\(\int \frac {(d+e x)^{9/2}}{(a^2+2 a b x+b^2 x^2)^3} \, dx\) [192]

Optimal result

Integrand size = 28, antiderivative size = 178 \[ \int \frac {(d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {63 e^4 \sqrt {d+e x}}{128 b^5 (a+b x)}-\frac {21 e^3 (d+e x)^{3/2}}{64 b^4 (a+b x)^2}-\frac {21 e^2 (d+e x)^{5/2}}{80 b^3 (a+b x)^3}-\frac {9 e (d+e x)^{7/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{9/2}}{5 b (a+b x)^5}-\frac {63 e^5 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{11/2} \sqrt {b d-a e}} \] Output:

-63/128*e^4*(e*x+d)^(1/2)/b^5/(b*x+a)-21/64*e^3*(e*x+d)^(3/2)/b^4/(b*x+a)^ 
2-21/80*e^2*(e*x+d)^(5/2)/b^3/(b*x+a)^3-9/40*e*(e*x+d)^(7/2)/b^2/(b*x+a)^4 
-1/5*(e*x+d)^(9/2)/b/(b*x+a)^5-63/128*e^5*arctanh(b^(1/2)*(e*x+d)^(1/2)/(- 
a*e+b*d)^(1/2))/b^(11/2)/(-a*e+b*d)^(1/2)
 

Mathematica [A] (verified)

Time = 1.38 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.19 \[ \int \frac {(d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {\sqrt {d+e x} \left (315 a^4 e^4+210 a^3 b e^3 (d+7 e x)+42 a^2 b^2 e^2 \left (4 d^2+23 d e x+64 e^2 x^2\right )+6 a b^3 e \left (24 d^3+128 d^2 e x+289 d e^2 x^2+395 e^3 x^3\right )+b^4 \left (128 d^4+656 d^3 e x+1368 d^2 e^2 x^2+1490 d e^3 x^3+965 e^4 x^4\right )\right )}{640 b^5 (a+b x)^5}+\frac {63 e^5 \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{128 b^{11/2} \sqrt {-b d+a e}} \] Input:

Integrate[(d + e*x)^(9/2)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]
 

Output:

-1/640*(Sqrt[d + e*x]*(315*a^4*e^4 + 210*a^3*b*e^3*(d + 7*e*x) + 42*a^2*b^ 
2*e^2*(4*d^2 + 23*d*e*x + 64*e^2*x^2) + 6*a*b^3*e*(24*d^3 + 128*d^2*e*x + 
289*d*e^2*x^2 + 395*e^3*x^3) + b^4*(128*d^4 + 656*d^3*e*x + 1368*d^2*e^2*x 
^2 + 1490*d*e^3*x^3 + 965*e^4*x^4)))/(b^5*(a + b*x)^5) + (63*e^5*ArcTan[(S 
qrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]])/(128*b^(11/2)*Sqrt[-(b*d) + a*e 
])
 

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.11, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {1098, 27, 51, 51, 51, 51, 51, 73, 221}

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.

\(\displaystyle \int \frac {(d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx\)

\(\Big \downarrow \) 1098

\(\displaystyle b^6 \int \frac {(d+e x)^{9/2}}{b^6 (a+b x)^6}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \int \frac {(d+e x)^{9/2}}{(a+b x)^6}dx\)

\(\Big \downarrow \) 51

\(\displaystyle \frac {9 e \int \frac {(d+e x)^{7/2}}{(a+b x)^5}dx}{10 b}-\frac {(d+e x)^{9/2}}{5 b (a+b x)^5}\)

\(\Big \downarrow \) 51

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

\(\Big \downarrow \) 51

\(\displaystyle \frac {9 e \left (\frac {7 e \left (\frac {5 e \int \frac {(d+e x)^{3/2}}{(a+b x)^3}dx}{6 b}-\frac {(d+e x)^{5/2}}{3 b (a+b x)^3}\right )}{8 b}-\frac {(d+e x)^{7/2}}{4 b (a+b x)^4}\right )}{10 b}-\frac {(d+e x)^{9/2}}{5 b (a+b x)^5}\)

\(\Big \downarrow \) 51

\(\displaystyle \frac {9 e \left (\frac {7 e \left (\frac {5 e \left (\frac {3 e \int \frac {\sqrt {d+e x}}{(a+b x)^2}dx}{4 b}-\frac {(d+e x)^{3/2}}{2 b (a+b x)^2}\right )}{6 b}-\frac {(d+e x)^{5/2}}{3 b (a+b x)^3}\right )}{8 b}-\frac {(d+e x)^{7/2}}{4 b (a+b x)^4}\right )}{10 b}-\frac {(d+e x)^{9/2}}{5 b (a+b x)^5}\)

\(\Big \downarrow \) 51

\(\displaystyle \frac {9 e \left (\frac {7 e \left (\frac {5 e \left (\frac {3 e \left (\frac {e \int \frac {1}{(a+b x) \sqrt {d+e x}}dx}{2 b}-\frac {\sqrt {d+e x}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{3/2}}{2 b (a+b x)^2}\right )}{6 b}-\frac {(d+e x)^{5/2}}{3 b (a+b x)^3}\right )}{8 b}-\frac {(d+e x)^{7/2}}{4 b (a+b x)^4}\right )}{10 b}-\frac {(d+e x)^{9/2}}{5 b (a+b x)^5}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {9 e \left (\frac {7 e \left (\frac {5 e \left (\frac {3 e \left (\frac {\int \frac {1}{a+\frac {b (d+e x)}{e}-\frac {b d}{e}}d\sqrt {d+e x}}{b}-\frac {\sqrt {d+e x}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{3/2}}{2 b (a+b x)^2}\right )}{6 b}-\frac {(d+e x)^{5/2}}{3 b (a+b x)^3}\right )}{8 b}-\frac {(d+e x)^{7/2}}{4 b (a+b x)^4}\right )}{10 b}-\frac {(d+e x)^{9/2}}{5 b (a+b x)^5}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {9 e \left (\frac {7 e \left (\frac {5 e \left (\frac {3 e \left (-\frac {e \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{3/2} \sqrt {b d-a e}}-\frac {\sqrt {d+e x}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{3/2}}{2 b (a+b x)^2}\right )}{6 b}-\frac {(d+e x)^{5/2}}{3 b (a+b x)^3}\right )}{8 b}-\frac {(d+e x)^{7/2}}{4 b (a+b x)^4}\right )}{10 b}-\frac {(d+e x)^{9/2}}{5 b (a+b x)^5}\)

Input:

Int[(d + e*x)^(9/2)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]
 

Output:

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

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 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) 
Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x 
] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 1098

Int[((d_.) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_ 
Symbol] :> Simp[1/c^p   Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[ 
{a, b, c, d, e, m}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
 

Maple [A] (verified)

Time = 3.57 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.19

method	result	size
pseudoelliptic	\(\frac {\frac {63 e^{5} \left (b x +a \right )^{5} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )}{128}-\frac {63 \sqrt {e x +d}\, \sqrt {b \left (a e -b d \right )}\, \left (\left (\frac {193}{63} e^{4} x^{4}+\frac {298}{63} d \,e^{3} x^{3}+\frac {152}{35} d^{2} e^{2} x^{2}+\frac {656}{315} d^{3} e x +\frac {128}{315} d^{4}\right ) b^{4}+\frac {16 e \left (\frac {395}{24} e^{3} x^{3}+\frac {289}{24} d \,e^{2} x^{2}+\frac {16}{3} d^{2} e x +d^{3}\right ) a \,b^{3}}{35}+\frac {8 e^{2} \left (16 e^{2} x^{2}+\frac {23}{4} d e x +d^{2}\right ) a^{2} b^{2}}{15}+\frac {2 a^{3} e^{3} \left (7 e x +d \right ) b}{3}+a^{4} e^{4}\right )}{128}}{b^{5} \left (b x +a \right )^{5} \sqrt {b \left (a e -b d \right )}}\)	\(211\)
derivativedivides	\(2 e^{5} \left (\frac {-\frac {193 \left (e x +d \right )^{\frac {9}{2}}}{256 b}-\frac {237 \left (a e -b d \right ) \left (e x +d \right )^{\frac {7}{2}}}{128 b^{2}}-\frac {21 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{10 b^{3}}-\frac {147 \left (e^{3} a^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}}{128 b^{4}}-\frac {63 \left (a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) \sqrt {e x +d}}{256 b^{5}}}{\left (b \left (e x +d \right )+a e -b d \right )^{5}}+\frac {63 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )}{256 b^{5} \sqrt {b \left (a e -b d \right )}}\right )\)	\(239\)
default	\(2 e^{5} \left (\frac {-\frac {193 \left (e x +d \right )^{\frac {9}{2}}}{256 b}-\frac {237 \left (a e -b d \right ) \left (e x +d \right )^{\frac {7}{2}}}{128 b^{2}}-\frac {21 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{10 b^{3}}-\frac {147 \left (e^{3} a^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}}{128 b^{4}}-\frac {63 \left (a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) \sqrt {e x +d}}{256 b^{5}}}{\left (b \left (e x +d \right )+a e -b d \right )^{5}}+\frac {63 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )}{256 b^{5} \sqrt {b \left (a e -b d \right )}}\right )\)	\(239\)

Input:

int((e*x+d)^(9/2)/(b^2*x^2+2*a*b*x+a^2)^3,x,method=_RETURNVERBOSE)
 

Output:

63/128*(e^5*(b*x+a)^5*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))-(e*x+d)^ 
(1/2)*(b*(a*e-b*d))^(1/2)*((193/63*e^4*x^4+298/63*d*e^3*x^3+152/35*d^2*e^2 
*x^2+656/315*d^3*e*x+128/315*d^4)*b^4+16/35*e*(395/24*e^3*x^3+289/24*d*e^2 
*x^2+16/3*d^2*e*x+d^3)*a*b^3+8/15*e^2*(16*e^2*x^2+23/4*d*e*x+d^2)*a^2*b^2+ 
2/3*a^3*e^3*(7*e*x+d)*b+a^4*e^4))/(b*(a*e-b*d))^(1/2)/b^5/(b*x+a)^5
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 495 vs. \(2 (146) = 292\).

Time = 0.13 (sec) , antiderivative size = 1003, normalized size of antiderivative = 5.63 \[ \int \frac {(d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx =\text {Too large to display} \] Input:

integrate((e*x+d)^(9/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")
 

Output:

[1/1280*(315*(b^5*e^5*x^5 + 5*a*b^4*e^5*x^4 + 10*a^2*b^3*e^5*x^3 + 10*a^3* 
b^2*e^5*x^2 + 5*a^4*b*e^5*x + a^5*e^5)*sqrt(b^2*d - a*b*e)*log((b*e*x + 2* 
b*d - a*e - 2*sqrt(b^2*d - a*b*e)*sqrt(e*x + d))/(b*x + a)) - 2*(128*b^6*d 
^5 + 16*a*b^5*d^4*e + 24*a^2*b^4*d^3*e^2 + 42*a^3*b^3*d^2*e^3 + 105*a^4*b^ 
2*d*e^4 - 315*a^5*b*e^5 + 965*(b^6*d*e^4 - a*b^5*e^5)*x^4 + 10*(149*b^6*d^ 
2*e^3 + 88*a*b^5*d*e^4 - 237*a^2*b^4*e^5)*x^3 + 6*(228*b^6*d^3*e^2 + 61*a* 
b^5*d^2*e^3 + 159*a^2*b^4*d*e^4 - 448*a^3*b^3*e^5)*x^2 + 2*(328*b^6*d^4*e 
+ 56*a*b^5*d^3*e^2 + 99*a^2*b^4*d^2*e^3 + 252*a^3*b^3*d*e^4 - 735*a^4*b^2* 
e^5)*x)*sqrt(e*x + d))/(a^5*b^7*d - a^6*b^6*e + (b^12*d - a*b^11*e)*x^5 + 
5*(a*b^11*d - a^2*b^10*e)*x^4 + 10*(a^2*b^10*d - a^3*b^9*e)*x^3 + 10*(a^3* 
b^9*d - a^4*b^8*e)*x^2 + 5*(a^4*b^8*d - a^5*b^7*e)*x), 1/640*(315*(b^5*e^5 
*x^5 + 5*a*b^4*e^5*x^4 + 10*a^2*b^3*e^5*x^3 + 10*a^3*b^2*e^5*x^2 + 5*a^4*b 
*e^5*x + a^5*e^5)*sqrt(-b^2*d + a*b*e)*arctan(sqrt(-b^2*d + a*b*e)*sqrt(e* 
x + d)/(b*e*x + b*d)) - (128*b^6*d^5 + 16*a*b^5*d^4*e + 24*a^2*b^4*d^3*e^2 
 + 42*a^3*b^3*d^2*e^3 + 105*a^4*b^2*d*e^4 - 315*a^5*b*e^5 + 965*(b^6*d*e^4 
 - a*b^5*e^5)*x^4 + 10*(149*b^6*d^2*e^3 + 88*a*b^5*d*e^4 - 237*a^2*b^4*e^5 
)*x^3 + 6*(228*b^6*d^3*e^2 + 61*a*b^5*d^2*e^3 + 159*a^2*b^4*d*e^4 - 448*a^ 
3*b^3*e^5)*x^2 + 2*(328*b^6*d^4*e + 56*a*b^5*d^3*e^2 + 99*a^2*b^4*d^2*e^3 
+ 252*a^3*b^3*d*e^4 - 735*a^4*b^2*e^5)*x)*sqrt(e*x + d))/(a^5*b^7*d - a^6* 
b^6*e + (b^12*d - a*b^11*e)*x^5 + 5*(a*b^11*d - a^2*b^10*e)*x^4 + 10*(a...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Timed out} \] Input:

integrate((e*x+d)**(9/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((e*x+d)^(9/2)/(b^2*x^2+2*a*b*x+a^2)^3,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
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 330 vs. \(2 (146) = 292\).

Time = 0.20 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.85 \[ \int \frac {(d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {63 \, e^{5} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{128 \, \sqrt {-b^{2} d + a b e} b^{5}} - \frac {965 \, {\left (e x + d\right )}^{\frac {9}{2}} b^{4} e^{5} - 2370 \, {\left (e x + d\right )}^{\frac {7}{2}} b^{4} d e^{5} + 2688 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{4} d^{2} e^{5} - 1470 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{4} d^{3} e^{5} + 315 \, \sqrt {e x + d} b^{4} d^{4} e^{5} + 2370 \, {\left (e x + d\right )}^{\frac {7}{2}} a b^{3} e^{6} - 5376 \, {\left (e x + d\right )}^{\frac {5}{2}} a b^{3} d e^{6} + 4410 \, {\left (e x + d\right )}^{\frac {3}{2}} a b^{3} d^{2} e^{6} - 1260 \, \sqrt {e x + d} a b^{3} d^{3} e^{6} + 2688 \, {\left (e x + d\right )}^{\frac {5}{2}} a^{2} b^{2} e^{7} - 4410 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{2} b^{2} d e^{7} + 1890 \, \sqrt {e x + d} a^{2} b^{2} d^{2} e^{7} + 1470 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{3} b e^{8} - 1260 \, \sqrt {e x + d} a^{3} b d e^{8} + 315 \, \sqrt {e x + d} a^{4} e^{9}}{640 \, {\left ({\left (e x + d\right )} b - b d + a e\right )}^{5} b^{5}} \] Input:

integrate((e*x+d)^(9/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")
 

Output:

63/128*e^5*arctan(sqrt(e*x + d)*b/sqrt(-b^2*d + a*b*e))/(sqrt(-b^2*d + a*b 
*e)*b^5) - 1/640*(965*(e*x + d)^(9/2)*b^4*e^5 - 2370*(e*x + d)^(7/2)*b^4*d 
*e^5 + 2688*(e*x + d)^(5/2)*b^4*d^2*e^5 - 1470*(e*x + d)^(3/2)*b^4*d^3*e^5 
 + 315*sqrt(e*x + d)*b^4*d^4*e^5 + 2370*(e*x + d)^(7/2)*a*b^3*e^6 - 5376*( 
e*x + d)^(5/2)*a*b^3*d*e^6 + 4410*(e*x + d)^(3/2)*a*b^3*d^2*e^6 - 1260*sqr 
t(e*x + d)*a*b^3*d^3*e^6 + 2688*(e*x + d)^(5/2)*a^2*b^2*e^7 - 4410*(e*x + 
d)^(3/2)*a^2*b^2*d*e^7 + 1890*sqrt(e*x + d)*a^2*b^2*d^2*e^7 + 1470*(e*x + 
d)^(3/2)*a^3*b*e^8 - 1260*sqrt(e*x + d)*a^3*b*d*e^8 + 315*sqrt(e*x + d)*a^ 
4*e^9)/(((e*x + d)*b - b*d + a*e)^5*b^5)
 

Mupad [B] (verification not implemented)

Time = 9.41 (sec) , antiderivative size = 480, normalized size of antiderivative = 2.70 \[ \int \frac {(d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {63\,e^5\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}}{\sqrt {a\,e-b\,d}}\right )}{128\,b^{11/2}\,\sqrt {a\,e-b\,d}}-\frac {\frac {193\,e^5\,{\left (d+e\,x\right )}^{9/2}}{128\,b}+\frac {63\,e^5\,\sqrt {d+e\,x}\,\left (a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4\right )}{128\,b^5}+\frac {21\,e^5\,{\left (d+e\,x\right )}^{5/2}\,\left (a^2\,e^2-2\,a\,b\,d\,e+b^2\,d^2\right )}{5\,b^3}+\frac {147\,e^5\,{\left (d+e\,x\right )}^{3/2}\,\left (a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3\right )}{64\,b^4}+\frac {237\,e^5\,\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{7/2}}{64\,b^2}}{\left (d+e\,x\right )\,\left (5\,a^4\,b\,e^4-20\,a^3\,b^2\,d\,e^3+30\,a^2\,b^3\,d^2\,e^2-20\,a\,b^4\,d^3\,e+5\,b^5\,d^4\right )-{\left (d+e\,x\right )}^2\,\left (-10\,a^3\,b^2\,e^3+30\,a^2\,b^3\,d\,e^2-30\,a\,b^4\,d^2\,e+10\,b^5\,d^3\right )+b^5\,{\left (d+e\,x\right )}^5-\left (5\,b^5\,d-5\,a\,b^4\,e\right )\,{\left (d+e\,x\right )}^4+a^5\,e^5-b^5\,d^5+{\left (d+e\,x\right )}^3\,\left (10\,a^2\,b^3\,e^2-20\,a\,b^4\,d\,e+10\,b^5\,d^2\right )-10\,a^2\,b^3\,d^3\,e^2+10\,a^3\,b^2\,d^2\,e^3+5\,a\,b^4\,d^4\,e-5\,a^4\,b\,d\,e^4} \] Input:

int((d + e*x)^(9/2)/(a^2 + b^2*x^2 + 2*a*b*x)^3,x)
 

Output:

(63*e^5*atan((b^(1/2)*(d + e*x)^(1/2))/(a*e - b*d)^(1/2)))/(128*b^(11/2)*( 
a*e - b*d)^(1/2)) - ((193*e^5*(d + e*x)^(9/2))/(128*b) + (63*e^5*(d + e*x) 
^(1/2)*(a^4*e^4 + b^4*d^4 + 6*a^2*b^2*d^2*e^2 - 4*a*b^3*d^3*e - 4*a^3*b*d* 
e^3))/(128*b^5) + (21*e^5*(d + e*x)^(5/2)*(a^2*e^2 + b^2*d^2 - 2*a*b*d*e)) 
/(5*b^3) + (147*e^5*(d + e*x)^(3/2)*(a^3*e^3 - b^3*d^3 + 3*a*b^2*d^2*e - 3 
*a^2*b*d*e^2))/(64*b^4) + (237*e^5*(a*e - b*d)*(d + e*x)^(7/2))/(64*b^2))/ 
((d + e*x)*(5*b^5*d^4 + 5*a^4*b*e^4 - 20*a^3*b^2*d*e^3 + 30*a^2*b^3*d^2*e^ 
2 - 20*a*b^4*d^3*e) - (d + e*x)^2*(10*b^5*d^3 - 10*a^3*b^2*e^3 + 30*a^2*b^ 
3*d*e^2 - 30*a*b^4*d^2*e) + b^5*(d + e*x)^5 - (5*b^5*d - 5*a*b^4*e)*(d + e 
*x)^4 + a^5*e^5 - b^5*d^5 + (d + e*x)^3*(10*b^5*d^2 + 10*a^2*b^3*e^2 - 20* 
a*b^4*d*e) - 10*a^2*b^3*d^3*e^2 + 10*a^3*b^2*d^2*e^3 + 5*a*b^4*d^4*e - 5*a 
^4*b*d*e^4)
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 779, normalized size of antiderivative = 4.38 \[ \int \frac {(d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {128 \sqrt {e x +d}\, b^{6} d^{5}-315 \sqrt {e x +d}\, a^{5} b \,e^{5}+315 \sqrt {b}\, \sqrt {a e -b d}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, b}{\sqrt {b}\, \sqrt {a e -b d}}\right ) a^{5} e^{5}+105 \sqrt {e x +d}\, a^{4} b^{2} d \,e^{4}-1470 \sqrt {e x +d}\, a^{4} b^{2} e^{5} x +42 \sqrt {e x +d}\, a^{3} b^{3} d^{2} e^{3}-2688 \sqrt {e x +d}\, a^{3} b^{3} e^{5} x^{2}+24 \sqrt {e x +d}\, a^{2} b^{4} d^{3} e^{2}-2370 \sqrt {e x +d}\, a^{2} b^{4} e^{5} x^{3}+16 \sqrt {e x +d}\, a \,b^{5} d^{4} e -965 \sqrt {e x +d}\, a \,b^{5} e^{5} x^{4}+656 \sqrt {e x +d}\, b^{6} d^{4} e x +1368 \sqrt {e x +d}\, b^{6} d^{3} e^{2} x^{2}+1490 \sqrt {e x +d}\, b^{6} d^{2} e^{3} x^{3}+965 \sqrt {e x +d}\, b^{6} d \,e^{4} x^{4}+315 \sqrt {b}\, \sqrt {a e -b d}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, b}{\sqrt {b}\, \sqrt {a e -b d}}\right ) b^{5} e^{5} x^{5}+1575 \sqrt {b}\, \sqrt {a e -b d}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, b}{\sqrt {b}\, \sqrt {a e -b d}}\right ) a \,b^{4} e^{5} x^{4}+1575 \sqrt {b}\, \sqrt {a e -b d}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, b}{\sqrt {b}\, \sqrt {a e -b d}}\right ) a^{4} b \,e^{5} x +3150 \sqrt {b}\, \sqrt {a e -b d}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, b}{\sqrt {b}\, \sqrt {a e -b d}}\right ) a^{3} b^{2} e^{5} x^{2}+3150 \sqrt {b}\, \sqrt {a e -b d}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, b}{\sqrt {b}\, \sqrt {a e -b d}}\right ) a^{2} b^{3} e^{5} x^{3}+504 \sqrt {e x +d}\, a^{3} b^{3} d \,e^{4} x +198 \sqrt {e x +d}\, a^{2} b^{4} d^{2} e^{3} x +954 \sqrt {e x +d}\, a^{2} b^{4} d \,e^{4} x^{2}+112 \sqrt {e x +d}\, a \,b^{5} d^{3} e^{2} x +366 \sqrt {e x +d}\, a \,b^{5} d^{2} e^{3} x^{2}+880 \sqrt {e x +d}\, a \,b^{5} d \,e^{4} x^{3}}{640 b^{6} \left (a \,b^{5} e \,x^{5}-b^{6} d \,x^{5}+5 a^{2} b^{4} e \,x^{4}-5 a \,b^{5} d \,x^{4}+10 a^{3} b^{3} e \,x^{3}-10 a^{2} b^{4} d \,x^{3}+10 a^{4} b^{2} e \,x^{2}-10 a^{3} b^{3} d \,x^{2}+5 a^{5} b e x -5 a^{4} b^{2} d x +a^{6} e -a^{5} b d \right )} \] Input:

int((e*x+d)^(9/2)/(b^2*x^2+2*a*b*x+a^2)^3,x)
 

Output:

(315*sqrt(b)*sqrt(a*e - b*d)*atan((sqrt(d + e*x)*b)/(sqrt(b)*sqrt(a*e - b* 
d)))*a**5*e**5 + 1575*sqrt(b)*sqrt(a*e - b*d)*atan((sqrt(d + e*x)*b)/(sqrt 
(b)*sqrt(a*e - b*d)))*a**4*b*e**5*x + 3150*sqrt(b)*sqrt(a*e - b*d)*atan((s 
qrt(d + e*x)*b)/(sqrt(b)*sqrt(a*e - b*d)))*a**3*b**2*e**5*x**2 + 3150*sqrt 
(b)*sqrt(a*e - b*d)*atan((sqrt(d + e*x)*b)/(sqrt(b)*sqrt(a*e - b*d)))*a**2 
*b**3*e**5*x**3 + 1575*sqrt(b)*sqrt(a*e - b*d)*atan((sqrt(d + e*x)*b)/(sqr 
t(b)*sqrt(a*e - b*d)))*a*b**4*e**5*x**4 + 315*sqrt(b)*sqrt(a*e - b*d)*atan 
((sqrt(d + e*x)*b)/(sqrt(b)*sqrt(a*e - b*d)))*b**5*e**5*x**5 - 315*sqrt(d 
+ e*x)*a**5*b*e**5 + 105*sqrt(d + e*x)*a**4*b**2*d*e**4 - 1470*sqrt(d + e* 
x)*a**4*b**2*e**5*x + 42*sqrt(d + e*x)*a**3*b**3*d**2*e**3 + 504*sqrt(d + 
e*x)*a**3*b**3*d*e**4*x - 2688*sqrt(d + e*x)*a**3*b**3*e**5*x**2 + 24*sqrt 
(d + e*x)*a**2*b**4*d**3*e**2 + 198*sqrt(d + e*x)*a**2*b**4*d**2*e**3*x + 
954*sqrt(d + e*x)*a**2*b**4*d*e**4*x**2 - 2370*sqrt(d + e*x)*a**2*b**4*e** 
5*x**3 + 16*sqrt(d + e*x)*a*b**5*d**4*e + 112*sqrt(d + e*x)*a*b**5*d**3*e* 
*2*x + 366*sqrt(d + e*x)*a*b**5*d**2*e**3*x**2 + 880*sqrt(d + e*x)*a*b**5* 
d*e**4*x**3 - 965*sqrt(d + e*x)*a*b**5*e**5*x**4 + 128*sqrt(d + e*x)*b**6* 
d**5 + 656*sqrt(d + e*x)*b**6*d**4*e*x + 1368*sqrt(d + e*x)*b**6*d**3*e**2 
*x**2 + 1490*sqrt(d + e*x)*b**6*d**2*e**3*x**3 + 965*sqrt(d + e*x)*b**6*d* 
e**4*x**4)/(640*b**6*(a**6*e - a**5*b*d + 5*a**5*b*e*x - 5*a**4*b**2*d*x + 
 10*a**4*b**2*e*x**2 - 10*a**3*b**3*d*x**2 + 10*a**3*b**3*e*x**3 - 10*a...