\(\int \frac {1}{(d+e x)^{7/2} (a^2+2 a b x+b^2 x^2)^2} \, dx\) [1663]

Optimal result

Integrand size = 28, antiderivative size = 229 \[ \int \frac {1}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=-\frac {231 e^3}{40 (b d-a e)^4 (d+e x)^{5/2}}-\frac {1}{3 (b d-a e) (a+b x)^3 (d+e x)^{5/2}}+\frac {11 e}{12 (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2}}-\frac {33 e^2}{8 (b d-a e)^3 (a+b x) (d+e x)^{5/2}}-\frac {77 b e^3}{8 (b d-a e)^5 (d+e x)^{3/2}}-\frac {231 b^2 e^3}{8 (b d-a e)^6 \sqrt {d+e x}}+\frac {231 b^{5/2} e^3 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 (b d-a e)^{13/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {27, 44, 53, 65, 214} \[ \int \frac {1}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {231 b^{5/2} e^3 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 (b d-a e)^{13/2}}-\frac {231 b^2 e^3}{8 \sqrt {d+e x} (b d-a e)^6}-\frac {77 b e^3}{8 (d+e x)^{3/2} (b d-a e)^5}-\frac {231 e^3}{40 (d+e x)^{5/2} (b d-a e)^4}-\frac {33 e^2}{8 (a+b x) (d+e x)^{5/2} (b d-a e)^3}+\frac {11 e}{12 (a+b x)^2 (d+e x)^{5/2} (b d-a e)^2}-\frac {1}{3 (a+b x)^3 (d+e x)^{5/2} (b d-a e)} \]

[In]

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

[Out]

(-231*e^3)/(40*(b*d - a*e)^4*(d + e*x)^(5/2)) - 1/(3*(b*d - a*e)*(a + b*x)^3*(d + e*x)^(5/2)) + (11*e)/(12*(b*
d - a*e)^2*(a + b*x)^2*(d + e*x)^(5/2)) - (33*e^2)/(8*(b*d - a*e)^3*(a + b*x)*(d + e*x)^(5/2)) - (77*b*e^3)/(8
*(b*d - a*e)^5*(d + e*x)^(3/2)) - (231*b^2*e^3)/(8*(b*d - a*e)^6*Sqrt[d + e*x]) + (231*b^(5/2)*e^3*ArcTanh[(Sq
rt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(8*(b*d - a*e)^(13/2))

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 44

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] - Dist[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] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 0]

Rule 53

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] - Dist[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] && NeQ[b*c - a*d, 0] && 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 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[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] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 214

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]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(a+b x)^4 (d+e x)^{7/2}} \, dx \\ & = -\frac {1}{3 (b d-a e) (a+b x)^3 (d+e x)^{5/2}}-\frac {(11 e) \int \frac {1}{(a+b x)^3 (d+e x)^{7/2}} \, dx}{6 (b d-a e)} \\ & = -\frac {1}{3 (b d-a e) (a+b x)^3 (d+e x)^{5/2}}+\frac {11 e}{12 (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2}}+\frac {\left (33 e^2\right ) \int \frac {1}{(a+b x)^2 (d+e x)^{7/2}} \, dx}{8 (b d-a e)^2} \\ & = -\frac {1}{3 (b d-a e) (a+b x)^3 (d+e x)^{5/2}}+\frac {11 e}{12 (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2}}-\frac {33 e^2}{8 (b d-a e)^3 (a+b x) (d+e x)^{5/2}}-\frac {\left (231 e^3\right ) \int \frac {1}{(a+b x) (d+e x)^{7/2}} \, dx}{16 (b d-a e)^3} \\ & = -\frac {231 e^3}{40 (b d-a e)^4 (d+e x)^{5/2}}-\frac {1}{3 (b d-a e) (a+b x)^3 (d+e x)^{5/2}}+\frac {11 e}{12 (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2}}-\frac {33 e^2}{8 (b d-a e)^3 (a+b x) (d+e x)^{5/2}}-\frac {\left (231 b e^3\right ) \int \frac {1}{(a+b x) (d+e x)^{5/2}} \, dx}{16 (b d-a e)^4} \\ & = -\frac {231 e^3}{40 (b d-a e)^4 (d+e x)^{5/2}}-\frac {1}{3 (b d-a e) (a+b x)^3 (d+e x)^{5/2}}+\frac {11 e}{12 (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2}}-\frac {33 e^2}{8 (b d-a e)^3 (a+b x) (d+e x)^{5/2}}-\frac {77 b e^3}{8 (b d-a e)^5 (d+e x)^{3/2}}-\frac {\left (231 b^2 e^3\right ) \int \frac {1}{(a+b x) (d+e x)^{3/2}} \, dx}{16 (b d-a e)^5} \\ & = -\frac {231 e^3}{40 (b d-a e)^4 (d+e x)^{5/2}}-\frac {1}{3 (b d-a e) (a+b x)^3 (d+e x)^{5/2}}+\frac {11 e}{12 (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2}}-\frac {33 e^2}{8 (b d-a e)^3 (a+b x) (d+e x)^{5/2}}-\frac {77 b e^3}{8 (b d-a e)^5 (d+e x)^{3/2}}-\frac {231 b^2 e^3}{8 (b d-a e)^6 \sqrt {d+e x}}-\frac {\left (231 b^3 e^3\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{16 (b d-a e)^6} \\ & = -\frac {231 e^3}{40 (b d-a e)^4 (d+e x)^{5/2}}-\frac {1}{3 (b d-a e) (a+b x)^3 (d+e x)^{5/2}}+\frac {11 e}{12 (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2}}-\frac {33 e^2}{8 (b d-a e)^3 (a+b x) (d+e x)^{5/2}}-\frac {77 b e^3}{8 (b d-a e)^5 (d+e x)^{3/2}}-\frac {231 b^2 e^3}{8 (b d-a e)^6 \sqrt {d+e x}}-\frac {\left (231 b^3 e^2\right ) \text {Subst}\left (\int \frac {1}{a-\frac {b d}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{8 (b d-a e)^6} \\ & = -\frac {231 e^3}{40 (b d-a e)^4 (d+e x)^{5/2}}-\frac {1}{3 (b d-a e) (a+b x)^3 (d+e x)^{5/2}}+\frac {11 e}{12 (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2}}-\frac {33 e^2}{8 (b d-a e)^3 (a+b x) (d+e x)^{5/2}}-\frac {77 b e^3}{8 (b d-a e)^5 (d+e x)^{3/2}}-\frac {231 b^2 e^3}{8 (b d-a e)^6 \sqrt {d+e x}}+\frac {231 b^{5/2} e^3 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 (b d-a e)^{13/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.05 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.24 \[ \int \frac {1}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {-48 a^5 e^5+16 a^4 b e^4 (26 d+11 e x)-16 a^3 b^2 e^3 \left (173 d^2+242 d e x+99 e^2 x^2\right )-3 a^2 b^3 e^2 \left (445 d^3+4103 d^2 e x+6039 d e^2 x^2+2541 e^3 x^3\right )-2 a b^4 e \left (-155 d^4+715 d^3 e x+7227 d^2 e^2 x^2+10857 d e^3 x^3+4620 e^4 x^4\right )-b^5 \left (40 d^5-110 d^4 e x+495 d^3 e^2 x^2+5313 d^2 e^3 x^3+8085 d e^4 x^4+3465 e^5 x^5\right )}{120 (b d-a e)^6 (a+b x)^3 (d+e x)^{5/2}}-\frac {231 b^{5/2} e^3 \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{8 (-b d+a e)^{13/2}} \]

[In]

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

[Out]

(-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))/(120*(b*d - a*e)^6*(a + b*x)^3*(d + e*x)^(5/2)) - (231*b^(5/2)*e^3*ArcTan[(Sqrt[b]*
Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]])/(8*(-(b*d) + a*e)^(13/2))

Maple [A] (verified)

Time = 2.52 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.87

method	result	size
derivativedivides	\(2 e^{3} \left (-\frac {b^{3} \left (\frac {\frac {71 b^{2} \left (e x +d \right )^{\frac {5}{2}}}{16}+\frac {59 \left (a e -b d \right ) b \left (e x +d \right )^{\frac {3}{2}}}{6}+\left (\frac {89}{16} a^{2} e^{2}-\frac {89}{8} a b d e +\frac {89}{16} b^{2} d^{2}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{3}}+\frac {231 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{16 \sqrt {\left (a e -b d \right ) b}}\right )}{\left (a e -b d \right )^{6}}-\frac {1}{5 \left (a e -b d \right )^{4} \left (e x +d \right )^{\frac {5}{2}}}-\frac {10 b^{2}}{\left (a e -b d \right )^{6} \sqrt {e x +d}}+\frac {4 b}{3 \left (a e -b d \right )^{5} \left (e x +d \right )^{\frac {3}{2}}}\right )\)	\(200\)
default	\(2 e^{3} \left (-\frac {b^{3} \left (\frac {\frac {71 b^{2} \left (e x +d \right )^{\frac {5}{2}}}{16}+\frac {59 \left (a e -b d \right ) b \left (e x +d \right )^{\frac {3}{2}}}{6}+\left (\frac {89}{16} a^{2} e^{2}-\frac {89}{8} a b d e +\frac {89}{16} b^{2} d^{2}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{3}}+\frac {231 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{16 \sqrt {\left (a e -b d \right ) b}}\right )}{\left (a e -b d \right )^{6}}-\frac {1}{5 \left (a e -b d \right )^{4} \left (e x +d \right )^{\frac {5}{2}}}-\frac {10 b^{2}}{\left (a e -b d \right )^{6} \sqrt {e x +d}}+\frac {4 b}{3 \left (a e -b d \right )^{5} \left (e x +d \right )^{\frac {3}{2}}}\right )\)	\(200\)
pseudoelliptic	\(-\frac {2 \left (\frac {1155 b^{3} e^{3} \left (e x +d \right )^{\frac {5}{2}} \left (b x +a \right )^{3} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{16}+\left (\left (\frac {1155}{16} e^{5} x^{5}-\frac {55}{24} d^{4} e x +\frac {2695}{16} x^{4} d \,e^{4}+\frac {5}{6} d^{5}+\frac {1771}{16} d^{2} e^{3} x^{3}+\frac {165}{16} d^{3} e^{2} x^{2}\right ) b^{5}-\frac {155 \left (-\frac {924}{31} e^{4} x^{4}-\frac {10857}{155} d \,e^{3} x^{3}-\frac {7227}{155} d^{2} e^{2} x^{2}-\frac {143}{31} d^{3} e x +d^{4}\right ) e a \,b^{4}}{24}+\frac {445 \left (\frac {2541}{445} e^{3} x^{3}+\frac {6039}{445} d \,e^{2} x^{2}+\frac {4103}{445} d^{2} e x +d^{3}\right ) e^{2} a^{2} b^{3}}{16}+\frac {173 \left (\frac {99}{173} x^{2} e^{2}+\frac {242}{173} d e x +d^{2}\right ) e^{3} a^{3} b^{2}}{3}-\frac {26 \left (\frac {11 e x}{26}+d \right ) e^{4} a^{4} b}{3}+a^{5} e^{5}\right ) \sqrt {\left (a e -b d \right ) b}\right )}{5 \left (e x +d \right )^{\frac {5}{2}} \sqrt {\left (a e -b d \right ) b}\, \left (b x +a \right )^{3} \left (a e -b d \right )^{6}}\)	\(289\)

[In]

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

[Out]

2*e^3*(-1/(a*e-b*d)^6*b^3*((71/16*b^2*(e*x+d)^(5/2)+59/6*(a*e-b*d)*b*(e*x+d)^(3/2)+(89/16*a^2*e^2-89/8*a*b*d*e
+89/16*b^2*d^2)*(e*x+d)^(1/2))/(b*(e*x+d)+a*e-b*d)^3+231/16/((a*e-b*d)*b)^(1/2)*arctan(b*(e*x+d)^(1/2)/((a*e-b
*d)*b)^(1/2)))-1/5/(a*e-b*d)^4/(e*x+d)^(5/2)-10/(a*e-b*d)^6*b^2/(e*x+d)^(1/2)+4/3/(a*e-b*d)^5*b/(e*x+d)^(3/2))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1270 vs. \(2 (193) = 386\).

Time = 1.03 (sec) , antiderivative size = 2550, normalized size of antiderivative = 11.14 \[ \int \frac {1}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[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^6*d^6*e^3 - 36*a^4*b^5*d^5*e^4 - 36*a^5*b^4*d^4*e^5 + 62*a^6*b^
3*d^3*e^6 - 30*a^7*b^2*d^2*e^7 + 3*a^8*b*d*e^8 + a^9*e^9)*x^3 + 3*(a*b^8*d^9 - 3*a^2*b^7*d^8*e - 2*a^3*b^6*d^7
*e^2 + 19*a^4*b^5*d^6*e^3 - 30*a^5*b^4*d^5*e^4 + 19*a^6*b^3*d^4*e^5 - 2*a^7*b^2*d^3*e^6 - 3*a^8*b*d^2*e^7 + a^
9*d*e^8)*x^2 + 3*(a^2*b^7*d^9 - 5*a^3*b^6*d^8*e + 9*a^4*b^5*d^7*e^2 - 5*a^5*b^4*d^6*e^3 - 5*a^6*b^3*d^5*e^4 +
9*a^7*b^2*d^4*e^5 - 5*a^8*b*d^3*e^6 + a^9*d^2*e^7)*x), 1/120*(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))*arctan(-(b*d - a*e)*sqrt(e*x + d)*sqrt(-b/(b*d - a*e))/(b*e*x +
b*d)) - (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^6*d^6*e^3 - 36*a^4*b^5*d^5*
e^4 - 36*a^5*b^4*d^4*e^5 + 62*a^6*b^3*d^3*e^6 - 30*a^7*b^2*d^2*e^7 + 3*a^8*b*d*e^8 + a^9*e^9)*x^3 + 3*(a*b^8*d
^9 - 3*a^2*b^7*d^8*e - 2*a^3*b^6*d^7*e^2 + 19*a^4*b^5*d^6*e^3 - 30*a^5*b^4*d^5*e^4 + 19*a^6*b^3*d^4*e^5 - 2*a^
7*b^2*d^3*e^6 - 3*a^8*b*d^2*e^7 + a^9*d*e^8)*x^2 + 3*(a^2*b^7*d^9 - 5*a^3*b^6*d^8*e + 9*a^4*b^5*d^7*e^2 - 5*a^
5*b^4*d^6*e^3 - 5*a^6*b^3*d^5*e^4 + 9*a^7*b^2*d^4*e^5 - 5*a^8*b*d^3*e^6 + a^9*d^2*e^7)*x)]

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(1/(e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^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. 480 vs. \(2 (193) = 386\).

Time = 0.28 (sec) , antiderivative size = 480, normalized size of antiderivative = 2.10 \[ \int \frac {1}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=-\frac {231 \, b^{3} e^{3} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{8 \, {\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} \sqrt {-b^{2} d + a b e}} - \frac {2 \, {\left (150 \, {\left (e x + d\right )}^{2} b^{2} e^{3} + 20 \, {\left (e x + d\right )} b^{2} d e^{3} + 3 \, b^{2} d^{2} e^{3} - 20 \, {\left (e x + d\right )} a b e^{4} - 6 \, a b d e^{4} + 3 \, a^{2} e^{5}\right )}}{15 \, {\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} {\left (e x + d\right )}^{\frac {5}{2}}} - \frac {213 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{5} e^{3} - 472 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{5} d e^{3} + 267 \, \sqrt {e x + d} b^{5} d^{2} e^{3} + 472 \, {\left (e x + d\right )}^{\frac {3}{2}} a b^{4} e^{4} - 534 \, \sqrt {e x + d} a b^{4} d e^{4} + 267 \, \sqrt {e x + d} a^{2} b^{3} e^{5}}{24 \, {\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} {\left ({\left (e x + d\right )} b - b d + a e\right )}^{3}} \]

[In]

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

[Out]

-231/8*b^3*e^3*arctan(sqrt(e*x + d)*b/sqrt(-b^2*d + a*b*e))/((b^6*d^6 - 6*a*b^5*d^5*e + 15*a^2*b^4*d^4*e^2 - 2
0*a^3*b^3*d^3*e^3 + 15*a^4*b^2*d^2*e^4 - 6*a^5*b*d*e^5 + a^6*e^6)*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
- 6*a*b^5*d^5*e + 15*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3 + 15*a^4*b^2*d^2*e^4 - 6*a^5*b*d*e^5 + a^6*e^6)*(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 + 472*(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
- 6*a*b^5*d^5*e + 15*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3 + 15*a^4*b^2*d^2*e^4 - 6*a^5*b*d*e^5 + a^6*e^6)*((e
*x + d)*b - b*d + a*e)^3)

Mupad [B] (verification not implemented)

Time = 9.83 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.63 \[ \int \frac {1}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=-\frac {\frac {2\,e^3}{5\,\left (a\,e-b\,d\right )}+\frac {66\,b^2\,e^3\,{\left (d+e\,x\right )}^2}{5\,{\left (a\,e-b\,d\right )}^3}+\frac {2541\,b^3\,e^3\,{\left (d+e\,x\right )}^3}{40\,{\left (a\,e-b\,d\right )}^4}+\frac {77\,b^4\,e^3\,{\left (d+e\,x\right )}^4}{{\left (a\,e-b\,d\right )}^5}+\frac {231\,b^5\,e^3\,{\left (d+e\,x\right )}^5}{8\,{\left (a\,e-b\,d\right )}^6}-\frac {22\,b\,e^3\,\left (d+e\,x\right )}{15\,{\left (a\,e-b\,d\right )}^2}}{{\left (d+e\,x\right )}^{5/2}\,\left (a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3\right )+b^3\,{\left (d+e\,x\right )}^{11/2}-\left (3\,b^3\,d-3\,a\,b^2\,e\right )\,{\left (d+e\,x\right )}^{9/2}+{\left (d+e\,x\right )}^{7/2}\,\left (3\,a^2\,b\,e^2-6\,a\,b^2\,d\,e+3\,b^3\,d^2\right )}-\frac {231\,b^{5/2}\,e^3\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}\,\left (a^6\,e^6-6\,a^5\,b\,d\,e^5+15\,a^4\,b^2\,d^2\,e^4-20\,a^3\,b^3\,d^3\,e^3+15\,a^2\,b^4\,d^4\,e^2-6\,a\,b^5\,d^5\,e+b^6\,d^6\right )}{{\left (a\,e-b\,d\right )}^{13/2}}\right )}{8\,{\left (a\,e-b\,d\right )}^{13/2}} \]

[In]

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

[Out]

- ((2*e^3)/(5*(a*e - b*d)) + (66*b^2*e^3*(d + e*x)^2)/(5*(a*e - b*d)^3) + (2541*b^3*e^3*(d + e*x)^3)/(40*(a*e
- b*d)^4) + (77*b^4*e^3*(d + e*x)^4)/(a*e - b*d)^5 + (231*b^5*e^3*(d + e*x)^5)/(8*(a*e - b*d)^6) - (22*b*e^3*(
d + e*x))/(15*(a*e - b*d)^2))/((d + e*x)^(5/2)*(a^3*e^3 - b^3*d^3 + 3*a*b^2*d^2*e - 3*a^2*b*d*e^2) + b^3*(d +
e*x)^(11/2) - (3*b^3*d - 3*a*b^2*e)*(d + e*x)^(9/2) + (d + e*x)^(7/2)*(3*b^3*d^2 + 3*a^2*b*e^2 - 6*a*b^2*d*e))
- (231*b^(5/2)*e^3*atan((b^(1/2)*(d + e*x)^(1/2)*(a^6*e^6 + b^6*d^6 + 15*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3
+ 15*a^4*b^2*d^2*e^4 - 6*a*b^5*d^5*e - 6*a^5*b*d*e^5))/(a*e - b*d)^(13/2)))/(8*(a*e - b*d)^(13/2))