Integrand size = 24, antiderivative size = 86 \[ \int \frac {(b d+2 c d x)^7}{a+b x+c x^2} \, dx=\left (b^2-4 a c\right )^2 d^7 (b+2 c x)^2+\frac {1}{2} \left (b^2-4 a c\right ) d^7 (b+2 c x)^4+\frac {1}{3} d^7 (b+2 c x)^6+\left (b^2-4 a c\right )^3 d^7 \log \left (a+b x+c x^2\right ) \]
(-4*a*c+b^2)^2*d^7*(2*c*x+b)^2+1/2*(-4*a*c+b^2)*d^7*(2*c*x+b)^4+1/3*d^7*(2 *c*x+b)^6+(-4*a*c+b^2)^3*d^7*ln(c*x^2+b*x+a)
Time = 0.06 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.28 \[ \int \frac {(b d+2 c d x)^7}{a+b x+c x^2} \, dx=d^7 \left (\frac {4}{3} c x (b+c x) \left (9 b^4+18 b^3 c x+8 b c^2 x \left (-3 a+4 c x^2\right )+b^2 \left (-36 a c+34 c^2 x^2\right )+8 c^2 \left (6 a^2-3 a c x^2+2 c^2 x^4\right )\right )+\left (b^2-4 a c\right )^3 \log (a+x (b+c x))\right ) \]
d^7*((4*c*x*(b + c*x)*(9*b^4 + 18*b^3*c*x + 8*b*c^2*x*(-3*a + 4*c*x^2) + b ^2*(-36*a*c + 34*c^2*x^2) + 8*c^2*(6*a^2 - 3*a*c*x^2 + 2*c^2*x^4)))/3 + (b ^2 - 4*a*c)^3*Log[a + x*(b + c*x)])
Time = 0.26 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.92, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {1116, 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.
\(\displaystyle \int \frac {(b d+2 c d x)^7}{a+b x+c x^2} \, dx\) |
\(\Big \downarrow \) 1116 |
\(\displaystyle d^2 \left (b^2-4 a c\right ) \int \frac {d^5 (b+2 c x)^5}{c x^2+b x+a}dx+\frac {1}{3} d^7 (b+2 c x)^6\) |
\(\Big \downarrow \) 27 |
\(\displaystyle d^7 \left (b^2-4 a c\right ) \int \frac {(b+2 c x)^5}{c x^2+b x+a}dx+\frac {1}{3} d^7 (b+2 c x)^6\) |
\(\Big \downarrow \) 1116 |
\(\displaystyle d^7 \left (b^2-4 a c\right ) \left (\left (b^2-4 a c\right ) \int \frac {(b+2 c x)^3}{c x^2+b x+a}dx+\frac {1}{2} (b+2 c x)^4\right )+\frac {1}{3} d^7 (b+2 c x)^6\) |
\(\Big \downarrow \) 1116 |
\(\displaystyle d^7 \left (b^2-4 a c\right ) \left (\left (b^2-4 a c\right ) \left (\left (b^2-4 a c\right ) \int \frac {b+2 c x}{c x^2+b x+a}dx+(b+2 c x)^2\right )+\frac {1}{2} (b+2 c x)^4\right )+\frac {1}{3} d^7 (b+2 c x)^6\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle d^7 \left (b^2-4 a c\right ) \left (\left (b^2-4 a c\right ) \left (\left (b^2-4 a c\right ) \log \left (a+b x+c x^2\right )+(b+2 c x)^2\right )+\frac {1}{2} (b+2 c x)^4\right )+\frac {1}{3} d^7 (b+2 c x)^6\) |
(d^7*(b + 2*c*x)^6)/3 + (b^2 - 4*a*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]))
3.12.55.3.1 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_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])
Leaf count of result is larger than twice the leaf count of optimal. \(166\) vs. \(2(82)=164\).
Time = 2.86 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.94
method | result | size |
default | \(d^{7} \left (\frac {64 c^{6} x^{6}}{3}+64 b \,c^{5} x^{5}-32 a \,c^{5} x^{4}+88 b^{2} c^{4} x^{4}-64 a b \,c^{4} x^{3}+\frac {208 x^{3} b^{3} c^{3}}{3}+64 a^{2} c^{4} x^{2}-80 a \,b^{2} c^{3} x^{2}+36 x^{2} b^{4} c^{2}+64 a^{2} b \,c^{3} x -48 x a \,b^{3} c^{2}+12 x \,b^{5} c +\left (-64 c^{3} a^{3}+48 a^{2} b^{2} c^{2}-12 a \,b^{4} c +b^{6}\right ) \ln \left (c \,x^{2}+b x +a \right )\right )\) | \(167\) |
norman | \(\left (-32 a \,d^{7} c^{5}+88 b^{2} d^{7} c^{4}\right ) x^{4}+\left (64 a^{2} d^{7} c^{4}-80 a \,b^{2} d^{7} c^{3}+36 b^{4} d^{7} c^{2}\right ) x^{2}+\frac {64 d^{7} c^{6} x^{6}}{3}+64 b \,d^{7} c^{5} x^{5}-\frac {16 b \,c^{3} d^{7} \left (12 a c -13 b^{2}\right ) x^{3}}{3}+4 b \,d^{7} c \left (16 a^{2} c^{2}-12 a \,b^{2} c +3 b^{4}\right ) x -d^{7} \left (64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) \ln \left (c \,x^{2}+b x +a \right )\) | \(188\) |
risch | \(\frac {64 d^{7} c^{6} x^{6}}{3}+64 b \,d^{7} c^{5} x^{5}-32 d^{7} c^{5} a \,x^{4}+88 d^{7} c^{4} b^{2} x^{4}-64 d^{7} c^{4} a b \,x^{3}+\frac {208 d^{7} c^{3} x^{3} b^{3}}{3}+64 d^{7} c^{4} a^{2} x^{2}-80 d^{7} c^{3} a \,b^{2} x^{2}+36 d^{7} c^{2} b^{4} x^{2}+64 d^{7} c^{3} a^{2} b x -48 d^{7} c^{2} a \,b^{3} x +12 d^{7} c \,b^{5} x -64 \ln \left (c \,x^{2}+b x +a \right ) a^{3} c^{3} d^{7}+48 \ln \left (c \,x^{2}+b x +a \right ) a^{2} b^{2} c^{2} d^{7}-12 \ln \left (c \,x^{2}+b x +a \right ) a \,b^{4} c \,d^{7}+\ln \left (c \,x^{2}+b x +a \right ) b^{6} d^{7}\) | \(243\) |
parallelrisch | \(\frac {64 d^{7} c^{6} x^{6}}{3}+64 b \,d^{7} c^{5} x^{5}-32 d^{7} c^{5} a \,x^{4}+88 d^{7} c^{4} b^{2} x^{4}-64 d^{7} c^{4} a b \,x^{3}+\frac {208 d^{7} c^{3} x^{3} b^{3}}{3}+64 d^{7} c^{4} a^{2} x^{2}-80 d^{7} c^{3} a \,b^{2} x^{2}+36 d^{7} c^{2} b^{4} x^{2}+64 d^{7} c^{3} a^{2} b x -48 d^{7} c^{2} a \,b^{3} x +12 d^{7} c \,b^{5} x -64 \ln \left (c \,x^{2}+b x +a \right ) a^{3} c^{3} d^{7}+48 \ln \left (c \,x^{2}+b x +a \right ) a^{2} b^{2} c^{2} d^{7}-12 \ln \left (c \,x^{2}+b x +a \right ) a \,b^{4} c \,d^{7}+\ln \left (c \,x^{2}+b x +a \right ) b^{6} d^{7}\) | \(243\) |
d^7*(64/3*c^6*x^6+64*b*c^5*x^5-32*a*c^5*x^4+88*b^2*c^4*x^4-64*a*b*c^4*x^3+ 208/3*x^3*b^3*c^3+64*a^2*c^4*x^2-80*a*b^2*c^3*x^2+36*x^2*b^4*c^2+64*a^2*b* c^3*x-48*x*a*b^3*c^2+12*x*b^5*c+(-64*a^3*c^3+48*a^2*b^2*c^2-12*a*b^4*c+b^6 )*ln(c*x^2+b*x+a))
Leaf count of result is larger than twice the leaf count of optimal. 181 vs. \(2 (82) = 164\).
Time = 0.31 (sec) , antiderivative size = 181, normalized size of antiderivative = 2.10 \[ \int \frac {(b d+2 c d x)^7}{a+b x+c x^2} \, dx=\frac {64}{3} \, c^{6} d^{7} x^{6} + 64 \, b c^{5} d^{7} x^{5} + 8 \, {\left (11 \, b^{2} c^{4} - 4 \, a c^{5}\right )} d^{7} x^{4} + \frac {16}{3} \, {\left (13 \, b^{3} c^{3} - 12 \, a b c^{4}\right )} d^{7} x^{3} + 4 \, {\left (9 \, b^{4} c^{2} - 20 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d^{7} x^{2} + 4 \, {\left (3 \, b^{5} c - 12 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} d^{7} x + {\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} d^{7} \log \left (c x^{2} + b x + a\right ) \]
64/3*c^6*d^7*x^6 + 64*b*c^5*d^7*x^5 + 8*(11*b^2*c^4 - 4*a*c^5)*d^7*x^4 + 1 6/3*(13*b^3*c^3 - 12*a*b*c^4)*d^7*x^3 + 4*(9*b^4*c^2 - 20*a*b^2*c^3 + 16*a ^2*c^4)*d^7*x^2 + 4*(3*b^5*c - 12*a*b^3*c^2 + 16*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*log(c*x^2 + b*x + a)
Leaf count of result is larger than twice the leaf count of optimal. 185 vs. \(2 (82) = 164\).
Time = 0.58 (sec) , antiderivative size = 185, normalized size of antiderivative = 2.15 \[ \int \frac {(b d+2 c d x)^7}{a+b x+c x^2} \, dx=64 b c^{5} d^{7} x^{5} + \frac {64 c^{6} d^{7} x^{6}}{3} - d^{7} \left (4 a c - b^{2}\right )^{3} \log {\left (a + b x + c x^{2} \right )} + x^{4} \left (- 32 a c^{5} d^{7} + 88 b^{2} c^{4} d^{7}\right ) + x^{3} \left (- 64 a b c^{4} d^{7} + \frac {208 b^{3} c^{3} d^{7}}{3}\right ) + x^{2} \cdot \left (64 a^{2} c^{4} d^{7} - 80 a b^{2} c^{3} d^{7} + 36 b^{4} c^{2} d^{7}\right ) + x \left (64 a^{2} b c^{3} d^{7} - 48 a b^{3} c^{2} d^{7} + 12 b^{5} c d^{7}\right ) \]
64*b*c**5*d**7*x**5 + 64*c**6*d**7*x**6/3 - d**7*(4*a*c - b**2)**3*log(a + b*x + c*x**2) + x**4*(-32*a*c**5*d**7 + 88*b**2*c**4*d**7) + x**3*(-64*a* b*c**4*d**7 + 208*b**3*c**3*d**7/3) + x**2*(64*a**2*c**4*d**7 - 80*a*b**2* c**3*d**7 + 36*b**4*c**2*d**7) + x*(64*a**2*b*c**3*d**7 - 48*a*b**3*c**2*d **7 + 12*b**5*c*d**7)
Leaf count of result is larger than twice the leaf count of optimal. 181 vs. \(2 (82) = 164\).
Time = 0.21 (sec) , antiderivative size = 181, normalized size of antiderivative = 2.10 \[ \int \frac {(b d+2 c d x)^7}{a+b x+c x^2} \, dx=\frac {64}{3} \, c^{6} d^{7} x^{6} + 64 \, b c^{5} d^{7} x^{5} + 8 \, {\left (11 \, b^{2} c^{4} - 4 \, a c^{5}\right )} d^{7} x^{4} + \frac {16}{3} \, {\left (13 \, b^{3} c^{3} - 12 \, a b c^{4}\right )} d^{7} x^{3} + 4 \, {\left (9 \, b^{4} c^{2} - 20 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d^{7} x^{2} + 4 \, {\left (3 \, b^{5} c - 12 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} d^{7} x + {\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} d^{7} \log \left (c x^{2} + b x + a\right ) \]
64/3*c^6*d^7*x^6 + 64*b*c^5*d^7*x^5 + 8*(11*b^2*c^4 - 4*a*c^5)*d^7*x^4 + 1 6/3*(13*b^3*c^3 - 12*a*b*c^4)*d^7*x^3 + 4*(9*b^4*c^2 - 20*a*b^2*c^3 + 16*a ^2*c^4)*d^7*x^2 + 4*(3*b^5*c - 12*a*b^3*c^2 + 16*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*log(c*x^2 + b*x + a)
Leaf count of result is larger than twice the leaf count of optimal. 219 vs. \(2 (82) = 164\).
Time = 0.29 (sec) , antiderivative size = 219, normalized size of antiderivative = 2.55 \[ \int \frac {(b d+2 c d x)^7}{a+b x+c x^2} \, dx={\left (b^{6} d^{7} - 12 \, a b^{4} c d^{7} + 48 \, a^{2} b^{2} c^{2} d^{7} - 64 \, a^{3} c^{3} d^{7}\right )} \log \left (c x^{2} + b x + a\right ) + \frac {4 \, {\left (16 \, c^{12} d^{7} x^{6} + 48 \, b c^{11} d^{7} x^{5} + 66 \, b^{2} c^{10} d^{7} x^{4} - 24 \, a c^{11} d^{7} x^{4} + 52 \, b^{3} c^{9} d^{7} x^{3} - 48 \, a b c^{10} d^{7} x^{3} + 27 \, b^{4} c^{8} d^{7} x^{2} - 60 \, a b^{2} c^{9} d^{7} x^{2} + 48 \, a^{2} c^{10} d^{7} x^{2} + 9 \, b^{5} c^{7} d^{7} x - 36 \, a b^{3} c^{8} d^{7} x + 48 \, a^{2} b c^{9} d^{7} x\right )}}{3 \, c^{6}} \]
(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)*log(c*x^2 + b*x + a) + 4/3*(16*c^12*d^7*x^6 + 48*b*c^11*d^7*x^5 + 66*b^2*c^10*d^7*x ^4 - 24*a*c^11*d^7*x^4 + 52*b^3*c^9*d^7*x^3 - 48*a*b*c^10*d^7*x^3 + 27*b^4 *c^8*d^7*x^2 - 60*a*b^2*c^9*d^7*x^2 + 48*a^2*c^10*d^7*x^2 + 9*b^5*c^7*d^7* x - 36*a*b^3*c^8*d^7*x + 48*a^2*b*c^9*d^7*x)/c^6
Time = 9.59 (sec) , antiderivative size = 418, normalized size of antiderivative = 4.86 \[ \int \frac {(b d+2 c d x)^7}{a+b x+c x^2} \, dx=x^2\,\left (140\,b^4\,c^2\,d^7-\frac {b\,\left (560\,b^3\,c^3\,d^7+\frac {b\,\left (128\,a\,c^5\,d^7-352\,b^2\,c^4\,d^7\right )}{c}-320\,a\,b\,c^4\,d^7\right )}{2\,c}+\frac {a\,\left (128\,a\,c^5\,d^7-352\,b^2\,c^4\,d^7\right )}{2\,c}\right )+\ln \left (c\,x^2+b\,x+a\right )\,\left (-64\,a^3\,c^3\,d^7+48\,a^2\,b^2\,c^2\,d^7-12\,a\,b^4\,c\,d^7+b^6\,d^7\right )-x^4\,\left (32\,a\,c^5\,d^7-88\,b^2\,c^4\,d^7\right )+x^3\,\left (\frac {560\,b^3\,c^3\,d^7}{3}+\frac {b\,\left (128\,a\,c^5\,d^7-352\,b^2\,c^4\,d^7\right )}{3\,c}-\frac {320\,a\,b\,c^4\,d^7}{3}\right )-x\,\left (\frac {a\,\left (560\,b^3\,c^3\,d^7+\frac {b\,\left (128\,a\,c^5\,d^7-352\,b^2\,c^4\,d^7\right )}{c}-320\,a\,b\,c^4\,d^7\right )}{c}-84\,b^5\,c\,d^7+\frac {b\,\left (280\,b^4\,c^2\,d^7-\frac {b\,\left (560\,b^3\,c^3\,d^7+\frac {b\,\left (128\,a\,c^5\,d^7-352\,b^2\,c^4\,d^7\right )}{c}-320\,a\,b\,c^4\,d^7\right )}{c}+\frac {a\,\left (128\,a\,c^5\,d^7-352\,b^2\,c^4\,d^7\right )}{c}\right )}{c}\right )+\frac {64\,c^6\,d^7\,x^6}{3}+64\,b\,c^5\,d^7\,x^5 \]
x^2*(140*b^4*c^2*d^7 - (b*(560*b^3*c^3*d^7 + (b*(128*a*c^5*d^7 - 352*b^2*c ^4*d^7))/c - 320*a*b*c^4*d^7))/(2*c) + (a*(128*a*c^5*d^7 - 352*b^2*c^4*d^7 ))/(2*c)) + log(a + b*x + c*x^2)*(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) - x^4*(32*a*c^5*d^7 - 88*b^2*c^4*d^7) + x^3*((560* b^3*c^3*d^7)/3 + (b*(128*a*c^5*d^7 - 352*b^2*c^4*d^7))/(3*c) - (320*a*b*c^ 4*d^7)/3) - x*((a*(560*b^3*c^3*d^7 + (b*(128*a*c^5*d^7 - 352*b^2*c^4*d^7)) /c - 320*a*b*c^4*d^7))/c - 84*b^5*c*d^7 + (b*(280*b^4*c^2*d^7 - (b*(560*b^ 3*c^3*d^7 + (b*(128*a*c^5*d^7 - 352*b^2*c^4*d^7))/c - 320*a*b*c^4*d^7))/c + (a*(128*a*c^5*d^7 - 352*b^2*c^4*d^7))/c))/c) + (64*c^6*d^7*x^6)/3 + 64*b *c^5*d^7*x^5