Integrand size = 28, antiderivative size = 46 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^7} \, dx=\frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{6 (b d-a e) (d+e x)^6} \] Output:
1/6*(b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(-a*e+b*d)/(e*x+d)^6
Leaf count is larger than twice the leaf count of optimal. \(218\) vs. \(2(46)=92\).
Time = 1.07 (sec) , antiderivative size = 218, normalized size of antiderivative = 4.74 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^7} \, dx=-\frac {\sqrt {(a+b x)^2} \left (a^5 e^5+a^4 b e^4 (d+6 e x)+a^3 b^2 e^3 \left (d^2+6 d e x+15 e^2 x^2\right )+a^2 b^3 e^2 \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )+a b^4 e \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )+b^5 \left (d^5+6 d^4 e x+15 d^3 e^2 x^2+20 d^2 e^3 x^3+15 d e^4 x^4+6 e^5 x^5\right )\right )}{6 e^6 (a+b x) (d+e x)^6} \] Input:
Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^7,x]
Output:
-1/6*(Sqrt[(a + b*x)^2]*(a^5*e^5 + a^4*b*e^4*(d + 6*e*x) + a^3*b^2*e^3*(d^ 2 + 6*d*e*x + 15*e^2*x^2) + a^2*b^3*e^2*(d^3 + 6*d^2*e*x + 15*d*e^2*x^2 + 20*e^3*x^3) + a*b^4*e*(d^4 + 6*d^3*e*x + 15*d^2*e^2*x^2 + 20*d*e^3*x^3 + 1 5*e^4*x^4) + b^5*(d^5 + 6*d^4*e*x + 15*d^3*e^2*x^2 + 20*d^2*e^3*x^3 + 15*d *e^4*x^4 + 6*e^5*x^5)))/(e^6*(a + b*x)*(d + e*x)^6)
Time = 0.30 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.04, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {1102, 27, 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.
\(\displaystyle \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^7} \, dx\) |
\(\Big \downarrow \) 1102 |
\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {b^5 (a+b x)^5}{(d+e x)^7}dx}{b^5 (a+b x)}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x)^5}{(d+e x)^7}dx}{a+b x}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle \frac {(a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{6 (d+e x)^6 (b d-a e)}\) |
Input:
Int[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^7,x]
Output:
((a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*(b*d - a*e)*(d + e*x)^6)
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[((d_.) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*F racPart[p])) Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(246\) vs. \(2(42)=84\).
Time = 3.04 (sec) , antiderivative size = 247, normalized size of antiderivative = 5.37
method | result | size |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {b^{5} x^{5}}{e}-\frac {5 b^{4} \left (a e +b d \right ) x^{4}}{2 e^{2}}-\frac {10 b^{3} \left (e^{2} a^{2}+a b d e +b^{2} d^{2}\right ) x^{3}}{3 e^{3}}-\frac {5 b^{2} \left (e^{3} a^{3}+a^{2} b d \,e^{2}+a \,b^{2} d^{2} e +b^{3} d^{3}\right ) x^{2}}{2 e^{4}}-\frac {b \left (a^{4} e^{4}+a^{3} b d \,e^{3}+a^{2} b^{2} d^{2} e^{2}+a \,b^{3} d^{3} e +b^{4} d^{4}\right ) x}{e^{5}}-\frac {e^{5} a^{5}+a^{4} b d \,e^{4}+a^{3} b^{2} d^{2} e^{3}+a^{2} b^{3} d^{3} e^{2}+a \,b^{4} d^{4} e +b^{5} d^{5}}{6 e^{6}}\right )}{\left (b x +a \right ) \left (e x +d \right )^{6}}\) | \(247\) |
gosper | \(-\frac {\left (6 x^{5} e^{5} b^{5}+15 x^{4} a \,b^{4} e^{5}+15 x^{4} b^{5} d \,e^{4}+20 x^{3} a^{2} b^{3} e^{5}+20 x^{3} a \,b^{4} d \,e^{4}+20 x^{3} b^{5} d^{2} e^{3}+15 x^{2} a^{3} b^{2} e^{5}+15 x^{2} a^{2} b^{3} d \,e^{4}+15 x^{2} a \,b^{4} d^{2} e^{3}+15 x^{2} b^{5} d^{3} e^{2}+6 a^{4} b \,e^{5} x +6 a^{3} b^{2} d \,e^{4} x +6 x \,a^{2} b^{3} d^{2} e^{3}+6 x a \,b^{4} d^{3} e^{2}+6 b^{5} d^{4} e x +e^{5} a^{5}+a^{4} b d \,e^{4}+a^{3} b^{2} d^{2} e^{3}+a^{2} b^{3} d^{3} e^{2}+a \,b^{4} d^{4} e +b^{5} d^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{6 \left (e x +d \right )^{6} e^{6} \left (b x +a \right )^{5}}\) | \(283\) |
default | \(-\frac {\left (6 x^{5} e^{5} b^{5}+15 x^{4} a \,b^{4} e^{5}+15 x^{4} b^{5} d \,e^{4}+20 x^{3} a^{2} b^{3} e^{5}+20 x^{3} a \,b^{4} d \,e^{4}+20 x^{3} b^{5} d^{2} e^{3}+15 x^{2} a^{3} b^{2} e^{5}+15 x^{2} a^{2} b^{3} d \,e^{4}+15 x^{2} a \,b^{4} d^{2} e^{3}+15 x^{2} b^{5} d^{3} e^{2}+6 a^{4} b \,e^{5} x +6 a^{3} b^{2} d \,e^{4} x +6 x \,a^{2} b^{3} d^{2} e^{3}+6 x a \,b^{4} d^{3} e^{2}+6 b^{5} d^{4} e x +e^{5} a^{5}+a^{4} b d \,e^{4}+a^{3} b^{2} d^{2} e^{3}+a^{2} b^{3} d^{3} e^{2}+a \,b^{4} d^{4} e +b^{5} d^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{6 \left (e x +d \right )^{6} e^{6} \left (b x +a \right )^{5}}\) | \(283\) |
orering | \(-\frac {\left (6 x^{5} e^{5} b^{5}+15 x^{4} a \,b^{4} e^{5}+15 x^{4} b^{5} d \,e^{4}+20 x^{3} a^{2} b^{3} e^{5}+20 x^{3} a \,b^{4} d \,e^{4}+20 x^{3} b^{5} d^{2} e^{3}+15 x^{2} a^{3} b^{2} e^{5}+15 x^{2} a^{2} b^{3} d \,e^{4}+15 x^{2} a \,b^{4} d^{2} e^{3}+15 x^{2} b^{5} d^{3} e^{2}+6 a^{4} b \,e^{5} x +6 a^{3} b^{2} d \,e^{4} x +6 x \,a^{2} b^{3} d^{2} e^{3}+6 x a \,b^{4} d^{3} e^{2}+6 b^{5} d^{4} e x +e^{5} a^{5}+a^{4} b d \,e^{4}+a^{3} b^{2} d^{2} e^{3}+a^{2} b^{3} d^{3} e^{2}+a \,b^{4} d^{4} e +b^{5} d^{5}\right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{\frac {5}{2}}}{6 e^{6} \left (b x +a \right )^{5} \left (e x +d \right )^{6}}\) | \(292\) |
Input:
int((b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^7,x,method=_RETURNVERBOSE)
Output:
((b*x+a)^2)^(1/2)/(b*x+a)*(-b^5/e*x^5-5/2*b^4*(a*e+b*d)/e^2*x^4-10/3*b^3*( a^2*e^2+a*b*d*e+b^2*d^2)/e^3*x^3-5/2*b^2*(a^3*e^3+a^2*b*d*e^2+a*b^2*d^2*e+ b^3*d^3)/e^4*x^2-b*(a^4*e^4+a^3*b*d*e^3+a^2*b^2*d^2*e^2+a*b^3*d^3*e+b^4*d^ 4)/e^5*x-1/6*(a^5*e^5+a^4*b*d*e^4+a^3*b^2*d^2*e^3+a^2*b^3*d^3*e^2+a*b^4*d^ 4*e+b^5*d^5)/e^6)/(e*x+d)^6
Leaf count of result is larger than twice the leaf count of optimal. 300 vs. \(2 (42) = 84\).
Time = 0.08 (sec) , antiderivative size = 300, normalized size of antiderivative = 6.52 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^7} \, dx=-\frac {6 \, b^{5} e^{5} x^{5} + b^{5} d^{5} + a b^{4} d^{4} e + a^{2} b^{3} d^{3} e^{2} + a^{3} b^{2} d^{2} e^{3} + a^{4} b d e^{4} + a^{5} e^{5} + 15 \, {\left (b^{5} d e^{4} + a b^{4} e^{5}\right )} x^{4} + 20 \, {\left (b^{5} d^{2} e^{3} + a b^{4} d e^{4} + a^{2} b^{3} e^{5}\right )} x^{3} + 15 \, {\left (b^{5} d^{3} e^{2} + a b^{4} d^{2} e^{3} + a^{2} b^{3} d e^{4} + a^{3} b^{2} e^{5}\right )} x^{2} + 6 \, {\left (b^{5} d^{4} e + a b^{4} d^{3} e^{2} + a^{2} b^{3} d^{2} e^{3} + a^{3} b^{2} d e^{4} + a^{4} b e^{5}\right )} x}{6 \, {\left (e^{12} x^{6} + 6 \, d e^{11} x^{5} + 15 \, d^{2} e^{10} x^{4} + 20 \, d^{3} e^{9} x^{3} + 15 \, d^{4} e^{8} x^{2} + 6 \, d^{5} e^{7} x + d^{6} e^{6}\right )}} \] Input:
integrate((b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^7,x, algorithm="fricas")
Output:
-1/6*(6*b^5*e^5*x^5 + b^5*d^5 + a*b^4*d^4*e + a^2*b^3*d^3*e^2 + a^3*b^2*d^ 2*e^3 + a^4*b*d*e^4 + a^5*e^5 + 15*(b^5*d*e^4 + a*b^4*e^5)*x^4 + 20*(b^5*d ^2*e^3 + a*b^4*d*e^4 + a^2*b^3*e^5)*x^3 + 15*(b^5*d^3*e^2 + a*b^4*d^2*e^3 + a^2*b^3*d*e^4 + a^3*b^2*e^5)*x^2 + 6*(b^5*d^4*e + a*b^4*d^3*e^2 + a^2*b^ 3*d^2*e^3 + a^3*b^2*d*e^4 + a^4*b*e^5)*x)/(e^12*x^6 + 6*d*e^11*x^5 + 15*d^ 2*e^10*x^4 + 20*d^3*e^9*x^3 + 15*d^4*e^8*x^2 + 6*d^5*e^7*x + d^6*e^6)
Exception generated. \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^7} \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:
integrate((b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**7,x)
Output:
Exception raised: HeuristicGCDFailed >> no luck
Exception generated. \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^7} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^7,x, algorithm="maxima")
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. 419 vs. \(2 (42) = 84\).
Time = 0.51 (sec) , antiderivative size = 419, normalized size of antiderivative = 9.11 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^7} \, dx=\frac {b^{6} \mathrm {sgn}\left (b x + a\right )}{6 \, {\left (b d e^{6} - a e^{7}\right )}} - \frac {6 \, b^{5} e^{5} x^{5} \mathrm {sgn}\left (b x + a\right ) + 15 \, b^{5} d e^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + 15 \, a b^{4} e^{5} x^{4} \mathrm {sgn}\left (b x + a\right ) + 20 \, b^{5} d^{2} e^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 20 \, a b^{4} d e^{4} x^{3} \mathrm {sgn}\left (b x + a\right ) + 20 \, a^{2} b^{3} e^{5} x^{3} \mathrm {sgn}\left (b x + a\right ) + 15 \, b^{5} d^{3} e^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 15 \, a b^{4} d^{2} e^{3} x^{2} \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{2} b^{3} d e^{4} x^{2} \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{3} b^{2} e^{5} x^{2} \mathrm {sgn}\left (b x + a\right ) + 6 \, b^{5} d^{4} e x \mathrm {sgn}\left (b x + a\right ) + 6 \, a b^{4} d^{3} e^{2} x \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{2} b^{3} d^{2} e^{3} x \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{3} b^{2} d e^{4} x \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{4} b e^{5} x \mathrm {sgn}\left (b x + a\right ) + b^{5} d^{5} \mathrm {sgn}\left (b x + a\right ) + a b^{4} d^{4} e \mathrm {sgn}\left (b x + a\right ) + a^{2} b^{3} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + a^{3} b^{2} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + a^{4} b d e^{4} \mathrm {sgn}\left (b x + a\right ) + a^{5} e^{5} \mathrm {sgn}\left (b x + a\right )}{6 \, {\left (e x + d\right )}^{6} e^{6}} \] Input:
integrate((b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^7,x, algorithm="giac")
Output:
1/6*b^6*sgn(b*x + a)/(b*d*e^6 - a*e^7) - 1/6*(6*b^5*e^5*x^5*sgn(b*x + a) + 15*b^5*d*e^4*x^4*sgn(b*x + a) + 15*a*b^4*e^5*x^4*sgn(b*x + a) + 20*b^5*d^ 2*e^3*x^3*sgn(b*x + a) + 20*a*b^4*d*e^4*x^3*sgn(b*x + a) + 20*a^2*b^3*e^5* x^3*sgn(b*x + a) + 15*b^5*d^3*e^2*x^2*sgn(b*x + a) + 15*a*b^4*d^2*e^3*x^2* sgn(b*x + a) + 15*a^2*b^3*d*e^4*x^2*sgn(b*x + a) + 15*a^3*b^2*e^5*x^2*sgn( b*x + a) + 6*b^5*d^4*e*x*sgn(b*x + a) + 6*a*b^4*d^3*e^2*x*sgn(b*x + a) + 6 *a^2*b^3*d^2*e^3*x*sgn(b*x + a) + 6*a^3*b^2*d*e^4*x*sgn(b*x + a) + 6*a^4*b *e^5*x*sgn(b*x + a) + b^5*d^5*sgn(b*x + a) + a*b^4*d^4*e*sgn(b*x + a) + a^ 2*b^3*d^3*e^2*sgn(b*x + a) + a^3*b^2*d^2*e^3*sgn(b*x + a) + a^4*b*d*e^4*sg n(b*x + a) + a^5*e^5*sgn(b*x + a))/((e*x + d)^6*e^6)
Time = 5.16 (sec) , antiderivative size = 687, normalized size of antiderivative = 14.93 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^7} \, dx=\frac {\left (\frac {4\,b^5\,d-5\,a\,b^4\,e}{2\,e^6}+\frac {b^5\,d}{2\,e^6}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^2}-\frac {\left (\frac {5\,a^4\,b\,e^4-10\,a^3\,b^2\,d\,e^3+10\,a^2\,b^3\,d^2\,e^2-5\,a\,b^4\,d^3\,e+b^5\,d^4}{5\,e^6}+\frac {d\,\left (\frac {-10\,a^3\,b^2\,e^4+10\,a^2\,b^3\,d\,e^3-5\,a\,b^4\,d^2\,e^2+b^5\,d^3\,e}{5\,e^6}+\frac {d\,\left (\frac {d\,\left (\frac {b^5\,d}{5\,e^3}-\frac {b^4\,\left (5\,a\,e-b\,d\right )}{5\,e^3}\right )}{e}+\frac {b^3\,\left (10\,a^2\,e^2-5\,a\,b\,d\,e+b^2\,d^2\right )}{5\,e^4}\right )}{e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^5}-\frac {\left (\frac {10\,a^2\,b^3\,e^2-15\,a\,b^4\,d\,e+6\,b^5\,d^2}{3\,e^6}+\frac {d\,\left (\frac {b^5\,d}{3\,e^5}-\frac {b^4\,\left (5\,a\,e-3\,b\,d\right )}{3\,e^5}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^3}-\frac {\left (\frac {a^5}{6\,e}-\frac {d\,\left (\frac {5\,a^4\,b}{6\,e}-\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {5\,a\,b^4}{6\,e}-\frac {b^5\,d}{6\,e^2}\right )}{e}-\frac {5\,a^2\,b^3}{3\,e}\right )}{e}+\frac {5\,a^3\,b^2}{3\,e}\right )}{e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^6}+\frac {\left (\frac {-10\,a^3\,b^2\,e^3+20\,a^2\,b^3\,d\,e^2-15\,a\,b^4\,d^2\,e+4\,b^5\,d^3}{4\,e^6}+\frac {d\,\left (\frac {d\,\left (\frac {b^5\,d}{4\,e^4}-\frac {b^4\,\left (5\,a\,e-2\,b\,d\right )}{4\,e^4}\right )}{e}+\frac {b^3\,\left (10\,a^2\,e^2-10\,a\,b\,d\,e+3\,b^2\,d^2\right )}{4\,e^5}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^4}-\frac {b^5\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{e^6\,\left (a+b\,x\right )\,\left (d+e\,x\right )} \] Input:
int((a^2 + b^2*x^2 + 2*a*b*x)^(5/2)/(d + e*x)^7,x)
Output:
(((4*b^5*d - 5*a*b^4*e)/(2*e^6) + (b^5*d)/(2*e^6))*(a^2 + b^2*x^2 + 2*a*b* x)^(1/2))/((a + b*x)*(d + e*x)^2) - (((b^5*d^4 + 5*a^4*b*e^4 - 10*a^3*b^2* d*e^3 + 10*a^2*b^3*d^2*e^2 - 5*a*b^4*d^3*e)/(5*e^6) + (d*((b^5*d^3*e - 10* a^3*b^2*e^4 - 5*a*b^4*d^2*e^2 + 10*a^2*b^3*d*e^3)/(5*e^6) + (d*((d*((b^5*d )/(5*e^3) - (b^4*(5*a*e - b*d))/(5*e^3)))/e + (b^3*(10*a^2*e^2 + b^2*d^2 - 5*a*b*d*e))/(5*e^4)))/e))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)* (d + e*x)^5) - (((6*b^5*d^2 + 10*a^2*b^3*e^2 - 15*a*b^4*d*e)/(3*e^6) + (d* ((b^5*d)/(3*e^5) - (b^4*(5*a*e - 3*b*d))/(3*e^5)))/e)*(a^2 + b^2*x^2 + 2*a *b*x)^(1/2))/((a + b*x)*(d + e*x)^3) - ((a^5/(6*e) - (d*((5*a^4*b)/(6*e) - (d*((d*((d*((5*a*b^4)/(6*e) - (b^5*d)/(6*e^2)))/e - (5*a^2*b^3)/(3*e)))/e + (5*a^3*b^2)/(3*e)))/e))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)* (d + e*x)^6) + (((4*b^5*d^3 - 10*a^3*b^2*e^3 + 20*a^2*b^3*d*e^2 - 15*a*b^4 *d^2*e)/(4*e^6) + (d*((d*((b^5*d)/(4*e^4) - (b^4*(5*a*e - 2*b*d))/(4*e^4)) )/e + (b^3*(10*a^2*e^2 + 3*b^2*d^2 - 10*a*b*d*e))/(4*e^5)))/e)*(a^2 + b^2* x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^4) - (b^5*(a^2 + b^2*x^2 + 2*a* b*x)^(1/2))/(e^6*(a + b*x)*(d + e*x))
Time = 0.27 (sec) , antiderivative size = 279, normalized size of antiderivative = 6.07 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^7} \, dx=\frac {b^{5} e^{5} x^{6}-15 a \,b^{4} d \,e^{4} x^{4}-20 a^{2} b^{3} d \,e^{4} x^{3}-20 a \,b^{4} d^{2} e^{3} x^{3}-15 a^{3} b^{2} d \,e^{4} x^{2}-15 a^{2} b^{3} d^{2} e^{3} x^{2}-15 a \,b^{4} d^{3} e^{2} x^{2}-6 a^{4} b d \,e^{4} x -6 a^{3} b^{2} d^{2} e^{3} x -6 a^{2} b^{3} d^{3} e^{2} x -6 a \,b^{4} d^{4} e x -a^{5} d \,e^{4}-a^{4} b \,d^{2} e^{3}-a^{3} b^{2} d^{3} e^{2}-a^{2} b^{3} d^{4} e -a \,b^{4} d^{5}}{6 d \,e^{5} \left (e^{6} x^{6}+6 d \,e^{5} x^{5}+15 d^{2} e^{4} x^{4}+20 d^{3} e^{3} x^{3}+15 d^{4} e^{2} x^{2}+6 d^{5} e x +d^{6}\right )} \] Input:
int((b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^7,x)
Output:
( - a**5*d*e**4 - a**4*b*d**2*e**3 - 6*a**4*b*d*e**4*x - a**3*b**2*d**3*e* *2 - 6*a**3*b**2*d**2*e**3*x - 15*a**3*b**2*d*e**4*x**2 - a**2*b**3*d**4*e - 6*a**2*b**3*d**3*e**2*x - 15*a**2*b**3*d**2*e**3*x**2 - 20*a**2*b**3*d* e**4*x**3 - a*b**4*d**5 - 6*a*b**4*d**4*e*x - 15*a*b**4*d**3*e**2*x**2 - 2 0*a*b**4*d**2*e**3*x**3 - 15*a*b**4*d*e**4*x**4 + b**5*e**5*x**6)/(6*d*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))