3.1612 \(\int \frac {1}{(d+e x)^2 (a^2+2 a b x+b^2 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=307 \[ \frac {e^4 (a+b x)}{\sqrt {a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^5}+\frac {5 b e^4 (a+b x) \log (a+b x)}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}-\frac {5 b e^4 (a+b x) \log (d+e x)}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}+\frac {4 b e^3}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}-\frac {3 b e^2}{2 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}+\frac {2 b e}{3 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac {b}{4 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2} \]

[Out]

4*b*e^3/(-a*e+b*d)^5/((b*x+a)^2)^(1/2)-1/4*b/(-a*e+b*d)^2/(b*x+a)^3/((b*x+a)^2)^(1/2)+2/3*b*e/(-a*e+b*d)^3/(b*
x+a)^2/((b*x+a)^2)^(1/2)-3/2*b*e^2/(-a*e+b*d)^4/(b*x+a)/((b*x+a)^2)^(1/2)+e^4*(b*x+a)/(-a*e+b*d)^5/(e*x+d)/((b
*x+a)^2)^(1/2)+5*b*e^4*(b*x+a)*ln(b*x+a)/(-a*e+b*d)^6/((b*x+a)^2)^(1/2)-5*b*e^4*(b*x+a)*ln(e*x+d)/(-a*e+b*d)^6
/((b*x+a)^2)^(1/2)

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Rubi [A]  time = 0.21, antiderivative size = 307, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {646, 44} \[ \frac {e^4 (a+b x)}{\sqrt {a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^5}+\frac {4 b e^3}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}-\frac {3 b e^2}{2 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}+\frac {5 b e^4 (a+b x) \log (a+b x)}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}-\frac {5 b e^4 (a+b x) \log (d+e x)}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}+\frac {2 b e}{3 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac {b}{4 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*b*e^3)/((b*d - a*e)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - b/(4*(b*d - a*e)^2*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x +
b^2*x^2]) + (2*b*e)/(3*(b*d - a*e)^3*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (3*b*e^2)/(2*(b*d - a*e)^4*(
a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (e^4*(a + b*x))/((b*d - a*e)^5*(d + e*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^
2]) + (5*b*e^4*(a + b*x)*Log[a + b*x])/((b*d - a*e)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (5*b*e^4*(a + b*x)*Log[
d + e*x])/((b*d - a*e)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 646

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*} \int \frac {1}{(d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx &=\frac {\left (b^4 \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right )^5 (d+e x)^2} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {\left (b^4 \left (a b+b^2 x\right )\right ) \int \left (\frac {1}{b^3 (b d-a e)^2 (a+b x)^5}-\frac {2 e}{b^3 (b d-a e)^3 (a+b x)^4}+\frac {3 e^2}{b^3 (b d-a e)^4 (a+b x)^3}-\frac {4 e^3}{b^3 (b d-a e)^5 (a+b x)^2}+\frac {5 e^4}{b^3 (b d-a e)^6 (a+b x)}-\frac {e^5}{b^5 (b d-a e)^5 (d+e x)^2}-\frac {5 e^5}{b^4 (b d-a e)^6 (d+e x)}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {4 b e^3}{(b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b}{4 (b d-a e)^2 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 b e}{3 (b d-a e)^3 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 b e^2}{2 (b d-a e)^4 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e^4 (a+b x)}{(b d-a e)^5 (d+e x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 b e^4 (a+b x) \log (a+b x)}{(b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 b e^4 (a+b x) \log (d+e x)}{(b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 167, normalized size = 0.54 \[ \frac {\frac {12 e^4 (a+b x)^3 (b d-a e)}{d+e x}-60 b e^4 (a+b x)^3 \log (d+e x)+48 b e^3 (a+b x)^2 (b d-a e)-18 b e^2 (a+b x) (b d-a e)^2-\frac {3 b (b d-a e)^4}{a+b x}+8 b e (b d-a e)^3+60 b e^4 (a+b x)^3 \log (a+b x)}{12 \left ((a+b x)^2\right )^{3/2} (b d-a e)^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(8*b*e*(b*d - a*e)^3 - (3*b*(b*d - a*e)^4)/(a + b*x) - 18*b*e^2*(b*d - a*e)^2*(a + b*x) + 48*b*e^3*(b*d - a*e)
*(a + b*x)^2 + (12*e^4*(b*d - a*e)*(a + b*x)^3)/(d + e*x) + 60*b*e^4*(a + b*x)^3*Log[a + b*x] - 60*b*e^4*(a +
b*x)^3*Log[d + e*x])/(12*(b*d - a*e)^6*((a + b*x)^2)^(3/2))

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fricas [B]  time = 0.94, size = 1083, normalized size = 3.53 \[ -\frac {3 \, b^{5} d^{5} - 20 \, a b^{4} d^{4} e + 60 \, a^{2} b^{3} d^{3} e^{2} - 120 \, a^{3} b^{2} d^{2} e^{3} + 65 \, a^{4} b d e^{4} + 12 \, a^{5} e^{5} - 60 \, {\left (b^{5} d e^{4} - a b^{4} e^{5}\right )} x^{4} - 30 \, {\left (b^{5} d^{2} e^{3} + 6 \, a b^{4} d e^{4} - 7 \, a^{2} b^{3} e^{5}\right )} x^{3} + 10 \, {\left (b^{5} d^{3} e^{2} - 12 \, a b^{4} d^{2} e^{3} - 15 \, a^{2} b^{3} d e^{4} + 26 \, a^{3} b^{2} e^{5}\right )} x^{2} - 5 \, {\left (b^{5} d^{4} e - 8 \, a b^{4} d^{3} e^{2} + 36 \, a^{2} b^{3} d^{2} e^{3} - 4 \, a^{3} b^{2} d e^{4} - 25 \, a^{4} b e^{5}\right )} x - 60 \, {\left (b^{5} e^{5} x^{5} + a^{4} b d e^{4} + {\left (b^{5} d e^{4} + 4 \, a b^{4} e^{5}\right )} x^{4} + 2 \, {\left (2 \, a b^{4} d e^{4} + 3 \, a^{2} b^{3} e^{5}\right )} x^{3} + 2 \, {\left (3 \, a^{2} b^{3} d e^{4} + 2 \, a^{3} b^{2} e^{5}\right )} x^{2} + {\left (4 \, a^{3} b^{2} d e^{4} + a^{4} b e^{5}\right )} x\right )} \log \left (b x + a\right ) + 60 \, {\left (b^{5} e^{5} x^{5} + a^{4} b d e^{4} + {\left (b^{5} d e^{4} + 4 \, a b^{4} e^{5}\right )} x^{4} + 2 \, {\left (2 \, a b^{4} d e^{4} + 3 \, a^{2} b^{3} e^{5}\right )} x^{3} + 2 \, {\left (3 \, a^{2} b^{3} d e^{4} + 2 \, a^{3} b^{2} e^{5}\right )} x^{2} + {\left (4 \, a^{3} b^{2} d e^{4} + a^{4} b e^{5}\right )} x\right )} \log \left (e x + d\right )}{12 \, {\left (a^{4} b^{6} d^{7} - 6 \, a^{5} b^{5} d^{6} e + 15 \, a^{6} b^{4} d^{5} e^{2} - 20 \, a^{7} b^{3} d^{4} e^{3} + 15 \, a^{8} b^{2} d^{3} e^{4} - 6 \, a^{9} b d^{2} e^{5} + a^{10} d e^{6} + {\left (b^{10} d^{6} e - 6 \, a b^{9} d^{5} e^{2} + 15 \, a^{2} b^{8} d^{4} e^{3} - 20 \, a^{3} b^{7} d^{3} e^{4} + 15 \, a^{4} b^{6} d^{2} e^{5} - 6 \, a^{5} b^{5} d e^{6} + a^{6} b^{4} e^{7}\right )} x^{5} + {\left (b^{10} d^{7} - 2 \, a b^{9} d^{6} e - 9 \, a^{2} b^{8} d^{5} e^{2} + 40 \, a^{3} b^{7} d^{4} e^{3} - 65 \, a^{4} b^{6} d^{3} e^{4} + 54 \, a^{5} b^{5} d^{2} e^{5} - 23 \, a^{6} b^{4} d e^{6} + 4 \, a^{7} b^{3} e^{7}\right )} x^{4} + 2 \, {\left (2 \, a b^{9} d^{7} - 9 \, a^{2} b^{8} d^{6} e + 12 \, a^{3} b^{7} d^{5} e^{2} + 5 \, a^{4} b^{6} d^{4} e^{3} - 30 \, a^{5} b^{5} d^{3} e^{4} + 33 \, a^{6} b^{4} d^{2} e^{5} - 16 \, a^{7} b^{3} d e^{6} + 3 \, a^{8} b^{2} e^{7}\right )} x^{3} + 2 \, {\left (3 \, a^{2} b^{8} d^{7} - 16 \, a^{3} b^{7} d^{6} e + 33 \, a^{4} b^{6} d^{5} e^{2} - 30 \, a^{5} b^{5} d^{4} e^{3} + 5 \, a^{6} b^{4} d^{3} e^{4} + 12 \, a^{7} b^{3} d^{2} e^{5} - 9 \, a^{8} b^{2} d e^{6} + 2 \, a^{9} b e^{7}\right )} x^{2} + {\left (4 \, a^{3} b^{7} d^{7} - 23 \, a^{4} b^{6} d^{6} e + 54 \, a^{5} b^{5} d^{5} e^{2} - 65 \, a^{6} b^{4} d^{4} e^{3} + 40 \, a^{7} b^{3} d^{3} e^{4} - 9 \, a^{8} b^{2} d^{2} e^{5} - 2 \, a^{9} b d e^{6} + a^{10} e^{7}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(3*b^5*d^5 - 20*a*b^4*d^4*e + 60*a^2*b^3*d^3*e^2 - 120*a^3*b^2*d^2*e^3 + 65*a^4*b*d*e^4 + 12*a^5*e^5 - 6
0*(b^5*d*e^4 - a*b^4*e^5)*x^4 - 30*(b^5*d^2*e^3 + 6*a*b^4*d*e^4 - 7*a^2*b^3*e^5)*x^3 + 10*(b^5*d^3*e^2 - 12*a*
b^4*d^2*e^3 - 15*a^2*b^3*d*e^4 + 26*a^3*b^2*e^5)*x^2 - 5*(b^5*d^4*e - 8*a*b^4*d^3*e^2 + 36*a^2*b^3*d^2*e^3 - 4
*a^3*b^2*d*e^4 - 25*a^4*b*e^5)*x - 60*(b^5*e^5*x^5 + a^4*b*d*e^4 + (b^5*d*e^4 + 4*a*b^4*e^5)*x^4 + 2*(2*a*b^4*
d*e^4 + 3*a^2*b^3*e^5)*x^3 + 2*(3*a^2*b^3*d*e^4 + 2*a^3*b^2*e^5)*x^2 + (4*a^3*b^2*d*e^4 + a^4*b*e^5)*x)*log(b*
x + a) + 60*(b^5*e^5*x^5 + a^4*b*d*e^4 + (b^5*d*e^4 + 4*a*b^4*e^5)*x^4 + 2*(2*a*b^4*d*e^4 + 3*a^2*b^3*e^5)*x^3
+ 2*(3*a^2*b^3*d*e^4 + 2*a^3*b^2*e^5)*x^2 + (4*a^3*b^2*d*e^4 + a^4*b*e^5)*x)*log(e*x + d))/(a^4*b^6*d^7 - 6*a
^5*b^5*d^6*e + 15*a^6*b^4*d^5*e^2 - 20*a^7*b^3*d^4*e^3 + 15*a^8*b^2*d^3*e^4 - 6*a^9*b*d^2*e^5 + a^10*d*e^6 + (
b^10*d^6*e - 6*a*b^9*d^5*e^2 + 15*a^2*b^8*d^4*e^3 - 20*a^3*b^7*d^3*e^4 + 15*a^4*b^6*d^2*e^5 - 6*a^5*b^5*d*e^6
+ a^6*b^4*e^7)*x^5 + (b^10*d^7 - 2*a*b^9*d^6*e - 9*a^2*b^8*d^5*e^2 + 40*a^3*b^7*d^4*e^3 - 65*a^4*b^6*d^3*e^4 +
54*a^5*b^5*d^2*e^5 - 23*a^6*b^4*d*e^6 + 4*a^7*b^3*e^7)*x^4 + 2*(2*a*b^9*d^7 - 9*a^2*b^8*d^6*e + 12*a^3*b^7*d^
5*e^2 + 5*a^4*b^6*d^4*e^3 - 30*a^5*b^5*d^3*e^4 + 33*a^6*b^4*d^2*e^5 - 16*a^7*b^3*d*e^6 + 3*a^8*b^2*e^7)*x^3 +
2*(3*a^2*b^8*d^7 - 16*a^3*b^7*d^6*e + 33*a^4*b^6*d^5*e^2 - 30*a^5*b^5*d^4*e^3 + 5*a^6*b^4*d^3*e^4 + 12*a^7*b^3
*d^2*e^5 - 9*a^8*b^2*d*e^6 + 2*a^9*b*e^7)*x^2 + (4*a^3*b^7*d^7 - 23*a^4*b^6*d^6*e + 54*a^5*b^5*d^5*e^2 - 65*a^
6*b^4*d^4*e^3 + 40*a^7*b^3*d^3*e^4 - 9*a^8*b^2*d^2*e^5 - 2*a^9*b*d*e^6 + a^10*e^7)*x)

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giac [B]  time = 0.71, size = 937, normalized size = 3.05 \[ -\frac {5 \, b e^{5} \log \left ({\left | -b + \frac {b d}{x e + d} - \frac {a e}{x e + d} \right |}\right )}{b^{6} d^{6} e \mathrm {sgn}\left (-\frac {b e}{x e + d} + \frac {b d e}{{\left (x e + d\right )}^{2}} - \frac {a e^{2}}{{\left (x e + d\right )}^{2}}\right ) - 6 \, a b^{5} d^{5} e^{2} \mathrm {sgn}\left (-\frac {b e}{x e + d} + \frac {b d e}{{\left (x e + d\right )}^{2}} - \frac {a e^{2}}{{\left (x e + d\right )}^{2}}\right ) + 15 \, a^{2} b^{4} d^{4} e^{3} \mathrm {sgn}\left (-\frac {b e}{x e + d} + \frac {b d e}{{\left (x e + d\right )}^{2}} - \frac {a e^{2}}{{\left (x e + d\right )}^{2}}\right ) - 20 \, a^{3} b^{3} d^{3} e^{4} \mathrm {sgn}\left (-\frac {b e}{x e + d} + \frac {b d e}{{\left (x e + d\right )}^{2}} - \frac {a e^{2}}{{\left (x e + d\right )}^{2}}\right ) + 15 \, a^{4} b^{2} d^{2} e^{5} \mathrm {sgn}\left (-\frac {b e}{x e + d} + \frac {b d e}{{\left (x e + d\right )}^{2}} - \frac {a e^{2}}{{\left (x e + d\right )}^{2}}\right ) - 6 \, a^{5} b d e^{6} \mathrm {sgn}\left (-\frac {b e}{x e + d} + \frac {b d e}{{\left (x e + d\right )}^{2}} - \frac {a e^{2}}{{\left (x e + d\right )}^{2}}\right ) + a^{6} e^{7} \mathrm {sgn}\left (-\frac {b e}{x e + d} + \frac {b d e}{{\left (x e + d\right )}^{2}} - \frac {a e^{2}}{{\left (x e + d\right )}^{2}}\right )} - \frac {e^{9}}{{\left (b^{5} d^{5} e^{5} \mathrm {sgn}\left (-\frac {b e}{x e + d} + \frac {b d e}{{\left (x e + d\right )}^{2}} - \frac {a e^{2}}{{\left (x e + d\right )}^{2}}\right ) - 5 \, a b^{4} d^{4} e^{6} \mathrm {sgn}\left (-\frac {b e}{x e + d} + \frac {b d e}{{\left (x e + d\right )}^{2}} - \frac {a e^{2}}{{\left (x e + d\right )}^{2}}\right ) + 10 \, a^{2} b^{3} d^{3} e^{7} \mathrm {sgn}\left (-\frac {b e}{x e + d} + \frac {b d e}{{\left (x e + d\right )}^{2}} - \frac {a e^{2}}{{\left (x e + d\right )}^{2}}\right ) - 10 \, a^{3} b^{2} d^{2} e^{8} \mathrm {sgn}\left (-\frac {b e}{x e + d} + \frac {b d e}{{\left (x e + d\right )}^{2}} - \frac {a e^{2}}{{\left (x e + d\right )}^{2}}\right ) + 5 \, a^{4} b d e^{9} \mathrm {sgn}\left (-\frac {b e}{x e + d} + \frac {b d e}{{\left (x e + d\right )}^{2}} - \frac {a e^{2}}{{\left (x e + d\right )}^{2}}\right ) - a^{5} e^{10} \mathrm {sgn}\left (-\frac {b e}{x e + d} + \frac {b d e}{{\left (x e + d\right )}^{2}} - \frac {a e^{2}}{{\left (x e + d\right )}^{2}}\right )\right )} {\left (x e + d\right )}} - \frac {77 \, b^{5} e^{4} - \frac {260 \, {\left (b^{5} d e^{5} - a b^{4} e^{6}\right )} e^{\left (-1\right )}}{x e + d} + \frac {300 \, {\left (b^{5} d^{2} e^{6} - 2 \, a b^{4} d e^{7} + a^{2} b^{3} e^{8}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} - \frac {120 \, {\left (b^{5} d^{3} e^{7} - 3 \, a b^{4} d^{2} e^{8} + 3 \, a^{2} b^{3} d e^{9} - a^{3} b^{2} e^{10}\right )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{3}}}{12 \, {\left (b d - a e\right )}^{6} {\left (b - \frac {b d}{x e + d} + \frac {a e}{x e + d}\right )}^{4} \mathrm {sgn}\left (-\frac {b e}{x e + d} + \frac {b d e}{{\left (x e + d\right )}^{2}} - \frac {a e^{2}}{{\left (x e + d\right )}^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5*b*e^5*log(abs(-b + b*d/(x*e + d) - a*e/(x*e + d)))/(b^6*d^6*e*sgn(-b*e/(x*e + d) + b*d*e/(x*e + d)^2 - a*e^
2/(x*e + d)^2) - 6*a*b^5*d^5*e^2*sgn(-b*e/(x*e + d) + b*d*e/(x*e + d)^2 - a*e^2/(x*e + d)^2) + 15*a^2*b^4*d^4*
e^3*sgn(-b*e/(x*e + d) + b*d*e/(x*e + d)^2 - a*e^2/(x*e + d)^2) - 20*a^3*b^3*d^3*e^4*sgn(-b*e/(x*e + d) + b*d*
e/(x*e + d)^2 - a*e^2/(x*e + d)^2) + 15*a^4*b^2*d^2*e^5*sgn(-b*e/(x*e + d) + b*d*e/(x*e + d)^2 - a*e^2/(x*e +
d)^2) - 6*a^5*b*d*e^6*sgn(-b*e/(x*e + d) + b*d*e/(x*e + d)^2 - a*e^2/(x*e + d)^2) + a^6*e^7*sgn(-b*e/(x*e + d)
+ b*d*e/(x*e + d)^2 - a*e^2/(x*e + d)^2)) - e^9/((b^5*d^5*e^5*sgn(-b*e/(x*e + d) + b*d*e/(x*e + d)^2 - a*e^2/
(x*e + d)^2) - 5*a*b^4*d^4*e^6*sgn(-b*e/(x*e + d) + b*d*e/(x*e + d)^2 - a*e^2/(x*e + d)^2) + 10*a^2*b^3*d^3*e^
7*sgn(-b*e/(x*e + d) + b*d*e/(x*e + d)^2 - a*e^2/(x*e + d)^2) - 10*a^3*b^2*d^2*e^8*sgn(-b*e/(x*e + d) + b*d*e/
(x*e + d)^2 - a*e^2/(x*e + d)^2) + 5*a^4*b*d*e^9*sgn(-b*e/(x*e + d) + b*d*e/(x*e + d)^2 - a*e^2/(x*e + d)^2) -
a^5*e^10*sgn(-b*e/(x*e + d) + b*d*e/(x*e + d)^2 - a*e^2/(x*e + d)^2))*(x*e + d)) - 1/12*(77*b^5*e^4 - 260*(b^
5*d*e^5 - a*b^4*e^6)*e^(-1)/(x*e + d) + 300*(b^5*d^2*e^6 - 2*a*b^4*d*e^7 + a^2*b^3*e^8)*e^(-2)/(x*e + d)^2 - 1
20*(b^5*d^3*e^7 - 3*a*b^4*d^2*e^8 + 3*a^2*b^3*d*e^9 - a^3*b^2*e^10)*e^(-3)/(x*e + d)^3)/((b*d - a*e)^6*(b - b*
d/(x*e + d) + a*e/(x*e + d))^4*sgn(-b*e/(x*e + d) + b*d*e/(x*e + d)^2 - a*e^2/(x*e + d)^2))

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maple [B]  time = 0.07, size = 651, normalized size = 2.12 \[ \frac {\left (60 b^{5} e^{5} x^{5} \ln \left (b x +a \right )-60 b^{5} e^{5} x^{5} \ln \left (e x +d \right )+240 a \,b^{4} e^{5} x^{4} \ln \left (b x +a \right )-240 a \,b^{4} e^{5} x^{4} \ln \left (e x +d \right )+60 b^{5} d \,e^{4} x^{4} \ln \left (b x +a \right )-60 b^{5} d \,e^{4} x^{4} \ln \left (e x +d \right )+360 a^{2} b^{3} e^{5} x^{3} \ln \left (b x +a \right )-360 a^{2} b^{3} e^{5} x^{3} \ln \left (e x +d \right )+240 a \,b^{4} d \,e^{4} x^{3} \ln \left (b x +a \right )-240 a \,b^{4} d \,e^{4} x^{3} \ln \left (e x +d \right )-60 a \,b^{4} e^{5} x^{4}+60 b^{5} d \,e^{4} x^{4}+240 a^{3} b^{2} e^{5} x^{2} \ln \left (b x +a \right )-240 a^{3} b^{2} e^{5} x^{2} \ln \left (e x +d \right )+360 a^{2} b^{3} d \,e^{4} x^{2} \ln \left (b x +a \right )-360 a^{2} b^{3} d \,e^{4} x^{2} \ln \left (e x +d \right )-210 a^{2} b^{3} e^{5} x^{3}+180 a \,b^{4} d \,e^{4} x^{3}+30 b^{5} d^{2} e^{3} x^{3}+60 a^{4} b \,e^{5} x \ln \left (b x +a \right )-60 a^{4} b \,e^{5} x \ln \left (e x +d \right )+240 a^{3} b^{2} d \,e^{4} x \ln \left (b x +a \right )-240 a^{3} b^{2} d \,e^{4} x \ln \left (e x +d \right )-260 a^{3} b^{2} e^{5} x^{2}+150 a^{2} b^{3} d \,e^{4} x^{2}+120 a \,b^{4} d^{2} e^{3} x^{2}-10 b^{5} d^{3} e^{2} x^{2}+60 a^{4} b d \,e^{4} \ln \left (b x +a \right )-60 a^{4} b d \,e^{4} \ln \left (e x +d \right )-125 a^{4} b \,e^{5} x -20 a^{3} b^{2} d \,e^{4} x +180 a^{2} b^{3} d^{2} e^{3} x -40 a \,b^{4} d^{3} e^{2} x +5 b^{5} d^{4} e x -12 a^{5} e^{5}-65 a^{4} b d \,e^{4}+120 a^{3} b^{2} d^{2} e^{3}-60 a^{2} b^{3} d^{3} e^{2}+20 a \,b^{4} d^{4} e -3 b^{5} d^{5}\right ) \left (b x +a \right )}{12 \left (e x +d \right ) \left (a e -b d \right )^{6} \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(-240*ln(e*x+d)*x*a^3*b^2*d*e^4-65*a^4*b*d*e^4+120*a^3*b^2*d^2*e^3-60*a^2*b^3*d^3*e^2+20*a*b^4*d^4*e-240*
a*b^4*d*e^4*x^3*ln(e*x+d)+240*a^3*b^2*d*e^4*x*ln(b*x+a)+240*a*b^4*d*e^4*x^3*ln(b*x+a)-360*a^2*b^3*d*e^4*x^2*ln
(e*x+d)+360*a^2*b^3*d*e^4*x^2*ln(b*x+a)-12*a^5*e^5-60*b^5*d*e^4*x^4*ln(e*x+d)-360*a^2*b^3*e^5*x^3*ln(e*x+d)+60
*ln(b*x+a)*x^5*b^5*e^5-3*b^5*d^5-60*b^5*e^5*x^5*ln(e*x+d)-260*a^3*b^2*e^5*x^2-10*b^5*d^3*e^2*x^2-125*a^4*b*e^5
*x+5*b^5*d^4*e*x-60*a*b^4*e^5*x^4+60*b^5*d*e^4*x^4-210*a^2*b^3*e^5*x^3+30*b^5*d^2*e^3*x^3-240*a^3*b^2*e^5*x^2*
ln(e*x+d)-240*a*b^4*e^5*x^4*ln(e*x+d)-20*a^3*b^2*d*e^4*x+180*a^2*b^3*d^2*e^3*x-40*a*b^4*d^3*e^2*x+150*a^2*b^3*
d*e^4*x^2+120*a*b^4*d^2*e^3*x^2-60*a^4*b*d*e^4*ln(e*x+d)+180*a*b^4*d*e^4*x^3+60*b^5*d*e^4*x^4*ln(b*x+a)-60*ln(
e*x+d)*x*a^4*b*e^5+360*a^2*b^3*e^5*x^3*ln(b*x+a)+240*a^3*b^2*e^5*x^2*ln(b*x+a)+240*a*b^4*e^5*x^4*ln(b*x+a)+60*
a^4*b*e^5*x*ln(b*x+a)+60*a^4*b*d*e^4*ln(b*x+a))*(b*x+a)/(e*x+d)/(a*e-b*d)^6/((b*x+a)^2)^(5/2)

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^2/(b^2*x^2+2*a*b*x+a^2)^(5/2),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 details)Is a*e-b*d zero or nonzero?

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (d+e\,x\right )}^2\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (d + e x\right )^{2} \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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