\(\int \frac {(d+e x)^{9/2}}{(a^2+2 a b x+b^2 x^2)^{5/2}} \, dx\) [1722]

Optimal result

Integrand size = 30, antiderivative size = 294 \[ \int \frac {(d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {315 e^4 (a+b x) \sqrt {d+e x}}{64 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {105 e^3 (d+e x)^{3/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {21 e^2 (d+e x)^{5/2}}{32 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 e (d+e x)^{7/2}}{8 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{9/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {315 e^4 \sqrt {b d-a e} (a+b x) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]

[Out]

-105/64*e^3*(e*x+d)^(3/2)/b^4/((b*x+a)^2)^(1/2)-21/32*e^2*(e*x+d)^(5/2)/b^3/(b*x+a)/((b*x+a)^2)^(1/2)-3/8*e*(e
*x+d)^(7/2)/b^2/(b*x+a)^2/((b*x+a)^2)^(1/2)-1/4*(e*x+d)^(9/2)/b/(b*x+a)^3/((b*x+a)^2)^(1/2)-315/64*e^4*(b*x+a)
*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))*(-a*e+b*d)^(1/2)/b^(11/2)/((b*x+a)^2)^(1/2)+315/64*e^4*(b*x+a
)*(e*x+d)^(1/2)/b^5/((b*x+a)^2)^(1/2)

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {660, 43, 52, 65, 214} \[ \int \frac {(d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {315 e^4 (a+b x) \sqrt {b d-a e} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{9/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 e (d+e x)^{7/2}}{8 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {315 e^4 (a+b x) \sqrt {d+e x}}{64 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {105 e^3 (d+e x)^{3/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {21 e^2 (d+e x)^{5/2}}{32 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}} \]

[In]

Int[(d + e*x)^(9/2)/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(315*e^4*(a + b*x)*Sqrt[d + e*x])/(64*b^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (105*e^3*(d + e*x)^(3/2))/(64*b^4*S
qrt[a^2 + 2*a*b*x + b^2*x^2]) - (21*e^2*(d + e*x)^(5/2))/(32*b^3*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (3
*e*(d + e*x)^(7/2))/(8*b^2*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (d + e*x)^(9/2)/(4*b*(a + b*x)^3*Sqrt[
a^2 + 2*a*b*x + b^2*x^2]) - (315*e^4*Sqrt[b*d - a*e]*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]
])/(64*b^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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]

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 {(d+e x)^{9/2}}{\left (a b+b^2 x\right )^5} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {(d+e x)^{9/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (9 b^2 e \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{7/2}}{\left (a b+b^2 x\right )^4} \, dx}{8 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {3 e (d+e x)^{7/2}}{8 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{9/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (21 e^2 \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{5/2}}{\left (a b+b^2 x\right )^3} \, dx}{16 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {21 e^2 (d+e x)^{5/2}}{32 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 e (d+e x)^{7/2}}{8 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{9/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (105 e^3 \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{3/2}}{\left (a b+b^2 x\right )^2} \, dx}{64 b^2 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {105 e^3 (d+e x)^{3/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {21 e^2 (d+e x)^{5/2}}{32 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 e (d+e x)^{7/2}}{8 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{9/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (315 e^4 \left (a b+b^2 x\right )\right ) \int \frac {\sqrt {d+e x}}{a b+b^2 x} \, dx}{128 b^4 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {315 e^4 (a+b x) \sqrt {d+e x}}{64 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {105 e^3 (d+e x)^{3/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {21 e^2 (d+e x)^{5/2}}{32 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 e (d+e x)^{7/2}}{8 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{9/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (315 e^4 \left (b^2 d-a b e\right ) \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right ) \sqrt {d+e x}} \, dx}{128 b^6 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {315 e^4 (a+b x) \sqrt {d+e x}}{64 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {105 e^3 (d+e x)^{3/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {21 e^2 (d+e x)^{5/2}}{32 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 e (d+e x)^{7/2}}{8 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{9/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (315 e^3 \left (b^2 d-a b e\right ) \left (a b+b^2 x\right )\right ) \text {Subst}\left (\int \frac {1}{a b-\frac {b^2 d}{e}+\frac {b^2 x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {315 e^4 (a+b x) \sqrt {d+e x}}{64 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {105 e^3 (d+e x)^{3/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {21 e^2 (d+e x)^{5/2}}{32 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 e (d+e x)^{7/2}}{8 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{9/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {315 e^4 \sqrt {b d-a e} (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.93 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.79 \[ \int \frac {(d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {-\sqrt {b} \sqrt {d+e x} \left (-315 a^4 e^4+105 a^3 b e^3 (d-11 e x)+21 a^2 b^2 e^2 \left (2 d^2+19 d e x-73 e^2 x^2\right )+3 a b^3 e \left (8 d^3+52 d^2 e x+185 d e^2 x^2-279 e^3 x^3\right )+b^4 \left (16 d^4+88 d^3 e x+210 d^2 e^2 x^2+325 d e^3 x^3-128 e^4 x^4\right )\right )-315 e^4 \sqrt {-b d+a e} (a+b x)^4 \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{64 b^{11/2} (a+b x)^3 \sqrt {(a+b x)^2}} \]

[In]

Integrate[(d + e*x)^(9/2)/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(-(Sqrt[b]*Sqrt[d + e*x]*(-315*a^4*e^4 + 105*a^3*b*e^3*(d - 11*e*x) + 21*a^2*b^2*e^2*(2*d^2 + 19*d*e*x - 73*e^
2*x^2) + 3*a*b^3*e*(8*d^3 + 52*d^2*e*x + 185*d*e^2*x^2 - 279*e^3*x^3) + b^4*(16*d^4 + 88*d^3*e*x + 210*d^2*e^2
*x^2 + 325*d*e^3*x^3 - 128*e^4*x^4))) - 315*e^4*Sqrt[-(b*d) + a*e]*(a + b*x)^4*ArcTan[(Sqrt[b]*Sqrt[d + e*x])/
Sqrt[-(b*d) + a*e]])/(64*b^(11/2)*(a + b*x)^3*Sqrt[(a + b*x)^2])

Maple [A] (verified)

Time = 2.44 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.79

method	result	size
risch	\(\frac {2 e^{4} \sqrt {e x +d}\, \sqrt {\left (b x +a \right )^{2}}}{b^{5} \left (b x +a \right )}-\frac {\left (2 a e -2 b d \right ) e^{4} \left (\frac {-\frac {325 \left (e x +d \right )^{\frac {7}{2}} b^{3}}{128}-\frac {765 \left (a e -b d \right ) b^{2} \left (e x +d \right )^{\frac {5}{2}}}{128}+\left (-\frac {643}{128} e^{2} a^{2} b +\frac {643}{64} a d e \,b^{2}-\frac {643}{128} b^{3} d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}+\left (-\frac {187}{128} a^{3} e^{3}+\frac {561}{128} a^{2} b d \,e^{2}-\frac {561}{128} a \,b^{2} d^{2} e +\frac {187}{128} b^{3} d^{3}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{4}}+\frac {315 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{128 \sqrt {\left (a e -b d \right ) b}}\right ) \sqrt {\left (b x +a \right )^{2}}}{b^{5} \left (b x +a \right )}\)	\(233\)
default	\(\frac {\left (-1530 \left (e x +d \right )^{\frac {5}{2}} \sqrt {\left (a e -b d \right ) b}\, a \,b^{3} d e -1929 \left (e x +d \right )^{\frac {3}{2}} \sqrt {\left (a e -b d \right ) b}\, a^{2} b^{2} d \,e^{2}+1929 \left (e x +d \right )^{\frac {3}{2}} \sqrt {\left (a e -b d \right ) b}\, a \,b^{3} d^{2} e -643 \left (e x +d \right )^{\frac {3}{2}} \sqrt {\left (a e -b d \right ) b}\, b^{4} d^{3}+765 \left (e x +d \right )^{\frac {5}{2}} \sqrt {\left (a e -b d \right ) b}\, a^{2} b^{2} e^{2}-325 \left (e x +d \right )^{\frac {7}{2}} \sqrt {\left (a e -b d \right ) b}\, b^{4} d +765 \left (e x +d \right )^{\frac {5}{2}} \sqrt {\left (a e -b d \right ) b}\, b^{4} d^{2}-1260 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a^{4} b \,e^{5} x +315 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a^{4} b d \,e^{4}-315 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a \,b^{4} e^{5} x^{4}+1260 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a^{3} b^{2} d \,e^{4} x +643 \left (e x +d \right )^{\frac {3}{2}} \sqrt {\left (a e -b d \right ) b}\, a^{3} b \,e^{3}-315 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a^{5} e^{5}+768 a^{2} b^{2} e^{4} x^{2} \sqrt {\left (a e -b d \right ) b}\, \sqrt {e x +d}+1260 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a \,b^{4} d \,e^{4} x^{3}+1890 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a^{2} b^{3} d \,e^{4} x^{2}+315 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) b^{5} d \,e^{4} x^{4}+325 \left (e x +d \right )^{\frac {7}{2}} \sqrt {\left (a e -b d \right ) b}\, a \,b^{3} e +512 \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, a^{3} b \,e^{4} x -748 \sqrt {e x +d}\, e^{3} d \,a^{3} b \sqrt {\left (a e -b d \right ) b}+1122 \sqrt {e x +d}\, e^{2} d^{2} a^{2} b^{2} \sqrt {\left (a e -b d \right ) b}-748 \sqrt {e x +d}\, a \,d^{3} e \,b^{3} \sqrt {\left (a e -b d \right ) b}+512 a \,b^{3} e^{4} x^{3} \sqrt {\left (a e -b d \right ) b}\, \sqrt {e x +d}+315 \sqrt {e x +d}\, e^{4} a^{4} \sqrt {\left (a e -b d \right ) b}+187 \sqrt {e x +d}\, d^{4} b^{4} \sqrt {\left (a e -b d \right ) b}+128 b^{4} e^{4} x^{4} \sqrt {\left (a e -b d \right ) b}\, \sqrt {e x +d}-1260 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a^{2} b^{3} e^{5} x^{3}-1890 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a^{3} b^{2} e^{5} x^{2}\right ) \left (b x +a \right )}{64 \sqrt {\left (a e -b d \right ) b}\, b^{5} \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}\)	\(892\)

[In]

int((e*x+d)^(9/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

2*e^4/b^5*(e*x+d)^(1/2)*((b*x+a)^2)^(1/2)/(b*x+a)-1/b^5*(2*a*e-2*b*d)*e^4*((-325/128*(e*x+d)^(7/2)*b^3-765/128
*(a*e-b*d)*b^2*(e*x+d)^(5/2)+(-643/128*e^2*a^2*b+643/64*a*d*e*b^2-643/128*b^3*d^2)*(e*x+d)^(3/2)+(-187/128*a^3
*e^3+561/128*a^2*b*d*e^2-561/128*a*b^2*d^2*e+187/128*b^3*d^3)*(e*x+d)^(1/2))/(b*(e*x+d)+a*e-b*d)^4+315/128/((a
*e-b*d)*b)^(1/2)*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)))*((b*x+a)^2)^(1/2)/(b*x+a)

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 680, normalized size of antiderivative = 2.31 \[ \int \frac {(d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\left [\frac {315 \, {\left (b^{4} e^{4} x^{4} + 4 \, a b^{3} e^{4} x^{3} + 6 \, a^{2} b^{2} e^{4} x^{2} + 4 \, a^{3} b e^{4} x + a^{4} e^{4}\right )} \sqrt {\frac {b d - a e}{b}} \log \left (\frac {b e x + 2 \, b d - a e - 2 \, \sqrt {e x + d} b \sqrt {\frac {b d - a e}{b}}}{b x + a}\right ) + 2 \, {\left (128 \, b^{4} e^{4} x^{4} - 16 \, b^{4} d^{4} - 24 \, a b^{3} d^{3} e - 42 \, a^{2} b^{2} d^{2} e^{2} - 105 \, a^{3} b d e^{3} + 315 \, a^{4} e^{4} - {\left (325 \, b^{4} d e^{3} - 837 \, a b^{3} e^{4}\right )} x^{3} - 3 \, {\left (70 \, b^{4} d^{2} e^{2} + 185 \, a b^{3} d e^{3} - 511 \, a^{2} b^{2} e^{4}\right )} x^{2} - {\left (88 \, b^{4} d^{3} e + 156 \, a b^{3} d^{2} e^{2} + 399 \, a^{2} b^{2} d e^{3} - 1155 \, a^{3} b e^{4}\right )} x\right )} \sqrt {e x + d}}{128 \, {\left (b^{9} x^{4} + 4 \, a b^{8} x^{3} + 6 \, a^{2} b^{7} x^{2} + 4 \, a^{3} b^{6} x + a^{4} b^{5}\right )}}, -\frac {315 \, {\left (b^{4} e^{4} x^{4} + 4 \, a b^{3} e^{4} x^{3} + 6 \, a^{2} b^{2} e^{4} x^{2} + 4 \, a^{3} b e^{4} x + a^{4} e^{4}\right )} \sqrt {-\frac {b d - a e}{b}} \arctan \left (-\frac {\sqrt {e x + d} b \sqrt {-\frac {b d - a e}{b}}}{b d - a e}\right ) - {\left (128 \, b^{4} e^{4} x^{4} - 16 \, b^{4} d^{4} - 24 \, a b^{3} d^{3} e - 42 \, a^{2} b^{2} d^{2} e^{2} - 105 \, a^{3} b d e^{3} + 315 \, a^{4} e^{4} - {\left (325 \, b^{4} d e^{3} - 837 \, a b^{3} e^{4}\right )} x^{3} - 3 \, {\left (70 \, b^{4} d^{2} e^{2} + 185 \, a b^{3} d e^{3} - 511 \, a^{2} b^{2} e^{4}\right )} x^{2} - {\left (88 \, b^{4} d^{3} e + 156 \, a b^{3} d^{2} e^{2} + 399 \, a^{2} b^{2} d e^{3} - 1155 \, a^{3} b e^{4}\right )} x\right )} \sqrt {e x + d}}{64 \, {\left (b^{9} x^{4} + 4 \, a b^{8} x^{3} + 6 \, a^{2} b^{7} x^{2} + 4 \, a^{3} b^{6} x + a^{4} b^{5}\right )}}\right ] \]

[In]

integrate((e*x+d)^(9/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")

[Out]

[1/128*(315*(b^4*e^4*x^4 + 4*a*b^3*e^4*x^3 + 6*a^2*b^2*e^4*x^2 + 4*a^3*b*e^4*x + a^4*e^4)*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*(128*b^4*e^4*x^4 - 16*b^4*d^4
- 24*a*b^3*d^3*e - 42*a^2*b^2*d^2*e^2 - 105*a^3*b*d*e^3 + 315*a^4*e^4 - (325*b^4*d*e^3 - 837*a*b^3*e^4)*x^3 -
3*(70*b^4*d^2*e^2 + 185*a*b^3*d*e^3 - 511*a^2*b^2*e^4)*x^2 - (88*b^4*d^3*e + 156*a*b^3*d^2*e^2 + 399*a^2*b^2*
d*e^3 - 1155*a^3*b*e^4)*x)*sqrt(e*x + d))/(b^9*x^4 + 4*a*b^8*x^3 + 6*a^2*b^7*x^2 + 4*a^3*b^6*x + a^4*b^5), -1/
64*(315*(b^4*e^4*x^4 + 4*a*b^3*e^4*x^3 + 6*a^2*b^2*e^4*x^2 + 4*a^3*b*e^4*x + a^4*e^4)*sqrt(-(b*d - a*e)/b)*arc
tan(-sqrt(e*x + d)*b*sqrt(-(b*d - a*e)/b)/(b*d - a*e)) - (128*b^4*e^4*x^4 - 16*b^4*d^4 - 24*a*b^3*d^3*e - 42*a
^2*b^2*d^2*e^2 - 105*a^3*b*d*e^3 + 315*a^4*e^4 - (325*b^4*d*e^3 - 837*a*b^3*e^4)*x^3 - 3*(70*b^4*d^2*e^2 + 185
*a*b^3*d*e^3 - 511*a^2*b^2*e^4)*x^2 - (88*b^4*d^3*e + 156*a*b^3*d^2*e^2 + 399*a^2*b^2*d*e^3 - 1155*a^3*b*e^4)*
x)*sqrt(e*x + d))/(b^9*x^4 + 4*a*b^8*x^3 + 6*a^2*b^7*x^2 + 4*a^3*b^6*x + a^4*b^5)]

Sympy [F(-1)]

Timed out. \[ \int \frac {(d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\text {Timed out} \]

[In]

integrate((e*x+d)**(9/2)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {9}{2}}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate((e*x+d)^(9/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="maxima")

[Out]

integrate((e*x + d)^(9/2)/(b^2*x^2 + 2*a*b*x + a^2)^(5/2), x)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.24 \[ \int \frac {(d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {2 \, \sqrt {e x + d} e^{4}}{b^{5} \mathrm {sgn}\left (b x + a\right )} + \frac {315 \, {\left (b d e^{4} - a e^{5}\right )} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{64 \, \sqrt {-b^{2} d + a b e} b^{5} \mathrm {sgn}\left (b x + a\right )} - \frac {325 \, {\left (e x + d\right )}^{\frac {7}{2}} b^{4} d e^{4} - 765 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{4} d^{2} e^{4} + 643 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{4} d^{3} e^{4} - 187 \, \sqrt {e x + d} b^{4} d^{4} e^{4} - 325 \, {\left (e x + d\right )}^{\frac {7}{2}} a b^{3} e^{5} + 1530 \, {\left (e x + d\right )}^{\frac {5}{2}} a b^{3} d e^{5} - 1929 \, {\left (e x + d\right )}^{\frac {3}{2}} a b^{3} d^{2} e^{5} + 748 \, \sqrt {e x + d} a b^{3} d^{3} e^{5} - 765 \, {\left (e x + d\right )}^{\frac {5}{2}} a^{2} b^{2} e^{6} + 1929 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{2} b^{2} d e^{6} - 1122 \, \sqrt {e x + d} a^{2} b^{2} d^{2} e^{6} - 643 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{3} b e^{7} + 748 \, \sqrt {e x + d} a^{3} b d e^{7} - 187 \, \sqrt {e x + d} a^{4} e^{8}}{64 \, {\left ({\left (e x + d\right )} b - b d + a e\right )}^{4} b^{5} \mathrm {sgn}\left (b x + a\right )} \]

[In]

integrate((e*x+d)^(9/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")

[Out]

2*sqrt(e*x + d)*e^4/(b^5*sgn(b*x + a)) + 315/64*(b*d*e^4 - a*e^5)*arctan(sqrt(e*x + d)*b/sqrt(-b^2*d + a*b*e))
/(sqrt(-b^2*d + a*b*e)*b^5*sgn(b*x + a)) - 1/64*(325*(e*x + d)^(7/2)*b^4*d*e^4 - 765*(e*x + d)^(5/2)*b^4*d^2*e
^4 + 643*(e*x + d)^(3/2)*b^4*d^3*e^4 - 187*sqrt(e*x + d)*b^4*d^4*e^4 - 325*(e*x + d)^(7/2)*a*b^3*e^5 + 1530*(e
*x + d)^(5/2)*a*b^3*d*e^5 - 1929*(e*x + d)^(3/2)*a*b^3*d^2*e^5 + 748*sqrt(e*x + d)*a*b^3*d^3*e^5 - 765*(e*x +
d)^(5/2)*a^2*b^2*e^6 + 1929*(e*x + d)^(3/2)*a^2*b^2*d*e^6 - 1122*sqrt(e*x + d)*a^2*b^2*d^2*e^6 - 643*(e*x + d)
^(3/2)*a^3*b*e^7 + 748*sqrt(e*x + d)*a^3*b*d*e^7 - 187*sqrt(e*x + d)*a^4*e^8)/(((e*x + d)*b - b*d + a*e)^4*b^5
*sgn(b*x + a))

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^{9/2}}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \]

[In]

int((d + e*x)^(9/2)/(a^2 + b^2*x^2 + 2*a*b*x)^(5/2),x)

[Out]

int((d + e*x)^(9/2)/(a^2 + b^2*x^2 + 2*a*b*x)^(5/2), x)