\(\int \frac {1}{(d+e x)^{7/2} (a^2+2 a b x+b^2 x^2)^{3/2}} \, dx\) [387]

Optimal result

Integrand size = 30, antiderivative size = 329 \[ \int \frac {1}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {9 e}{4 (b d-a e)^2 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{2 (b d-a e) (a+b x) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {63 e^2 (a+b x)}{20 (b d-a e)^3 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {21 b e^2 (a+b x)}{4 (b d-a e)^4 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {63 b^2 e^2 (a+b x)}{4 (b d-a e)^5 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {63 b^{5/2} e^2 (a+b x) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 (b d-a e)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}} \] Output:

9/4*e/(-a*e+b*d)^2/(e*x+d)^(5/2)/((b*x+a)^2)^(1/2)-1/2/(-a*e+b*d)/(b*x+a)/ 
(e*x+d)^(5/2)/((b*x+a)^2)^(1/2)+63/20*e^2*(b*x+a)/(-a*e+b*d)^3/(e*x+d)^(5/ 
2)/((b*x+a)^2)^(1/2)+21/4*b*e^2*(b*x+a)/(-a*e+b*d)^4/(e*x+d)^(3/2)/((b*x+a 
)^2)^(1/2)+63/4*b^2*e^2*(b*x+a)/(-a*e+b*d)^5/(e*x+d)^(1/2)/((b*x+a)^2)^(1/ 
2)-63/4*b^(5/2)*e^2*(b*x+a)*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2) 
)/(-a*e+b*d)^(11/2)/((b*x+a)^2)^(1/2)
 

Mathematica [A] (verified)

Time = 0.87 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.73 \[ \int \frac {1}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {e^2 (a+b x)^3 \left (\frac {-8 a^4 e^4+8 a^3 b e^3 (7 d+3 e x)-24 a^2 b^2 e^2 \left (12 d^2+17 d e x+7 e^2 x^2\right )-a b^3 e \left (85 d^3+831 d^2 e x+1239 d e^2 x^2+525 e^3 x^3\right )+b^4 \left (10 d^4-45 d^3 e x-483 d^2 e^2 x^2-735 d e^3 x^3-315 e^4 x^4\right )}{e^2 (-b d+a e)^5 (a+b x)^2 (d+e x)^{5/2}}-\frac {315 b^{5/2} \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{(-b d+a e)^{11/2}}\right )}{20 \left ((a+b x)^2\right )^{3/2}} \] Input:

Integrate[1/((d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2)),x]
 

Output:

(e^2*(a + b*x)^3*((-8*a^4*e^4 + 8*a^3*b*e^3*(7*d + 3*e*x) - 24*a^2*b^2*e^2 
*(12*d^2 + 17*d*e*x + 7*e^2*x^2) - a*b^3*e*(85*d^3 + 831*d^2*e*x + 1239*d* 
e^2*x^2 + 525*e^3*x^3) + b^4*(10*d^4 - 45*d^3*e*x - 483*d^2*e^2*x^2 - 735* 
d*e^3*x^3 - 315*e^4*x^4))/(e^2*(-(b*d) + a*e)^5*(a + b*x)^2*(d + e*x)^(5/2 
)) - (315*b^(5/2)*ArcTan[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]])/(-(b 
*d) + a*e)^(11/2)))/(20*((a + b*x)^2)^(3/2))
 

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 257, normalized size of antiderivative = 0.78, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1102, 27, 52, 52, 61, 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.

\(\displaystyle \int \frac {1}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2} (d+e x)^{7/2}} \, dx\)

\(\Big \downarrow \) 1102

\(\displaystyle \frac {b^3 (a+b x) \int \frac {1}{b^3 (a+b x)^3 (d+e x)^{7/2}}dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(a+b x) \int \frac {1}{(a+b x)^3 (d+e x)^{7/2}}dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\)

\(\Big \downarrow \) 52

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

\(\Big \downarrow \) 52

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

\(\Big \downarrow \) 61

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

\(\Big \downarrow \) 61

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

\(\Big \downarrow \) 61

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

Int[1/((d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2)),x]
 

Output:

((a + b*x)*(-1/2*1/((b*d - a*e)*(a + b*x)^2*(d + e*x)^(5/2)) - (9*e*(-(1/( 
(b*d - a*e)*(a + b*x)*(d + e*x)^(5/2))) - (7*e*(2/(5*(b*d - a*e)*(d + e*x) 
^(5/2)) + (b*(2/(3*(b*d - a*e)*(d + e*x)^(3/2)) + (b*(2/((b*d - a*e)*Sqrt[ 
d + e*x]) - (2*Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/( 
b*d - a*e)^(3/2)))/(b*d - a*e)))/(b*d - a*e)))/(2*(b*d - a*e))))/(4*(b*d - 
 a*e))))/Sqrt[a^2 + 2*a*b*x + b^2*x^2]
 

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 1102

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(517\) vs. \(2(231)=462\).

Time = 1.51 (sec) , antiderivative size = 518, normalized size of antiderivative = 1.57

method	result	size
default	\(-\frac {\left (315 \left (e x +d \right )^{\frac {5}{2}} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) b^{5} e^{2} x^{2}+630 \left (e x +d \right )^{\frac {5}{2}} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a \,b^{4} e^{2} x +315 \left (e x +d \right )^{\frac {5}{2}} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a^{2} b^{3} e^{2}+315 \sqrt {b \left (a e -b d \right )}\, b^{4} e^{4} x^{4}+525 \sqrt {b \left (a e -b d \right )}\, a \,b^{3} e^{4} x^{3}+735 \sqrt {b \left (a e -b d \right )}\, b^{4} d \,e^{3} x^{3}+168 \sqrt {b \left (a e -b d \right )}\, a^{2} b^{2} e^{4} x^{2}+1239 \sqrt {b \left (a e -b d \right )}\, a \,b^{3} d \,e^{3} x^{2}+483 \sqrt {b \left (a e -b d \right )}\, b^{4} d^{2} e^{2} x^{2}-24 \sqrt {b \left (a e -b d \right )}\, a^{3} b \,e^{4} x +408 \sqrt {b \left (a e -b d \right )}\, a^{2} b^{2} d \,e^{3} x +831 \sqrt {b \left (a e -b d \right )}\, a \,b^{3} d^{2} e^{2} x +45 \sqrt {b \left (a e -b d \right )}\, b^{4} d^{3} e x +8 \sqrt {b \left (a e -b d \right )}\, a^{4} e^{4}-56 \sqrt {b \left (a e -b d \right )}\, a^{3} b d \,e^{3}+288 \sqrt {b \left (a e -b d \right )}\, a^{2} b^{2} d^{2} e^{2}+85 \sqrt {b \left (a e -b d \right )}\, a \,b^{3} d^{3} e -10 \sqrt {b \left (a e -b d \right )}\, b^{4} d^{4}\right ) \left (b x +a \right )}{20 \left (e x +d \right )^{\frac {5}{2}} \sqrt {b \left (a e -b d \right )}\, \left (a e -b d \right )^{5} \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}\)	\(518\)

Input:

int(1/(e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

-1/20*(315*(e*x+d)^(5/2)*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*b^5*e 
^2*x^2+630*(e*x+d)^(5/2)*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*a*b^4 
*e^2*x+315*(e*x+d)^(5/2)*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*a^2*b 
^3*e^2+315*(b*(a*e-b*d))^(1/2)*b^4*e^4*x^4+525*(b*(a*e-b*d))^(1/2)*a*b^3*e 
^4*x^3+735*(b*(a*e-b*d))^(1/2)*b^4*d*e^3*x^3+168*(b*(a*e-b*d))^(1/2)*a^2*b 
^2*e^4*x^2+1239*(b*(a*e-b*d))^(1/2)*a*b^3*d*e^3*x^2+483*(b*(a*e-b*d))^(1/2 
)*b^4*d^2*e^2*x^2-24*(b*(a*e-b*d))^(1/2)*a^3*b*e^4*x+408*(b*(a*e-b*d))^(1/ 
2)*a^2*b^2*d*e^3*x+831*(b*(a*e-b*d))^(1/2)*a*b^3*d^2*e^2*x+45*(b*(a*e-b*d) 
)^(1/2)*b^4*d^3*e*x+8*(b*(a*e-b*d))^(1/2)*a^4*e^4-56*(b*(a*e-b*d))^(1/2)*a 
^3*b*d*e^3+288*(b*(a*e-b*d))^(1/2)*a^2*b^2*d^2*e^2+85*(b*(a*e-b*d))^(1/2)* 
a*b^3*d^3*e-10*(b*(a*e-b*d))^(1/2)*b^4*d^4)*(b*x+a)/(e*x+d)^(5/2)/(b*(a*e- 
b*d))^(1/2)/(a*e-b*d)^5/((b*x+a)^2)^(3/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 904 vs. \(2 (231) = 462\).

Time = 0.21 (sec) , antiderivative size = 1838, normalized size of antiderivative = 5.59 \[ \int \frac {1}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:

integrate(1/(e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="fricas 
")
 

Output:

[-1/40*(315*(b^4*e^5*x^5 + a^2*b^2*d^3*e^2 + (3*b^4*d*e^4 + 2*a*b^3*e^5)*x 
^4 + (3*b^4*d^2*e^3 + 6*a*b^3*d*e^4 + a^2*b^2*e^5)*x^3 + (b^4*d^3*e^2 + 6* 
a*b^3*d^2*e^3 + 3*a^2*b^2*d*e^4)*x^2 + (2*a*b^3*d^3*e^2 + 3*a^2*b^2*d^2*e^ 
3)*x)*sqrt(b/(b*d - a*e))*log((b*e*x + 2*b*d - a*e + 2*(b*d - a*e)*sqrt(e* 
x + d)*sqrt(b/(b*d - a*e)))/(b*x + a)) - 2*(315*b^4*e^4*x^4 - 10*b^4*d^4 + 
 85*a*b^3*d^3*e + 288*a^2*b^2*d^2*e^2 - 56*a^3*b*d*e^3 + 8*a^4*e^4 + 105*( 
7*b^4*d*e^3 + 5*a*b^3*e^4)*x^3 + 21*(23*b^4*d^2*e^2 + 59*a*b^3*d*e^3 + 8*a 
^2*b^2*e^4)*x^2 + 3*(15*b^4*d^3*e + 277*a*b^3*d^2*e^2 + 136*a^2*b^2*d*e^3 
- 8*a^3*b*e^4)*x)*sqrt(e*x + d))/(a^2*b^5*d^8 - 5*a^3*b^4*d^7*e + 10*a^4*b 
^3*d^6*e^2 - 10*a^5*b^2*d^5*e^3 + 5*a^6*b*d^4*e^4 - a^7*d^3*e^5 + (b^7*d^5 
*e^3 - 5*a*b^6*d^4*e^4 + 10*a^2*b^5*d^3*e^5 - 10*a^3*b^4*d^2*e^6 + 5*a^4*b 
^3*d*e^7 - a^5*b^2*e^8)*x^5 + (3*b^7*d^6*e^2 - 13*a*b^6*d^5*e^3 + 20*a^2*b 
^5*d^4*e^4 - 10*a^3*b^4*d^3*e^5 - 5*a^4*b^3*d^2*e^6 + 7*a^5*b^2*d*e^7 - 2* 
a^6*b*e^8)*x^4 + (3*b^7*d^7*e - 9*a*b^6*d^6*e^2 + a^2*b^5*d^5*e^3 + 25*a^3 
*b^4*d^4*e^4 - 35*a^4*b^3*d^3*e^5 + 17*a^5*b^2*d^2*e^6 - a^6*b*d*e^7 - a^7 
*e^8)*x^3 + (b^7*d^8 + a*b^6*d^7*e - 17*a^2*b^5*d^6*e^2 + 35*a^3*b^4*d^5*e 
^3 - 25*a^4*b^3*d^4*e^4 - a^5*b^2*d^3*e^5 + 9*a^6*b*d^2*e^6 - 3*a^7*d*e^7) 
*x^2 + (2*a*b^6*d^8 - 7*a^2*b^5*d^7*e + 5*a^3*b^4*d^6*e^2 + 10*a^4*b^3*d^5 
*e^3 - 20*a^5*b^2*d^4*e^4 + 13*a^6*b*d^3*e^5 - 3*a^7*d^2*e^6)*x), 1/20*(31 
5*(b^4*e^5*x^5 + a^2*b^2*d^3*e^2 + (3*b^4*d*e^4 + 2*a*b^3*e^5)*x^4 + (3...
 

Sympy [F]

\[ \int \frac {1}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\int \frac {1}{\left (d + e x\right )^{\frac {7}{2}} \left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(1/(e*x+d)**(7/2)/(b**2*x**2+2*a*b*x+a**2)**(3/2),x)
 

Output:

Integral(1/((d + e*x)**(7/2)*((a + b*x)**2)**(3/2)), x)
 

Maxima [F]

\[ \int \frac {1}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate(1/(e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="maxima 
")
 

Output:

integrate(1/((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(e*x + d)^(7/2)), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 494 vs. \(2 (231) = 462\).

Time = 1.35 (sec) , antiderivative size = 494, normalized size of antiderivative = 1.50 \[ \int \frac {1}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {63 \, b^{3} e^{2} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{4 \, {\left (b^{5} d^{5} \mathrm {sgn}\left (b x + a\right ) - 5 \, a b^{4} d^{4} e \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{2} b^{3} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) - 10 \, a^{3} b^{2} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} b d e^{4} \mathrm {sgn}\left (b x + a\right ) - a^{5} e^{5} \mathrm {sgn}\left (b x + a\right )\right )} \sqrt {-b^{2} d + a b e}} + \frac {15 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{4} e^{2} - 17 \, \sqrt {e x + d} b^{4} d e^{2} + 17 \, \sqrt {e x + d} a b^{3} e^{3}}{4 \, {\left (b^{5} d^{5} \mathrm {sgn}\left (b x + a\right ) - 5 \, a b^{4} d^{4} e \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{2} b^{3} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) - 10 \, a^{3} b^{2} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} b d e^{4} \mathrm {sgn}\left (b x + a\right ) - a^{5} e^{5} \mathrm {sgn}\left (b x + a\right )\right )} {\left ({\left (e x + d\right )} b - b d + a e\right )}^{2}} + \frac {2 \, {\left (30 \, {\left (e x + d\right )}^{2} b^{2} e^{2} + 5 \, {\left (e x + d\right )} b^{2} d e^{2} + b^{2} d^{2} e^{2} - 5 \, {\left (e x + d\right )} a b e^{3} - 2 \, a b d e^{3} + a^{2} e^{4}\right )}}{5 \, {\left (b^{5} d^{5} \mathrm {sgn}\left (b x + a\right ) - 5 \, a b^{4} d^{4} e \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{2} b^{3} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) - 10 \, a^{3} b^{2} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} b d e^{4} \mathrm {sgn}\left (b x + a\right ) - a^{5} e^{5} \mathrm {sgn}\left (b x + a\right )\right )} {\left (e x + d\right )}^{\frac {5}{2}}} \] Input:

integrate(1/(e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="giac")
 

Output:

63/4*b^3*e^2*arctan(sqrt(e*x + d)*b/sqrt(-b^2*d + a*b*e))/((b^5*d^5*sgn(b* 
x + a) - 5*a*b^4*d^4*e*sgn(b*x + a) + 10*a^2*b^3*d^3*e^2*sgn(b*x + a) - 10 
*a^3*b^2*d^2*e^3*sgn(b*x + a) + 5*a^4*b*d*e^4*sgn(b*x + a) - a^5*e^5*sgn(b 
*x + a))*sqrt(-b^2*d + a*b*e)) + 1/4*(15*(e*x + d)^(3/2)*b^4*e^2 - 17*sqrt 
(e*x + d)*b^4*d*e^2 + 17*sqrt(e*x + d)*a*b^3*e^3)/((b^5*d^5*sgn(b*x + a) - 
 5*a*b^4*d^4*e*sgn(b*x + a) + 10*a^2*b^3*d^3*e^2*sgn(b*x + a) - 10*a^3*b^2 
*d^2*e^3*sgn(b*x + a) + 5*a^4*b*d*e^4*sgn(b*x + a) - a^5*e^5*sgn(b*x + a)) 
*((e*x + d)*b - b*d + a*e)^2) + 2/5*(30*(e*x + d)^2*b^2*e^2 + 5*(e*x + d)* 
b^2*d*e^2 + b^2*d^2*e^2 - 5*(e*x + d)*a*b*e^3 - 2*a*b*d*e^3 + a^2*e^4)/((b 
^5*d^5*sgn(b*x + a) - 5*a*b^4*d^4*e*sgn(b*x + a) + 10*a^2*b^3*d^3*e^2*sgn( 
b*x + a) - 10*a^3*b^2*d^2*e^3*sgn(b*x + a) + 5*a^4*b*d*e^4*sgn(b*x + a) - 
a^5*e^5*sgn(b*x + a))*(e*x + d)^(5/2))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\int \frac {1}{{\left (d+e\,x\right )}^{7/2}\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}} \,d x \] Input:

int(1/((d + e*x)^(7/2)*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2)),x)
 

Output:

int(1/((d + e*x)^(7/2)*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 1299, normalized size of antiderivative = 3.95 \[ \int \frac {1}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

int(1/(e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^(3/2),x)
 

Output:

( - 315*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**2*d**2*e**2 - 630*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**2*d* 
e**3*x - 315*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**2*e**4*x**2 - 630*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**3* 
d**2*e**2*x - 1260*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**3*d*e**3*x**2 - 630*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**3*e**4*x**3 - 315*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**4*d**2*e**2*x**2 - 630*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**4*d*e**3*x**3 - 315*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**4*e**4*x**4 - 8*a**5*e**5 + 64*a* 
*4*b*d*e**4 + 24*a**4*b*e**5*x - 344*a**3*b**2*d**2*e**3 - 432*a**3*b**2*d 
*e**4*x - 168*a**3*b**2*e**5*x**2 + 203*a**2*b**3*d**3*e**2 - 423*a**2*b** 
3*d**2*e**3*x - 1071*a**2*b**3*d*e**4*x**2 - 525*a**2*b**3*e**5*x**3 + 95* 
a*b**4*d**4*e + 786*a*b**4*d**3*e**2*x + 756*a*b**4*d**2*e**3*x**2 - 210*a 
*b**4*d*e**4*x**3 - 315*a*b**4*e**5*x**4 - 10*b**5*d**5 + 45*b**5*d**4*e*x 
 + 483*b**5*d**3*e**2*x**2 + 735*b**5*d**2*e**3*x**3 + 315*b**5*d*e**4*...