Integrand size = 33, antiderivative size = 254 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^{12}} \, dx=-\frac {(b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^5 (a+b x) (d+e x)^{11}}+\frac {2 b (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^5 (a+b x) (d+e x)^{10}}-\frac {2 b^2 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^5 (a+b x) (d+e x)^9}+\frac {b^3 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^5 (a+b x) (d+e x)^8}-\frac {b^4 \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^5 (a+b x) (d+e x)^7} \]
-1/11*(-a*e+b*d)^4*((b*x+a)^2)^(1/2)/e^5/(b*x+a)/(e*x+d)^11+2/5*b*(-a*e+b* d)^3*((b*x+a)^2)^(1/2)/e^5/(b*x+a)/(e*x+d)^10-2/3*b^2*(-a*e+b*d)^2*((b*x+a )^2)^(1/2)/e^5/(b*x+a)/(e*x+d)^9+1/2*b^3*(-a*e+b*d)*((b*x+a)^2)^(1/2)/e^5/ (b*x+a)/(e*x+d)^8-1/7*b^4*((b*x+a)^2)^(1/2)/e^5/(b*x+a)/(e*x+d)^7
Time = 1.05 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.64 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^{12}} \, dx=-\frac {\sqrt {(a+b x)^2} \left (210 a^4 e^4+84 a^3 b e^3 (d+11 e x)+28 a^2 b^2 e^2 \left (d^2+11 d e x+55 e^2 x^2\right )+7 a b^3 e \left (d^3+11 d^2 e x+55 d e^2 x^2+165 e^3 x^3\right )+b^4 \left (d^4+11 d^3 e x+55 d^2 e^2 x^2+165 d e^3 x^3+330 e^4 x^4\right )\right )}{2310 e^5 (a+b x) (d+e x)^{11}} \]
-1/2310*(Sqrt[(a + b*x)^2]*(210*a^4*e^4 + 84*a^3*b*e^3*(d + 11*e*x) + 28*a ^2*b^2*e^2*(d^2 + 11*d*e*x + 55*e^2*x^2) + 7*a*b^3*e*(d^3 + 11*d^2*e*x + 5 5*d*e^2*x^2 + 165*e^3*x^3) + b^4*(d^4 + 11*d^3*e*x + 55*d^2*e^2*x^2 + 165* d*e^3*x^3 + 330*e^4*x^4)))/(e^5*(a + b*x)*(d + e*x)^11)
Time = 0.32 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.58, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {1187, 27, 53, 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.
| \(\displaystyle \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^{12}} \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {b^3 (a+b x)^4}{(d+e x)^{12}}dx}{b^3 (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x)^4}{(d+e x)^{12}}dx}{a+b x}\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {b^4}{e^4 (d+e x)^8}-\frac {4 (b d-a e) b^3}{e^4 (d+e x)^9}+\frac {6 (b d-a e)^2 b^2}{e^4 (d+e x)^{10}}-\frac {4 (b d-a e)^3 b}{e^4 (d+e x)^{11}}+\frac {(a e-b d)^4}{e^4 (d+e x)^{12}}\right )dx}{a+b x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {b^3 (b d-a e)}{2 e^5 (d+e x)^8}-\frac {2 b^2 (b d-a e)^2}{3 e^5 (d+e x)^9}+\frac {2 b (b d-a e)^3}{5 e^5 (d+e x)^{10}}-\frac {(b d-a e)^4}{11 e^5 (d+e x)^{11}}-\frac {b^4}{7 e^5 (d+e x)^7}\right )}{a+b x}\) |
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-1/11*(b*d - a*e)^4/(e^5*(d + e*x)^11) + ( 2*b*(b*d - a*e)^3)/(5*e^5*(d + e*x)^10) - (2*b^2*(b*d - a*e)^2)/(3*e^5*(d + e*x)^9) + (b^3*(b*d - a*e))/(2*e^5*(d + e*x)^8) - b^4/(7*e^5*(d + e*x)^7 )))/(a + b*x)
3.20.87.3.1 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[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x + c*x^2)^FracPart[p]/(c^ IntPart[p]*(b/2 + c*x)^(2*FracPart[p])) Int[(d + e*x)^m*(f + g*x)^n*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p]
Time = 3.49 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.74
| method | result | size |
| risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {b^{4} x^{4}}{7 e}-\frac {b^{3} \left (7 a e +b d \right ) x^{3}}{14 e^{2}}-\frac {b^{2} \left (28 e^{2} a^{2}+7 a b d e +b^{2} d^{2}\right ) x^{2}}{42 e^{3}}-\frac {b \left (84 a^{3} e^{3}+28 a^{2} b d \,e^{2}+7 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) x}{210 e^{4}}-\frac {210 e^{4} a^{4}+84 b d \,e^{3} a^{3}+28 b^{2} d^{2} e^{2} a^{2}+7 b^{3} d^{3} e a +b^{4} d^{4}}{2310 e^{5}}\right )}{\left (b x +a \right ) \left (e x +d \right )^{11}}\) | \(187\) |
| gosper | \(-\frac {\left (330 e^{4} x^{4} b^{4}+1155 x^{3} a \,b^{3} e^{4}+165 x^{3} b^{4} d \,e^{3}+1540 x^{2} a^{2} b^{2} e^{4}+385 x^{2} a \,b^{3} d \,e^{3}+55 x^{2} b^{4} d^{2} e^{2}+924 x \,a^{3} b \,e^{4}+308 x \,a^{2} b^{2} d \,e^{3}+77 x a \,b^{3} d^{2} e^{2}+11 x \,b^{4} d^{3} e +210 e^{4} a^{4}+84 b d \,e^{3} a^{3}+28 b^{2} d^{2} e^{2} a^{2}+7 b^{3} d^{3} e a +b^{4} d^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{2310 e^{5} \left (e x +d \right )^{11} \left (b x +a \right )^{3}}\) | \(201\) |
| default | \(-\frac {\left (330 e^{4} x^{4} b^{4}+1155 x^{3} a \,b^{3} e^{4}+165 x^{3} b^{4} d \,e^{3}+1540 x^{2} a^{2} b^{2} e^{4}+385 x^{2} a \,b^{3} d \,e^{3}+55 x^{2} b^{4} d^{2} e^{2}+924 x \,a^{3} b \,e^{4}+308 x \,a^{2} b^{2} d \,e^{3}+77 x a \,b^{3} d^{2} e^{2}+11 x \,b^{4} d^{3} e +210 e^{4} a^{4}+84 b d \,e^{3} a^{3}+28 b^{2} d^{2} e^{2} a^{2}+7 b^{3} d^{3} e a +b^{4} d^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{2310 e^{5} \left (e x +d \right )^{11} \left (b x +a \right )^{3}}\) | \(201\) |
((b*x+a)^2)^(1/2)/(b*x+a)*(-1/7*b^4/e*x^4-1/14*b^3/e^2*(7*a*e+b*d)*x^3-1/4 2*b^2/e^3*(28*a^2*e^2+7*a*b*d*e+b^2*d^2)*x^2-1/210*b/e^4*(84*a^3*e^3+28*a^ 2*b*d*e^2+7*a*b^2*d^2*e+b^3*d^3)*x-1/2310/e^5*(210*a^4*e^4+84*a^3*b*d*e^3+ 28*a^2*b^2*d^2*e^2+7*a*b^3*d^3*e+b^4*d^4))/(e*x+d)^11
Time = 0.33 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.15 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^{12}} \, dx=-\frac {330 \, b^{4} e^{4} x^{4} + b^{4} d^{4} + 7 \, a b^{3} d^{3} e + 28 \, a^{2} b^{2} d^{2} e^{2} + 84 \, a^{3} b d e^{3} + 210 \, a^{4} e^{4} + 165 \, {\left (b^{4} d e^{3} + 7 \, a b^{3} e^{4}\right )} x^{3} + 55 \, {\left (b^{4} d^{2} e^{2} + 7 \, a b^{3} d e^{3} + 28 \, a^{2} b^{2} e^{4}\right )} x^{2} + 11 \, {\left (b^{4} d^{3} e + 7 \, a b^{3} d^{2} e^{2} + 28 \, a^{2} b^{2} d e^{3} + 84 \, a^{3} b e^{4}\right )} x}{2310 \, {\left (e^{16} x^{11} + 11 \, d e^{15} x^{10} + 55 \, d^{2} e^{14} x^{9} + 165 \, d^{3} e^{13} x^{8} + 330 \, d^{4} e^{12} x^{7} + 462 \, d^{5} e^{11} x^{6} + 462 \, d^{6} e^{10} x^{5} + 330 \, d^{7} e^{9} x^{4} + 165 \, d^{8} e^{8} x^{3} + 55 \, d^{9} e^{7} x^{2} + 11 \, d^{10} e^{6} x + d^{11} e^{5}\right )}} \]
-1/2310*(330*b^4*e^4*x^4 + b^4*d^4 + 7*a*b^3*d^3*e + 28*a^2*b^2*d^2*e^2 + 84*a^3*b*d*e^3 + 210*a^4*e^4 + 165*(b^4*d*e^3 + 7*a*b^3*e^4)*x^3 + 55*(b^4 *d^2*e^2 + 7*a*b^3*d*e^3 + 28*a^2*b^2*e^4)*x^2 + 11*(b^4*d^3*e + 7*a*b^3*d ^2*e^2 + 28*a^2*b^2*d*e^3 + 84*a^3*b*e^4)*x)/(e^16*x^11 + 11*d*e^15*x^10 + 55*d^2*e^14*x^9 + 165*d^3*e^13*x^8 + 330*d^4*e^12*x^7 + 462*d^5*e^11*x^6 + 462*d^6*e^10*x^5 + 330*d^7*e^9*x^4 + 165*d^8*e^8*x^3 + 55*d^9*e^7*x^2 + 11*d^10*e^6*x + d^11*e^5)
Timed out. \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^{12}} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^{12}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a*e-b*d>0)', see `assume?` for m ore detail
Leaf count of result is larger than twice the leaf count of optimal. 385 vs. \(2 (189) = 378\).
Time = 0.26 (sec) , antiderivative size = 385, normalized size of antiderivative = 1.52 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^{12}} \, dx=\frac {b^{11} \mathrm {sgn}\left (b x + a\right )}{2310 \, {\left (b^{7} d^{7} e^{5} - 7 \, a b^{6} d^{6} e^{6} + 21 \, a^{2} b^{5} d^{5} e^{7} - 35 \, a^{3} b^{4} d^{4} e^{8} + 35 \, a^{4} b^{3} d^{3} e^{9} - 21 \, a^{5} b^{2} d^{2} e^{10} + 7 \, a^{6} b d e^{11} - a^{7} e^{12}\right )}} - \frac {330 \, b^{4} e^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + 165 \, b^{4} d e^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 1155 \, a b^{3} e^{4} x^{3} \mathrm {sgn}\left (b x + a\right ) + 55 \, b^{4} d^{2} e^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 385 \, a b^{3} d e^{3} x^{2} \mathrm {sgn}\left (b x + a\right ) + 1540 \, a^{2} b^{2} e^{4} x^{2} \mathrm {sgn}\left (b x + a\right ) + 11 \, b^{4} d^{3} e x \mathrm {sgn}\left (b x + a\right ) + 77 \, a b^{3} d^{2} e^{2} x \mathrm {sgn}\left (b x + a\right ) + 308 \, a^{2} b^{2} d e^{3} x \mathrm {sgn}\left (b x + a\right ) + 924 \, a^{3} b e^{4} x \mathrm {sgn}\left (b x + a\right ) + b^{4} d^{4} \mathrm {sgn}\left (b x + a\right ) + 7 \, a b^{3} d^{3} e \mathrm {sgn}\left (b x + a\right ) + 28 \, a^{2} b^{2} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 84 \, a^{3} b d e^{3} \mathrm {sgn}\left (b x + a\right ) + 210 \, a^{4} e^{4} \mathrm {sgn}\left (b x + a\right )}{2310 \, {\left (e x + d\right )}^{11} e^{5}} \]
1/2310*b^11*sgn(b*x + a)/(b^7*d^7*e^5 - 7*a*b^6*d^6*e^6 + 21*a^2*b^5*d^5*e ^7 - 35*a^3*b^4*d^4*e^8 + 35*a^4*b^3*d^3*e^9 - 21*a^5*b^2*d^2*e^10 + 7*a^6 *b*d*e^11 - a^7*e^12) - 1/2310*(330*b^4*e^4*x^4*sgn(b*x + a) + 165*b^4*d*e ^3*x^3*sgn(b*x + a) + 1155*a*b^3*e^4*x^3*sgn(b*x + a) + 55*b^4*d^2*e^2*x^2 *sgn(b*x + a) + 385*a*b^3*d*e^3*x^2*sgn(b*x + a) + 1540*a^2*b^2*e^4*x^2*sg n(b*x + a) + 11*b^4*d^3*e*x*sgn(b*x + a) + 77*a*b^3*d^2*e^2*x*sgn(b*x + a) + 308*a^2*b^2*d*e^3*x*sgn(b*x + a) + 924*a^3*b*e^4*x*sgn(b*x + a) + b^4*d ^4*sgn(b*x + a) + 7*a*b^3*d^3*e*sgn(b*x + a) + 28*a^2*b^2*d^2*e^2*sgn(b*x + a) + 84*a^3*b*d*e^3*sgn(b*x + a) + 210*a^4*e^4*sgn(b*x + a))/((e*x + d)^ 11*e^5)
Time = 11.13 (sec) , antiderivative size = 449, normalized size of antiderivative = 1.77 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^{12}} \, dx=\frac {\left (\frac {-4\,a^3\,b\,e^3+6\,a^2\,b^2\,d\,e^2-4\,a\,b^3\,d^2\,e+b^4\,d^3}{10\,e^5}+\frac {d\,\left (\frac {d\,\left (\frac {b^4\,d}{10\,e^3}-\frac {b^3\,\left (4\,a\,e-b\,d\right )}{10\,e^3}\right )}{e}+\frac {b^2\,\left (6\,a^2\,e^2-4\,a\,b\,d\,e+b^2\,d^2\right )}{10\,e^4}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^{10}}-\frac {\left (\frac {a^4}{11\,e}-\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {4\,a\,b^3}{11\,e}-\frac {b^4\,d}{11\,e^2}\right )}{e}-\frac {6\,a^2\,b^2}{11\,e}\right )}{e}+\frac {4\,a^3\,b}{11\,e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^{11}}-\frac {\left (\frac {6\,a^2\,b^2\,e^2-8\,a\,b^3\,d\,e+3\,b^4\,d^2}{9\,e^5}+\frac {d\,\left (\frac {b^4\,d}{9\,e^4}-\frac {2\,b^3\,\left (2\,a\,e-b\,d\right )}{9\,e^4}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^9}+\frac {\left (\frac {3\,b^4\,d-4\,a\,b^3\,e}{8\,e^5}+\frac {b^4\,d}{8\,e^5}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^8}-\frac {b^4\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{7\,e^5\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^7} \]
(((b^4*d^3 - 4*a^3*b*e^3 + 6*a^2*b^2*d*e^2 - 4*a*b^3*d^2*e)/(10*e^5) + (d* ((d*((b^4*d)/(10*e^3) - (b^3*(4*a*e - b*d))/(10*e^3)))/e + (b^2*(6*a^2*e^2 + b^2*d^2 - 4*a*b*d*e))/(10*e^4)))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(( a + b*x)*(d + e*x)^10) - ((a^4/(11*e) - (d*((d*((d*((4*a*b^3)/(11*e) - (b^ 4*d)/(11*e^2)))/e - (6*a^2*b^2)/(11*e)))/e + (4*a^3*b)/(11*e)))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^11) - (((3*b^4*d^2 + 6*a^2* b^2*e^2 - 8*a*b^3*d*e)/(9*e^5) + (d*((b^4*d)/(9*e^4) - (2*b^3*(2*a*e - b*d ))/(9*e^4)))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^9) + (((3*b^4*d - 4*a*b^3*e)/(8*e^5) + (b^4*d)/(8*e^5))*(a^2 + b^2*x^2 + 2*a*b *x)^(1/2))/((a + b*x)*(d + e*x)^8) - (b^4*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2)) /(7*e^5*(a + b*x)*(d + e*x)^7)