Integrand size = 33, antiderivative size = 154 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{\sqrt {d+e x}} \, dx=-\frac {2 (b d-a e)^5 \sqrt {d+e x}}{e^6}+\frac {10 b (b d-a e)^4 (d+e x)^{3/2}}{3 e^6}-\frac {4 b^2 (b d-a e)^3 (d+e x)^{5/2}}{e^6}+\frac {20 b^3 (b d-a e)^2 (d+e x)^{7/2}}{7 e^6}-\frac {10 b^4 (b d-a e) (d+e x)^{9/2}}{9 e^6}+\frac {2 b^5 (d+e x)^{11/2}}{11 e^6} \]
10/3*b*(-a*e+b*d)^4*(e*x+d)^(3/2)/e^6-4*b^2*(-a*e+b*d)^3*(e*x+d)^(5/2)/e^6 +20/7*b^3*(-a*e+b*d)^2*(e*x+d)^(7/2)/e^6-10/9*b^4*(-a*e+b*d)*(e*x+d)^(9/2) /e^6+2/11*b^5*(e*x+d)^(11/2)/e^6-2*(-a*e+b*d)^5*(e*x+d)^(1/2)/e^6
Time = 0.04 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.40 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{\sqrt {d+e x}} \, dx=\frac {2 \sqrt {d+e x} \left (693 a^5 e^5+1155 a^4 b e^4 (-2 d+e x)+462 a^3 b^2 e^3 \left (8 d^2-4 d e x+3 e^2 x^2\right )+198 a^2 b^3 e^2 \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )+11 a b^4 e \left (128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right )+b^5 \left (-256 d^5+128 d^4 e x-96 d^3 e^2 x^2+80 d^2 e^3 x^3-70 d e^4 x^4+63 e^5 x^5\right )\right )}{693 e^6} \]
(2*Sqrt[d + e*x]*(693*a^5*e^5 + 1155*a^4*b*e^4*(-2*d + e*x) + 462*a^3*b^2* e^3*(8*d^2 - 4*d*e*x + 3*e^2*x^2) + 198*a^2*b^3*e^2*(-16*d^3 + 8*d^2*e*x - 6*d*e^2*x^2 + 5*e^3*x^3) + 11*a*b^4*e*(128*d^4 - 64*d^3*e*x + 48*d^2*e^2* x^2 - 40*d*e^3*x^3 + 35*e^4*x^4) + b^5*(-256*d^5 + 128*d^4*e*x - 96*d^3*e^ 2*x^2 + 80*d^2*e^3*x^3 - 70*d*e^4*x^4 + 63*e^5*x^5)))/(693*e^6)
Time = 0.30 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {1184, 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.
| \(\displaystyle \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{\sqrt {d+e x}} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle \frac {\int \frac {b^4 (a+b x)^5}{\sqrt {d+e x}}dx}{b^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {(a+b x)^5}{\sqrt {d+e x}}dx\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \int \left (-\frac {5 b^4 (d+e x)^{7/2} (b d-a e)}{e^5}+\frac {10 b^3 (d+e x)^{5/2} (b d-a e)^2}{e^5}-\frac {10 b^2 (d+e x)^{3/2} (b d-a e)^3}{e^5}+\frac {5 b \sqrt {d+e x} (b d-a e)^4}{e^5}+\frac {(a e-b d)^5}{e^5 \sqrt {d+e x}}+\frac {b^5 (d+e x)^{9/2}}{e^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {10 b^4 (d+e x)^{9/2} (b d-a e)}{9 e^6}+\frac {20 b^3 (d+e x)^{7/2} (b d-a e)^2}{7 e^6}-\frac {4 b^2 (d+e x)^{5/2} (b d-a e)^3}{e^6}+\frac {10 b (d+e x)^{3/2} (b d-a e)^4}{3 e^6}-\frac {2 \sqrt {d+e x} (b d-a e)^5}{e^6}+\frac {2 b^5 (d+e x)^{11/2}}{11 e^6}\) |
(-2*(b*d - a*e)^5*Sqrt[d + e*x])/e^6 + (10*b*(b*d - a*e)^4*(d + e*x)^(3/2) )/(3*e^6) - (4*b^2*(b*d - a*e)^3*(d + e*x)^(5/2))/e^6 + (20*b^3*(b*d - a*e )^2*(d + e*x)^(7/2))/(7*e^6) - (10*b^4*(b*d - a*e)*(d + e*x)^(9/2))/(9*e^6 ) + (2*b^5*(d + e*x)^(11/2))/(11*e^6)
3.21.54.3.1 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[1/c^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}, x] && E qQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Time = 0.32 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.32
| method | result | size |
| pseudoelliptic | \(\frac {2 \left (\left (\frac {1}{11} b^{5} x^{5}+\frac {5}{9} a \,b^{4} x^{4}+\frac {10}{7} a^{2} b^{3} x^{3}+2 a^{3} b^{2} x^{2}+\frac {5}{3} a^{4} b x +a^{5}\right ) e^{5}-\frac {10 b \left (\frac {1}{33} x^{4} b^{4}+\frac {4}{21} a \,b^{3} x^{3}+\frac {18}{35} x^{2} b^{2} a^{2}+\frac {4}{5} b \,a^{3} x +a^{4}\right ) d \,e^{4}}{3}+\frac {16 b^{2} \left (\frac {5}{231} x^{3} b^{3}+\frac {1}{7} a \,b^{2} x^{2}+\frac {3}{7} b \,a^{2} x +a^{3}\right ) d^{2} e^{3}}{3}-\frac {32 \left (\frac {1}{33} b^{2} x^{2}+\frac {2}{9} a b x +a^{2}\right ) b^{3} d^{3} e^{2}}{7}+\frac {128 b^{4} \left (\frac {b x}{11}+a \right ) d^{4} e}{63}-\frac {256 b^{5} d^{5}}{693}\right ) \sqrt {e x +d}}{e^{6}}\) | \(204\) |
| gosper | \(\frac {2 \left (63 x^{5} b^{5} e^{5}+385 x^{4} a \,b^{4} e^{5}-70 x^{4} b^{5} d \,e^{4}+990 x^{3} a^{2} b^{3} e^{5}-440 x^{3} a \,b^{4} d \,e^{4}+80 x^{3} b^{5} d^{2} e^{3}+1386 x^{2} a^{3} b^{2} e^{5}-1188 x^{2} a^{2} b^{3} d \,e^{4}+528 x^{2} a \,b^{4} d^{2} e^{3}-96 x^{2} b^{5} d^{3} e^{2}+1155 x \,a^{4} b \,e^{5}-1848 x \,a^{3} b^{2} d \,e^{4}+1584 x \,a^{2} b^{3} d^{2} e^{3}-704 x a \,b^{4} d^{3} e^{2}+128 x \,b^{5} d^{4} e +693 e^{5} a^{5}-2310 b d \,e^{4} a^{4}+3696 b^{2} d^{2} e^{3} a^{3}-3168 b^{3} d^{3} e^{2} a^{2}+1408 b^{4} d^{4} e a -256 b^{5} d^{5}\right ) \sqrt {e x +d}}{693 e^{6}}\) | \(273\) |
| trager | \(\frac {2 \left (63 x^{5} b^{5} e^{5}+385 x^{4} a \,b^{4} e^{5}-70 x^{4} b^{5} d \,e^{4}+990 x^{3} a^{2} b^{3} e^{5}-440 x^{3} a \,b^{4} d \,e^{4}+80 x^{3} b^{5} d^{2} e^{3}+1386 x^{2} a^{3} b^{2} e^{5}-1188 x^{2} a^{2} b^{3} d \,e^{4}+528 x^{2} a \,b^{4} d^{2} e^{3}-96 x^{2} b^{5} d^{3} e^{2}+1155 x \,a^{4} b \,e^{5}-1848 x \,a^{3} b^{2} d \,e^{4}+1584 x \,a^{2} b^{3} d^{2} e^{3}-704 x a \,b^{4} d^{3} e^{2}+128 x \,b^{5} d^{4} e +693 e^{5} a^{5}-2310 b d \,e^{4} a^{4}+3696 b^{2} d^{2} e^{3} a^{3}-3168 b^{3} d^{3} e^{2} a^{2}+1408 b^{4} d^{4} e a -256 b^{5} d^{5}\right ) \sqrt {e x +d}}{693 e^{6}}\) | \(273\) |
| risch | \(\frac {2 \left (63 x^{5} b^{5} e^{5}+385 x^{4} a \,b^{4} e^{5}-70 x^{4} b^{5} d \,e^{4}+990 x^{3} a^{2} b^{3} e^{5}-440 x^{3} a \,b^{4} d \,e^{4}+80 x^{3} b^{5} d^{2} e^{3}+1386 x^{2} a^{3} b^{2} e^{5}-1188 x^{2} a^{2} b^{3} d \,e^{4}+528 x^{2} a \,b^{4} d^{2} e^{3}-96 x^{2} b^{5} d^{3} e^{2}+1155 x \,a^{4} b \,e^{5}-1848 x \,a^{3} b^{2} d \,e^{4}+1584 x \,a^{2} b^{3} d^{2} e^{3}-704 x a \,b^{4} d^{3} e^{2}+128 x \,b^{5} d^{4} e +693 e^{5} a^{5}-2310 b d \,e^{4} a^{4}+3696 b^{2} d^{2} e^{3} a^{3}-3168 b^{3} d^{3} e^{2} a^{2}+1408 b^{4} d^{4} e a -256 b^{5} d^{5}\right ) \sqrt {e x +d}}{693 e^{6}}\) | \(273\) |
| derivativedivides | \(\frac {\frac {2 b^{5} \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (a e -b d \right ) b^{4}+2 b^{3} \left (2 a b e -2 b^{2} d \right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (2 \left (a e -b d \right ) \left (2 a b e -2 b^{2} d \right ) b^{2}+b \left (2 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right )\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (\left (a e -b d \right ) \left (2 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right )+2 b \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a b e -2 b^{2} d \right )\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (2 \left (a e -b d \right ) \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a b e -2 b^{2} d \right )+b \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{3}+2 \left (a e -b d \right ) \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{2} \sqrt {e x +d}}{e^{6}}\) | \(349\) |
| default | \(\frac {\frac {2 b^{5} \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (a e -b d \right ) b^{4}+2 b^{3} \left (2 a b e -2 b^{2} d \right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (2 \left (a e -b d \right ) \left (2 a b e -2 b^{2} d \right ) b^{2}+b \left (2 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right )\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (\left (a e -b d \right ) \left (2 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right )+2 b \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a b e -2 b^{2} d \right )\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (2 \left (a e -b d \right ) \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a b e -2 b^{2} d \right )+b \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{3}+2 \left (a e -b d \right ) \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{2} \sqrt {e x +d}}{e^{6}}\) | \(349\) |
2*((1/11*b^5*x^5+5/9*a*b^4*x^4+10/7*a^2*b^3*x^3+2*a^3*b^2*x^2+5/3*a^4*b*x+ a^5)*e^5-10/3*b*(1/33*x^4*b^4+4/21*a*b^3*x^3+18/35*x^2*b^2*a^2+4/5*b*a^3*x +a^4)*d*e^4+16/3*b^2*(5/231*x^3*b^3+1/7*a*b^2*x^2+3/7*b*a^2*x+a^3)*d^2*e^3 -32/7*(1/33*b^2*x^2+2/9*a*b*x+a^2)*b^3*d^3*e^2+128/63*b^4*(1/11*b*x+a)*d^4 *e-256/693*b^5*d^5)*(e*x+d)^(1/2)/e^6
Time = 0.64 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.69 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (63 \, b^{5} e^{5} x^{5} - 256 \, b^{5} d^{5} + 1408 \, a b^{4} d^{4} e - 3168 \, a^{2} b^{3} d^{3} e^{2} + 3696 \, a^{3} b^{2} d^{2} e^{3} - 2310 \, a^{4} b d e^{4} + 693 \, a^{5} e^{5} - 35 \, {\left (2 \, b^{5} d e^{4} - 11 \, a b^{4} e^{5}\right )} x^{4} + 10 \, {\left (8 \, b^{5} d^{2} e^{3} - 44 \, a b^{4} d e^{4} + 99 \, a^{2} b^{3} e^{5}\right )} x^{3} - 6 \, {\left (16 \, b^{5} d^{3} e^{2} - 88 \, a b^{4} d^{2} e^{3} + 198 \, a^{2} b^{3} d e^{4} - 231 \, a^{3} b^{2} e^{5}\right )} x^{2} + {\left (128 \, b^{5} d^{4} e - 704 \, a b^{4} d^{3} e^{2} + 1584 \, a^{2} b^{3} d^{2} e^{3} - 1848 \, a^{3} b^{2} d e^{4} + 1155 \, a^{4} b e^{5}\right )} x\right )} \sqrt {e x + d}}{693 \, e^{6}} \]
2/693*(63*b^5*e^5*x^5 - 256*b^5*d^5 + 1408*a*b^4*d^4*e - 3168*a^2*b^3*d^3* e^2 + 3696*a^3*b^2*d^2*e^3 - 2310*a^4*b*d*e^4 + 693*a^5*e^5 - 35*(2*b^5*d* e^4 - 11*a*b^4*e^5)*x^4 + 10*(8*b^5*d^2*e^3 - 44*a*b^4*d*e^4 + 99*a^2*b^3* e^5)*x^3 - 6*(16*b^5*d^3*e^2 - 88*a*b^4*d^2*e^3 + 198*a^2*b^3*d*e^4 - 231* a^3*b^2*e^5)*x^2 + (128*b^5*d^4*e - 704*a*b^4*d^3*e^2 + 1584*a^2*b^3*d^2*e ^3 - 1848*a^3*b^2*d*e^4 + 1155*a^4*b*e^5)*x)*sqrt(e*x + d)/e^6
Leaf count of result is larger than twice the leaf count of optimal. 345 vs. \(2 (143) = 286\).
Time = 1.44 (sec) , antiderivative size = 345, normalized size of antiderivative = 2.24 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{\sqrt {d+e x}} \, dx=\begin {cases} \frac {2 \left (\frac {b^{5} \left (d + e x\right )^{\frac {11}{2}}}{11 e^{5}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \cdot \left (5 a b^{4} e - 5 b^{5} d\right )}{9 e^{5}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \cdot \left (10 a^{2} b^{3} e^{2} - 20 a b^{4} d e + 10 b^{5} d^{2}\right )}{7 e^{5}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \left (10 a^{3} b^{2} e^{3} - 30 a^{2} b^{3} d e^{2} + 30 a b^{4} d^{2} e - 10 b^{5} d^{3}\right )}{5 e^{5}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \cdot \left (5 a^{4} b e^{4} - 20 a^{3} b^{2} d e^{3} + 30 a^{2} b^{3} d^{2} e^{2} - 20 a b^{4} d^{3} e + 5 b^{5} d^{4}\right )}{3 e^{5}} + \frac {\sqrt {d + e x} \left (a^{5} e^{5} - 5 a^{4} b d e^{4} + 10 a^{3} b^{2} d^{2} e^{3} - 10 a^{2} b^{3} d^{3} e^{2} + 5 a b^{4} d^{4} e - b^{5} d^{5}\right )}{e^{5}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {\begin {cases} a^{5} x & \text {for}\: b = 0 \\\frac {\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{3}}{6 b} & \text {otherwise} \end {cases}}{\sqrt {d}} & \text {otherwise} \end {cases} \]
Piecewise((2*(b**5*(d + e*x)**(11/2)/(11*e**5) + (d + e*x)**(9/2)*(5*a*b** 4*e - 5*b**5*d)/(9*e**5) + (d + e*x)**(7/2)*(10*a**2*b**3*e**2 - 20*a*b**4 *d*e + 10*b**5*d**2)/(7*e**5) + (d + e*x)**(5/2)*(10*a**3*b**2*e**3 - 30*a **2*b**3*d*e**2 + 30*a*b**4*d**2*e - 10*b**5*d**3)/(5*e**5) + (d + e*x)**( 3/2)*(5*a**4*b*e**4 - 20*a**3*b**2*d*e**3 + 30*a**2*b**3*d**2*e**2 - 20*a* b**4*d**3*e + 5*b**5*d**4)/(3*e**5) + sqrt(d + e*x)*(a**5*e**5 - 5*a**4*b* d*e**4 + 10*a**3*b**2*d**2*e**3 - 10*a**2*b**3*d**3*e**2 + 5*a*b**4*d**4*e - b**5*d**5)/e**5)/e, Ne(e, 0)), (Piecewise((a**5*x, Eq(b, 0)), ((a**2 + 2*a*b*x + b**2*x**2)**3/(6*b), True))/sqrt(d), True))
Time = 0.19 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.68 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (63 \, {\left (e x + d\right )}^{\frac {11}{2}} b^{5} - 385 \, {\left (b^{5} d - a b^{4} e\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 990 \, {\left (b^{5} d^{2} - 2 \, a b^{4} d e + a^{2} b^{3} e^{2}\right )} {\left (e x + d\right )}^{\frac {7}{2}} - 1386 \, {\left (b^{5} d^{3} - 3 \, a b^{4} d^{2} e + 3 \, a^{2} b^{3} d e^{2} - a^{3} b^{2} e^{3}\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 1155 \, {\left (b^{5} d^{4} - 4 \, a b^{4} d^{3} e + 6 \, a^{2} b^{3} d^{2} e^{2} - 4 \, a^{3} b^{2} d e^{3} + a^{4} b e^{4}\right )} {\left (e x + d\right )}^{\frac {3}{2}} - 693 \, {\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\right )} \sqrt {e x + d}\right )}}{693 \, e^{6}} \]
2/693*(63*(e*x + d)^(11/2)*b^5 - 385*(b^5*d - a*b^4*e)*(e*x + d)^(9/2) + 9 90*(b^5*d^2 - 2*a*b^4*d*e + a^2*b^3*e^2)*(e*x + d)^(7/2) - 1386*(b^5*d^3 - 3*a*b^4*d^2*e + 3*a^2*b^3*d*e^2 - a^3*b^2*e^3)*(e*x + d)^(5/2) + 1155*(b^ 5*d^4 - 4*a*b^4*d^3*e + 6*a^2*b^3*d^2*e^2 - 4*a^3*b^2*d*e^3 + a^4*b*e^4)*( e*x + d)^(3/2) - 693*(b^5*d^5 - 5*a*b^4*d^4*e + 10*a^2*b^3*d^3*e^2 - 10*a^ 3*b^2*d^2*e^3 + 5*a^4*b*d*e^4 - a^5*e^5)*sqrt(e*x + d))/e^6
Leaf count of result is larger than twice the leaf count of optimal. 283 vs. \(2 (134) = 268\).
Time = 0.27 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.84 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (693 \, \sqrt {e x + d} a^{5} + \frac {1155 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} a^{4} b}{e} + \frac {462 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} a^{3} b^{2}}{e^{2}} + \frac {198 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} a^{2} b^{3}}{e^{3}} + \frac {11 \, {\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} - 180 \, {\left (e x + d\right )}^{\frac {7}{2}} d + 378 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {e x + d} d^{4}\right )} a b^{4}}{e^{4}} + \frac {{\left (63 \, {\left (e x + d\right )}^{\frac {11}{2}} - 385 \, {\left (e x + d\right )}^{\frac {9}{2}} d + 990 \, {\left (e x + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {e x + d} d^{5}\right )} b^{5}}{e^{5}}\right )}}{693 \, e} \]
2/693*(693*sqrt(e*x + d)*a^5 + 1155*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)* a^4*b/e + 462*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d) *d^2)*a^3*b^2/e^2 + 198*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e* x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*a^2*b^3/e^3 + 11*(35*(e*x + d)^(9 /2) - 180*(e*x + d)^(7/2)*d + 378*(e*x + d)^(5/2)*d^2 - 420*(e*x + d)^(3/2 )*d^3 + 315*sqrt(e*x + d)*d^4)*a*b^4/e^4 + (63*(e*x + d)^(11/2) - 385*(e*x + d)^(9/2)*d + 990*(e*x + d)^(7/2)*d^2 - 1386*(e*x + d)^(5/2)*d^3 + 1155* (e*x + d)^(3/2)*d^4 - 693*sqrt(e*x + d)*d^5)*b^5/e^5)/e
Time = 0.05 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.89 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{\sqrt {d+e x}} \, dx=\frac {2\,b^5\,{\left (d+e\,x\right )}^{11/2}}{11\,e^6}-\frac {\left (10\,b^5\,d-10\,a\,b^4\,e\right )\,{\left (d+e\,x\right )}^{9/2}}{9\,e^6}+\frac {2\,{\left (a\,e-b\,d\right )}^5\,\sqrt {d+e\,x}}{e^6}+\frac {4\,b^2\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{5/2}}{e^6}+\frac {20\,b^3\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{7/2}}{7\,e^6}+\frac {10\,b\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{3/2}}{3\,e^6} \]