Integrand size = 35, antiderivative size = 368 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=-\frac {2 (b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) \sqrt {d+e x}}-\frac {12 b (b d-a e)^5 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x)}+\frac {10 b^2 (b d-a e)^4 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x)}-\frac {8 b^3 (b d-a e)^3 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x)}+\frac {30 b^4 (b d-a e)^2 (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x)}-\frac {4 b^5 (b d-a e) (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x)}+\frac {2 b^6 (d+e x)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x)} \] Output:
-2*(-a*e+b*d)^6*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^(1/2)-12*b*(-a*e+b*d )^5*(e*x+d)^(1/2)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)+10*b^2*(-a*e+b*d)^4*(e*x+d )^(3/2)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)-8*b^3*(-a*e+b*d)^3*(e*x+d)^(5/2)*((b *x+a)^2)^(1/2)/e^7/(b*x+a)+30/7*b^4*(-a*e+b*d)^2*(e*x+d)^(7/2)*((b*x+a)^2) ^(1/2)/e^7/(b*x+a)-4/3*b^5*(-a*e+b*d)*(e*x+d)^(9/2)*((b*x+a)^2)^(1/2)/e^7/ (b*x+a)+2/11*b^6*(e*x+d)^(11/2)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)
Time = 0.26 (sec) , antiderivative size = 307, normalized size of antiderivative = 0.83 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=-\frac {2 \sqrt {(a+b x)^2} \left (231 a^6 e^6-1386 a^5 b e^5 (2 d+e x)+1155 a^4 b^2 e^4 \left (8 d^2+4 d e x-e^2 x^2\right )-924 a^3 b^3 e^3 \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )+99 a^2 b^4 e^2 \left (128 d^4+64 d^3 e x-16 d^2 e^2 x^2+8 d e^3 x^3-5 e^4 x^4\right )-22 a b^5 e \left (256 d^5+128 d^4 e x-32 d^3 e^2 x^2+16 d^2 e^3 x^3-10 d e^4 x^4+7 e^5 x^5\right )+b^6 \left (1024 d^6+512 d^5 e x-128 d^4 e^2 x^2+64 d^3 e^3 x^3-40 d^2 e^4 x^4+28 d e^5 x^5-21 e^6 x^6\right )\right )}{231 e^7 (a+b x) \sqrt {d+e x}} \] Input:
Integrate[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^(3/2),x]
Output:
(-2*Sqrt[(a + b*x)^2]*(231*a^6*e^6 - 1386*a^5*b*e^5*(2*d + e*x) + 1155*a^4 *b^2*e^4*(8*d^2 + 4*d*e*x - e^2*x^2) - 924*a^3*b^3*e^3*(16*d^3 + 8*d^2*e*x - 2*d*e^2*x^2 + e^3*x^3) + 99*a^2*b^4*e^2*(128*d^4 + 64*d^3*e*x - 16*d^2* e^2*x^2 + 8*d*e^3*x^3 - 5*e^4*x^4) - 22*a*b^5*e*(256*d^5 + 128*d^4*e*x - 3 2*d^3*e^2*x^2 + 16*d^2*e^3*x^3 - 10*d*e^4*x^4 + 7*e^5*x^5) + b^6*(1024*d^6 + 512*d^5*e*x - 128*d^4*e^2*x^2 + 64*d^3*e^3*x^3 - 40*d^2*e^4*x^4 + 28*d* e^5*x^5 - 21*e^6*x^6)))/(231*e^7*(a + b*x)*Sqrt[d + e*x])
Time = 0.60 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.56, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, 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.
| \(\displaystyle \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {b^5 (a+b x)^6}{(d+e x)^{3/2}}dx}{b^5 (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x)^6}{(d+e x)^{3/2}}dx}{a+b x}\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {(d+e x)^{9/2} b^6}{e^6}-\frac {6 (b d-a e) (d+e x)^{7/2} b^5}{e^6}+\frac {15 (b d-a e)^2 (d+e x)^{5/2} b^4}{e^6}-\frac {20 (b d-a e)^3 (d+e x)^{3/2} b^3}{e^6}+\frac {15 (b d-a e)^4 \sqrt {d+e x} b^2}{e^6}-\frac {6 (b d-a e)^5 b}{e^6 \sqrt {d+e x}}+\frac {(a e-b d)^6}{e^6 (d+e x)^{3/2}}\right )dx}{a+b x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (-\frac {4 b^5 (d+e x)^{9/2} (b d-a e)}{3 e^7}+\frac {30 b^4 (d+e x)^{7/2} (b d-a e)^2}{7 e^7}-\frac {8 b^3 (d+e x)^{5/2} (b d-a e)^3}{e^7}+\frac {10 b^2 (d+e x)^{3/2} (b d-a e)^4}{e^7}-\frac {12 b \sqrt {d+e x} (b d-a e)^5}{e^7}-\frac {2 (b d-a e)^6}{e^7 \sqrt {d+e x}}+\frac {2 b^6 (d+e x)^{11/2}}{11 e^7}\right )}{a+b x}\) |
Input:
Int[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^(3/2),x]
Output:
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*((-2*(b*d - a*e)^6)/(e^7*Sqrt[d + e*x]) - ( 12*b*(b*d - a*e)^5*Sqrt[d + e*x])/e^7 + (10*b^2*(b*d - a*e)^4*(d + e*x)^(3 /2))/e^7 - (8*b^3*(b*d - a*e)^3*(d + e*x)^(5/2))/e^7 + (30*b^4*(b*d - a*e) ^2*(d + e*x)^(7/2))/(7*e^7) - (4*b^5*(b*d - a*e)*(d + e*x)^(9/2))/(3*e^7) + (2*b^6*(d + e*x)^(11/2))/(11*e^7)))/(a + b*x)
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 = 1.38 (sec) , antiderivative size = 393, normalized size of antiderivative = 1.07
| method | result | size |
| gosper | \(-\frac {2 \left (-21 b^{6} x^{6} e^{6}-154 x^{5} a \,b^{5} e^{6}+28 x^{5} b^{6} d \,e^{5}-495 x^{4} a^{2} b^{4} e^{6}+220 x^{4} a \,b^{5} d \,e^{5}-40 x^{4} b^{6} d^{2} e^{4}-924 x^{3} a^{3} b^{3} e^{6}+792 x^{3} a^{2} b^{4} d \,e^{5}-352 x^{3} a \,b^{5} d^{2} e^{4}+64 x^{3} b^{6} d^{3} e^{3}-1155 x^{2} a^{4} b^{2} e^{6}+1848 x^{2} a^{3} b^{3} d \,e^{5}-1584 x^{2} a^{2} b^{4} d^{2} e^{4}+704 x^{2} a \,b^{5} d^{3} e^{3}-128 x^{2} b^{6} d^{4} e^{2}-1386 x \,a^{5} b \,e^{6}+4620 x \,a^{4} b^{2} d \,e^{5}-7392 x \,a^{3} b^{3} d^{2} e^{4}+6336 x \,a^{2} b^{4} d^{3} e^{3}-2816 x a \,b^{5} d^{4} e^{2}+512 x \,b^{6} d^{5} e +231 a^{6} e^{6}-2772 a^{5} b d \,e^{5}+9240 a^{4} b^{2} d^{2} e^{4}-14784 a^{3} b^{3} d^{3} e^{3}+12672 a^{2} b^{4} d^{4} e^{2}-5632 a \,b^{5} d^{5} e +1024 b^{6} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{231 \sqrt {e x +d}\, e^{7} \left (b x +a \right )^{5}}\) | \(393\) |
| default | \(-\frac {2 \left (-21 b^{6} x^{6} e^{6}-154 x^{5} a \,b^{5} e^{6}+28 x^{5} b^{6} d \,e^{5}-495 x^{4} a^{2} b^{4} e^{6}+220 x^{4} a \,b^{5} d \,e^{5}-40 x^{4} b^{6} d^{2} e^{4}-924 x^{3} a^{3} b^{3} e^{6}+792 x^{3} a^{2} b^{4} d \,e^{5}-352 x^{3} a \,b^{5} d^{2} e^{4}+64 x^{3} b^{6} d^{3} e^{3}-1155 x^{2} a^{4} b^{2} e^{6}+1848 x^{2} a^{3} b^{3} d \,e^{5}-1584 x^{2} a^{2} b^{4} d^{2} e^{4}+704 x^{2} a \,b^{5} d^{3} e^{3}-128 x^{2} b^{6} d^{4} e^{2}-1386 x \,a^{5} b \,e^{6}+4620 x \,a^{4} b^{2} d \,e^{5}-7392 x \,a^{3} b^{3} d^{2} e^{4}+6336 x \,a^{2} b^{4} d^{3} e^{3}-2816 x a \,b^{5} d^{4} e^{2}+512 x \,b^{6} d^{5} e +231 a^{6} e^{6}-2772 a^{5} b d \,e^{5}+9240 a^{4} b^{2} d^{2} e^{4}-14784 a^{3} b^{3} d^{3} e^{3}+12672 a^{2} b^{4} d^{4} e^{2}-5632 a \,b^{5} d^{5} e +1024 b^{6} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{231 \sqrt {e x +d}\, e^{7} \left (b x +a \right )^{5}}\) | \(393\) |
| risch | \(\frac {2 b \left (21 x^{5} e^{5} b^{5}+154 x^{4} a \,b^{4} e^{5}-49 x^{4} b^{5} d \,e^{4}+495 x^{3} a^{2} b^{3} e^{5}-374 x^{3} a \,b^{4} d \,e^{4}+89 x^{3} b^{5} d^{2} e^{3}+924 x^{2} a^{3} b^{2} e^{5}-1287 x^{2} a^{2} b^{3} d \,e^{4}+726 x^{2} a \,b^{4} d^{2} e^{3}-153 x^{2} b^{5} d^{3} e^{2}+1155 a^{4} b \,e^{5} x -2772 a^{3} b^{2} d \,e^{4} x +2871 x \,a^{2} b^{3} d^{2} e^{3}-1430 x a \,b^{4} d^{3} e^{2}+281 b^{5} d^{4} e x +1386 e^{5} a^{5}-5775 a^{4} b d \,e^{4}+10164 a^{3} b^{2} d^{2} e^{3}-9207 a^{2} b^{3} d^{3} e^{2}+4246 a \,b^{4} d^{4} e -793 b^{5} d^{5}\right ) \sqrt {e x +d}\, \sqrt {\left (b x +a \right )^{2}}}{231 e^{7} \left (b x +a \right )}-\frac {2 \left (a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}\right ) \sqrt {\left (b x +a \right )^{2}}}{e^{7} \sqrt {e x +d}\, \left (b x +a \right )}\) | \(396\) |
| orering | \(-\frac {2 \left (-21 b^{6} x^{6} e^{6}-154 x^{5} a \,b^{5} e^{6}+28 x^{5} b^{6} d \,e^{5}-495 x^{4} a^{2} b^{4} e^{6}+220 x^{4} a \,b^{5} d \,e^{5}-40 x^{4} b^{6} d^{2} e^{4}-924 x^{3} a^{3} b^{3} e^{6}+792 x^{3} a^{2} b^{4} d \,e^{5}-352 x^{3} a \,b^{5} d^{2} e^{4}+64 x^{3} b^{6} d^{3} e^{3}-1155 x^{2} a^{4} b^{2} e^{6}+1848 x^{2} a^{3} b^{3} d \,e^{5}-1584 x^{2} a^{2} b^{4} d^{2} e^{4}+704 x^{2} a \,b^{5} d^{3} e^{3}-128 x^{2} b^{6} d^{4} e^{2}-1386 x \,a^{5} b \,e^{6}+4620 x \,a^{4} b^{2} d \,e^{5}-7392 x \,a^{3} b^{3} d^{2} e^{4}+6336 x \,a^{2} b^{4} d^{3} e^{3}-2816 x a \,b^{5} d^{4} e^{2}+512 x \,b^{6} d^{5} e +231 a^{6} e^{6}-2772 a^{5} b d \,e^{5}+9240 a^{4} b^{2} d^{2} e^{4}-14784 a^{3} b^{3} d^{3} e^{3}+12672 a^{2} b^{4} d^{4} e^{2}-5632 a \,b^{5} d^{5} e +1024 b^{6} d^{6}\right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{\frac {5}{2}}}{231 e^{7} \left (b x +a \right )^{5} \sqrt {e x +d}}\) | \(402\) |
Input:
int((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^(3/2),x,method=_RETURNVERB OSE)
Output:
-2/231/(e*x+d)^(1/2)*(-21*b^6*e^6*x^6-154*a*b^5*e^6*x^5+28*b^6*d*e^5*x^5-4 95*a^2*b^4*e^6*x^4+220*a*b^5*d*e^5*x^4-40*b^6*d^2*e^4*x^4-924*a^3*b^3*e^6* x^3+792*a^2*b^4*d*e^5*x^3-352*a*b^5*d^2*e^4*x^3+64*b^6*d^3*e^3*x^3-1155*a^ 4*b^2*e^6*x^2+1848*a^3*b^3*d*e^5*x^2-1584*a^2*b^4*d^2*e^4*x^2+704*a*b^5*d^ 3*e^3*x^2-128*b^6*d^4*e^2*x^2-1386*a^5*b*e^6*x+4620*a^4*b^2*d*e^5*x-7392*a ^3*b^3*d^2*e^4*x+6336*a^2*b^4*d^3*e^3*x-2816*a*b^5*d^4*e^2*x+512*b^6*d^5*e *x+231*a^6*e^6-2772*a^5*b*d*e^5+9240*a^4*b^2*d^2*e^4-14784*a^3*b^3*d^3*e^3 +12672*a^2*b^4*d^4*e^2-5632*a*b^5*d^5*e+1024*b^6*d^6)*((b*x+a)^2)^(5/2)/e^ 7/(b*x+a)^5
Time = 0.08 (sec) , antiderivative size = 365, normalized size of antiderivative = 0.99 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (21 \, b^{6} e^{6} x^{6} - 1024 \, b^{6} d^{6} + 5632 \, a b^{5} d^{5} e - 12672 \, a^{2} b^{4} d^{4} e^{2} + 14784 \, a^{3} b^{3} d^{3} e^{3} - 9240 \, a^{4} b^{2} d^{2} e^{4} + 2772 \, a^{5} b d e^{5} - 231 \, a^{6} e^{6} - 14 \, {\left (2 \, b^{6} d e^{5} - 11 \, a b^{5} e^{6}\right )} x^{5} + 5 \, {\left (8 \, b^{6} d^{2} e^{4} - 44 \, a b^{5} d e^{5} + 99 \, a^{2} b^{4} e^{6}\right )} x^{4} - 4 \, {\left (16 \, b^{6} d^{3} e^{3} - 88 \, a b^{5} d^{2} e^{4} + 198 \, a^{2} b^{4} d e^{5} - 231 \, a^{3} b^{3} e^{6}\right )} x^{3} + {\left (128 \, b^{6} d^{4} e^{2} - 704 \, a b^{5} d^{3} e^{3} + 1584 \, a^{2} b^{4} d^{2} e^{4} - 1848 \, a^{3} b^{3} d e^{5} + 1155 \, a^{4} b^{2} e^{6}\right )} x^{2} - 2 \, {\left (256 \, b^{6} d^{5} e - 1408 \, a b^{5} d^{4} e^{2} + 3168 \, a^{2} b^{4} d^{3} e^{3} - 3696 \, a^{3} b^{3} d^{2} e^{4} + 2310 \, a^{4} b^{2} d e^{5} - 693 \, a^{5} b e^{6}\right )} x\right )} \sqrt {e x + d}}{231 \, {\left (e^{8} x + d e^{7}\right )}} \] Input:
integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^(3/2),x, algorithm=" fricas")
Output:
2/231*(21*b^6*e^6*x^6 - 1024*b^6*d^6 + 5632*a*b^5*d^5*e - 12672*a^2*b^4*d^ 4*e^2 + 14784*a^3*b^3*d^3*e^3 - 9240*a^4*b^2*d^2*e^4 + 2772*a^5*b*d*e^5 - 231*a^6*e^6 - 14*(2*b^6*d*e^5 - 11*a*b^5*e^6)*x^5 + 5*(8*b^6*d^2*e^4 - 44* a*b^5*d*e^5 + 99*a^2*b^4*e^6)*x^4 - 4*(16*b^6*d^3*e^3 - 88*a*b^5*d^2*e^4 + 198*a^2*b^4*d*e^5 - 231*a^3*b^3*e^6)*x^3 + (128*b^6*d^4*e^2 - 704*a*b^5*d ^3*e^3 + 1584*a^2*b^4*d^2*e^4 - 1848*a^3*b^3*d*e^5 + 1155*a^4*b^2*e^6)*x^2 - 2*(256*b^6*d^5*e - 1408*a*b^5*d^4*e^2 + 3168*a^2*b^4*d^3*e^3 - 3696*a^3 *b^3*d^2*e^4 + 2310*a^4*b^2*d*e^5 - 693*a^5*b*e^6)*x)*sqrt(e*x + d)/(e^8*x + d*e^7)
Timed out. \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=\text {Timed out} \] Input:
integrate((b*x+a)*(b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**(3/2),x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 603 vs. \(2 (271) = 542\).
Time = 0.10 (sec) , antiderivative size = 603, normalized size of antiderivative = 1.64 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (7 \, b^{5} e^{5} x^{5} + 256 \, b^{5} d^{5} - 1152 \, a b^{4} d^{4} e + 2016 \, a^{2} b^{3} d^{3} e^{2} - 1680 \, a^{3} b^{2} d^{2} e^{3} + 630 \, a^{4} b d e^{4} - 63 \, a^{5} e^{5} - 5 \, {\left (2 \, b^{5} d e^{4} - 9 \, a b^{4} e^{5}\right )} x^{4} + 2 \, {\left (8 \, b^{5} d^{2} e^{3} - 36 \, a b^{4} d e^{4} + 63 \, a^{2} b^{3} e^{5}\right )} x^{3} - 2 \, {\left (16 \, b^{5} d^{3} e^{2} - 72 \, a b^{4} d^{2} e^{3} + 126 \, a^{2} b^{3} d e^{4} - 105 \, a^{3} b^{2} e^{5}\right )} x^{2} + {\left (128 \, b^{5} d^{4} e - 576 \, a b^{4} d^{3} e^{2} + 1008 \, a^{2} b^{3} d^{2} e^{3} - 840 \, a^{3} b^{2} d e^{4} + 315 \, a^{4} b e^{5}\right )} x\right )} a}{63 \, \sqrt {e x + d} e^{6}} + \frac {2 \, {\left (63 \, b^{5} e^{6} x^{6} - 3072 \, b^{5} d^{6} + 14080 \, a b^{4} d^{5} e - 25344 \, a^{2} b^{3} d^{4} e^{2} + 22176 \, a^{3} b^{2} d^{3} e^{3} - 9240 \, a^{4} b d^{2} e^{4} + 1386 \, a^{5} d e^{5} - 7 \, {\left (12 \, b^{5} d e^{5} - 55 \, a b^{4} e^{6}\right )} x^{5} + 10 \, {\left (12 \, b^{5} d^{2} e^{4} - 55 \, a b^{4} d e^{5} + 99 \, a^{2} b^{3} e^{6}\right )} x^{4} - 2 \, {\left (96 \, b^{5} d^{3} e^{3} - 440 \, a b^{4} d^{2} e^{4} + 792 \, a^{2} b^{3} d e^{5} - 693 \, a^{3} b^{2} e^{6}\right )} x^{3} + {\left (384 \, b^{5} d^{4} e^{2} - 1760 \, a b^{4} d^{3} e^{3} + 3168 \, a^{2} b^{3} d^{2} e^{4} - 2772 \, a^{3} b^{2} d e^{5} + 1155 \, a^{4} b e^{6}\right )} x^{2} - {\left (1536 \, b^{5} d^{5} e - 7040 \, a b^{4} d^{4} e^{2} + 12672 \, a^{2} b^{3} d^{3} e^{3} - 11088 \, a^{3} b^{2} d^{2} e^{4} + 4620 \, a^{4} b d e^{5} - 693 \, a^{5} e^{6}\right )} x\right )} b}{693 \, \sqrt {e x + d} e^{7}} \] Input:
integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^(3/2),x, algorithm=" maxima")
Output:
2/63*(7*b^5*e^5*x^5 + 256*b^5*d^5 - 1152*a*b^4*d^4*e + 2016*a^2*b^3*d^3*e^ 2 - 1680*a^3*b^2*d^2*e^3 + 630*a^4*b*d*e^4 - 63*a^5*e^5 - 5*(2*b^5*d*e^4 - 9*a*b^4*e^5)*x^4 + 2*(8*b^5*d^2*e^3 - 36*a*b^4*d*e^4 + 63*a^2*b^3*e^5)*x^ 3 - 2*(16*b^5*d^3*e^2 - 72*a*b^4*d^2*e^3 + 126*a^2*b^3*d*e^4 - 105*a^3*b^2 *e^5)*x^2 + (128*b^5*d^4*e - 576*a*b^4*d^3*e^2 + 1008*a^2*b^3*d^2*e^3 - 84 0*a^3*b^2*d*e^4 + 315*a^4*b*e^5)*x)*a/(sqrt(e*x + d)*e^6) + 2/693*(63*b^5* e^6*x^6 - 3072*b^5*d^6 + 14080*a*b^4*d^5*e - 25344*a^2*b^3*d^4*e^2 + 22176 *a^3*b^2*d^3*e^3 - 9240*a^4*b*d^2*e^4 + 1386*a^5*d*e^5 - 7*(12*b^5*d*e^5 - 55*a*b^4*e^6)*x^5 + 10*(12*b^5*d^2*e^4 - 55*a*b^4*d*e^5 + 99*a^2*b^3*e^6) *x^4 - 2*(96*b^5*d^3*e^3 - 440*a*b^4*d^2*e^4 + 792*a^2*b^3*d*e^5 - 693*a^3 *b^2*e^6)*x^3 + (384*b^5*d^4*e^2 - 1760*a*b^4*d^3*e^3 + 3168*a^2*b^3*d^2*e ^4 - 2772*a^3*b^2*d*e^5 + 1155*a^4*b*e^6)*x^2 - (1536*b^5*d^5*e - 7040*a*b ^4*d^4*e^2 + 12672*a^2*b^3*d^3*e^3 - 11088*a^3*b^2*d^2*e^4 + 4620*a^4*b*d* e^5 - 693*a^5*e^6)*x)*b/(sqrt(e*x + d)*e^7)
Leaf count of result is larger than twice the leaf count of optimal. 647 vs. \(2 (271) = 542\).
Time = 0.19 (sec) , antiderivative size = 647, normalized size of antiderivative = 1.76 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx =\text {Too large to display} \] Input:
integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^(3/2),x, algorithm=" giac")
Output:
-2*(b^6*d^6*sgn(b*x + a) - 6*a*b^5*d^5*e*sgn(b*x + a) + 15*a^2*b^4*d^4*e^2 *sgn(b*x + a) - 20*a^3*b^3*d^3*e^3*sgn(b*x + a) + 15*a^4*b^2*d^2*e^4*sgn(b *x + a) - 6*a^5*b*d*e^5*sgn(b*x + a) + a^6*e^6*sgn(b*x + a))/(sqrt(e*x + d )*e^7) + 2/231*(21*(e*x + d)^(11/2)*b^6*e^70*sgn(b*x + a) - 154*(e*x + d)^ (9/2)*b^6*d*e^70*sgn(b*x + a) + 495*(e*x + d)^(7/2)*b^6*d^2*e^70*sgn(b*x + a) - 924*(e*x + d)^(5/2)*b^6*d^3*e^70*sgn(b*x + a) + 1155*(e*x + d)^(3/2) *b^6*d^4*e^70*sgn(b*x + a) - 1386*sqrt(e*x + d)*b^6*d^5*e^70*sgn(b*x + a) + 154*(e*x + d)^(9/2)*a*b^5*e^71*sgn(b*x + a) - 990*(e*x + d)^(7/2)*a*b^5* d*e^71*sgn(b*x + a) + 2772*(e*x + d)^(5/2)*a*b^5*d^2*e^71*sgn(b*x + a) - 4 620*(e*x + d)^(3/2)*a*b^5*d^3*e^71*sgn(b*x + a) + 6930*sqrt(e*x + d)*a*b^5 *d^4*e^71*sgn(b*x + a) + 495*(e*x + d)^(7/2)*a^2*b^4*e^72*sgn(b*x + a) - 2 772*(e*x + d)^(5/2)*a^2*b^4*d*e^72*sgn(b*x + a) + 6930*(e*x + d)^(3/2)*a^2 *b^4*d^2*e^72*sgn(b*x + a) - 13860*sqrt(e*x + d)*a^2*b^4*d^3*e^72*sgn(b*x + a) + 924*(e*x + d)^(5/2)*a^3*b^3*e^73*sgn(b*x + a) - 4620*(e*x + d)^(3/2 )*a^3*b^3*d*e^73*sgn(b*x + a) + 13860*sqrt(e*x + d)*a^3*b^3*d^2*e^73*sgn(b *x + a) + 1155*(e*x + d)^(3/2)*a^4*b^2*e^74*sgn(b*x + a) - 6930*sqrt(e*x + d)*a^4*b^2*d*e^74*sgn(b*x + a) + 1386*sqrt(e*x + d)*a^5*b*e^75*sgn(b*x + a))/e^77
Time = 12.57 (sec) , antiderivative size = 396, normalized size of antiderivative = 1.08 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=\frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (\frac {2\,b^5\,x^6}{11\,e}-\frac {462\,a^6\,e^6-5544\,a^5\,b\,d\,e^5+18480\,a^4\,b^2\,d^2\,e^4-29568\,a^3\,b^3\,d^3\,e^3+25344\,a^2\,b^4\,d^4\,e^2-11264\,a\,b^5\,d^5\,e+2048\,b^6\,d^6}{231\,b\,e^7}+\frac {x\,\left (2772\,a^5\,b\,e^6-9240\,a^4\,b^2\,d\,e^5+14784\,a^3\,b^3\,d^2\,e^4-12672\,a^2\,b^4\,d^3\,e^3+5632\,a\,b^5\,d^4\,e^2-1024\,b^6\,d^5\,e\right )}{231\,b\,e^7}+\frac {8\,b^2\,x^3\,\left (231\,a^3\,e^3-198\,a^2\,b\,d\,e^2+88\,a\,b^2\,d^2\,e-16\,b^3\,d^3\right )}{231\,e^4}+\frac {4\,b^4\,x^5\,\left (11\,a\,e-2\,b\,d\right )}{33\,e^2}+\frac {10\,b^3\,x^4\,\left (99\,a^2\,e^2-44\,a\,b\,d\,e+8\,b^2\,d^2\right )}{231\,e^3}+\frac {x^2\,\left (2310\,a^4\,b^2\,e^6-3696\,a^3\,b^3\,d\,e^5+3168\,a^2\,b^4\,d^2\,e^4-1408\,a\,b^5\,d^3\,e^3+256\,b^6\,d^4\,e^2\right )}{231\,b\,e^7}\right )}{x\,\sqrt {d+e\,x}+\frac {a\,\sqrt {d+e\,x}}{b}} \] Input:
int(((a + b*x)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2))/(d + e*x)^(3/2),x)
Output:
((a^2 + b^2*x^2 + 2*a*b*x)^(1/2)*((2*b^5*x^6)/(11*e) - (462*a^6*e^6 + 2048 *b^6*d^6 + 25344*a^2*b^4*d^4*e^2 - 29568*a^3*b^3*d^3*e^3 + 18480*a^4*b^2*d ^2*e^4 - 11264*a*b^5*d^5*e - 5544*a^5*b*d*e^5)/(231*b*e^7) + (x*(2772*a^5* b*e^6 - 1024*b^6*d^5*e + 5632*a*b^5*d^4*e^2 - 9240*a^4*b^2*d*e^5 - 12672*a ^2*b^4*d^3*e^3 + 14784*a^3*b^3*d^2*e^4))/(231*b*e^7) + (8*b^2*x^3*(231*a^3 *e^3 - 16*b^3*d^3 + 88*a*b^2*d^2*e - 198*a^2*b*d*e^2))/(231*e^4) + (4*b^4* x^5*(11*a*e - 2*b*d))/(33*e^2) + (10*b^3*x^4*(99*a^2*e^2 + 8*b^2*d^2 - 44* a*b*d*e))/(231*e^3) + (x^2*(2310*a^4*b^2*e^6 + 256*b^6*d^4*e^2 - 1408*a*b^ 5*d^3*e^3 - 3696*a^3*b^3*d*e^5 + 3168*a^2*b^4*d^2*e^4))/(231*b*e^7)))/(x*( d + e*x)^(1/2) + (a*(d + e*x)^(1/2))/b)
Time = 0.21 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.02 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=\frac {24 a^{5} b d \,e^{5}+12 a^{5} b \,e^{6} x -80 a^{4} b^{2} d^{2} e^{4}+10 a^{4} b^{2} e^{6} x^{2}+128 a^{3} b^{3} d^{3} e^{3}+8 a^{3} b^{3} e^{6} x^{3}-\frac {768}{7} a^{2} b^{4} d^{4} e^{2}+\frac {1024}{21} a \,b^{5} d^{5} e +\frac {4}{3} a \,b^{5} e^{6} x^{5}-\frac {1024}{231} b^{6} d^{5} e x +\frac {256}{231} b^{6} d^{4} e^{2} x^{2}-\frac {128}{231} b^{6} d^{3} e^{3} x^{3}-\frac {8}{33} b^{6} d \,e^{5} x^{5}+\frac {2}{11} b^{6} e^{6} x^{6}-40 a^{4} b^{2} d \,e^{5} x +64 a^{3} b^{3} d^{2} e^{4} x -16 a^{3} b^{3} d \,e^{5} x^{2}-\frac {384}{7} a^{2} b^{4} d^{3} e^{3} x +\frac {96}{7} a^{2} b^{4} d^{2} e^{4} x^{2}-\frac {48}{7} a^{2} b^{4} d \,e^{5} x^{3}+\frac {512}{21} a \,b^{5} d^{4} e^{2} x -\frac {128}{21} a \,b^{5} d^{3} e^{3} x^{2}+\frac {64}{21} a \,b^{5} d^{2} e^{4} x^{3}+\frac {30}{7} a^{2} b^{4} e^{6} x^{4}+\frac {80}{231} b^{6} d^{2} e^{4} x^{4}-\frac {40}{21} a \,b^{5} d \,e^{5} x^{4}-2 a^{6} e^{6}-\frac {2048}{231} b^{6} d^{6}}{\sqrt {e x +d}\, e^{7}} \] Input:
int((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^(3/2),x)
Output:
(2*( - 231*a**6*e**6 + 2772*a**5*b*d*e**5 + 1386*a**5*b*e**6*x - 9240*a**4 *b**2*d**2*e**4 - 4620*a**4*b**2*d*e**5*x + 1155*a**4*b**2*e**6*x**2 + 147 84*a**3*b**3*d**3*e**3 + 7392*a**3*b**3*d**2*e**4*x - 1848*a**3*b**3*d*e** 5*x**2 + 924*a**3*b**3*e**6*x**3 - 12672*a**2*b**4*d**4*e**2 - 6336*a**2*b **4*d**3*e**3*x + 1584*a**2*b**4*d**2*e**4*x**2 - 792*a**2*b**4*d*e**5*x** 3 + 495*a**2*b**4*e**6*x**4 + 5632*a*b**5*d**5*e + 2816*a*b**5*d**4*e**2*x - 704*a*b**5*d**3*e**3*x**2 + 352*a*b**5*d**2*e**4*x**3 - 220*a*b**5*d*e* *5*x**4 + 154*a*b**5*e**6*x**5 - 1024*b**6*d**6 - 512*b**6*d**5*e*x + 128* b**6*d**4*e**2*x**2 - 64*b**6*d**3*e**3*x**3 + 40*b**6*d**2*e**4*x**4 - 28 *b**6*d*e**5*x**5 + 21*b**6*e**6*x**6))/(231*sqrt(d + e*x)*e**7)