\(\int \frac {1}{(d+e x)^{7/2} (a^2+2 a b x+b^2 x^2)} \, dx\) [1653]

Optimal result

Integrand size = 28, antiderivative size = 151 \[ \int \frac {1}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )} \, dx=-\frac {7 e}{5 (b d-a e)^2 (d+e x)^{5/2}}-\frac {1}{(b d-a e) (a+b x) (d+e x)^{5/2}}-\frac {7 b e}{3 (b d-a e)^3 (d+e x)^{3/2}}-\frac {7 b^2 e}{(b d-a e)^4 \sqrt {d+e x}}+\frac {7 b^{5/2} e \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{9/2}} \]

[Out]

-7/5*e/(-a*e+b*d)^2/(e*x+d)^(5/2)-1/(-a*e+b*d)/(b*x+a)/(e*x+d)^(5/2)-7/3*b*e/(-a*e+b*d)^3/(e*x+d)^(3/2)+7*b^(5
/2)*e*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))/(-a*e+b*d)^(9/2)-7*b^2*e/(-a*e+b*d)^4/(e*x+d)^(1/2)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {27, 44, 53, 65, 214} \[ \int \frac {1}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )} \, dx=\frac {7 b^{5/2} e \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{9/2}}-\frac {7 b^2 e}{\sqrt {d+e x} (b d-a e)^4}-\frac {7 b e}{3 (d+e x)^{3/2} (b d-a e)^3}-\frac {1}{(a+b x) (d+e x)^{5/2} (b d-a e)}-\frac {7 e}{5 (d+e x)^{5/2} (b d-a e)^2} \]

[In]

Int[1/((d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)),x]

[Out]

(-7*e)/(5*(b*d - a*e)^2*(d + e*x)^(5/2)) - 1/((b*d - a*e)*(a + b*x)*(d + e*x)^(5/2)) - (7*b*e)/(3*(b*d - a*e)^
3*(d + e*x)^(3/2)) - (7*b^2*e)/((b*d - a*e)^4*Sqrt[d + e*x]) + (7*b^(5/2)*e*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sq
rt[b*d - a*e]])/(b*d - a*e)^(9/2)

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 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 53

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] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && 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]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(a+b x)^2 (d+e x)^{7/2}} \, dx \\ & = -\frac {1}{(b d-a e) (a+b x) (d+e x)^{5/2}}-\frac {(7 e) \int \frac {1}{(a+b x) (d+e x)^{7/2}} \, dx}{2 (b d-a e)} \\ & = -\frac {7 e}{5 (b d-a e)^2 (d+e x)^{5/2}}-\frac {1}{(b d-a e) (a+b x) (d+e x)^{5/2}}-\frac {(7 b e) \int \frac {1}{(a+b x) (d+e x)^{5/2}} \, dx}{2 (b d-a e)^2} \\ & = -\frac {7 e}{5 (b d-a e)^2 (d+e x)^{5/2}}-\frac {1}{(b d-a e) (a+b x) (d+e x)^{5/2}}-\frac {7 b e}{3 (b d-a e)^3 (d+e x)^{3/2}}-\frac {\left (7 b^2 e\right ) \int \frac {1}{(a+b x) (d+e x)^{3/2}} \, dx}{2 (b d-a e)^3} \\ & = -\frac {7 e}{5 (b d-a e)^2 (d+e x)^{5/2}}-\frac {1}{(b d-a e) (a+b x) (d+e x)^{5/2}}-\frac {7 b e}{3 (b d-a e)^3 (d+e x)^{3/2}}-\frac {7 b^2 e}{(b d-a e)^4 \sqrt {d+e x}}-\frac {\left (7 b^3 e\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{2 (b d-a e)^4} \\ & = -\frac {7 e}{5 (b d-a e)^2 (d+e x)^{5/2}}-\frac {1}{(b d-a e) (a+b x) (d+e x)^{5/2}}-\frac {7 b e}{3 (b d-a e)^3 (d+e x)^{3/2}}-\frac {7 b^2 e}{(b d-a e)^4 \sqrt {d+e x}}-\frac {\left (7 b^3\right ) \text {Subst}\left (\int \frac {1}{a-\frac {b d}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{(b d-a e)^4} \\ & = -\frac {7 e}{5 (b d-a e)^2 (d+e x)^{5/2}}-\frac {1}{(b d-a e) (a+b x) (d+e x)^{5/2}}-\frac {7 b e}{3 (b d-a e)^3 (d+e x)^{3/2}}-\frac {7 b^2 e}{(b d-a e)^4 \sqrt {d+e x}}+\frac {7 b^{5/2} e \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{9/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )} \, dx=\frac {-6 a^3 e^3+2 a^2 b e^2 (16 d+7 e x)-2 a b^2 e \left (58 d^2+84 d e x+35 e^2 x^2\right )-b^3 \left (15 d^3+161 d^2 e x+245 d e^2 x^2+105 e^3 x^3\right )}{15 (b d-a e)^4 (a+b x) (d+e x)^{5/2}}-\frac {7 b^{5/2} e \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{(-b d+a e)^{9/2}} \]

[In]

Integrate[1/((d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)),x]

[Out]

(-6*a^3*e^3 + 2*a^2*b*e^2*(16*d + 7*e*x) - 2*a*b^2*e*(58*d^2 + 84*d*e*x + 35*e^2*x^2) - b^3*(15*d^3 + 161*d^2*
e*x + 245*d*e^2*x^2 + 105*e^3*x^3))/(15*(b*d - a*e)^4*(a + b*x)*(d + e*x)^(5/2)) - (7*b^(5/2)*e*ArcTan[(Sqrt[b
]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]])/(-(b*d) + a*e)^(9/2)

Maple [A] (verified)

Time = 2.35 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.95

method	result	size
derivativedivides	\(2 e \left (-\frac {b^{3} \left (\frac {\sqrt {e x +d}}{2 b \left (e x +d \right )+2 a e -2 b d}+\frac {7 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{2 \sqrt {\left (a e -b d \right ) b}}\right )}{\left (a e -b d \right )^{4}}-\frac {1}{5 \left (a e -b d \right )^{2} \left (e x +d \right )^{\frac {5}{2}}}-\frac {3 b^{2}}{\left (a e -b d \right )^{4} \sqrt {e x +d}}+\frac {2 b}{3 \left (a e -b d \right )^{3} \left (e x +d \right )^{\frac {3}{2}}}\right )\)	\(144\)
default	\(2 e \left (-\frac {b^{3} \left (\frac {\sqrt {e x +d}}{2 b \left (e x +d \right )+2 a e -2 b d}+\frac {7 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{2 \sqrt {\left (a e -b d \right ) b}}\right )}{\left (a e -b d \right )^{4}}-\frac {1}{5 \left (a e -b d \right )^{2} \left (e x +d \right )^{\frac {5}{2}}}-\frac {3 b^{2}}{\left (a e -b d \right )^{4} \sqrt {e x +d}}+\frac {2 b}{3 \left (a e -b d \right )^{3} \left (e x +d \right )^{\frac {3}{2}}}\right )\)	\(144\)
pseudoelliptic	\(-\frac {2 \left (\frac {35 b^{3} e \left (e x +d \right )^{\frac {5}{2}} \left (b x +a \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{2}+\sqrt {\left (a e -b d \right ) b}\, \left (\frac {\left (35 e^{3} x^{3}+\frac {245}{3} d \,e^{2} x^{2}+\frac {161}{3} d^{2} e x +5 d^{3}\right ) b^{3}}{2}+\frac {58 \left (\frac {35}{58} x^{2} e^{2}+\frac {42}{29} d e x +d^{2}\right ) e a \,b^{2}}{3}-\frac {16 \left (\frac {7 e x}{16}+d \right ) e^{2} a^{2} b}{3}+a^{3} e^{3}\right )\right )}{5 \left (e x +d \right )^{\frac {5}{2}} \sqrt {\left (a e -b d \right ) b}\, \left (b x +a \right ) \left (a e -b d \right )^{4}}\)	\(175\)

[In]

int(1/(e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2),x,method=_RETURNVERBOSE)

[Out]

2*e*(-1/(a*e-b*d)^4*b^3*(1/2*(e*x+d)^(1/2)/(b*(e*x+d)+a*e-b*d)+7/2/((a*e-b*d)*b)^(1/2)*arctan(b*(e*x+d)^(1/2)/
((a*e-b*d)*b)^(1/2)))-1/5/(a*e-b*d)^2/(e*x+d)^(5/2)-3/(a*e-b*d)^4*b^2/(e*x+d)^(1/2)+2/3/(a*e-b*d)^3*b/(e*x+d)^
(3/2))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 604 vs. \(2 (129) = 258\).

Time = 0.34 (sec) , antiderivative size = 1218, normalized size of antiderivative = 8.07 \[ \int \frac {1}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )} \, dx=\left [\frac {105 \, {\left (b^{3} e^{4} x^{4} + a b^{2} d^{3} e + {\left (3 \, b^{3} d e^{3} + a b^{2} e^{4}\right )} x^{3} + 3 \, {\left (b^{3} d^{2} e^{2} + a b^{2} d e^{3}\right )} x^{2} + {\left (b^{3} d^{3} e + 3 \, a b^{2} d^{2} e^{2}\right )} x\right )} \sqrt {\frac {b}{b d - a e}} \log \left (\frac {b e x + 2 \, b d - a e + 2 \, {\left (b d - a e\right )} \sqrt {e x + d} \sqrt {\frac {b}{b d - a e}}}{b x + a}\right ) - 2 \, {\left (105 \, b^{3} e^{3} x^{3} + 15 \, b^{3} d^{3} + 116 \, a b^{2} d^{2} e - 32 \, a^{2} b d e^{2} + 6 \, a^{3} e^{3} + 35 \, {\left (7 \, b^{3} d e^{2} + 2 \, a b^{2} e^{3}\right )} x^{2} + 7 \, {\left (23 \, b^{3} d^{2} e + 24 \, a b^{2} d e^{2} - 2 \, a^{2} b e^{3}\right )} x\right )} \sqrt {e x + d}}{30 \, {\left (a b^{4} d^{7} - 4 \, a^{2} b^{3} d^{6} e + 6 \, a^{3} b^{2} d^{5} e^{2} - 4 \, a^{4} b d^{4} e^{3} + a^{5} d^{3} e^{4} + {\left (b^{5} d^{4} e^{3} - 4 \, a b^{4} d^{3} e^{4} + 6 \, a^{2} b^{3} d^{2} e^{5} - 4 \, a^{3} b^{2} d e^{6} + a^{4} b e^{7}\right )} x^{4} + {\left (3 \, b^{5} d^{5} e^{2} - 11 \, a b^{4} d^{4} e^{3} + 14 \, a^{2} b^{3} d^{3} e^{4} - 6 \, a^{3} b^{2} d^{2} e^{5} - a^{4} b d e^{6} + a^{5} e^{7}\right )} x^{3} + 3 \, {\left (b^{5} d^{6} e - 3 \, a b^{4} d^{5} e^{2} + 2 \, a^{2} b^{3} d^{4} e^{3} + 2 \, a^{3} b^{2} d^{3} e^{4} - 3 \, a^{4} b d^{2} e^{5} + a^{5} d e^{6}\right )} x^{2} + {\left (b^{5} d^{7} - a b^{4} d^{6} e - 6 \, a^{2} b^{3} d^{5} e^{2} + 14 \, a^{3} b^{2} d^{4} e^{3} - 11 \, a^{4} b d^{3} e^{4} + 3 \, a^{5} d^{2} e^{5}\right )} x\right )}}, \frac {105 \, {\left (b^{3} e^{4} x^{4} + a b^{2} d^{3} e + {\left (3 \, b^{3} d e^{3} + a b^{2} e^{4}\right )} x^{3} + 3 \, {\left (b^{3} d^{2} e^{2} + a b^{2} d e^{3}\right )} x^{2} + {\left (b^{3} d^{3} e + 3 \, a b^{2} d^{2} e^{2}\right )} x\right )} \sqrt {-\frac {b}{b d - a e}} \arctan \left (-\frac {{\left (b d - a e\right )} \sqrt {e x + d} \sqrt {-\frac {b}{b d - a e}}}{b e x + b d}\right ) - {\left (105 \, b^{3} e^{3} x^{3} + 15 \, b^{3} d^{3} + 116 \, a b^{2} d^{2} e - 32 \, a^{2} b d e^{2} + 6 \, a^{3} e^{3} + 35 \, {\left (7 \, b^{3} d e^{2} + 2 \, a b^{2} e^{3}\right )} x^{2} + 7 \, {\left (23 \, b^{3} d^{2} e + 24 \, a b^{2} d e^{2} - 2 \, a^{2} b e^{3}\right )} x\right )} \sqrt {e x + d}}{15 \, {\left (a b^{4} d^{7} - 4 \, a^{2} b^{3} d^{6} e + 6 \, a^{3} b^{2} d^{5} e^{2} - 4 \, a^{4} b d^{4} e^{3} + a^{5} d^{3} e^{4} + {\left (b^{5} d^{4} e^{3} - 4 \, a b^{4} d^{3} e^{4} + 6 \, a^{2} b^{3} d^{2} e^{5} - 4 \, a^{3} b^{2} d e^{6} + a^{4} b e^{7}\right )} x^{4} + {\left (3 \, b^{5} d^{5} e^{2} - 11 \, a b^{4} d^{4} e^{3} + 14 \, a^{2} b^{3} d^{3} e^{4} - 6 \, a^{3} b^{2} d^{2} e^{5} - a^{4} b d e^{6} + a^{5} e^{7}\right )} x^{3} + 3 \, {\left (b^{5} d^{6} e - 3 \, a b^{4} d^{5} e^{2} + 2 \, a^{2} b^{3} d^{4} e^{3} + 2 \, a^{3} b^{2} d^{3} e^{4} - 3 \, a^{4} b d^{2} e^{5} + a^{5} d e^{6}\right )} x^{2} + {\left (b^{5} d^{7} - a b^{4} d^{6} e - 6 \, a^{2} b^{3} d^{5} e^{2} + 14 \, a^{3} b^{2} d^{4} e^{3} - 11 \, a^{4} b d^{3} e^{4} + 3 \, a^{5} d^{2} e^{5}\right )} x\right )}}\right ] \]

[In]

integrate(1/(e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2),x, algorithm="fricas")

[Out]

[1/30*(105*(b^3*e^4*x^4 + a*b^2*d^3*e + (3*b^3*d*e^3 + a*b^2*e^4)*x^3 + 3*(b^3*d^2*e^2 + a*b^2*d*e^3)*x^2 + (b
^3*d^3*e + 3*a*b^2*d^2*e^2)*x)*sqrt(b/(b*d - a*e))*log((b*e*x + 2*b*d - a*e + 2*(b*d - a*e)*sqrt(e*x + d)*sqrt
(b/(b*d - a*e)))/(b*x + a)) - 2*(105*b^3*e^3*x^3 + 15*b^3*d^3 + 116*a*b^2*d^2*e - 32*a^2*b*d*e^2 + 6*a^3*e^3 +
35*(7*b^3*d*e^2 + 2*a*b^2*e^3)*x^2 + 7*(23*b^3*d^2*e + 24*a*b^2*d*e^2 - 2*a^2*b*e^3)*x)*sqrt(e*x + d))/(a*b^4
*d^7 - 4*a^2*b^3*d^6*e + 6*a^3*b^2*d^5*e^2 - 4*a^4*b*d^4*e^3 + a^5*d^3*e^4 + (b^5*d^4*e^3 - 4*a*b^4*d^3*e^4 +
6*a^2*b^3*d^2*e^5 - 4*a^3*b^2*d*e^6 + a^4*b*e^7)*x^4 + (3*b^5*d^5*e^2 - 11*a*b^4*d^4*e^3 + 14*a^2*b^3*d^3*e^4
- 6*a^3*b^2*d^2*e^5 - a^4*b*d*e^6 + a^5*e^7)*x^3 + 3*(b^5*d^6*e - 3*a*b^4*d^5*e^2 + 2*a^2*b^3*d^4*e^3 + 2*a^3*
b^2*d^3*e^4 - 3*a^4*b*d^2*e^5 + a^5*d*e^6)*x^2 + (b^5*d^7 - a*b^4*d^6*e - 6*a^2*b^3*d^5*e^2 + 14*a^3*b^2*d^4*e
^3 - 11*a^4*b*d^3*e^4 + 3*a^5*d^2*e^5)*x), 1/15*(105*(b^3*e^4*x^4 + a*b^2*d^3*e + (3*b^3*d*e^3 + a*b^2*e^4)*x^
3 + 3*(b^3*d^2*e^2 + a*b^2*d*e^3)*x^2 + (b^3*d^3*e + 3*a*b^2*d^2*e^2)*x)*sqrt(-b/(b*d - a*e))*arctan(-(b*d - a
*e)*sqrt(e*x + d)*sqrt(-b/(b*d - a*e))/(b*e*x + b*d)) - (105*b^3*e^3*x^3 + 15*b^3*d^3 + 116*a*b^2*d^2*e - 32*a
^2*b*d*e^2 + 6*a^3*e^3 + 35*(7*b^3*d*e^2 + 2*a*b^2*e^3)*x^2 + 7*(23*b^3*d^2*e + 24*a*b^2*d*e^2 - 2*a^2*b*e^3)*
x)*sqrt(e*x + d))/(a*b^4*d^7 - 4*a^2*b^3*d^6*e + 6*a^3*b^2*d^5*e^2 - 4*a^4*b*d^4*e^3 + a^5*d^3*e^4 + (b^5*d^4*
e^3 - 4*a*b^4*d^3*e^4 + 6*a^2*b^3*d^2*e^5 - 4*a^3*b^2*d*e^6 + a^4*b*e^7)*x^4 + (3*b^5*d^5*e^2 - 11*a*b^4*d^4*e
^3 + 14*a^2*b^3*d^3*e^4 - 6*a^3*b^2*d^2*e^5 - a^4*b*d*e^6 + a^5*e^7)*x^3 + 3*(b^5*d^6*e - 3*a*b^4*d^5*e^2 + 2*
a^2*b^3*d^4*e^3 + 2*a^3*b^2*d^3*e^4 - 3*a^4*b*d^2*e^5 + a^5*d*e^6)*x^2 + (b^5*d^7 - a*b^4*d^6*e - 6*a^2*b^3*d^
5*e^2 + 14*a^3*b^2*d^4*e^3 - 11*a^4*b*d^3*e^4 + 3*a^5*d^2*e^5)*x)]

Sympy [F]

\[ \int \frac {1}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )} \, dx=\int \frac {1}{\left (a + b x\right )^{2} \left (d + e x\right )^{\frac {7}{2}}}\, dx \]

[In]

integrate(1/(e*x+d)**(7/2)/(b**2*x**2+2*a*b*x+a**2),x)

[Out]

Integral(1/((a + b*x)**2*(d + e*x)**(7/2)), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(1/(e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 298 vs. \(2 (129) = 258\).

Time = 0.28 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.97 \[ \int \frac {1}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )} \, dx=-\frac {7 \, b^{3} e \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} \sqrt {-b^{2} d + a b e}} - \frac {\sqrt {e x + d} b^{3} e}{{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} {\left ({\left (e x + d\right )} b - b d + a e\right )}} - \frac {2 \, {\left (45 \, {\left (e x + d\right )}^{2} b^{2} e + 10 \, {\left (e x + d\right )} b^{2} d e + 3 \, b^{2} d^{2} e - 10 \, {\left (e x + d\right )} a b e^{2} - 6 \, a b d e^{2} + 3 \, a^{2} e^{3}\right )}}{15 \, {\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} {\left (e x + d\right )}^{\frac {5}{2}}} \]

[In]

integrate(1/(e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2),x, algorithm="giac")

[Out]

-7*b^3*e*arctan(sqrt(e*x + d)*b/sqrt(-b^2*d + a*b*e))/((b^4*d^4 - 4*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 - 4*a^3*b*
d*e^3 + a^4*e^4)*sqrt(-b^2*d + a*b*e)) - sqrt(e*x + d)*b^3*e/((b^4*d^4 - 4*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 - 4
*a^3*b*d*e^3 + a^4*e^4)*((e*x + d)*b - b*d + a*e)) - 2/15*(45*(e*x + d)^2*b^2*e + 10*(e*x + d)*b^2*d*e + 3*b^2
*d^2*e - 10*(e*x + d)*a*b*e^2 - 6*a*b*d*e^2 + 3*a^2*e^3)/((b^4*d^4 - 4*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 - 4*a^3
*b*d*e^3 + a^4*e^4)*(e*x + d)^(5/2))

Mupad [B] (verification not implemented)

Time = 9.52 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.31 \[ \int \frac {1}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )} \, dx=-\frac {\frac {2\,e}{5\,\left (a\,e-b\,d\right )}-\frac {14\,b\,e\,\left (d+e\,x\right )}{15\,{\left (a\,e-b\,d\right )}^2}+\frac {14\,b^2\,e\,{\left (d+e\,x\right )}^2}{3\,{\left (a\,e-b\,d\right )}^3}+\frac {7\,b^3\,e\,{\left (d+e\,x\right )}^3}{{\left (a\,e-b\,d\right )}^4}}{b\,{\left (d+e\,x\right )}^{7/2}+\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{5/2}}-\frac {7\,b^{5/2}\,e\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}\,\left (a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4\right )}{{\left (a\,e-b\,d\right )}^{9/2}}\right )}{{\left (a\,e-b\,d\right )}^{9/2}} \]

[In]

int(1/((d + e*x)^(7/2)*(a^2 + b^2*x^2 + 2*a*b*x)),x)

[Out]

- ((2*e)/(5*(a*e - b*d)) - (14*b*e*(d + e*x))/(15*(a*e - b*d)^2) + (14*b^2*e*(d + e*x)^2)/(3*(a*e - b*d)^3) +
(7*b^3*e*(d + e*x)^3)/(a*e - b*d)^4)/(b*(d + e*x)^(7/2) + (a*e - b*d)*(d + e*x)^(5/2)) - (7*b^(5/2)*e*atan((b^
(1/2)*(d + e*x)^(1/2)*(a^4*e^4 + b^4*d^4 + 6*a^2*b^2*d^2*e^2 - 4*a*b^3*d^3*e - 4*a^3*b*d*e^3))/(a*e - b*d)^(9/
2)))/(a*e - b*d)^(9/2)