\(\int (a+b x) (d+e x)^6 (a^2+2 a b x+b^2 x^2)^{3/2} \, dx\) [128]

Optimal result

Integrand size = 33, antiderivative size = 254 \[ \int (a+b x) (d+e x)^6 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {(b d-a e)^4 (d+e x)^7 \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^5 (a+b x)}-\frac {b (b d-a e)^3 (d+e x)^8 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^5 (a+b x)}+\frac {2 b^2 (b d-a e)^2 (d+e x)^9 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^5 (a+b x)}-\frac {2 b^3 (b d-a e) (d+e x)^{10} \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^5 (a+b x)}+\frac {b^4 (d+e x)^{11} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^5 (a+b x)} \] Output:

1/7*(-a*e+b*d)^4*(e*x+d)^7*((b*x+a)^2)^(1/2)/e^5/(b*x+a)-1/2*b*(-a*e+b*d)^ 
3*(e*x+d)^8*((b*x+a)^2)^(1/2)/e^5/(b*x+a)+2/3*b^2*(-a*e+b*d)^2*(e*x+d)^9*( 
(b*x+a)^2)^(1/2)/e^5/(b*x+a)-2/5*b^3*(-a*e+b*d)*(e*x+d)^10*((b*x+a)^2)^(1/ 
2)/e^5/(b*x+a)+1/11*b^4*(e*x+d)^11*((b*x+a)^2)^(1/2)/e^5/(b*x+a)
 

Mathematica [A] (verified)

Time = 1.14 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.48 \[ \int (a+b x) (d+e x)^6 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {x \sqrt {(a+b x)^2} \left (330 a^4 \left (7 d^6+21 d^5 e x+35 d^4 e^2 x^2+35 d^3 e^3 x^3+21 d^2 e^4 x^4+7 d e^5 x^5+e^6 x^6\right )+165 a^3 b x \left (28 d^6+112 d^5 e x+210 d^4 e^2 x^2+224 d^3 e^3 x^3+140 d^2 e^4 x^4+48 d e^5 x^5+7 e^6 x^6\right )+55 a^2 b^2 x^2 \left (84 d^6+378 d^5 e x+756 d^4 e^2 x^2+840 d^3 e^3 x^3+540 d^2 e^4 x^4+189 d e^5 x^5+28 e^6 x^6\right )+11 a b^3 x^3 \left (210 d^6+1008 d^5 e x+2100 d^4 e^2 x^2+2400 d^3 e^3 x^3+1575 d^2 e^4 x^4+560 d e^5 x^5+84 e^6 x^6\right )+b^4 x^4 \left (462 d^6+2310 d^5 e x+4950 d^4 e^2 x^2+5775 d^3 e^3 x^3+3850 d^2 e^4 x^4+1386 d e^5 x^5+210 e^6 x^6\right )\right )}{2310 (a+b x)} \] Input:

Integrate[(a + b*x)*(d + e*x)^6*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]
 

Output:

(x*Sqrt[(a + b*x)^2]*(330*a^4*(7*d^6 + 21*d^5*e*x + 35*d^4*e^2*x^2 + 35*d^ 
3*e^3*x^3 + 21*d^2*e^4*x^4 + 7*d*e^5*x^5 + e^6*x^6) + 165*a^3*b*x*(28*d^6 
+ 112*d^5*e*x + 210*d^4*e^2*x^2 + 224*d^3*e^3*x^3 + 140*d^2*e^4*x^4 + 48*d 
*e^5*x^5 + 7*e^6*x^6) + 55*a^2*b^2*x^2*(84*d^6 + 378*d^5*e*x + 756*d^4*e^2 
*x^2 + 840*d^3*e^3*x^3 + 540*d^2*e^4*x^4 + 189*d*e^5*x^5 + 28*e^6*x^6) + 1 
1*a*b^3*x^3*(210*d^6 + 1008*d^5*e*x + 2100*d^4*e^2*x^2 + 2400*d^3*e^3*x^3 
+ 1575*d^2*e^4*x^4 + 560*d*e^5*x^5 + 84*e^6*x^6) + b^4*x^4*(462*d^6 + 2310 
*d^5*e*x + 4950*d^4*e^2*x^2 + 5775*d^3*e^3*x^3 + 3850*d^2*e^4*x^4 + 1386*d 
*e^5*x^5 + 210*e^6*x^6)))/(2310*(a + b*x))
 

Rubi [A] (verified)

Time = 0.76 (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, 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.

\(\displaystyle \int (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2} (d+e x)^6 \, dx\)

\(\Big \downarrow \) 1187

\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int b^3 (a+b x)^4 (d+e x)^6dx}{b^3 (a+b x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int (a+b x)^4 (d+e x)^6dx}{a+b x}\)

\(\Big \downarrow \) 49

\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {b^4 (d+e x)^{10}}{e^4}-\frac {4 b^3 (b d-a e) (d+e x)^9}{e^4}+\frac {6 b^2 (b d-a e)^2 (d+e x)^8}{e^4}-\frac {4 b (b d-a e)^3 (d+e x)^7}{e^4}+\frac {(a e-b d)^4 (d+e x)^6}{e^4}\right )dx}{a+b x}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (-\frac {2 b^3 (d+e x)^{10} (b d-a e)}{5 e^5}+\frac {2 b^2 (d+e x)^9 (b d-a e)^2}{3 e^5}-\frac {b (d+e x)^8 (b d-a e)^3}{2 e^5}+\frac {(d+e x)^7 (b d-a e)^4}{7 e^5}+\frac {b^4 (d+e x)^{11}}{11 e^5}\right )}{a+b x}\)

Input:

Int[(a + b*x)*(d + e*x)^6*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]
 

Output:

(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(((b*d - a*e)^4*(d + e*x)^7)/(7*e^5) - (b*( 
b*d - a*e)^3*(d + e*x)^8)/(2*e^5) + (2*b^2*(b*d - a*e)^2*(d + e*x)^9)/(3*e 
^5) - (2*b^3*(b*d - a*e)*(d + e*x)^10)/(5*e^5) + (b^4*(d + e*x)^11)/(11*e^ 
5)))/(a + b*x)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 49

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]
 

rule 1187

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]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(488\) vs. \(2(189)=378\).

Time = 1.76 (sec) , antiderivative size = 489, normalized size of antiderivative = 1.93

method	result	size
gosper	\(\frac {x \left (210 b^{4} e^{6} x^{10}+924 x^{9} a \,b^{3} e^{6}+1386 x^{9} b^{4} d \,e^{5}+1540 x^{8} a^{2} b^{2} e^{6}+6160 x^{8} a \,b^{3} d \,e^{5}+3850 x^{8} b^{4} d^{2} e^{4}+1155 x^{7} a^{3} b \,e^{6}+10395 x^{7} a^{2} b^{2} d \,e^{5}+17325 x^{7} a \,b^{3} d^{2} e^{4}+5775 x^{7} b^{4} d^{3} e^{3}+330 x^{6} a^{4} e^{6}+7920 x^{6} a^{3} b d \,e^{5}+29700 x^{6} a^{2} b^{2} d^{2} e^{4}+26400 x^{6} a \,b^{3} d^{3} e^{3}+4950 x^{6} b^{4} d^{4} e^{2}+2310 a^{4} d \,e^{5} x^{5}+23100 a^{3} b \,d^{2} e^{4} x^{5}+46200 a^{2} b^{2} d^{3} e^{3} x^{5}+23100 a \,b^{3} d^{4} e^{2} x^{5}+2310 b^{4} d^{5} e \,x^{5}+6930 x^{4} a^{4} d^{2} e^{4}+36960 x^{4} a^{3} b \,d^{3} e^{3}+41580 x^{4} a^{2} b^{2} d^{4} e^{2}+11088 x^{4} a \,b^{3} d^{5} e +462 x^{4} b^{4} d^{6}+11550 a^{4} d^{3} e^{3} x^{3}+34650 a^{3} b \,d^{4} e^{2} x^{3}+20790 a^{2} b^{2} d^{5} e \,x^{3}+2310 a \,b^{3} d^{6} x^{3}+11550 a^{4} d^{4} e^{2} x^{2}+18480 a^{3} b \,d^{5} e \,x^{2}+4620 a^{2} b^{2} d^{6} x^{2}+6930 a^{4} d^{5} e x +4620 a^{3} b \,d^{6} x +2310 d^{6} a^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{2310 \left (b x +a \right )^{3}}\)	\(489\)
default	\(\frac {x \left (210 b^{4} e^{6} x^{10}+924 x^{9} a \,b^{3} e^{6}+1386 x^{9} b^{4} d \,e^{5}+1540 x^{8} a^{2} b^{2} e^{6}+6160 x^{8} a \,b^{3} d \,e^{5}+3850 x^{8} b^{4} d^{2} e^{4}+1155 x^{7} a^{3} b \,e^{6}+10395 x^{7} a^{2} b^{2} d \,e^{5}+17325 x^{7} a \,b^{3} d^{2} e^{4}+5775 x^{7} b^{4} d^{3} e^{3}+330 x^{6} a^{4} e^{6}+7920 x^{6} a^{3} b d \,e^{5}+29700 x^{6} a^{2} b^{2} d^{2} e^{4}+26400 x^{6} a \,b^{3} d^{3} e^{3}+4950 x^{6} b^{4} d^{4} e^{2}+2310 a^{4} d \,e^{5} x^{5}+23100 a^{3} b \,d^{2} e^{4} x^{5}+46200 a^{2} b^{2} d^{3} e^{3} x^{5}+23100 a \,b^{3} d^{4} e^{2} x^{5}+2310 b^{4} d^{5} e \,x^{5}+6930 x^{4} a^{4} d^{2} e^{4}+36960 x^{4} a^{3} b \,d^{3} e^{3}+41580 x^{4} a^{2} b^{2} d^{4} e^{2}+11088 x^{4} a \,b^{3} d^{5} e +462 x^{4} b^{4} d^{6}+11550 a^{4} d^{3} e^{3} x^{3}+34650 a^{3} b \,d^{4} e^{2} x^{3}+20790 a^{2} b^{2} d^{5} e \,x^{3}+2310 a \,b^{3} d^{6} x^{3}+11550 a^{4} d^{4} e^{2} x^{2}+18480 a^{3} b \,d^{5} e \,x^{2}+4620 a^{2} b^{2} d^{6} x^{2}+6930 a^{4} d^{5} e x +4620 a^{3} b \,d^{6} x +2310 d^{6} a^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{2310 \left (b x +a \right )^{3}}\)	\(489\)
orering	\(\frac {x \left (210 b^{4} e^{6} x^{10}+924 x^{9} a \,b^{3} e^{6}+1386 x^{9} b^{4} d \,e^{5}+1540 x^{8} a^{2} b^{2} e^{6}+6160 x^{8} a \,b^{3} d \,e^{5}+3850 x^{8} b^{4} d^{2} e^{4}+1155 x^{7} a^{3} b \,e^{6}+10395 x^{7} a^{2} b^{2} d \,e^{5}+17325 x^{7} a \,b^{3} d^{2} e^{4}+5775 x^{7} b^{4} d^{3} e^{3}+330 x^{6} a^{4} e^{6}+7920 x^{6} a^{3} b d \,e^{5}+29700 x^{6} a^{2} b^{2} d^{2} e^{4}+26400 x^{6} a \,b^{3} d^{3} e^{3}+4950 x^{6} b^{4} d^{4} e^{2}+2310 a^{4} d \,e^{5} x^{5}+23100 a^{3} b \,d^{2} e^{4} x^{5}+46200 a^{2} b^{2} d^{3} e^{3} x^{5}+23100 a \,b^{3} d^{4} e^{2} x^{5}+2310 b^{4} d^{5} e \,x^{5}+6930 x^{4} a^{4} d^{2} e^{4}+36960 x^{4} a^{3} b \,d^{3} e^{3}+41580 x^{4} a^{2} b^{2} d^{4} e^{2}+11088 x^{4} a \,b^{3} d^{5} e +462 x^{4} b^{4} d^{6}+11550 a^{4} d^{3} e^{3} x^{3}+34650 a^{3} b \,d^{4} e^{2} x^{3}+20790 a^{2} b^{2} d^{5} e \,x^{3}+2310 a \,b^{3} d^{6} x^{3}+11550 a^{4} d^{4} e^{2} x^{2}+18480 a^{3} b \,d^{5} e \,x^{2}+4620 a^{2} b^{2} d^{6} x^{2}+6930 a^{4} d^{5} e x +4620 a^{3} b \,d^{6} x +2310 d^{6} a^{4}\right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{\frac {3}{2}}}{2310 \left (b x +a \right )^{3}}\)	\(498\)
risch	\(\frac {\sqrt {\left (b x +a \right )^{2}}\, b^{4} e^{6} x^{11}}{11 b x +11 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (4 a \,b^{3} e^{6}+6 b^{4} d \,e^{5}\right ) x^{10}}{10 b x +10 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (6 a^{2} b^{2} e^{6}+24 a \,b^{3} d \,e^{5}+15 b^{4} d^{2} e^{4}\right ) x^{9}}{9 b x +9 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (4 a^{3} b \,e^{6}+36 a^{2} b^{2} d \,e^{5}+60 a \,b^{3} d^{2} e^{4}+20 b^{4} d^{3} e^{3}\right ) x^{8}}{8 b x +8 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (a^{4} e^{6}+24 a^{3} b d \,e^{5}+90 a^{2} b^{2} d^{2} e^{4}+80 a \,b^{3} d^{3} e^{3}+15 b^{4} d^{4} e^{2}\right ) x^{7}}{7 b x +7 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (6 a^{4} d \,e^{5}+60 a^{3} b \,d^{2} e^{4}+120 a^{2} b^{2} e^{3} d^{3}+60 a \,b^{3} d^{4} e^{2}+6 b^{4} d^{5} e \right ) x^{6}}{6 b x +6 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (15 a^{4} d^{2} e^{4}+80 a^{3} b \,d^{3} e^{3}+90 a^{2} b^{2} d^{4} e^{2}+24 a \,b^{3} d^{5} e +b^{4} d^{6}\right ) x^{5}}{5 b x +5 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (20 a^{4} e^{3} d^{3}+60 a^{3} b \,d^{4} e^{2}+36 a^{2} b^{2} d^{5} e +4 a \,b^{3} d^{6}\right ) x^{4}}{4 b x +4 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (15 a^{4} d^{4} e^{2}+24 a^{3} d^{5} e b +6 a^{2} b^{2} d^{6}\right ) x^{3}}{3 b x +3 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (6 a^{4} d^{5} e +4 a^{3} b \,d^{6}\right ) x^{2}}{2 b x +2 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, d^{6} a^{4} x}{b x +a}\)	\(603\)

Input:

int((b*x+a)*(e*x+d)^6*(b^2*x^2+2*a*b*x+a^2)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

1/2310*x*(210*b^4*e^6*x^10+924*a*b^3*e^6*x^9+1386*b^4*d*e^5*x^9+1540*a^2*b 
^2*e^6*x^8+6160*a*b^3*d*e^5*x^8+3850*b^4*d^2*e^4*x^8+1155*a^3*b*e^6*x^7+10 
395*a^2*b^2*d*e^5*x^7+17325*a*b^3*d^2*e^4*x^7+5775*b^4*d^3*e^3*x^7+330*a^4 
*e^6*x^6+7920*a^3*b*d*e^5*x^6+29700*a^2*b^2*d^2*e^4*x^6+26400*a*b^3*d^3*e^ 
3*x^6+4950*b^4*d^4*e^2*x^6+2310*a^4*d*e^5*x^5+23100*a^3*b*d^2*e^4*x^5+4620 
0*a^2*b^2*d^3*e^3*x^5+23100*a*b^3*d^4*e^2*x^5+2310*b^4*d^5*e*x^5+6930*a^4* 
d^2*e^4*x^4+36960*a^3*b*d^3*e^3*x^4+41580*a^2*b^2*d^4*e^2*x^4+11088*a*b^3* 
d^5*e*x^4+462*b^4*d^6*x^4+11550*a^4*d^3*e^3*x^3+34650*a^3*b*d^4*e^2*x^3+20 
790*a^2*b^2*d^5*e*x^3+2310*a*b^3*d^6*x^3+11550*a^4*d^4*e^2*x^2+18480*a^3*b 
*d^5*e*x^2+4620*a^2*b^2*d^6*x^2+6930*a^4*d^5*e*x+4620*a^3*b*d^6*x+2310*a^4 
*d^6)*((b*x+a)^2)^(3/2)/(b*x+a)^3
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 418 vs. \(2 (189) = 378\).

Time = 0.07 (sec) , antiderivative size = 418, normalized size of antiderivative = 1.65 \[ \int (a+b x) (d+e x)^6 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {1}{11} \, b^{4} e^{6} x^{11} + a^{4} d^{6} x + \frac {1}{5} \, {\left (3 \, b^{4} d e^{5} + 2 \, a b^{3} e^{6}\right )} x^{10} + \frac {1}{3} \, {\left (5 \, b^{4} d^{2} e^{4} + 8 \, a b^{3} d e^{5} + 2 \, a^{2} b^{2} e^{6}\right )} x^{9} + \frac {1}{2} \, {\left (5 \, b^{4} d^{3} e^{3} + 15 \, a b^{3} d^{2} e^{4} + 9 \, a^{2} b^{2} d e^{5} + a^{3} b e^{6}\right )} x^{8} + \frac {1}{7} \, {\left (15 \, b^{4} d^{4} e^{2} + 80 \, a b^{3} d^{3} e^{3} + 90 \, a^{2} b^{2} d^{2} e^{4} + 24 \, a^{3} b d e^{5} + a^{4} e^{6}\right )} x^{7} + {\left (b^{4} d^{5} e + 10 \, a b^{3} d^{4} e^{2} + 20 \, a^{2} b^{2} d^{3} e^{3} + 10 \, a^{3} b d^{2} e^{4} + a^{4} d e^{5}\right )} x^{6} + \frac {1}{5} \, {\left (b^{4} d^{6} + 24 \, a b^{3} d^{5} e + 90 \, a^{2} b^{2} d^{4} e^{2} + 80 \, a^{3} b d^{3} e^{3} + 15 \, a^{4} d^{2} e^{4}\right )} x^{5} + {\left (a b^{3} d^{6} + 9 \, a^{2} b^{2} d^{5} e + 15 \, a^{3} b d^{4} e^{2} + 5 \, a^{4} d^{3} e^{3}\right )} x^{4} + {\left (2 \, a^{2} b^{2} d^{6} + 8 \, a^{3} b d^{5} e + 5 \, a^{4} d^{4} e^{2}\right )} x^{3} + {\left (2 \, a^{3} b d^{6} + 3 \, a^{4} d^{5} e\right )} x^{2} \] Input:

integrate((b*x+a)*(e*x+d)^6*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="fric 
as")
 

Output:

1/11*b^4*e^6*x^11 + a^4*d^6*x + 1/5*(3*b^4*d*e^5 + 2*a*b^3*e^6)*x^10 + 1/3 
*(5*b^4*d^2*e^4 + 8*a*b^3*d*e^5 + 2*a^2*b^2*e^6)*x^9 + 1/2*(5*b^4*d^3*e^3 
+ 15*a*b^3*d^2*e^4 + 9*a^2*b^2*d*e^5 + a^3*b*e^6)*x^8 + 1/7*(15*b^4*d^4*e^ 
2 + 80*a*b^3*d^3*e^3 + 90*a^2*b^2*d^2*e^4 + 24*a^3*b*d*e^5 + a^4*e^6)*x^7 
+ (b^4*d^5*e + 10*a*b^3*d^4*e^2 + 20*a^2*b^2*d^3*e^3 + 10*a^3*b*d^2*e^4 + 
a^4*d*e^5)*x^6 + 1/5*(b^4*d^6 + 24*a*b^3*d^5*e + 90*a^2*b^2*d^4*e^2 + 80*a 
^3*b*d^3*e^3 + 15*a^4*d^2*e^4)*x^5 + (a*b^3*d^6 + 9*a^2*b^2*d^5*e + 15*a^3 
*b*d^4*e^2 + 5*a^4*d^3*e^3)*x^4 + (2*a^2*b^2*d^6 + 8*a^3*b*d^5*e + 5*a^4*d 
^4*e^2)*x^3 + (2*a^3*b*d^6 + 3*a^4*d^5*e)*x^2
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 29211 vs. \(2 (182) = 364\).

Time = 1.90 (sec) , antiderivative size = 29211, normalized size of antiderivative = 115.00 \[ \int (a+b x) (d+e x)^6 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)*(e*x+d)**6*(b**2*x**2+2*a*b*x+a**2)**(3/2),x)
 

Output:

Piecewise((sqrt(a**2 + 2*a*b*x + b**2*x**2)*(b**3*e**6*x**10/11 + x**9*(34 
*a*b**4*e**6/11 + 6*b**5*d*e**5)/(10*b**2) + x**8*(100*a**2*b**3*e**6/11 + 
 30*a*b**4*d*e**5 - 19*a*(34*a*b**4*e**6/11 + 6*b**5*d*e**5)/(10*b) + 15*b 
**5*d**2*e**4)/(9*b**2) + x**7*(10*a**3*b**2*e**6 + 60*a**2*b**3*d*e**5 - 
9*a**2*(34*a*b**4*e**6/11 + 6*b**5*d*e**5)/(10*b**2) + 75*a*b**4*d**2*e**4 
 - 17*a*(100*a**2*b**3*e**6/11 + 30*a*b**4*d*e**5 - 19*a*(34*a*b**4*e**6/1 
1 + 6*b**5*d*e**5)/(10*b) + 15*b**5*d**2*e**4)/(9*b) + 20*b**5*d**3*e**3)/ 
(8*b**2) + x**6*(5*a**4*b*e**6 + 60*a**3*b**2*d*e**5 + 150*a**2*b**3*d**2* 
e**4 - 8*a**2*(100*a**2*b**3*e**6/11 + 30*a*b**4*d*e**5 - 19*a*(34*a*b**4* 
e**6/11 + 6*b**5*d*e**5)/(10*b) + 15*b**5*d**2*e**4)/(9*b**2) + 100*a*b**4 
*d**3*e**3 - 15*a*(10*a**3*b**2*e**6 + 60*a**2*b**3*d*e**5 - 9*a**2*(34*a* 
b**4*e**6/11 + 6*b**5*d*e**5)/(10*b**2) + 75*a*b**4*d**2*e**4 - 17*a*(100* 
a**2*b**3*e**6/11 + 30*a*b**4*d*e**5 - 19*a*(34*a*b**4*e**6/11 + 6*b**5*d* 
e**5)/(10*b) + 15*b**5*d**2*e**4)/(9*b) + 20*b**5*d**3*e**3)/(8*b) + 15*b* 
*5*d**4*e**2)/(7*b**2) + x**5*(a**5*e**6 + 30*a**4*b*d*e**5 + 150*a**3*b** 
2*d**2*e**4 + 200*a**2*b**3*d**3*e**3 - 7*a**2*(10*a**3*b**2*e**6 + 60*a** 
2*b**3*d*e**5 - 9*a**2*(34*a*b**4*e**6/11 + 6*b**5*d*e**5)/(10*b**2) + 75* 
a*b**4*d**2*e**4 - 17*a*(100*a**2*b**3*e**6/11 + 30*a*b**4*d*e**5 - 19*a*( 
34*a*b**4*e**6/11 + 6*b**5*d*e**5)/(10*b) + 15*b**5*d**2*e**4)/(9*b) + 20* 
b**5*d**3*e**3)/(8*b**2) + 75*a*b**4*d**4*e**2 - 13*a*(5*a**4*b*e**6 + ...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1736 vs. \(2 (189) = 378\).

Time = 0.05 (sec) , antiderivative size = 1736, normalized size of antiderivative = 6.83 \[ \int (a+b x) (d+e x)^6 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)*(e*x+d)^6*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="maxi 
ma")
 

Output:

1/11*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*e^6*x^6/b - 17/110*(b^2*x^2 + 2*a*b*x 
 + a^2)^(5/2)*a*e^6*x^5/b^2 + 13/66*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2*e^ 
6*x^4/b^3 - 59/264*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^3*e^6*x^3/b^4 + 1/4*( 
b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a*d^6*x - 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/ 
2)*a^7*e^6*x/b^6 + 21/88*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^4*e^6*x^2/b^5 + 
 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^2*d^6/b - 1/4*(b^2*x^2 + 2*a*b*x + 
a^2)^(3/2)*a^8*e^6/b^7 - 65/264*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^5*e^6*x/ 
b^6 + 329/1320*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^6*e^6/b^7 + 1/10*(6*b*d*e 
^5 + a*e^6)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*x^5/b^2 - 1/6*(6*b*d*e^5 + a*e 
^6)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a*x^4/b^3 + 1/3*(5*b*d^2*e^4 + 2*a*d*e 
^5)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*x^4/b^2 + 5/24*(6*b*d*e^5 + a*e^6)*(b^ 
2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2*x^3/b^4 - 13/24*(5*b*d^2*e^4 + 2*a*d*e^5) 
*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a*x^3/b^3 + 5/8*(4*b*d^3*e^3 + 3*a*d^2*e^ 
4)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*x^3/b^2 + 1/4*(6*b*d*e^5 + a*e^6)*(b^2* 
x^2 + 2*a*b*x + a^2)^(3/2)*a^6*x/b^6 - 3/4*(5*b*d^2*e^4 + 2*a*d*e^5)*(b^2* 
x^2 + 2*a*b*x + a^2)^(3/2)*a^5*x/b^5 + 5/4*(4*b*d^3*e^3 + 3*a*d^2*e^4)*(b^ 
2*x^2 + 2*a*b*x + a^2)^(3/2)*a^4*x/b^4 - 5/4*(3*b*d^4*e^2 + 4*a*d^3*e^3)*( 
b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^3*x/b^3 + 3/4*(2*b*d^5*e + 5*a*d^4*e^2)*( 
b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^2*x/b^2 - 1/4*(b*d^6 + 6*a*d^5*e)*(b^2*x^ 
2 + 2*a*b*x + a^2)^(3/2)*a*x/b - 13/56*(6*b*d*e^5 + a*e^6)*(b^2*x^2 + 2...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 774 vs. \(2 (189) = 378\).

Time = 0.23 (sec) , antiderivative size = 774, normalized size of antiderivative = 3.05 \[ \int (a+b x) (d+e x)^6 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx =\text {Too large to display} \] Input:

integrate((b*x+a)*(e*x+d)^6*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="giac 
")
 

Output:

1/11*b^4*e^6*x^11*sgn(b*x + a) + 3/5*b^4*d*e^5*x^10*sgn(b*x + a) + 2/5*a*b 
^3*e^6*x^10*sgn(b*x + a) + 5/3*b^4*d^2*e^4*x^9*sgn(b*x + a) + 8/3*a*b^3*d* 
e^5*x^9*sgn(b*x + a) + 2/3*a^2*b^2*e^6*x^9*sgn(b*x + a) + 5/2*b^4*d^3*e^3* 
x^8*sgn(b*x + a) + 15/2*a*b^3*d^2*e^4*x^8*sgn(b*x + a) + 9/2*a^2*b^2*d*e^5 
*x^8*sgn(b*x + a) + 1/2*a^3*b*e^6*x^8*sgn(b*x + a) + 15/7*b^4*d^4*e^2*x^7* 
sgn(b*x + a) + 80/7*a*b^3*d^3*e^3*x^7*sgn(b*x + a) + 90/7*a^2*b^2*d^2*e^4* 
x^7*sgn(b*x + a) + 24/7*a^3*b*d*e^5*x^7*sgn(b*x + a) + 1/7*a^4*e^6*x^7*sgn 
(b*x + a) + b^4*d^5*e*x^6*sgn(b*x + a) + 10*a*b^3*d^4*e^2*x^6*sgn(b*x + a) 
 + 20*a^2*b^2*d^3*e^3*x^6*sgn(b*x + a) + 10*a^3*b*d^2*e^4*x^6*sgn(b*x + a) 
 + a^4*d*e^5*x^6*sgn(b*x + a) + 1/5*b^4*d^6*x^5*sgn(b*x + a) + 24/5*a*b^3* 
d^5*e*x^5*sgn(b*x + a) + 18*a^2*b^2*d^4*e^2*x^5*sgn(b*x + a) + 16*a^3*b*d^ 
3*e^3*x^5*sgn(b*x + a) + 3*a^4*d^2*e^4*x^5*sgn(b*x + a) + a*b^3*d^6*x^4*sg 
n(b*x + a) + 9*a^2*b^2*d^5*e*x^4*sgn(b*x + a) + 15*a^3*b*d^4*e^2*x^4*sgn(b 
*x + a) + 5*a^4*d^3*e^3*x^4*sgn(b*x + a) + 2*a^2*b^2*d^6*x^3*sgn(b*x + a) 
+ 8*a^3*b*d^5*e*x^3*sgn(b*x + a) + 5*a^4*d^4*e^2*x^3*sgn(b*x + a) + 2*a^3* 
b*d^6*x^2*sgn(b*x + a) + 3*a^4*d^5*e*x^2*sgn(b*x + a) + a^4*d^6*x*sgn(b*x 
+ a) + 1/2310*(462*a^5*b^6*d^6 - 462*a^6*b^5*d^5*e + 330*a^7*b^4*d^4*e^2 - 
 165*a^8*b^3*d^3*e^3 + 55*a^9*b^2*d^2*e^4 - 11*a^10*b*d*e^5 + a^11*e^6)*sg 
n(b*x + a)/b^7
 

Mupad [F(-1)]

Timed out. \[ \int (a+b x) (d+e x)^6 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\int \left (a+b\,x\right )\,{\left (d+e\,x\right )}^6\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2} \,d x \] Input:

int((a + b*x)*(d + e*x)^6*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2),x)
 

Output:

int((a + b*x)*(d + e*x)^6*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 472, normalized size of antiderivative = 1.86 \[ \int (a+b x) (d+e x)^6 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {x \left (210 b^{4} e^{6} x^{10}+924 a \,b^{3} e^{6} x^{9}+1386 b^{4} d \,e^{5} x^{9}+1540 a^{2} b^{2} e^{6} x^{8}+6160 a \,b^{3} d \,e^{5} x^{8}+3850 b^{4} d^{2} e^{4} x^{8}+1155 a^{3} b \,e^{6} x^{7}+10395 a^{2} b^{2} d \,e^{5} x^{7}+17325 a \,b^{3} d^{2} e^{4} x^{7}+5775 b^{4} d^{3} e^{3} x^{7}+330 a^{4} e^{6} x^{6}+7920 a^{3} b d \,e^{5} x^{6}+29700 a^{2} b^{2} d^{2} e^{4} x^{6}+26400 a \,b^{3} d^{3} e^{3} x^{6}+4950 b^{4} d^{4} e^{2} x^{6}+2310 a^{4} d \,e^{5} x^{5}+23100 a^{3} b \,d^{2} e^{4} x^{5}+46200 a^{2} b^{2} d^{3} e^{3} x^{5}+23100 a \,b^{3} d^{4} e^{2} x^{5}+2310 b^{4} d^{5} e \,x^{5}+6930 a^{4} d^{2} e^{4} x^{4}+36960 a^{3} b \,d^{3} e^{3} x^{4}+41580 a^{2} b^{2} d^{4} e^{2} x^{4}+11088 a \,b^{3} d^{5} e \,x^{4}+462 b^{4} d^{6} x^{4}+11550 a^{4} d^{3} e^{3} x^{3}+34650 a^{3} b \,d^{4} e^{2} x^{3}+20790 a^{2} b^{2} d^{5} e \,x^{3}+2310 a \,b^{3} d^{6} x^{3}+11550 a^{4} d^{4} e^{2} x^{2}+18480 a^{3} b \,d^{5} e \,x^{2}+4620 a^{2} b^{2} d^{6} x^{2}+6930 a^{4} d^{5} e x +4620 a^{3} b \,d^{6} x +2310 a^{4} d^{6}\right )}{2310} \] Input:

int((b*x+a)*(e*x+d)^6*(b^2*x^2+2*a*b*x+a^2)^(3/2),x)
 

Output:

(x*(2310*a**4*d**6 + 6930*a**4*d**5*e*x + 11550*a**4*d**4*e**2*x**2 + 1155 
0*a**4*d**3*e**3*x**3 + 6930*a**4*d**2*e**4*x**4 + 2310*a**4*d*e**5*x**5 + 
 330*a**4*e**6*x**6 + 4620*a**3*b*d**6*x + 18480*a**3*b*d**5*e*x**2 + 3465 
0*a**3*b*d**4*e**2*x**3 + 36960*a**3*b*d**3*e**3*x**4 + 23100*a**3*b*d**2* 
e**4*x**5 + 7920*a**3*b*d*e**5*x**6 + 1155*a**3*b*e**6*x**7 + 4620*a**2*b* 
*2*d**6*x**2 + 20790*a**2*b**2*d**5*e*x**3 + 41580*a**2*b**2*d**4*e**2*x** 
4 + 46200*a**2*b**2*d**3*e**3*x**5 + 29700*a**2*b**2*d**2*e**4*x**6 + 1039 
5*a**2*b**2*d*e**5*x**7 + 1540*a**2*b**2*e**6*x**8 + 2310*a*b**3*d**6*x**3 
 + 11088*a*b**3*d**5*e*x**4 + 23100*a*b**3*d**4*e**2*x**5 + 26400*a*b**3*d 
**3*e**3*x**6 + 17325*a*b**3*d**2*e**4*x**7 + 6160*a*b**3*d*e**5*x**8 + 92 
4*a*b**3*e**6*x**9 + 462*b**4*d**6*x**4 + 2310*b**4*d**5*e*x**5 + 4950*b** 
4*d**4*e**2*x**6 + 5775*b**4*d**3*e**3*x**7 + 3850*b**4*d**2*e**4*x**8 + 1 
386*b**4*d*e**5*x**9 + 210*b**4*e**6*x**10))/2310