3.17.12 \(\int \frac {1}{(d+e x)^2 (a^2+2 a b x+b^2 x^2)^{5/2}} \, dx\) [1612]

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

[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 = 0.15, 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 = {660, 46} \begin {gather*} \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} \end {gather*}

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 46

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] && Lt
Q[m + n + 2, 0])

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*} \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 = 0.08, size = 167, normalized size = 0.54 \begin {gather*} \frac {8 b e (b d-a e)^3-\frac {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+\frac {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 \left ((a+b x)^2\right )^{3/2}} \end {gather*}

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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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(650\) vs. \(2(224)=448\).
time = 0.71, size = 651, normalized size = 2.12

method	result	size
default	\(\frac {\left (180 a \,b^{4} d \,e^{4} x^{3}-40 a \,b^{4} d^{3} e^{2} x -240 \ln \left (e x +d \right ) a \,b^{4} d \,e^{4} x^{3}-60 \ln \left (e x +d \right ) a^{4} b \,e^{5} x -240 \ln \left (e x +d \right ) a^{3} b^{2} e^{5} x^{2}+60 \ln \left (b x +a \right ) a^{4} b d \,e^{4}+240 \ln \left (b x +a \right ) a^{3} b^{2} e^{5} x^{2}+60 \ln \left (b x +a \right ) a^{4} b \,e^{5} x -3 b^{5} d^{5}-60 \ln \left (e x +d \right ) a^{4} b d \,e^{4}+150 a^{2} b^{3} d \,e^{4} x^{2}+120 a \,b^{4} d^{2} e^{3} x^{2}-20 a^{3} b^{2} d \,e^{4} x +180 a^{2} b^{3} d^{2} e^{3} x -12 a^{5} e^{5}-360 \ln \left (e x +d \right ) a^{2} b^{3} d \,e^{4} x^{2}-240 \ln \left (e x +d \right ) a^{3} b^{2} d \,e^{4} x +240 \ln \left (b x +a \right ) a \,b^{4} d \,e^{4} x^{3}+360 \ln \left (b x +a \right ) a^{2} b^{3} d \,e^{4} x^{2}+240 \ln \left (b x +a \right ) a^{3} b^{2} d \,e^{4} x -240 \ln \left (e x +d \right ) a \,b^{4} e^{5} x^{4}-60 \ln \left (e x +d \right ) b^{5} d \,e^{4} x^{4}+360 \ln \left (b x +a \right ) a^{2} b^{3} e^{5} x^{3}+240 \ln \left (b x +a \right ) a \,b^{4} e^{5} x^{4}+60 \ln \left (b x +a \right ) b^{5} d \,e^{4} x^{4}-60 a \,b^{4} e^{5} x^{4}+60 b^{5} d \,e^{4} x^{4}-210 a^{2} b^{3} e^{5} x^{3}+30 b^{5} d^{2} e^{3} x^{3}-260 a^{3} b^{2} e^{5} x^{2}-10 b^{5} d^{3} e^{2} x^{2}-125 a^{4} b \,e^{5} x +5 b^{5} d^{4} e x -360 \ln \left (e x +d \right ) a^{2} b^{3} e^{5} x^{3}+60 \ln \left (b x +a \right ) b^{5} e^{5} x^{5}-60 \ln \left (e x +d \right ) b^{5} e^{5} x^{5}-65 a^{4} b d \,e^{4}+120 a^{3} b^{2} d^{2} e^{3}-60 a^{2} b^{3} d^{3} e^{2}+20 a \,b^{4} d^{4} e \right ) \left (b x +a \right )}{12 \left (e x +d \right ) \left (a e -b d \right )^{6} \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}\)	\(651\)
risch	\(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {5 b^{4} e^{4} x^{4}}{a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}}-\frac {5 b^{3} \left (7 a e +b d \right ) e^{3} x^{3}}{2 \left (a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}-\frac {5 b^{2} e^{2} \left (26 a^{2} e^{2}+11 a b d e -b^{2} d^{2}\right ) x^{2}}{6 \left (a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}-\frac {5 \left (25 e^{3} a^{3}+29 a^{2} b d \,e^{2}-7 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) b e x}{12 \left (a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}-\frac {12 e^{4} a^{4}+77 a^{3} b d \,e^{3}-43 a^{2} b^{2} d^{2} e^{2}+17 a \,b^{3} d^{3} e -3 b^{4} d^{4}}{12 \left (a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}\right )}{\left (b x +a \right )^{5} \left (e x +d \right )}+\frac {5 \sqrt {\left (b x +a \right )^{2}}\, e^{4} b \ln \left (-b x -a \right )}{\left (b x +a \right ) \left (a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 b^{2} d^{2} e^{4} a^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 d^{4} b^{4} e^{2} a^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}\right )}-\frac {5 \sqrt {\left (b x +a \right )^{2}}\, e^{4} b \ln \left (e x +d \right )}{\left (b x +a \right ) \left (a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 b^{2} d^{2} e^{4} a^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 d^{4} b^{4} e^{2} a^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}\right )}\)	\(732\)

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,method=_RETURNVERBOSE)

[Out]

1/12*(180*a*b^4*d*e^4*x^3-40*a*b^4*d^3*e^2*x-240*ln(e*x+d)*a*b^4*d*e^4*x^3-60*ln(e*x+d)*a^4*b*e^5*x-240*ln(e*x
+d)*a^3*b^2*e^5*x^2+60*ln(b*x+a)*a^4*b*d*e^4+240*ln(b*x+a)*a^3*b^2*e^5*x^2+60*ln(b*x+a)*a^4*b*e^5*x-3*b^5*d^5-
60*ln(e*x+d)*a^4*b*d*e^4+150*a^2*b^3*d*e^4*x^2+120*a*b^4*d^2*e^3*x^2-20*a^3*b^2*d*e^4*x+180*a^2*b^3*d^2*e^3*x-
12*a^5*e^5-360*ln(e*x+d)*a^2*b^3*d*e^4*x^2-240*ln(e*x+d)*a^3*b^2*d*e^4*x+240*ln(b*x+a)*a*b^4*d*e^4*x^3+360*ln(
b*x+a)*a^2*b^3*d*e^4*x^2+240*ln(b*x+a)*a^3*b^2*d*e^4*x-240*ln(e*x+d)*a*b^4*e^5*x^4-60*ln(e*x+d)*b^5*d*e^4*x^4+
360*ln(b*x+a)*a^2*b^3*e^5*x^3+240*ln(b*x+a)*a*b^4*e^5*x^4+60*ln(b*x+a)*b^5*d*e^4*x^4-60*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^3-260*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-360*ln(e*x+d)*a^2*b^3*e^5*x^3+60*ln(b*x+a)*b^5*e^5*x^5-60*ln(e*x+d)*b^5*e^5*x^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)*(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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

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(b*d-%e*a>0)', see `assume?` fo
r more detai

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1021 vs. \(2 (229) = 458\).
time = 2.36, size = 1021, normalized size = 3.33 \begin {gather*} -\frac {3 \, b^{5} d^{5} + {\left (60 \, a b^{4} x^{4} + 210 \, a^{2} b^{3} x^{3} + 260 \, a^{3} b^{2} x^{2} + 125 \, a^{4} b x + 12 \, a^{5}\right )} e^{5} - 5 \, {\left (12 \, b^{5} d x^{4} + 36 \, a b^{4} d x^{3} + 30 \, a^{2} b^{3} d x^{2} - 4 \, a^{3} b^{2} d x - 13 \, a^{4} b d\right )} e^{4} - 30 \, {\left (b^{5} d^{2} x^{3} + 4 \, a b^{4} d^{2} x^{2} + 6 \, a^{2} b^{3} d^{2} x + 4 \, a^{3} b^{2} d^{2}\right )} e^{3} + 10 \, {\left (b^{5} d^{3} x^{2} + 4 \, a b^{4} d^{3} x + 6 \, a^{2} b^{3} d^{3}\right )} e^{2} - 5 \, {\left (b^{5} d^{4} x + 4 \, a b^{4} d^{4}\right )} e - 60 \, {\left ({\left (b^{5} x^{5} + 4 \, a b^{4} x^{4} + 6 \, a^{2} b^{3} x^{3} + 4 \, a^{3} b^{2} x^{2} + a^{4} b x\right )} e^{5} + {\left (b^{5} d x^{4} + 4 \, a b^{4} d x^{3} + 6 \, a^{2} b^{3} d x^{2} + 4 \, a^{3} b^{2} d x + a^{4} b d\right )} e^{4}\right )} \log \left (b x + a\right ) + 60 \, {\left ({\left (b^{5} x^{5} + 4 \, a b^{4} x^{4} + 6 \, a^{2} b^{3} x^{3} + 4 \, a^{3} b^{2} x^{2} + a^{4} b x\right )} e^{5} + {\left (b^{5} d x^{4} + 4 \, a b^{4} d x^{3} + 6 \, a^{2} b^{3} d x^{2} + 4 \, a^{3} b^{2} d x + a^{4} b d\right )} e^{4}\right )} \log \left (x e + d\right )}{12 \, {\left (b^{10} d^{7} x^{4} + 4 \, a b^{9} d^{7} x^{3} + 6 \, a^{2} b^{8} d^{7} x^{2} + 4 \, a^{3} b^{7} d^{7} x + a^{4} b^{6} d^{7} + {\left (a^{6} b^{4} x^{5} + 4 \, a^{7} b^{3} x^{4} + 6 \, a^{8} b^{2} x^{3} + 4 \, a^{9} b x^{2} + a^{10} x\right )} e^{7} - {\left (6 \, a^{5} b^{5} d x^{5} + 23 \, a^{6} b^{4} d x^{4} + 32 \, a^{7} b^{3} d x^{3} + 18 \, a^{8} b^{2} d x^{2} + 2 \, a^{9} b d x - a^{10} d\right )} e^{6} + 3 \, {\left (5 \, a^{4} b^{6} d^{2} x^{5} + 18 \, a^{5} b^{5} d^{2} x^{4} + 22 \, a^{6} b^{4} d^{2} x^{3} + 8 \, a^{7} b^{3} d^{2} x^{2} - 3 \, a^{8} b^{2} d^{2} x - 2 \, a^{9} b d^{2}\right )} e^{5} - 5 \, {\left (4 \, a^{3} b^{7} d^{3} x^{5} + 13 \, a^{4} b^{6} d^{3} x^{4} + 12 \, a^{5} b^{5} d^{3} x^{3} - 2 \, a^{6} b^{4} d^{3} x^{2} - 8 \, a^{7} b^{3} d^{3} x - 3 \, a^{8} b^{2} d^{3}\right )} e^{4} + 5 \, {\left (3 \, a^{2} b^{8} d^{4} x^{5} + 8 \, a^{3} b^{7} d^{4} x^{4} + 2 \, a^{4} b^{6} d^{4} x^{3} - 12 \, a^{5} b^{5} d^{4} x^{2} - 13 \, a^{6} b^{4} d^{4} x - 4 \, a^{7} b^{3} d^{4}\right )} e^{3} - 3 \, {\left (2 \, a b^{9} d^{5} x^{5} + 3 \, a^{2} b^{8} d^{5} x^{4} - 8 \, a^{3} b^{7} d^{5} x^{3} - 22 \, a^{4} b^{6} d^{5} x^{2} - 18 \, a^{5} b^{5} d^{5} x - 5 \, a^{6} b^{4} d^{5}\right )} e^{2} + {\left (b^{10} d^{6} x^{5} - 2 \, a b^{9} d^{6} x^{4} - 18 \, a^{2} b^{8} d^{6} x^{3} - 32 \, a^{3} b^{7} d^{6} x^{2} - 23 \, a^{4} b^{6} d^{6} x - 6 \, a^{5} b^{5} d^{6}\right )} e\right )}} \end {gather*}

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 + (60*a*b^4*x^4 + 210*a^2*b^3*x^3 + 260*a^3*b^2*x^2 + 125*a^4*b*x + 12*a^5)*e^5 - 5*(12*b^5*d
*x^4 + 36*a*b^4*d*x^3 + 30*a^2*b^3*d*x^2 - 4*a^3*b^2*d*x - 13*a^4*b*d)*e^4 - 30*(b^5*d^2*x^3 + 4*a*b^4*d^2*x^2
+ 6*a^2*b^3*d^2*x + 4*a^3*b^2*d^2)*e^3 + 10*(b^5*d^3*x^2 + 4*a*b^4*d^3*x + 6*a^2*b^3*d^3)*e^2 - 5*(b^5*d^4*x
+ 4*a*b^4*d^4)*e - 60*((b^5*x^5 + 4*a*b^4*x^4 + 6*a^2*b^3*x^3 + 4*a^3*b^2*x^2 + a^4*b*x)*e^5 + (b^5*d*x^4 + 4*
a*b^4*d*x^3 + 6*a^2*b^3*d*x^2 + 4*a^3*b^2*d*x + a^4*b*d)*e^4)*log(b*x + a) + 60*((b^5*x^5 + 4*a*b^4*x^4 + 6*a^
2*b^3*x^3 + 4*a^3*b^2*x^2 + a^4*b*x)*e^5 + (b^5*d*x^4 + 4*a*b^4*d*x^3 + 6*a^2*b^3*d*x^2 + 4*a^3*b^2*d*x + a^4*
b*d)*e^4)*log(x*e + d))/(b^10*d^7*x^4 + 4*a*b^9*d^7*x^3 + 6*a^2*b^8*d^7*x^2 + 4*a^3*b^7*d^7*x + a^4*b^6*d^7 +
(a^6*b^4*x^5 + 4*a^7*b^3*x^4 + 6*a^8*b^2*x^3 + 4*a^9*b*x^2 + a^10*x)*e^7 - (6*a^5*b^5*d*x^5 + 23*a^6*b^4*d*x^4
+ 32*a^7*b^3*d*x^3 + 18*a^8*b^2*d*x^2 + 2*a^9*b*d*x - a^10*d)*e^6 + 3*(5*a^4*b^6*d^2*x^5 + 18*a^5*b^5*d^2*x^4
+ 22*a^6*b^4*d^2*x^3 + 8*a^7*b^3*d^2*x^2 - 3*a^8*b^2*d^2*x - 2*a^9*b*d^2)*e^5 - 5*(4*a^3*b^7*d^3*x^5 + 13*a^4
*b^6*d^3*x^4 + 12*a^5*b^5*d^3*x^3 - 2*a^6*b^4*d^3*x^2 - 8*a^7*b^3*d^3*x - 3*a^8*b^2*d^3)*e^4 + 5*(3*a^2*b^8*d^
4*x^5 + 8*a^3*b^7*d^4*x^4 + 2*a^4*b^6*d^4*x^3 - 12*a^5*b^5*d^4*x^2 - 13*a^6*b^4*d^4*x - 4*a^7*b^3*d^4)*e^3 - 3
*(2*a*b^9*d^5*x^5 + 3*a^2*b^8*d^5*x^4 - 8*a^3*b^7*d^5*x^3 - 22*a^4*b^6*d^5*x^2 - 18*a^5*b^5*d^5*x - 5*a^6*b^4*
d^5)*e^2 + (b^10*d^6*x^5 - 2*a*b^9*d^6*x^4 - 18*a^2*b^8*d^6*x^3 - 32*a^3*b^7*d^6*x^2 - 23*a^4*b^6*d^6*x - 6*a^
5*b^5*d^6)*e)

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

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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 523 vs. \(2 (229) = 458\).
time = 0.71, size = 523, normalized size = 1.70 \begin {gather*} \frac {5 \, b^{2} e^{4} \log \left ({\left | b x + a \right |}\right )}{b^{7} d^{6} \mathrm {sgn}\left (b x + a\right ) - 6 \, a b^{6} d^{5} e \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{2} b^{5} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) - 20 \, a^{3} b^{4} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{4} b^{3} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) - 6 \, a^{5} b^{2} d e^{5} \mathrm {sgn}\left (b x + a\right ) + a^{6} b e^{6} \mathrm {sgn}\left (b x + a\right )} - \frac {5 \, b e^{5} \log \left ({\left | x e + d \right |}\right )}{b^{6} d^{6} e \mathrm {sgn}\left (b x + a\right ) - 6 \, a b^{5} d^{5} e^{2} \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{2} b^{4} d^{4} e^{3} \mathrm {sgn}\left (b x + a\right ) - 20 \, a^{3} b^{3} d^{3} e^{4} \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{4} b^{2} d^{2} e^{5} \mathrm {sgn}\left (b x + a\right ) - 6 \, a^{5} b d e^{6} \mathrm {sgn}\left (b x + a\right ) + a^{6} e^{7} \mathrm {sgn}\left (b x + a\right )} - \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}{12 \, {\left (b d - a e\right )}^{6} {\left (b x + a\right )}^{4} {\left (x e + d\right )} \mathrm {sgn}\left (b x + a\right )} \end {gather*}

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^2*e^4*log(abs(b*x + a))/(b^7*d^6*sgn(b*x + a) - 6*a*b^6*d^5*e*sgn(b*x + a) + 15*a^2*b^5*d^4*e^2*sgn(b*x +
a) - 20*a^3*b^4*d^3*e^3*sgn(b*x + a) + 15*a^4*b^3*d^2*e^4*sgn(b*x + a) - 6*a^5*b^2*d*e^5*sgn(b*x + a) + a^6*b*
e^6*sgn(b*x + a)) - 5*b*e^5*log(abs(x*e + d))/(b^6*d^6*e*sgn(b*x + a) - 6*a*b^5*d^5*e^2*sgn(b*x + a) + 15*a^2*
b^4*d^4*e^3*sgn(b*x + a) - 20*a^3*b^3*d^3*e^4*sgn(b*x + a) + 15*a^4*b^2*d^2*e^5*sgn(b*x + a) - 6*a^5*b*d*e^6*s
gn(b*x + a) + a^6*e^7*sgn(b*x + a)) - 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 - 60*(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)/((b*d - a*e)^6*(b*x + a)^4*(x*e + d)*sg
n(b*x + a))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 \end {gather*}

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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