Integrand size = 33, antiderivative size = 161 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^7} \, dx=\frac {b (A b-a B) (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{4 (b d-a e)^3 (d+e x)^4}+\frac {(B d-A e) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{6 (b d-a e)^2 (d+e x)^6}+\frac {(b B d-7 A b e+6 a B e) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{30 (b d-a e)^3 (d+e x)^5} \] Output:
1/4*b*(A*b-B*a)*(b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(3/2)/(-a*e+b*d)^3/(e*x+d)^4 +1/6*(-A*e+B*d)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(-a*e+b*d)^2/(e*x+d)^6+1/30*(- 7*A*b*e+6*B*a*e+B*b*d)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(-a*e+b*d)^3/(e*x+d)^5
Time = 1.11 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.42 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^7} \, dx=-\frac {\sqrt {(a+b x)^2} \left (2 a^3 e^3 (5 A e+B (d+6 e x))+3 a^2 b e^2 \left (2 A e (d+6 e x)+B \left (d^2+6 d e x+15 e^2 x^2\right )\right )+3 a b^2 e \left (A e \left (d^2+6 d e x+15 e^2 x^2\right )+B \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )\right )+b^3 \left (A e \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )+2 B \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )\right )\right )}{60 e^5 (a+b x) (d+e x)^6} \] Input:
Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(d + e*x)^7,x]
Output:
-1/60*(Sqrt[(a + b*x)^2]*(2*a^3*e^3*(5*A*e + B*(d + 6*e*x)) + 3*a^2*b*e^2* (2*A*e*(d + 6*e*x) + B*(d^2 + 6*d*e*x + 15*e^2*x^2)) + 3*a*b^2*e*(A*e*(d^2 + 6*d*e*x + 15*e^2*x^2) + B*(d^3 + 6*d^2*e*x + 15*d*e^2*x^2 + 20*e^3*x^3) ) + b^3*(A*e*(d^3 + 6*d^2*e*x + 15*d*e^2*x^2 + 20*e^3*x^3) + 2*B*(d^4 + 6* d^3*e*x + 15*d^2*e^2*x^2 + 20*d*e^3*x^3 + 15*e^4*x^4))))/(e^5*(a + b*x)*(d + e*x)^6)
Time = 0.46 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.98, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {1187, 27, 87, 55, 48}
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.
| \(\displaystyle \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2} (A+B x)}{(d+e x)^7} \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {b^3 (a+b x)^3 (A+B x)}{(d+e x)^7}dx}{b^3 (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^7}dx}{a+b x}\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {(-3 a B e+A b e+2 b B d) \int \frac {(a+b x)^3}{(d+e x)^6}dx}{3 e (b d-a e)}-\frac {(a+b x)^4 (B d-A e)}{6 e (d+e x)^6 (b d-a e)}\right )}{a+b x}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {(-3 a B e+A b e+2 b B d) \left (\frac {b \int \frac {(a+b x)^3}{(d+e x)^5}dx}{5 (b d-a e)}+\frac {(a+b x)^4}{5 (d+e x)^5 (b d-a e)}\right )}{3 e (b d-a e)}-\frac {(a+b x)^4 (B d-A e)}{6 e (d+e x)^6 (b d-a e)}\right )}{a+b x}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {\left (\frac {b (a+b x)^4}{20 (d+e x)^4 (b d-a e)^2}+\frac {(a+b x)^4}{5 (d+e x)^5 (b d-a e)}\right ) (-3 a B e+A b e+2 b B d)}{3 e (b d-a e)}-\frac {(a+b x)^4 (B d-A e)}{6 e (d+e x)^6 (b d-a e)}\right )}{a+b x}\) |
Input:
Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(d + e*x)^7,x]
Output:
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-1/6*((B*d - A*e)*(a + b*x)^4)/(e*(b*d - a *e)*(d + e*x)^6) + ((2*b*B*d + A*b*e - 3*a*B*e)*((a + b*x)^4/(5*(b*d - a*e )*(d + e*x)^5) + (b*(a + b*x)^4)/(20*(b*d - a*e)^2*(d + e*x)^4)))/(3*e*(b* d - a*e))))/(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] :> Simp [(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(S implify[m + n + 2]/((b*c - a*d)*(m + 1))) Int[(a + b*x)^Simplify[m + 1]*( c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 2], 0] && NeQ[m, -1] && !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] || !SumSimp lerQ[n, 1])
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
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 = 2.26 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.75
| method | result | size |
| risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {B \,b^{3} x^{4}}{2 e}-\frac {b^{2} \left (A b e +3 B a e +2 B b d \right ) x^{3}}{3 e^{2}}-\frac {b \left (3 A a b \,e^{2}+A \,b^{2} d e +3 B \,e^{2} a^{2}+3 B a b d e +2 B \,b^{2} d^{2}\right ) x^{2}}{4 e^{3}}-\frac {\left (6 A \,a^{2} b \,e^{3}+3 A a \,b^{2} d \,e^{2}+A \,b^{3} d^{2} e +2 B \,e^{3} a^{3}+3 B \,a^{2} b d \,e^{2}+3 B a \,b^{2} d^{2} e +2 B \,b^{3} d^{3}\right ) x}{10 e^{4}}-\frac {10 A \,a^{3} e^{4}+6 A \,a^{2} b d \,e^{3}+3 A a \,b^{2} d^{2} e^{2}+A \,b^{3} d^{3} e +2 B \,a^{3} d \,e^{3}+3 B \,a^{2} b \,d^{2} e^{2}+3 B a \,b^{2} d^{3} e +2 B \,b^{3} d^{4}}{60 e^{5}}\right )}{\left (b x +a \right ) \left (e x +d \right )^{6}}\) | \(282\) |
| gosper | \(-\frac {\left (30 B \,b^{3} e^{4} x^{4}+20 A \,b^{3} e^{4} x^{3}+60 B a \,b^{2} e^{4} x^{3}+40 B \,b^{3} d \,e^{3} x^{3}+45 A a \,b^{2} e^{4} x^{2}+15 A \,b^{3} d \,e^{3} x^{2}+45 B \,a^{2} b \,e^{4} x^{2}+45 B a \,b^{2} d \,e^{3} x^{2}+30 B \,b^{3} d^{2} e^{2} x^{2}+36 A \,a^{2} b \,e^{4} x +18 A a \,b^{2} d \,e^{3} x +6 A \,b^{3} d^{2} e^{2} x +12 B \,a^{3} e^{4} x +18 B \,a^{2} b d \,e^{3} x +18 B a \,b^{2} d^{2} e^{2} x +12 B \,b^{3} d^{3} e x +10 A \,a^{3} e^{4}+6 A \,a^{2} b d \,e^{3}+3 A a \,b^{2} d^{2} e^{2}+A \,b^{3} d^{3} e +2 B \,a^{3} d \,e^{3}+3 B \,a^{2} b \,d^{2} e^{2}+3 B a \,b^{2} d^{3} e +2 B \,b^{3} d^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{60 e^{5} \left (e x +d \right )^{6} \left (b x +a \right )^{3}}\) | \(316\) |
| default | \(-\frac {\left (30 B \,b^{3} e^{4} x^{4}+20 A \,b^{3} e^{4} x^{3}+60 B a \,b^{2} e^{4} x^{3}+40 B \,b^{3} d \,e^{3} x^{3}+45 A a \,b^{2} e^{4} x^{2}+15 A \,b^{3} d \,e^{3} x^{2}+45 B \,a^{2} b \,e^{4} x^{2}+45 B a \,b^{2} d \,e^{3} x^{2}+30 B \,b^{3} d^{2} e^{2} x^{2}+36 A \,a^{2} b \,e^{4} x +18 A a \,b^{2} d \,e^{3} x +6 A \,b^{3} d^{2} e^{2} x +12 B \,a^{3} e^{4} x +18 B \,a^{2} b d \,e^{3} x +18 B a \,b^{2} d^{2} e^{2} x +12 B \,b^{3} d^{3} e x +10 A \,a^{3} e^{4}+6 A \,a^{2} b d \,e^{3}+3 A a \,b^{2} d^{2} e^{2}+A \,b^{3} d^{3} e +2 B \,a^{3} d \,e^{3}+3 B \,a^{2} b \,d^{2} e^{2}+3 B a \,b^{2} d^{3} e +2 B \,b^{3} d^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{60 e^{5} \left (e x +d \right )^{6} \left (b x +a \right )^{3}}\) | \(316\) |
| orering | \(-\frac {\left (30 B \,b^{3} e^{4} x^{4}+20 A \,b^{3} e^{4} x^{3}+60 B a \,b^{2} e^{4} x^{3}+40 B \,b^{3} d \,e^{3} x^{3}+45 A a \,b^{2} e^{4} x^{2}+15 A \,b^{3} d \,e^{3} x^{2}+45 B \,a^{2} b \,e^{4} x^{2}+45 B a \,b^{2} d \,e^{3} x^{2}+30 B \,b^{3} d^{2} e^{2} x^{2}+36 A \,a^{2} b \,e^{4} x +18 A a \,b^{2} d \,e^{3} x +6 A \,b^{3} d^{2} e^{2} x +12 B \,a^{3} e^{4} x +18 B \,a^{2} b d \,e^{3} x +18 B a \,b^{2} d^{2} e^{2} x +12 B \,b^{3} d^{3} e x +10 A \,a^{3} e^{4}+6 A \,a^{2} b d \,e^{3}+3 A a \,b^{2} d^{2} e^{2}+A \,b^{3} d^{3} e +2 B \,a^{3} d \,e^{3}+3 B \,a^{2} b \,d^{2} e^{2}+3 B a \,b^{2} d^{3} e +2 B \,b^{3} d^{4}\right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{\frac {3}{2}}}{60 e^{5} \left (b x +a \right )^{3} \left (e x +d \right )^{6}}\) | \(325\) |
Input:
int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(3/2)/(e*x+d)^7,x,method=_RETURNVERBOSE)
Output:
((b*x+a)^2)^(1/2)/(b*x+a)*(-1/2*B*b^3/e*x^4-1/3/e^2*b^2*(A*b*e+3*B*a*e+2*B *b*d)*x^3-1/4*b/e^3*(3*A*a*b*e^2+A*b^2*d*e+3*B*a^2*e^2+3*B*a*b*d*e+2*B*b^2 *d^2)*x^2-1/10/e^4*(6*A*a^2*b*e^3+3*A*a*b^2*d*e^2+A*b^3*d^2*e+2*B*a^3*e^3+ 3*B*a^2*b*d*e^2+3*B*a*b^2*d^2*e+2*B*b^3*d^3)*x-1/60/e^5*(10*A*a^3*e^4+6*A* a^2*b*d*e^3+3*A*a*b^2*d^2*e^2+A*b^3*d^3*e+2*B*a^3*d*e^3+3*B*a^2*b*d^2*e^2+ 3*B*a*b^2*d^3*e+2*B*b^3*d^4))/(e*x+d)^6
Leaf count of result is larger than twice the leaf count of optimal. 317 vs. \(2 (149) = 298\).
Time = 0.08 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.97 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^7} \, dx=-\frac {30 \, B b^{3} e^{4} x^{4} + 2 \, B b^{3} d^{4} + 10 \, A a^{3} e^{4} + {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 3 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} + 2 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} + 20 \, {\left (2 \, B b^{3} d e^{3} + {\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 15 \, {\left (2 \, B b^{3} d^{2} e^{2} + {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 3 \, {\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} + 6 \, {\left (2 \, B b^{3} d^{3} e + {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 3 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{3} + 2 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x}{60 \, {\left (e^{11} x^{6} + 6 \, d e^{10} x^{5} + 15 \, d^{2} e^{9} x^{4} + 20 \, d^{3} e^{8} x^{3} + 15 \, d^{4} e^{7} x^{2} + 6 \, d^{5} e^{6} x + d^{6} e^{5}\right )}} \] Input:
integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(3/2)/(e*x+d)^7,x, algorithm="fric as")
Output:
-1/60*(30*B*b^3*e^4*x^4 + 2*B*b^3*d^4 + 10*A*a^3*e^4 + (3*B*a*b^2 + A*b^3) *d^3*e + 3*(B*a^2*b + A*a*b^2)*d^2*e^2 + 2*(B*a^3 + 3*A*a^2*b)*d*e^3 + 20* (2*B*b^3*d*e^3 + (3*B*a*b^2 + A*b^3)*e^4)*x^3 + 15*(2*B*b^3*d^2*e^2 + (3*B *a*b^2 + A*b^3)*d*e^3 + 3*(B*a^2*b + A*a*b^2)*e^4)*x^2 + 6*(2*B*b^3*d^3*e + (3*B*a*b^2 + A*b^3)*d^2*e^2 + 3*(B*a^2*b + A*a*b^2)*d*e^3 + 2*(B*a^3 + 3 *A*a^2*b)*e^4)*x)/(e^11*x^6 + 6*d*e^10*x^5 + 15*d^2*e^9*x^4 + 20*d^3*e^8*x ^3 + 15*d^4*e^7*x^2 + 6*d^5*e^6*x + d^6*e^5)
\[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^7} \, dx=\int \frac {\left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}{\left (d + e x\right )^{7}}\, dx \] Input:
integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**(3/2)/(e*x+d)**7,x)
Output:
Integral((A + B*x)*((a + b*x)**2)**(3/2)/(d + e*x)**7, x)
Exception generated. \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^7} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(3/2)/(e*x+d)^7,x, algorithm="maxi ma")
Output:
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. 517 vs. \(2 (149) = 298\).
Time = 0.17 (sec) , antiderivative size = 517, normalized size of antiderivative = 3.21 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^7} \, dx=\frac {{\left (2 \, B b^{6} d - 3 \, B a b^{5} e + A b^{6} e\right )} \mathrm {sgn}\left (b x + a\right )}{60 \, {\left (b^{3} d^{3} e^{5} - 3 \, a b^{2} d^{2} e^{6} + 3 \, a^{2} b d e^{7} - a^{3} e^{8}\right )}} - \frac {30 \, B b^{3} e^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + 40 \, B b^{3} d e^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 60 \, B a b^{2} e^{4} x^{3} \mathrm {sgn}\left (b x + a\right ) + 20 \, A b^{3} e^{4} x^{3} \mathrm {sgn}\left (b x + a\right ) + 30 \, B b^{3} d^{2} e^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 45 \, B a b^{2} d e^{3} x^{2} \mathrm {sgn}\left (b x + a\right ) + 15 \, A b^{3} d e^{3} x^{2} \mathrm {sgn}\left (b x + a\right ) + 45 \, B a^{2} b e^{4} x^{2} \mathrm {sgn}\left (b x + a\right ) + 45 \, A a b^{2} e^{4} x^{2} \mathrm {sgn}\left (b x + a\right ) + 12 \, B b^{3} d^{3} e x \mathrm {sgn}\left (b x + a\right ) + 18 \, B a b^{2} d^{2} e^{2} x \mathrm {sgn}\left (b x + a\right ) + 6 \, A b^{3} d^{2} e^{2} x \mathrm {sgn}\left (b x + a\right ) + 18 \, B a^{2} b d e^{3} x \mathrm {sgn}\left (b x + a\right ) + 18 \, A a b^{2} d e^{3} x \mathrm {sgn}\left (b x + a\right ) + 12 \, B a^{3} e^{4} x \mathrm {sgn}\left (b x + a\right ) + 36 \, A a^{2} b e^{4} x \mathrm {sgn}\left (b x + a\right ) + 2 \, B b^{3} d^{4} \mathrm {sgn}\left (b x + a\right ) + 3 \, B a b^{2} d^{3} e \mathrm {sgn}\left (b x + a\right ) + A b^{3} d^{3} e \mathrm {sgn}\left (b x + a\right ) + 3 \, B a^{2} b d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 3 \, A a b^{2} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 2 \, B a^{3} d e^{3} \mathrm {sgn}\left (b x + a\right ) + 6 \, A a^{2} b d e^{3} \mathrm {sgn}\left (b x + a\right ) + 10 \, A a^{3} e^{4} \mathrm {sgn}\left (b x + a\right )}{60 \, {\left (e x + d\right )}^{6} e^{5}} \] Input:
integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(3/2)/(e*x+d)^7,x, algorithm="giac ")
Output:
1/60*(2*B*b^6*d - 3*B*a*b^5*e + A*b^6*e)*sgn(b*x + a)/(b^3*d^3*e^5 - 3*a*b ^2*d^2*e^6 + 3*a^2*b*d*e^7 - a^3*e^8) - 1/60*(30*B*b^3*e^4*x^4*sgn(b*x + a ) + 40*B*b^3*d*e^3*x^3*sgn(b*x + a) + 60*B*a*b^2*e^4*x^3*sgn(b*x + a) + 20 *A*b^3*e^4*x^3*sgn(b*x + a) + 30*B*b^3*d^2*e^2*x^2*sgn(b*x + a) + 45*B*a*b ^2*d*e^3*x^2*sgn(b*x + a) + 15*A*b^3*d*e^3*x^2*sgn(b*x + a) + 45*B*a^2*b*e ^4*x^2*sgn(b*x + a) + 45*A*a*b^2*e^4*x^2*sgn(b*x + a) + 12*B*b^3*d^3*e*x*s gn(b*x + a) + 18*B*a*b^2*d^2*e^2*x*sgn(b*x + a) + 6*A*b^3*d^2*e^2*x*sgn(b* x + a) + 18*B*a^2*b*d*e^3*x*sgn(b*x + a) + 18*A*a*b^2*d*e^3*x*sgn(b*x + a) + 12*B*a^3*e^4*x*sgn(b*x + a) + 36*A*a^2*b*e^4*x*sgn(b*x + a) + 2*B*b^3*d ^4*sgn(b*x + a) + 3*B*a*b^2*d^3*e*sgn(b*x + a) + A*b^3*d^3*e*sgn(b*x + a) + 3*B*a^2*b*d^2*e^2*sgn(b*x + a) + 3*A*a*b^2*d^2*e^2*sgn(b*x + a) + 2*B*a^ 3*d*e^3*sgn(b*x + a) + 6*A*a^2*b*d*e^3*sgn(b*x + a) + 10*A*a^3*e^4*sgn(b*x + a))/((e*x + d)^6*e^5)
Time = 11.68 (sec) , antiderivative size = 577, normalized size of antiderivative = 3.58 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^7} \, dx=-\frac {\left (\frac {A\,b^3\,e-3\,B\,b^3\,d+3\,B\,a\,b^2\,e}{3\,e^5}-\frac {B\,b^3\,d}{3\,e^5}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^3}-\frac {\left (\frac {A\,a^3}{6\,e}-\frac {d\,\left (\frac {B\,a^3+3\,A\,b\,a^2}{6\,e}+\frac {d\,\left (\frac {d\,\left (\frac {A\,b^3+3\,B\,a\,b^2}{6\,e}-\frac {B\,b^3\,d}{6\,e^2}\right )}{e}-\frac {a\,b\,\left (A\,b+B\,a\right )}{2\,e}\right )}{e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^6}-\frac {\left (\frac {B\,a^3\,e^3-3\,B\,a^2\,b\,d\,e^2+3\,A\,a^2\,b\,e^3+3\,B\,a\,b^2\,d^2\,e-3\,A\,a\,b^2\,d\,e^2-B\,b^3\,d^3+A\,b^3\,d^2\,e}{5\,e^5}-\frac {d\,\left (\frac {3\,B\,a^2\,b\,e^3-3\,B\,a\,b^2\,d\,e^2+3\,A\,a\,b^2\,e^3+B\,b^3\,d^2\,e-A\,b^3\,d\,e^2}{5\,e^5}-\frac {d\,\left (\frac {b^2\,\left (A\,b\,e+3\,B\,a\,e-B\,b\,d\right )}{5\,e^3}-\frac {B\,b^3\,d}{5\,e^3}\right )}{e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^5}-\frac {\left (\frac {3\,B\,a^2\,b\,e^2-6\,B\,a\,b^2\,d\,e+3\,A\,a\,b^2\,e^2+3\,B\,b^3\,d^2-2\,A\,b^3\,d\,e}{4\,e^5}-\frac {d\,\left (\frac {b^2\,\left (A\,b\,e+3\,B\,a\,e-2\,B\,b\,d\right )}{4\,e^4}-\frac {B\,b^3\,d}{4\,e^4}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^4}-\frac {B\,b^3\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{2\,e^5\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^2} \] Input:
int(((A + B*x)*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2))/(d + e*x)^7,x)
Output:
- (((A*b^3*e - 3*B*b^3*d + 3*B*a*b^2*e)/(3*e^5) - (B*b^3*d)/(3*e^5))*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^3) - (((A*a^3)/(6*e) - (d *((B*a^3 + 3*A*a^2*b)/(6*e) + (d*((d*((A*b^3 + 3*B*a*b^2)/(6*e) - (B*b^3*d )/(6*e^2)))/e - (a*b*(A*b + B*a))/(2*e)))/e))/e)*(a^2 + b^2*x^2 + 2*a*b*x) ^(1/2))/((a + b*x)*(d + e*x)^6) - (((B*a^3*e^3 - B*b^3*d^3 + 3*A*a^2*b*e^3 + A*b^3*d^2*e - 3*A*a*b^2*d*e^2 + 3*B*a*b^2*d^2*e - 3*B*a^2*b*d*e^2)/(5*e ^5) - (d*((3*A*a*b^2*e^3 + 3*B*a^2*b*e^3 - A*b^3*d*e^2 + B*b^3*d^2*e - 3*B *a*b^2*d*e^2)/(5*e^5) - (d*((b^2*(A*b*e + 3*B*a*e - B*b*d))/(5*e^3) - (B*b ^3*d)/(5*e^3)))/e))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e* x)^5) - (((3*B*b^3*d^2 - 2*A*b^3*d*e + 3*A*a*b^2*e^2 + 3*B*a^2*b*e^2 - 6*B *a*b^2*d*e)/(4*e^5) - (d*((b^2*(A*b*e + 3*B*a*e - 2*B*b*d))/(4*e^4) - (B*b ^3*d)/(4*e^4)))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^4 ) - (B*b^3*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(2*e^5*(a + b*x)*(d + e*x)^2)
Time = 0.23 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.49 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^7} \, dx=\frac {-15 b^{4} e^{4} x^{4}-40 a \,b^{3} e^{4} x^{3}-20 b^{4} d \,e^{3} x^{3}-45 a^{2} b^{2} e^{4} x^{2}-30 a \,b^{3} d \,e^{3} x^{2}-15 b^{4} d^{2} e^{2} x^{2}-24 a^{3} b \,e^{4} x -18 a^{2} b^{2} d \,e^{3} x -12 a \,b^{3} d^{2} e^{2} x -6 b^{4} d^{3} e x -5 a^{4} e^{4}-4 a^{3} b d \,e^{3}-3 a^{2} b^{2} d^{2} e^{2}-2 a \,b^{3} d^{3} e -b^{4} d^{4}}{30 e^{5} \left (e^{6} x^{6}+6 d \,e^{5} x^{5}+15 d^{2} e^{4} x^{4}+20 d^{3} e^{3} x^{3}+15 d^{4} e^{2} x^{2}+6 d^{5} e x +d^{6}\right )} \] Input:
int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(3/2)/(e*x+d)^7,x)
Output:
( - 5*a**4*e**4 - 4*a**3*b*d*e**3 - 24*a**3*b*e**4*x - 3*a**2*b**2*d**2*e* *2 - 18*a**2*b**2*d*e**3*x - 45*a**2*b**2*e**4*x**2 - 2*a*b**3*d**3*e - 12 *a*b**3*d**2*e**2*x - 30*a*b**3*d*e**3*x**2 - 40*a*b**3*e**4*x**3 - b**4*d **4 - 6*b**4*d**3*e*x - 15*b**4*d**2*e**2*x**2 - 20*b**4*d*e**3*x**3 - 15* b**4*e**4*x**4)/(30*e**5*(d**6 + 6*d**5*e*x + 15*d**4*e**2*x**2 + 20*d**3* e**3*x**3 + 15*d**2*e**4*x**4 + 6*d*e**5*x**5 + e**6*x**6))