Integrand size = 24, antiderivative size = 89 \[ \int \frac {(b d+2 c d x)^7}{\left (a+b x+c x^2\right )^2} \, dx=12 c \left (b^2-4 a c\right ) d^7 (b+2 c x)^2+6 c d^7 (b+2 c x)^4-\frac {d^7 (b+2 c x)^6}{a+b x+c x^2}+12 c \left (b^2-4 a c\right )^2 d^7 \log \left (a+b x+c x^2\right ) \]
12*c*(-4*a*c+b^2)*d^7*(2*c*x+b)^2+6*c*d^7*(2*c*x+b)^4-d^7*(2*c*x+b)^6/(c*x ^2+b*x+a)+12*c*(-4*a*c+b^2)^2*d^7*ln(c*x^2+b*x+a)
Time = 0.09 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.16 \[ \int \frac {(b d+2 c d x)^7}{\left (a+b x+c x^2\right )^2} \, dx=d^7 \left (16 b c^2 \left (3 b^2-8 a c\right ) x-16 c^3 \left (-5 b^2+8 a c\right ) x^2+64 b c^4 x^3+32 c^5 x^4-\frac {\left (b^2-4 a c\right )^3}{a+x (b+c x)}+12 c \left (b^2-4 a c\right )^2 \log (a+x (b+c x))\right ) \]
d^7*(16*b*c^2*(3*b^2 - 8*a*c)*x - 16*c^3*(-5*b^2 + 8*a*c)*x^2 + 64*b*c^4*x ^3 + 32*c^5*x^4 - (b^2 - 4*a*c)^3/(a + x*(b + c*x)) + 12*c*(b^2 - 4*a*c)^2 *Log[a + x*(b + c*x)])
Time = 0.26 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.93, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {1110, 27, 1116, 1116, 1103}
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.
\(\displaystyle \int \frac {(b d+2 c d x)^7}{\left (a+b x+c x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 1110 |
\(\displaystyle 12 c d^2 \int \frac {d^5 (b+2 c x)^5}{c x^2+b x+a}dx-\frac {d^7 (b+2 c x)^6}{a+b x+c x^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 12 c d^7 \int \frac {(b+2 c x)^5}{c x^2+b x+a}dx-\frac {d^7 (b+2 c x)^6}{a+b x+c x^2}\) |
\(\Big \downarrow \) 1116 |
\(\displaystyle 12 c d^7 \left (\left (b^2-4 a c\right ) \int \frac {(b+2 c x)^3}{c x^2+b x+a}dx+\frac {1}{2} (b+2 c x)^4\right )-\frac {d^7 (b+2 c x)^6}{a+b x+c x^2}\) |
\(\Big \downarrow \) 1116 |
\(\displaystyle 12 c d^7 \left (\left (b^2-4 a c\right ) \left (\left (b^2-4 a c\right ) \int \frac {b+2 c x}{c x^2+b x+a}dx+(b+2 c x)^2\right )+\frac {1}{2} (b+2 c x)^4\right )-\frac {d^7 (b+2 c x)^6}{a+b x+c x^2}\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle 12 c d^7 \left (\left (b^2-4 a c\right ) \left (\left (b^2-4 a c\right ) \log \left (a+b x+c x^2\right )+(b+2 c x)^2\right )+\frac {1}{2} (b+2 c x)^4\right )-\frac {d^7 (b+2 c x)^6}{a+b x+c x^2}\) |
-((d^7*(b + 2*c*x)^6)/(a + b*x + c*x^2)) + 12*c*d^7*((b + 2*c*x)^4/2 + (b^ 2 - 4*a*c)*((b + 2*c*x)^2 + (b^2 - 4*a*c)*Log[a + b*x + c*x^2]))
3.12.67.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Sy mbol] :> Simp[d*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(b*(p + 1))), x] - Simp[d*e*((m - 1)/(b*(p + 1))) Int[(d + e*x)^(m - 2)*(a + b*x + c*x^ 2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0] && N eQ[m + 2*p + 3, 0] && LtQ[p, -1] && GtQ[m, 1] && IntegerQ[2*p]
Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_S ymbol] :> Simp[2*d*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] + Simp[d^2*(m - 1)*((b^2 - 4*a*c)/(b^2*(m + 2*p + 1))) Int[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && (IntegerQ[2*p] || (IntegerQ[m] && RationalQ[p]) || OddQ[m])
Time = 2.98 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.55
method | result | size |
default | \(d^{7} \left (32 c^{5} x^{4}+64 b \,c^{4} x^{3}-128 a \,c^{4} x^{2}+80 c^{3} b^{2} x^{2}-128 a b \,c^{3} x +48 b^{3} c^{2} x +\frac {64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}{c \,x^{2}+b x +a}+12 c \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \ln \left (c \,x^{2}+b x +a \right )\right )\) | \(138\) |
norman | \(\frac {\left (-96 a \,d^{7} c^{5}+144 b^{2} d^{7} c^{4}\right ) x^{4}+32 d^{7} c^{6} x^{6}+96 b \,d^{7} c^{5} x^{5}-\frac {b \left (192 a^{3} c^{3} d^{7}-96 a^{2} b^{2} c^{2} d^{7}+12 a \,b^{4} c \,d^{7}-b^{6} d^{7}\right ) x}{a}-\frac {c \left (192 a^{3} c^{3} d^{7}-36 a \,b^{4} c \,d^{7}-b^{6} d^{7}\right ) x^{2}}{a}-64 b \,c^{3} d^{7} \left (3 a c -2 b^{2}\right ) x^{3}}{c \,x^{2}+b x +a}+\left (192 a^{2} d^{7} c^{3}-96 a \,b^{2} d^{7} c^{2}+12 b^{4} d^{7} c \right ) \ln \left (c \,x^{2}+b x +a \right )\) | \(221\) |
risch | \(32 c^{5} d^{7} x^{4}+64 b \,c^{4} d^{7} x^{3}-128 a \,c^{4} d^{7} x^{2}+80 b^{2} c^{3} d^{7} x^{2}-128 a b \,c^{3} d^{7} x +48 b^{3} c^{2} d^{7} x +128 a^{2} d^{7} c^{3}-96 a \,b^{2} d^{7} c^{2}+18 b^{4} d^{7} c +\frac {64 d^{7} c^{3} a^{3}}{c \,x^{2}+b x +a}-\frac {48 d^{7} a^{2} b^{2} c^{2}}{c \,x^{2}+b x +a}+\frac {12 d^{7} a \,b^{4} c}{c \,x^{2}+b x +a}-\frac {d^{7} b^{6}}{c \,x^{2}+b x +a}+192 \ln \left (c \,x^{2}+b x +a \right ) a^{2} c^{3} d^{7}-96 \ln \left (c \,x^{2}+b x +a \right ) a \,b^{2} c^{2} d^{7}+12 \ln \left (c \,x^{2}+b x +a \right ) b^{4} c \,d^{7}\) | \(262\) |
parallelrisch | \(\frac {32 d^{7} c^{7} x^{6}+96 b \,d^{7} c^{6} x^{5}-96 x^{4} a \,c^{6} d^{7}+144 x^{4} b^{2} c^{5} d^{7}+192 \ln \left (c \,x^{2}+b x +a \right ) x^{2} a^{2} c^{5} d^{7}-96 \ln \left (c \,x^{2}+b x +a \right ) x^{2} a \,b^{2} c^{4} d^{7}+12 \ln \left (c \,x^{2}+b x +a \right ) x^{2} b^{4} c^{3} d^{7}-192 x^{3} a b \,c^{5} d^{7}+128 x^{3} b^{3} c^{4} d^{7}+192 \ln \left (c \,x^{2}+b x +a \right ) x \,a^{2} b \,c^{4} d^{7}-96 \ln \left (c \,x^{2}+b x +a \right ) x a \,b^{3} c^{3} d^{7}+12 \ln \left (c \,x^{2}+b x +a \right ) x \,b^{5} c^{2} d^{7}+192 \ln \left (c \,x^{2}+b x +a \right ) a^{3} c^{4} d^{7}-96 \ln \left (c \,x^{2}+b x +a \right ) a^{2} b^{2} c^{3} d^{7}+12 \ln \left (c \,x^{2}+b x +a \right ) a \,b^{4} c^{2} d^{7}+96 x a \,b^{3} c^{3} d^{7}-48 x \,b^{5} c^{2} d^{7}+192 a^{3} c^{4} d^{7}-36 a \,b^{4} c^{2} d^{7}-b^{6} c \,d^{7}}{c \left (c \,x^{2}+b x +a \right )}\) | \(368\) |
d^7*(32*c^5*x^4+64*b*c^4*x^3-128*a*c^4*x^2+80*c^3*b^2*x^2-128*a*b*c^3*x+48 *b^3*c^2*x+(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)/(c*x^2+b*x+a)+12*c*( 16*a^2*c^2-8*a*b^2*c+b^4)*ln(c*x^2+b*x+a))
Leaf count of result is larger than twice the leaf count of optimal. 283 vs. \(2 (89) = 178\).
Time = 0.29 (sec) , antiderivative size = 283, normalized size of antiderivative = 3.18 \[ \int \frac {(b d+2 c d x)^7}{\left (a+b x+c x^2\right )^2} \, dx=\frac {32 \, c^{6} d^{7} x^{6} + 96 \, b c^{5} d^{7} x^{5} + 48 \, {\left (3 \, b^{2} c^{4} - 2 \, a c^{5}\right )} d^{7} x^{4} + 64 \, {\left (2 \, b^{3} c^{3} - 3 \, a b c^{4}\right )} d^{7} x^{3} + 16 \, {\left (3 \, b^{4} c^{2} - 3 \, a b^{2} c^{3} - 8 \, a^{2} c^{4}\right )} d^{7} x^{2} + 16 \, {\left (3 \, a b^{3} c^{2} - 8 \, a^{2} b c^{3}\right )} d^{7} x - {\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} d^{7} + 12 \, {\left ({\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d^{7} x^{2} + {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} d^{7} x + {\left (a b^{4} c - 8 \, a^{2} b^{2} c^{2} + 16 \, a^{3} c^{3}\right )} d^{7}\right )} \log \left (c x^{2} + b x + a\right )}{c x^{2} + b x + a} \]
(32*c^6*d^7*x^6 + 96*b*c^5*d^7*x^5 + 48*(3*b^2*c^4 - 2*a*c^5)*d^7*x^4 + 64 *(2*b^3*c^3 - 3*a*b*c^4)*d^7*x^3 + 16*(3*b^4*c^2 - 3*a*b^2*c^3 - 8*a^2*c^4 )*d^7*x^2 + 16*(3*a*b^3*c^2 - 8*a^2*b*c^3)*d^7*x - (b^6 - 12*a*b^4*c + 48* a^2*b^2*c^2 - 64*a^3*c^3)*d^7 + 12*((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d ^7*x^2 + (b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^7*x + (a*b^4*c - 8*a^2*b^2 *c^2 + 16*a^3*c^3)*d^7)*log(c*x^2 + b*x + a))/(c*x^2 + b*x + a)
Time = 1.58 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.80 \[ \int \frac {(b d+2 c d x)^7}{\left (a+b x+c x^2\right )^2} \, dx=64 b c^{4} d^{7} x^{3} + 32 c^{5} d^{7} x^{4} + 12 c d^{7} \left (4 a c - b^{2}\right )^{2} \log {\left (a + b x + c x^{2} \right )} + x^{2} \left (- 128 a c^{4} d^{7} + 80 b^{2} c^{3} d^{7}\right ) + x \left (- 128 a b c^{3} d^{7} + 48 b^{3} c^{2} d^{7}\right ) + \frac {64 a^{3} c^{3} d^{7} - 48 a^{2} b^{2} c^{2} d^{7} + 12 a b^{4} c d^{7} - b^{6} d^{7}}{a + b x + c x^{2}} \]
64*b*c**4*d**7*x**3 + 32*c**5*d**7*x**4 + 12*c*d**7*(4*a*c - b**2)**2*log( a + b*x + c*x**2) + x**2*(-128*a*c**4*d**7 + 80*b**2*c**3*d**7) + x*(-128* a*b*c**3*d**7 + 48*b**3*c**2*d**7) + (64*a**3*c**3*d**7 - 48*a**2*b**2*c** 2*d**7 + 12*a*b**4*c*d**7 - b**6*d**7)/(a + b*x + c*x**2)
Time = 0.21 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.74 \[ \int \frac {(b d+2 c d x)^7}{\left (a+b x+c x^2\right )^2} \, dx=32 \, c^{5} d^{7} x^{4} + 64 \, b c^{4} d^{7} x^{3} + 16 \, {\left (5 \, b^{2} c^{3} - 8 \, a c^{4}\right )} d^{7} x^{2} + 16 \, {\left (3 \, b^{3} c^{2} - 8 \, a b c^{3}\right )} d^{7} x + 12 \, {\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} d^{7} \log \left (c x^{2} + b x + a\right ) - \frac {{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} d^{7}}{c x^{2} + b x + a} \]
32*c^5*d^7*x^4 + 64*b*c^4*d^7*x^3 + 16*(5*b^2*c^3 - 8*a*c^4)*d^7*x^2 + 16* (3*b^3*c^2 - 8*a*b*c^3)*d^7*x + 12*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^7* log(c*x^2 + b*x + a) - (b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3)*d^ 7/(c*x^2 + b*x + a)
Leaf count of result is larger than twice the leaf count of optimal. 181 vs. \(2 (89) = 178\).
Time = 0.32 (sec) , antiderivative size = 181, normalized size of antiderivative = 2.03 \[ \int \frac {(b d+2 c d x)^7}{\left (a+b x+c x^2\right )^2} \, dx=12 \, {\left (b^{4} c d^{7} - 8 \, a b^{2} c^{2} d^{7} + 16 \, a^{2} c^{3} d^{7}\right )} \log \left (c x^{2} + b x + a\right ) - \frac {b^{6} d^{7} - 12 \, a b^{4} c d^{7} + 48 \, a^{2} b^{2} c^{2} d^{7} - 64 \, a^{3} c^{3} d^{7}}{c x^{2} + b x + a} + \frac {16 \, {\left (2 \, c^{13} d^{7} x^{4} + 4 \, b c^{12} d^{7} x^{3} + 5 \, b^{2} c^{11} d^{7} x^{2} - 8 \, a c^{12} d^{7} x^{2} + 3 \, b^{3} c^{10} d^{7} x - 8 \, a b c^{11} d^{7} x\right )}}{c^{8}} \]
12*(b^4*c*d^7 - 8*a*b^2*c^2*d^7 + 16*a^2*c^3*d^7)*log(c*x^2 + b*x + a) - ( b^6*d^7 - 12*a*b^4*c*d^7 + 48*a^2*b^2*c^2*d^7 - 64*a^3*c^3*d^7)/(c*x^2 + b *x + a) + 16*(2*c^13*d^7*x^4 + 4*b*c^12*d^7*x^3 + 5*b^2*c^11*d^7*x^2 - 8*a *c^12*d^7*x^2 + 3*b^3*c^10*d^7*x - 8*a*b*c^11*d^7*x)/c^8
Time = 9.35 (sec) , antiderivative size = 234, normalized size of antiderivative = 2.63 \[ \int \frac {(b d+2 c d x)^7}{\left (a+b x+c x^2\right )^2} \, dx=\ln \left (c\,x^2+b\,x+a\right )\,\left (192\,a^2\,c^3\,d^7-96\,a\,b^2\,c^2\,d^7+12\,b^4\,c\,d^7\right )-x^2\,\left (64\,c^3\,d^7\,\left (b^2+2\,a\,c\right )-144\,b^2\,c^3\,d^7\right )-\frac {-64\,a^3\,c^3\,d^7+48\,a^2\,b^2\,c^2\,d^7-12\,a\,b^4\,c\,d^7+b^6\,d^7}{c\,x^2+b\,x+a}+x\,\left (560\,b^3\,c^2\,d^7+\frac {2\,b\,\left (128\,c^3\,d^7\,\left (b^2+2\,a\,c\right )-288\,b^2\,c^3\,d^7\right )}{c}-192\,b\,c^2\,d^7\,\left (b^2+2\,a\,c\right )-256\,a\,b\,c^3\,d^7\right )+32\,c^5\,d^7\,x^4+64\,b\,c^4\,d^7\,x^3 \]
log(a + b*x + c*x^2)*(12*b^4*c*d^7 + 192*a^2*c^3*d^7 - 96*a*b^2*c^2*d^7) - x^2*(64*c^3*d^7*(2*a*c + b^2) - 144*b^2*c^3*d^7) - (b^6*d^7 - 64*a^3*c^3* d^7 + 48*a^2*b^2*c^2*d^7 - 12*a*b^4*c*d^7)/(a + b*x + c*x^2) + x*(560*b^3* c^2*d^7 + (2*b*(128*c^3*d^7*(2*a*c + b^2) - 288*b^2*c^3*d^7))/c - 192*b*c^ 2*d^7*(2*a*c + b^2) - 256*a*b*c^3*d^7) + 32*c^5*d^7*x^4 + 64*b*c^4*d^7*x^3