\(\int \frac {(a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^{11}} \, dx\) [1585]

Optimal result

Integrand size = 28, antiderivative size = 308 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{11}} \, dx=\frac {(b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{10 e^6 (a+b x) (d+e x)^{10}}-\frac {5 b (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^6 (a+b x) (d+e x)^9}+\frac {5 b^2 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^6 (a+b x) (d+e x)^8}-\frac {10 b^3 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^6 (a+b x) (d+e x)^7}+\frac {5 b^4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{6 e^6 (a+b x) (d+e x)^6}-\frac {b^5 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^6 (a+b x) (d+e x)^5} \]

[Out]

1/10*(-a*e+b*d)^5*((b*x+a)^2)^(1/2)/e^6/(b*x+a)/(e*x+d)^10-5/9*b*(-a*e+b*d)^4*((b*x+a)^2)^(1/2)/e^6/(b*x+a)/(e
*x+d)^9+5/4*b^2*(-a*e+b*d)^3*((b*x+a)^2)^(1/2)/e^6/(b*x+a)/(e*x+d)^8-10/7*b^3*(-a*e+b*d)^2*((b*x+a)^2)^(1/2)/e
^6/(b*x+a)/(e*x+d)^7+5/6*b^4*(-a*e+b*d)*((b*x+a)^2)^(1/2)/e^6/(b*x+a)/(e*x+d)^6-1/5*b^5*((b*x+a)^2)^(1/2)/e^6/
(b*x+a)/(e*x+d)^5

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {660, 45} \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{11}} \, dx=\frac {5 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{4 e^6 (a+b x) (d+e x)^8}-\frac {5 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{9 e^6 (a+b x) (d+e x)^9}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{10 e^6 (a+b x) (d+e x)^{10}}-\frac {b^5 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^6 (a+b x) (d+e x)^5}+\frac {5 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{6 e^6 (a+b x) (d+e x)^6}-\frac {10 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{7 e^6 (a+b x) (d+e x)^7} \]

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^11,x]

[Out]

((b*d - a*e)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(10*e^6*(a + b*x)*(d + e*x)^10) - (5*b*(b*d - a*e)^4*Sqrt[a^2 +
2*a*b*x + b^2*x^2])/(9*e^6*(a + b*x)*(d + e*x)^9) + (5*b^2*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*e^6
*(a + b*x)*(d + e*x)^8) - (10*b^3*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^6*(a + b*x)*(d + e*x)^7) +
(5*b^4*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*e^6*(a + b*x)*(d + e*x)^6) - (b^5*Sqrt[a^2 + 2*a*b*x + b
^2*x^2])/(5*e^6*(a + b*x)*(d + e*x)^5)

Rule 45

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] && NeQ[b*c - a*d, 0] && 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])

Rule 660

Int[((d_.) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(a + b*x + c*x^2)^Fra
cPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[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] &&  !IntegerQ[p] && NeQ[2*c*d - b*e, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^5}{(d+e x)^{11}} \, dx}{b^4 \left (a b+b^2 x\right )} \\ & = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (-\frac {b^5 (b d-a e)^5}{e^5 (d+e x)^{11}}+\frac {5 b^6 (b d-a e)^4}{e^5 (d+e x)^{10}}-\frac {10 b^7 (b d-a e)^3}{e^5 (d+e x)^9}+\frac {10 b^8 (b d-a e)^2}{e^5 (d+e x)^8}-\frac {5 b^9 (b d-a e)}{e^5 (d+e x)^7}+\frac {b^{10}}{e^5 (d+e x)^6}\right ) \, dx}{b^4 \left (a b+b^2 x\right )} \\ & = \frac {(b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{10 e^6 (a+b x) (d+e x)^{10}}-\frac {5 b (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^6 (a+b x) (d+e x)^9}+\frac {5 b^2 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^6 (a+b x) (d+e x)^8}-\frac {10 b^3 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^6 (a+b x) (d+e x)^7}+\frac {5 b^4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{6 e^6 (a+b x) (d+e x)^6}-\frac {b^5 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^6 (a+b x) (d+e x)^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.05 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.72 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{11}} \, dx=-\frac {\sqrt {(a+b x)^2} \left (126 a^5 e^5+70 a^4 b e^4 (d+10 e x)+35 a^3 b^2 e^3 \left (d^2+10 d e x+45 e^2 x^2\right )+15 a^2 b^3 e^2 \left (d^3+10 d^2 e x+45 d e^2 x^2+120 e^3 x^3\right )+5 a b^4 e \left (d^4+10 d^3 e x+45 d^2 e^2 x^2+120 d e^3 x^3+210 e^4 x^4\right )+b^5 \left (d^5+10 d^4 e x+45 d^3 e^2 x^2+120 d^2 e^3 x^3+210 d e^4 x^4+252 e^5 x^5\right )\right )}{1260 e^6 (a+b x) (d+e x)^{10}} \]

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^11,x]

[Out]

-1/1260*(Sqrt[(a + b*x)^2]*(126*a^5*e^5 + 70*a^4*b*e^4*(d + 10*e*x) + 35*a^3*b^2*e^3*(d^2 + 10*d*e*x + 45*e^2*
x^2) + 15*a^2*b^3*e^2*(d^3 + 10*d^2*e*x + 45*d*e^2*x^2 + 120*e^3*x^3) + 5*a*b^4*e*(d^4 + 10*d^3*e*x + 45*d^2*e
^2*x^2 + 120*d*e^3*x^3 + 210*e^4*x^4) + b^5*(d^5 + 10*d^4*e*x + 45*d^3*e^2*x^2 + 120*d^2*e^3*x^3 + 210*d*e^4*x
^4 + 252*e^5*x^5)))/(e^6*(a + b*x)*(d + e*x)^10)

Maple [A] (verified)

Time = 7.77 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.85

method	result	size
risch	\(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {b^{5} x^{5}}{5 e}-\frac {b^{4} \left (5 a e +b d \right ) x^{4}}{6 e^{2}}-\frac {2 b^{3} \left (15 a^{2} e^{2}+5 a b d e +b^{2} d^{2}\right ) x^{3}}{21 e^{3}}-\frac {b^{2} \left (35 a^{3} e^{3}+15 a^{2} b d \,e^{2}+5 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) x^{2}}{28 e^{4}}-\frac {b \left (70 e^{4} a^{4}+35 b \,e^{3} d \,a^{3}+15 b^{2} e^{2} d^{2} a^{2}+5 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) x}{126 e^{5}}-\frac {126 a^{5} e^{5}+70 a^{4} b d \,e^{4}+35 a^{3} b^{2} d^{2} e^{3}+15 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e +b^{5} d^{5}}{1260 e^{6}}\right )}{\left (b x +a \right ) \left (e x +d \right )^{10}}\)	\(262\)
gosper	\(-\frac {\left (252 x^{5} e^{5} b^{5}+1050 x^{4} a \,b^{4} e^{5}+210 x^{4} b^{5} d \,e^{4}+1800 x^{3} a^{2} b^{3} e^{5}+600 x^{3} a \,b^{4} d \,e^{4}+120 x^{3} b^{5} d^{2} e^{3}+1575 x^{2} a^{3} b^{2} e^{5}+675 x^{2} a^{2} b^{3} d \,e^{4}+225 x^{2} a \,b^{4} d^{2} e^{3}+45 x^{2} b^{5} d^{3} e^{2}+700 a^{4} b \,e^{5} x +350 a^{3} b^{2} d \,e^{4} x +150 x \,a^{2} b^{3} d^{2} e^{3}+50 x a \,b^{4} d^{3} e^{2}+10 b^{5} d^{4} e x +126 a^{5} e^{5}+70 a^{4} b d \,e^{4}+35 a^{3} b^{2} d^{2} e^{3}+15 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e +b^{5} d^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{1260 e^{6} \left (e x +d \right )^{10} \left (b x +a \right )^{5}}\)	\(288\)
default	\(-\frac {\left (252 x^{5} e^{5} b^{5}+1050 x^{4} a \,b^{4} e^{5}+210 x^{4} b^{5} d \,e^{4}+1800 x^{3} a^{2} b^{3} e^{5}+600 x^{3} a \,b^{4} d \,e^{4}+120 x^{3} b^{5} d^{2} e^{3}+1575 x^{2} a^{3} b^{2} e^{5}+675 x^{2} a^{2} b^{3} d \,e^{4}+225 x^{2} a \,b^{4} d^{2} e^{3}+45 x^{2} b^{5} d^{3} e^{2}+700 a^{4} b \,e^{5} x +350 a^{3} b^{2} d \,e^{4} x +150 x \,a^{2} b^{3} d^{2} e^{3}+50 x a \,b^{4} d^{3} e^{2}+10 b^{5} d^{4} e x +126 a^{5} e^{5}+70 a^{4} b d \,e^{4}+35 a^{3} b^{2} d^{2} e^{3}+15 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e +b^{5} d^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{1260 e^{6} \left (e x +d \right )^{10} \left (b x +a \right )^{5}}\)	\(288\)

[In]

int((b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^11,x,method=_RETURNVERBOSE)

[Out]

((b*x+a)^2)^(1/2)/(b*x+a)*(-1/5*b^5/e*x^5-1/6*b^4/e^2*(5*a*e+b*d)*x^4-2/21*b^3/e^3*(15*a^2*e^2+5*a*b*d*e+b^2*d
^2)*x^3-1/28*b^2/e^4*(35*a^3*e^3+15*a^2*b*d*e^2+5*a*b^2*d^2*e+b^3*d^3)*x^2-1/126*b/e^5*(70*a^4*e^4+35*a^3*b*d*
e^3+15*a^2*b^2*d^2*e^2+5*a*b^3*d^3*e+b^4*d^4)*x-1/1260/e^6*(126*a^5*e^5+70*a^4*b*d*e^4+35*a^3*b^2*d^2*e^3+15*a
^2*b^3*d^3*e^2+5*a*b^4*d^4*e+b^5*d^5))/(e*x+d)^10

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.17 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{11}} \, dx=-\frac {252 \, b^{5} e^{5} x^{5} + b^{5} d^{5} + 5 \, a b^{4} d^{4} e + 15 \, a^{2} b^{3} d^{3} e^{2} + 35 \, a^{3} b^{2} d^{2} e^{3} + 70 \, a^{4} b d e^{4} + 126 \, a^{5} e^{5} + 210 \, {\left (b^{5} d e^{4} + 5 \, a b^{4} e^{5}\right )} x^{4} + 120 \, {\left (b^{5} d^{2} e^{3} + 5 \, a b^{4} d e^{4} + 15 \, a^{2} b^{3} e^{5}\right )} x^{3} + 45 \, {\left (b^{5} d^{3} e^{2} + 5 \, a b^{4} d^{2} e^{3} + 15 \, a^{2} b^{3} d e^{4} + 35 \, a^{3} b^{2} e^{5}\right )} x^{2} + 10 \, {\left (b^{5} d^{4} e + 5 \, a b^{4} d^{3} e^{2} + 15 \, a^{2} b^{3} d^{2} e^{3} + 35 \, a^{3} b^{2} d e^{4} + 70 \, a^{4} b e^{5}\right )} x}{1260 \, {\left (e^{16} x^{10} + 10 \, d e^{15} x^{9} + 45 \, d^{2} e^{14} x^{8} + 120 \, d^{3} e^{13} x^{7} + 210 \, d^{4} e^{12} x^{6} + 252 \, d^{5} e^{11} x^{5} + 210 \, d^{6} e^{10} x^{4} + 120 \, d^{7} e^{9} x^{3} + 45 \, d^{8} e^{8} x^{2} + 10 \, d^{9} e^{7} x + d^{10} e^{6}\right )}} \]

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^11,x, algorithm="fricas")

[Out]

-1/1260*(252*b^5*e^5*x^5 + b^5*d^5 + 5*a*b^4*d^4*e + 15*a^2*b^3*d^3*e^2 + 35*a^3*b^2*d^2*e^3 + 70*a^4*b*d*e^4
+ 126*a^5*e^5 + 210*(b^5*d*e^4 + 5*a*b^4*e^5)*x^4 + 120*(b^5*d^2*e^3 + 5*a*b^4*d*e^4 + 15*a^2*b^3*e^5)*x^3 + 4
5*(b^5*d^3*e^2 + 5*a*b^4*d^2*e^3 + 15*a^2*b^3*d*e^4 + 35*a^3*b^2*e^5)*x^2 + 10*(b^5*d^4*e + 5*a*b^4*d^3*e^2 +
15*a^2*b^3*d^2*e^3 + 35*a^3*b^2*d*e^4 + 70*a^4*b*e^5)*x)/(e^16*x^10 + 10*d*e^15*x^9 + 45*d^2*e^14*x^8 + 120*d^
3*e^13*x^7 + 210*d^4*e^12*x^6 + 252*d^5*e^11*x^5 + 210*d^6*e^10*x^4 + 120*d^7*e^9*x^3 + 45*d^8*e^8*x^2 + 10*d^
9*e^7*x + d^10*e^6)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{11}} \, dx=\text {Timed out} \]

[In]

integrate((b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**11,x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{11}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^11,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*e-b*d>0)', see `assume?` for
more detail

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 480 vs. \(2 (230) = 460\).

Time = 0.28 (sec) , antiderivative size = 480, normalized size of antiderivative = 1.56 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{11}} \, dx=\frac {b^{10} \mathrm {sgn}\left (b x + a\right )}{1260 \, {\left (b^{5} d^{5} e^{6} - 5 \, a b^{4} d^{4} e^{7} + 10 \, a^{2} b^{3} d^{3} e^{8} - 10 \, a^{3} b^{2} d^{2} e^{9} + 5 \, a^{4} b d e^{10} - a^{5} e^{11}\right )}} - \frac {252 \, b^{5} e^{5} x^{5} \mathrm {sgn}\left (b x + a\right ) + 210 \, b^{5} d e^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + 1050 \, a b^{4} e^{5} x^{4} \mathrm {sgn}\left (b x + a\right ) + 120 \, b^{5} d^{2} e^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 600 \, a b^{4} d e^{4} x^{3} \mathrm {sgn}\left (b x + a\right ) + 1800 \, a^{2} b^{3} e^{5} x^{3} \mathrm {sgn}\left (b x + a\right ) + 45 \, b^{5} d^{3} e^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 225 \, a b^{4} d^{2} e^{3} x^{2} \mathrm {sgn}\left (b x + a\right ) + 675 \, a^{2} b^{3} d e^{4} x^{2} \mathrm {sgn}\left (b x + a\right ) + 1575 \, a^{3} b^{2} e^{5} x^{2} \mathrm {sgn}\left (b x + a\right ) + 10 \, b^{5} d^{4} e x \mathrm {sgn}\left (b x + a\right ) + 50 \, a b^{4} d^{3} e^{2} x \mathrm {sgn}\left (b x + a\right ) + 150 \, a^{2} b^{3} d^{2} e^{3} x \mathrm {sgn}\left (b x + a\right ) + 350 \, a^{3} b^{2} d e^{4} x \mathrm {sgn}\left (b x + a\right ) + 700 \, a^{4} b e^{5} x \mathrm {sgn}\left (b x + a\right ) + b^{5} d^{5} \mathrm {sgn}\left (b x + a\right ) + 5 \, a b^{4} d^{4} e \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{2} b^{3} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + 35 \, a^{3} b^{2} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 70 \, a^{4} b d e^{4} \mathrm {sgn}\left (b x + a\right ) + 126 \, a^{5} e^{5} \mathrm {sgn}\left (b x + a\right )}{1260 \, {\left (e x + d\right )}^{10} e^{6}} \]

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^11,x, algorithm="giac")

[Out]

1/1260*b^10*sgn(b*x + a)/(b^5*d^5*e^6 - 5*a*b^4*d^4*e^7 + 10*a^2*b^3*d^3*e^8 - 10*a^3*b^2*d^2*e^9 + 5*a^4*b*d*
e^10 - a^5*e^11) - 1/1260*(252*b^5*e^5*x^5*sgn(b*x + a) + 210*b^5*d*e^4*x^4*sgn(b*x + a) + 1050*a*b^4*e^5*x^4*
sgn(b*x + a) + 120*b^5*d^2*e^3*x^3*sgn(b*x + a) + 600*a*b^4*d*e^4*x^3*sgn(b*x + a) + 1800*a^2*b^3*e^5*x^3*sgn(
b*x + a) + 45*b^5*d^3*e^2*x^2*sgn(b*x + a) + 225*a*b^4*d^2*e^3*x^2*sgn(b*x + a) + 675*a^2*b^3*d*e^4*x^2*sgn(b*
x + a) + 1575*a^3*b^2*e^5*x^2*sgn(b*x + a) + 10*b^5*d^4*e*x*sgn(b*x + a) + 50*a*b^4*d^3*e^2*x*sgn(b*x + a) + 1
50*a^2*b^3*d^2*e^3*x*sgn(b*x + a) + 350*a^3*b^2*d*e^4*x*sgn(b*x + a) + 700*a^4*b*e^5*x*sgn(b*x + a) + b^5*d^5*
sgn(b*x + a) + 5*a*b^4*d^4*e*sgn(b*x + a) + 15*a^2*b^3*d^3*e^2*sgn(b*x + a) + 35*a^3*b^2*d^2*e^3*sgn(b*x + a)
+ 70*a^4*b*d*e^4*sgn(b*x + a) + 126*a^5*e^5*sgn(b*x + a))/((e*x + d)^10*e^6)

Mupad [B] (verification not implemented)

Time = 9.64 (sec) , antiderivative size = 686, normalized size of antiderivative = 2.23 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{11}} \, dx=\frac {\left (\frac {4\,b^5\,d-5\,a\,b^4\,e}{6\,e^6}+\frac {b^5\,d}{6\,e^6}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^6}-\frac {\left (\frac {5\,a^4\,b\,e^4-10\,a^3\,b^2\,d\,e^3+10\,a^2\,b^3\,d^2\,e^2-5\,a\,b^4\,d^3\,e+b^5\,d^4}{9\,e^6}+\frac {d\,\left (\frac {-10\,a^3\,b^2\,e^4+10\,a^2\,b^3\,d\,e^3-5\,a\,b^4\,d^2\,e^2+b^5\,d^3\,e}{9\,e^6}+\frac {d\,\left (\frac {d\,\left (\frac {b^5\,d}{9\,e^3}-\frac {b^4\,\left (5\,a\,e-b\,d\right )}{9\,e^3}\right )}{e}+\frac {b^3\,\left (10\,a^2\,e^2-5\,a\,b\,d\,e+b^2\,d^2\right )}{9\,e^4}\right )}{e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^9}-\frac {\left (\frac {10\,a^2\,b^3\,e^2-15\,a\,b^4\,d\,e+6\,b^5\,d^2}{7\,e^6}+\frac {d\,\left (\frac {b^5\,d}{7\,e^5}-\frac {b^4\,\left (5\,a\,e-3\,b\,d\right )}{7\,e^5}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^7}-\frac {\left (\frac {a^5}{10\,e}-\frac {d\,\left (\frac {a^4\,b}{2\,e}-\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {a\,b^4}{2\,e}-\frac {b^5\,d}{10\,e^2}\right )}{e}-\frac {a^2\,b^3}{e}\right )}{e}+\frac {a^3\,b^2}{e}\right )}{e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^{10}}+\frac {\left (\frac {-10\,a^3\,b^2\,e^3+20\,a^2\,b^3\,d\,e^2-15\,a\,b^4\,d^2\,e+4\,b^5\,d^3}{8\,e^6}+\frac {d\,\left (\frac {d\,\left (\frac {b^5\,d}{8\,e^4}-\frac {b^4\,\left (5\,a\,e-2\,b\,d\right )}{8\,e^4}\right )}{e}+\frac {b^3\,\left (10\,a^2\,e^2-10\,a\,b\,d\,e+3\,b^2\,d^2\right )}{8\,e^5}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^8}-\frac {b^5\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{5\,e^6\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^5} \]

[In]

int((a^2 + b^2*x^2 + 2*a*b*x)^(5/2)/(d + e*x)^11,x)

[Out]

(((4*b^5*d - 5*a*b^4*e)/(6*e^6) + (b^5*d)/(6*e^6))*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^6) -
(((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)/(9*e^6) + (d*((b^5*d^3*e - 1
0*a^3*b^2*e^4 - 5*a*b^4*d^2*e^2 + 10*a^2*b^3*d*e^3)/(9*e^6) + (d*((d*((b^5*d)/(9*e^3) - (b^4*(5*a*e - b*d))/(9
*e^3)))/e + (b^3*(10*a^2*e^2 + b^2*d^2 - 5*a*b*d*e))/(9*e^4)))/e))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b
*x)*(d + e*x)^9) - (((6*b^5*d^2 + 10*a^2*b^3*e^2 - 15*a*b^4*d*e)/(7*e^6) + (d*((b^5*d)/(7*e^5) - (b^4*(5*a*e -
3*b*d))/(7*e^5)))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^7) - ((a^5/(10*e) - (d*((a^4*b)/(2
*e) - (d*((d*((d*((a*b^4)/(2*e) - (b^5*d)/(10*e^2)))/e - (a^2*b^3)/e))/e + (a^3*b^2)/e))/e))/e)*(a^2 + b^2*x^2
+ 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^10) + (((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)/(8*e^6) + (d*((d*((b^5*d)/(8*e^4) - (b^4*(5*a*e - 2*b*d))/(8*e^4)))/e + (b^3*(10*a^2*e^2 + 3*b^2*d^2 - 10*a
*b*d*e))/(8*e^5)))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^8) - (b^5*(a^2 + b^2*x^2 + 2*a*b*x
)^(1/2))/(5*e^6*(a + b*x)*(d + e*x)^5)