3.17.13 \(\int \frac {1}{(d+e x)^3 (a^2+2 a b x+b^2 x^2)^{5/2}} \, dx\) [1613]

Optimal. Leaf size=365 \[ \frac {10 b^2 e^3}{(b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b^2}{4 (b d-a e)^3 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b^2 e}{(b d-a e)^4 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 b^2 e^2}{(b d-a e)^5 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e^4 (a+b x)}{2 (b d-a e)^5 (d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 b e^4 (a+b x)}{(b d-a e)^6 (d+e x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {15 b^2 e^4 (a+b x) \log (a+b x)}{(b d-a e)^7 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {15 b^2 e^4 (a+b x) \log (d+e x)}{(b d-a e)^7 \sqrt {a^2+2 a b x+b^2 x^2}} \]

[Out]

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

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Rubi [A]
time = 0.19, antiderivative size = 365, 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 {5 b e^4 (a+b x)}{\sqrt {a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^6}+\frac {e^4 (a+b x)}{2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^5}+\frac {15 b^2 e^4 (a+b x) \log (a+b x)}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^7}-\frac {15 b^2 e^4 (a+b x) \log (d+e x)}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^7}+\frac {10 b^2 e^3}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}-\frac {3 b^2 e^2}{(a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}+\frac {b^2 e}{(a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}-\frac {b^2}{4 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]

[Out]

(10*b^2*e^3)/((b*d - a*e)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - b^2/(4*(b*d - a*e)^3*(a + b*x)^3*Sqrt[a^2 + 2*a*b
*x + b^2*x^2]) + (b^2*e)/((b*d - a*e)^4*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (3*b^2*e^2)/((b*d - a*e)^
5*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (e^4*(a + b*x))/(2*(b*d - a*e)^5*(d + e*x)^2*Sqrt[a^2 + 2*a*b*x +
b^2*x^2]) + (5*b*e^4*(a + b*x))/((b*d - a*e)^6*(d + e*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (15*b^2*e^4*(a + b*
x)*Log[a + b*x])/((b*d - a*e)^7*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (15*b^2*e^4*(a + b*x)*Log[d + e*x])/((b*d - a
*e)^7*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)^3 \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)^3} \, 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^2 (b d-a e)^3 (a+b x)^5}-\frac {3 e}{b^2 (b d-a e)^4 (a+b x)^4}+\frac {6 e^2}{b^2 (b d-a e)^5 (a+b x)^3}-\frac {10 e^3}{b^2 (b d-a e)^6 (a+b x)^2}+\frac {15 e^4}{b^2 (b d-a e)^7 (a+b x)}-\frac {e^5}{b^5 (b d-a e)^5 (d+e x)^3}-\frac {5 e^5}{b^4 (b d-a e)^6 (d+e x)^2}-\frac {15 e^5}{b^3 (b d-a e)^7 (d+e x)}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {10 b^2 e^3}{(b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b^2}{4 (b d-a e)^3 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b^2 e}{(b d-a e)^4 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 b^2 e^2}{(b d-a e)^5 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e^4 (a+b x)}{2 (b d-a e)^5 (d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 b e^4 (a+b x)}{(b d-a e)^6 (d+e x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {15 b^2 e^4 (a+b x) \log (a+b x)}{(b d-a e)^7 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {15 b^2 e^4 (a+b x) \log (d+e x)}{(b d-a e)^7 \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.10, size = 209, normalized size = 0.57 \begin {gather*} \frac {4 b^2 e (b d-a e)^3-\frac {b^2 (b d-a e)^4}{a+b x}-12 b^2 e^2 (b d-a e)^2 (a+b x)+40 b^2 e^3 (b d-a e) (a+b x)^2+\frac {2 e^4 (b d-a e)^2 (a+b x)^3}{(d+e x)^2}+\frac {20 b e^4 (b d-a e) (a+b x)^3}{d+e x}+60 b^2 e^4 (a+b x)^3 \log (a+b x)-60 b^2 e^4 (a+b x)^3 \log (d+e x)}{4 (b d-a e)^7 \left ((a+b x)^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]

[Out]

(4*b^2*e*(b*d - a*e)^3 - (b^2*(b*d - a*e)^4)/(a + b*x) - 12*b^2*e^2*(b*d - a*e)^2*(a + b*x) + 40*b^2*e^3*(b*d
- a*e)*(a + b*x)^2 + (2*e^4*(b*d - a*e)^2*(a + b*x)^3)/(d + e*x)^2 + (20*b*e^4*(b*d - a*e)*(a + b*x)^3)/(d + e
*x) + 60*b^2*e^4*(a + b*x)^3*Log[a + b*x] - 60*b^2*e^4*(a + b*x)^3*Log[d + e*x])/(4*(b*d - a*e)^7*((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. \(982\) vs. \(2(273)=546\).
time = 0.85, size = 983, normalized size = 2.69

method	result	size
default	\(-\frac {\left (-b^{6} d^{6}+80 a^{3} b^{3} d^{3} e^{3}-360 \ln \left (e x +d \right ) a^{2} b^{4} e^{6} x^{4}-60 \ln \left (e x +d \right ) b^{6} d^{2} e^{4} x^{4}-240 \ln \left (e x +d \right ) a^{3} b^{3} e^{6} x^{3}-60 \ln \left (e x +d \right ) a^{4} b^{2} e^{6} x^{2}+240 \ln \left (b x +a \right ) a \,b^{5} e^{6} x^{5}+120 \ln \left (b x +a \right ) b^{6} d \,e^{5} x^{5}-240 \ln \left (e x +d \right ) a \,b^{5} e^{6} x^{5}+120 a \,b^{5} d \,e^{5} x^{4}-190 a^{4} b^{2} d \,e^{5} x -5 b^{6} d^{4} e^{2} x^{2}+2 a^{6} e^{6}-60 a^{2} b^{4} d \,e^{5} x^{3}-280 a^{3} b^{3} d \,e^{5} x^{2}-60 \ln \left (e x +d \right ) a^{4} b^{2} d^{2} e^{4}-30 d^{4} b^{4} e^{2} a^{2}+60 \ln \left (b x +a \right ) a^{4} b^{2} e^{6} x^{2}+60 \ln \left (b x +a \right ) a^{4} b^{2} d^{2} e^{4}+8 a \,b^{5} d^{5} e -125 a^{4} b^{2} e^{6} x^{2}-24 a^{5} b d \,e^{5}-210 a^{2} b^{4} e^{6} x^{4}-260 a^{3} b^{3} e^{6} x^{3}+20 b^{6} d^{3} e^{3} x^{3}-35 b^{2} d^{2} e^{4} a^{4}+90 b^{6} d^{2} e^{4} x^{4}+60 \ln \left (b x +a \right ) b^{6} e^{6} x^{6}-60 \ln \left (e x +d \right ) b^{6} e^{6} x^{6}+360 \ln \left (b x +a \right ) a^{2} b^{4} e^{6} x^{4}+60 \ln \left (b x +a \right ) b^{6} d^{2} e^{4} x^{4}+240 \ln \left (b x +a \right ) a^{3} b^{3} e^{6} x^{3}-480 \ln \left (e x +d \right ) a \,b^{5} d \,e^{5} x^{4}+300 a \,b^{5} d^{2} e^{4} x^{3}+330 a^{2} b^{4} d^{2} e^{4} x^{2}+80 a \,b^{5} d^{3} e^{3} x^{2}-12 a^{5} b \,e^{6} x +2 b^{6} d^{5} e x +100 a^{3} b^{3} d^{2} e^{4} x +120 a^{2} b^{4} d^{3} e^{3} x -20 a \,b^{5} d^{4} e^{2} x -720 \ln \left (e x +d \right ) a^{2} b^{4} d \,e^{5} x^{3}-240 \ln \left (e x +d \right ) a \,b^{5} d^{2} e^{4} x^{3}-480 \ln \left (e x +d \right ) a^{3} b^{3} d \,e^{5} x^{2}-360 \ln \left (e x +d \right ) a^{2} b^{4} d^{2} e^{4} x^{2}-120 \ln \left (e x +d \right ) a^{4} b^{2} d \,e^{5} x -240 \ln \left (e x +d \right ) a^{3} b^{3} d^{2} e^{4} x +60 b^{6} d \,e^{5} x^{5}-60 a \,b^{5} e^{6} x^{5}+120 \ln \left (b x +a \right ) a^{4} b^{2} d \,e^{5} x +240 \ln \left (b x +a \right ) a^{3} b^{3} d^{2} e^{4} x -120 \ln \left (e x +d \right ) b^{6} d \,e^{5} x^{5}+480 \ln \left (b x +a \right ) a \,b^{5} d \,e^{5} x^{4}+720 \ln \left (b x +a \right ) a^{2} b^{4} d \,e^{5} x^{3}+240 \ln \left (b x +a \right ) a \,b^{5} d^{2} e^{4} x^{3}+480 \ln \left (b x +a \right ) a^{3} b^{3} d \,e^{5} x^{2}+360 \ln \left (b x +a \right ) a^{2} b^{4} d^{2} e^{4} x^{2}\right ) \left (b x +a \right )}{4 \left (e x +d \right )^{2} \left (a e -b d \right )^{7} \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}\)	\(983\)
risch	\(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (\frac {15 b^{5} e^{5} x^{5}}{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}}+\frac {15 e^{4} b^{4} \left (7 a e +3 b d \right ) x^{4}}{2 \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 b^{3} e^{3} \left (13 a^{2} e^{2}+16 a b d e +b^{2} d^{2}\right ) x^{3}}{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}}+\frac {5 b^{2} e^{2} \left (25 e^{3} a^{3}+81 a^{2} b d \,e^{2}+15 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) x^{2}}{4 \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 {\left (6 e^{4} a^{4}+101 a^{3} b d \,e^{3}+51 a^{2} b^{2} d^{2} e^{2}-9 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) b e x}{2 a^{6} e^{6}-12 a^{5} b d \,e^{5}+30 b^{2} d^{2} e^{4} a^{4}-40 a^{3} b^{3} d^{3} e^{3}+30 d^{4} b^{4} e^{2} a^{2}-12 a \,b^{5} d^{5} e +2 b^{6} d^{6}}-\frac {2 a^{5} e^{5}-22 a^{4} b d \,e^{4}-57 a^{3} b^{2} d^{2} e^{3}+23 a^{2} b^{3} d^{3} e^{2}-7 a \,b^{4} d^{4} e +b^{5} d^{5}}{4 \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 )}\right )}{\left (b x +a \right )^{5} \left (e x +d \right )^{2}}+\frac {15 \sqrt {\left (b x +a \right )^{2}}\, b^{2} e^{4} \ln \left (-e x -d \right )}{\left (b x +a \right ) \left (a^{7} e^{7}-7 a^{6} b d \,e^{6}+21 a^{5} b^{2} d^{2} e^{5}-35 a^{4} b^{3} d^{3} e^{4}+35 b^{4} d^{4} e^{3} a^{3}-21 a^{2} b^{5} d^{5} e^{2}+7 a \,b^{6} d^{6} e -b^{7} d^{7}\right )}-\frac {15 \sqrt {\left (b x +a \right )^{2}}\, b^{2} e^{4} \ln \left (b x +a \right )}{\left (b x +a \right ) \left (a^{7} e^{7}-7 a^{6} b d \,e^{6}+21 a^{5} b^{2} d^{2} e^{5}-35 a^{4} b^{3} d^{3} e^{4}+35 b^{4} d^{4} e^{3} a^{3}-21 a^{2} b^{5} d^{5} e^{2}+7 a \,b^{6} d^{6} e -b^{7} d^{7}\right )}\)	\(985\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^3/(b^2*x^2+2*a*b*x+a^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-1/4*(-b^6*d^6+80*a^3*b^3*d^3*e^3-360*ln(e*x+d)*a^2*b^4*e^6*x^4-60*ln(e*x+d)*b^6*d^2*e^4*x^4-240*ln(e*x+d)*a^3
*b^3*e^6*x^3-60*ln(e*x+d)*a^4*b^2*e^6*x^2+240*ln(b*x+a)*a*b^5*e^6*x^5+120*ln(b*x+a)*b^6*d*e^5*x^5-240*ln(e*x+d
)*a*b^5*e^6*x^5+120*a*b^5*d*e^5*x^4-190*a^4*b^2*d*e^5*x-5*b^6*d^4*e^2*x^2+2*a^6*e^6-60*a^2*b^4*d*e^5*x^3-280*a
^3*b^3*d*e^5*x^2-60*ln(e*x+d)*a^4*b^2*d^2*e^4-30*d^4*b^4*e^2*a^2+60*ln(b*x+a)*a^4*b^2*e^6*x^2+60*ln(b*x+a)*a^4
*b^2*d^2*e^4+8*a*b^5*d^5*e-125*a^4*b^2*e^6*x^2-24*a^5*b*d*e^5-210*a^2*b^4*e^6*x^4-260*a^3*b^3*e^6*x^3+20*b^6*d
^3*e^3*x^3-35*b^2*d^2*e^4*a^4+90*b^6*d^2*e^4*x^4+60*ln(b*x+a)*b^6*e^6*x^6-60*ln(e*x+d)*b^6*e^6*x^6+360*ln(b*x+
a)*a^2*b^4*e^6*x^4+60*ln(b*x+a)*b^6*d^2*e^4*x^4+240*ln(b*x+a)*a^3*b^3*e^6*x^3-480*ln(e*x+d)*a*b^5*d*e^5*x^4+30
0*a*b^5*d^2*e^4*x^3+330*a^2*b^4*d^2*e^4*x^2+80*a*b^5*d^3*e^3*x^2-12*a^5*b*e^6*x+2*b^6*d^5*e*x+100*a^3*b^3*d^2*
e^4*x+120*a^2*b^4*d^3*e^3*x-20*a*b^5*d^4*e^2*x-720*ln(e*x+d)*a^2*b^4*d*e^5*x^3-240*ln(e*x+d)*a*b^5*d^2*e^4*x^3
-480*ln(e*x+d)*a^3*b^3*d*e^5*x^2-360*ln(e*x+d)*a^2*b^4*d^2*e^4*x^2-120*ln(e*x+d)*a^4*b^2*d*e^5*x-240*ln(e*x+d)
*a^3*b^3*d^2*e^4*x+60*b^6*d*e^5*x^5-60*a*b^5*e^6*x^5+120*ln(b*x+a)*a^4*b^2*d*e^5*x+240*ln(b*x+a)*a^3*b^3*d^2*e
^4*x-120*ln(e*x+d)*b^6*d*e^5*x^5+480*ln(b*x+a)*a*b^5*d*e^5*x^4+720*ln(b*x+a)*a^2*b^4*d*e^5*x^3+240*ln(b*x+a)*a
*b^5*d^2*e^4*x^3+480*ln(b*x+a)*a^3*b^3*d*e^5*x^2+360*ln(b*x+a)*a^2*b^4*d^2*e^4*x^2)*(b*x+a)/(e*x+d)^2/(a*e-b*d
)^7/((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)^3/(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. 1494 vs. \(2 (279) = 558\).
time = 4.76, size = 1494, normalized size = 4.09 \begin {gather*} -\frac {b^{6} d^{6} + {\left (60 \, a b^{5} x^{5} + 210 \, a^{2} b^{4} x^{4} + 260 \, a^{3} b^{3} x^{3} + 125 \, a^{4} b^{2} x^{2} + 12 \, a^{5} b x - 2 \, a^{6}\right )} e^{6} - 2 \, {\left (30 \, b^{6} d x^{5} + 60 \, a b^{5} d x^{4} - 30 \, a^{2} b^{4} d x^{3} - 140 \, a^{3} b^{3} d x^{2} - 95 \, a^{4} b^{2} d x - 12 \, a^{5} b d\right )} e^{5} - 5 \, {\left (18 \, b^{6} d^{2} x^{4} + 60 \, a b^{5} d^{2} x^{3} + 66 \, a^{2} b^{4} d^{2} x^{2} + 20 \, a^{3} b^{3} d^{2} x - 7 \, a^{4} b^{2} d^{2}\right )} e^{4} - 20 \, {\left (b^{6} d^{3} x^{3} + 4 \, a b^{5} d^{3} x^{2} + 6 \, a^{2} b^{4} d^{3} x + 4 \, a^{3} b^{3} d^{3}\right )} e^{3} + 5 \, {\left (b^{6} d^{4} x^{2} + 4 \, a b^{5} d^{4} x + 6 \, a^{2} b^{4} d^{4}\right )} e^{2} - 2 \, {\left (b^{6} d^{5} x + 4 \, a b^{5} d^{5}\right )} e - 60 \, {\left ({\left (b^{6} x^{6} + 4 \, a b^{5} x^{5} + 6 \, a^{2} b^{4} x^{4} + 4 \, a^{3} b^{3} x^{3} + a^{4} b^{2} x^{2}\right )} e^{6} + 2 \, {\left (b^{6} d x^{5} + 4 \, a b^{5} d x^{4} + 6 \, a^{2} b^{4} d x^{3} + 4 \, a^{3} b^{3} d x^{2} + a^{4} b^{2} d x\right )} e^{5} + {\left (b^{6} d^{2} x^{4} + 4 \, a b^{5} d^{2} x^{3} + 6 \, a^{2} b^{4} d^{2} x^{2} + 4 \, a^{3} b^{3} d^{2} x + a^{4} b^{2} d^{2}\right )} e^{4}\right )} \log \left (b x + a\right ) + 60 \, {\left ({\left (b^{6} x^{6} + 4 \, a b^{5} x^{5} + 6 \, a^{2} b^{4} x^{4} + 4 \, a^{3} b^{3} x^{3} + a^{4} b^{2} x^{2}\right )} e^{6} + 2 \, {\left (b^{6} d x^{5} + 4 \, a b^{5} d x^{4} + 6 \, a^{2} b^{4} d x^{3} + 4 \, a^{3} b^{3} d x^{2} + a^{4} b^{2} d x\right )} e^{5} + {\left (b^{6} d^{2} x^{4} + 4 \, a b^{5} d^{2} x^{3} + 6 \, a^{2} b^{4} d^{2} x^{2} + 4 \, a^{3} b^{3} d^{2} x + a^{4} b^{2} d^{2}\right )} e^{4}\right )} \log \left (x e + d\right )}{4 \, {\left (b^{11} d^{9} x^{4} + 4 \, a b^{10} d^{9} x^{3} + 6 \, a^{2} b^{9} d^{9} x^{2} + 4 \, a^{3} b^{8} d^{9} x + a^{4} b^{7} d^{9} - {\left (a^{7} b^{4} x^{6} + 4 \, a^{8} b^{3} x^{5} + 6 \, a^{9} b^{2} x^{4} + 4 \, a^{10} b x^{3} + a^{11} x^{2}\right )} e^{9} + {\left (7 \, a^{6} b^{5} d x^{6} + 26 \, a^{7} b^{4} d x^{5} + 34 \, a^{8} b^{3} d x^{4} + 16 \, a^{9} b^{2} d x^{3} - a^{10} b d x^{2} - 2 \, a^{11} d x\right )} e^{8} - {\left (21 \, a^{5} b^{6} d^{2} x^{6} + 70 \, a^{6} b^{5} d^{2} x^{5} + 71 \, a^{7} b^{4} d^{2} x^{4} + 4 \, a^{8} b^{3} d^{2} x^{3} - 29 \, a^{9} b^{2} d^{2} x^{2} - 10 \, a^{10} b d^{2} x + a^{11} d^{2}\right )} e^{7} + 7 \, {\left (5 \, a^{4} b^{7} d^{3} x^{6} + 14 \, a^{5} b^{6} d^{3} x^{5} + 7 \, a^{6} b^{5} d^{3} x^{4} - 12 \, a^{7} b^{4} d^{3} x^{3} - 13 \, a^{8} b^{3} d^{3} x^{2} - 2 \, a^{9} b^{2} d^{3} x + a^{10} b d^{3}\right )} e^{6} - 7 \, {\left (5 \, a^{3} b^{8} d^{4} x^{6} + 10 \, a^{4} b^{7} d^{4} x^{5} - 7 \, a^{5} b^{6} d^{4} x^{4} - 28 \, a^{6} b^{5} d^{4} x^{3} - 17 \, a^{7} b^{4} d^{4} x^{2} + 2 \, a^{8} b^{3} d^{4} x + 3 \, a^{9} b^{2} d^{4}\right )} e^{5} + 7 \, {\left (3 \, a^{2} b^{9} d^{5} x^{6} + 2 \, a^{3} b^{8} d^{5} x^{5} - 17 \, a^{4} b^{7} d^{5} x^{4} - 28 \, a^{5} b^{6} d^{5} x^{3} - 7 \, a^{6} b^{5} d^{5} x^{2} + 10 \, a^{7} b^{4} d^{5} x + 5 \, a^{8} b^{3} d^{5}\right )} e^{4} - 7 \, {\left (a b^{10} d^{6} x^{6} - 2 \, a^{2} b^{9} d^{6} x^{5} - 13 \, a^{3} b^{8} d^{6} x^{4} - 12 \, a^{4} b^{7} d^{6} x^{3} + 7 \, a^{5} b^{6} d^{6} x^{2} + 14 \, a^{6} b^{5} d^{6} x + 5 \, a^{7} b^{4} d^{6}\right )} e^{3} + {\left (b^{11} d^{7} x^{6} - 10 \, a b^{10} d^{7} x^{5} - 29 \, a^{2} b^{9} d^{7} x^{4} + 4 \, a^{3} b^{8} d^{7} x^{3} + 71 \, a^{4} b^{7} d^{7} x^{2} + 70 \, a^{5} b^{6} d^{7} x + 21 \, a^{6} b^{5} d^{7}\right )} e^{2} + {\left (2 \, b^{11} d^{8} x^{5} + a b^{10} d^{8} x^{4} - 16 \, a^{2} b^{9} d^{8} x^{3} - 34 \, a^{3} b^{8} d^{8} x^{2} - 26 \, a^{4} b^{7} d^{8} x - 7 \, a^{5} b^{6} d^{8}\right )} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^3/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")

[Out]

-1/4*(b^6*d^6 + (60*a*b^5*x^5 + 210*a^2*b^4*x^4 + 260*a^3*b^3*x^3 + 125*a^4*b^2*x^2 + 12*a^5*b*x - 2*a^6)*e^6
- 2*(30*b^6*d*x^5 + 60*a*b^5*d*x^4 - 30*a^2*b^4*d*x^3 - 140*a^3*b^3*d*x^2 - 95*a^4*b^2*d*x - 12*a^5*b*d)*e^5 -
5*(18*b^6*d^2*x^4 + 60*a*b^5*d^2*x^3 + 66*a^2*b^4*d^2*x^2 + 20*a^3*b^3*d^2*x - 7*a^4*b^2*d^2)*e^4 - 20*(b^6*d
^3*x^3 + 4*a*b^5*d^3*x^2 + 6*a^2*b^4*d^3*x + 4*a^3*b^3*d^3)*e^3 + 5*(b^6*d^4*x^2 + 4*a*b^5*d^4*x + 6*a^2*b^4*d
^4)*e^2 - 2*(b^6*d^5*x + 4*a*b^5*d^5)*e - 60*((b^6*x^6 + 4*a*b^5*x^5 + 6*a^2*b^4*x^4 + 4*a^3*b^3*x^3 + a^4*b^2
*x^2)*e^6 + 2*(b^6*d*x^5 + 4*a*b^5*d*x^4 + 6*a^2*b^4*d*x^3 + 4*a^3*b^3*d*x^2 + a^4*b^2*d*x)*e^5 + (b^6*d^2*x^4
+ 4*a*b^5*d^2*x^3 + 6*a^2*b^4*d^2*x^2 + 4*a^3*b^3*d^2*x + a^4*b^2*d^2)*e^4)*log(b*x + a) + 60*((b^6*x^6 + 4*a
*b^5*x^5 + 6*a^2*b^4*x^4 + 4*a^3*b^3*x^3 + a^4*b^2*x^2)*e^6 + 2*(b^6*d*x^5 + 4*a*b^5*d*x^4 + 6*a^2*b^4*d*x^3 +
4*a^3*b^3*d*x^2 + a^4*b^2*d*x)*e^5 + (b^6*d^2*x^4 + 4*a*b^5*d^2*x^3 + 6*a^2*b^4*d^2*x^2 + 4*a^3*b^3*d^2*x + a
^4*b^2*d^2)*e^4)*log(x*e + d))/(b^11*d^9*x^4 + 4*a*b^10*d^9*x^3 + 6*a^2*b^9*d^9*x^2 + 4*a^3*b^8*d^9*x + a^4*b^
7*d^9 - (a^7*b^4*x^6 + 4*a^8*b^3*x^5 + 6*a^9*b^2*x^4 + 4*a^10*b*x^3 + a^11*x^2)*e^9 + (7*a^6*b^5*d*x^6 + 26*a^
7*b^4*d*x^5 + 34*a^8*b^3*d*x^4 + 16*a^9*b^2*d*x^3 - a^10*b*d*x^2 - 2*a^11*d*x)*e^8 - (21*a^5*b^6*d^2*x^6 + 70*
a^6*b^5*d^2*x^5 + 71*a^7*b^4*d^2*x^4 + 4*a^8*b^3*d^2*x^3 - 29*a^9*b^2*d^2*x^2 - 10*a^10*b*d^2*x + a^11*d^2)*e^
7 + 7*(5*a^4*b^7*d^3*x^6 + 14*a^5*b^6*d^3*x^5 + 7*a^6*b^5*d^3*x^4 - 12*a^7*b^4*d^3*x^3 - 13*a^8*b^3*d^3*x^2 -
2*a^9*b^2*d^3*x + a^10*b*d^3)*e^6 - 7*(5*a^3*b^8*d^4*x^6 + 10*a^4*b^7*d^4*x^5 - 7*a^5*b^6*d^4*x^4 - 28*a^6*b^5
*d^4*x^3 - 17*a^7*b^4*d^4*x^2 + 2*a^8*b^3*d^4*x + 3*a^9*b^2*d^4)*e^5 + 7*(3*a^2*b^9*d^5*x^6 + 2*a^3*b^8*d^5*x^
5 - 17*a^4*b^7*d^5*x^4 - 28*a^5*b^6*d^5*x^3 - 7*a^6*b^5*d^5*x^2 + 10*a^7*b^4*d^5*x + 5*a^8*b^3*d^5)*e^4 - 7*(a
*b^10*d^6*x^6 - 2*a^2*b^9*d^6*x^5 - 13*a^3*b^8*d^6*x^4 - 12*a^4*b^7*d^6*x^3 + 7*a^5*b^6*d^6*x^2 + 14*a^6*b^5*d
^6*x + 5*a^7*b^4*d^6)*e^3 + (b^11*d^7*x^6 - 10*a*b^10*d^7*x^5 - 29*a^2*b^9*d^7*x^4 + 4*a^3*b^8*d^7*x^3 + 71*a^
4*b^7*d^7*x^2 + 70*a^5*b^6*d^7*x + 21*a^6*b^5*d^7)*e^2 + (2*b^11*d^8*x^5 + a*b^10*d^8*x^4 - 16*a^2*b^9*d^8*x^3
- 34*a^3*b^8*d^8*x^2 - 26*a^4*b^7*d^8*x - 7*a^5*b^6*d^8)*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 )^{3} \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)**3/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Integral(1/((d + e*x)**3*((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. 651 vs. \(2 (279) = 558\).
time = 0.71, size = 651, normalized size = 1.78 \begin {gather*} \frac {15 \, b^{3} e^{4} \log \left ({\left | b x + a \right |}\right )}{b^{8} d^{7} \mathrm {sgn}\left (b x + a\right ) - 7 \, a b^{7} d^{6} e \mathrm {sgn}\left (b x + a\right ) + 21 \, a^{2} b^{6} d^{5} e^{2} \mathrm {sgn}\left (b x + a\right ) - 35 \, a^{3} b^{5} d^{4} e^{3} \mathrm {sgn}\left (b x + a\right ) + 35 \, a^{4} b^{4} d^{3} e^{4} \mathrm {sgn}\left (b x + a\right ) - 21 \, a^{5} b^{3} d^{2} e^{5} \mathrm {sgn}\left (b x + a\right ) + 7 \, a^{6} b^{2} d e^{6} \mathrm {sgn}\left (b x + a\right ) - a^{7} b e^{7} \mathrm {sgn}\left (b x + a\right )} - \frac {15 \, b^{2} e^{5} \log \left ({\left | x e + d \right |}\right )}{b^{7} d^{7} e \mathrm {sgn}\left (b x + a\right ) - 7 \, a b^{6} d^{6} e^{2} \mathrm {sgn}\left (b x + a\right ) + 21 \, a^{2} b^{5} d^{5} e^{3} \mathrm {sgn}\left (b x + a\right ) - 35 \, a^{3} b^{4} d^{4} e^{4} \mathrm {sgn}\left (b x + a\right ) + 35 \, a^{4} b^{3} d^{3} e^{5} \mathrm {sgn}\left (b x + a\right ) - 21 \, a^{5} b^{2} d^{2} e^{6} \mathrm {sgn}\left (b x + a\right ) + 7 \, a^{6} b d e^{7} \mathrm {sgn}\left (b x + a\right ) - a^{7} e^{8} \mathrm {sgn}\left (b x + a\right )} - \frac {b^{6} d^{6} - 8 \, a b^{5} d^{5} e + 30 \, a^{2} b^{4} d^{4} e^{2} - 80 \, a^{3} b^{3} d^{3} e^{3} + 35 \, a^{4} b^{2} d^{2} e^{4} + 24 \, a^{5} b d e^{5} - 2 \, a^{6} e^{6} - 60 \, {\left (b^{6} d e^{5} - a b^{5} e^{6}\right )} x^{5} - 30 \, {\left (3 \, b^{6} d^{2} e^{4} + 4 \, a b^{5} d e^{5} - 7 \, a^{2} b^{4} e^{6}\right )} x^{4} - 20 \, {\left (b^{6} d^{3} e^{3} + 15 \, a b^{5} d^{2} e^{4} - 3 \, a^{2} b^{4} d e^{5} - 13 \, a^{3} b^{3} e^{6}\right )} x^{3} + 5 \, {\left (b^{6} d^{4} e^{2} - 16 \, a b^{5} d^{3} e^{3} - 66 \, a^{2} b^{4} d^{2} e^{4} + 56 \, a^{3} b^{3} d e^{5} + 25 \, a^{4} b^{2} e^{6}\right )} x^{2} - 2 \, {\left (b^{6} d^{5} e - 10 \, a b^{5} d^{4} e^{2} + 60 \, a^{2} b^{4} d^{3} e^{3} + 50 \, a^{3} b^{3} d^{2} e^{4} - 95 \, a^{4} b^{2} d e^{5} - 6 \, a^{5} b e^{6}\right )} x}{4 \, {\left (b d - a e\right )}^{7} {\left (b x + a\right )}^{4} {\left (x e + d\right )}^{2} \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)^3/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")

[Out]

15*b^3*e^4*log(abs(b*x + a))/(b^8*d^7*sgn(b*x + a) - 7*a*b^7*d^6*e*sgn(b*x + a) + 21*a^2*b^6*d^5*e^2*sgn(b*x +
a) - 35*a^3*b^5*d^4*e^3*sgn(b*x + a) + 35*a^4*b^4*d^3*e^4*sgn(b*x + a) - 21*a^5*b^3*d^2*e^5*sgn(b*x + a) + 7*
a^6*b^2*d*e^6*sgn(b*x + a) - a^7*b*e^7*sgn(b*x + a)) - 15*b^2*e^5*log(abs(x*e + d))/(b^7*d^7*e*sgn(b*x + a) -
7*a*b^6*d^6*e^2*sgn(b*x + a) + 21*a^2*b^5*d^5*e^3*sgn(b*x + a) - 35*a^3*b^4*d^4*e^4*sgn(b*x + a) + 35*a^4*b^3*
d^3*e^5*sgn(b*x + a) - 21*a^5*b^2*d^2*e^6*sgn(b*x + a) + 7*a^6*b*d*e^7*sgn(b*x + a) - a^7*e^8*sgn(b*x + a)) -
1/4*(b^6*d^6 - 8*a*b^5*d^5*e + 30*a^2*b^4*d^4*e^2 - 80*a^3*b^3*d^3*e^3 + 35*a^4*b^2*d^2*e^4 + 24*a^5*b*d*e^5 -
2*a^6*e^6 - 60*(b^6*d*e^5 - a*b^5*e^6)*x^5 - 30*(3*b^6*d^2*e^4 + 4*a*b^5*d*e^5 - 7*a^2*b^4*e^6)*x^4 - 20*(b^6
*d^3*e^3 + 15*a*b^5*d^2*e^4 - 3*a^2*b^4*d*e^5 - 13*a^3*b^3*e^6)*x^3 + 5*(b^6*d^4*e^2 - 16*a*b^5*d^3*e^3 - 66*a
^2*b^4*d^2*e^4 + 56*a^3*b^3*d*e^5 + 25*a^4*b^2*e^6)*x^2 - 2*(b^6*d^5*e - 10*a*b^5*d^4*e^2 + 60*a^2*b^4*d^3*e^3
+ 50*a^3*b^3*d^2*e^4 - 95*a^4*b^2*d*e^5 - 6*a^5*b*e^6)*x)/((b*d - a*e)^7*(b*x + a)^4*(x*e + d)^2*sgn(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 )}^3\,{\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)^3*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2)),x)

[Out]

int(1/((d + e*x)^3*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2)), x)

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