\frac {(c+d x)^{n+2} \operatorname {Hypergeometric2F1}\left (1,n+2,n+3,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d}\right )}{3 b^{2/3} (n+2) \left (\sqrt [3]{-1} \sqrt [3]{a} d+\sqrt [3]{b} c\right )}-\frac {(c+d x)^{n+2} \operatorname {Hypergeometric2F1}\left (1,n+2,n+3,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right )}{3 b^{2/3} (n+2) \left (\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d\right )}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7276

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE 
xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ 
[n, 0]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((d*x + c)^(n + 1)*x^2/(b*x^3 + a), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^2 (c+d x)^{1+n}}{a+b x^3} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^2 (c+d x)^{1+n}}{a+b x^3} \, dx=\frac {2 \left (d x +c \right )^{n} c n +\left (d x +c \right )^{n} c +\left (d x +c \right )^{n} d n x -2 \left (\int \frac {\left (d x +c \right )^{n}}{b d \,x^{4}+b c \,x^{3}+a d x +a c}d x \right ) a c d \,n^{2}-2 \left (\int \frac {\left (d x +c \right )^{n}}{b d \,x^{4}+b c \,x^{3}+a d x +a c}d x \right ) a c d n +\left (\int \frac {\left (d x +c \right )^{n} x^{2}}{b d \,x^{4}+b c \,x^{3}+a d x +a c}d x \right ) b \,c^{2} n^{2}+\left (\int \frac {\left (d x +c \right )^{n} x^{2}}{b d \,x^{4}+b c \,x^{3}+a d x +a c}d x \right ) b \,c^{2} n -\left (\int \frac {\left (d x +c \right )^{n} x}{b d \,x^{4}+b c \,x^{3}+a d x +a c}d x \right ) a \,d^{2} n^{2}-\left (\int \frac {\left (d x +c \right )^{n} x}{b d \,x^{4}+b c \,x^{3}+a d x +a c}d x \right ) a \,d^{2} n}{b n \left (n +1\right )} \] Input:

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

Output:

(2*(c + d*x)**n*c*n + (c + d*x)**n*c + (c + d*x)**n*d*n*x - 2*int((c + d*x 
)**n/(a*c + a*d*x + b*c*x**3 + b*d*x**4),x)*a*c*d*n**2 - 2*int((c + d*x)** 
n/(a*c + a*d*x + b*c*x**3 + b*d*x**4),x)*a*c*d*n + int(((c + d*x)**n*x**2) 
/(a*c + a*d*x + b*c*x**3 + b*d*x**4),x)*b*c**2*n**2 + int(((c + d*x)**n*x* 
*2)/(a*c + a*d*x + b*c*x**3 + b*d*x**4),x)*b*c**2*n - int(((c + d*x)**n*x) 
/(a*c + a*d*x + b*c*x**3 + b*d*x**4),x)*a*d**2*n**2 - int(((c + d*x)**n*x) 
/(a*c + a*d*x + b*c*x**3 + b*d*x**4),x)*a*d**2*n)/(b*n*(n + 1))