Integrand size = 18, antiderivative size = 215 \[ \int \frac {x \left (c+d x^4\right )^2}{(a+b x)^2} \, dx=-\frac {8 a^3 d \left (b^4 c+a^4 d\right ) x}{b^9}+\frac {a^2 d \left (6 b^4 c+7 a^4 d\right ) x^2}{2 b^8}-\frac {2 a d \left (2 b^4 c+3 a^4 d\right ) x^3}{3 b^7}+\frac {d \left (2 b^4 c+5 a^4 d\right ) x^4}{4 b^6}-\frac {4 a^3 d^2 x^5}{5 b^5}+\frac {a^2 d^2 x^6}{2 b^4}-\frac {2 a d^2 x^7}{7 b^3}+\frac {d^2 x^8}{8 b^2}+\frac {a \left (b^4 c+a^4 d\right )^2}{b^{10} (a+b x)}+\frac {\left (b^4 c+a^4 d\right ) \left (b^4 c+9 a^4 d\right ) \log (a+b x)}{b^{10}} \] Output:
-8*a^3*d*(a^4*d+b^4*c)*x/b^9+1/2*a^2*d*(7*a^4*d+6*b^4*c)*x^2/b^8-2/3*a*d*( 3*a^4*d+2*b^4*c)*x^3/b^7+1/4*d*(5*a^4*d+2*b^4*c)*x^4/b^6-4/5*a^3*d^2*x^5/b ^5+1/2*a^2*d^2*x^6/b^4-2/7*a*d^2*x^7/b^3+1/8*d^2*x^8/b^2+a*(a^4*d+b^4*c)^2 /b^10/(b*x+a)+(a^4*d+b^4*c)*(9*a^4*d+b^4*c)*ln(b*x+a)/b^10
Time = 0.06 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.01 \[ \int \frac {x \left (c+d x^4\right )^2}{(a+b x)^2} \, dx=-\frac {8 a^3 d \left (b^4 c+a^4 d\right ) x}{b^9}+\frac {a^2 d \left (6 b^4 c+7 a^4 d\right ) x^2}{2 b^8}-\frac {2 a d \left (2 b^4 c+3 a^4 d\right ) x^3}{3 b^7}+\frac {d \left (2 b^4 c+5 a^4 d\right ) x^4}{4 b^6}-\frac {4 a^3 d^2 x^5}{5 b^5}+\frac {a^2 d^2 x^6}{2 b^4}-\frac {2 a d^2 x^7}{7 b^3}+\frac {d^2 x^8}{8 b^2}+\frac {a \left (b^4 c+a^4 d\right )^2}{b^{10} (a+b x)}+\frac {\left (b^8 c^2+10 a^4 b^4 c d+9 a^8 d^2\right ) \log (a+b x)}{b^{10}} \] Input:
Integrate[(x*(c + d*x^4)^2)/(a + b*x)^2,x]
Output:
(-8*a^3*d*(b^4*c + a^4*d)*x)/b^9 + (a^2*d*(6*b^4*c + 7*a^4*d)*x^2)/(2*b^8) - (2*a*d*(2*b^4*c + 3*a^4*d)*x^3)/(3*b^7) + (d*(2*b^4*c + 5*a^4*d)*x^4)/( 4*b^6) - (4*a^3*d^2*x^5)/(5*b^5) + (a^2*d^2*x^6)/(2*b^4) - (2*a*d^2*x^7)/( 7*b^3) + (d^2*x^8)/(8*b^2) + (a*(b^4*c + a^4*d)^2)/(b^10*(a + b*x)) + ((b^ 8*c^2 + 10*a^4*b^4*c*d + 9*a^8*d^2)*Log[a + b*x])/b^10
Time = 0.51 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2123, 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.
| \(\displaystyle \int \frac {x \left (c+d x^4\right )^2}{(a+b x)^2} \, dx\) |
| \(\Big \downarrow \) 2123 |
| \(\displaystyle \int \left (\frac {\left (a^4 d+b^4 c\right ) \left (9 a^4 d+b^4 c\right )}{b^9 (a+b x)}-\frac {a \left (a^4 d+b^4 c\right )^2}{b^9 (a+b x)^2}-\frac {2 a d x^2 \left (3 a^4 d+2 b^4 c\right )}{b^7}+\frac {d x^3 \left (5 a^4 d+2 b^4 c\right )}{b^6}-\frac {4 a^3 d^2 x^4}{b^5}+\frac {3 a^2 d^2 x^5}{b^4}-\frac {8 a^3 d \left (a^4 d+b^4 c\right )}{b^9}+\frac {a^2 d x \left (7 a^4 d+6 b^4 c\right )}{b^8}-\frac {2 a d^2 x^6}{b^3}+\frac {d^2 x^7}{b^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a \left (a^4 d+b^4 c\right )^2}{b^{10} (a+b x)}+\frac {\left (a^4 d+b^4 c\right ) \left (9 a^4 d+b^4 c\right ) \log (a+b x)}{b^{10}}-\frac {2 a d x^3 \left (3 a^4 d+2 b^4 c\right )}{3 b^7}+\frac {d x^4 \left (5 a^4 d+2 b^4 c\right )}{4 b^6}-\frac {4 a^3 d^2 x^5}{5 b^5}+\frac {a^2 d^2 x^6}{2 b^4}-\frac {8 a^3 d x \left (a^4 d+b^4 c\right )}{b^9}+\frac {a^2 d x^2 \left (7 a^4 d+6 b^4 c\right )}{2 b^8}-\frac {2 a d^2 x^7}{7 b^3}+\frac {d^2 x^8}{8 b^2}\) |
Input:
Int[(x*(c + d*x^4)^2)/(a + b*x)^2,x]
Output:
(-8*a^3*d*(b^4*c + a^4*d)*x)/b^9 + (a^2*d*(6*b^4*c + 7*a^4*d)*x^2)/(2*b^8) - (2*a*d*(2*b^4*c + 3*a^4*d)*x^3)/(3*b^7) + (d*(2*b^4*c + 5*a^4*d)*x^4)/( 4*b^6) - (4*a^3*d^2*x^5)/(5*b^5) + (a^2*d^2*x^6)/(2*b^4) - (2*a*d^2*x^7)/( 7*b^3) + (d^2*x^8)/(8*b^2) + (a*(b^4*c + a^4*d)^2)/(b^10*(a + b*x)) + ((b^ 4*c + a^4*d)*(b^4*c + 9*a^4*d)*Log[a + b*x])/b^10
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c , d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2])
Time = 0.38 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.96
| method | result | size |
| default | \(-\frac {d \left (-\frac {1}{8} d \,x^{8} b^{7}+\frac {2}{7} a d \,x^{7} b^{6}-\frac {1}{2} a^{2} d \,x^{6} b^{5}+\frac {4}{5} a^{3} d \,x^{5} b^{4}-\frac {5}{4} a^{4} b^{3} d \,x^{4}-\frac {1}{2} b^{7} c \,x^{4}+2 a^{5} b^{2} d \,x^{3}+\frac {4}{3} a \,b^{6} c \,x^{3}-\frac {7}{2} a^{6} b d \,x^{2}-3 a^{2} b^{5} c \,x^{2}+8 a^{7} d x +8 a^{3} b^{4} c x \right )}{b^{9}}+\frac {a \left (a^{8} d^{2}+2 a^{4} b^{4} c d +b^{8} c^{2}\right )}{b^{10} \left (b x +a \right )}+\frac {\left (9 a^{8} d^{2}+10 a^{4} b^{4} c d +b^{8} c^{2}\right ) \ln \left (b x +a \right )}{b^{10}}\) | \(207\) |
| norman | \(\frac {\frac {d^{2} x^{9}}{8 b}-\frac {9 a \,d^{2} x^{8}}{56 b^{2}}+\frac {3 a^{2} d^{2} x^{7}}{14 b^{3}}-\frac {3 a^{3} d^{2} x^{6}}{10 b^{4}}+\frac {d \left (9 a^{4} d +10 b^{4} c \right ) x^{5}}{20 b^{5}}-\frac {\left (9 a^{9} d^{2}+10 a^{5} b^{4} c d +a \,b^{8} c^{2}\right ) x}{a \,b^{9}}-\frac {a d \left (9 a^{4} d +10 b^{4} c \right ) x^{4}}{12 b^{6}}+\frac {a^{2} d \left (9 a^{4} d +10 b^{4} c \right ) x^{3}}{6 b^{7}}-\frac {a^{3} d \left (9 a^{4} d +10 b^{4} c \right ) x^{2}}{2 b^{8}}}{b x +a}+\frac {\left (9 a^{8} d^{2}+10 a^{4} b^{4} c d +b^{8} c^{2}\right ) \ln \left (b x +a \right )}{b^{10}}\) | \(229\) |
| risch | \(\frac {d^{2} x^{8}}{8 b^{2}}-\frac {2 a \,d^{2} x^{7}}{7 b^{3}}+\frac {a^{2} d^{2} x^{6}}{2 b^{4}}-\frac {4 a^{3} d^{2} x^{5}}{5 b^{5}}+\frac {5 d^{2} a^{4} x^{4}}{4 b^{6}}+\frac {d c \,x^{4}}{2 b^{2}}-\frac {2 d^{2} a^{5} x^{3}}{b^{7}}-\frac {4 d a c \,x^{3}}{3 b^{3}}+\frac {7 d^{2} a^{6} x^{2}}{2 b^{8}}+\frac {3 d \,a^{2} c \,x^{2}}{b^{4}}-\frac {8 d^{2} a^{7} x}{b^{9}}-\frac {8 d \,a^{3} c x}{b^{5}}+\frac {a^{9} d^{2}}{b^{10} \left (b x +a \right )}+\frac {2 a^{5} c d}{b^{6} \left (b x +a \right )}+\frac {a \,c^{2}}{b^{2} \left (b x +a \right )}+\frac {9 \ln \left (b x +a \right ) a^{8} d^{2}}{b^{10}}+\frac {10 \ln \left (b x +a \right ) a^{4} c d}{b^{6}}+\frac {\ln \left (b x +a \right ) c^{2}}{b^{2}}\) | \(247\) |
| parallelrisch | \(\frac {105 x^{9} d^{2} b^{9}-135 a \,d^{2} x^{8} b^{8}+180 a^{2} d^{2} x^{7} b^{7}-252 a^{3} d^{2} x^{6} b^{6}+378 x^{5} a^{4} b^{5} d^{2}+420 x^{5} b^{9} c d -630 x^{4} a^{5} b^{4} d^{2}-700 x^{4} a \,b^{8} c d +1260 x^{3} a^{6} b^{3} d^{2}+1400 x^{3} a^{2} b^{7} c d +7560 \ln \left (b x +a \right ) x \,a^{8} b \,d^{2}+8400 \ln \left (b x +a \right ) x \,a^{4} b^{5} c d +840 \ln \left (b x +a \right ) x \,b^{9} c^{2}-3780 x^{2} a^{7} b^{2} d^{2}-4200 x^{2} a^{3} b^{6} c d +7560 \ln \left (b x +a \right ) a^{9} d^{2}+8400 \ln \left (b x +a \right ) a^{5} b^{4} c d +840 \ln \left (b x +a \right ) a \,b^{8} c^{2}+7560 a^{9} d^{2}+8400 a^{5} b^{4} c d +840 a \,b^{8} c^{2}}{840 b^{10} \left (b x +a \right )}\) | \(288\) |
Input:
int(x*(d*x^4+c)^2/(b*x+a)^2,x,method=_RETURNVERBOSE)
Output:
-d/b^9*(-1/8*d*x^8*b^7+2/7*a*d*x^7*b^6-1/2*a^2*d*x^6*b^5+4/5*a^3*d*x^5*b^4 -5/4*a^4*b^3*d*x^4-1/2*b^7*c*x^4+2*a^5*b^2*d*x^3+4/3*a*b^6*c*x^3-7/2*a^6*b *d*x^2-3*a^2*b^5*c*x^2+8*a^7*d*x+8*a^3*b^4*c*x)+a*(a^8*d^2+2*a^4*b^4*c*d+b ^8*c^2)/b^10/(b*x+a)+1/b^10*(9*a^8*d^2+10*a^4*b^4*c*d+b^8*c^2)*ln(b*x+a)
Time = 0.07 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.31 \[ \int \frac {x \left (c+d x^4\right )^2}{(a+b x)^2} \, dx=\frac {105 \, b^{9} d^{2} x^{9} - 135 \, a b^{8} d^{2} x^{8} + 180 \, a^{2} b^{7} d^{2} x^{7} - 252 \, a^{3} b^{6} d^{2} x^{6} + 840 \, a b^{8} c^{2} + 1680 \, a^{5} b^{4} c d + 840 \, a^{9} d^{2} + 42 \, {\left (10 \, b^{9} c d + 9 \, a^{4} b^{5} d^{2}\right )} x^{5} - 70 \, {\left (10 \, a b^{8} c d + 9 \, a^{5} b^{4} d^{2}\right )} x^{4} + 140 \, {\left (10 \, a^{2} b^{7} c d + 9 \, a^{6} b^{3} d^{2}\right )} x^{3} - 420 \, {\left (10 \, a^{3} b^{6} c d + 9 \, a^{7} b^{2} d^{2}\right )} x^{2} - 6720 \, {\left (a^{4} b^{5} c d + a^{8} b d^{2}\right )} x + 840 \, {\left (a b^{8} c^{2} + 10 \, a^{5} b^{4} c d + 9 \, a^{9} d^{2} + {\left (b^{9} c^{2} + 10 \, a^{4} b^{5} c d + 9 \, a^{8} b d^{2}\right )} x\right )} \log \left (b x + a\right )}{840 \, {\left (b^{11} x + a b^{10}\right )}} \] Input:
integrate(x*(d*x^4+c)^2/(b*x+a)^2,x, algorithm="fricas")
Output:
1/840*(105*b^9*d^2*x^9 - 135*a*b^8*d^2*x^8 + 180*a^2*b^7*d^2*x^7 - 252*a^3 *b^6*d^2*x^6 + 840*a*b^8*c^2 + 1680*a^5*b^4*c*d + 840*a^9*d^2 + 42*(10*b^9 *c*d + 9*a^4*b^5*d^2)*x^5 - 70*(10*a*b^8*c*d + 9*a^5*b^4*d^2)*x^4 + 140*(1 0*a^2*b^7*c*d + 9*a^6*b^3*d^2)*x^3 - 420*(10*a^3*b^6*c*d + 9*a^7*b^2*d^2)* x^2 - 6720*(a^4*b^5*c*d + a^8*b*d^2)*x + 840*(a*b^8*c^2 + 10*a^5*b^4*c*d + 9*a^9*d^2 + (b^9*c^2 + 10*a^4*b^5*c*d + 9*a^8*b*d^2)*x)*log(b*x + a))/(b^ 11*x + a*b^10)
Time = 0.35 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.11 \[ \int \frac {x \left (c+d x^4\right )^2}{(a+b x)^2} \, dx=- \frac {4 a^{3} d^{2} x^{5}}{5 b^{5}} + \frac {a^{2} d^{2} x^{6}}{2 b^{4}} - \frac {2 a d^{2} x^{7}}{7 b^{3}} + x^{4} \cdot \left (\frac {5 a^{4} d^{2}}{4 b^{6}} + \frac {c d}{2 b^{2}}\right ) + x^{3} \left (- \frac {2 a^{5} d^{2}}{b^{7}} - \frac {4 a c d}{3 b^{3}}\right ) + x^{2} \cdot \left (\frac {7 a^{6} d^{2}}{2 b^{8}} + \frac {3 a^{2} c d}{b^{4}}\right ) + x \left (- \frac {8 a^{7} d^{2}}{b^{9}} - \frac {8 a^{3} c d}{b^{5}}\right ) + \frac {a^{9} d^{2} + 2 a^{5} b^{4} c d + a b^{8} c^{2}}{a b^{10} + b^{11} x} + \frac {d^{2} x^{8}}{8 b^{2}} + \frac {\left (a^{4} d + b^{4} c\right ) \left (9 a^{4} d + b^{4} c\right ) \log {\left (a + b x \right )}}{b^{10}} \] Input:
integrate(x*(d*x**4+c)**2/(b*x+a)**2,x)
Output:
-4*a**3*d**2*x**5/(5*b**5) + a**2*d**2*x**6/(2*b**4) - 2*a*d**2*x**7/(7*b* *3) + x**4*(5*a**4*d**2/(4*b**6) + c*d/(2*b**2)) + x**3*(-2*a**5*d**2/b**7 - 4*a*c*d/(3*b**3)) + x**2*(7*a**6*d**2/(2*b**8) + 3*a**2*c*d/b**4) + x*( -8*a**7*d**2/b**9 - 8*a**3*c*d/b**5) + (a**9*d**2 + 2*a**5*b**4*c*d + a*b* *8*c**2)/(a*b**10 + b**11*x) + d**2*x**8/(8*b**2) + (a**4*d + b**4*c)*(9*a **4*d + b**4*c)*log(a + b*x)/b**10
Time = 0.03 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.06 \[ \int \frac {x \left (c+d x^4\right )^2}{(a+b x)^2} \, dx=\frac {a b^{8} c^{2} + 2 \, a^{5} b^{4} c d + a^{9} d^{2}}{b^{11} x + a b^{10}} + \frac {105 \, b^{7} d^{2} x^{8} - 240 \, a b^{6} d^{2} x^{7} + 420 \, a^{2} b^{5} d^{2} x^{6} - 672 \, a^{3} b^{4} d^{2} x^{5} + 210 \, {\left (2 \, b^{7} c d + 5 \, a^{4} b^{3} d^{2}\right )} x^{4} - 560 \, {\left (2 \, a b^{6} c d + 3 \, a^{5} b^{2} d^{2}\right )} x^{3} + 420 \, {\left (6 \, a^{2} b^{5} c d + 7 \, a^{6} b d^{2}\right )} x^{2} - 6720 \, {\left (a^{3} b^{4} c d + a^{7} d^{2}\right )} x}{840 \, b^{9}} + \frac {{\left (b^{8} c^{2} + 10 \, a^{4} b^{4} c d + 9 \, a^{8} d^{2}\right )} \log \left (b x + a\right )}{b^{10}} \] Input:
integrate(x*(d*x^4+c)^2/(b*x+a)^2,x, algorithm="maxima")
Output:
(a*b^8*c^2 + 2*a^5*b^4*c*d + a^9*d^2)/(b^11*x + a*b^10) + 1/840*(105*b^7*d ^2*x^8 - 240*a*b^6*d^2*x^7 + 420*a^2*b^5*d^2*x^6 - 672*a^3*b^4*d^2*x^5 + 2 10*(2*b^7*c*d + 5*a^4*b^3*d^2)*x^4 - 560*(2*a*b^6*c*d + 3*a^5*b^2*d^2)*x^3 + 420*(6*a^2*b^5*c*d + 7*a^6*b*d^2)*x^2 - 6720*(a^3*b^4*c*d + a^7*d^2)*x) /b^9 + (b^8*c^2 + 10*a^4*b^4*c*d + 9*a^8*d^2)*log(b*x + a)/b^10
Time = 0.11 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.40 \[ \int \frac {x \left (c+d x^4\right )^2}{(a+b x)^2} \, dx=-\frac {\frac {{\left (\frac {1080 \, a d^{2}}{b x + a} - \frac {5040 \, a^{2} d^{2}}{{\left (b x + a\right )}^{2}} + \frac {14112 \, a^{3} d^{2}}{{\left (b x + a\right )}^{3}} - 105 \, d^{2} - \frac {420 \, {\left (b^{8} c d + 63 \, a^{4} b^{4} d^{2}\right )}}{{\left (b x + a\right )}^{4} b^{4}} + \frac {560 \, {\left (5 \, a b^{9} c d + 63 \, a^{5} b^{5} d^{2}\right )}}{{\left (b x + a\right )}^{5} b^{5}} - \frac {1680 \, {\left (5 \, a^{2} b^{10} c d + 21 \, a^{6} b^{6} d^{2}\right )}}{{\left (b x + a\right )}^{6} b^{6}} + \frac {3360 \, {\left (5 \, a^{3} b^{11} c d + 9 \, a^{7} b^{7} d^{2}\right )}}{{\left (b x + a\right )}^{7} b^{7}}\right )} {\left (b x + a\right )}^{8}}{b^{9}} + \frac {840 \, {\left (b^{8} c^{2} + 10 \, a^{4} b^{4} c d + 9 \, a^{8} d^{2}\right )} \log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b^{9}} - \frac {840 \, {\left (\frac {a b^{16} c^{2}}{b x + a} + \frac {2 \, a^{5} b^{12} c d}{b x + a} + \frac {a^{9} b^{8} d^{2}}{b x + a}\right )}}{b^{17}}}{840 \, b} \] Input:
integrate(x*(d*x^4+c)^2/(b*x+a)^2,x, algorithm="giac")
Output:
-1/840*((1080*a*d^2/(b*x + a) - 5040*a^2*d^2/(b*x + a)^2 + 14112*a^3*d^2/( b*x + a)^3 - 105*d^2 - 420*(b^8*c*d + 63*a^4*b^4*d^2)/((b*x + a)^4*b^4) + 560*(5*a*b^9*c*d + 63*a^5*b^5*d^2)/((b*x + a)^5*b^5) - 1680*(5*a^2*b^10*c* d + 21*a^6*b^6*d^2)/((b*x + a)^6*b^6) + 3360*(5*a^3*b^11*c*d + 9*a^7*b^7*d ^2)/((b*x + a)^7*b^7))*(b*x + a)^8/b^9 + 840*(b^8*c^2 + 10*a^4*b^4*c*d + 9 *a^8*d^2)*log(abs(b*x + a)/((b*x + a)^2*abs(b)))/b^9 - 840*(a*b^16*c^2/(b* x + a) + 2*a^5*b^12*c*d/(b*x + a) + a^9*b^8*d^2/(b*x + a))/b^17)/b
Time = 0.06 (sec) , antiderivative size = 395, normalized size of antiderivative = 1.84 \[ \int \frac {x \left (c+d x^4\right )^2}{(a+b x)^2} \, dx=x^4\,\left (\frac {5\,a^4\,d^2}{4\,b^6}+\frac {c\,d}{2\,b^2}\right )+x\,\left (\frac {a^2\,\left (\frac {2\,a\,\left (\frac {5\,a^4\,d^2}{b^6}+\frac {2\,c\,d}{b^2}\right )}{b}-\frac {4\,a^5\,d^2}{b^7}\right )}{b^2}+\frac {2\,a\,\left (\frac {a^2\,\left (\frac {5\,a^4\,d^2}{b^6}+\frac {2\,c\,d}{b^2}\right )}{b^2}-\frac {2\,a\,\left (\frac {2\,a\,\left (\frac {5\,a^4\,d^2}{b^6}+\frac {2\,c\,d}{b^2}\right )}{b}-\frac {4\,a^5\,d^2}{b^7}\right )}{b}\right )}{b}\right )-x^2\,\left (\frac {a^2\,\left (\frac {5\,a^4\,d^2}{b^6}+\frac {2\,c\,d}{b^2}\right )}{2\,b^2}-\frac {a\,\left (\frac {2\,a\,\left (\frac {5\,a^4\,d^2}{b^6}+\frac {2\,c\,d}{b^2}\right )}{b}-\frac {4\,a^5\,d^2}{b^7}\right )}{b}\right )-x^3\,\left (\frac {2\,a\,\left (\frac {5\,a^4\,d^2}{b^6}+\frac {2\,c\,d}{b^2}\right )}{3\,b}-\frac {4\,a^5\,d^2}{3\,b^7}\right )+\frac {a^9\,d^2+2\,a^5\,b^4\,c\,d+a\,b^8\,c^2}{b\,\left (x\,b^{10}+a\,b^9\right )}+\frac {\ln \left (a+b\,x\right )\,\left (9\,a^8\,d^2+10\,a^4\,b^4\,c\,d+b^8\,c^2\right )}{b^{10}}+\frac {d^2\,x^8}{8\,b^2}-\frac {2\,a\,d^2\,x^7}{7\,b^3}+\frac {a^2\,d^2\,x^6}{2\,b^4}-\frac {4\,a^3\,d^2\,x^5}{5\,b^5} \] Input:
int((x*(c + d*x^4)^2)/(a + b*x)^2,x)
Output:
x^4*((5*a^4*d^2)/(4*b^6) + (c*d)/(2*b^2)) + x*((a^2*((2*a*((5*a^4*d^2)/b^6 + (2*c*d)/b^2))/b - (4*a^5*d^2)/b^7))/b^2 + (2*a*((a^2*((5*a^4*d^2)/b^6 + (2*c*d)/b^2))/b^2 - (2*a*((2*a*((5*a^4*d^2)/b^6 + (2*c*d)/b^2))/b - (4*a^ 5*d^2)/b^7))/b))/b) - x^2*((a^2*((5*a^4*d^2)/b^6 + (2*c*d)/b^2))/(2*b^2) - (a*((2*a*((5*a^4*d^2)/b^6 + (2*c*d)/b^2))/b - (4*a^5*d^2)/b^7))/b) - x^3* ((2*a*((5*a^4*d^2)/b^6 + (2*c*d)/b^2))/(3*b) - (4*a^5*d^2)/(3*b^7)) + (a^9 *d^2 + a*b^8*c^2 + 2*a^5*b^4*c*d)/(b*(a*b^9 + b^10*x)) + (log(a + b*x)*(9* a^8*d^2 + b^8*c^2 + 10*a^4*b^4*c*d))/b^10 + (d^2*x^8)/(8*b^2) - (2*a*d^2*x ^7)/(7*b^3) + (a^2*d^2*x^6)/(2*b^4) - (4*a^3*d^2*x^5)/(5*b^5)
Time = 0.14 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.35 \[ \int \frac {x \left (c+d x^4\right )^2}{(a+b x)^2} \, dx=\frac {7560 \,\mathrm {log}\left (b x +a \right ) a^{9} d^{2}+7560 \,\mathrm {log}\left (b x +a \right ) a^{8} b \,d^{2} x +8400 \,\mathrm {log}\left (b x +a \right ) a^{5} b^{4} c d +8400 \,\mathrm {log}\left (b x +a \right ) a^{4} b^{5} c d x +840 \,\mathrm {log}\left (b x +a \right ) a \,b^{8} c^{2}+840 \,\mathrm {log}\left (b x +a \right ) b^{9} c^{2} x -7560 a^{8} b \,d^{2} x -3780 a^{7} b^{2} d^{2} x^{2}+1260 a^{6} b^{3} d^{2} x^{3}-630 a^{5} b^{4} d^{2} x^{4}-8400 a^{4} b^{5} c d x +378 a^{4} b^{5} d^{2} x^{5}-4200 a^{3} b^{6} c d \,x^{2}-252 a^{3} b^{6} d^{2} x^{6}+1400 a^{2} b^{7} c d \,x^{3}+180 a^{2} b^{7} d^{2} x^{7}-700 a \,b^{8} c d \,x^{4}-135 a \,b^{8} d^{2} x^{8}-840 b^{9} c^{2} x +420 b^{9} c d \,x^{5}+105 b^{9} d^{2} x^{9}}{840 b^{10} \left (b x +a \right )} \] Input:
int(x*(d*x^4+c)^2/(b*x+a)^2,x)
Output:
(7560*log(a + b*x)*a**9*d**2 + 7560*log(a + b*x)*a**8*b*d**2*x + 8400*log( a + b*x)*a**5*b**4*c*d + 8400*log(a + b*x)*a**4*b**5*c*d*x + 840*log(a + b *x)*a*b**8*c**2 + 840*log(a + b*x)*b**9*c**2*x - 7560*a**8*b*d**2*x - 3780 *a**7*b**2*d**2*x**2 + 1260*a**6*b**3*d**2*x**3 - 630*a**5*b**4*d**2*x**4 - 8400*a**4*b**5*c*d*x + 378*a**4*b**5*d**2*x**5 - 4200*a**3*b**6*c*d*x**2 - 252*a**3*b**6*d**2*x**6 + 1400*a**2*b**7*c*d*x**3 + 180*a**2*b**7*d**2* x**7 - 700*a*b**8*c*d*x**4 - 135*a*b**8*d**2*x**8 - 840*b**9*c**2*x + 420* b**9*c*d*x**5 + 105*b**9*d**2*x**9)/(840*b**10*(a + b*x))