\(\int \frac {a+b x}{(d+e x)^{7/2} (a^2+2 a b x+b^2 x^2)^{5/2}} \, dx\) [254]

Optimal result

Integrand size = 35, antiderivative size = 382 \[ \int \frac {a+b x}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {33 e^2}{8 (b d-a e)^3 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{3 (b d-a e) (a+b x)^2 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {11 e}{12 (b d-a e)^2 (a+b x) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {231 e^3 (a+b x)}{40 (b d-a e)^4 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {77 b e^3 (a+b x)}{8 (b d-a e)^5 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {231 b^2 e^3 (a+b x)}{8 (b d-a e)^6 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {231 b^{5/2} e^3 (a+b x) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 (b d-a e)^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}} \] Output:

-33/8*e^2/(-a*e+b*d)^3/(e*x+d)^(5/2)/((b*x+a)^2)^(1/2)-1/3/(-a*e+b*d)/(b*x 
+a)^2/(e*x+d)^(5/2)/((b*x+a)^2)^(1/2)+11/12*e/(-a*e+b*d)^2/(b*x+a)/(e*x+d) 
^(5/2)/((b*x+a)^2)^(1/2)-231/40*e^3*(b*x+a)/(-a*e+b*d)^4/(e*x+d)^(5/2)/((b 
*x+a)^2)^(1/2)-77/8*b*e^3*(b*x+a)/(-a*e+b*d)^5/(e*x+d)^(3/2)/((b*x+a)^2)^( 
1/2)-231/8*b^2*e^3*(b*x+a)/(-a*e+b*d)^6/(e*x+d)^(1/2)/((b*x+a)^2)^(1/2)+23 
1/8*b^(5/2)*e^3*(b*x+a)*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))/(- 
a*e+b*d)^(13/2)/((b*x+a)^2)^(1/2)
 

Mathematica [A] (verified)

Time = 1.95 (sec) , antiderivative size = 303, normalized size of antiderivative = 0.79 \[ \int \frac {a+b x}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {e^3 (a+b x) \left (\frac {-48 a^5 e^5+16 a^4 b e^4 (26 d+11 e x)-16 a^3 b^2 e^3 \left (173 d^2+242 d e x+99 e^2 x^2\right )-3 a^2 b^3 e^2 \left (445 d^3+4103 d^2 e x+6039 d e^2 x^2+2541 e^3 x^3\right )-2 a b^4 e \left (-155 d^4+715 d^3 e x+7227 d^2 e^2 x^2+10857 d e^3 x^3+4620 e^4 x^4\right )-b^5 \left (40 d^5-110 d^4 e x+495 d^3 e^2 x^2+5313 d^2 e^3 x^3+8085 d e^4 x^4+3465 e^5 x^5\right )}{e^3 (b d-a e)^6 (a+b x)^3 (d+e x)^{5/2}}-\frac {3465 b^{5/2} \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{(-b d+a e)^{13/2}}\right )}{120 \sqrt {(a+b x)^2}} \] Input:

Integrate[(a + b*x)/((d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]
 

Output:

(e^3*(a + b*x)*((-48*a^5*e^5 + 16*a^4*b*e^4*(26*d + 11*e*x) - 16*a^3*b^2*e 
^3*(173*d^2 + 242*d*e*x + 99*e^2*x^2) - 3*a^2*b^3*e^2*(445*d^3 + 4103*d^2* 
e*x + 6039*d*e^2*x^2 + 2541*e^3*x^3) - 2*a*b^4*e*(-155*d^4 + 715*d^3*e*x + 
 7227*d^2*e^2*x^2 + 10857*d*e^3*x^3 + 4620*e^4*x^4) - b^5*(40*d^5 - 110*d^ 
4*e*x + 495*d^3*e^2*x^2 + 5313*d^2*e^3*x^3 + 8085*d*e^4*x^4 + 3465*e^5*x^5 
))/(e^3*(b*d - a*e)^6*(a + b*x)^3*(d + e*x)^(5/2)) - (3465*b^(5/2)*ArcTan[ 
(Sqrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]])/(-(b*d) + a*e)^(13/2)))/(120* 
Sqrt[(a + b*x)^2])
 

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 303, normalized size of antiderivative = 0.79, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {1187, 27, 52, 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.

\(\displaystyle \int \frac {a+b x}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2} (d+e x)^{7/2}} \, dx\)

\(\Big \downarrow \) 1187

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 52

\(\displaystyle \frac {(a+b x) \left (-\frac {11 e \int \frac {1}{(a+b x)^3 (d+e x)^{7/2}}dx}{6 (b d-a e)}-\frac {1}{3 (a+b x)^3 (d+e x)^{5/2} (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\)

\(\Big \downarrow \) 52

\(\displaystyle \frac {(a+b x) \left (-\frac {11 e \left (-\frac {9 e \int \frac {1}{(a+b x)^2 (d+e x)^{7/2}}dx}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 (d+e x)^{5/2} (b d-a e)}\right )}{6 (b d-a e)}-\frac {1}{3 (a+b x)^3 (d+e x)^{5/2} (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\)

\(\Big \downarrow \) 52

\(\displaystyle \frac {(a+b x) \left (-\frac {11 e \left (-\frac {9 e \left (-\frac {7 e \int \frac {1}{(a+b x) (d+e x)^{7/2}}dx}{2 (b d-a e)}-\frac {1}{(a+b x) (d+e x)^{5/2} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 (d+e x)^{5/2} (b d-a e)}\right )}{6 (b d-a e)}-\frac {1}{3 (a+b x)^3 (d+e x)^{5/2} (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\)

\(\Big \downarrow \) 61

\(\displaystyle \frac {(a+b x) \left (-\frac {11 e \left (-\frac {9 e \left (-\frac {7 e \left (\frac {b \int \frac {1}{(a+b x) (d+e x)^{5/2}}dx}{b d-a e}+\frac {2}{5 (d+e x)^{5/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{(a+b x) (d+e x)^{5/2} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 (d+e x)^{5/2} (b d-a e)}\right )}{6 (b d-a e)}-\frac {1}{3 (a+b x)^3 (d+e x)^{5/2} (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\)

\(\Big \downarrow \) 61

\(\displaystyle \frac {(a+b x) \left (-\frac {11 e \left (-\frac {9 e \left (-\frac {7 e \left (\frac {b \left (\frac {b \int \frac {1}{(a+b x) (d+e x)^{3/2}}dx}{b d-a e}+\frac {2}{3 (d+e x)^{3/2} (b d-a e)}\right )}{b d-a e}+\frac {2}{5 (d+e x)^{5/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{(a+b x) (d+e x)^{5/2} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 (d+e x)^{5/2} (b d-a e)}\right )}{6 (b d-a e)}-\frac {1}{3 (a+b x)^3 (d+e x)^{5/2} (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\)

\(\Big \downarrow \) 61

\(\displaystyle \frac {(a+b x) \left (-\frac {11 e \left (-\frac {9 e \left (-\frac {7 e \left (\frac {b \left (\frac {b \left (\frac {b \int \frac {1}{(a+b x) \sqrt {d+e x}}dx}{b d-a e}+\frac {2}{\sqrt {d+e x} (b d-a e)}\right )}{b d-a e}+\frac {2}{3 (d+e x)^{3/2} (b d-a e)}\right )}{b d-a e}+\frac {2}{5 (d+e x)^{5/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{(a+b x) (d+e x)^{5/2} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 (d+e x)^{5/2} (b d-a e)}\right )}{6 (b d-a e)}-\frac {1}{3 (a+b x)^3 (d+e x)^{5/2} (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {(a+b x) \left (-\frac {11 e \left (-\frac {9 e \left (-\frac {7 e \left (\frac {b \left (\frac {b \left (\frac {2 b \int \frac {1}{a+\frac {b (d+e x)}{e}-\frac {b d}{e}}d\sqrt {d+e x}}{e (b d-a e)}+\frac {2}{\sqrt {d+e x} (b d-a e)}\right )}{b d-a e}+\frac {2}{3 (d+e x)^{3/2} (b d-a e)}\right )}{b d-a e}+\frac {2}{5 (d+e x)^{5/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{(a+b x) (d+e x)^{5/2} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 (d+e x)^{5/2} (b d-a e)}\right )}{6 (b d-a e)}-\frac {1}{3 (a+b x)^3 (d+e x)^{5/2} (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {(a+b x) \left (-\frac {11 e \left (-\frac {9 e \left (-\frac {7 e \left (\frac {b \left (\frac {b \left (\frac {2}{\sqrt {d+e x} (b d-a e)}-\frac {2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{3/2}}\right )}{b d-a e}+\frac {2}{3 (d+e x)^{3/2} (b d-a e)}\right )}{b d-a e}+\frac {2}{5 (d+e x)^{5/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{(a+b x) (d+e x)^{5/2} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 (d+e x)^{5/2} (b d-a e)}\right )}{6 (b d-a e)}-\frac {1}{3 (a+b x)^3 (d+e x)^{5/2} (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\)

Input:

Int[(a + b*x)/((d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]
 

Output:

((a + b*x)*(-1/3*1/((b*d - a*e)*(a + b*x)^3*(d + e*x)^(5/2)) - (11*e*(-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*S 
qrt[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))))/(6*(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 1187

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ 
) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x + c*x^2)^FracPart[p]/(c^ 
IntPart[p]*(b/2 + c*x)^(2*FracPart[p]))   Int[(d + e*x)^m*(f + g*x)^n*(b/2 
+ c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && EqQ[b^2 
 - 4*a*c, 0] &&  !IntegerQ[p]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(721\) vs. \(2(269)=538\).

Time = 1.44 (sec) , antiderivative size = 722, normalized size of antiderivative = 1.89

method	result	size
default	\(-\frac {\left (9240 \sqrt {b \left (a e -b d \right )}\, a \,b^{4} e^{5} x^{4}+8085 \sqrt {b \left (a e -b d \right )}\, b^{5} d \,e^{4} x^{4}-110 \sqrt {b \left (a e -b d \right )}\, b^{5} d^{4} e x +3872 \sqrt {b \left (a e -b d \right )}\, a^{3} b^{2} d \,e^{4} x -416 \sqrt {b \left (a e -b d \right )}\, a^{4} b d \,e^{4}+10395 \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^{5} e^{3} x^{2}+10395 \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^{4} e^{3} x +1584 \sqrt {b \left (a e -b d \right )}\, a^{3} b^{2} e^{5} x^{2}+3465 \sqrt {b \left (a e -b d \right )}\, b^{5} e^{5} x^{5}+3465 \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^{6} e^{3} x^{3}+7623 \sqrt {b \left (a e -b d \right )}\, a^{2} b^{3} e^{5} x^{3}+14454 \sqrt {b \left (a e -b d \right )}\, a \,b^{4} d^{2} e^{3} x^{2}-176 \sqrt {b \left (a e -b d \right )}\, a^{4} b \,e^{5} x +18117 \sqrt {b \left (a e -b d \right )}\, a^{2} b^{3} d \,e^{4} x^{2}+1430 \sqrt {b \left (a e -b d \right )}\, a \,b^{4} d^{3} e^{2} x +21714 \sqrt {b \left (a e -b d \right )}\, a \,b^{4} d \,e^{4} x^{3}+5313 \sqrt {b \left (a e -b d \right )}\, b^{5} d^{2} e^{3} x^{3}+495 \sqrt {b \left (a e -b d \right )}\, b^{5} d^{3} e^{2} x^{2}+12309 \sqrt {b \left (a e -b d \right )}\, a^{2} b^{3} d^{2} e^{3} x +2768 \sqrt {b \left (a e -b d \right )}\, a^{3} b^{2} d^{2} e^{3}+1335 \sqrt {b \left (a e -b d \right )}\, a^{2} b^{3} d^{3} e^{2}-310 \sqrt {b \left (a e -b d \right )}\, a \,b^{4} d^{4} e +3465 \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^{3} b^{3} e^{3}+48 \sqrt {b \left (a e -b d \right )}\, a^{5} e^{5}+40 \sqrt {b \left (a e -b d \right )}\, b^{5} d^{5}\right ) \left (b x +a \right )^{2}}{120 \sqrt {b \left (a e -b d \right )}\, \left (e x +d \right )^{\frac {5}{2}} \left (a e -b d \right )^{6} \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}\)	\(722\)

Input:

int((b*x+a)/(e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x,method=_RETURNVERB 
OSE)
 

Output:

-1/120*(9240*(b*(a*e-b*d))^(1/2)*a*b^4*e^5*x^4+8085*(b*(a*e-b*d))^(1/2)*b^ 
5*d*e^4*x^4-110*(b*(a*e-b*d))^(1/2)*b^5*d^4*e*x+3872*(b*(a*e-b*d))^(1/2)*a 
^3*b^2*d*e^4*x-416*(b*(a*e-b*d))^(1/2)*a^4*b*d*e^4+10395*(e*x+d)^(5/2)*arc 
tan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*a*b^5*e^3*x^2+10395*(e*x+d)^(5/2) 
*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*a^2*b^4*e^3*x+1584*(b*(a*e-b* 
d))^(1/2)*a^3*b^2*e^5*x^2+3465*(b*(a*e-b*d))^(1/2)*b^5*e^5*x^5+3465*(e*x+d 
)^(5/2)*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*b^6*e^3*x^3+7623*(b*(a 
*e-b*d))^(1/2)*a^2*b^3*e^5*x^3+14454*(b*(a*e-b*d))^(1/2)*a*b^4*d^2*e^3*x^2 
-176*(b*(a*e-b*d))^(1/2)*a^4*b*e^5*x+18117*(b*(a*e-b*d))^(1/2)*a^2*b^3*d*e 
^4*x^2+1430*(b*(a*e-b*d))^(1/2)*a*b^4*d^3*e^2*x+21714*(b*(a*e-b*d))^(1/2)* 
a*b^4*d*e^4*x^3+5313*(b*(a*e-b*d))^(1/2)*b^5*d^2*e^3*x^3+495*(b*(a*e-b*d)) 
^(1/2)*b^5*d^3*e^2*x^2+12309*(b*(a*e-b*d))^(1/2)*a^2*b^3*d^2*e^3*x+2768*(b 
*(a*e-b*d))^(1/2)*a^3*b^2*d^2*e^3+1335*(b*(a*e-b*d))^(1/2)*a^2*b^3*d^3*e^2 
-310*(b*(a*e-b*d))^(1/2)*a*b^4*d^4*e+3465*(e*x+d)^(5/2)*arctan(b*(e*x+d)^( 
1/2)/(b*(a*e-b*d))^(1/2))*a^3*b^3*e^3+48*(b*(a*e-b*d))^(1/2)*a^5*e^5+40*(b 
*(a*e-b*d))^(1/2)*b^5*d^5)*(b*x+a)^2/(b*(a*e-b*d))^(1/2)/(e*x+d)^(5/2)/(a* 
e-b*d)^6/((b*x+a)^2)^(5/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1250 vs. \(2 (269) = 538\).

Time = 0.35 (sec) , antiderivative size = 2530, normalized size of antiderivative = 6.62 \[ \int \frac {a+b x}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)/(e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm=" 
fricas")
 

Output:

[1/240*(3465*(b^5*e^6*x^6 + a^3*b^2*d^3*e^3 + 3*(b^5*d*e^5 + a*b^4*e^6)*x^ 
5 + 3*(b^5*d^2*e^4 + 3*a*b^4*d*e^5 + a^2*b^3*e^6)*x^4 + (b^5*d^3*e^3 + 9*a 
*b^4*d^2*e^4 + 9*a^2*b^3*d*e^5 + a^3*b^2*e^6)*x^3 + 3*(a*b^4*d^3*e^3 + 3*a 
^2*b^3*d^2*e^4 + a^3*b^2*d*e^5)*x^2 + 3*(a^2*b^3*d^3*e^3 + a^3*b^2*d^2*e^4 
)*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*(3465*b^5*e^5*x^5 + 40*b^5*d^5 - 
 310*a*b^4*d^4*e + 1335*a^2*b^3*d^3*e^2 + 2768*a^3*b^2*d^2*e^3 - 416*a^4*b 
*d*e^4 + 48*a^5*e^5 + 1155*(7*b^5*d*e^4 + 8*a*b^4*e^5)*x^4 + 231*(23*b^5*d 
^2*e^3 + 94*a*b^4*d*e^4 + 33*a^2*b^3*e^5)*x^3 + 99*(5*b^5*d^3*e^2 + 146*a* 
b^4*d^2*e^3 + 183*a^2*b^3*d*e^4 + 16*a^3*b^2*e^5)*x^2 - 11*(10*b^5*d^4*e - 
 130*a*b^4*d^3*e^2 - 1119*a^2*b^3*d^2*e^3 - 352*a^3*b^2*d*e^4 + 16*a^4*b*e 
^5)*x)*sqrt(e*x + d))/(a^3*b^6*d^9 - 6*a^4*b^5*d^8*e + 15*a^5*b^4*d^7*e^2 
- 20*a^6*b^3*d^6*e^3 + 15*a^7*b^2*d^5*e^4 - 6*a^8*b*d^4*e^5 + a^9*d^3*e^6 
+ (b^9*d^6*e^3 - 6*a*b^8*d^5*e^4 + 15*a^2*b^7*d^4*e^5 - 20*a^3*b^6*d^3*e^6 
 + 15*a^4*b^5*d^2*e^7 - 6*a^5*b^4*d*e^8 + a^6*b^3*e^9)*x^6 + 3*(b^9*d^7*e^ 
2 - 5*a*b^8*d^6*e^3 + 9*a^2*b^7*d^5*e^4 - 5*a^3*b^6*d^4*e^5 - 5*a^4*b^5*d^ 
3*e^6 + 9*a^5*b^4*d^2*e^7 - 5*a^6*b^3*d*e^8 + a^7*b^2*e^9)*x^5 + 3*(b^9*d^ 
8*e - 3*a*b^8*d^7*e^2 - 2*a^2*b^7*d^6*e^3 + 19*a^3*b^6*d^5*e^4 - 30*a^4*b^ 
5*d^4*e^5 + 19*a^5*b^4*d^3*e^6 - 2*a^6*b^3*d^2*e^7 - 3*a^7*b^2*d*e^8 + a^8 
*b*e^9)*x^4 + (b^9*d^9 + 3*a*b^8*d^8*e - 30*a^2*b^7*d^7*e^2 + 62*a^3*b^...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {a+b x}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:

integrate((b*x+a)/(e*x+d)**(7/2)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {a+b x}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int { \frac {b x + a}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (e x + d\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((b*x+a)/(e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm=" 
maxima")
 

Output:

integrate((b*x + a)/((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(e*x + d)^(7/2)), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 606 vs. \(2 (269) = 538\).

Time = 0.21 (sec) , antiderivative size = 606, normalized size of antiderivative = 1.59 \[ \int \frac {a+b x}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

integrate((b*x+a)/(e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm=" 
giac")
 

Output:

-231/8*b^3*e^3*arctan(sqrt(e*x + d)*b/sqrt(-b^2*d + a*b*e))/((b^6*d^6*sgn( 
b*x + a) - 6*a*b^5*d^5*e*sgn(b*x + a) + 15*a^2*b^4*d^4*e^2*sgn(b*x + a) - 
20*a^3*b^3*d^3*e^3*sgn(b*x + a) + 15*a^4*b^2*d^2*e^4*sgn(b*x + a) - 6*a^5* 
b*d*e^5*sgn(b*x + a) + a^6*e^6*sgn(b*x + a))*sqrt(-b^2*d + a*b*e)) - 2/15* 
(150*(e*x + d)^2*b^2*e^3 + 20*(e*x + d)*b^2*d*e^3 + 3*b^2*d^2*e^3 - 20*(e* 
x + d)*a*b*e^4 - 6*a*b*d*e^4 + 3*a^2*e^5)/((b^6*d^6*sgn(b*x + a) - 6*a*b^5 
*d^5*e*sgn(b*x + a) + 15*a^2*b^4*d^4*e^2*sgn(b*x + a) - 20*a^3*b^3*d^3*e^3 
*sgn(b*x + a) + 15*a^4*b^2*d^2*e^4*sgn(b*x + a) - 6*a^5*b*d*e^5*sgn(b*x + 
a) + a^6*e^6*sgn(b*x + a))*(e*x + d)^(5/2)) - 1/24*(213*(e*x + d)^(5/2)*b^ 
5*e^3 - 472*(e*x + d)^(3/2)*b^5*d*e^3 + 267*sqrt(e*x + d)*b^5*d^2*e^3 + 47 
2*(e*x + d)^(3/2)*a*b^4*e^4 - 534*sqrt(e*x + d)*a*b^4*d*e^4 + 267*sqrt(e*x 
 + d)*a^2*b^3*e^5)/((b^6*d^6*sgn(b*x + a) - 6*a*b^5*d^5*e*sgn(b*x + a) + 1 
5*a^2*b^4*d^4*e^2*sgn(b*x + a) - 20*a^3*b^3*d^3*e^3*sgn(b*x + a) + 15*a^4* 
b^2*d^2*e^4*sgn(b*x + a) - 6*a^5*b*d*e^5*sgn(b*x + a) + a^6*e^6*sgn(b*x + 
a))*((e*x + d)*b - b*d + a*e)^3)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b x}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int \frac {a+b\,x}{{\left (d+e\,x\right )}^{7/2}\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \] Input:

int((a + b*x)/((d + e*x)^(7/2)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2)),x)
 

Output:

int((a + b*x)/((d + e*x)^(7/2)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 1814, normalized size of antiderivative = 4.75 \[ \int \frac {a+b x}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

int((b*x+a)/(e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x)
 

Output:

( - 3465*sqrt(b)*sqrt(d + e*x)*sqrt(a*e - b*d)*atan((sqrt(d + e*x)*b)/(sqr 
t(b)*sqrt(a*e - b*d)))*a**3*b**2*d**2*e**3 - 6930*sqrt(b)*sqrt(d + e*x)*sq 
rt(a*e - b*d)*atan((sqrt(d + e*x)*b)/(sqrt(b)*sqrt(a*e - b*d)))*a**3*b**2* 
d*e**4*x - 3465*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**2*e**5*x**2 - 10395*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**3*d**2*e**3*x - 20790*sqrt(b)*sqrt(d + e*x)*sqrt(a*e - b*d)*atan((sq 
rt(d + e*x)*b)/(sqrt(b)*sqrt(a*e - b*d)))*a**2*b**3*d*e**4*x**2 - 10395*sq 
rt(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**3*e**5*x**3 - 10395*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**4*d**2*e**3*x** 
2 - 20790*sqrt(b)*sqrt(d + e*x)*sqrt(a*e - b*d)*atan((sqrt(d + e*x)*b)/(sq 
rt(b)*sqrt(a*e - b*d)))*a*b**4*d*e**4*x**3 - 10395*sqrt(b)*sqrt(d + e*x)*s 
qrt(a*e - b*d)*atan((sqrt(d + e*x)*b)/(sqrt(b)*sqrt(a*e - b*d)))*a*b**4*e* 
*5*x**4 - 3465*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**5*d**2*e**3*x**3 - 6930*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**5 
*d*e**4*x**4 - 3465*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**5*e**5*x**5 - 48*a**6*e**6 + 464*a**5 
*b*d*e**5 + 176*a**5*b*e**6*x - 3184*a**4*b**2*d**2*e**4 - 4048*a**4*b*...