\(\int \frac {(d+e x)^{3/2}}{(a^2+2 a b x+b^2 x^2)^{5/2}} \, dx\) [1725]

Optimal result

Integrand size = 30, antiderivative size = 270 \[ \int \frac {(d+e x)^{3/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {3 e^3 \sqrt {d+e x}}{64 b^2 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e \sqrt {d+e x}}{8 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e^2 \sqrt {d+e x}}{32 b^2 (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{3/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 e^4 (a+b x) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{5/2} (b d-a e)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]

[Out]

-1/4*(e*x+d)^(3/2)/b/(b*x+a)^3/((b*x+a)^2)^(1/2)-3/64*e^4*(b*x+a)*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/
2))/b^(5/2)/(-a*e+b*d)^(5/2)/((b*x+a)^2)^(1/2)+3/64*e^3*(e*x+d)^(1/2)/b^2/(-a*e+b*d)^2/((b*x+a)^2)^(1/2)-1/8*e
*(e*x+d)^(1/2)/b^2/(b*x+a)^2/((b*x+a)^2)^(1/2)-1/32*e^2*(e*x+d)^(1/2)/b^2/(-a*e+b*d)/(b*x+a)/((b*x+a)^2)^(1/2)

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {660, 43, 44, 65, 214} \[ \int \frac {(d+e x)^{3/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {3 e^4 (a+b x) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{5/2} \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^{5/2}}+\frac {3 e^3 \sqrt {d+e x}}{64 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac {e^2 \sqrt {d+e x}}{32 b^2 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac {e \sqrt {d+e x}}{8 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{3/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}} \]

[In]

Int[(d + e*x)^(3/2)/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(3*e^3*Sqrt[d + e*x])/(64*b^2*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (e*Sqrt[d + e*x])/(8*b^2*(a + b*x
)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (e^2*Sqrt[d + e*x])/(32*b^2*(b*d - a*e)*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^
2*x^2]) - (d + e*x)^(3/2)/(4*b*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (3*e^4*(a + b*x)*ArcTanh[(Sqrt[b]*
Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(64*b^(5/2)*(b*d - a*e)^(5/2)*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 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 {(d+e x)^{3/2}}{\left (a b+b^2 x\right )^5} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {(d+e x)^{3/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (3 b^2 e \left (a b+b^2 x\right )\right ) \int \frac {\sqrt {d+e x}}{\left (a b+b^2 x\right )^4} \, dx}{8 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {e \sqrt {d+e x}}{8 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{3/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (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}{16 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {e \sqrt {d+e x}}{8 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e^2 \sqrt {d+e x}}{32 b^2 (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{3/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (3 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 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {3 e^3 \sqrt {d+e x}}{64 b^2 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e \sqrt {d+e x}}{8 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e^2 \sqrt {d+e x}}{32 b^2 (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{3/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (3 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^2 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {3 e^3 \sqrt {d+e x}}{64 b^2 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e \sqrt {d+e x}}{8 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e^2 \sqrt {d+e x}}{32 b^2 (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{3/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (3 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^2 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {3 e^3 \sqrt {d+e x}}{64 b^2 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e \sqrt {d+e x}}{8 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e^2 \sqrt {d+e x}}{32 b^2 (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{3/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 e^4 (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{5/2} (b d-a e)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.82 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.72 \[ \int \frac {(d+e x)^{3/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {e^4 (a+b x)^5 \left (-\frac {\sqrt {b} \sqrt {d+e x} \left (3 a^3 e^3+a^2 b e^2 (2 d+11 e x)-a b^2 e \left (24 d^2+44 d e x+11 e^2 x^2\right )+b^3 \left (16 d^3+24 d^2 e x+2 d e^2 x^2-3 e^3 x^3\right )\right )}{e^4 (b d-a e)^2 (a+b x)^4}+\frac {3 \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{(-b d+a e)^{5/2}}\right )}{64 b^{5/2} \left ((a+b x)^2\right )^{5/2}} \]

[In]

Integrate[(d + e*x)^(3/2)/(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]*(3*a^3*e^3 + a^2*b*e^2*(2*d + 11*e*x) - a*b^2*e*(24*d^2 + 44*d*e*x
+ 11*e^2*x^2) + b^3*(16*d^3 + 24*d^2*e*x + 2*d*e^2*x^2 - 3*e^3*x^3)))/(e^4*(b*d - a*e)^2*(a + b*x)^4)) + (3*Ar
cTan[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]])/(-(b*d) + a*e)^(5/2)))/(64*b^(5/2)*((a + b*x)^2)^(5/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(476\) vs. \(2(187)=374\).

Time = 2.26 (sec) , antiderivative size = 477, normalized size of antiderivative = 1.77

method	result	size
default	\(\frac {\left (b x +a \right ) \left (3 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) b^{4} e^{4} x^{4}+12 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a \,b^{3} e^{4} x^{3}+3 \sqrt {\left (a e -b d \right ) b}\, \left (e x +d \right )^{\frac {7}{2}} b^{3}+18 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a^{2} b^{2} e^{4} x^{2}+11 \left (e x +d \right )^{\frac {5}{2}} \sqrt {\left (a e -b d \right ) b}\, a \,b^{2} e -11 \left (e x +d \right )^{\frac {5}{2}} \sqrt {\left (a e -b d \right ) b}\, b^{3} d +12 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a^{3} b \,e^{4} x -11 \sqrt {\left (a e -b d \right ) b}\, \left (e x +d \right )^{\frac {3}{2}} a^{2} b \,e^{2}+22 \sqrt {\left (a e -b d \right ) b}\, \left (e x +d \right )^{\frac {3}{2}} a \,b^{2} d e -11 \sqrt {\left (a e -b d \right ) b}\, \left (e x +d \right )^{\frac {3}{2}} b^{3} d^{2}+3 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a^{4} e^{4}-3 \sqrt {e x +d}\, a^{3} e^{3} \sqrt {\left (a e -b d \right ) b}+9 \sqrt {e x +d}\, a^{2} d \,e^{2} b \sqrt {\left (a e -b d \right ) b}-9 \sqrt {e x +d}\, a \,d^{2} e \,b^{2} \sqrt {\left (a e -b d \right ) b}+3 \sqrt {e x +d}\, d^{3} b^{3} \sqrt {\left (a e -b d \right ) b}\right )}{64 \sqrt {\left (a e -b d \right ) b}\, b^{2} \left (a e -b d \right )^{2} \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}\)	\(477\)

[In]

int((e*x+d)^(3/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/64*(b*x+a)*(3*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))*b^4*e^4*x^4+12*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b
)^(1/2))*a*b^3*e^4*x^3+3*((a*e-b*d)*b)^(1/2)*(e*x+d)^(7/2)*b^3+18*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))*
a^2*b^2*e^4*x^2+11*(e*x+d)^(5/2)*((a*e-b*d)*b)^(1/2)*a*b^2*e-11*(e*x+d)^(5/2)*((a*e-b*d)*b)^(1/2)*b^3*d+12*arc
tan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))*a^3*b*e^4*x-11*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*a^2*b*e^2+22*((a*e-b
*d)*b)^(1/2)*(e*x+d)^(3/2)*a*b^2*d*e-11*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*b^3*d^2+3*arctan(b*(e*x+d)^(1/2)/((a
*e-b*d)*b)^(1/2))*a^4*e^4-3*(e*x+d)^(1/2)*a^3*e^3*((a*e-b*d)*b)^(1/2)+9*(e*x+d)^(1/2)*a^2*d*e^2*b*((a*e-b*d)*b
)^(1/2)-9*(e*x+d)^(1/2)*a*d^2*e*b^2*((a*e-b*d)*b)^(1/2)+3*(e*x+d)^(1/2)*d^3*b^3*((a*e-b*d)*b)^(1/2))/((a*e-b*d
)*b)^(1/2)/b^2/(a*e-b*d)^2/((b*x+a)^2)^(5/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 515 vs. \(2 (187) = 374\).

Time = 0.30 (sec) , antiderivative size = 1043, normalized size of antiderivative = 3.86 \[ \int \frac {(d+e x)^{3/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\left [\frac {3 \, {\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 {b^{2} d - a b e} \log \left (\frac {b e x + 2 \, b d - a e - 2 \, \sqrt {b^{2} d - a b e} \sqrt {e x + d}}{b x + a}\right ) - 2 \, {\left (16 \, b^{5} d^{4} - 40 \, a b^{4} d^{3} e + 26 \, a^{2} b^{3} d^{2} e^{2} + a^{3} b^{2} d e^{3} - 3 \, a^{4} b e^{4} - 3 \, {\left (b^{5} d e^{3} - a b^{4} e^{4}\right )} x^{3} + {\left (2 \, b^{5} d^{2} e^{2} - 13 \, a b^{4} d e^{3} + 11 \, a^{2} b^{3} e^{4}\right )} x^{2} + {\left (24 \, b^{5} d^{3} e - 68 \, a b^{4} d^{2} e^{2} + 55 \, a^{2} b^{3} d e^{3} - 11 \, a^{3} b^{2} e^{4}\right )} x\right )} \sqrt {e x + d}}{128 \, {\left (a^{4} b^{6} d^{3} - 3 \, a^{5} b^{5} d^{2} e + 3 \, a^{6} b^{4} d e^{2} - a^{7} b^{3} e^{3} + {\left (b^{10} d^{3} - 3 \, a b^{9} d^{2} e + 3 \, a^{2} b^{8} d e^{2} - a^{3} b^{7} e^{3}\right )} x^{4} + 4 \, {\left (a b^{9} d^{3} - 3 \, a^{2} b^{8} d^{2} e + 3 \, a^{3} b^{7} d e^{2} - a^{4} b^{6} e^{3}\right )} x^{3} + 6 \, {\left (a^{2} b^{8} d^{3} - 3 \, a^{3} b^{7} d^{2} e + 3 \, a^{4} b^{6} d e^{2} - a^{5} b^{5} e^{3}\right )} x^{2} + 4 \, {\left (a^{3} b^{7} d^{3} - 3 \, a^{4} b^{6} d^{2} e + 3 \, a^{5} b^{5} d e^{2} - a^{6} b^{4} e^{3}\right )} x\right )}}, \frac {3 \, {\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 {-b^{2} d + a b e} \arctan \left (\frac {\sqrt {-b^{2} d + a b e} \sqrt {e x + d}}{b e x + b d}\right ) - {\left (16 \, b^{5} d^{4} - 40 \, a b^{4} d^{3} e + 26 \, a^{2} b^{3} d^{2} e^{2} + a^{3} b^{2} d e^{3} - 3 \, a^{4} b e^{4} - 3 \, {\left (b^{5} d e^{3} - a b^{4} e^{4}\right )} x^{3} + {\left (2 \, b^{5} d^{2} e^{2} - 13 \, a b^{4} d e^{3} + 11 \, a^{2} b^{3} e^{4}\right )} x^{2} + {\left (24 \, b^{5} d^{3} e - 68 \, a b^{4} d^{2} e^{2} + 55 \, a^{2} b^{3} d e^{3} - 11 \, a^{3} b^{2} e^{4}\right )} x\right )} \sqrt {e x + d}}{64 \, {\left (a^{4} b^{6} d^{3} - 3 \, a^{5} b^{5} d^{2} e + 3 \, a^{6} b^{4} d e^{2} - a^{7} b^{3} e^{3} + {\left (b^{10} d^{3} - 3 \, a b^{9} d^{2} e + 3 \, a^{2} b^{8} d e^{2} - a^{3} b^{7} e^{3}\right )} x^{4} + 4 \, {\left (a b^{9} d^{3} - 3 \, a^{2} b^{8} d^{2} e + 3 \, a^{3} b^{7} d e^{2} - a^{4} b^{6} e^{3}\right )} x^{3} + 6 \, {\left (a^{2} b^{8} d^{3} - 3 \, a^{3} b^{7} d^{2} e + 3 \, a^{4} b^{6} d e^{2} - a^{5} b^{5} e^{3}\right )} x^{2} + 4 \, {\left (a^{3} b^{7} d^{3} - 3 \, a^{4} b^{6} d^{2} e + 3 \, a^{5} b^{5} d e^{2} - a^{6} b^{4} e^{3}\right )} x\right )}}\right ] \]

[In]

integrate((e*x+d)^(3/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")

[Out]

[1/128*(3*(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)*lo
g((b*e*x + 2*b*d - a*e - 2*sqrt(b^2*d - a*b*e)*sqrt(e*x + d))/(b*x + a)) - 2*(16*b^5*d^4 - 40*a*b^4*d^3*e + 26
*a^2*b^3*d^2*e^2 + a^3*b^2*d*e^3 - 3*a^4*b*e^4 - 3*(b^5*d*e^3 - a*b^4*e^4)*x^3 + (2*b^5*d^2*e^2 - 13*a*b^4*d*e
^3 + 11*a^2*b^3*e^4)*x^2 + (24*b^5*d^3*e - 68*a*b^4*d^2*e^2 + 55*a^2*b^3*d*e^3 - 11*a^3*b^2*e^4)*x)*sqrt(e*x +
d))/(a^4*b^6*d^3 - 3*a^5*b^5*d^2*e + 3*a^6*b^4*d*e^2 - a^7*b^3*e^3 + (b^10*d^3 - 3*a*b^9*d^2*e + 3*a^2*b^8*d*
e^2 - a^3*b^7*e^3)*x^4 + 4*(a*b^9*d^3 - 3*a^2*b^8*d^2*e + 3*a^3*b^7*d*e^2 - a^4*b^6*e^3)*x^3 + 6*(a^2*b^8*d^3
- 3*a^3*b^7*d^2*e + 3*a^4*b^6*d*e^2 - a^5*b^5*e^3)*x^2 + 4*(a^3*b^7*d^3 - 3*a^4*b^6*d^2*e + 3*a^5*b^5*d*e^2 -
a^6*b^4*e^3)*x), 1/64*(3*(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)) - (16*b^5*d^4 - 40*a*b^4*d^3*e + 26*a^2
*b^3*d^2*e^2 + a^3*b^2*d*e^3 - 3*a^4*b*e^4 - 3*(b^5*d*e^3 - a*b^4*e^4)*x^3 + (2*b^5*d^2*e^2 - 13*a*b^4*d*e^3 +
11*a^2*b^3*e^4)*x^2 + (24*b^5*d^3*e - 68*a*b^4*d^2*e^2 + 55*a^2*b^3*d*e^3 - 11*a^3*b^2*e^4)*x)*sqrt(e*x + d))
/(a^4*b^6*d^3 - 3*a^5*b^5*d^2*e + 3*a^6*b^4*d*e^2 - a^7*b^3*e^3 + (b^10*d^3 - 3*a*b^9*d^2*e + 3*a^2*b^8*d*e^2
- a^3*b^7*e^3)*x^4 + 4*(a*b^9*d^3 - 3*a^2*b^8*d^2*e + 3*a^3*b^7*d*e^2 - a^4*b^6*e^3)*x^3 + 6*(a^2*b^8*d^3 - 3*
a^3*b^7*d^2*e + 3*a^4*b^6*d*e^2 - a^5*b^5*e^3)*x^2 + 4*(a^3*b^7*d^3 - 3*a^4*b^6*d^2*e + 3*a^5*b^5*d*e^2 - a^6*
b^4*e^3)*x)]

Sympy [F(-1)]

Timed out. \[ \int \frac {(d+e x)^{3/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\text {Timed out} \]

[In]

integrate((e*x+d)**(3/2)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(d+e x)^{3/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {3}{2}}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate((e*x+d)^(3/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="maxima")

[Out]

integrate((e*x + d)^(3/2)/(b^2*x^2 + 2*a*b*x + a^2)^(5/2), x)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.19 \[ \int \frac {(d+e x)^{3/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {3 \, e^{4} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{64 \, {\left (b^{4} d^{2} \mathrm {sgn}\left (b x + a\right ) - 2 \, a b^{3} d e \mathrm {sgn}\left (b x + a\right ) + a^{2} b^{2} e^{2} \mathrm {sgn}\left (b x + a\right )\right )} \sqrt {-b^{2} d + a b e}} + \frac {3 \, {\left (e x + d\right )}^{\frac {7}{2}} b^{3} e^{4} - 11 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{3} d e^{4} - 11 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{3} d^{2} e^{4} + 3 \, \sqrt {e x + d} b^{3} d^{3} e^{4} + 11 \, {\left (e x + d\right )}^{\frac {5}{2}} a b^{2} e^{5} + 22 \, {\left (e x + d\right )}^{\frac {3}{2}} a b^{2} d e^{5} - 9 \, \sqrt {e x + d} a b^{2} d^{2} e^{5} - 11 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{2} b e^{6} + 9 \, \sqrt {e x + d} a^{2} b d e^{6} - 3 \, \sqrt {e x + d} a^{3} e^{7}}{64 \, {\left (b^{4} d^{2} \mathrm {sgn}\left (b x + a\right ) - 2 \, a b^{3} d e \mathrm {sgn}\left (b x + a\right ) + a^{2} b^{2} e^{2} \mathrm {sgn}\left (b x + a\right )\right )} {\left ({\left (e x + d\right )} b - b d + a e\right )}^{4}} \]

[In]

integrate((e*x+d)^(3/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")

[Out]

3/64*e^4*arctan(sqrt(e*x + d)*b/sqrt(-b^2*d + a*b*e))/((b^4*d^2*sgn(b*x + a) - 2*a*b^3*d*e*sgn(b*x + a) + a^2*
b^2*e^2*sgn(b*x + a))*sqrt(-b^2*d + a*b*e)) + 1/64*(3*(e*x + d)^(7/2)*b^3*e^4 - 11*(e*x + d)^(5/2)*b^3*d*e^4 -
11*(e*x + d)^(3/2)*b^3*d^2*e^4 + 3*sqrt(e*x + d)*b^3*d^3*e^4 + 11*(e*x + d)^(5/2)*a*b^2*e^5 + 22*(e*x + d)^(3
/2)*a*b^2*d*e^5 - 9*sqrt(e*x + d)*a*b^2*d^2*e^5 - 11*(e*x + d)^(3/2)*a^2*b*e^6 + 9*sqrt(e*x + d)*a^2*b*d*e^6 -
3*sqrt(e*x + d)*a^3*e^7)/((b^4*d^2*sgn(b*x + a) - 2*a*b^3*d*e*sgn(b*x + a) + a^2*b^2*e^2*sgn(b*x + a))*((e*x
+ d)*b - b*d + a*e)^4)

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+e x)^{3/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^{3/2}}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \]

[In]

int((d + e*x)^(3/2)/(a^2 + b^2*x^2 + 2*a*b*x)^(5/2),x)

[Out]

int((d + e*x)^(3/2)/(a^2 + b^2*x^2 + 2*a*b*x)^(5/2), x)