Integrand size = 17, antiderivative size = 184 \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^7} \, dx=-\frac {\left (c d^2+a e^2\right )^3}{6 e^7 (d+e x)^6}+\frac {6 c d \left (c d^2+a e^2\right )^2}{5 e^7 (d+e x)^5}-\frac {3 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right )}{4 e^7 (d+e x)^4}+\frac {4 c^2 d \left (5 c d^2+3 a e^2\right )}{3 e^7 (d+e x)^3}-\frac {3 c^2 \left (5 c d^2+a e^2\right )}{2 e^7 (d+e x)^2}+\frac {6 c^3 d}{e^7 (d+e x)}+\frac {c^3 \log (d+e x)}{e^7} \]
-1/6*(a*e^2+c*d^2)^3/e^7/(e*x+d)^6+6/5*c*d*(a*e^2+c*d^2)^2/e^7/(e*x+d)^5-3 /4*c*(a*e^2+c*d^2)*(a*e^2+5*c*d^2)/e^7/(e*x+d)^4+4/3*c^2*d*(3*a*e^2+5*c*d^ 2)/e^7/(e*x+d)^3-3/2*c^2*(a*e^2+5*c*d^2)/e^7/(e*x+d)^2+6*c^3*d/e^7/(e*x+d) +c^3*ln(e*x+d)/e^7
Time = 0.06 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.93 \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^7} \, dx=\frac {-10 a^3 e^6-3 a^2 c e^4 \left (d^2+6 d e x+15 e^2 x^2\right )-6 a c^2 e^2 \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )+c^3 d \left (147 d^5+822 d^4 e x+1875 d^3 e^2 x^2+2200 d^2 e^3 x^3+1350 d e^4 x^4+360 e^5 x^5\right )+60 c^3 (d+e x)^6 \log (d+e x)}{60 e^7 (d+e x)^6} \]
(-10*a^3*e^6 - 3*a^2*c*e^4*(d^2 + 6*d*e*x + 15*e^2*x^2) - 6*a*c^2*e^2*(d^4 + 6*d^3*e*x + 15*d^2*e^2*x^2 + 20*d*e^3*x^3 + 15*e^4*x^4) + c^3*d*(147*d^ 5 + 822*d^4*e*x + 1875*d^3*e^2*x^2 + 2200*d^2*e^3*x^3 + 1350*d*e^4*x^4 + 3 60*e^5*x^5) + 60*c^3*(d + e*x)^6*Log[d + e*x])/(60*e^7*(d + e*x)^6)
Time = 0.36 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {476, 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.
| \(\displaystyle \int \frac {\left (a+c x^2\right )^3}{(d+e x)^7} \, dx\) |
| \(\Big \downarrow \) 476 |
| \(\displaystyle \int \left (\frac {3 c^2 \left (a e^2+5 c d^2\right )}{e^6 (d+e x)^3}-\frac {4 c^2 d \left (3 a e^2+5 c d^2\right )}{e^6 (d+e x)^4}+\frac {3 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{e^6 (d+e x)^5}-\frac {6 c d \left (a e^2+c d^2\right )^2}{e^6 (d+e x)^6}+\frac {\left (a e^2+c d^2\right )^3}{e^6 (d+e x)^7}+\frac {c^3}{e^6 (d+e x)}-\frac {6 c^3 d}{e^6 (d+e x)^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 c^2 \left (a e^2+5 c d^2\right )}{2 e^7 (d+e x)^2}+\frac {4 c^2 d \left (3 a e^2+5 c d^2\right )}{3 e^7 (d+e x)^3}-\frac {3 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{4 e^7 (d+e x)^4}+\frac {6 c d \left (a e^2+c d^2\right )^2}{5 e^7 (d+e x)^5}-\frac {\left (a e^2+c d^2\right )^3}{6 e^7 (d+e x)^6}+\frac {6 c^3 d}{e^7 (d+e x)}+\frac {c^3 \log (d+e x)}{e^7}\) |
-1/6*(c*d^2 + a*e^2)^3/(e^7*(d + e*x)^6) + (6*c*d*(c*d^2 + a*e^2)^2)/(5*e^ 7*(d + e*x)^5) - (3*c*(c*d^2 + a*e^2)*(5*c*d^2 + a*e^2))/(4*e^7*(d + e*x)^ 4) + (4*c^2*d*(5*c*d^2 + 3*a*e^2))/(3*e^7*(d + e*x)^3) - (3*c^2*(5*c*d^2 + a*e^2))/(2*e^7*(d + e*x)^2) + (6*c^3*d)/(e^7*(d + e*x)) + (c^3*Log[d + e* x])/e^7
3.5.84.3.1 Defintions of rubi rules used
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ ExpandIntegrand[(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[p, 0]
Time = 2.15 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.09
| method | result | size |
| risch | \(\frac {\frac {6 c^{3} d \,x^{5}}{e^{2}}-\frac {3 c^{2} \left (e^{2} a -15 c \,d^{2}\right ) x^{4}}{2 e^{3}}-\frac {2 c^{2} d \left (3 e^{2} a -55 c \,d^{2}\right ) x^{3}}{3 e^{4}}-\frac {c \left (3 a^{2} e^{4}+6 a c \,d^{2} e^{2}-125 c^{2} d^{4}\right ) x^{2}}{4 e^{5}}-\frac {c d \left (3 a^{2} e^{4}+6 a c \,d^{2} e^{2}-137 c^{2} d^{4}\right ) x}{10 e^{6}}-\frac {10 e^{6} a^{3}+3 d^{2} e^{4} a^{2} c +6 d^{4} e^{2} c^{2} a -147 c^{3} d^{6}}{60 e^{7}}}{\left (e x +d \right )^{6}}+\frac {c^{3} \ln \left (e x +d \right )}{e^{7}}\) | \(201\) |
| norman | \(\frac {-\frac {10 e^{6} a^{3}+3 d^{2} e^{4} a^{2} c +6 d^{4} e^{2} c^{2} a -147 c^{3} d^{6}}{60 e^{7}}-\frac {3 \left (e^{2} c^{2} a -15 c^{3} d^{2}\right ) x^{4}}{2 e^{3}}-\frac {\left (3 e^{4} a^{2} c +6 d^{2} e^{2} c^{2} a -125 d^{4} c^{3}\right ) x^{2}}{4 e^{5}}+\frac {6 c^{3} d \,x^{5}}{e^{2}}-\frac {2 d \left (3 e^{2} c^{2} a -55 c^{3} d^{2}\right ) x^{3}}{3 e^{4}}-\frac {d \left (3 e^{4} a^{2} c +6 d^{2} e^{2} c^{2} a -137 d^{4} c^{3}\right ) x}{10 e^{6}}}{\left (e x +d \right )^{6}}+\frac {c^{3} \ln \left (e x +d \right )}{e^{7}}\) | \(209\) |
| default | \(-\frac {e^{6} a^{3}+3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a +c^{3} d^{6}}{6 e^{7} \left (e x +d \right )^{6}}+\frac {6 c^{3} d}{e^{7} \left (e x +d \right )}+\frac {4 c^{2} d \left (3 e^{2} a +5 c \,d^{2}\right )}{3 e^{7} \left (e x +d \right )^{3}}-\frac {3 c \left (a^{2} e^{4}+6 a c \,d^{2} e^{2}+5 c^{2} d^{4}\right )}{4 e^{7} \left (e x +d \right )^{4}}-\frac {3 c^{2} \left (e^{2} a +5 c \,d^{2}\right )}{2 e^{7} \left (e x +d \right )^{2}}+\frac {6 c d \left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{5 e^{7} \left (e x +d \right )^{5}}+\frac {c^{3} \ln \left (e x +d \right )}{e^{7}}\) | \(216\) |
| parallelrisch | \(\frac {-6 d^{4} e^{2} c^{2} a -3 d^{2} e^{4} a^{2} c +1350 x^{4} c^{3} d^{2} e^{4}+2200 x^{3} c^{3} d^{3} e^{3}+1875 x^{2} c^{3} d^{4} e^{2}+822 x \,c^{3} d^{5} e +147 c^{3} d^{6}+900 \ln \left (e x +d \right ) x^{4} c^{3} d^{2} e^{4}-18 x \,a^{2} c d \,e^{5}-36 x a \,c^{2} d^{3} e^{3}+360 \ln \left (e x +d \right ) x \,c^{3} d^{5} e -10 e^{6} a^{3}+900 \ln \left (e x +d \right ) x^{2} c^{3} d^{4} e^{2}-120 x^{3} a \,c^{2} d \,e^{5}-90 x^{2} a \,c^{2} d^{2} e^{4}+1200 \ln \left (e x +d \right ) x^{3} c^{3} d^{3} e^{3}+360 \ln \left (e x +d \right ) x^{5} c^{3} d \,e^{5}+360 x^{5} c^{3} d \,e^{5}-90 x^{4} a \,c^{2} e^{6}-45 x^{2} a^{2} c \,e^{6}+60 \ln \left (e x +d \right ) x^{6} c^{3} e^{6}+60 \ln \left (e x +d \right ) c^{3} d^{6}}{60 e^{7} \left (e x +d \right )^{6}}\) | \(319\) |
(6*c^3*d*x^5/e^2-3/2*c^2*(a*e^2-15*c*d^2)/e^3*x^4-2/3*c^2*d*(3*a*e^2-55*c* d^2)/e^4*x^3-1/4*c*(3*a^2*e^4+6*a*c*d^2*e^2-125*c^2*d^4)/e^5*x^2-1/10*c*d* (3*a^2*e^4+6*a*c*d^2*e^2-137*c^2*d^4)/e^6*x-1/60*(10*a^3*e^6+3*a^2*c*d^2*e ^4+6*a*c^2*d^4*e^2-147*c^3*d^6)/e^7)/(e*x+d)^6+c^3*ln(e*x+d)/e^7
Time = 0.27 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.84 \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^7} \, dx=\frac {360 \, c^{3} d e^{5} x^{5} + 147 \, c^{3} d^{6} - 6 \, a c^{2} d^{4} e^{2} - 3 \, a^{2} c d^{2} e^{4} - 10 \, a^{3} e^{6} + 90 \, {\left (15 \, c^{3} d^{2} e^{4} - a c^{2} e^{6}\right )} x^{4} + 40 \, {\left (55 \, c^{3} d^{3} e^{3} - 3 \, a c^{2} d e^{5}\right )} x^{3} + 15 \, {\left (125 \, c^{3} d^{4} e^{2} - 6 \, a c^{2} d^{2} e^{4} - 3 \, a^{2} c e^{6}\right )} x^{2} + 6 \, {\left (137 \, c^{3} d^{5} e - 6 \, a c^{2} d^{3} e^{3} - 3 \, a^{2} c d e^{5}\right )} x + 60 \, {\left (c^{3} e^{6} x^{6} + 6 \, c^{3} d e^{5} x^{5} + 15 \, c^{3} d^{2} e^{4} x^{4} + 20 \, c^{3} d^{3} e^{3} x^{3} + 15 \, c^{3} d^{4} e^{2} x^{2} + 6 \, c^{3} d^{5} e x + c^{3} d^{6}\right )} \log \left (e x + d\right )}{60 \, {\left (e^{13} x^{6} + 6 \, d e^{12} x^{5} + 15 \, d^{2} e^{11} x^{4} + 20 \, d^{3} e^{10} x^{3} + 15 \, d^{4} e^{9} x^{2} + 6 \, d^{5} e^{8} x + d^{6} e^{7}\right )}} \]
1/60*(360*c^3*d*e^5*x^5 + 147*c^3*d^6 - 6*a*c^2*d^4*e^2 - 3*a^2*c*d^2*e^4 - 10*a^3*e^6 + 90*(15*c^3*d^2*e^4 - a*c^2*e^6)*x^4 + 40*(55*c^3*d^3*e^3 - 3*a*c^2*d*e^5)*x^3 + 15*(125*c^3*d^4*e^2 - 6*a*c^2*d^2*e^4 - 3*a^2*c*e^6)* x^2 + 6*(137*c^3*d^5*e - 6*a*c^2*d^3*e^3 - 3*a^2*c*d*e^5)*x + 60*(c^3*e^6* x^6 + 6*c^3*d*e^5*x^5 + 15*c^3*d^2*e^4*x^4 + 20*c^3*d^3*e^3*x^3 + 15*c^3*d ^4*e^2*x^2 + 6*c^3*d^5*e*x + c^3*d^6)*log(e*x + d))/(e^13*x^6 + 6*d*e^12*x ^5 + 15*d^2*e^11*x^4 + 20*d^3*e^10*x^3 + 15*d^4*e^9*x^2 + 6*d^5*e^8*x + d^ 6*e^7)
Time = 24.87 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.48 \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^7} \, dx=\frac {c^{3} \log {\left (d + e x \right )}}{e^{7}} + \frac {- 10 a^{3} e^{6} - 3 a^{2} c d^{2} e^{4} - 6 a c^{2} d^{4} e^{2} + 147 c^{3} d^{6} + 360 c^{3} d e^{5} x^{5} + x^{4} \left (- 90 a c^{2} e^{6} + 1350 c^{3} d^{2} e^{4}\right ) + x^{3} \left (- 120 a c^{2} d e^{5} + 2200 c^{3} d^{3} e^{3}\right ) + x^{2} \left (- 45 a^{2} c e^{6} - 90 a c^{2} d^{2} e^{4} + 1875 c^{3} d^{4} e^{2}\right ) + x \left (- 18 a^{2} c d e^{5} - 36 a c^{2} d^{3} e^{3} + 822 c^{3} d^{5} e\right )}{60 d^{6} e^{7} + 360 d^{5} e^{8} x + 900 d^{4} e^{9} x^{2} + 1200 d^{3} e^{10} x^{3} + 900 d^{2} e^{11} x^{4} + 360 d e^{12} x^{5} + 60 e^{13} x^{6}} \]
c**3*log(d + e*x)/e**7 + (-10*a**3*e**6 - 3*a**2*c*d**2*e**4 - 6*a*c**2*d* *4*e**2 + 147*c**3*d**6 + 360*c**3*d*e**5*x**5 + x**4*(-90*a*c**2*e**6 + 1 350*c**3*d**2*e**4) + x**3*(-120*a*c**2*d*e**5 + 2200*c**3*d**3*e**3) + x* *2*(-45*a**2*c*e**6 - 90*a*c**2*d**2*e**4 + 1875*c**3*d**4*e**2) + x*(-18* a**2*c*d*e**5 - 36*a*c**2*d**3*e**3 + 822*c**3*d**5*e))/(60*d**6*e**7 + 36 0*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11 *x**4 + 360*d*e**12*x**5 + 60*e**13*x**6)
Time = 0.20 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.43 \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^7} \, dx=\frac {360 \, c^{3} d e^{5} x^{5} + 147 \, c^{3} d^{6} - 6 \, a c^{2} d^{4} e^{2} - 3 \, a^{2} c d^{2} e^{4} - 10 \, a^{3} e^{6} + 90 \, {\left (15 \, c^{3} d^{2} e^{4} - a c^{2} e^{6}\right )} x^{4} + 40 \, {\left (55 \, c^{3} d^{3} e^{3} - 3 \, a c^{2} d e^{5}\right )} x^{3} + 15 \, {\left (125 \, c^{3} d^{4} e^{2} - 6 \, a c^{2} d^{2} e^{4} - 3 \, a^{2} c e^{6}\right )} x^{2} + 6 \, {\left (137 \, c^{3} d^{5} e - 6 \, a c^{2} d^{3} e^{3} - 3 \, a^{2} c d e^{5}\right )} x}{60 \, {\left (e^{13} x^{6} + 6 \, d e^{12} x^{5} + 15 \, d^{2} e^{11} x^{4} + 20 \, d^{3} e^{10} x^{3} + 15 \, d^{4} e^{9} x^{2} + 6 \, d^{5} e^{8} x + d^{6} e^{7}\right )}} + \frac {c^{3} \log \left (e x + d\right )}{e^{7}} \]
1/60*(360*c^3*d*e^5*x^5 + 147*c^3*d^6 - 6*a*c^2*d^4*e^2 - 3*a^2*c*d^2*e^4 - 10*a^3*e^6 + 90*(15*c^3*d^2*e^4 - a*c^2*e^6)*x^4 + 40*(55*c^3*d^3*e^3 - 3*a*c^2*d*e^5)*x^3 + 15*(125*c^3*d^4*e^2 - 6*a*c^2*d^2*e^4 - 3*a^2*c*e^6)* x^2 + 6*(137*c^3*d^5*e - 6*a*c^2*d^3*e^3 - 3*a^2*c*d*e^5)*x)/(e^13*x^6 + 6 *d*e^12*x^5 + 15*d^2*e^11*x^4 + 20*d^3*e^10*x^3 + 15*d^4*e^9*x^2 + 6*d^5*e ^8*x + d^6*e^7) + c^3*log(e*x + d)/e^7
Time = 0.26 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.13 \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^7} \, dx=\frac {c^{3} \log \left ({\left | e x + d \right |}\right )}{e^{7}} + \frac {360 \, c^{3} d e^{4} x^{5} + 90 \, {\left (15 \, c^{3} d^{2} e^{3} - a c^{2} e^{5}\right )} x^{4} + 40 \, {\left (55 \, c^{3} d^{3} e^{2} - 3 \, a c^{2} d e^{4}\right )} x^{3} + 15 \, {\left (125 \, c^{3} d^{4} e - 6 \, a c^{2} d^{2} e^{3} - 3 \, a^{2} c e^{5}\right )} x^{2} + 6 \, {\left (137 \, c^{3} d^{5} - 6 \, a c^{2} d^{3} e^{2} - 3 \, a^{2} c d e^{4}\right )} x + \frac {147 \, c^{3} d^{6} - 6 \, a c^{2} d^{4} e^{2} - 3 \, a^{2} c d^{2} e^{4} - 10 \, a^{3} e^{6}}{e}}{60 \, {\left (e x + d\right )}^{6} e^{6}} \]
c^3*log(abs(e*x + d))/e^7 + 1/60*(360*c^3*d*e^4*x^5 + 90*(15*c^3*d^2*e^3 - a*c^2*e^5)*x^4 + 40*(55*c^3*d^3*e^2 - 3*a*c^2*d*e^4)*x^3 + 15*(125*c^3*d^ 4*e - 6*a*c^2*d^2*e^3 - 3*a^2*c*e^5)*x^2 + 6*(137*c^3*d^5 - 6*a*c^2*d^3*e^ 2 - 3*a^2*c*d*e^4)*x + (147*c^3*d^6 - 6*a*c^2*d^4*e^2 - 3*a^2*c*d^2*e^4 - 10*a^3*e^6)/e)/((e*x + d)^6*e^6)
Time = 9.58 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.42 \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^7} \, dx=\frac {c^3\,\ln \left (d+e\,x\right )}{e^7}-\frac {\frac {10\,a^3\,e^6+3\,a^2\,c\,d^2\,e^4+6\,a\,c^2\,d^4\,e^2-147\,c^3\,d^6}{60\,e^7}+\frac {x^2\,\left (3\,a^2\,c\,e^4+6\,a\,c^2\,d^2\,e^2-125\,c^3\,d^4\right )}{4\,e^5}+\frac {x\,\left (3\,a^2\,c\,d\,e^4+6\,a\,c^2\,d^3\,e^2-137\,c^3\,d^5\right )}{10\,e^6}-\frac {2\,x^3\,\left (55\,c^3\,d^3-3\,a\,c^2\,d\,e^2\right )}{3\,e^4}-\frac {6\,c^3\,d\,x^5}{e^2}+\frac {3\,c^2\,x^4\,\left (a\,e^2-15\,c\,d^2\right )}{2\,e^3}}{d^6+6\,d^5\,e\,x+15\,d^4\,e^2\,x^2+20\,d^3\,e^3\,x^3+15\,d^2\,e^4\,x^4+6\,d\,e^5\,x^5+e^6\,x^6} \]
(c^3*log(d + e*x))/e^7 - ((10*a^3*e^6 - 147*c^3*d^6 + 6*a*c^2*d^4*e^2 + 3* a^2*c*d^2*e^4)/(60*e^7) + (x^2*(3*a^2*c*e^4 - 125*c^3*d^4 + 6*a*c^2*d^2*e^ 2))/(4*e^5) + (x*(6*a*c^2*d^3*e^2 - 137*c^3*d^5 + 3*a^2*c*d*e^4))/(10*e^6) - (2*x^3*(55*c^3*d^3 - 3*a*c^2*d*e^2))/(3*e^4) - (6*c^3*d*x^5)/e^2 + (3*c ^2*x^4*(a*e^2 - 15*c*d^2))/(2*e^3))/(d^6 + e^6*x^6 + 6*d*e^5*x^5 + 15*d^4* e^2*x^2 + 20*d^3*e^3*x^3 + 15*d^2*e^4*x^4 + 6*d^5*e*x)