\(\int \frac {(d+e x)^{9/2}}{a^2+2 a b x+b^2 x^2} \, dx\) [1645]

Optimal result

Integrand size = 28, antiderivative size = 162 \[ \int \frac {(d+e x)^{9/2}}{a^2+2 a b x+b^2 x^2} \, dx=\frac {9 e (b d-a e)^3 \sqrt {d+e x}}{b^5}+\frac {3 e (b d-a e)^2 (d+e x)^{3/2}}{b^4}+\frac {9 e (b d-a e) (d+e x)^{5/2}}{5 b^3}+\frac {9 e (d+e x)^{7/2}}{7 b^2}-\frac {(d+e x)^{9/2}}{b (a+b x)}-\frac {9 e (b d-a e)^{7/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{11/2}} \]

[Out]

3*e*(-a*e+b*d)^2*(e*x+d)^(3/2)/b^4+9/5*e*(-a*e+b*d)*(e*x+d)^(5/2)/b^3+9/7*e*(e*x+d)^(7/2)/b^2-(e*x+d)^(9/2)/b/
(b*x+a)-9*e*(-a*e+b*d)^(7/2)*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))/b^(11/2)+9*e*(-a*e+b*d)^3*(e*x+d)
^(1/2)/b^5

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {27, 43, 52, 65, 214} \[ \int \frac {(d+e x)^{9/2}}{a^2+2 a b x+b^2 x^2} \, dx=-\frac {9 e (b d-a e)^{7/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{11/2}}+\frac {9 e \sqrt {d+e x} (b d-a e)^3}{b^5}+\frac {3 e (d+e x)^{3/2} (b d-a e)^2}{b^4}+\frac {9 e (d+e x)^{5/2} (b d-a e)}{5 b^3}-\frac {(d+e x)^{9/2}}{b (a+b x)}+\frac {9 e (d+e x)^{7/2}}{7 b^2} \]

[In]

Int[(d + e*x)^(9/2)/(a^2 + 2*a*b*x + b^2*x^2),x]

[Out]

(9*e*(b*d - a*e)^3*Sqrt[d + e*x])/b^5 + (3*e*(b*d - a*e)^2*(d + e*x)^(3/2))/b^4 + (9*e*(b*d - a*e)*(d + e*x)^(
5/2))/(5*b^3) + (9*e*(d + e*x)^(7/2))/(7*b^2) - (d + e*x)^(9/2)/(b*(a + b*x)) - (9*e*(b*d - a*e)^(7/2)*ArcTanh
[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/b^(11/2)

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 43

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + 1))), x] - Dist[d*(n/(b*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n
}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && GtQ[n, 0]

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && 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 {(d+e x)^{9/2}}{(a+b x)^2} \, dx \\ & = -\frac {(d+e x)^{9/2}}{b (a+b x)}+\frac {(9 e) \int \frac {(d+e x)^{7/2}}{a+b x} \, dx}{2 b} \\ & = \frac {9 e (d+e x)^{7/2}}{7 b^2}-\frac {(d+e x)^{9/2}}{b (a+b x)}+\frac {(9 e (b d-a e)) \int \frac {(d+e x)^{5/2}}{a+b x} \, dx}{2 b^2} \\ & = \frac {9 e (b d-a e) (d+e x)^{5/2}}{5 b^3}+\frac {9 e (d+e x)^{7/2}}{7 b^2}-\frac {(d+e x)^{9/2}}{b (a+b x)}+\frac {\left (9 e (b d-a e)^2\right ) \int \frac {(d+e x)^{3/2}}{a+b x} \, dx}{2 b^3} \\ & = \frac {3 e (b d-a e)^2 (d+e x)^{3/2}}{b^4}+\frac {9 e (b d-a e) (d+e x)^{5/2}}{5 b^3}+\frac {9 e (d+e x)^{7/2}}{7 b^2}-\frac {(d+e x)^{9/2}}{b (a+b x)}+\frac {\left (9 e (b d-a e)^3\right ) \int \frac {\sqrt {d+e x}}{a+b x} \, dx}{2 b^4} \\ & = \frac {9 e (b d-a e)^3 \sqrt {d+e x}}{b^5}+\frac {3 e (b d-a e)^2 (d+e x)^{3/2}}{b^4}+\frac {9 e (b d-a e) (d+e x)^{5/2}}{5 b^3}+\frac {9 e (d+e x)^{7/2}}{7 b^2}-\frac {(d+e x)^{9/2}}{b (a+b x)}+\frac {\left (9 e (b d-a e)^4\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{2 b^5} \\ & = \frac {9 e (b d-a e)^3 \sqrt {d+e x}}{b^5}+\frac {3 e (b d-a e)^2 (d+e x)^{3/2}}{b^4}+\frac {9 e (b d-a e) (d+e x)^{5/2}}{5 b^3}+\frac {9 e (d+e x)^{7/2}}{7 b^2}-\frac {(d+e x)^{9/2}}{b (a+b x)}+\frac {\left (9 (b d-a e)^4\right ) \text {Subst}\left (\int \frac {1}{a-\frac {b d}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{b^5} \\ & = \frac {9 e (b d-a e)^3 \sqrt {d+e x}}{b^5}+\frac {3 e (b d-a e)^2 (d+e x)^{3/2}}{b^4}+\frac {9 e (b d-a e) (d+e x)^{5/2}}{5 b^3}+\frac {9 e (d+e x)^{7/2}}{7 b^2}-\frac {(d+e x)^{9/2}}{b (a+b x)}-\frac {9 e (b d-a e)^{7/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{11/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.28 \[ \int \frac {(d+e x)^{9/2}}{a^2+2 a b x+b^2 x^2} \, dx=-\frac {\sqrt {d+e x} \left (315 a^4 e^4+210 a^3 b e^3 (-5 d+e x)-42 a^2 b^2 e^2 \left (-29 d^2+17 d e x+e^2 x^2\right )+6 a b^3 e \left (-88 d^3+142 d^2 e x+23 d e^2 x^2+3 e^3 x^3\right )+b^4 \left (35 d^4-388 d^3 e x-156 d^2 e^2 x^2-58 d e^3 x^3-10 e^4 x^4\right )\right )}{35 b^5 (a+b x)}+\frac {9 e (-b d+a e)^{7/2} \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{b^{11/2}} \]

[In]

Integrate[(d + e*x)^(9/2)/(a^2 + 2*a*b*x + b^2*x^2),x]

[Out]

-1/35*(Sqrt[d + e*x]*(315*a^4*e^4 + 210*a^3*b*e^3*(-5*d + e*x) - 42*a^2*b^2*e^2*(-29*d^2 + 17*d*e*x + e^2*x^2)
+ 6*a*b^3*e*(-88*d^3 + 142*d^2*e*x + 23*d*e^2*x^2 + 3*e^3*x^3) + b^4*(35*d^4 - 388*d^3*e*x - 156*d^2*e^2*x^2
- 58*d*e^3*x^3 - 10*e^4*x^4)))/(b^5*(a + b*x)) + (9*e*(-(b*d) + a*e)^(7/2)*ArcTan[(Sqrt[b]*Sqrt[d + e*x])/Sqrt
[-(b*d) + a*e]])/b^(11/2)

Maple [A] (verified)

Time = 2.57 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.34

method	result	size
pseudoelliptic	\(\frac {9 e \left (a e -b d \right )^{4} \left (b x +a \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )-9 \sqrt {e x +d}\, \left (\left (-\frac {2}{63} e^{4} x^{4}-\frac {58}{315} d \,e^{3} x^{3}-\frac {52}{105} d^{2} e^{2} x^{2}-\frac {388}{315} d^{3} e x +\frac {1}{9} d^{4}\right ) b^{4}-\frac {176 \left (-\frac {3}{88} e^{3} x^{3}-\frac {23}{88} d \,e^{2} x^{2}-\frac {71}{44} d^{2} e x +d^{3}\right ) e a \,b^{3}}{105}+\frac {58 \left (-\frac {1}{29} x^{2} e^{2}-\frac {17}{29} d e x +d^{2}\right ) e^{2} a^{2} b^{2}}{15}-\frac {10 e^{3} \left (-\frac {e x}{5}+d \right ) a^{3} b}{3}+e^{4} a^{4}\right ) \sqrt {\left (a e -b d \right ) b}}{b^{5} \left (b x +a \right ) \sqrt {\left (a e -b d \right ) b}}\)	\(217\)
risch	\(-\frac {2 e \left (-5 e^{3} x^{3} b^{3}+14 x^{2} a \,b^{2} e^{3}-29 x^{2} b^{3} d \,e^{2}-35 a^{2} b \,e^{3} x +98 x a \,b^{2} d \,e^{2}-78 b^{3} d^{2} e x +140 a^{3} e^{3}-455 a^{2} b d \,e^{2}+504 a \,b^{2} d^{2} e -194 b^{3} d^{3}\right ) \sqrt {e x +d}}{35 b^{5}}+\frac {\left (2 e^{4} a^{4}-8 b \,e^{3} d \,a^{3}+12 b^{2} e^{2} d^{2} a^{2}-8 a \,b^{3} d^{3} e +2 b^{4} d^{4}\right ) e \left (-\frac {\sqrt {e x +d}}{2 \left (b \left (e x +d \right )+a e -b d \right )}+\frac {9 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{2 \sqrt {\left (a e -b d \right ) b}}\right )}{b^{5}}\)	\(237\)
derivativedivides	\(2 e \left (-\frac {-\frac {\left (e x +d \right )^{\frac {7}{2}} b^{3}}{7}+\frac {2 a \,b^{2} e \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {2 b^{3} d \left (e x +d \right )^{\frac {5}{2}}}{5}-a^{2} b \,e^{2} \left (e x +d \right )^{\frac {3}{2}}+2 a \,b^{2} d e \left (e x +d \right )^{\frac {3}{2}}-b^{3} d^{2} \left (e x +d \right )^{\frac {3}{2}}+4 a^{3} e^{3} \sqrt {e x +d}-12 a^{2} d \,e^{2} b \sqrt {e x +d}+12 a \,d^{2} e \,b^{2} \sqrt {e x +d}-4 d^{3} b^{3} \sqrt {e x +d}}{b^{5}}+\frac {\frac {\left (-\frac {1}{2} e^{4} a^{4}+2 b \,e^{3} d \,a^{3}-3 b^{2} e^{2} d^{2} a^{2}+2 a \,b^{3} d^{3} e -\frac {1}{2} b^{4} d^{4}\right ) \sqrt {e x +d}}{b \left (e x +d \right )+a e -b d}+\frac {9 \left (e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{2 \sqrt {\left (a e -b d \right ) b}}}{b^{5}}\right )\)	\(326\)
default	\(2 e \left (-\frac {-\frac {\left (e x +d \right )^{\frac {7}{2}} b^{3}}{7}+\frac {2 a \,b^{2} e \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {2 b^{3} d \left (e x +d \right )^{\frac {5}{2}}}{5}-a^{2} b \,e^{2} \left (e x +d \right )^{\frac {3}{2}}+2 a \,b^{2} d e \left (e x +d \right )^{\frac {3}{2}}-b^{3} d^{2} \left (e x +d \right )^{\frac {3}{2}}+4 a^{3} e^{3} \sqrt {e x +d}-12 a^{2} d \,e^{2} b \sqrt {e x +d}+12 a \,d^{2} e \,b^{2} \sqrt {e x +d}-4 d^{3} b^{3} \sqrt {e x +d}}{b^{5}}+\frac {\frac {\left (-\frac {1}{2} e^{4} a^{4}+2 b \,e^{3} d \,a^{3}-3 b^{2} e^{2} d^{2} a^{2}+2 a \,b^{3} d^{3} e -\frac {1}{2} b^{4} d^{4}\right ) \sqrt {e x +d}}{b \left (e x +d \right )+a e -b d}+\frac {9 \left (e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{2 \sqrt {\left (a e -b d \right ) b}}}{b^{5}}\right )\)	\(326\)

[In]

int((e*x+d)^(9/2)/(b^2*x^2+2*a*b*x+a^2),x,method=_RETURNVERBOSE)

[Out]

9/((a*e-b*d)*b)^(1/2)*(e*(a*e-b*d)^4*(b*x+a)*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))-(e*x+d)^(1/2)*((-2/63
*e^4*x^4-58/315*d*e^3*x^3-52/105*d^2*e^2*x^2-388/315*d^3*e*x+1/9*d^4)*b^4-176/105*(-3/88*e^3*x^3-23/88*d*e^2*x
^2-71/44*d^2*e*x+d^3)*e*a*b^3+58/15*(-1/29*x^2*e^2-17/29*d*e*x+d^2)*e^2*a^2*b^2-10/3*e^3*(-1/5*e*x+d)*a^3*b+e^
4*a^4)*((a*e-b*d)*b)^(1/2))/b^5/(b*x+a)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 334 vs. \(2 (138) = 276\).

Time = 0.34 (sec) , antiderivative size = 678, normalized size of antiderivative = 4.19 \[ \int \frac {(d+e x)^{9/2}}{a^2+2 a b x+b^2 x^2} \, dx=\left [-\frac {315 \, {\left (a b^{3} d^{3} e - 3 \, a^{2} b^{2} d^{2} e^{2} + 3 \, a^{3} b d e^{3} - a^{4} e^{4} + {\left (b^{4} d^{3} e - 3 \, a b^{3} d^{2} e^{2} + 3 \, a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x\right )} \sqrt {\frac {b d - a e}{b}} \log \left (\frac {b e x + 2 \, b d - a e + 2 \, \sqrt {e x + d} b \sqrt {\frac {b d - a e}{b}}}{b x + a}\right ) - 2 \, {\left (10 \, b^{4} e^{4} x^{4} - 35 \, b^{4} d^{4} + 528 \, a b^{3} d^{3} e - 1218 \, a^{2} b^{2} d^{2} e^{2} + 1050 \, a^{3} b d e^{3} - 315 \, a^{4} e^{4} + 2 \, {\left (29 \, b^{4} d e^{3} - 9 \, a b^{3} e^{4}\right )} x^{3} + 6 \, {\left (26 \, b^{4} d^{2} e^{2} - 23 \, a b^{3} d e^{3} + 7 \, a^{2} b^{2} e^{4}\right )} x^{2} + 2 \, {\left (194 \, b^{4} d^{3} e - 426 \, a b^{3} d^{2} e^{2} + 357 \, a^{2} b^{2} d e^{3} - 105 \, a^{3} b e^{4}\right )} x\right )} \sqrt {e x + d}}{70 \, {\left (b^{6} x + a b^{5}\right )}}, -\frac {315 \, {\left (a b^{3} d^{3} e - 3 \, a^{2} b^{2} d^{2} e^{2} + 3 \, a^{3} b d e^{3} - a^{4} e^{4} + {\left (b^{4} d^{3} e - 3 \, a b^{3} d^{2} e^{2} + 3 \, a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x\right )} \sqrt {-\frac {b d - a e}{b}} \arctan \left (-\frac {\sqrt {e x + d} b \sqrt {-\frac {b d - a e}{b}}}{b d - a e}\right ) - {\left (10 \, b^{4} e^{4} x^{4} - 35 \, b^{4} d^{4} + 528 \, a b^{3} d^{3} e - 1218 \, a^{2} b^{2} d^{2} e^{2} + 1050 \, a^{3} b d e^{3} - 315 \, a^{4} e^{4} + 2 \, {\left (29 \, b^{4} d e^{3} - 9 \, a b^{3} e^{4}\right )} x^{3} + 6 \, {\left (26 \, b^{4} d^{2} e^{2} - 23 \, a b^{3} d e^{3} + 7 \, a^{2} b^{2} e^{4}\right )} x^{2} + 2 \, {\left (194 \, b^{4} d^{3} e - 426 \, a b^{3} d^{2} e^{2} + 357 \, a^{2} b^{2} d e^{3} - 105 \, a^{3} b e^{4}\right )} x\right )} \sqrt {e x + d}}{35 \, {\left (b^{6} x + a b^{5}\right )}}\right ] \]

[In]

integrate((e*x+d)^(9/2)/(b^2*x^2+2*a*b*x+a^2),x, algorithm="fricas")

[Out]

[-1/70*(315*(a*b^3*d^3*e - 3*a^2*b^2*d^2*e^2 + 3*a^3*b*d*e^3 - a^4*e^4 + (b^4*d^3*e - 3*a*b^3*d^2*e^2 + 3*a^2*
b^2*d*e^3 - a^3*b*e^4)*x)*sqrt((b*d - a*e)/b)*log((b*e*x + 2*b*d - a*e + 2*sqrt(e*x + d)*b*sqrt((b*d - a*e)/b)
)/(b*x + a)) - 2*(10*b^4*e^4*x^4 - 35*b^4*d^4 + 528*a*b^3*d^3*e - 1218*a^2*b^2*d^2*e^2 + 1050*a^3*b*d*e^3 - 31
5*a^4*e^4 + 2*(29*b^4*d*e^3 - 9*a*b^3*e^4)*x^3 + 6*(26*b^4*d^2*e^2 - 23*a*b^3*d*e^3 + 7*a^2*b^2*e^4)*x^2 + 2*(
194*b^4*d^3*e - 426*a*b^3*d^2*e^2 + 357*a^2*b^2*d*e^3 - 105*a^3*b*e^4)*x)*sqrt(e*x + d))/(b^6*x + a*b^5), -1/3
5*(315*(a*b^3*d^3*e - 3*a^2*b^2*d^2*e^2 + 3*a^3*b*d*e^3 - a^4*e^4 + (b^4*d^3*e - 3*a*b^3*d^2*e^2 + 3*a^2*b^2*d
*e^3 - a^3*b*e^4)*x)*sqrt(-(b*d - a*e)/b)*arctan(-sqrt(e*x + d)*b*sqrt(-(b*d - a*e)/b)/(b*d - a*e)) - (10*b^4*
e^4*x^4 - 35*b^4*d^4 + 528*a*b^3*d^3*e - 1218*a^2*b^2*d^2*e^2 + 1050*a^3*b*d*e^3 - 315*a^4*e^4 + 2*(29*b^4*d*e
^3 - 9*a*b^3*e^4)*x^3 + 6*(26*b^4*d^2*e^2 - 23*a*b^3*d*e^3 + 7*a^2*b^2*e^4)*x^2 + 2*(194*b^4*d^3*e - 426*a*b^3
*d^2*e^2 + 357*a^2*b^2*d*e^3 - 105*a^3*b*e^4)*x)*sqrt(e*x + d))/(b^6*x + a*b^5)]

Sympy [F(-1)]

Timed out. \[ \int \frac {(d+e x)^{9/2}}{a^2+2 a b x+b^2 x^2} \, dx=\text {Timed out} \]

[In]

integrate((e*x+d)**(9/2)/(b**2*x**2+2*a*b*x+a**2),x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {(d+e x)^{9/2}}{a^2+2 a b x+b^2 x^2} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((e*x+d)^(9/2)/(b^2*x^2+2*a*b*x+a^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. 375 vs. \(2 (138) = 276\).

Time = 0.27 (sec) , antiderivative size = 375, normalized size of antiderivative = 2.31 \[ \int \frac {(d+e x)^{9/2}}{a^2+2 a b x+b^2 x^2} \, dx=\frac {9 \, {\left (b^{4} d^{4} e - 4 \, a b^{3} d^{3} e^{2} + 6 \, a^{2} b^{2} d^{2} e^{3} - 4 \, a^{3} b d e^{4} + a^{4} e^{5}\right )} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{\sqrt {-b^{2} d + a b e} b^{5}} - \frac {\sqrt {e x + d} b^{4} d^{4} e - 4 \, \sqrt {e x + d} a b^{3} d^{3} e^{2} + 6 \, \sqrt {e x + d} a^{2} b^{2} d^{2} e^{3} - 4 \, \sqrt {e x + d} a^{3} b d e^{4} + \sqrt {e x + d} a^{4} e^{5}}{{\left ({\left (e x + d\right )} b - b d + a e\right )} b^{5}} + \frac {2 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} b^{12} e + 14 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{12} d e + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{12} d^{2} e + 140 \, \sqrt {e x + d} b^{12} d^{3} e - 14 \, {\left (e x + d\right )}^{\frac {5}{2}} a b^{11} e^{2} - 70 \, {\left (e x + d\right )}^{\frac {3}{2}} a b^{11} d e^{2} - 420 \, \sqrt {e x + d} a b^{11} d^{2} e^{2} + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{2} b^{10} e^{3} + 420 \, \sqrt {e x + d} a^{2} b^{10} d e^{3} - 140 \, \sqrt {e x + d} a^{3} b^{9} e^{4}\right )}}{35 \, b^{14}} \]

[In]

integrate((e*x+d)^(9/2)/(b^2*x^2+2*a*b*x+a^2),x, algorithm="giac")

[Out]

9*(b^4*d^4*e - 4*a*b^3*d^3*e^2 + 6*a^2*b^2*d^2*e^3 - 4*a^3*b*d*e^4 + a^4*e^5)*arctan(sqrt(e*x + d)*b/sqrt(-b^2
*d + a*b*e))/(sqrt(-b^2*d + a*b*e)*b^5) - (sqrt(e*x + d)*b^4*d^4*e - 4*sqrt(e*x + d)*a*b^3*d^3*e^2 + 6*sqrt(e*
x + d)*a^2*b^2*d^2*e^3 - 4*sqrt(e*x + d)*a^3*b*d*e^4 + sqrt(e*x + d)*a^4*e^5)/(((e*x + d)*b - b*d + a*e)*b^5)
+ 2/35*(5*(e*x + d)^(7/2)*b^12*e + 14*(e*x + d)^(5/2)*b^12*d*e + 35*(e*x + d)^(3/2)*b^12*d^2*e + 140*sqrt(e*x
+ d)*b^12*d^3*e - 14*(e*x + d)^(5/2)*a*b^11*e^2 - 70*(e*x + d)^(3/2)*a*b^11*d*e^2 - 420*sqrt(e*x + d)*a*b^11*d
^2*e^2 + 35*(e*x + d)^(3/2)*a^2*b^10*e^3 + 420*sqrt(e*x + d)*a^2*b^10*d*e^3 - 140*sqrt(e*x + d)*a^3*b^9*e^4)/b
^14

Mupad [B] (verification not implemented)

Time = 9.45 (sec) , antiderivative size = 352, normalized size of antiderivative = 2.17 \[ \int \frac {(d+e x)^{9/2}}{a^2+2 a b x+b^2 x^2} \, dx=\left (\frac {\left (\frac {2\,e\,{\left (2\,b^2\,d-2\,a\,b\,e\right )}^2}{b^6}-\frac {2\,e\,{\left (a\,e-b\,d\right )}^2}{b^4}\right )\,\left (2\,b^2\,d-2\,a\,b\,e\right )}{b^2}-\frac {2\,e\,\left (2\,b^2\,d-2\,a\,b\,e\right )\,{\left (a\,e-b\,d\right )}^2}{b^6}\right )\,\sqrt {d+e\,x}+\left (\frac {2\,e\,{\left (2\,b^2\,d-2\,a\,b\,e\right )}^2}{3\,b^6}-\frac {2\,e\,{\left (a\,e-b\,d\right )}^2}{3\,b^4}\right )\,{\left (d+e\,x\right )}^{3/2}-\frac {\sqrt {d+e\,x}\,\left (a^4\,e^5-4\,a^3\,b\,d\,e^4+6\,a^2\,b^2\,d^2\,e^3-4\,a\,b^3\,d^3\,e^2+b^4\,d^4\,e\right )}{b^6\,\left (d+e\,x\right )-b^6\,d+a\,b^5\,e}+\frac {2\,e\,{\left (d+e\,x\right )}^{7/2}}{7\,b^2}+\frac {2\,e\,\left (2\,b^2\,d-2\,a\,b\,e\right )\,{\left (d+e\,x\right )}^{5/2}}{5\,b^4}+\frac {9\,e\,\mathrm {atan}\left (\frac {\sqrt {b}\,e\,{\left (a\,e-b\,d\right )}^{7/2}\,\sqrt {d+e\,x}}{a^4\,e^5-4\,a^3\,b\,d\,e^4+6\,a^2\,b^2\,d^2\,e^3-4\,a\,b^3\,d^3\,e^2+b^4\,d^4\,e}\right )\,{\left (a\,e-b\,d\right )}^{7/2}}{b^{11/2}} \]

[In]

int((d + e*x)^(9/2)/(a^2 + b^2*x^2 + 2*a*b*x),x)

[Out]

((((2*e*(2*b^2*d - 2*a*b*e)^2)/b^6 - (2*e*(a*e - b*d)^2)/b^4)*(2*b^2*d - 2*a*b*e))/b^2 - (2*e*(2*b^2*d - 2*a*b
*e)*(a*e - b*d)^2)/b^6)*(d + e*x)^(1/2) + ((2*e*(2*b^2*d - 2*a*b*e)^2)/(3*b^6) - (2*e*(a*e - b*d)^2)/(3*b^4))*
(d + e*x)^(3/2) - ((d + e*x)^(1/2)*(a^4*e^5 + b^4*d^4*e - 4*a*b^3*d^3*e^2 + 6*a^2*b^2*d^2*e^3 - 4*a^3*b*d*e^4)
)/(b^6*(d + e*x) - b^6*d + a*b^5*e) + (2*e*(d + e*x)^(7/2))/(7*b^2) + (2*e*(2*b^2*d - 2*a*b*e)*(d + e*x)^(5/2)
)/(5*b^4) + (9*e*atan((b^(1/2)*e*(a*e - b*d)^(7/2)*(d + e*x)^(1/2))/(a^4*e^5 + b^4*d^4*e - 4*a*b^3*d^3*e^2 + 6
*a^2*b^2*d^2*e^3 - 4*a^3*b*d*e^4))*(a*e - b*d)^(7/2))/b^(11/2)