Integrand size = 22, antiderivative size = 253 \[ \int \frac {x^2 (c+d x)^{1+n}}{a+b x^3} \, dx=-\frac {(c+d x)^{2+n} \operatorname {Hypergeometric2F1}\left (1,2+n,3+n,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 b^{2/3} \left (\sqrt [3]{b} c-\sqrt [3]{a} d\right ) (2+n)}-\frac {(c+d x)^{2+n} \operatorname {Hypergeometric2F1}\left (1,2+n,3+n,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d}\right )}{3 b^{2/3} \left (\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d\right ) (2+n)}-\frac {(c+d x)^{2+n} \operatorname {Hypergeometric2F1}\left (1,2+n,3+n,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right )}{3 b^{2/3} \left (\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d\right ) (2+n)} \]
-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)
Time = 0.33 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.84 \[ \int \frac {x^2 (c+d x)^{1+n}}{a+b x^3} \, dx=\frac {(c+d x)^{2+n} \left (-\frac {\operatorname {Hypergeometric2F1}\left (1,2+n,3+n,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{\sqrt [3]{b} c-\sqrt [3]{a} d}-\frac {\operatorname {Hypergeometric2F1}\left (1,2+n,3+n,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d}\right )}{\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d}-\frac {\operatorname {Hypergeometric2F1}\left (1,2+n,3+n,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right )}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right )}{3 b^{2/3} (2+n)} \]
((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))
Time = 0.72 (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 |
\(\displaystyle -\frac {(c+d x)^{n+2} \operatorname {Hypergeometric2F1}\left (1,n+2,n+3,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 b^{2/3} (n+2) \left (\sqrt [3]{b} c-\sqrt [3]{a} d\right )}-\frac {(c+d x)^{n+2} \operatorname {Hypergeometric2F1}\left (1,n+2,n+3,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d}\right )}{3 b^{2/3} (n+2) \left (\sqrt [3]{-1} \sqrt [3]{a} d+\sqrt [3]{b} c\right )}-\frac {(c+d x)^{n+2} \operatorname {Hypergeometric2F1}\left (1,n+2,n+3,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right )}{3 b^{2/3} (n+2) \left (\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d\right )}\) |
-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))
3.2.94.3.1 Defintions of rubi rules used
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]
\[\int \frac {x^{2} \left (d x +c \right )^{1+n}}{b \,x^{3}+a}d x\]
\[ \int \frac {x^2 (c+d x)^{1+n}}{a+b x^3} \, dx=\int { \frac {{\left (d x + c\right )}^{n + 1} x^{2}}{b x^{3} + a} \,d x } \]
Timed out. \[ \int \frac {x^2 (c+d x)^{1+n}}{a+b x^3} \, dx=\text {Timed out} \]
\[ \int \frac {x^2 (c+d x)^{1+n}}{a+b x^3} \, dx=\int { \frac {{\left (d x + c\right )}^{n + 1} x^{2}}{b x^{3} + a} \,d x } \]
\[ \int \frac {x^2 (c+d x)^{1+n}}{a+b x^3} \, dx=\int { \frac {{\left (d x + c\right )}^{n + 1} x^{2}}{b x^{3} + a} \,d x } \]
Timed out. \[ \int \frac {x^2 (c+d x)^{1+n}}{a+b x^3} \, dx=\int \frac {x^2\,{\left (c+d\,x\right )}^{n+1}}{b\,x^3+a} \,d x \]