Integrand size = 33, antiderivative size = 362 \[ \int (a+b x) (d+e x)^7 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {(b d-a e)^6 (d+e x)^8 \sqrt {a^2+2 a b x+b^2 x^2}}{8 e^7 (a+b x)}-\frac {2 b (b d-a e)^5 (d+e x)^9 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x)}+\frac {3 b^2 (b d-a e)^4 (d+e x)^{10} \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^7 (a+b x)}-\frac {20 b^3 (b d-a e)^3 (d+e x)^{11} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x)}+\frac {5 b^4 (b d-a e)^2 (d+e x)^{12} \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^7 (a+b x)}-\frac {6 b^5 (b d-a e) (d+e x)^{13} \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^7 (a+b x)}+\frac {b^6 (d+e x)^{14} \sqrt {a^2+2 a b x+b^2 x^2}}{14 e^7 (a+b x)} \] Output:
1/8*(-a*e+b*d)^6*(e*x+d)^8*((b*x+a)^2)^(1/2)/e^7/(b*x+a)-2/3*b*(-a*e+b*d)^ 5*(e*x+d)^9*((b*x+a)^2)^(1/2)/e^7/(b*x+a)+3/2*b^2*(-a*e+b*d)^4*(e*x+d)^10* ((b*x+a)^2)^(1/2)/e^7/(b*x+a)-20/11*b^3*(-a*e+b*d)^3*(e*x+d)^11*((b*x+a)^2 )^(1/2)/e^7/(b*x+a)+5/4*b^4*(-a*e+b*d)^2*(e*x+d)^12*((b*x+a)^2)^(1/2)/e^7/ (b*x+a)-6/13*b^5*(-a*e+b*d)*(e*x+d)^13*((b*x+a)^2)^(1/2)/e^7/(b*x+a)+1/14* b^6*(e*x+d)^14*((b*x+a)^2)^(1/2)/e^7/(b*x+a)
Time = 1.23 (sec) , antiderivative size = 602, normalized size of antiderivative = 1.66 \[ \int (a+b x) (d+e x)^7 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {x \sqrt {(a+b x)^2} \left (3003 a^6 \left (8 d^7+28 d^6 e x+56 d^5 e^2 x^2+70 d^4 e^3 x^3+56 d^3 e^4 x^4+28 d^2 e^5 x^5+8 d e^6 x^6+e^7 x^7\right )+2002 a^5 b x \left (36 d^7+168 d^6 e x+378 d^5 e^2 x^2+504 d^4 e^3 x^3+420 d^3 e^4 x^4+216 d^2 e^5 x^5+63 d e^6 x^6+8 e^7 x^7\right )+1001 a^4 b^2 x^2 \left (120 d^7+630 d^6 e x+1512 d^5 e^2 x^2+2100 d^4 e^3 x^3+1800 d^3 e^4 x^4+945 d^2 e^5 x^5+280 d e^6 x^6+36 e^7 x^7\right )+364 a^3 b^3 x^3 \left (330 d^7+1848 d^6 e x+4620 d^5 e^2 x^2+6600 d^4 e^3 x^3+5775 d^3 e^4 x^4+3080 d^2 e^5 x^5+924 d e^6 x^6+120 e^7 x^7\right )+91 a^2 b^4 x^4 \left (792 d^7+4620 d^6 e x+11880 d^5 e^2 x^2+17325 d^4 e^3 x^3+15400 d^3 e^4 x^4+8316 d^2 e^5 x^5+2520 d e^6 x^6+330 e^7 x^7\right )+14 a b^5 x^5 \left (1716 d^7+10296 d^6 e x+27027 d^5 e^2 x^2+40040 d^4 e^3 x^3+36036 d^3 e^4 x^4+19656 d^2 e^5 x^5+6006 d e^6 x^6+792 e^7 x^7\right )+b^6 x^6 \left (3432 d^7+21021 d^6 e x+56056 d^5 e^2 x^2+84084 d^4 e^3 x^3+76440 d^3 e^4 x^4+42042 d^2 e^5 x^5+12936 d e^6 x^6+1716 e^7 x^7\right )\right )}{24024 (a+b x)} \] Input:
Integrate[(a + b*x)*(d + e*x)^7*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]
Output:
(x*Sqrt[(a + b*x)^2]*(3003*a^6*(8*d^7 + 28*d^6*e*x + 56*d^5*e^2*x^2 + 70*d ^4*e^3*x^3 + 56*d^3*e^4*x^4 + 28*d^2*e^5*x^5 + 8*d*e^6*x^6 + e^7*x^7) + 20 02*a^5*b*x*(36*d^7 + 168*d^6*e*x + 378*d^5*e^2*x^2 + 504*d^4*e^3*x^3 + 420 *d^3*e^4*x^4 + 216*d^2*e^5*x^5 + 63*d*e^6*x^6 + 8*e^7*x^7) + 1001*a^4*b^2* x^2*(120*d^7 + 630*d^6*e*x + 1512*d^5*e^2*x^2 + 2100*d^4*e^3*x^3 + 1800*d^ 3*e^4*x^4 + 945*d^2*e^5*x^5 + 280*d*e^6*x^6 + 36*e^7*x^7) + 364*a^3*b^3*x^ 3*(330*d^7 + 1848*d^6*e*x + 4620*d^5*e^2*x^2 + 6600*d^4*e^3*x^3 + 5775*d^3 *e^4*x^4 + 3080*d^2*e^5*x^5 + 924*d*e^6*x^6 + 120*e^7*x^7) + 91*a^2*b^4*x^ 4*(792*d^7 + 4620*d^6*e*x + 11880*d^5*e^2*x^2 + 17325*d^4*e^3*x^3 + 15400* d^3*e^4*x^4 + 8316*d^2*e^5*x^5 + 2520*d*e^6*x^6 + 330*e^7*x^7) + 14*a*b^5* x^5*(1716*d^7 + 10296*d^6*e*x + 27027*d^5*e^2*x^2 + 40040*d^4*e^3*x^3 + 36 036*d^3*e^4*x^4 + 19656*d^2*e^5*x^5 + 6006*d*e^6*x^6 + 792*e^7*x^7) + b^6* x^6*(3432*d^7 + 21021*d^6*e*x + 56056*d^5*e^2*x^2 + 84084*d^4*e^3*x^3 + 76 440*d^3*e^4*x^4 + 42042*d^2*e^5*x^5 + 12936*d*e^6*x^6 + 1716*e^7*x^7)))/(2 4024*(a + b*x))
Time = 1.07 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.56, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {1187, 27, 49, 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.
| \(\displaystyle \int (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} (d+e x)^7 \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int b^5 (a+b x)^6 (d+e x)^7dx}{b^5 (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int (a+b x)^6 (d+e x)^7dx}{a+b x}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {b^6 (d+e x)^{13}}{e^6}-\frac {6 b^5 (b d-a e) (d+e x)^{12}}{e^6}+\frac {15 b^4 (b d-a e)^2 (d+e x)^{11}}{e^6}-\frac {20 b^3 (b d-a e)^3 (d+e x)^{10}}{e^6}+\frac {15 b^2 (b d-a e)^4 (d+e x)^9}{e^6}-\frac {6 b (b d-a e)^5 (d+e x)^8}{e^6}+\frac {(a e-b d)^6 (d+e x)^7}{e^6}\right )dx}{a+b x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (-\frac {6 b^5 (d+e x)^{13} (b d-a e)}{13 e^7}+\frac {5 b^4 (d+e x)^{12} (b d-a e)^2}{4 e^7}-\frac {20 b^3 (d+e x)^{11} (b d-a e)^3}{11 e^7}+\frac {3 b^2 (d+e x)^{10} (b d-a e)^4}{2 e^7}-\frac {2 b (d+e x)^9 (b d-a e)^5}{3 e^7}+\frac {(d+e x)^8 (b d-a e)^6}{8 e^7}+\frac {b^6 (d+e x)^{14}}{14 e^7}\right )}{a+b x}\) |
Input:
Int[(a + b*x)*(d + e*x)^7*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]
Output:
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(((b*d - a*e)^6*(d + e*x)^8)/(8*e^7) - (2*b *(b*d - a*e)^5*(d + e*x)^9)/(3*e^7) + (3*b^2*(b*d - a*e)^4*(d + e*x)^10)/( 2*e^7) - (20*b^3*(b*d - a*e)^3*(d + e*x)^11)/(11*e^7) + (5*b^4*(b*d - a*e) ^2*(d + e*x)^12)/(4*e^7) - (6*b^5*(b*d - a*e)*(d + e*x)^13)/(13*e^7) + (b^ 6*(d + e*x)^14)/(14*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}, x] && IGtQ[m, 0] && IGtQ[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]
Leaf count of result is larger than twice the leaf count of optimal. \(815\) vs. \(2(271)=542\).
Time = 2.10 (sec) , antiderivative size = 816, normalized size of antiderivative = 2.25
| method | result | size |
| gosper | \(\frac {x \left (1716 b^{6} e^{7} x^{13}+11088 x^{12} a \,b^{5} e^{7}+12936 x^{12} b^{6} d \,e^{6}+30030 x^{11} a^{2} b^{4} e^{7}+84084 x^{11} a \,b^{5} d \,e^{6}+42042 x^{11} b^{6} d^{2} e^{5}+43680 x^{10} a^{3} b^{3} e^{7}+229320 x^{10} a^{2} b^{4} d \,e^{6}+275184 x^{10} a \,b^{5} d^{2} e^{5}+76440 x^{10} b^{6} d^{3} e^{4}+36036 x^{9} a^{4} b^{2} e^{7}+336336 x^{9} a^{3} b^{3} d \,e^{6}+756756 x^{9} a^{2} b^{4} d^{2} e^{5}+504504 x^{9} a \,b^{5} d^{3} e^{4}+84084 x^{9} b^{6} e^{3} d^{4}+16016 x^{8} a^{5} b \,e^{7}+280280 x^{8} a^{4} b^{2} d \,e^{6}+1121120 x^{8} a^{3} b^{3} d^{2} e^{5}+1401400 x^{8} a^{2} b^{4} d^{3} e^{4}+560560 x^{8} a \,b^{5} e^{3} d^{4}+56056 x^{8} b^{6} d^{5} e^{2}+3003 x^{7} a^{6} e^{7}+126126 x^{7} a^{5} b d \,e^{6}+945945 x^{7} a^{4} b^{2} d^{2} e^{5}+2102100 x^{7} a^{3} b^{3} d^{3} e^{4}+1576575 x^{7} a^{2} b^{4} e^{3} d^{4}+378378 x^{7} a \,b^{5} d^{5} e^{2}+21021 x^{7} b^{6} d^{6} e +24024 x^{6} a^{6} d \,e^{6}+432432 x^{6} a^{5} b \,d^{2} e^{5}+1801800 x^{6} a^{4} b^{2} d^{3} e^{4}+2402400 x^{6} a^{3} b^{3} e^{3} d^{4}+1081080 x^{6} a^{2} b^{4} d^{5} e^{2}+144144 x^{6} a \,b^{5} d^{6} e +3432 x^{6} b^{6} d^{7}+84084 x^{5} a^{6} d^{2} e^{5}+840840 x^{5} a^{5} b \,d^{3} e^{4}+2102100 x^{5} a^{4} b^{2} e^{3} d^{4}+1681680 x^{5} a^{3} b^{3} d^{5} e^{2}+420420 x^{5} a^{2} b^{4} d^{6} e +24024 x^{5} a \,b^{5} d^{7}+168168 a^{6} d^{3} e^{4} x^{4}+1009008 a^{5} b \,d^{4} e^{3} x^{4}+1513512 a^{4} b^{2} d^{5} e^{2} x^{4}+672672 a^{3} b^{3} d^{6} e \,x^{4}+72072 a^{2} b^{4} d^{7} x^{4}+210210 x^{3} a^{6} e^{3} d^{4}+756756 x^{3} a^{5} b \,d^{5} e^{2}+630630 x^{3} a^{4} b^{2} d^{6} e +120120 x^{3} a^{3} b^{3} d^{7}+168168 a^{6} d^{5} e^{2} x^{2}+336336 a^{5} b \,d^{6} e \,x^{2}+120120 a^{4} b^{2} d^{7} x^{2}+84084 x \,a^{6} d^{6} e +72072 x \,a^{5} b \,d^{7}+24024 a^{6} d^{7}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{24024 \left (b x +a \right )^{5}}\) | \(816\) |
| default | \(\frac {x \left (1716 b^{6} e^{7} x^{13}+11088 x^{12} a \,b^{5} e^{7}+12936 x^{12} b^{6} d \,e^{6}+30030 x^{11} a^{2} b^{4} e^{7}+84084 x^{11} a \,b^{5} d \,e^{6}+42042 x^{11} b^{6} d^{2} e^{5}+43680 x^{10} a^{3} b^{3} e^{7}+229320 x^{10} a^{2} b^{4} d \,e^{6}+275184 x^{10} a \,b^{5} d^{2} e^{5}+76440 x^{10} b^{6} d^{3} e^{4}+36036 x^{9} a^{4} b^{2} e^{7}+336336 x^{9} a^{3} b^{3} d \,e^{6}+756756 x^{9} a^{2} b^{4} d^{2} e^{5}+504504 x^{9} a \,b^{5} d^{3} e^{4}+84084 x^{9} b^{6} e^{3} d^{4}+16016 x^{8} a^{5} b \,e^{7}+280280 x^{8} a^{4} b^{2} d \,e^{6}+1121120 x^{8} a^{3} b^{3} d^{2} e^{5}+1401400 x^{8} a^{2} b^{4} d^{3} e^{4}+560560 x^{8} a \,b^{5} e^{3} d^{4}+56056 x^{8} b^{6} d^{5} e^{2}+3003 x^{7} a^{6} e^{7}+126126 x^{7} a^{5} b d \,e^{6}+945945 x^{7} a^{4} b^{2} d^{2} e^{5}+2102100 x^{7} a^{3} b^{3} d^{3} e^{4}+1576575 x^{7} a^{2} b^{4} e^{3} d^{4}+378378 x^{7} a \,b^{5} d^{5} e^{2}+21021 x^{7} b^{6} d^{6} e +24024 x^{6} a^{6} d \,e^{6}+432432 x^{6} a^{5} b \,d^{2} e^{5}+1801800 x^{6} a^{4} b^{2} d^{3} e^{4}+2402400 x^{6} a^{3} b^{3} e^{3} d^{4}+1081080 x^{6} a^{2} b^{4} d^{5} e^{2}+144144 x^{6} a \,b^{5} d^{6} e +3432 x^{6} b^{6} d^{7}+84084 x^{5} a^{6} d^{2} e^{5}+840840 x^{5} a^{5} b \,d^{3} e^{4}+2102100 x^{5} a^{4} b^{2} e^{3} d^{4}+1681680 x^{5} a^{3} b^{3} d^{5} e^{2}+420420 x^{5} a^{2} b^{4} d^{6} e +24024 x^{5} a \,b^{5} d^{7}+168168 a^{6} d^{3} e^{4} x^{4}+1009008 a^{5} b \,d^{4} e^{3} x^{4}+1513512 a^{4} b^{2} d^{5} e^{2} x^{4}+672672 a^{3} b^{3} d^{6} e \,x^{4}+72072 a^{2} b^{4} d^{7} x^{4}+210210 x^{3} a^{6} e^{3} d^{4}+756756 x^{3} a^{5} b \,d^{5} e^{2}+630630 x^{3} a^{4} b^{2} d^{6} e +120120 x^{3} a^{3} b^{3} d^{7}+168168 a^{6} d^{5} e^{2} x^{2}+336336 a^{5} b \,d^{6} e \,x^{2}+120120 a^{4} b^{2} d^{7} x^{2}+84084 x \,a^{6} d^{6} e +72072 x \,a^{5} b \,d^{7}+24024 a^{6} d^{7}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{24024 \left (b x +a \right )^{5}}\) | \(816\) |
| orering | \(\frac {x \left (1716 b^{6} e^{7} x^{13}+11088 x^{12} a \,b^{5} e^{7}+12936 x^{12} b^{6} d \,e^{6}+30030 x^{11} a^{2} b^{4} e^{7}+84084 x^{11} a \,b^{5} d \,e^{6}+42042 x^{11} b^{6} d^{2} e^{5}+43680 x^{10} a^{3} b^{3} e^{7}+229320 x^{10} a^{2} b^{4} d \,e^{6}+275184 x^{10} a \,b^{5} d^{2} e^{5}+76440 x^{10} b^{6} d^{3} e^{4}+36036 x^{9} a^{4} b^{2} e^{7}+336336 x^{9} a^{3} b^{3} d \,e^{6}+756756 x^{9} a^{2} b^{4} d^{2} e^{5}+504504 x^{9} a \,b^{5} d^{3} e^{4}+84084 x^{9} b^{6} e^{3} d^{4}+16016 x^{8} a^{5} b \,e^{7}+280280 x^{8} a^{4} b^{2} d \,e^{6}+1121120 x^{8} a^{3} b^{3} d^{2} e^{5}+1401400 x^{8} a^{2} b^{4} d^{3} e^{4}+560560 x^{8} a \,b^{5} e^{3} d^{4}+56056 x^{8} b^{6} d^{5} e^{2}+3003 x^{7} a^{6} e^{7}+126126 x^{7} a^{5} b d \,e^{6}+945945 x^{7} a^{4} b^{2} d^{2} e^{5}+2102100 x^{7} a^{3} b^{3} d^{3} e^{4}+1576575 x^{7} a^{2} b^{4} e^{3} d^{4}+378378 x^{7} a \,b^{5} d^{5} e^{2}+21021 x^{7} b^{6} d^{6} e +24024 x^{6} a^{6} d \,e^{6}+432432 x^{6} a^{5} b \,d^{2} e^{5}+1801800 x^{6} a^{4} b^{2} d^{3} e^{4}+2402400 x^{6} a^{3} b^{3} e^{3} d^{4}+1081080 x^{6} a^{2} b^{4} d^{5} e^{2}+144144 x^{6} a \,b^{5} d^{6} e +3432 x^{6} b^{6} d^{7}+84084 x^{5} a^{6} d^{2} e^{5}+840840 x^{5} a^{5} b \,d^{3} e^{4}+2102100 x^{5} a^{4} b^{2} e^{3} d^{4}+1681680 x^{5} a^{3} b^{3} d^{5} e^{2}+420420 x^{5} a^{2} b^{4} d^{6} e +24024 x^{5} a \,b^{5} d^{7}+168168 a^{6} d^{3} e^{4} x^{4}+1009008 a^{5} b \,d^{4} e^{3} x^{4}+1513512 a^{4} b^{2} d^{5} e^{2} x^{4}+672672 a^{3} b^{3} d^{6} e \,x^{4}+72072 a^{2} b^{4} d^{7} x^{4}+210210 x^{3} a^{6} e^{3} d^{4}+756756 x^{3} a^{5} b \,d^{5} e^{2}+630630 x^{3} a^{4} b^{2} d^{6} e +120120 x^{3} a^{3} b^{3} d^{7}+168168 a^{6} d^{5} e^{2} x^{2}+336336 a^{5} b \,d^{6} e \,x^{2}+120120 a^{4} b^{2} d^{7} x^{2}+84084 x \,a^{6} d^{6} e +72072 x \,a^{5} b \,d^{7}+24024 a^{6} d^{7}\right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{\frac {5}{2}}}{24024 \left (b x +a \right )^{5}}\) | \(825\) |
| risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, b^{6} e^{7} x^{14}}{14 b x +14 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (6 a \,b^{5} e^{7}+7 b^{6} d \,e^{6}\right ) x^{13}}{13 b x +13 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (15 a^{2} b^{4} e^{7}+42 a \,b^{5} d \,e^{6}+21 b^{6} d^{2} e^{5}\right ) x^{12}}{12 b x +12 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (20 a^{3} b^{3} e^{7}+105 a^{2} b^{4} d \,e^{6}+126 a \,b^{5} d^{2} e^{5}+35 b^{6} d^{3} e^{4}\right ) x^{11}}{11 b x +11 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (15 a^{4} b^{2} e^{7}+140 a^{3} b^{3} d \,e^{6}+315 a^{2} b^{4} d^{2} e^{5}+210 a \,b^{5} d^{3} e^{4}+35 b^{6} e^{3} d^{4}\right ) x^{10}}{10 b x +10 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (6 a^{5} b \,e^{7}+105 a^{4} b^{2} d \,e^{6}+420 a^{3} b^{3} d^{2} e^{5}+525 a^{2} b^{4} d^{3} e^{4}+210 a \,b^{5} e^{3} d^{4}+21 b^{6} d^{5} e^{2}\right ) x^{9}}{9 b x +9 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (a^{6} e^{7}+42 a^{5} b d \,e^{6}+315 a^{4} b^{2} d^{2} e^{5}+700 a^{3} b^{3} d^{3} e^{4}+525 a^{2} b^{4} e^{3} d^{4}+126 a \,b^{5} d^{5} e^{2}+7 b^{6} d^{6} e \right ) x^{8}}{8 b x +8 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (7 a^{6} d \,e^{6}+126 a^{5} b \,d^{2} e^{5}+525 a^{4} b^{2} d^{3} e^{4}+700 a^{3} b^{3} e^{3} d^{4}+315 a^{2} b^{4} d^{5} e^{2}+42 a \,b^{5} d^{6} e +b^{6} d^{7}\right ) x^{7}}{7 b x +7 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (21 a^{6} d^{2} e^{5}+210 a^{5} b \,d^{3} e^{4}+525 a^{4} b^{2} e^{3} d^{4}+420 a^{3} b^{3} d^{5} e^{2}+105 a^{2} b^{4} d^{6} e +6 a \,b^{5} d^{7}\right ) x^{6}}{6 b x +6 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (35 a^{6} d^{3} e^{4}+210 a^{5} b \,e^{3} d^{4}+315 a^{4} b^{2} d^{5} e^{2}+140 a^{3} b^{3} d^{6} e +15 a^{2} b^{4} d^{7}\right ) x^{5}}{5 b x +5 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (35 a^{6} e^{3} d^{4}+126 a^{5} b \,d^{5} e^{2}+105 a^{4} b^{2} d^{6} e +20 a^{3} b^{3} d^{7}\right ) x^{4}}{4 b x +4 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (21 a^{6} d^{5} e^{2}+42 a^{5} b \,d^{6} e +15 a^{4} b^{2} d^{7}\right ) x^{3}}{3 b x +3 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (7 a^{6} d^{6} e +6 a^{5} b \,d^{7}\right ) x^{2}}{2 b x +2 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, a^{6} d^{7} x}{b x +a}\) | \(933\) |
Input:
int((b*x+a)*(e*x+d)^7*(b^2*x^2+2*a*b*x+a^2)^(5/2),x,method=_RETURNVERBOSE)
Output:
1/24024*x*(1716*b^6*e^7*x^13+11088*a*b^5*e^7*x^12+12936*b^6*d*e^6*x^12+300 30*a^2*b^4*e^7*x^11+84084*a*b^5*d*e^6*x^11+42042*b^6*d^2*e^5*x^11+43680*a^ 3*b^3*e^7*x^10+229320*a^2*b^4*d*e^6*x^10+275184*a*b^5*d^2*e^5*x^10+76440*b ^6*d^3*e^4*x^10+36036*a^4*b^2*e^7*x^9+336336*a^3*b^3*d*e^6*x^9+756756*a^2* b^4*d^2*e^5*x^9+504504*a*b^5*d^3*e^4*x^9+84084*b^6*d^4*e^3*x^9+16016*a^5*b *e^7*x^8+280280*a^4*b^2*d*e^6*x^8+1121120*a^3*b^3*d^2*e^5*x^8+1401400*a^2* b^4*d^3*e^4*x^8+560560*a*b^5*d^4*e^3*x^8+56056*b^6*d^5*e^2*x^8+3003*a^6*e^ 7*x^7+126126*a^5*b*d*e^6*x^7+945945*a^4*b^2*d^2*e^5*x^7+2102100*a^3*b^3*d^ 3*e^4*x^7+1576575*a^2*b^4*d^4*e^3*x^7+378378*a*b^5*d^5*e^2*x^7+21021*b^6*d ^6*e*x^7+24024*a^6*d*e^6*x^6+432432*a^5*b*d^2*e^5*x^6+1801800*a^4*b^2*d^3* e^4*x^6+2402400*a^3*b^3*d^4*e^3*x^6+1081080*a^2*b^4*d^5*e^2*x^6+144144*a*b ^5*d^6*e*x^6+3432*b^6*d^7*x^6+84084*a^6*d^2*e^5*x^5+840840*a^5*b*d^3*e^4*x ^5+2102100*a^4*b^2*d^4*e^3*x^5+1681680*a^3*b^3*d^5*e^2*x^5+420420*a^2*b^4* d^6*e*x^5+24024*a*b^5*d^7*x^5+168168*a^6*d^3*e^4*x^4+1009008*a^5*b*d^4*e^3 *x^4+1513512*a^4*b^2*d^5*e^2*x^4+672672*a^3*b^3*d^6*e*x^4+72072*a^2*b^4*d^ 7*x^4+210210*a^6*d^4*e^3*x^3+756756*a^5*b*d^5*e^2*x^3+630630*a^4*b^2*d^6*e *x^3+120120*a^3*b^3*d^7*x^3+168168*a^6*d^5*e^2*x^2+336336*a^5*b*d^6*e*x^2+ 120120*a^4*b^2*d^7*x^2+84084*a^6*d^6*e*x+72072*a^5*b*d^7*x+24024*a^6*d^7)* ((b*x+a)^2)^(5/2)/(b*x+a)^5
Leaf count of result is larger than twice the leaf count of optimal. 706 vs. \(2 (271) = 542\).
Time = 0.07 (sec) , antiderivative size = 706, normalized size of antiderivative = 1.95 \[ \int (a+b x) (d+e x)^7 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {1}{14} \, b^{6} e^{7} x^{14} + a^{6} d^{7} x + \frac {1}{13} \, {\left (7 \, b^{6} d e^{6} + 6 \, a b^{5} e^{7}\right )} x^{13} + \frac {1}{4} \, {\left (7 \, b^{6} d^{2} e^{5} + 14 \, a b^{5} d e^{6} + 5 \, a^{2} b^{4} e^{7}\right )} x^{12} + \frac {1}{11} \, {\left (35 \, b^{6} d^{3} e^{4} + 126 \, a b^{5} d^{2} e^{5} + 105 \, a^{2} b^{4} d e^{6} + 20 \, a^{3} b^{3} e^{7}\right )} x^{11} + \frac {1}{2} \, {\left (7 \, b^{6} d^{4} e^{3} + 42 \, a b^{5} d^{3} e^{4} + 63 \, a^{2} b^{4} d^{2} e^{5} + 28 \, a^{3} b^{3} d e^{6} + 3 \, a^{4} b^{2} e^{7}\right )} x^{10} + \frac {1}{3} \, {\left (7 \, b^{6} d^{5} e^{2} + 70 \, a b^{5} d^{4} e^{3} + 175 \, a^{2} b^{4} d^{3} e^{4} + 140 \, a^{3} b^{3} d^{2} e^{5} + 35 \, a^{4} b^{2} d e^{6} + 2 \, a^{5} b e^{7}\right )} x^{9} + \frac {1}{8} \, {\left (7 \, b^{6} d^{6} e + 126 \, a b^{5} d^{5} e^{2} + 525 \, a^{2} b^{4} d^{4} e^{3} + 700 \, a^{3} b^{3} d^{3} e^{4} + 315 \, a^{4} b^{2} d^{2} e^{5} + 42 \, a^{5} b d e^{6} + a^{6} e^{7}\right )} x^{8} + \frac {1}{7} \, {\left (b^{6} d^{7} + 42 \, a b^{5} d^{6} e + 315 \, a^{2} b^{4} d^{5} e^{2} + 700 \, a^{3} b^{3} d^{4} e^{3} + 525 \, a^{4} b^{2} d^{3} e^{4} + 126 \, a^{5} b d^{2} e^{5} + 7 \, a^{6} d e^{6}\right )} x^{7} + \frac {1}{2} \, {\left (2 \, a b^{5} d^{7} + 35 \, a^{2} b^{4} d^{6} e + 140 \, a^{3} b^{3} d^{5} e^{2} + 175 \, a^{4} b^{2} d^{4} e^{3} + 70 \, a^{5} b d^{3} e^{4} + 7 \, a^{6} d^{2} e^{5}\right )} x^{6} + {\left (3 \, a^{2} b^{4} d^{7} + 28 \, a^{3} b^{3} d^{6} e + 63 \, a^{4} b^{2} d^{5} e^{2} + 42 \, a^{5} b d^{4} e^{3} + 7 \, a^{6} d^{3} e^{4}\right )} x^{5} + \frac {1}{4} \, {\left (20 \, a^{3} b^{3} d^{7} + 105 \, a^{4} b^{2} d^{6} e + 126 \, a^{5} b d^{5} e^{2} + 35 \, a^{6} d^{4} e^{3}\right )} x^{4} + {\left (5 \, a^{4} b^{2} d^{7} + 14 \, a^{5} b d^{6} e + 7 \, a^{6} d^{5} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (6 \, a^{5} b d^{7} + 7 \, a^{6} d^{6} e\right )} x^{2} \] Input:
integrate((b*x+a)*(e*x+d)^7*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fric as")
Output:
1/14*b^6*e^7*x^14 + a^6*d^7*x + 1/13*(7*b^6*d*e^6 + 6*a*b^5*e^7)*x^13 + 1/ 4*(7*b^6*d^2*e^5 + 14*a*b^5*d*e^6 + 5*a^2*b^4*e^7)*x^12 + 1/11*(35*b^6*d^3 *e^4 + 126*a*b^5*d^2*e^5 + 105*a^2*b^4*d*e^6 + 20*a^3*b^3*e^7)*x^11 + 1/2* (7*b^6*d^4*e^3 + 42*a*b^5*d^3*e^4 + 63*a^2*b^4*d^2*e^5 + 28*a^3*b^3*d*e^6 + 3*a^4*b^2*e^7)*x^10 + 1/3*(7*b^6*d^5*e^2 + 70*a*b^5*d^4*e^3 + 175*a^2*b^ 4*d^3*e^4 + 140*a^3*b^3*d^2*e^5 + 35*a^4*b^2*d*e^6 + 2*a^5*b*e^7)*x^9 + 1/ 8*(7*b^6*d^6*e + 126*a*b^5*d^5*e^2 + 525*a^2*b^4*d^4*e^3 + 700*a^3*b^3*d^3 *e^4 + 315*a^4*b^2*d^2*e^5 + 42*a^5*b*d*e^6 + a^6*e^7)*x^8 + 1/7*(b^6*d^7 + 42*a*b^5*d^6*e + 315*a^2*b^4*d^5*e^2 + 700*a^3*b^3*d^4*e^3 + 525*a^4*b^2 *d^3*e^4 + 126*a^5*b*d^2*e^5 + 7*a^6*d*e^6)*x^7 + 1/2*(2*a*b^5*d^7 + 35*a^ 2*b^4*d^6*e + 140*a^3*b^3*d^5*e^2 + 175*a^4*b^2*d^4*e^3 + 70*a^5*b*d^3*e^4 + 7*a^6*d^2*e^5)*x^6 + (3*a^2*b^4*d^7 + 28*a^3*b^3*d^6*e + 63*a^4*b^2*d^5 *e^2 + 42*a^5*b*d^4*e^3 + 7*a^6*d^3*e^4)*x^5 + 1/4*(20*a^3*b^3*d^7 + 105*a ^4*b^2*d^6*e + 126*a^5*b*d^5*e^2 + 35*a^6*d^4*e^3)*x^4 + (5*a^4*b^2*d^7 + 14*a^5*b*d^6*e + 7*a^6*d^5*e^2)*x^3 + 1/2*(6*a^5*b*d^7 + 7*a^6*d^6*e)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 132085 vs. \(2 (265) = 530\).
Time = 2.26 (sec) , antiderivative size = 132085, normalized size of antiderivative = 364.88 \[ \int (a+b x) (d+e x)^7 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)*(e*x+d)**7*(b**2*x**2+2*a*b*x+a**2)**(5/2),x)
Output:
Piecewise((sqrt(a**2 + 2*a*b*x + b**2*x**2)*(b**5*e**7*x**13/14 + x**12*(7 1*a*b**6*e**7/14 + 7*b**7*d*e**6)/(13*b**2) + x**11*(281*a**2*b**5*e**7/14 + 49*a*b**6*d*e**6 - 25*a*(71*a*b**6*e**7/14 + 7*b**7*d*e**6)/(13*b) + 21 *b**7*d**2*e**5)/(12*b**2) + x**10*(35*a**3*b**4*e**7 + 147*a**2*b**5*d*e* *6 - 12*a**2*(71*a*b**6*e**7/14 + 7*b**7*d*e**6)/(13*b**2) + 147*a*b**6*d* *2*e**5 - 23*a*(281*a**2*b**5*e**7/14 + 49*a*b**6*d*e**6 - 25*a*(71*a*b**6 *e**7/14 + 7*b**7*d*e**6)/(13*b) + 21*b**7*d**2*e**5)/(12*b) + 35*b**7*d** 3*e**4)/(11*b**2) + x**9*(35*a**4*b**3*e**7 + 245*a**3*b**4*d*e**6 + 441*a **2*b**5*d**2*e**5 - 11*a**2*(281*a**2*b**5*e**7/14 + 49*a*b**6*d*e**6 - 2 5*a*(71*a*b**6*e**7/14 + 7*b**7*d*e**6)/(13*b) + 21*b**7*d**2*e**5)/(12*b* *2) + 245*a*b**6*d**3*e**4 - 21*a*(35*a**3*b**4*e**7 + 147*a**2*b**5*d*e** 6 - 12*a**2*(71*a*b**6*e**7/14 + 7*b**7*d*e**6)/(13*b**2) + 147*a*b**6*d** 2*e**5 - 23*a*(281*a**2*b**5*e**7/14 + 49*a*b**6*d*e**6 - 25*a*(71*a*b**6* e**7/14 + 7*b**7*d*e**6)/(13*b) + 21*b**7*d**2*e**5)/(12*b) + 35*b**7*d**3 *e**4)/(11*b) + 35*b**7*d**4*e**3)/(10*b**2) + x**8*(21*a**5*b**2*e**7 + 2 45*a**4*b**3*d*e**6 + 735*a**3*b**4*d**2*e**5 + 735*a**2*b**5*d**3*e**4 - 10*a**2*(35*a**3*b**4*e**7 + 147*a**2*b**5*d*e**6 - 12*a**2*(71*a*b**6*e** 7/14 + 7*b**7*d*e**6)/(13*b**2) + 147*a*b**6*d**2*e**5 - 23*a*(281*a**2*b* *5*e**7/14 + 49*a*b**6*d*e**6 - 25*a*(71*a*b**6*e**7/14 + 7*b**7*d*e**6)/( 13*b) + 21*b**7*d**2*e**5)/(12*b) + 35*b**7*d**3*e**4)/(11*b**2) + 245*...
Leaf count of result is larger than twice the leaf count of optimal. 2153 vs. \(2 (271) = 542\).
Time = 0.05 (sec) , antiderivative size = 2153, normalized size of antiderivative = 5.95 \[ \int (a+b x) (d+e x)^7 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)*(e*x+d)^7*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="maxi ma")
Output:
1/14*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*e^7*x^7/b - 3/26*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a*e^7*x^6/b^2 + 11/78*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^2*e^7* x^5/b^3 - 133/858*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^3*e^7*x^4/b^4 + 139/85 8*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^4*e^7*x^3/b^5 + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a*d^7*x + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^8*e^7*x/b^7 - 425/2574*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^5*e^7*x^2/b^6 + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2*d^7/b + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^9 *e^7/b^8 + 214/1287*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^6*e^7*x/b^7 - 1501/9 009*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^7*e^7/b^8 + 1/13*(7*b*d*e^6 + a*e^7) *(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*x^6/b^2 - 19/156*(7*b*d*e^6 + a*e^7)*(b^2 *x^2 + 2*a*b*x + a^2)^(7/2)*a*x^5/b^3 + 7/12*(3*b*d^2*e^5 + a*d*e^6)*(b^2* x^2 + 2*a*b*x + a^2)^(7/2)*x^5/b^2 + 251/1716*(7*b*d*e^6 + a*e^7)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^2*x^4/b^4 - 119/132*(3*b*d^2*e^5 + a*d*e^6)*(b^2 *x^2 + 2*a*b*x + a^2)^(7/2)*a*x^4/b^3 + 7/11*(5*b*d^3*e^4 + 3*a*d^2*e^5)*( b^2*x^2 + 2*a*b*x + a^2)^(7/2)*x^4/b^2 - 68/429*(7*b*d*e^6 + a*e^7)*(b^2*x ^2 + 2*a*b*x + a^2)^(7/2)*a^3*x^3/b^5 + 35/33*(3*b*d^2*e^5 + a*d*e^6)*(b^2 *x^2 + 2*a*b*x + a^2)^(7/2)*a^2*x^3/b^4 - 21/22*(5*b*d^3*e^4 + 3*a*d^2*e^5 )*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a*x^3/b^3 + 7/2*(b*d^4*e^3 + a*d^3*e^4)* (b^2*x^2 + 2*a*b*x + a^2)^(7/2)*x^3/b^2 - 1/6*(7*b*d*e^6 + a*e^7)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^7*x/b^7 + 7/6*(3*b*d^2*e^5 + a*d*e^6)*(b^2*x^...
Leaf count of result is larger than twice the leaf count of optimal. 1243 vs. \(2 (271) = 542\).
Time = 0.19 (sec) , antiderivative size = 1243, normalized size of antiderivative = 3.43 \[ \int (a+b x) (d+e x)^7 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)*(e*x+d)^7*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac ")
Output:
1/14*b^6*e^7*x^14*sgn(b*x + a) + 7/13*b^6*d*e^6*x^13*sgn(b*x + a) + 6/13*a *b^5*e^7*x^13*sgn(b*x + a) + 7/4*b^6*d^2*e^5*x^12*sgn(b*x + a) + 7/2*a*b^5 *d*e^6*x^12*sgn(b*x + a) + 5/4*a^2*b^4*e^7*x^12*sgn(b*x + a) + 35/11*b^6*d ^3*e^4*x^11*sgn(b*x + a) + 126/11*a*b^5*d^2*e^5*x^11*sgn(b*x + a) + 105/11 *a^2*b^4*d*e^6*x^11*sgn(b*x + a) + 20/11*a^3*b^3*e^7*x^11*sgn(b*x + a) + 7 /2*b^6*d^4*e^3*x^10*sgn(b*x + a) + 21*a*b^5*d^3*e^4*x^10*sgn(b*x + a) + 63 /2*a^2*b^4*d^2*e^5*x^10*sgn(b*x + a) + 14*a^3*b^3*d*e^6*x^10*sgn(b*x + a) + 3/2*a^4*b^2*e^7*x^10*sgn(b*x + a) + 7/3*b^6*d^5*e^2*x^9*sgn(b*x + a) + 7 0/3*a*b^5*d^4*e^3*x^9*sgn(b*x + a) + 175/3*a^2*b^4*d^3*e^4*x^9*sgn(b*x + a ) + 140/3*a^3*b^3*d^2*e^5*x^9*sgn(b*x + a) + 35/3*a^4*b^2*d*e^6*x^9*sgn(b* x + a) + 2/3*a^5*b*e^7*x^9*sgn(b*x + a) + 7/8*b^6*d^6*e*x^8*sgn(b*x + a) + 63/4*a*b^5*d^5*e^2*x^8*sgn(b*x + a) + 525/8*a^2*b^4*d^4*e^3*x^8*sgn(b*x + a) + 175/2*a^3*b^3*d^3*e^4*x^8*sgn(b*x + a) + 315/8*a^4*b^2*d^2*e^5*x^8*s gn(b*x + a) + 21/4*a^5*b*d*e^6*x^8*sgn(b*x + a) + 1/8*a^6*e^7*x^8*sgn(b*x + a) + 1/7*b^6*d^7*x^7*sgn(b*x + a) + 6*a*b^5*d^6*e*x^7*sgn(b*x + a) + 45* a^2*b^4*d^5*e^2*x^7*sgn(b*x + a) + 100*a^3*b^3*d^4*e^3*x^7*sgn(b*x + a) + 75*a^4*b^2*d^3*e^4*x^7*sgn(b*x + a) + 18*a^5*b*d^2*e^5*x^7*sgn(b*x + a) + a^6*d*e^6*x^7*sgn(b*x + a) + a*b^5*d^7*x^6*sgn(b*x + a) + 35/2*a^2*b^4*d^6 *e*x^6*sgn(b*x + a) + 70*a^3*b^3*d^5*e^2*x^6*sgn(b*x + a) + 175/2*a^4*b^2* d^4*e^3*x^6*sgn(b*x + a) + 35*a^5*b*d^3*e^4*x^6*sgn(b*x + a) + 7/2*a^6*...
Timed out. \[ \int (a+b x) (d+e x)^7 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\int \left (a+b\,x\right )\,{\left (d+e\,x\right )}^7\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2} \,d x \] Input:
int((a + b*x)*(d + e*x)^7*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2),x)
Output:
int((a + b*x)*(d + e*x)^7*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2), x)
Time = 0.25 (sec) , antiderivative size = 799, normalized size of antiderivative = 2.21 \[ \int (a+b x) (d+e x)^7 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {x \left (1716 b^{6} e^{7} x^{13}+11088 a \,b^{5} e^{7} x^{12}+12936 b^{6} d \,e^{6} x^{12}+30030 a^{2} b^{4} e^{7} x^{11}+84084 a \,b^{5} d \,e^{6} x^{11}+42042 b^{6} d^{2} e^{5} x^{11}+43680 a^{3} b^{3} e^{7} x^{10}+229320 a^{2} b^{4} d \,e^{6} x^{10}+275184 a \,b^{5} d^{2} e^{5} x^{10}+76440 b^{6} d^{3} e^{4} x^{10}+36036 a^{4} b^{2} e^{7} x^{9}+336336 a^{3} b^{3} d \,e^{6} x^{9}+756756 a^{2} b^{4} d^{2} e^{5} x^{9}+504504 a \,b^{5} d^{3} e^{4} x^{9}+84084 b^{6} d^{4} e^{3} x^{9}+16016 a^{5} b \,e^{7} x^{8}+280280 a^{4} b^{2} d \,e^{6} x^{8}+1121120 a^{3} b^{3} d^{2} e^{5} x^{8}+1401400 a^{2} b^{4} d^{3} e^{4} x^{8}+560560 a \,b^{5} d^{4} e^{3} x^{8}+56056 b^{6} d^{5} e^{2} x^{8}+3003 a^{6} e^{7} x^{7}+126126 a^{5} b d \,e^{6} x^{7}+945945 a^{4} b^{2} d^{2} e^{5} x^{7}+2102100 a^{3} b^{3} d^{3} e^{4} x^{7}+1576575 a^{2} b^{4} d^{4} e^{3} x^{7}+378378 a \,b^{5} d^{5} e^{2} x^{7}+21021 b^{6} d^{6} e \,x^{7}+24024 a^{6} d \,e^{6} x^{6}+432432 a^{5} b \,d^{2} e^{5} x^{6}+1801800 a^{4} b^{2} d^{3} e^{4} x^{6}+2402400 a^{3} b^{3} d^{4} e^{3} x^{6}+1081080 a^{2} b^{4} d^{5} e^{2} x^{6}+144144 a \,b^{5} d^{6} e \,x^{6}+3432 b^{6} d^{7} x^{6}+84084 a^{6} d^{2} e^{5} x^{5}+840840 a^{5} b \,d^{3} e^{4} x^{5}+2102100 a^{4} b^{2} d^{4} e^{3} x^{5}+1681680 a^{3} b^{3} d^{5} e^{2} x^{5}+420420 a^{2} b^{4} d^{6} e \,x^{5}+24024 a \,b^{5} d^{7} x^{5}+168168 a^{6} d^{3} e^{4} x^{4}+1009008 a^{5} b \,d^{4} e^{3} x^{4}+1513512 a^{4} b^{2} d^{5} e^{2} x^{4}+672672 a^{3} b^{3} d^{6} e \,x^{4}+72072 a^{2} b^{4} d^{7} x^{4}+210210 a^{6} d^{4} e^{3} x^{3}+756756 a^{5} b \,d^{5} e^{2} x^{3}+630630 a^{4} b^{2} d^{6} e \,x^{3}+120120 a^{3} b^{3} d^{7} x^{3}+168168 a^{6} d^{5} e^{2} x^{2}+336336 a^{5} b \,d^{6} e \,x^{2}+120120 a^{4} b^{2} d^{7} x^{2}+84084 a^{6} d^{6} e x +72072 a^{5} b \,d^{7} x +24024 a^{6} d^{7}\right )}{24024} \] Input:
int((b*x+a)*(e*x+d)^7*(b^2*x^2+2*a*b*x+a^2)^(5/2),x)
Output:
(x*(24024*a**6*d**7 + 84084*a**6*d**6*e*x + 168168*a**6*d**5*e**2*x**2 + 2 10210*a**6*d**4*e**3*x**3 + 168168*a**6*d**3*e**4*x**4 + 84084*a**6*d**2*e **5*x**5 + 24024*a**6*d*e**6*x**6 + 3003*a**6*e**7*x**7 + 72072*a**5*b*d** 7*x + 336336*a**5*b*d**6*e*x**2 + 756756*a**5*b*d**5*e**2*x**3 + 1009008*a **5*b*d**4*e**3*x**4 + 840840*a**5*b*d**3*e**4*x**5 + 432432*a**5*b*d**2*e **5*x**6 + 126126*a**5*b*d*e**6*x**7 + 16016*a**5*b*e**7*x**8 + 120120*a** 4*b**2*d**7*x**2 + 630630*a**4*b**2*d**6*e*x**3 + 1513512*a**4*b**2*d**5*e **2*x**4 + 2102100*a**4*b**2*d**4*e**3*x**5 + 1801800*a**4*b**2*d**3*e**4* x**6 + 945945*a**4*b**2*d**2*e**5*x**7 + 280280*a**4*b**2*d*e**6*x**8 + 36 036*a**4*b**2*e**7*x**9 + 120120*a**3*b**3*d**7*x**3 + 672672*a**3*b**3*d* *6*e*x**4 + 1681680*a**3*b**3*d**5*e**2*x**5 + 2402400*a**3*b**3*d**4*e**3 *x**6 + 2102100*a**3*b**3*d**3*e**4*x**7 + 1121120*a**3*b**3*d**2*e**5*x** 8 + 336336*a**3*b**3*d*e**6*x**9 + 43680*a**3*b**3*e**7*x**10 + 72072*a**2 *b**4*d**7*x**4 + 420420*a**2*b**4*d**6*e*x**5 + 1081080*a**2*b**4*d**5*e* *2*x**6 + 1576575*a**2*b**4*d**4*e**3*x**7 + 1401400*a**2*b**4*d**3*e**4*x **8 + 756756*a**2*b**4*d**2*e**5*x**9 + 229320*a**2*b**4*d*e**6*x**10 + 30 030*a**2*b**4*e**7*x**11 + 24024*a*b**5*d**7*x**5 + 144144*a*b**5*d**6*e*x **6 + 378378*a*b**5*d**5*e**2*x**7 + 560560*a*b**5*d**4*e**3*x**8 + 504504 *a*b**5*d**3*e**4*x**9 + 275184*a*b**5*d**2*e**5*x**10 + 84084*a*b**5*d*e* *6*x**11 + 11088*a*b**5*e**7*x**12 + 3432*b**6*d**7*x**6 + 21021*b**6*d...