\(\int \frac {(d+e x)^{3/2}}{(a^2+2 a b x+b^2 x^2)^2} \, dx\) [1658]

Optimal result

Integrand size = 28, antiderivative size = 136 \[ \int \frac {(d+e x)^{3/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=-\frac {e \sqrt {d+e x}}{4 b^2 (a+b x)^2}-\frac {e^2 \sqrt {d+e x}}{8 b^2 (b d-a e) (a+b x)}-\frac {(d+e x)^{3/2}}{3 b (a+b x)^3}+\frac {e^3 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 b^{5/2} (b d-a e)^{3/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {27, 43, 44, 65, 214} \[ \int \frac {(d+e x)^{3/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {e^3 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 b^{5/2} (b d-a e)^{3/2}}-\frac {e^2 \sqrt {d+e x}}{8 b^2 (a+b x) (b d-a e)}-\frac {e \sqrt {d+e x}}{4 b^2 (a+b x)^2}-\frac {(d+e x)^{3/2}}{3 b (a+b x)^3} \]

[In]

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

[Out]

-1/4*(e*Sqrt[d + e*x])/(b^2*(a + b*x)^2) - (e^2*Sqrt[d + e*x])/(8*b^2*(b*d - a*e)*(a + b*x)) - (d + e*x)^(3/2)
/(3*b*(a + b*x)^3) + (e^3*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(8*b^(5/2)*(b*d - a*e)^(3/2))

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 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]

Rubi steps \begin{align*} \text {integral}& = \int \frac {(d+e x)^{3/2}}{(a+b x)^4} \, dx \\ & = -\frac {(d+e x)^{3/2}}{3 b (a+b x)^3}+\frac {e \int \frac {\sqrt {d+e x}}{(a+b x)^3} \, dx}{2 b} \\ & = -\frac {e \sqrt {d+e x}}{4 b^2 (a+b x)^2}-\frac {(d+e x)^{3/2}}{3 b (a+b x)^3}+\frac {e^2 \int \frac {1}{(a+b x)^2 \sqrt {d+e x}} \, dx}{8 b^2} \\ & = -\frac {e \sqrt {d+e x}}{4 b^2 (a+b x)^2}-\frac {e^2 \sqrt {d+e x}}{8 b^2 (b d-a e) (a+b x)}-\frac {(d+e x)^{3/2}}{3 b (a+b x)^3}-\frac {e^3 \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{16 b^2 (b d-a e)} \\ & = -\frac {e \sqrt {d+e x}}{4 b^2 (a+b x)^2}-\frac {e^2 \sqrt {d+e x}}{8 b^2 (b d-a e) (a+b x)}-\frac {(d+e x)^{3/2}}{3 b (a+b x)^3}-\frac {e^2 \text {Subst}\left (\int \frac {1}{a-\frac {b d}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{8 b^2 (b d-a e)} \\ & = -\frac {e \sqrt {d+e x}}{4 b^2 (a+b x)^2}-\frac {e^2 \sqrt {d+e x}}{8 b^2 (b d-a e) (a+b x)}-\frac {(d+e x)^{3/2}}{3 b (a+b x)^3}+\frac {e^3 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 b^{5/2} (b d-a e)^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.51 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.95 \[ \int \frac {(d+e x)^{3/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {\sqrt {d+e x} \left (-3 a^2 e^2-2 a b e (d+4 e x)+b^2 \left (8 d^2+14 d e x+3 e^2 x^2\right )\right )}{24 b^2 (-b d+a e) (a+b x)^3}+\frac {e^3 \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{8 b^{5/2} (-b d+a e)^{3/2}} \]

[In]

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

[Out]

(Sqrt[d + e*x]*(-3*a^2*e^2 - 2*a*b*e*(d + 4*e*x) + b^2*(8*d^2 + 14*d*e*x + 3*e^2*x^2)))/(24*b^2*(-(b*d) + a*e)
*(a + b*x)^3) + (e^3*ArcTan[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]])/(8*b^(5/2)*(-(b*d) + a*e)^(3/2))

Maple [A] (verified)

Time = 2.48 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.88

method	result	size
pseudoelliptic	\(\frac {e^{3} \left (b x +a \right )^{3} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )-\left (\frac {\left (-e x -4 d \right ) b}{3}+a e \right ) \sqrt {e x +d}\, \left (\left (3 e x +2 d \right ) b +a e \right ) \sqrt {\left (a e -b d \right ) b}}{8 \sqrt {\left (a e -b d \right ) b}\, \left (a e -b d \right ) b^{2} \left (b x +a \right )^{3}}\)	\(119\)
derivativedivides	\(2 e^{3} \left (\frac {\frac {\left (e x +d \right )^{\frac {5}{2}}}{16 a e -16 b d}-\frac {\left (e x +d \right )^{\frac {3}{2}}}{6 b}-\frac {\left (a e -b d \right ) \sqrt {e x +d}}{16 b^{2}}}{\left (b \left (e x +d \right )+a e -b d \right )^{3}}+\frac {\arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{16 \left (a e -b d \right ) b^{2} \sqrt {\left (a e -b d \right ) b}}\right )\)	\(126\)
default	\(2 e^{3} \left (\frac {\frac {\left (e x +d \right )^{\frac {5}{2}}}{16 a e -16 b d}-\frac {\left (e x +d \right )^{\frac {3}{2}}}{6 b}-\frac {\left (a e -b d \right ) \sqrt {e x +d}}{16 b^{2}}}{\left (b \left (e x +d \right )+a e -b d \right )^{3}}+\frac {\arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{16 \left (a e -b d \right ) b^{2} \sqrt {\left (a e -b d \right ) b}}\right )\)	\(126\)

[In]

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

[Out]

1/8*(e^3*(b*x+a)^3*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))-(1/3*(-e*x-4*d)*b+a*e)*(e*x+d)^(1/2)*((3*e*x+2*
d)*b+a*e)*((a*e-b*d)*b)^(1/2))/((a*e-b*d)*b)^(1/2)/(a*e-b*d)/b^2/(b*x+a)^3

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 326 vs. \(2 (112) = 224\).

Time = 0.30 (sec) , antiderivative size = 666, normalized size of antiderivative = 4.90 \[ \int \frac {(d+e x)^{3/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\left [-\frac {3 \, {\left (b^{3} e^{3} x^{3} + 3 \, a b^{2} e^{3} x^{2} + 3 \, a^{2} b e^{3} x + a^{3} e^{3}\right )} \sqrt {b^{2} d - a b e} \log \left (\frac {b e x + 2 \, b d - a e - 2 \, \sqrt {b^{2} d - a b e} \sqrt {e x + d}}{b x + a}\right ) + 2 \, {\left (8 \, b^{4} d^{3} - 10 \, a b^{3} d^{2} e - a^{2} b^{2} d e^{2} + 3 \, a^{3} b e^{3} + 3 \, {\left (b^{4} d e^{2} - a b^{3} e^{3}\right )} x^{2} + 2 \, {\left (7 \, b^{4} d^{2} e - 11 \, a b^{3} d e^{2} + 4 \, a^{2} b^{2} e^{3}\right )} x\right )} \sqrt {e x + d}}{48 \, {\left (a^{3} b^{5} d^{2} - 2 \, a^{4} b^{4} d e + a^{5} b^{3} e^{2} + {\left (b^{8} d^{2} - 2 \, a b^{7} d e + a^{2} b^{6} e^{2}\right )} x^{3} + 3 \, {\left (a b^{7} d^{2} - 2 \, a^{2} b^{6} d e + a^{3} b^{5} e^{2}\right )} x^{2} + 3 \, {\left (a^{2} b^{6} d^{2} - 2 \, a^{3} b^{5} d e + a^{4} b^{4} e^{2}\right )} x\right )}}, -\frac {3 \, {\left (b^{3} e^{3} x^{3} + 3 \, a b^{2} e^{3} x^{2} + 3 \, a^{2} b e^{3} x + a^{3} e^{3}\right )} \sqrt {-b^{2} d + a b e} \arctan \left (\frac {\sqrt {-b^{2} d + a b e} \sqrt {e x + d}}{b e x + b d}\right ) + {\left (8 \, b^{4} d^{3} - 10 \, a b^{3} d^{2} e - a^{2} b^{2} d e^{2} + 3 \, a^{3} b e^{3} + 3 \, {\left (b^{4} d e^{2} - a b^{3} e^{3}\right )} x^{2} + 2 \, {\left (7 \, b^{4} d^{2} e - 11 \, a b^{3} d e^{2} + 4 \, a^{2} b^{2} e^{3}\right )} x\right )} \sqrt {e x + d}}{24 \, {\left (a^{3} b^{5} d^{2} - 2 \, a^{4} b^{4} d e + a^{5} b^{3} e^{2} + {\left (b^{8} d^{2} - 2 \, a b^{7} d e + a^{2} b^{6} e^{2}\right )} x^{3} + 3 \, {\left (a b^{7} d^{2} - 2 \, a^{2} b^{6} d e + a^{3} b^{5} e^{2}\right )} x^{2} + 3 \, {\left (a^{2} b^{6} d^{2} - 2 \, a^{3} b^{5} d e + a^{4} b^{4} e^{2}\right )} x\right )}}\right ] \]

[In]

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

[Out]

[-1/48*(3*(b^3*e^3*x^3 + 3*a*b^2*e^3*x^2 + 3*a^2*b*e^3*x + a^3*e^3)*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*(8*b^4*d^3 - 10*a*b^3*d^2*e - a^2*b^2*d*e^2 + 3*a^3*b
*e^3 + 3*(b^4*d*e^2 - a*b^3*e^3)*x^2 + 2*(7*b^4*d^2*e - 11*a*b^3*d*e^2 + 4*a^2*b^2*e^3)*x)*sqrt(e*x + d))/(a^3
*b^5*d^2 - 2*a^4*b^4*d*e + a^5*b^3*e^2 + (b^8*d^2 - 2*a*b^7*d*e + a^2*b^6*e^2)*x^3 + 3*(a*b^7*d^2 - 2*a^2*b^6*
d*e + a^3*b^5*e^2)*x^2 + 3*(a^2*b^6*d^2 - 2*a^3*b^5*d*e + a^4*b^4*e^2)*x), -1/24*(3*(b^3*e^3*x^3 + 3*a*b^2*e^3
*x^2 + 3*a^2*b*e^3*x + a^3*e^3)*sqrt(-b^2*d + a*b*e)*arctan(sqrt(-b^2*d + a*b*e)*sqrt(e*x + d)/(b*e*x + b*d))
+ (8*b^4*d^3 - 10*a*b^3*d^2*e - a^2*b^2*d*e^2 + 3*a^3*b*e^3 + 3*(b^4*d*e^2 - a*b^3*e^3)*x^2 + 2*(7*b^4*d^2*e -
11*a*b^3*d*e^2 + 4*a^2*b^2*e^3)*x)*sqrt(e*x + d))/(a^3*b^5*d^2 - 2*a^4*b^4*d*e + a^5*b^3*e^2 + (b^8*d^2 - 2*a
*b^7*d*e + a^2*b^6*e^2)*x^3 + 3*(a*b^7*d^2 - 2*a^2*b^6*d*e + a^3*b^5*e^2)*x^2 + 3*(a^2*b^6*d^2 - 2*a^3*b^5*d*e
+ a^4*b^4*e^2)*x)]

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {(d+e x)^{3/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((e*x+d)^(3/2)/(b^2*x^2+2*a*b*x+a^2)^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 [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.36 \[ \int \frac {(d+e x)^{3/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=-\frac {e^{3} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{8 \, {\left (b^{3} d - a b^{2} e\right )} \sqrt {-b^{2} d + a b e}} - \frac {3 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{2} e^{3} + 8 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{2} d e^{3} - 3 \, \sqrt {e x + d} b^{2} d^{2} e^{3} - 8 \, {\left (e x + d\right )}^{\frac {3}{2}} a b e^{4} + 6 \, \sqrt {e x + d} a b d e^{4} - 3 \, \sqrt {e x + d} a^{2} e^{5}}{24 \, {\left (b^{3} d - a b^{2} e\right )} {\left ({\left (e x + d\right )} b - b d + a e\right )}^{3}} \]

[In]

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

[Out]

-1/8*e^3*arctan(sqrt(e*x + d)*b/sqrt(-b^2*d + a*b*e))/((b^3*d - a*b^2*e)*sqrt(-b^2*d + a*b*e)) - 1/24*(3*(e*x
+ d)^(5/2)*b^2*e^3 + 8*(e*x + d)^(3/2)*b^2*d*e^3 - 3*sqrt(e*x + d)*b^2*d^2*e^3 - 8*(e*x + d)^(3/2)*a*b*e^4 + 6
*sqrt(e*x + d)*a*b*d*e^4 - 3*sqrt(e*x + d)*a^2*e^5)/((b^3*d - a*b^2*e)*((e*x + d)*b - b*d + a*e)^3)

Mupad [B] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.54 \[ \int \frac {(d+e x)^{3/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {e^3\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}}{\sqrt {a\,e-b\,d}}\right )}{8\,b^{5/2}\,{\left (a\,e-b\,d\right )}^{3/2}}-\frac {\frac {e^3\,{\left (d+e\,x\right )}^{3/2}}{3\,b}-\frac {e^3\,{\left (d+e\,x\right )}^{5/2}}{8\,\left (a\,e-b\,d\right )}+\frac {e^3\,\left (a\,e-b\,d\right )\,\sqrt {d+e\,x}}{8\,b^2}}{\left (d+e\,x\right )\,\left (3\,a^2\,b\,e^2-6\,a\,b^2\,d\,e+3\,b^3\,d^2\right )+b^3\,{\left (d+e\,x\right )}^3-\left (3\,b^3\,d-3\,a\,b^2\,e\right )\,{\left (d+e\,x\right )}^2+a^3\,e^3-b^3\,d^3+3\,a\,b^2\,d^2\,e-3\,a^2\,b\,d\,e^2} \]

[In]

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

[Out]

(e^3*atan((b^(1/2)*(d + e*x)^(1/2))/(a*e - b*d)^(1/2)))/(8*b^(5/2)*(a*e - b*d)^(3/2)) - ((e^3*(d + e*x)^(3/2))
/(3*b) - (e^3*(d + e*x)^(5/2))/(8*(a*e - b*d)) + (e^3*(a*e - b*d)*(d + e*x)^(1/2))/(8*b^2))/((d + e*x)*(3*b^3*
d^2 + 3*a^2*b*e^2 - 6*a*b^2*d*e) + b^3*(d + e*x)^3 - (3*b^3*d - 3*a*b^2*e)*(d + e*x)^2 + a^3*e^3 - b^3*d^3 + 3
*a*b^2*d^2*e - 3*a^2*b*d*e^2)