Integrand size = 26, antiderivative size = 90 \[ \int \frac {x^{-1+2 n} \left (a+b x^n\right )^2}{c+d x^n} \, dx=\frac {(b c-a d)^2 x^n}{d^3 n}-\frac {b (b c-2 a d) x^{2 n}}{2 d^2 n}+\frac {b^2 x^{3 n}}{3 d n}-\frac {c (b c-a d)^2 \log \left (c+d x^n\right )}{d^4 n} \] Output:
(-a*d+b*c)^2*x^n/d^3/n-1/2*b*(-2*a*d+b*c)*x^(2*n)/d^2/n+1/3*b^2*x^(3*n)/d/ n-c*(-a*d+b*c)^2*ln(c+d*x^n)/d^4/n
Time = 0.09 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.97 \[ \int \frac {x^{-1+2 n} \left (a+b x^n\right )^2}{c+d x^n} \, dx=\frac {d x^n \left (6 a^2 d^2+6 a b d \left (-2 c+d x^n\right )+b^2 \left (6 c^2-3 c d x^n+2 d^2 x^{2 n}\right )\right )-6 c (b c-a d)^2 \log \left (c+d x^n\right )}{6 d^4 n} \] Input:
Integrate[(x^(-1 + 2*n)*(a + b*x^n)^2)/(c + d*x^n),x]
Output:
(d*x^n*(6*a^2*d^2 + 6*a*b*d*(-2*c + d*x^n) + b^2*(6*c^2 - 3*c*d*x^n + 2*d^ 2*x^(2*n))) - 6*c*(b*c - a*d)^2*Log[c + d*x^n])/(6*d^4*n)
Time = 0.43 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.91, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {948, 86, 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.
\(\displaystyle \int \frac {x^{2 n-1} \left (a+b x^n\right )^2}{c+d x^n} \, dx\) |
\(\Big \downarrow \) 948 |
\(\displaystyle \frac {\int \frac {x^n \left (b x^n+a\right )^2}{d x^n+c}dx^n}{n}\) |
\(\Big \downarrow \) 86 |
\(\displaystyle \frac {\int \left (-\frac {b (b c-2 a d) x^n}{d^2}+\frac {b^2 x^{2 n}}{d}+\frac {(a d-b c)^2}{d^3}-\frac {c (b c-a d)^2}{d^3 \left (d x^n+c\right )}\right )dx^n}{n}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-\frac {c (b c-a d)^2 \log \left (c+d x^n\right )}{d^4}+\frac {x^n (b c-a d)^2}{d^3}-\frac {b x^{2 n} (b c-2 a d)}{2 d^2}+\frac {b^2 x^{3 n}}{3 d}}{n}\) |
Input:
Int[(x^(-1 + 2*n)*(a + b*x^n)^2)/(c + d*x^n),x]
Output:
(((b*c - a*d)^2*x^n)/d^3 - (b*(b*c - 2*a*d)*x^(2*n))/(2*d^2) + (b^2*x^(3*n ))/(3*d) - (c*(b*c - a*d)^2*Log[c + d*x^n])/d^4)/n
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. ), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^ p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ [b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
Time = 0.38 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.31
method | result | size |
norman | \(\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) {\mathrm e}^{n \ln \left (x \right )}}{d^{3} n}+\frac {b^{2} {\mathrm e}^{3 n \ln \left (x \right )}}{3 d n}+\frac {b \left (2 a d -c b \right ) {\mathrm e}^{2 n \ln \left (x \right )}}{2 d^{2} n}-\frac {c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (c +d \,{\mathrm e}^{n \ln \left (x \right )}\right )}{d^{4} n}\) | \(118\) |
risch | \(\frac {b^{2} x^{3 n}}{3 d n}+\frac {b \,x^{2 n} a}{d n}-\frac {b^{2} x^{2 n} c}{2 d^{2} n}+\frac {x^{n} a^{2}}{d n}-\frac {2 x^{n} a b c}{d^{2} n}+\frac {x^{n} b^{2} c^{2}}{d^{3} n}-\frac {c \ln \left (x^{n}+\frac {c}{d}\right ) a^{2}}{d^{2} n}+\frac {2 c^{2} \ln \left (x^{n}+\frac {c}{d}\right ) a b}{d^{3} n}-\frac {c^{3} \ln \left (x^{n}+\frac {c}{d}\right ) b^{2}}{d^{4} n}\) | \(161\) |
Input:
int(x^(-1+2*n)*(a+b*x^n)^2/(c+d*x^n),x,method=_RETURNVERBOSE)
Output:
1/d^3*(a^2*d^2-2*a*b*c*d+b^2*c^2)/n*exp(n*ln(x))+1/3*b^2/d/n*exp(n*ln(x))^ 3+1/2*b*(2*a*d-b*c)/d^2/n*exp(n*ln(x))^2-c*(a^2*d^2-2*a*b*c*d+b^2*c^2)/d^4 /n*ln(c+d*exp(n*ln(x)))
Time = 0.11 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.20 \[ \int \frac {x^{-1+2 n} \left (a+b x^n\right )^2}{c+d x^n} \, dx=\frac {2 \, b^{2} d^{3} x^{3 \, n} - 3 \, {\left (b^{2} c d^{2} - 2 \, a b d^{3}\right )} x^{2 \, n} + 6 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x^{n} - 6 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} \log \left (d x^{n} + c\right )}{6 \, d^{4} n} \] Input:
integrate(x^(-1+2*n)*(a+b*x^n)^2/(c+d*x^n),x, algorithm="fricas")
Output:
1/6*(2*b^2*d^3*x^(3*n) - 3*(b^2*c*d^2 - 2*a*b*d^3)*x^(2*n) + 6*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*x^n - 6*(b^2*c^3 - 2*a*b*c^2*d + a^2*c*d^2)*log(d *x^n + c))/(d^4*n)
Leaf count of result is larger than twice the leaf count of optimal. 221 vs. \(2 (75) = 150\).
Time = 3.22 (sec) , antiderivative size = 221, normalized size of antiderivative = 2.46 \[ \int \frac {x^{-1+2 n} \left (a+b x^n\right )^2}{c+d x^n} \, dx=\begin {cases} \frac {\left (a + b\right )^{2} \log {\left (x \right )}}{c} & \text {for}\: d = 0 \wedge n = 0 \\\frac {\frac {a^{2} x x^{2 n - 1}}{2 n} + \frac {2 a b x x^{n} x^{2 n - 1}}{3 n} + \frac {b^{2} x x^{2 n} x^{2 n - 1}}{4 n}}{c} & \text {for}\: d = 0 \\\frac {\left (a + b\right )^{2} \log {\left (x \right )}}{c + d} & \text {for}\: n = 0 \\- \frac {a^{2} c \log {\left (\frac {c}{d} + x^{n} \right )}}{d^{2} n} + \frac {a^{2} x^{n}}{d n} + \frac {2 a b c^{2} \log {\left (\frac {c}{d} + x^{n} \right )}}{d^{3} n} - \frac {2 a b c x^{n}}{d^{2} n} + \frac {a b x^{2 n}}{d n} - \frac {b^{2} c^{3} \log {\left (\frac {c}{d} + x^{n} \right )}}{d^{4} n} + \frac {b^{2} c^{2} x^{n}}{d^{3} n} - \frac {b^{2} c x^{2 n}}{2 d^{2} n} + \frac {b^{2} x^{3 n}}{3 d n} & \text {otherwise} \end {cases} \] Input:
integrate(x**(-1+2*n)*(a+b*x**n)**2/(c+d*x**n),x)
Output:
Piecewise(((a + b)**2*log(x)/c, Eq(d, 0) & Eq(n, 0)), ((a**2*x*x**(2*n - 1 )/(2*n) + 2*a*b*x*x**n*x**(2*n - 1)/(3*n) + b**2*x*x**(2*n)*x**(2*n - 1)/( 4*n))/c, Eq(d, 0)), ((a + b)**2*log(x)/(c + d), Eq(n, 0)), (-a**2*c*log(c/ d + x**n)/(d**2*n) + a**2*x**n/(d*n) + 2*a*b*c**2*log(c/d + x**n)/(d**3*n) - 2*a*b*c*x**n/(d**2*n) + a*b*x**(2*n)/(d*n) - b**2*c**3*log(c/d + x**n)/ (d**4*n) + b**2*c**2*x**n/(d**3*n) - b**2*c*x**(2*n)/(2*d**2*n) + b**2*x** (3*n)/(3*d*n), True))
Time = 0.04 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.67 \[ \int \frac {x^{-1+2 n} \left (a+b x^n\right )^2}{c+d x^n} \, dx=a^{2} {\left (\frac {x^{n}}{d n} - \frac {c \log \left (\frac {d x^{n} + c}{d}\right )}{d^{2} n}\right )} - \frac {1}{6} \, b^{2} {\left (\frac {6 \, c^{3} \log \left (\frac {d x^{n} + c}{d}\right )}{d^{4} n} - \frac {2 \, d^{2} x^{3 \, n} - 3 \, c d x^{2 \, n} + 6 \, c^{2} x^{n}}{d^{3} n}\right )} + a b {\left (\frac {2 \, c^{2} \log \left (\frac {d x^{n} + c}{d}\right )}{d^{3} n} + \frac {d x^{2 \, n} - 2 \, c x^{n}}{d^{2} n}\right )} \] Input:
integrate(x^(-1+2*n)*(a+b*x^n)^2/(c+d*x^n),x, algorithm="maxima")
Output:
a^2*(x^n/(d*n) - c*log((d*x^n + c)/d)/(d^2*n)) - 1/6*b^2*(6*c^3*log((d*x^n + c)/d)/(d^4*n) - (2*d^2*x^(3*n) - 3*c*d*x^(2*n) + 6*c^2*x^n)/(d^3*n)) + a*b*(2*c^2*log((d*x^n + c)/d)/(d^3*n) + (d*x^(2*n) - 2*c*x^n)/(d^2*n))
\[ \int \frac {x^{-1+2 n} \left (a+b x^n\right )^2}{c+d x^n} \, dx=\int { \frac {{\left (b x^{n} + a\right )}^{2} x^{2 \, n - 1}}{d x^{n} + c} \,d x } \] Input:
integrate(x^(-1+2*n)*(a+b*x^n)^2/(c+d*x^n),x, algorithm="giac")
Output:
integrate((b*x^n + a)^2*x^(2*n - 1)/(d*x^n + c), x)
Timed out. \[ \int \frac {x^{-1+2 n} \left (a+b x^n\right )^2}{c+d x^n} \, dx=\int \frac {x^{2\,n-1}\,{\left (a+b\,x^n\right )}^2}{c+d\,x^n} \,d x \] Input:
int((x^(2*n - 1)*(a + b*x^n)^2)/(c + d*x^n),x)
Output:
int((x^(2*n - 1)*(a + b*x^n)^2)/(c + d*x^n), x)
Time = 0.20 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.46 \[ \int \frac {x^{-1+2 n} \left (a+b x^n\right )^2}{c+d x^n} \, dx=\frac {2 x^{3 n} b^{2} d^{3}+6 x^{2 n} a b \,d^{3}-3 x^{2 n} b^{2} c \,d^{2}+6 x^{n} a^{2} d^{3}-12 x^{n} a b c \,d^{2}+6 x^{n} b^{2} c^{2} d -6 \,\mathrm {log}\left (x^{n} d +c \right ) a^{2} c \,d^{2}+12 \,\mathrm {log}\left (x^{n} d +c \right ) a b \,c^{2} d -6 \,\mathrm {log}\left (x^{n} d +c \right ) b^{2} c^{3}}{6 d^{4} n} \] Input:
int(x^(-1+2*n)*(a+b*x^n)^2/(c+d*x^n),x)
Output:
(2*x**(3*n)*b**2*d**3 + 6*x**(2*n)*a*b*d**3 - 3*x**(2*n)*b**2*c*d**2 + 6*x **n*a**2*d**3 - 12*x**n*a*b*c*d**2 + 6*x**n*b**2*c**2*d - 6*log(x**n*d + c )*a**2*c*d**2 + 12*log(x**n*d + c)*a*b*c**2*d - 6*log(x**n*d + c)*b**2*c** 3)/(6*d**4*n)