\(\int \frac {(d+e x)^{11/2}}{(a^2+2 a b x+b^2 x^2)^{5/2}} \, dx\) [1721]

Optimal result

Integrand size = 30, antiderivative size = 346 \[ \int \frac {(d+e x)^{11/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {1155 e^4 (b d-a e) (a+b x) \sqrt {d+e x}}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {385 e^4 (a+b x) (d+e x)^{3/2}}{64 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {231 e^3 (d+e x)^{5/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {33 e^2 (d+e x)^{7/2}}{32 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {11 e (d+e x)^{9/2}}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{11/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1155 e^4 (b d-a e)^{3/2} (a+b x) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]

[Out]

385/64*e^4*(b*x+a)*(e*x+d)^(3/2)/b^5/((b*x+a)^2)^(1/2)-231/64*e^3*(e*x+d)^(5/2)/b^4/((b*x+a)^2)^(1/2)-33/32*e^
2*(e*x+d)^(7/2)/b^3/(b*x+a)/((b*x+a)^2)^(1/2)-11/24*e*(e*x+d)^(9/2)/b^2/(b*x+a)^2/((b*x+a)^2)^(1/2)-1/4*(e*x+d
)^(11/2)/b/(b*x+a)^3/((b*x+a)^2)^(1/2)-1155/64*e^4*(-a*e+b*d)^(3/2)*(b*x+a)*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*
e+b*d)^(1/2))/b^(13/2)/((b*x+a)^2)^(1/2)+1155/64*e^4*(-a*e+b*d)*(b*x+a)*(e*x+d)^(1/2)/b^6/((b*x+a)^2)^(1/2)

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {660, 43, 52, 65, 214} \[ \int \frac {(d+e x)^{11/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {1155 e^4 (a+b x) (b d-a e)^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{11/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {11 e (d+e x)^{9/2}}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {1155 e^4 (a+b x) \sqrt {d+e x} (b d-a e)}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {385 e^4 (a+b x) (d+e x)^{3/2}}{64 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {231 e^3 (d+e x)^{5/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {33 e^2 (d+e x)^{7/2}}{32 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}} \]

[In]

Int[(d + e*x)^(11/2)/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(1155*e^4*(b*d - a*e)*(a + b*x)*Sqrt[d + e*x])/(64*b^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (385*e^4*(a + b*x)*(d
+ e*x)^(3/2))/(64*b^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (231*e^3*(d + e*x)^(5/2))/(64*b^4*Sqrt[a^2 + 2*a*b*x +
b^2*x^2]) - (33*e^2*(d + e*x)^(7/2))/(32*b^3*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (11*e*(d + e*x)^(9/2))
/(24*b^2*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (d + e*x)^(11/2)/(4*b*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b
^2*x^2]) - (1155*e^4*(b*d - a*e)^(3/2)*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(64*b^(13/2
)*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)^{11/2}}{\left (a b+b^2 x\right )^5} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {(d+e x)^{11/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (11 b^2 e \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{9/2}}{\left (a b+b^2 x\right )^4} \, dx}{8 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {11 e (d+e x)^{9/2}}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{11/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (33 e^2 \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{7/2}}{\left (a b+b^2 x\right )^3} \, dx}{16 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {33 e^2 (d+e x)^{7/2}}{32 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {11 e (d+e x)^{9/2}}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{11/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (231 e^3 \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{5/2}}{\left (a b+b^2 x\right )^2} \, dx}{64 b^2 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {231 e^3 (d+e x)^{5/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {33 e^2 (d+e x)^{7/2}}{32 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {11 e (d+e x)^{9/2}}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{11/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (1155 e^4 \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{3/2}}{a b+b^2 x} \, dx}{128 b^4 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {385 e^4 (a+b x) (d+e x)^{3/2}}{64 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {231 e^3 (d+e x)^{5/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {33 e^2 (d+e x)^{7/2}}{32 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {11 e (d+e x)^{9/2}}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{11/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (1155 e^4 \left (b^2 d-a b e\right ) \left (a b+b^2 x\right )\right ) \int \frac {\sqrt {d+e x}}{a b+b^2 x} \, dx}{128 b^6 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {1155 e^4 (b d-a e) (a+b x) \sqrt {d+e x}}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {385 e^4 (a+b x) (d+e x)^{3/2}}{64 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {231 e^3 (d+e x)^{5/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {33 e^2 (d+e x)^{7/2}}{32 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {11 e (d+e x)^{9/2}}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{11/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (1155 e^4 \left (b^2 d-a b e\right )^2 \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^8 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {1155 e^4 (b d-a e) (a+b x) \sqrt {d+e x}}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {385 e^4 (a+b x) (d+e x)^{3/2}}{64 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {231 e^3 (d+e x)^{5/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {33 e^2 (d+e x)^{7/2}}{32 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {11 e (d+e x)^{9/2}}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{11/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (1155 e^3 \left (b^2 d-a b e\right )^2 \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^8 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {1155 e^4 (b d-a e) (a+b x) \sqrt {d+e x}}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {385 e^4 (a+b x) (d+e x)^{3/2}}{64 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {231 e^3 (d+e x)^{5/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {33 e^2 (d+e x)^{7/2}}{32 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {11 e (d+e x)^{9/2}}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{11/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1155 e^4 (b d-a e)^{3/2} (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.27 (sec) , antiderivative size = 300, normalized size of antiderivative = 0.87 \[ \int \frac {(d+e x)^{11/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {e^4 (a+b x)^5 \left (-\frac {\sqrt {b} \sqrt {d+e x} \left (3465 a^5 e^5+1155 a^4 b e^4 (-4 d+11 e x)+231 a^3 b^2 e^3 \left (3 d^2-74 d e x+73 e^2 x^2\right )+99 a^2 b^3 e^2 \left (2 d^3+27 d^2 e x-232 d e^2 x^2+93 e^3 x^3\right )+11 a b^4 e \left (8 d^4+68 d^3 e x+345 d^2 e^2 x^2-1162 d e^3 x^3+128 e^4 x^4\right )+b^5 \left (48 d^5+328 d^4 e x+1030 d^3 e^2 x^2+2295 d^2 e^3 x^3-2048 d e^4 x^4-128 e^5 x^5\right )\right )}{e^4 (a+b x)^4}+3465 (-b d+a e)^{3/2} \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )\right )}{192 b^{13/2} \left ((a+b x)^2\right )^{5/2}} \]

[In]

Integrate[(d + e*x)^(11/2)/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(e^4*(a + b*x)^5*(-((Sqrt[b]*Sqrt[d + e*x]*(3465*a^5*e^5 + 1155*a^4*b*e^4*(-4*d + 11*e*x) + 231*a^3*b^2*e^3*(3
*d^2 - 74*d*e*x + 73*e^2*x^2) + 99*a^2*b^3*e^2*(2*d^3 + 27*d^2*e*x - 232*d*e^2*x^2 + 93*e^3*x^3) + 11*a*b^4*e*
(8*d^4 + 68*d^3*e*x + 345*d^2*e^2*x^2 - 1162*d*e^3*x^3 + 128*e^4*x^4) + b^5*(48*d^5 + 328*d^4*e*x + 1030*d^3*e
^2*x^2 + 2295*d^2*e^3*x^3 - 2048*d*e^4*x^4 - 128*e^5*x^5)))/(e^4*(a + b*x)^4)) + 3465*(-(b*d) + a*e)^(3/2)*Arc
Tan[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]]))/(192*b^(13/2)*((a + b*x)^2)^(5/2))

Maple [A] (verified)

Time = 2.31 (sec) , antiderivative size = 260, normalized size of antiderivative = 0.75

method	result	size
risch	\(-\frac {2 e^{4} \left (-b e x +15 a e -16 b d \right ) \sqrt {e x +d}\, \sqrt {\left (b x +a \right )^{2}}}{3 b^{6} \left (b x +a \right )}+\frac {\left (2 a^{2} e^{2}-4 a b d e +2 b^{2} d^{2}\right ) e^{4} \left (\frac {-\frac {765 \left (e x +d \right )^{\frac {7}{2}} b^{3}}{128}-\frac {5855 \left (a e -b d \right ) b^{2} \left (e x +d \right )^{\frac {5}{2}}}{384}+\left (-\frac {5153}{384} e^{2} a^{2} b +\frac {5153}{192} a d e \,b^{2}-\frac {5153}{384} b^{3} d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}+\left (-\frac {515}{128} a^{3} e^{3}+\frac {1545}{128} a^{2} b d \,e^{2}-\frac {1545}{128} a \,b^{2} d^{2} e +\frac {515}{128} b^{3} d^{3}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{4}}+\frac {1155 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{128 \sqrt {\left (a e -b d \right ) b}}\right ) \sqrt {\left (b x +a \right )^{2}}}{b^{6} \left (b x +a \right )}\)	\(260\)
default	\(\text {Expression too large to display}\)	\(1471\)

[In]

int((e*x+d)^(11/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-2/3*e^4*(-b*e*x+15*a*e-16*b*d)*(e*x+d)^(1/2)/b^6*((b*x+a)^2)^(1/2)/(b*x+a)+1/b^6*(2*a^2*e^2-4*a*b*d*e+2*b^2*d
^2)*e^4*((-765/128*(e*x+d)^(7/2)*b^3-5855/384*(a*e-b*d)*b^2*(e*x+d)^(5/2)+(-5153/384*e^2*a^2*b+5153/192*a*d*e*
b^2-5153/384*b^3*d^2)*(e*x+d)^(3/2)+(-515/128*a^3*e^3+1545/128*a^2*b*d*e^2-1545/128*a*b^2*d^2*e+515/128*b^3*d^
3)*(e*x+d)^(1/2))/(b*(e*x+d)+a*e-b*d)^4+1155/128/((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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 479 vs. \(2 (233) = 466\).

Time = 0.39 (sec) , antiderivative size = 968, normalized size of antiderivative = 2.80 \[ \int \frac {(d+e x)^{11/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\left [-\frac {3465 \, {\left (a^{4} b d e^{4} - a^{5} e^{5} + {\left (b^{5} d e^{4} - a b^{4} e^{5}\right )} x^{4} + 4 \, {\left (a b^{4} d e^{4} - a^{2} b^{3} e^{5}\right )} x^{3} + 6 \, {\left (a^{2} b^{3} d e^{4} - a^{3} b^{2} e^{5}\right )} x^{2} + 4 \, {\left (a^{3} b^{2} d e^{4} - a^{4} b e^{5}\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 (128 \, b^{5} e^{5} x^{5} - 48 \, b^{5} d^{5} - 88 \, a b^{4} d^{4} e - 198 \, a^{2} b^{3} d^{3} e^{2} - 693 \, a^{3} b^{2} d^{2} e^{3} + 4620 \, a^{4} b d e^{4} - 3465 \, a^{5} e^{5} + 128 \, {\left (16 \, b^{5} d e^{4} - 11 \, a b^{4} e^{5}\right )} x^{4} - {\left (2295 \, b^{5} d^{2} e^{3} - 12782 \, a b^{4} d e^{4} + 9207 \, a^{2} b^{3} e^{5}\right )} x^{3} - {\left (1030 \, b^{5} d^{3} e^{2} + 3795 \, a b^{4} d^{2} e^{3} - 22968 \, a^{2} b^{3} d e^{4} + 16863 \, a^{3} b^{2} e^{5}\right )} x^{2} - {\left (328 \, b^{5} d^{4} e + 748 \, a b^{4} d^{3} e^{2} + 2673 \, a^{2} b^{3} d^{2} e^{3} - 17094 \, a^{3} b^{2} d e^{4} + 12705 \, a^{4} b e^{5}\right )} x\right )} \sqrt {e x + d}}{384 \, {\left (b^{10} x^{4} + 4 \, a b^{9} x^{3} + 6 \, a^{2} b^{8} x^{2} + 4 \, a^{3} b^{7} x + a^{4} b^{6}\right )}}, -\frac {3465 \, {\left (a^{4} b d e^{4} - a^{5} e^{5} + {\left (b^{5} d e^{4} - a b^{4} e^{5}\right )} x^{4} + 4 \, {\left (a b^{4} d e^{4} - a^{2} b^{3} e^{5}\right )} x^{3} + 6 \, {\left (a^{2} b^{3} d e^{4} - a^{3} b^{2} e^{5}\right )} x^{2} + 4 \, {\left (a^{3} b^{2} d e^{4} - a^{4} b e^{5}\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 (128 \, b^{5} e^{5} x^{5} - 48 \, b^{5} d^{5} - 88 \, a b^{4} d^{4} e - 198 \, a^{2} b^{3} d^{3} e^{2} - 693 \, a^{3} b^{2} d^{2} e^{3} + 4620 \, a^{4} b d e^{4} - 3465 \, a^{5} e^{5} + 128 \, {\left (16 \, b^{5} d e^{4} - 11 \, a b^{4} e^{5}\right )} x^{4} - {\left (2295 \, b^{5} d^{2} e^{3} - 12782 \, a b^{4} d e^{4} + 9207 \, a^{2} b^{3} e^{5}\right )} x^{3} - {\left (1030 \, b^{5} d^{3} e^{2} + 3795 \, a b^{4} d^{2} e^{3} - 22968 \, a^{2} b^{3} d e^{4} + 16863 \, a^{3} b^{2} e^{5}\right )} x^{2} - {\left (328 \, b^{5} d^{4} e + 748 \, a b^{4} d^{3} e^{2} + 2673 \, a^{2} b^{3} d^{2} e^{3} - 17094 \, a^{3} b^{2} d e^{4} + 12705 \, a^{4} b e^{5}\right )} x\right )} \sqrt {e x + d}}{192 \, {\left (b^{10} x^{4} + 4 \, a b^{9} x^{3} + 6 \, a^{2} b^{8} x^{2} + 4 \, a^{3} b^{7} x + a^{4} b^{6}\right )}}\right ] \]

[In]

integrate((e*x+d)^(11/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")

[Out]

[-1/384*(3465*(a^4*b*d*e^4 - a^5*e^5 + (b^5*d*e^4 - a*b^4*e^5)*x^4 + 4*(a*b^4*d*e^4 - a^2*b^3*e^5)*x^3 + 6*(a^
2*b^3*d*e^4 - a^3*b^2*e^5)*x^2 + 4*(a^3*b^2*d*e^4 - a^4*b*e^5)*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*(128*b^5*e^5*x^5 - 48*b^5*d^5 - 88*a*b^4*d^4*e - 198*
a^2*b^3*d^3*e^2 - 693*a^3*b^2*d^2*e^3 + 4620*a^4*b*d*e^4 - 3465*a^5*e^5 + 128*(16*b^5*d*e^4 - 11*a*b^4*e^5)*x^
4 - (2295*b^5*d^2*e^3 - 12782*a*b^4*d*e^4 + 9207*a^2*b^3*e^5)*x^3 - (1030*b^5*d^3*e^2 + 3795*a*b^4*d^2*e^3 - 2
2968*a^2*b^3*d*e^4 + 16863*a^3*b^2*e^5)*x^2 - (328*b^5*d^4*e + 748*a*b^4*d^3*e^2 + 2673*a^2*b^3*d^2*e^3 - 1709
4*a^3*b^2*d*e^4 + 12705*a^4*b*e^5)*x)*sqrt(e*x + d))/(b^10*x^4 + 4*a*b^9*x^3 + 6*a^2*b^8*x^2 + 4*a^3*b^7*x + a
^4*b^6), -1/192*(3465*(a^4*b*d*e^4 - a^5*e^5 + (b^5*d*e^4 - a*b^4*e^5)*x^4 + 4*(a*b^4*d*e^4 - a^2*b^3*e^5)*x^3
+ 6*(a^2*b^3*d*e^4 - a^3*b^2*e^5)*x^2 + 4*(a^3*b^2*d*e^4 - a^4*b*e^5)*x)*sqrt(-(b*d - a*e)/b)*arctan(-sqrt(e*
x + d)*b*sqrt(-(b*d - a*e)/b)/(b*d - a*e)) - (128*b^5*e^5*x^5 - 48*b^5*d^5 - 88*a*b^4*d^4*e - 198*a^2*b^3*d^3*
e^2 - 693*a^3*b^2*d^2*e^3 + 4620*a^4*b*d*e^4 - 3465*a^5*e^5 + 128*(16*b^5*d*e^4 - 11*a*b^4*e^5)*x^4 - (2295*b^
5*d^2*e^3 - 12782*a*b^4*d*e^4 + 9207*a^2*b^3*e^5)*x^3 - (1030*b^5*d^3*e^2 + 3795*a*b^4*d^2*e^3 - 22968*a^2*b^3
*d*e^4 + 16863*a^3*b^2*e^5)*x^2 - (328*b^5*d^4*e + 748*a*b^4*d^3*e^2 + 2673*a^2*b^3*d^2*e^3 - 17094*a^3*b^2*d*
e^4 + 12705*a^4*b*e^5)*x)*sqrt(e*x + d))/(b^10*x^4 + 4*a*b^9*x^3 + 6*a^2*b^8*x^2 + 4*a^3*b^7*x + a^4*b^6)]

Sympy [F(-1)]

Timed out. \[ \int \frac {(d+e x)^{11/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\text {Timed out} \]

[In]

integrate((e*x+d)**(11/2)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(d+e x)^{11/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {11}{2}}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate((e*x+d)^(11/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="maxima")

[Out]

integrate((e*x + d)^(11/2)/(b^2*x^2 + 2*a*b*x + a^2)^(5/2), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 498 vs. \(2 (233) = 466\).

Time = 0.32 (sec) , antiderivative size = 498, normalized size of antiderivative = 1.44 \[ \int \frac {(d+e x)^{11/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {1155 \, {\left (b^{2} d^{2} e^{4} - 2 \, a b d e^{5} + a^{2} e^{6}\right )} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{64 \, \sqrt {-b^{2} d + a b e} b^{6} \mathrm {sgn}\left (b x + a\right )} - \frac {2295 \, {\left (e x + d\right )}^{\frac {7}{2}} b^{5} d^{2} e^{4} - 5855 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{5} d^{3} e^{4} + 5153 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{5} d^{4} e^{4} - 1545 \, \sqrt {e x + d} b^{5} d^{5} e^{4} - 4590 \, {\left (e x + d\right )}^{\frac {7}{2}} a b^{4} d e^{5} + 17565 \, {\left (e x + d\right )}^{\frac {5}{2}} a b^{4} d^{2} e^{5} - 20612 \, {\left (e x + d\right )}^{\frac {3}{2}} a b^{4} d^{3} e^{5} + 7725 \, \sqrt {e x + d} a b^{4} d^{4} e^{5} + 2295 \, {\left (e x + d\right )}^{\frac {7}{2}} a^{2} b^{3} e^{6} - 17565 \, {\left (e x + d\right )}^{\frac {5}{2}} a^{2} b^{3} d e^{6} + 30918 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{2} b^{3} d^{2} e^{6} - 15450 \, \sqrt {e x + d} a^{2} b^{3} d^{3} e^{6} + 5855 \, {\left (e x + d\right )}^{\frac {5}{2}} a^{3} b^{2} e^{7} - 20612 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{3} b^{2} d e^{7} + 15450 \, \sqrt {e x + d} a^{3} b^{2} d^{2} e^{7} + 5153 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{4} b e^{8} - 7725 \, \sqrt {e x + d} a^{4} b d e^{8} + 1545 \, \sqrt {e x + d} a^{5} e^{9}}{192 \, {\left ({\left (e x + d\right )} b - b d + a e\right )}^{4} b^{6} \mathrm {sgn}\left (b x + a\right )} + \frac {2 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} b^{10} e^{4} + 15 \, \sqrt {e x + d} b^{10} d e^{4} - 15 \, \sqrt {e x + d} a b^{9} e^{5}\right )}}{3 \, b^{15} \mathrm {sgn}\left (b x + a\right )} \]

[In]

integrate((e*x+d)^(11/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")

[Out]

1155/64*(b^2*d^2*e^4 - 2*a*b*d*e^5 + a^2*e^6)*arctan(sqrt(e*x + d)*b/sqrt(-b^2*d + a*b*e))/(sqrt(-b^2*d + a*b*
e)*b^6*sgn(b*x + a)) - 1/192*(2295*(e*x + d)^(7/2)*b^5*d^2*e^4 - 5855*(e*x + d)^(5/2)*b^5*d^3*e^4 + 5153*(e*x
+ d)^(3/2)*b^5*d^4*e^4 - 1545*sqrt(e*x + d)*b^5*d^5*e^4 - 4590*(e*x + d)^(7/2)*a*b^4*d*e^5 + 17565*(e*x + d)^(
5/2)*a*b^4*d^2*e^5 - 20612*(e*x + d)^(3/2)*a*b^4*d^3*e^5 + 7725*sqrt(e*x + d)*a*b^4*d^4*e^5 + 2295*(e*x + d)^(
7/2)*a^2*b^3*e^6 - 17565*(e*x + d)^(5/2)*a^2*b^3*d*e^6 + 30918*(e*x + d)^(3/2)*a^2*b^3*d^2*e^6 - 15450*sqrt(e*
x + d)*a^2*b^3*d^3*e^6 + 5855*(e*x + d)^(5/2)*a^3*b^2*e^7 - 20612*(e*x + d)^(3/2)*a^3*b^2*d*e^7 + 15450*sqrt(e
*x + d)*a^3*b^2*d^2*e^7 + 5153*(e*x + d)^(3/2)*a^4*b*e^8 - 7725*sqrt(e*x + d)*a^4*b*d*e^8 + 1545*sqrt(e*x + d)
*a^5*e^9)/(((e*x + d)*b - b*d + a*e)^4*b^6*sgn(b*x + a)) + 2/3*((e*x + d)^(3/2)*b^10*e^4 + 15*sqrt(e*x + d)*b^
10*d*e^4 - 15*sqrt(e*x + d)*a*b^9*e^5)/(b^15*sgn(b*x + a))

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+e x)^{11/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^{11/2}}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \]

[In]

int((d + e*x)^(11/2)/(a^2 + b^2*x^2 + 2*a*b*x)^(5/2),x)

[Out]

int((d + e*x)^(11/2)/(a^2 + b^2*x^2 + 2*a*b*x)^(5/2), x)