\(\int \frac {1}{(d+e x)^{5/2} (a^2+2 a b x+b^2 x^2)^3} \, dx\) [199]

Optimal result

Integrand size = 28, antiderivative size = 266 \[ \int \frac {1}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {1001 e^5}{128 (b d-a e)^6 (d+e x)^{3/2}}-\frac {1}{5 (b d-a e) (a+b x)^5 (d+e x)^{3/2}}+\frac {13 e}{40 (b d-a e)^2 (a+b x)^4 (d+e x)^{3/2}}-\frac {143 e^2}{240 (b d-a e)^3 (a+b x)^3 (d+e x)^{3/2}}+\frac {429 e^3}{320 (b d-a e)^4 (a+b x)^2 (d+e x)^{3/2}}-\frac {3003 e^4}{640 (b d-a e)^5 (a+b x) (d+e x)^{3/2}}-\frac {3003 b e^5}{128 (b d-a e)^7 \sqrt {d+e x}}+\frac {3003 b^{3/2} e^5 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 (b d-a e)^{15/2}} \] Output:

-1001/128*e^5/(-a*e+b*d)^6/(e*x+d)^(3/2)-1/5/(-a*e+b*d)/(b*x+a)^5/(e*x+d)^ 
(3/2)+13/40*e/(-a*e+b*d)^2/(b*x+a)^4/(e*x+d)^(3/2)-143/240*e^2/(-a*e+b*d)^ 
3/(b*x+a)^3/(e*x+d)^(3/2)+429/320*e^3/(-a*e+b*d)^4/(b*x+a)^2/(e*x+d)^(3/2) 
-3003/640*e^4/(-a*e+b*d)^5/(b*x+a)/(e*x+d)^(3/2)-3003/128*b*e^5/(-a*e+b*d) 
^7/(e*x+d)^(1/2)+3003/128*b^(3/2)*e^5*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+ 
b*d)^(1/2))/(-a*e+b*d)^(15/2)
 

Mathematica [A] (verified)

Time = 2.43 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.34 \[ \int \frac {1}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {\frac {-1280 a^6 e^6+1280 a^5 b e^5 (19 d+13 e x)+5 a^4 b^2 e^4 \left (7119 d^2+38558 d e x+27599 e^2 x^2\right )+10 a^3 b^3 e^3 \left (-2107 d^3+7917 d^2 e x+46475 d e^2 x^2+33891 e^3 x^3\right )+2 a^2 b^4 e^2 \left (5012 d^4-11557 d^3 e x+42042 d^2 e^2 x^2+260403 d e^3 x^3+192192 e^4 x^4\right )+2 a b^5 e \left (-1464 d^5+2704 d^4 e x-6149 d^3 e^2 x^2+21879 d^2 e^3 x^3+141141 d e^4 x^4+105105 e^5 x^5\right )+b^6 \left (384 d^6-624 d^5 e x+1144 d^4 e^2 x^2-2574 d^3 e^3 x^3+9009 d^2 e^4 x^4+60060 d e^5 x^5+45045 e^6 x^6\right )}{(-b d+a e)^7 (a+b x)^5 (d+e x)^{3/2}}+\frac {45045 b^{3/2} e^5 \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{(-b d+a e)^{15/2}}}{1920} \] Input:

Integrate[1/((d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^3),x]
 

Output:

((-1280*a^6*e^6 + 1280*a^5*b*e^5*(19*d + 13*e*x) + 5*a^4*b^2*e^4*(7119*d^2 
 + 38558*d*e*x + 27599*e^2*x^2) + 10*a^3*b^3*e^3*(-2107*d^3 + 7917*d^2*e*x 
 + 46475*d*e^2*x^2 + 33891*e^3*x^3) + 2*a^2*b^4*e^2*(5012*d^4 - 11557*d^3* 
e*x + 42042*d^2*e^2*x^2 + 260403*d*e^3*x^3 + 192192*e^4*x^4) + 2*a*b^5*e*( 
-1464*d^5 + 2704*d^4*e*x - 6149*d^3*e^2*x^2 + 21879*d^2*e^3*x^3 + 141141*d 
*e^4*x^4 + 105105*e^5*x^5) + b^6*(384*d^6 - 624*d^5*e*x + 1144*d^4*e^2*x^2 
 - 2574*d^3*e^3*x^3 + 9009*d^2*e^4*x^4 + 60060*d*e^5*x^5 + 45045*e^6*x^6)) 
/((-(b*d) + a*e)^7*(a + b*x)^5*(d + e*x)^(3/2)) + (45045*b^(3/2)*e^5*ArcTa 
n[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]])/(-(b*d) + a*e)^(15/2))/1920
 

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.25, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {1098, 27, 52, 52, 52, 52, 52, 61, 61, 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.

\(\displaystyle \int \frac {1}{\left (a^2+2 a b x+b^2 x^2\right )^3 (d+e x)^{5/2}} \, dx\)

\(\Big \downarrow \) 1098

\(\displaystyle b^6 \int \frac {1}{b^6 (a+b x)^6 (d+e x)^{5/2}}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \int \frac {1}{(a+b x)^6 (d+e x)^{5/2}}dx\)

\(\Big \downarrow \) 52

\(\displaystyle -\frac {13 e \int \frac {1}{(a+b x)^5 (d+e x)^{5/2}}dx}{10 (b d-a e)}-\frac {1}{5 (a+b x)^5 (d+e x)^{3/2} (b d-a e)}\)

\(\Big \downarrow \) 52

\(\displaystyle -\frac {13 e \left (-\frac {11 e \int \frac {1}{(a+b x)^4 (d+e x)^{5/2}}dx}{8 (b d-a e)}-\frac {1}{4 (a+b x)^4 (d+e x)^{3/2} (b d-a e)}\right )}{10 (b d-a e)}-\frac {1}{5 (a+b x)^5 (d+e x)^{3/2} (b d-a e)}\)

\(\Big \downarrow \) 52

\(\displaystyle -\frac {13 e \left (-\frac {11 e \left (-\frac {3 e \int \frac {1}{(a+b x)^3 (d+e x)^{5/2}}dx}{2 (b d-a e)}-\frac {1}{3 (a+b x)^3 (d+e x)^{3/2} (b d-a e)}\right )}{8 (b d-a e)}-\frac {1}{4 (a+b x)^4 (d+e x)^{3/2} (b d-a e)}\right )}{10 (b d-a e)}-\frac {1}{5 (a+b x)^5 (d+e x)^{3/2} (b d-a e)}\)

\(\Big \downarrow \) 52

\(\displaystyle -\frac {13 e \left (-\frac {11 e \left (-\frac {3 e \left (-\frac {7 e \int \frac {1}{(a+b x)^2 (d+e x)^{5/2}}dx}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 (d+e x)^{3/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{3 (a+b x)^3 (d+e x)^{3/2} (b d-a e)}\right )}{8 (b d-a e)}-\frac {1}{4 (a+b x)^4 (d+e x)^{3/2} (b d-a e)}\right )}{10 (b d-a e)}-\frac {1}{5 (a+b x)^5 (d+e x)^{3/2} (b d-a e)}\)

\(\Big \downarrow \) 52

\(\displaystyle -\frac {13 e \left (-\frac {11 e \left (-\frac {3 e \left (-\frac {7 e \left (-\frac {5 e \int \frac {1}{(a+b x) (d+e x)^{5/2}}dx}{2 (b d-a e)}-\frac {1}{(a+b x) (d+e x)^{3/2} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 (d+e x)^{3/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{3 (a+b x)^3 (d+e x)^{3/2} (b d-a e)}\right )}{8 (b d-a e)}-\frac {1}{4 (a+b x)^4 (d+e x)^{3/2} (b d-a e)}\right )}{10 (b d-a e)}-\frac {1}{5 (a+b x)^5 (d+e x)^{3/2} (b d-a e)}\)

\(\Big \downarrow \) 61

\(\displaystyle -\frac {13 e \left (-\frac {11 e \left (-\frac {3 e \left (-\frac {7 e \left (-\frac {5 e \left (\frac {b \int \frac {1}{(a+b x) (d+e x)^{3/2}}dx}{b d-a e}+\frac {2}{3 (d+e x)^{3/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{(a+b x) (d+e x)^{3/2} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 (d+e x)^{3/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{3 (a+b x)^3 (d+e x)^{3/2} (b d-a e)}\right )}{8 (b d-a e)}-\frac {1}{4 (a+b x)^4 (d+e x)^{3/2} (b d-a e)}\right )}{10 (b d-a e)}-\frac {1}{5 (a+b x)^5 (d+e x)^{3/2} (b d-a e)}\)

\(\Big \downarrow \) 61

\(\displaystyle -\frac {13 e \left (-\frac {11 e \left (-\frac {3 e \left (-\frac {7 e \left (-\frac {5 e \left (\frac {b \left (\frac {b \int \frac {1}{(a+b x) \sqrt {d+e x}}dx}{b d-a e}+\frac {2}{\sqrt {d+e x} (b d-a e)}\right )}{b d-a e}+\frac {2}{3 (d+e x)^{3/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{(a+b x) (d+e x)^{3/2} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 (d+e x)^{3/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{3 (a+b x)^3 (d+e x)^{3/2} (b d-a e)}\right )}{8 (b d-a e)}-\frac {1}{4 (a+b x)^4 (d+e x)^{3/2} (b d-a e)}\right )}{10 (b d-a e)}-\frac {1}{5 (a+b x)^5 (d+e x)^{3/2} (b d-a e)}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {13 e \left (-\frac {11 e \left (-\frac {3 e \left (-\frac {7 e \left (-\frac {5 e \left (\frac {b \left (\frac {2 b \int \frac {1}{a+\frac {b (d+e x)}{e}-\frac {b d}{e}}d\sqrt {d+e x}}{e (b d-a e)}+\frac {2}{\sqrt {d+e x} (b d-a e)}\right )}{b d-a e}+\frac {2}{3 (d+e x)^{3/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{(a+b x) (d+e x)^{3/2} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 (d+e x)^{3/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{3 (a+b x)^3 (d+e x)^{3/2} (b d-a e)}\right )}{8 (b d-a e)}-\frac {1}{4 (a+b x)^4 (d+e x)^{3/2} (b d-a e)}\right )}{10 (b d-a e)}-\frac {1}{5 (a+b x)^5 (d+e x)^{3/2} (b d-a e)}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {13 e \left (-\frac {11 e \left (-\frac {3 e \left (-\frac {7 e \left (-\frac {5 e \left (\frac {b \left (\frac {2}{\sqrt {d+e x} (b d-a e)}-\frac {2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{3/2}}\right )}{b d-a e}+\frac {2}{3 (d+e x)^{3/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{(a+b x) (d+e x)^{3/2} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 (d+e x)^{3/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{3 (a+b x)^3 (d+e x)^{3/2} (b d-a e)}\right )}{8 (b d-a e)}-\frac {1}{4 (a+b x)^4 (d+e x)^{3/2} (b d-a e)}\right )}{10 (b d-a e)}-\frac {1}{5 (a+b x)^5 (d+e x)^{3/2} (b d-a e)}\)

Input:

Int[1/((d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^3),x]
 

Output:

-1/5*1/((b*d - a*e)*(a + b*x)^5*(d + e*x)^(3/2)) - (13*e*(-1/4*1/((b*d - a 
*e)*(a + b*x)^4*(d + e*x)^(3/2)) - (11*e*(-1/3*1/((b*d - a*e)*(a + b*x)^3* 
(d + e*x)^(3/2)) - (3*e*(-1/2*1/((b*d - a*e)*(a + b*x)^2*(d + e*x)^(3/2)) 
- (7*e*(-(1/((b*d - a*e)*(a + b*x)*(d + e*x)^(3/2))) - (5*e*(2/(3*(b*d - a 
*e)*(d + e*x)^(3/2)) + (b*(2/((b*d - a*e)*Sqrt[d + e*x]) - (2*Sqrt[b]*ArcT 
anh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(b*d - a*e)^(3/2)))/(b*d - a 
*e)))/(2*(b*d - a*e))))/(4*(b*d - a*e))))/(2*(b*d - a*e))))/(8*(b*d - a*e) 
)))/(10*(b*d - a*e))
 

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 52

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*(( 
m + n + 2)/((b*c - a*d)*(m + 1)))   Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], 
x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
 

rule 61

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*(( 
m + n + 2)/((b*c - a*d)*(m + 1)))   Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], 
x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0 
] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d 
, m, n, x]
 

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 = 1.71 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.09

method	result	size
derivativedivides	\(2 e^{5} \left (-\frac {1}{3 \left (a e -b d \right )^{6} \left (e x +d \right )^{\frac {3}{2}}}+\frac {6 b}{\left (a e -b d \right )^{7} \sqrt {e x +d}}+\frac {b^{2} \left (\frac {\frac {1467 b^{4} \left (e x +d \right )^{\frac {9}{2}}}{256}+\frac {9629 b^{3} \left (a e -b d \right ) \left (e x +d \right )^{\frac {7}{2}}}{384}+\left (\frac {1253}{30} a^{2} b^{2} e^{2}-\frac {1253}{15} a \,b^{3} d e +\frac {1253}{30} b^{4} d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}+\left (\frac {12131}{384} a^{3} b \,e^{3}-\frac {12131}{128} a^{2} b^{2} d \,e^{2}+\frac {12131}{128} d^{2} e a \,b^{3}-\frac {12131}{384} b^{4} d^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}+\left (\frac {2373}{256} a^{4} e^{4}-\frac {2373}{64} a^{3} b d \,e^{3}+\frac {7119}{128} a^{2} b^{2} d^{2} e^{2}-\frac {2373}{64} a \,b^{3} d^{3} e +\frac {2373}{256} b^{4} d^{4}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{5}}+\frac {3003 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )}{256 \sqrt {b \left (a e -b d \right )}}\right )}{\left (a e -b d \right )^{7}}\right )\)	\(291\)
default	\(2 e^{5} \left (-\frac {1}{3 \left (a e -b d \right )^{6} \left (e x +d \right )^{\frac {3}{2}}}+\frac {6 b}{\left (a e -b d \right )^{7} \sqrt {e x +d}}+\frac {b^{2} \left (\frac {\frac {1467 b^{4} \left (e x +d \right )^{\frac {9}{2}}}{256}+\frac {9629 b^{3} \left (a e -b d \right ) \left (e x +d \right )^{\frac {7}{2}}}{384}+\left (\frac {1253}{30} a^{2} b^{2} e^{2}-\frac {1253}{15} a \,b^{3} d e +\frac {1253}{30} b^{4} d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}+\left (\frac {12131}{384} a^{3} b \,e^{3}-\frac {12131}{128} a^{2} b^{2} d \,e^{2}+\frac {12131}{128} d^{2} e a \,b^{3}-\frac {12131}{384} b^{4} d^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}+\left (\frac {2373}{256} a^{4} e^{4}-\frac {2373}{64} a^{3} b d \,e^{3}+\frac {7119}{128} a^{2} b^{2} d^{2} e^{2}-\frac {2373}{64} a \,b^{3} d^{3} e +\frac {2373}{256} b^{4} d^{4}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{5}}+\frac {3003 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )}{256 \sqrt {b \left (a e -b d \right )}}\right )}{\left (a e -b d \right )^{7}}\right )\)	\(291\)
pseudoelliptic	\(-\frac {2 \left (-\frac {9009 b^{2} e^{5} \left (e x +d \right )^{\frac {3}{2}} \left (b x +a \right )^{5} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )}{256}+\left (\left (-\frac {3003}{64} d \,e^{5} x^{5}-\frac {9009}{1280} d^{2} e^{4} x^{4}+\frac {1287}{640} d^{3} e^{3} x^{3}-\frac {143}{160} d^{4} e^{2} x^{2}+\frac {39}{80} d^{5} e x -\frac {9009}{256} e^{6} x^{6}-\frac {3}{10} d^{6}\right ) b^{6}+\frac {183 e \left (-\frac {35035}{488} e^{5} x^{5}-\frac {47047}{488} d \,e^{4} x^{4}-\frac {7293}{488} d^{2} e^{3} x^{3}+\frac {6149}{1464} d^{3} e^{2} x^{2}-\frac {338}{183} d^{4} e x +d^{5}\right ) a \,b^{5}}{80}-\frac {1253 \left (\frac {6864}{179} e^{4} x^{4}+\frac {260403}{5012} d \,e^{3} x^{3}+\frac {3003}{358} d^{2} e^{2} x^{2}-\frac {1651}{716} d^{3} e x +d^{4}\right ) e^{2} a^{2} b^{4}}{160}+\frac {2107 e^{3} \left (-\frac {33891}{2107} e^{3} x^{3}-\frac {46475}{2107} d \,e^{2} x^{2}-\frac {1131}{301} d^{2} e x +d^{3}\right ) a^{3} b^{3}}{128}-\frac {7119 e^{4} \left (\frac {27599}{7119} e^{2} x^{2}+\frac {38558}{7119} d e x +d^{2}\right ) a^{4} b^{2}}{256}-19 e^{5} \left (\frac {13 e x}{19}+d \right ) a^{5} b +a^{6} e^{6}\right ) \sqrt {b \left (a e -b d \right )}\right )}{3 \left (e x +d \right )^{\frac {3}{2}} \sqrt {b \left (a e -b d \right )}\, \left (a e -b d \right )^{7} \left (b x +a \right )^{5}}\)	\(361\)

Input:

int(1/(e*x+d)^(5/2)/(b^2*x^2+2*a*b*x+a^2)^3,x,method=_RETURNVERBOSE)
 

Output:

2*e^5*(-1/3/(a*e-b*d)^6/(e*x+d)^(3/2)+6/(a*e-b*d)^7*b/(e*x+d)^(1/2)+1/(a*e 
-b*d)^7*b^2*((1467/256*b^4*(e*x+d)^(9/2)+9629/384*b^3*(a*e-b*d)*(e*x+d)^(7 
/2)+(1253/30*a^2*b^2*e^2-1253/15*a*b^3*d*e+1253/30*b^4*d^2)*(e*x+d)^(5/2)+ 
(12131/384*a^3*b*e^3-12131/128*a^2*b^2*d*e^2+12131/128*d^2*e*a*b^3-12131/3 
84*b^4*d^3)*(e*x+d)^(3/2)+(2373/256*a^4*e^4-2373/64*a^3*b*d*e^3+7119/128*a 
^2*b^2*d^2*e^2-2373/64*a*b^3*d^3*e+2373/256*b^4*d^4)*(e*x+d)^(1/2))/(b*(e* 
x+d)+a*e-b*d)^5+3003/256/(b*(a*e-b*d))^(1/2)*arctan(b*(e*x+d)^(1/2)/(b*(a* 
e-b*d))^(1/2))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1597 vs. \(2 (226) = 452\).

Time = 0.75 (sec) , antiderivative size = 3224, normalized size of antiderivative = 12.12 \[ \int \frac {1}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate(1/(e*x+d)^(5/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Timed out} \] Input:

integrate(1/(e*x+d)**(5/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(1/(e*x+d)^(5/2)/(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. 649 vs. \(2 (226) = 452\).

Time = 0.14 (sec) , antiderivative size = 649, normalized size of antiderivative = 2.44 \[ \int \frac {1}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {3003 \, b^{2} e^{5} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{128 \, {\left (b^{7} d^{7} - 7 \, a b^{6} d^{6} e + 21 \, a^{2} b^{5} d^{5} e^{2} - 35 \, a^{3} b^{4} d^{4} e^{3} + 35 \, a^{4} b^{3} d^{3} e^{4} - 21 \, a^{5} b^{2} d^{2} e^{5} + 7 \, a^{6} b d e^{6} - a^{7} e^{7}\right )} \sqrt {-b^{2} d + a b e}} - \frac {2 \, {\left (18 \, {\left (e x + d\right )} b e^{5} + b d e^{5} - a e^{6}\right )}}{3 \, {\left (b^{7} d^{7} - 7 \, a b^{6} d^{6} e + 21 \, a^{2} b^{5} d^{5} e^{2} - 35 \, a^{3} b^{4} d^{4} e^{3} + 35 \, a^{4} b^{3} d^{3} e^{4} - 21 \, a^{5} b^{2} d^{2} e^{5} + 7 \, a^{6} b d e^{6} - a^{7} e^{7}\right )} {\left (e x + d\right )}^{\frac {3}{2}}} - \frac {22005 \, {\left (e x + d\right )}^{\frac {9}{2}} b^{6} e^{5} - 96290 \, {\left (e x + d\right )}^{\frac {7}{2}} b^{6} d e^{5} + 160384 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{6} d^{2} e^{5} - 121310 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{6} d^{3} e^{5} + 35595 \, \sqrt {e x + d} b^{6} d^{4} e^{5} + 96290 \, {\left (e x + d\right )}^{\frac {7}{2}} a b^{5} e^{6} - 320768 \, {\left (e x + d\right )}^{\frac {5}{2}} a b^{5} d e^{6} + 363930 \, {\left (e x + d\right )}^{\frac {3}{2}} a b^{5} d^{2} e^{6} - 142380 \, \sqrt {e x + d} a b^{5} d^{3} e^{6} + 160384 \, {\left (e x + d\right )}^{\frac {5}{2}} a^{2} b^{4} e^{7} - 363930 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{2} b^{4} d e^{7} + 213570 \, \sqrt {e x + d} a^{2} b^{4} d^{2} e^{7} + 121310 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{3} b^{3} e^{8} - 142380 \, \sqrt {e x + d} a^{3} b^{3} d e^{8} + 35595 \, \sqrt {e x + d} a^{4} b^{2} e^{9}}{1920 \, {\left (b^{7} d^{7} - 7 \, a b^{6} d^{6} e + 21 \, a^{2} b^{5} d^{5} e^{2} - 35 \, a^{3} b^{4} d^{4} e^{3} + 35 \, a^{4} b^{3} d^{3} e^{4} - 21 \, a^{5} b^{2} d^{2} e^{5} + 7 \, a^{6} b d e^{6} - a^{7} e^{7}\right )} {\left ({\left (e x + d\right )} b - b d + a e\right )}^{5}} \] Input:

integrate(1/(e*x+d)^(5/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")
 

Output:

-3003/128*b^2*e^5*arctan(sqrt(e*x + d)*b/sqrt(-b^2*d + a*b*e))/((b^7*d^7 - 
 7*a*b^6*d^6*e + 21*a^2*b^5*d^5*e^2 - 35*a^3*b^4*d^4*e^3 + 35*a^4*b^3*d^3* 
e^4 - 21*a^5*b^2*d^2*e^5 + 7*a^6*b*d*e^6 - a^7*e^7)*sqrt(-b^2*d + a*b*e)) 
- 2/3*(18*(e*x + d)*b*e^5 + b*d*e^5 - a*e^6)/((b^7*d^7 - 7*a*b^6*d^6*e + 2 
1*a^2*b^5*d^5*e^2 - 35*a^3*b^4*d^4*e^3 + 35*a^4*b^3*d^3*e^4 - 21*a^5*b^2*d 
^2*e^5 + 7*a^6*b*d*e^6 - a^7*e^7)*(e*x + d)^(3/2)) - 1/1920*(22005*(e*x + 
d)^(9/2)*b^6*e^5 - 96290*(e*x + d)^(7/2)*b^6*d*e^5 + 160384*(e*x + d)^(5/2 
)*b^6*d^2*e^5 - 121310*(e*x + d)^(3/2)*b^6*d^3*e^5 + 35595*sqrt(e*x + d)*b 
^6*d^4*e^5 + 96290*(e*x + d)^(7/2)*a*b^5*e^6 - 320768*(e*x + d)^(5/2)*a*b^ 
5*d*e^6 + 363930*(e*x + d)^(3/2)*a*b^5*d^2*e^6 - 142380*sqrt(e*x + d)*a*b^ 
5*d^3*e^6 + 160384*(e*x + d)^(5/2)*a^2*b^4*e^7 - 363930*(e*x + d)^(3/2)*a^ 
2*b^4*d*e^7 + 213570*sqrt(e*x + d)*a^2*b^4*d^2*e^7 + 121310*(e*x + d)^(3/2 
)*a^3*b^3*e^8 - 142380*sqrt(e*x + d)*a^3*b^3*d*e^8 + 35595*sqrt(e*x + d)*a 
^4*b^2*e^9)/((b^7*d^7 - 7*a*b^6*d^6*e + 21*a^2*b^5*d^5*e^2 - 35*a^3*b^4*d^ 
4*e^3 + 35*a^4*b^3*d^3*e^4 - 21*a^5*b^2*d^2*e^5 + 7*a^6*b*d*e^6 - a^7*e^7) 
*((e*x + d)*b - b*d + a*e)^5)
 

Mupad [B] (verification not implemented)

Time = 10.16 (sec) , antiderivative size = 555, normalized size of antiderivative = 2.09 \[ \int \frac {1}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {\frac {27599\,b^2\,e^5\,{\left (d+e\,x\right )}^2}{384\,{\left (a\,e-b\,d\right )}^3}-\frac {2\,e^5}{3\,\left (a\,e-b\,d\right )}+\frac {11297\,b^3\,e^5\,{\left (d+e\,x\right )}^3}{64\,{\left (a\,e-b\,d\right )}^4}+\frac {1001\,b^4\,e^5\,{\left (d+e\,x\right )}^4}{5\,{\left (a\,e-b\,d\right )}^5}+\frac {7007\,b^5\,e^5\,{\left (d+e\,x\right )}^5}{64\,{\left (a\,e-b\,d\right )}^6}+\frac {3003\,b^6\,e^5\,{\left (d+e\,x\right )}^6}{128\,{\left (a\,e-b\,d\right )}^7}+\frac {26\,b\,e^5\,\left (d+e\,x\right )}{3\,{\left (a\,e-b\,d\right )}^2}}{{\left (d+e\,x\right )}^{3/2}\,\left (a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5\right )-{\left (d+e\,x\right )}^{7/2}\,\left (-10\,a^3\,b^2\,e^3+30\,a^2\,b^3\,d\,e^2-30\,a\,b^4\,d^2\,e+10\,b^5\,d^3\right )+{\left (d+e\,x\right )}^{5/2}\,\left (5\,a^4\,b\,e^4-20\,a^3\,b^2\,d\,e^3+30\,a^2\,b^3\,d^2\,e^2-20\,a\,b^4\,d^3\,e+5\,b^5\,d^4\right )+b^5\,{\left (d+e\,x\right )}^{13/2}-\left (5\,b^5\,d-5\,a\,b^4\,e\right )\,{\left (d+e\,x\right )}^{11/2}+{\left (d+e\,x\right )}^{9/2}\,\left (10\,a^2\,b^3\,e^2-20\,a\,b^4\,d\,e+10\,b^5\,d^2\right )}+\frac {3003\,b^{3/2}\,e^5\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}\,\left (a^7\,e^7-7\,a^6\,b\,d\,e^6+21\,a^5\,b^2\,d^2\,e^5-35\,a^4\,b^3\,d^3\,e^4+35\,a^3\,b^4\,d^4\,e^3-21\,a^2\,b^5\,d^5\,e^2+7\,a\,b^6\,d^6\,e-b^7\,d^7\right )}{{\left (a\,e-b\,d\right )}^{15/2}}\right )}{128\,{\left (a\,e-b\,d\right )}^{15/2}} \] Input:

int(1/((d + e*x)^(5/2)*(a^2 + b^2*x^2 + 2*a*b*x)^3),x)
 

Output:

((27599*b^2*e^5*(d + e*x)^2)/(384*(a*e - b*d)^3) - (2*e^5)/(3*(a*e - b*d)) 
 + (11297*b^3*e^5*(d + e*x)^3)/(64*(a*e - b*d)^4) + (1001*b^4*e^5*(d + e*x 
)^4)/(5*(a*e - b*d)^5) + (7007*b^5*e^5*(d + e*x)^5)/(64*(a*e - b*d)^6) + ( 
3003*b^6*e^5*(d + e*x)^6)/(128*(a*e - b*d)^7) + (26*b*e^5*(d + e*x))/(3*(a 
*e - b*d)^2))/((d + e*x)^(3/2)*(a^5*e^5 - b^5*d^5 - 10*a^2*b^3*d^3*e^2 + 1 
0*a^3*b^2*d^2*e^3 + 5*a*b^4*d^4*e - 5*a^4*b*d*e^4) - (d + e*x)^(7/2)*(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) + (d + e*x)^( 
5/2)*(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) + b^5*(d + e*x)^(13/2) - (5*b^5*d - 5*a*b^4*e)*(d + e*x)^(11 
/2) + (d + e*x)^(9/2)*(10*b^5*d^2 + 10*a^2*b^3*e^2 - 20*a*b^4*d*e)) + (300 
3*b^(3/2)*e^5*atan((b^(1/2)*(d + e*x)^(1/2)*(a^7*e^7 - b^7*d^7 - 21*a^2*b^ 
5*d^5*e^2 + 35*a^3*b^4*d^4*e^3 - 35*a^4*b^3*d^3*e^4 + 21*a^5*b^2*d^2*e^5 + 
 7*a*b^6*d^6*e - 7*a^6*b*d*e^6))/(a*e - b*d)^(15/2)))/(128*(a*e - b*d)^(15 
/2))
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 2113, normalized size of antiderivative = 7.94 \[ \int \frac {1}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx =\text {Too large to display} \] Input:

int(1/(e*x+d)^(5/2)/(b^2*x^2+2*a*b*x+a^2)^3,x)
 

Output:

(45045*sqrt(b)*sqrt(d + e*x)*sqrt(a*e - b*d)*atan((sqrt(d + e*x)*b)/(sqrt( 
b)*sqrt(a*e - b*d)))*a**5*b*d*e**5 + 45045*sqrt(b)*sqrt(d + e*x)*sqrt(a*e 
- b*d)*atan((sqrt(d + e*x)*b)/(sqrt(b)*sqrt(a*e - b*d)))*a**5*b*e**6*x + 2 
25225*sqrt(b)*sqrt(d + e*x)*sqrt(a*e - b*d)*atan((sqrt(d + e*x)*b)/(sqrt(b 
)*sqrt(a*e - b*d)))*a**4*b**2*d*e**5*x + 225225*sqrt(b)*sqrt(d + e*x)*sqrt 
(a*e - b*d)*atan((sqrt(d + e*x)*b)/(sqrt(b)*sqrt(a*e - b*d)))*a**4*b**2*e* 
*6*x**2 + 450450*sqrt(b)*sqrt(d + e*x)*sqrt(a*e - b*d)*atan((sqrt(d + e*x) 
*b)/(sqrt(b)*sqrt(a*e - b*d)))*a**3*b**3*d*e**5*x**2 + 450450*sqrt(b)*sqrt 
(d + e*x)*sqrt(a*e - b*d)*atan((sqrt(d + e*x)*b)/(sqrt(b)*sqrt(a*e - b*d)) 
)*a**3*b**3*e**6*x**3 + 450450*sqrt(b)*sqrt(d + e*x)*sqrt(a*e - b*d)*atan( 
(sqrt(d + e*x)*b)/(sqrt(b)*sqrt(a*e - b*d)))*a**2*b**4*d*e**5*x**3 + 45045 
0*sqrt(b)*sqrt(d + e*x)*sqrt(a*e - b*d)*atan((sqrt(d + e*x)*b)/(sqrt(b)*sq 
rt(a*e - b*d)))*a**2*b**4*e**6*x**4 + 225225*sqrt(b)*sqrt(d + e*x)*sqrt(a* 
e - b*d)*atan((sqrt(d + e*x)*b)/(sqrt(b)*sqrt(a*e - b*d)))*a*b**5*d*e**5*x 
**4 + 225225*sqrt(b)*sqrt(d + e*x)*sqrt(a*e - b*d)*atan((sqrt(d + e*x)*b)/ 
(sqrt(b)*sqrt(a*e - b*d)))*a*b**5*e**6*x**5 + 45045*sqrt(b)*sqrt(d + e*x)* 
sqrt(a*e - b*d)*atan((sqrt(d + e*x)*b)/(sqrt(b)*sqrt(a*e - b*d)))*b**6*d*e 
**5*x**5 + 45045*sqrt(b)*sqrt(d + e*x)*sqrt(a*e - b*d)*atan((sqrt(d + e*x) 
*b)/(sqrt(b)*sqrt(a*e - b*d)))*b**6*e**6*x**6 - 1280*a**7*e**7 + 25600*a** 
6*b*d*e**6 + 16640*a**6*b*e**7*x + 11275*a**5*b**2*d**2*e**5 + 176150*a...