\(\int \frac {1}{\sqrt {d+e x} (a^2+2 a b x+b^2 x^2)^{5/2}} \, dx\) [1727]

Optimal result

Integrand size = 30, antiderivative size = 278 \[ \int \frac {1}{\sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {35 e^3 \sqrt {d+e x}}{64 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\sqrt {d+e x}}{4 (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e \sqrt {d+e x}}{24 (b d-a e)^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 e^2 \sqrt {d+e x}}{96 (b d-a e)^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 e^4 (a+b x) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 \sqrt {b} (b d-a e)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]

[Out]

-35/64*e^4*(b*x+a)*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))/(-a*e+b*d)^(9/2)/b^(1/2)/((b*x+a)^2)^(1/2)+
35/64*e^3*(e*x+d)^(1/2)/(-a*e+b*d)^4/((b*x+a)^2)^(1/2)-1/4*(e*x+d)^(1/2)/(-a*e+b*d)/(b*x+a)^3/((b*x+a)^2)^(1/2
)+7/24*e*(e*x+d)^(1/2)/(-a*e+b*d)^2/(b*x+a)^2/((b*x+a)^2)^(1/2)-35/96*e^2*(e*x+d)^(1/2)/(-a*e+b*d)^3/(b*x+a)/(
(b*x+a)^2)^(1/2)

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {660, 44, 65, 214} \[ \int \frac {1}{\sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {35 e^4 (a+b x) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 \sqrt {b} \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^{9/2}}+\frac {35 e^3 \sqrt {d+e x}}{64 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}-\frac {35 e^2 \sqrt {d+e x}}{96 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}+\frac {7 e \sqrt {d+e x}}{24 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac {\sqrt {d+e x}}{4 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)} \]

[In]

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

[Out]

(35*e^3*Sqrt[d + e*x])/(64*(b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - Sqrt[d + e*x]/(4*(b*d - a*e)*(a + b*
x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (7*e*Sqrt[d + e*x])/(24*(b*d - a*e)^2*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b
^2*x^2]) - (35*e^2*Sqrt[d + e*x])/(96*(b*d - a*e)^3*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (35*e^4*(a + b*
x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(64*Sqrt[b]*(b*d - a*e)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^
2])

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 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 {1}{\left (a b+b^2 x\right )^5 \sqrt {d+e x}} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {\sqrt {d+e x}}{4 (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (7 b^3 e \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right )^4 \sqrt {d+e x}} \, dx}{8 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {\sqrt {d+e x}}{4 (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e \sqrt {d+e x}}{24 (b d-a e)^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (35 b^2 e^2 \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right )^3 \sqrt {d+e x}} \, dx}{48 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {\sqrt {d+e x}}{4 (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e \sqrt {d+e x}}{24 (b d-a e)^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 e^2 \sqrt {d+e x}}{96 (b d-a e)^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (35 b e^3 \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right )^2 \sqrt {d+e x}} \, dx}{64 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {35 e^3 \sqrt {d+e x}}{64 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\sqrt {d+e x}}{4 (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e \sqrt {d+e x}}{24 (b d-a e)^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 e^2 \sqrt {d+e x}}{96 (b d-a e)^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (35 e^4 \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 d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {35 e^3 \sqrt {d+e x}}{64 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\sqrt {d+e x}}{4 (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e \sqrt {d+e x}}{24 (b d-a e)^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 e^2 \sqrt {d+e x}}{96 (b d-a e)^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (35 e^3 \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 d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {35 e^3 \sqrt {d+e x}}{64 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\sqrt {d+e x}}{4 (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e \sqrt {d+e x}}{24 (b d-a e)^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 e^2 \sqrt {d+e x}}{96 (b d-a e)^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 e^4 (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 \sqrt {b} (b d-a e)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.48 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.67 \[ \int \frac {1}{\sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {e^4 (a+b x)^5 \left (\frac {\sqrt {d+e x} \left (279 a^3 e^3+a^2 b e^2 (-326 d+511 e x)+a b^2 e \left (200 d^2-252 d e x+385 e^2 x^2\right )+b^3 \left (-48 d^3+56 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right )\right )}{e^4 (b d-a e)^4 (a+b x)^4}+\frac {105 \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{\sqrt {b} (-b d+a e)^{9/2}}\right )}{192 \left ((a+b x)^2\right )^{5/2}} \]

[In]

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

[Out]

(e^4*(a + b*x)^5*((Sqrt[d + e*x]*(279*a^3*e^3 + a^2*b*e^2*(-326*d + 511*e*x) + a*b^2*e*(200*d^2 - 252*d*e*x +
385*e^2*x^2) + b^3*(-48*d^3 + 56*d^2*e*x - 70*d*e^2*x^2 + 105*e^3*x^3)))/(e^4*(b*d - a*e)^4*(a + b*x)^4) + (10
5*ArcTan[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]])/(Sqrt[b]*(-(b*d) + a*e)^(9/2))))/(192*((a + b*x)^2)^(5/2
))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(496\) vs. \(2(195)=390\).

Time = 2.28 (sec) , antiderivative size = 497, normalized size of antiderivative = 1.79

method	result	size
default	\(\frac {\left (105 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) b^{4} e^{4} x^{4}+420 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a \,b^{3} e^{4} x^{3}+105 b^{3} e^{3} x^{3} \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}+630 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a^{2} b^{2} e^{4} x^{2}+385 a \,b^{2} e^{3} x^{2} \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}-70 b^{3} d \,e^{2} x^{2} \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}+420 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a^{3} b \,e^{4} x +511 \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, a^{2} b \,e^{3} x -252 \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, a \,b^{2} d \,e^{2} x +56 \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, b^{3} d^{2} e x +105 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a^{4} e^{4}+279 \sqrt {e x +d}\, a^{3} e^{3} \sqrt {\left (a e -b d \right ) b}-326 \sqrt {e x +d}\, a^{2} d \,e^{2} b \sqrt {\left (a e -b d \right ) b}+200 \sqrt {e x +d}\, a \,d^{2} e \,b^{2} \sqrt {\left (a e -b d \right ) b}-48 \sqrt {e x +d}\, d^{3} b^{3} \sqrt {\left (a e -b d \right ) b}\right ) \left (b x +a \right )}{192 \sqrt {\left (a e -b d \right ) b}\, \left (a e -b d \right )^{4} \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}\)	\(497\)

[In]

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

[Out]

1/192*(105*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))*b^4*e^4*x^4+420*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1
/2))*a*b^3*e^4*x^3+105*b^3*e^3*x^3*(e*x+d)^(1/2)*((a*e-b*d)*b)^(1/2)+630*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^
(1/2))*a^2*b^2*e^4*x^2+385*a*b^2*e^3*x^2*(e*x+d)^(1/2)*((a*e-b*d)*b)^(1/2)-70*b^3*d*e^2*x^2*(e*x+d)^(1/2)*((a*
e-b*d)*b)^(1/2)+420*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))*a^3*b*e^4*x+511*(e*x+d)^(1/2)*((a*e-b*d)*b)^(1
/2)*a^2*b*e^3*x-252*(e*x+d)^(1/2)*((a*e-b*d)*b)^(1/2)*a*b^2*d*e^2*x+56*(e*x+d)^(1/2)*((a*e-b*d)*b)^(1/2)*b^3*d
^2*e*x+105*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))*a^4*e^4+279*(e*x+d)^(1/2)*a^3*e^3*((a*e-b*d)*b)^(1/2)-3
26*(e*x+d)^(1/2)*a^2*d*e^2*b*((a*e-b*d)*b)^(1/2)+200*(e*x+d)^(1/2)*a*d^2*e*b^2*((a*e-b*d)*b)^(1/2)-48*(e*x+d)^
(1/2)*d^3*b^3*((a*e-b*d)*b)^(1/2))*(b*x+a)/((a*e-b*d)*b)^(1/2)/(a*e-b*d)^4/((b*x+a)^2)^(5/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 656 vs. \(2 (195) = 390\).

Time = 0.35 (sec) , antiderivative size = 1325, normalized size of antiderivative = 4.77 \[ \int \frac {1}{\sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[1/384*(105*(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^2*d - a*b*e)*
log((b*e*x + 2*b*d - a*e - 2*sqrt(b^2*d - a*b*e)*sqrt(e*x + d))/(b*x + a)) - 2*(48*b^5*d^4 - 248*a*b^4*d^3*e +
526*a^2*b^3*d^2*e^2 - 605*a^3*b^2*d*e^3 + 279*a^4*b*e^4 - 105*(b^5*d*e^3 - a*b^4*e^4)*x^3 + 35*(2*b^5*d^2*e^2
- 13*a*b^4*d*e^3 + 11*a^2*b^3*e^4)*x^2 - 7*(8*b^5*d^3*e - 44*a*b^4*d^2*e^2 + 109*a^2*b^3*d*e^3 - 73*a^3*b^2*e
^4)*x)*sqrt(e*x + d))/(a^4*b^6*d^5 - 5*a^5*b^5*d^4*e + 10*a^6*b^4*d^3*e^2 - 10*a^7*b^3*d^2*e^3 + 5*a^8*b^2*d*e
^4 - a^9*b*e^5 + (b^10*d^5 - 5*a*b^9*d^4*e + 10*a^2*b^8*d^3*e^2 - 10*a^3*b^7*d^2*e^3 + 5*a^4*b^6*d*e^4 - a^5*b
^5*e^5)*x^4 + 4*(a*b^9*d^5 - 5*a^2*b^8*d^4*e + 10*a^3*b^7*d^3*e^2 - 10*a^4*b^6*d^2*e^3 + 5*a^5*b^5*d*e^4 - a^6
*b^4*e^5)*x^3 + 6*(a^2*b^8*d^5 - 5*a^3*b^7*d^4*e + 10*a^4*b^6*d^3*e^2 - 10*a^5*b^5*d^2*e^3 + 5*a^6*b^4*d*e^4 -
a^7*b^3*e^5)*x^2 + 4*(a^3*b^7*d^5 - 5*a^4*b^6*d^4*e + 10*a^5*b^5*d^3*e^2 - 10*a^6*b^4*d^2*e^3 + 5*a^7*b^3*d*e
^4 - a^8*b^2*e^5)*x), 1/192*(105*(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^2*d + a*b*e)*arctan(sqrt(-b^2*d + a*b*e)*sqrt(e*x + d)/(b*e*x + b*d)) - (48*b^5*d^4 - 248*a*b^4*d^3*e
+ 526*a^2*b^3*d^2*e^2 - 605*a^3*b^2*d*e^3 + 279*a^4*b*e^4 - 105*(b^5*d*e^3 - a*b^4*e^4)*x^3 + 35*(2*b^5*d^2*e
^2 - 13*a*b^4*d*e^3 + 11*a^2*b^3*e^4)*x^2 - 7*(8*b^5*d^3*e - 44*a*b^4*d^2*e^2 + 109*a^2*b^3*d*e^3 - 73*a^3*b^2
*e^4)*x)*sqrt(e*x + d))/(a^4*b^6*d^5 - 5*a^5*b^5*d^4*e + 10*a^6*b^4*d^3*e^2 - 10*a^7*b^3*d^2*e^3 + 5*a^8*b^2*d
*e^4 - a^9*b*e^5 + (b^10*d^5 - 5*a*b^9*d^4*e + 10*a^2*b^8*d^3*e^2 - 10*a^3*b^7*d^2*e^3 + 5*a^4*b^6*d*e^4 - a^5
*b^5*e^5)*x^4 + 4*(a*b^9*d^5 - 5*a^2*b^8*d^4*e + 10*a^3*b^7*d^3*e^2 - 10*a^4*b^6*d^2*e^3 + 5*a^5*b^5*d*e^4 - a
^6*b^4*e^5)*x^3 + 6*(a^2*b^8*d^5 - 5*a^3*b^7*d^4*e + 10*a^4*b^6*d^3*e^2 - 10*a^5*b^5*d^2*e^3 + 5*a^6*b^4*d*e^4
- a^7*b^3*e^5)*x^2 + 4*(a^3*b^7*d^5 - 5*a^4*b^6*d^4*e + 10*a^5*b^5*d^3*e^2 - 10*a^6*b^4*d^2*e^3 + 5*a^7*b^3*d
*e^4 - a^8*b^2*e^5)*x)]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 391 vs. \(2 (195) = 390\).

Time = 0.30 (sec) , antiderivative size = 391, normalized size of antiderivative = 1.41 \[ \int \frac {1}{\sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {35 \, e^{4} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{64 \, {\left (b^{4} d^{4} \mathrm {sgn}\left (b x + a\right ) - 4 \, a b^{3} d^{3} e \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{2} b^{2} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) - 4 \, a^{3} b d e^{3} \mathrm {sgn}\left (b x + a\right ) + a^{4} e^{4} \mathrm {sgn}\left (b x + a\right )\right )} \sqrt {-b^{2} d + a b e}} + \frac {105 \, {\left (e x + d\right )}^{\frac {7}{2}} b^{3} e^{4} - 385 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{3} d e^{4} + 511 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{3} d^{2} e^{4} - 279 \, \sqrt {e x + d} b^{3} d^{3} e^{4} + 385 \, {\left (e x + d\right )}^{\frac {5}{2}} a b^{2} e^{5} - 1022 \, {\left (e x + d\right )}^{\frac {3}{2}} a b^{2} d e^{5} + 837 \, \sqrt {e x + d} a b^{2} d^{2} e^{5} + 511 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{2} b e^{6} - 837 \, \sqrt {e x + d} a^{2} b d e^{6} + 279 \, \sqrt {e x + d} a^{3} e^{7}}{192 \, {\left (b^{4} d^{4} \mathrm {sgn}\left (b x + a\right ) - 4 \, a b^{3} d^{3} e \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{2} b^{2} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) - 4 \, a^{3} b d e^{3} \mathrm {sgn}\left (b x + a\right ) + a^{4} e^{4} \mathrm {sgn}\left (b x + a\right )\right )} {\left ({\left (e x + d\right )} b - b d + a e\right )}^{4}} \]

[In]

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

[Out]

35/64*e^4*arctan(sqrt(e*x + d)*b/sqrt(-b^2*d + a*b*e))/((b^4*d^4*sgn(b*x + a) - 4*a*b^3*d^3*e*sgn(b*x + a) + 6
*a^2*b^2*d^2*e^2*sgn(b*x + a) - 4*a^3*b*d*e^3*sgn(b*x + a) + a^4*e^4*sgn(b*x + a))*sqrt(-b^2*d + a*b*e)) + 1/1
92*(105*(e*x + d)^(7/2)*b^3*e^4 - 385*(e*x + d)^(5/2)*b^3*d*e^4 + 511*(e*x + d)^(3/2)*b^3*d^2*e^4 - 279*sqrt(e
*x + d)*b^3*d^3*e^4 + 385*(e*x + d)^(5/2)*a*b^2*e^5 - 1022*(e*x + d)^(3/2)*a*b^2*d*e^5 + 837*sqrt(e*x + d)*a*b
^2*d^2*e^5 + 511*(e*x + d)^(3/2)*a^2*b*e^6 - 837*sqrt(e*x + d)*a^2*b*d*e^6 + 279*sqrt(e*x + d)*a^3*e^7)/((b^4*
d^4*sgn(b*x + a) - 4*a*b^3*d^3*e*sgn(b*x + a) + 6*a^2*b^2*d^2*e^2*sgn(b*x + a) - 4*a^3*b*d*e^3*sgn(b*x + a) +
a^4*e^4*sgn(b*x + a))*((e*x + d)*b - b*d + a*e)^4)

Mupad [F(-1)]

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

[In]

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

[Out]

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