Integrand size = 19, antiderivative size = 167 \[ \int \frac {\left (a+b x^n\right )^2}{\left (c+d x^n\right )^3} \, dx=-\frac {(b c-a d) x \left (a+b x^n\right )}{2 c d n \left (c+d x^n\right )^2}-\frac {b (a d (1-n)-b c (1+n)) x}{2 c d^2 (1-n) n \left (c+d x^n\right )}+\frac {\left (2 a b c d (1-n)-b^2 c^2 (1+n)-a^2 d^2 \left (1-3 n+2 n^2\right )\right ) x \operatorname {Hypergeometric2F1}\left (2,\frac {1}{n},1+\frac {1}{n},-\frac {d x^n}{c}\right )}{2 c^3 d^2 (1-n) n} \] Output:
-1/2*(-a*d+b*c)*x*(a+b*x^n)/c/d/n/(c+d*x^n)^2-1/2*b*(a*d*(1-n)-b*c*(1+n))* x/c/d^2/(1-n)/n/(c+d*x^n)+1/2*(2*a*b*c*d*(1-n)-b^2*c^2*(1+n)-a^2*d^2*(2*n^ 2-3*n+1))*x*hypergeom([2, 1/n],[1+1/n],-d*x^n/c)/c^3/d^2/(1-n)/n
Time = 0.22 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.80 \[ \int \frac {\left (a+b x^n\right )^2}{\left (c+d x^n\right )^3} \, dx=\frac {x \left (\frac {c^2 (b c-a d)^2 n}{\left (c+d x^n\right )^2}-\frac {c (b c-a d) (a d (-1+2 n)+b (c+2 c n))}{c+d x^n}+\left (2 a b c d (-1+n)+b^2 c^2 (1+n)+a^2 d^2 \left (1-3 n+2 n^2\right )\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {d x^n}{c}\right )\right )}{2 c^3 d^2 n^2} \] Input:
Integrate[(a + b*x^n)^2/(c + d*x^n)^3,x]
Output:
(x*((c^2*(b*c - a*d)^2*n)/(c + d*x^n)^2 - (c*(b*c - a*d)*(a*d*(-1 + 2*n) + b*(c + 2*c*n)))/(c + d*x^n) + (2*a*b*c*d*(-1 + n) + b^2*c^2*(1 + n) + a^2 *d^2*(1 - 3*n + 2*n^2))*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((d*x^n) /c)]))/(2*c^3*d^2*n^2)
Time = 0.58 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.01, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {930, 910, 778}
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.
| \(\displaystyle \int \frac {\left (a+b x^n\right )^2}{\left (c+d x^n\right )^3} \, dx\) |
| \(\Big \downarrow \) 930 |
| \(\displaystyle \frac {\int \frac {a (b c-a d (1-2 n))-b (a d (1-n)-b c (n+1)) x^n}{\left (d x^n+c\right )^2}dx}{2 c d n}-\frac {x (b c-a d) \left (a+b x^n\right )}{2 c d n \left (c+d x^n\right )^2}\) |
| \(\Big \downarrow \) 910 |
| \(\displaystyle \frac {\frac {x (b c-a d) (a d (1-2 n)-b c (n+1))}{c d n \left (c+d x^n\right )}-\frac {\left (-a^2 d^2 \left (2 n^2-3 n+1\right )+2 a b c d (1-n)-b^2 c^2 (n+1)\right ) \int \frac {1}{d x^n+c}dx}{c d n}}{2 c d n}-\frac {x (b c-a d) \left (a+b x^n\right )}{2 c d n \left (c+d x^n\right )^2}\) |
| \(\Big \downarrow \) 778 |
| \(\displaystyle \frac {\frac {x (b c-a d) (a d (1-2 n)-b c (n+1))}{c d n \left (c+d x^n\right )}-\frac {x \left (-a^2 d^2 \left (2 n^2-3 n+1\right )+2 a b c d (1-n)-b^2 c^2 (n+1)\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {d x^n}{c}\right )}{c^2 d n}}{2 c d n}-\frac {x (b c-a d) \left (a+b x^n\right )}{2 c d n \left (c+d x^n\right )^2}\) |
Input:
Int[(a + b*x^n)^2/(c + d*x^n)^3,x]
Output:
-1/2*((b*c - a*d)*x*(a + b*x^n))/(c*d*n*(c + d*x^n)^2) + (((b*c - a*d)*(a* d*(1 - 2*n) - b*c*(1 + n))*x)/(c*d*n*(c + d*x^n)) - ((2*a*b*c*d*(1 - n) - b^2*c^2*(1 + n) - a^2*d^2*(1 - 3*n + 2*n^2))*x*Hypergeometric2F1[1, n^(-1) , 1 + n^(-1), -((d*x^n)/c)])/(c^2*d*n))/(2*c*d*n)
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F 1[-p, 1/n, 1/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, n, p}, x] && !IGtQ[p , 0] && !IntegerQ[1/n] && !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p] || GtQ[a, 0])
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Si mp[(-(b*c - a*d))*x*((a + b*x^n)^(p + 1)/(a*b*n*(p + 1))), x] - Simp[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)) Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/ n + p, 0])
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(a*d - c*b)*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q - 1)/(a*b*n*(p + 1))), x] - Simp[1/(a*b*n*(p + 1)) Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 2)*Simp[c*(a*d - c*b*(n*(p + 1) + 1)) + d*(a*d*(n*(q - 1) + 1) - b*c*(n*( p + q) + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, n, p, q, x]
\[\int \frac {\left (a +b \,x^{n}\right )^{2}}{\left (c +d \,x^{n}\right )^{3}}d x\]
Input:
int((a+b*x^n)^2/(c+d*x^n)^3,x)
Output:
int((a+b*x^n)^2/(c+d*x^n)^3,x)
\[ \int \frac {\left (a+b x^n\right )^2}{\left (c+d x^n\right )^3} \, dx=\int { \frac {{\left (b x^{n} + a\right )}^{2}}{{\left (d x^{n} + c\right )}^{3}} \,d x } \] Input:
integrate((a+b*x^n)^2/(c+d*x^n)^3,x, algorithm="fricas")
Output:
integral((b^2*x^(2*n) + 2*a*b*x^n + a^2)/(d^3*x^(3*n) + 3*c*d^2*x^(2*n) + 3*c^2*d*x^n + c^3), x)
\[ \int \frac {\left (a+b x^n\right )^2}{\left (c+d x^n\right )^3} \, dx=\int \frac {\left (a + b x^{n}\right )^{2}}{\left (c + d x^{n}\right )^{3}}\, dx \] Input:
integrate((a+b*x**n)**2/(c+d*x**n)**3,x)
Output:
Integral((a + b*x**n)**2/(c + d*x**n)**3, x)
\[ \int \frac {\left (a+b x^n\right )^2}{\left (c+d x^n\right )^3} \, dx=\int { \frac {{\left (b x^{n} + a\right )}^{2}}{{\left (d x^{n} + c\right )}^{3}} \,d x } \] Input:
integrate((a+b*x^n)^2/(c+d*x^n)^3,x, algorithm="maxima")
Output:
((2*n^2 - 3*n + 1)*a^2*d^2 + b^2*c^2*(n + 1) + 2*a*b*c*d*(n - 1))*integrat e(1/2/(c^2*d^3*n^2*x^n + c^3*d^2*n^2), x) - 1/2*((b^2*c^2*d*(2*n + 1) - a^ 2*d^3*(2*n - 1) - 2*a*b*c*d^2)*x*x^n - (a^2*c*d^2*(3*n - 1) - b^2*c^3*(n + 1) - 2*a*b*c^2*d*(n - 1))*x)/(c^2*d^4*n^2*x^(2*n) + 2*c^3*d^3*n^2*x^n + c ^4*d^2*n^2)
\[ \int \frac {\left (a+b x^n\right )^2}{\left (c+d x^n\right )^3} \, dx=\int { \frac {{\left (b x^{n} + a\right )}^{2}}{{\left (d x^{n} + c\right )}^{3}} \,d x } \] Input:
integrate((a+b*x^n)^2/(c+d*x^n)^3,x, algorithm="giac")
Output:
integrate((b*x^n + a)^2/(d*x^n + c)^3, x)
Timed out. \[ \int \frac {\left (a+b x^n\right )^2}{\left (c+d x^n\right )^3} \, dx=\int \frac {{\left (a+b\,x^n\right )}^2}{{\left (c+d\,x^n\right )}^3} \,d x \] Input:
int((a + b*x^n)^2/(c + d*x^n)^3,x)
Output:
int((a + b*x^n)^2/(c + d*x^n)^3, x)
\[ \int \frac {\left (a+b x^n\right )^2}{\left (c+d x^n\right )^3} \, dx=\text {too large to display} \] Input:
int((a+b*x^n)^2/(c+d*x^n)^3,x)
Output:
(2*x**(2*n)*int(x**(2*n)/(x**(3*n)*d**3*n + x**(3*n)*d**3 + 3*x**(2*n)*c*d **2*n + 3*x**(2*n)*c*d**2 + 3*x**n*c**2*d*n + 3*x**n*c**2*d + c**3*n + c** 3),x)*a**2*d**4*n**3 - x**(2*n)*int(x**(2*n)/(x**(3*n)*d**3*n + x**(3*n)*d **3 + 3*x**(2*n)*c*d**2*n + 3*x**(2*n)*c*d**2 + 3*x**n*c**2*d*n + 3*x**n*c **2*d + c**3*n + c**3),x)*a**2*d**4*n**2 - 2*x**(2*n)*int(x**(2*n)/(x**(3* n)*d**3*n + x**(3*n)*d**3 + 3*x**(2*n)*c*d**2*n + 3*x**(2*n)*c*d**2 + 3*x* *n*c**2*d*n + 3*x**n*c**2*d + c**3*n + c**3),x)*a**2*d**4*n + x**(2*n)*int (x**(2*n)/(x**(3*n)*d**3*n + x**(3*n)*d**3 + 3*x**(2*n)*c*d**2*n + 3*x**(2 *n)*c*d**2 + 3*x**n*c**2*d*n + 3*x**n*c**2*d + c**3*n + c**3),x)*a**2*d**4 + 2*x**(2*n)*int(x**(2*n)/(x**(3*n)*d**3*n + x**(3*n)*d**3 + 3*x**(2*n)*c *d**2*n + 3*x**(2*n)*c*d**2 + 3*x**n*c**2*d*n + 3*x**n*c**2*d + c**3*n + c **3),x)*a*b*c*d**3*n**2 - 2*x**(2*n)*int(x**(2*n)/(x**(3*n)*d**3*n + x**(3 *n)*d**3 + 3*x**(2*n)*c*d**2*n + 3*x**(2*n)*c*d**2 + 3*x**n*c**2*d*n + 3*x **n*c**2*d + c**3*n + c**3),x)*a*b*c*d**3 + x**(2*n)*int(x**(2*n)/(x**(3*n )*d**3*n + x**(3*n)*d**3 + 3*x**(2*n)*c*d**2*n + 3*x**(2*n)*c*d**2 + 3*x** n*c**2*d*n + 3*x**n*c**2*d + c**3*n + c**3),x)*b**2*c**2*d**2*n**2 + 2*x** (2*n)*int(x**(2*n)/(x**(3*n)*d**3*n + x**(3*n)*d**3 + 3*x**(2*n)*c*d**2*n + 3*x**(2*n)*c*d**2 + 3*x**n*c**2*d*n + 3*x**n*c**2*d + c**3*n + c**3),x)* b**2*c**2*d**2*n + x**(2*n)*int(x**(2*n)/(x**(3*n)*d**3*n + x**(3*n)*d**3 + 3*x**(2*n)*c*d**2*n + 3*x**(2*n)*c*d**2 + 3*x**n*c**2*d*n + 3*x**n*c*...