\(\int \frac {1}{(d+e x)^3 (a^2+2 a b x+b^2 x^2)^{5/2}} \, dx\) [1613]

Optimal result

Integrand size = 28, antiderivative size = 365 \[ \int \frac {1}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {10 b^2 e^3}{(b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b^2}{4 (b d-a e)^3 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b^2 e}{(b d-a e)^4 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 b^2 e^2}{(b d-a e)^5 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e^4 (a+b x)}{2 (b d-a e)^5 (d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 b e^4 (a+b x)}{(b d-a e)^6 (d+e x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {15 b^2 e^4 (a+b x) \log (a+b x)}{(b d-a e)^7 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {15 b^2 e^4 (a+b x) \log (d+e x)}{(b d-a e)^7 \sqrt {a^2+2 a b x+b^2 x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 365, 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, 46} \[ \int \frac {1}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {5 b e^4 (a+b x)}{\sqrt {a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^6}+\frac {e^4 (a+b x)}{2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^5}+\frac {15 b^2 e^4 (a+b x) \log (a+b x)}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^7}-\frac {15 b^2 e^4 (a+b x) \log (d+e x)}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^7}+\frac {10 b^2 e^3}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}-\frac {3 b^2 e^2}{(a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}+\frac {b^2 e}{(a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}-\frac {b^2}{4 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3} \]

[In]

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

[Out]

(10*b^2*e^3)/((b*d - a*e)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - b^2/(4*(b*d - a*e)^3*(a + b*x)^3*Sqrt[a^2 + 2*a*b
*x + b^2*x^2]) + (b^2*e)/((b*d - a*e)^4*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (3*b^2*e^2)/((b*d - a*e)^
5*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (e^4*(a + b*x))/(2*(b*d - a*e)^5*(d + e*x)^2*Sqrt[a^2 + 2*a*b*x +
b^2*x^2]) + (5*b*e^4*(a + b*x))/((b*d - a*e)^6*(d + e*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (15*b^2*e^4*(a + b*
x)*Log[a + b*x])/((b*d - a*e)^7*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (15*b^2*e^4*(a + b*x)*Log[d + e*x])/((b*d - a
*e)^7*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

Rule 46

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[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 {\left (b^4 \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right )^5 (d+e x)^3} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {\left (b^4 \left (a b+b^2 x\right )\right ) \int \left (\frac {1}{b^2 (b d-a e)^3 (a+b x)^5}-\frac {3 e}{b^2 (b d-a e)^4 (a+b x)^4}+\frac {6 e^2}{b^2 (b d-a e)^5 (a+b x)^3}-\frac {10 e^3}{b^2 (b d-a e)^6 (a+b x)^2}+\frac {15 e^4}{b^2 (b d-a e)^7 (a+b x)}-\frac {e^5}{b^5 (b d-a e)^5 (d+e x)^3}-\frac {5 e^5}{b^4 (b d-a e)^6 (d+e x)^2}-\frac {15 e^5}{b^3 (b d-a e)^7 (d+e x)}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {10 b^2 e^3}{(b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b^2}{4 (b d-a e)^3 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b^2 e}{(b d-a e)^4 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 b^2 e^2}{(b d-a e)^5 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e^4 (a+b x)}{2 (b d-a e)^5 (d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 b e^4 (a+b x)}{(b d-a e)^6 (d+e x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {15 b^2 e^4 (a+b x) \log (a+b x)}{(b d-a e)^7 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {15 b^2 e^4 (a+b x) \log (d+e x)}{(b d-a e)^7 \sqrt {a^2+2 a b x+b^2 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.09 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.57 \[ \int \frac {1}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {4 b^2 e (b d-a e)^3-\frac {b^2 (b d-a e)^4}{a+b x}-12 b^2 e^2 (b d-a e)^2 (a+b x)+40 b^2 e^3 (b d-a e) (a+b x)^2+\frac {2 e^4 (b d-a e)^2 (a+b x)^3}{(d+e x)^2}+\frac {20 b e^4 (b d-a e) (a+b x)^3}{d+e x}+60 b^2 e^4 (a+b x)^3 \log (a+b x)-60 b^2 e^4 (a+b x)^3 \log (d+e x)}{4 (b d-a e)^7 \left ((a+b x)^2\right )^{3/2}} \]

[In]

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

[Out]

(4*b^2*e*(b*d - a*e)^3 - (b^2*(b*d - a*e)^4)/(a + b*x) - 12*b^2*e^2*(b*d - a*e)^2*(a + b*x) + 40*b^2*e^3*(b*d
- a*e)*(a + b*x)^2 + (2*e^4*(b*d - a*e)^2*(a + b*x)^3)/(d + e*x)^2 + (20*b*e^4*(b*d - a*e)*(a + b*x)^3)/(d + e
*x) + 60*b^2*e^4*(a + b*x)^3*Log[a + b*x] - 60*b^2*e^4*(a + b*x)^3*Log[d + e*x])/(4*(b*d - a*e)^7*((a + b*x)^2
)^(3/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(982\) vs. \(2(273)=546\).

Time = 2.60 (sec) , antiderivative size = 983, normalized size of antiderivative = 2.69

method	result	size
default	\(-\frac {\left (-12 x \,a^{5} b \,e^{6}+2 x \,b^{6} d^{5} e -60 x^{5} a \,b^{5} e^{6}+60 x^{5} b^{6} d \,e^{5}-210 x^{4} a^{2} b^{4} e^{6}+90 x^{4} b^{6} d^{2} e^{4}-260 x^{3} a^{3} b^{3} e^{6}+20 x^{3} b^{6} d^{3} e^{3}-125 x^{2} a^{4} b^{2} e^{6}+480 \ln \left (b x +a \right ) a \,b^{5} d \,e^{5} x^{4}+60 \ln \left (b x +a \right ) b^{6} d^{2} e^{4} x^{4}+2 a^{6} e^{6}-b^{6} d^{6}+8 a \,b^{5} d^{5} e -35 a^{4} b^{2} d^{2} e^{4}+80 a^{3} b^{3} d^{3} e^{3}-30 a^{2} b^{4} d^{4} e^{2}-24 a^{5} b d \,e^{5}-360 \ln \left (e x +d \right ) x^{2} a^{2} b^{4} d^{2} e^{4}-60 \ln \left (e x +d \right ) a^{4} b^{2} e^{6} x^{2}+60 \ln \left (b x +a \right ) a^{4} b^{2} e^{6} x^{2}-720 \ln \left (e x +d \right ) x^{3} a^{2} b^{4} d \,e^{5}-480 \ln \left (e x +d \right ) x^{4} a \,b^{5} d \,e^{5}+360 \ln \left (b x +a \right ) a^{2} b^{4} e^{6} x^{4}+60 \ln \left (b x +a \right ) b^{6} e^{6} x^{6}-480 \ln \left (e x +d \right ) a^{3} b^{3} d \,e^{5} x^{2}+240 \ln \left (b x +a \right ) a^{3} b^{3} e^{6} x^{3}+120 \ln \left (b x +a \right ) a^{4} b^{2} d \,e^{5} x +240 \ln \left (b x +a \right ) a^{3} b^{3} d^{2} e^{4} x -240 \ln \left (e x +d \right ) a \,b^{5} e^{6} x^{5}-120 \ln \left (e x +d \right ) a^{4} b^{2} d \,e^{5} x -240 \ln \left (e x +d \right ) a^{3} b^{3} d^{2} e^{4} x +720 \ln \left (b x +a \right ) a^{2} b^{4} d \,e^{5} x^{3}+240 \ln \left (b x +a \right ) a \,b^{5} d^{2} e^{4} x^{3}+480 \ln \left (b x +a \right ) a^{3} b^{3} d \,e^{5} x^{2}+360 \ln \left (b x +a \right ) a^{2} b^{4} d^{2} e^{4} x^{2}-240 \ln \left (e x +d \right ) x^{3} a \,b^{5} d^{2} e^{4}+60 \ln \left (b x +a \right ) a^{4} b^{2} d^{2} e^{4}+240 \ln \left (b x +a \right ) a \,b^{5} e^{6} x^{5}+120 \ln \left (b x +a \right ) b^{6} d \,e^{5} x^{5}-240 \ln \left (e x +d \right ) a^{3} b^{3} e^{6} x^{3}-60 \ln \left (e x +d \right ) x^{6} b^{6} e^{6}-60 \ln \left (e x +d \right ) a^{4} b^{2} d^{2} e^{4}-360 \ln \left (e x +d \right ) x^{4} a^{2} b^{4} e^{6}-60 \ln \left (e x +d \right ) x^{4} b^{6} d^{2} e^{4}-120 \ln \left (e x +d \right ) x^{5} b^{6} d \,e^{5}-5 x^{2} b^{6} d^{4} e^{2}+120 x^{4} a \,b^{5} d \,e^{5}-60 x^{3} a^{2} b^{4} d \,e^{5}+300 x^{3} a \,b^{5} d^{2} e^{4}-280 x^{2} a^{3} b^{3} d \,e^{5}+330 x^{2} a^{2} b^{4} d^{2} e^{4}+80 x^{2} a \,b^{5} d^{3} e^{3}-190 x \,a^{4} b^{2} d \,e^{5}+100 x \,a^{3} b^{3} d^{2} e^{4}+120 x \,a^{2} b^{4} d^{3} e^{3}-20 x a \,b^{5} d^{4} e^{2}\right ) \left (b x +a \right )}{4 \left (e x +d \right )^{2} \left (a e -b d \right )^{7} \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}\)	\(983\)
risch	\(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (\frac {15 b^{5} e^{5} x^{5}}{a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}}+\frac {15 b^{4} e^{4} \left (7 a e +3 b d \right ) x^{4}}{2 \left (a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}\right )}+\frac {5 b^{3} e^{3} \left (13 a^{2} e^{2}+16 a b d e +b^{2} d^{2}\right ) x^{3}}{a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}}+\frac {5 b^{2} e^{2} \left (25 a^{3} e^{3}+81 a^{2} b d \,e^{2}+15 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) x^{2}}{4 \left (a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}\right )}+\frac {\left (6 e^{4} a^{4}+101 b \,e^{3} d \,a^{3}+51 b^{2} e^{2} d^{2} a^{2}-9 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) b e x}{2 a^{6} e^{6}-12 a^{5} b d \,e^{5}+30 a^{4} b^{2} d^{2} e^{4}-40 a^{3} b^{3} d^{3} e^{3}+30 a^{2} b^{4} d^{4} e^{2}-12 a \,b^{5} d^{5} e +2 b^{6} d^{6}}-\frac {2 a^{5} e^{5}-22 a^{4} b d \,e^{4}-57 a^{3} b^{2} d^{2} e^{3}+23 a^{2} b^{3} d^{3} e^{2}-7 a \,b^{4} d^{4} e +b^{5} d^{5}}{4 \left (a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}\right )}\right )}{\left (b x +a \right )^{5} \left (e x +d \right )^{2}}+\frac {15 \sqrt {\left (b x +a \right )^{2}}\, b^{2} e^{4} \ln \left (-e x -d \right )}{\left (b x +a \right ) \left (a^{7} e^{7}-7 a^{6} b d \,e^{6}+21 a^{5} b^{2} d^{2} e^{5}-35 b^{3} d^{3} e^{4} a^{4}+35 b^{4} d^{4} e^{3} a^{3}-21 b^{5} d^{5} e^{2} a^{2}+7 b^{6} d^{6} e a -b^{7} d^{7}\right )}-\frac {15 \sqrt {\left (b x +a \right )^{2}}\, b^{2} e^{4} \ln \left (b x +a \right )}{\left (b x +a \right ) \left (a^{7} e^{7}-7 a^{6} b d \,e^{6}+21 a^{5} b^{2} d^{2} e^{5}-35 b^{3} d^{3} e^{4} a^{4}+35 b^{4} d^{4} e^{3} a^{3}-21 b^{5} d^{5} e^{2} a^{2}+7 b^{6} d^{6} e a -b^{7} d^{7}\right )}\)	\(985\)

[In]

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

[Out]

-1/4*(-12*x*a^5*b*e^6+2*x*b^6*d^5*e-60*x^5*a*b^5*e^6+60*x^5*b^6*d*e^5-210*x^4*a^2*b^4*e^6+90*x^4*b^6*d^2*e^4-2
60*x^3*a^3*b^3*e^6+20*x^3*b^6*d^3*e^3-125*x^2*a^4*b^2*e^6+480*ln(b*x+a)*a*b^5*d*e^5*x^4+60*ln(b*x+a)*b^6*d^2*e
^4*x^4+2*a^6*e^6-b^6*d^6+8*a*b^5*d^5*e-35*a^4*b^2*d^2*e^4+80*a^3*b^3*d^3*e^3-30*a^2*b^4*d^4*e^2-24*a^5*b*d*e^5
-360*ln(e*x+d)*x^2*a^2*b^4*d^2*e^4-60*ln(e*x+d)*a^4*b^2*e^6*x^2+60*ln(b*x+a)*a^4*b^2*e^6*x^2-720*ln(e*x+d)*x^3
*a^2*b^4*d*e^5-480*ln(e*x+d)*x^4*a*b^5*d*e^5+360*ln(b*x+a)*a^2*b^4*e^6*x^4+60*ln(b*x+a)*b^6*e^6*x^6-480*ln(e*x
+d)*a^3*b^3*d*e^5*x^2+240*ln(b*x+a)*a^3*b^3*e^6*x^3+120*ln(b*x+a)*a^4*b^2*d*e^5*x+240*ln(b*x+a)*a^3*b^3*d^2*e^
4*x-240*ln(e*x+d)*a*b^5*e^6*x^5-120*ln(e*x+d)*a^4*b^2*d*e^5*x-240*ln(e*x+d)*a^3*b^3*d^2*e^4*x+720*ln(b*x+a)*a^
2*b^4*d*e^5*x^3+240*ln(b*x+a)*a*b^5*d^2*e^4*x^3+480*ln(b*x+a)*a^3*b^3*d*e^5*x^2+360*ln(b*x+a)*a^2*b^4*d^2*e^4*
x^2-240*ln(e*x+d)*x^3*a*b^5*d^2*e^4+60*ln(b*x+a)*a^4*b^2*d^2*e^4+240*ln(b*x+a)*a*b^5*e^6*x^5+120*ln(b*x+a)*b^6
*d*e^5*x^5-240*ln(e*x+d)*a^3*b^3*e^6*x^3-60*ln(e*x+d)*x^6*b^6*e^6-60*ln(e*x+d)*a^4*b^2*d^2*e^4-360*ln(e*x+d)*x
^4*a^2*b^4*e^6-60*ln(e*x+d)*x^4*b^6*d^2*e^4-120*ln(e*x+d)*x^5*b^6*d*e^5-5*x^2*b^6*d^4*e^2+120*x^4*a*b^5*d*e^5-
60*x^3*a^2*b^4*d*e^5+300*x^3*a*b^5*d^2*e^4-280*x^2*a^3*b^3*d*e^5+330*x^2*a^2*b^4*d^2*e^4+80*x^2*a*b^5*d^3*e^3-
190*x*a^4*b^2*d*e^5+100*x*a^3*b^3*d^2*e^4+120*x*a^2*b^4*d^3*e^3-20*x*a*b^5*d^4*e^2)*(b*x+a)/(e*x+d)^2/(a*e-b*d
)^7/((b*x+a)^2)^(5/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1565 vs. \(2 (273) = 546\).

Time = 0.44 (sec) , antiderivative size = 1565, normalized size of antiderivative = 4.29 \[ \int \frac {1}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/4*(b^6*d^6 - 8*a*b^5*d^5*e + 30*a^2*b^4*d^4*e^2 - 80*a^3*b^3*d^3*e^3 + 35*a^4*b^2*d^2*e^4 + 24*a^5*b*d*e^5
- 2*a^6*e^6 - 60*(b^6*d*e^5 - a*b^5*e^6)*x^5 - 30*(3*b^6*d^2*e^4 + 4*a*b^5*d*e^5 - 7*a^2*b^4*e^6)*x^4 - 20*(b^
6*d^3*e^3 + 15*a*b^5*d^2*e^4 - 3*a^2*b^4*d*e^5 - 13*a^3*b^3*e^6)*x^3 + 5*(b^6*d^4*e^2 - 16*a*b^5*d^3*e^3 - 66*
a^2*b^4*d^2*e^4 + 56*a^3*b^3*d*e^5 + 25*a^4*b^2*e^6)*x^2 - 2*(b^6*d^5*e - 10*a*b^5*d^4*e^2 + 60*a^2*b^4*d^3*e^
3 + 50*a^3*b^3*d^2*e^4 - 95*a^4*b^2*d*e^5 - 6*a^5*b*e^6)*x - 60*(b^6*e^6*x^6 + a^4*b^2*d^2*e^4 + 2*(b^6*d*e^5
+ 2*a*b^5*e^6)*x^5 + (b^6*d^2*e^4 + 8*a*b^5*d*e^5 + 6*a^2*b^4*e^6)*x^4 + 4*(a*b^5*d^2*e^4 + 3*a^2*b^4*d*e^5 +
a^3*b^3*e^6)*x^3 + (6*a^2*b^4*d^2*e^4 + 8*a^3*b^3*d*e^5 + a^4*b^2*e^6)*x^2 + 2*(2*a^3*b^3*d^2*e^4 + a^4*b^2*d*
e^5)*x)*log(b*x + a) + 60*(b^6*e^6*x^6 + a^4*b^2*d^2*e^4 + 2*(b^6*d*e^5 + 2*a*b^5*e^6)*x^5 + (b^6*d^2*e^4 + 8*
a*b^5*d*e^5 + 6*a^2*b^4*e^6)*x^4 + 4*(a*b^5*d^2*e^4 + 3*a^2*b^4*d*e^5 + a^3*b^3*e^6)*x^3 + (6*a^2*b^4*d^2*e^4
+ 8*a^3*b^3*d*e^5 + a^4*b^2*e^6)*x^2 + 2*(2*a^3*b^3*d^2*e^4 + a^4*b^2*d*e^5)*x)*log(e*x + d))/(a^4*b^7*d^9 - 7
*a^5*b^6*d^8*e + 21*a^6*b^5*d^7*e^2 - 35*a^7*b^4*d^6*e^3 + 35*a^8*b^3*d^5*e^4 - 21*a^9*b^2*d^4*e^5 + 7*a^10*b*
d^3*e^6 - a^11*d^2*e^7 + (b^11*d^7*e^2 - 7*a*b^10*d^6*e^3 + 21*a^2*b^9*d^5*e^4 - 35*a^3*b^8*d^4*e^5 + 35*a^4*b
^7*d^3*e^6 - 21*a^5*b^6*d^2*e^7 + 7*a^6*b^5*d*e^8 - a^7*b^4*e^9)*x^6 + 2*(b^11*d^8*e - 5*a*b^10*d^7*e^2 + 7*a^
2*b^9*d^6*e^3 + 7*a^3*b^8*d^5*e^4 - 35*a^4*b^7*d^4*e^5 + 49*a^5*b^6*d^3*e^6 - 35*a^6*b^5*d^2*e^7 + 13*a^7*b^4*
d*e^8 - 2*a^8*b^3*e^9)*x^5 + (b^11*d^9 + a*b^10*d^8*e - 29*a^2*b^9*d^7*e^2 + 91*a^3*b^8*d^6*e^3 - 119*a^4*b^7*
d^5*e^4 + 49*a^5*b^6*d^4*e^5 + 49*a^6*b^5*d^3*e^6 - 71*a^7*b^4*d^2*e^7 + 34*a^8*b^3*d*e^8 - 6*a^9*b^2*e^9)*x^4
+ 4*(a*b^10*d^9 - 4*a^2*b^9*d^8*e + a^3*b^8*d^7*e^2 + 21*a^4*b^7*d^6*e^3 - 49*a^5*b^6*d^5*e^4 + 49*a^6*b^5*d^
4*e^5 - 21*a^7*b^4*d^3*e^6 - a^8*b^3*d^2*e^7 + 4*a^9*b^2*d*e^8 - a^10*b*e^9)*x^3 + (6*a^2*b^9*d^9 - 34*a^3*b^8
*d^8*e + 71*a^4*b^7*d^7*e^2 - 49*a^5*b^6*d^6*e^3 - 49*a^6*b^5*d^5*e^4 + 119*a^7*b^4*d^4*e^5 - 91*a^8*b^3*d^3*e
^6 + 29*a^9*b^2*d^2*e^7 - a^10*b*d*e^8 - a^11*e^9)*x^2 + 2*(2*a^3*b^8*d^9 - 13*a^4*b^7*d^8*e + 35*a^5*b^6*d^7*
e^2 - 49*a^6*b^5*d^6*e^3 + 35*a^7*b^4*d^5*e^4 - 7*a^8*b^3*d^4*e^5 - 7*a^9*b^2*d^3*e^6 + 5*a^10*b*d^2*e^7 - a^1
1*d*e^8)*x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

integrate(1/(e*x+d)^3/(b^2*x^2+2*a*b*x+a^2)^(5/2),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. 683 vs. \(2 (273) = 546\).

Time = 0.28 (sec) , antiderivative size = 683, normalized size of antiderivative = 1.87 \[ \int \frac {1}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {15 \, b^{3} e^{4} \log \left ({\left | b x + a \right |}\right )}{b^{8} d^{7} \mathrm {sgn}\left (b x + a\right ) - 7 \, a b^{7} d^{6} e \mathrm {sgn}\left (b x + a\right ) + 21 \, a^{2} b^{6} d^{5} e^{2} \mathrm {sgn}\left (b x + a\right ) - 35 \, a^{3} b^{5} d^{4} e^{3} \mathrm {sgn}\left (b x + a\right ) + 35 \, a^{4} b^{4} d^{3} e^{4} \mathrm {sgn}\left (b x + a\right ) - 21 \, a^{5} b^{3} d^{2} e^{5} \mathrm {sgn}\left (b x + a\right ) + 7 \, a^{6} b^{2} d e^{6} \mathrm {sgn}\left (b x + a\right ) - a^{7} b e^{7} \mathrm {sgn}\left (b x + a\right )} - \frac {15 \, b^{2} e^{5} \log \left ({\left | e x + d \right |}\right )}{b^{7} d^{7} e \mathrm {sgn}\left (b x + a\right ) - 7 \, a b^{6} d^{6} e^{2} \mathrm {sgn}\left (b x + a\right ) + 21 \, a^{2} b^{5} d^{5} e^{3} \mathrm {sgn}\left (b x + a\right ) - 35 \, a^{3} b^{4} d^{4} e^{4} \mathrm {sgn}\left (b x + a\right ) + 35 \, a^{4} b^{3} d^{3} e^{5} \mathrm {sgn}\left (b x + a\right ) - 21 \, a^{5} b^{2} d^{2} e^{6} \mathrm {sgn}\left (b x + a\right ) + 7 \, a^{6} b d e^{7} \mathrm {sgn}\left (b x + a\right ) - a^{7} e^{8} \mathrm {sgn}\left (b x + a\right )} - \frac {b^{6} d^{6} - 8 \, a b^{5} d^{5} e + 30 \, a^{2} b^{4} d^{4} e^{2} - 80 \, a^{3} b^{3} d^{3} e^{3} + 35 \, a^{4} b^{2} d^{2} e^{4} + 24 \, a^{5} b d e^{5} - 2 \, a^{6} e^{6} - 60 \, {\left (b^{6} d e^{5} - a b^{5} e^{6}\right )} x^{5} - 30 \, {\left (3 \, b^{6} d^{2} e^{4} + 4 \, a b^{5} d e^{5} - 7 \, a^{2} b^{4} e^{6}\right )} x^{4} - 20 \, {\left (b^{6} d^{3} e^{3} + 15 \, a b^{5} d^{2} e^{4} - 3 \, a^{2} b^{4} d e^{5} - 13 \, a^{3} b^{3} e^{6}\right )} x^{3} + 5 \, {\left (b^{6} d^{4} e^{2} - 16 \, a b^{5} d^{3} e^{3} - 66 \, a^{2} b^{4} d^{2} e^{4} + 56 \, a^{3} b^{3} d e^{5} + 25 \, a^{4} b^{2} e^{6}\right )} x^{2} - 2 \, {\left (b^{6} d^{5} e - 10 \, a b^{5} d^{4} e^{2} + 60 \, a^{2} b^{4} d^{3} e^{3} + 50 \, a^{3} b^{3} d^{2} e^{4} - 95 \, a^{4} b^{2} d e^{5} - 6 \, a^{5} b e^{6}\right )} x}{4 \, {\left (b d - a e\right )}^{7} {\left (b x + a\right )}^{4} {\left (e x + d\right )}^{2} \mathrm {sgn}\left (b x + a\right )} \]

[In]

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

[Out]

15*b^3*e^4*log(abs(b*x + a))/(b^8*d^7*sgn(b*x + a) - 7*a*b^7*d^6*e*sgn(b*x + a) + 21*a^2*b^6*d^5*e^2*sgn(b*x +
a) - 35*a^3*b^5*d^4*e^3*sgn(b*x + a) + 35*a^4*b^4*d^3*e^4*sgn(b*x + a) - 21*a^5*b^3*d^2*e^5*sgn(b*x + a) + 7*
a^6*b^2*d*e^6*sgn(b*x + a) - a^7*b*e^7*sgn(b*x + a)) - 15*b^2*e^5*log(abs(e*x + d))/(b^7*d^7*e*sgn(b*x + a) -
7*a*b^6*d^6*e^2*sgn(b*x + a) + 21*a^2*b^5*d^5*e^3*sgn(b*x + a) - 35*a^3*b^4*d^4*e^4*sgn(b*x + a) + 35*a^4*b^3*
d^3*e^5*sgn(b*x + a) - 21*a^5*b^2*d^2*e^6*sgn(b*x + a) + 7*a^6*b*d*e^7*sgn(b*x + a) - a^7*e^8*sgn(b*x + a)) -
1/4*(b^6*d^6 - 8*a*b^5*d^5*e + 30*a^2*b^4*d^4*e^2 - 80*a^3*b^3*d^3*e^3 + 35*a^4*b^2*d^2*e^4 + 24*a^5*b*d*e^5 -
2*a^6*e^6 - 60*(b^6*d*e^5 - a*b^5*e^6)*x^5 - 30*(3*b^6*d^2*e^4 + 4*a*b^5*d*e^5 - 7*a^2*b^4*e^6)*x^4 - 20*(b^6
*d^3*e^3 + 15*a*b^5*d^2*e^4 - 3*a^2*b^4*d*e^5 - 13*a^3*b^3*e^6)*x^3 + 5*(b^6*d^4*e^2 - 16*a*b^5*d^3*e^3 - 66*a
^2*b^4*d^2*e^4 + 56*a^3*b^3*d*e^5 + 25*a^4*b^2*e^6)*x^2 - 2*(b^6*d^5*e - 10*a*b^5*d^4*e^2 + 60*a^2*b^4*d^3*e^3
+ 50*a^3*b^3*d^2*e^4 - 95*a^4*b^2*d*e^5 - 6*a^5*b*e^6)*x)/((b*d - a*e)^7*(b*x + a)^4*(e*x + d)^2*sgn(b*x + a)
)

Mupad [F(-1)]

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

[In]

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

[Out]

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