\(\int \frac {(d+e x)^6}{(a^2+2 a b x+b^2 x^2)^{5/2}} \, dx\) [1604]

Optimal result

Integrand size = 28, antiderivative size = 302 \[ \int \frac {(d+e x)^6}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {20 e^3 (b d-a e)^3}{b^7 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(b d-a e)^6}{4 b^7 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 e (b d-a e)^5}{b^7 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {15 e^2 (b d-a e)^4}{2 b^7 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e^5 (6 b d-5 a e) x (a+b x)}{b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e^6 x^2 (a+b x)}{2 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {15 e^4 (b d-a e)^2 (a+b x) \log (a+b x)}{b^7 \sqrt {a^2+2 a b x+b^2 x^2}} \]

[Out]

-20*e^3*(-a*e+b*d)^3/b^7/((b*x+a)^2)^(1/2)-1/4*(-a*e+b*d)^6/b^7/(b*x+a)^3/((b*x+a)^2)^(1/2)-2*e*(-a*e+b*d)^5/b
^7/(b*x+a)^2/((b*x+a)^2)^(1/2)-15/2*e^2*(-a*e+b*d)^4/b^7/(b*x+a)/((b*x+a)^2)^(1/2)+e^5*(-5*a*e+6*b*d)*x*(b*x+a
)/b^6/((b*x+a)^2)^(1/2)+1/2*e^6*x^2*(b*x+a)/b^5/((b*x+a)^2)^(1/2)+15*e^4*(-a*e+b*d)^2*(b*x+a)*ln(b*x+a)/b^7/((
b*x+a)^2)^(1/2)

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {660, 45} \[ \int \frac {(d+e x)^6}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {15 e^4 (a+b x) (b d-a e)^2 \log (a+b x)}{b^7 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {20 e^3 (b d-a e)^3}{b^7 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {15 e^2 (b d-a e)^4}{2 b^7 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 e (b d-a e)^5}{b^7 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(b d-a e)^6}{4 b^7 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e^5 x (a+b x) (6 b d-5 a e)}{b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e^6 x^2 (a+b x)}{2 b^5 \sqrt {a^2+2 a b x+b^2 x^2}} \]

[In]

Int[(d + e*x)^6/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(-20*e^3*(b*d - a*e)^3)/(b^7*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (b*d - a*e)^6/(4*b^7*(a + b*x)^3*Sqrt[a^2 + 2*a*
b*x + b^2*x^2]) - (2*e*(b*d - a*e)^5)/(b^7*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (15*e^2*(b*d - a*e)^4)
/(2*b^7*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (e^5*(6*b*d - 5*a*e)*x*(a + b*x))/(b^6*Sqrt[a^2 + 2*a*b*x +
b^2*x^2]) + (e^6*x^2*(a + b*x))/(2*b^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (15*e^4*(b*d - a*e)^2*(a + b*x)*Log[a
+ b*x])/(b^7*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

Rule 45

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[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*} \text {integral}& = \frac {\left (b^4 \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^6}{\left (a b+b^2 x\right )^5} \, 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 {e^5 (6 b d-5 a e)}{b^{11}}+\frac {e^6 x}{b^{10}}+\frac {(b d-a e)^6}{b^{11} (a+b x)^5}+\frac {6 e (b d-a e)^5}{b^{11} (a+b x)^4}+\frac {15 e^2 (b d-a e)^4}{b^{11} (a+b x)^3}+\frac {20 e^3 (b d-a e)^3}{b^{11} (a+b x)^2}+\frac {15 e^4 (b d-a e)^2}{b^{11} (a+b x)}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {20 e^3 (b d-a e)^3}{b^7 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(b d-a e)^6}{4 b^7 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 e (b d-a e)^5}{b^7 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {15 e^2 (b d-a e)^4}{2 b^7 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e^5 (6 b d-5 a e) x (a+b x)}{b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e^6 x^2 (a+b x)}{2 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {15 e^4 (b d-a e)^2 (a+b x) \log (a+b x)}{b^7 \sqrt {a^2+2 a b x+b^2 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.10 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.04 \[ \int \frac {(d+e x)^6}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {57 a^6 e^6+14 a^5 b e^5 (-11 d+12 e x)+a^4 b^2 e^4 \left (125 d^2-496 d e x+132 e^2 x^2\right )-4 a^3 b^3 e^3 \left (5 d^3-110 d^2 e x+126 d e^2 x^2+8 e^3 x^3\right )-a^2 b^4 e^2 \left (5 d^4+80 d^3 e x-540 d^2 e^2 x^2+96 d e^3 x^3+68 e^4 x^4\right )-2 a b^5 e \left (d^5+10 d^4 e x+60 d^3 e^2 x^2-120 d^2 e^3 x^3-48 d e^4 x^4+6 e^5 x^5\right )-b^6 \left (d^6+8 d^5 e x+30 d^4 e^2 x^2+80 d^3 e^3 x^3-24 d e^5 x^5-2 e^6 x^6\right )+60 e^4 (b d-a e)^2 (a+b x)^4 \log (a+b x)}{4 b^7 (a+b x)^3 \sqrt {(a+b x)^2}} \]

[In]

Integrate[(d + e*x)^6/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(57*a^6*e^6 + 14*a^5*b*e^5*(-11*d + 12*e*x) + a^4*b^2*e^4*(125*d^2 - 496*d*e*x + 132*e^2*x^2) - 4*a^3*b^3*e^3*
(5*d^3 - 110*d^2*e*x + 126*d*e^2*x^2 + 8*e^3*x^3) - a^2*b^4*e^2*(5*d^4 + 80*d^3*e*x - 540*d^2*e^2*x^2 + 96*d*e
^3*x^3 + 68*e^4*x^4) - 2*a*b^5*e*(d^5 + 10*d^4*e*x + 60*d^3*e^2*x^2 - 120*d^2*e^3*x^3 - 48*d*e^4*x^4 + 6*e^5*x
^5) - b^6*(d^6 + 8*d^5*e*x + 30*d^4*e^2*x^2 + 80*d^3*e^3*x^3 - 24*d*e^5*x^5 - 2*e^6*x^6) + 60*e^4*(b*d - a*e)^
2*(a + b*x)^4*Log[a + b*x])/(4*b^7*(a + b*x)^3*Sqrt[(a + b*x)^2])

Maple [A] (verified)

Time = 2.85 (sec) , antiderivative size = 379, normalized size of antiderivative = 1.25

method	result	size
risch	\(-\frac {\sqrt {\left (b x +a \right )^{2}}\, e^{5} \left (-\frac {1}{2} b e \,x^{2}+5 a e x -6 b d x \right )}{\left (b x +a \right ) b^{6}}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (\left (20 a^{3} b^{2} e^{6}-60 a^{2} d \,e^{5} b^{3}+60 a \,b^{4} d^{2} e^{4}-20 d^{3} e^{3} b^{5}\right ) x^{3}+\frac {15 b \,e^{2} \left (7 e^{4} a^{4}-20 b \,e^{3} d \,a^{3}+18 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e -b^{4} d^{4}\right ) x^{2}}{2}+e \left (47 a^{5} e^{5}-130 a^{4} b d \,e^{4}+110 a^{3} b^{2} d^{2} e^{3}-20 a^{2} b^{3} d^{3} e^{2}-5 a \,b^{4} d^{4} e -2 b^{5} d^{5}\right ) x +\frac {57 a^{6} e^{6}-154 a^{5} b d \,e^{5}+125 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}-5 a^{2} b^{4} d^{4} e^{2}-2 a \,b^{5} d^{5} e -b^{6} d^{6}}{4 b}\right )}{\left (b x +a \right )^{5} b^{6}}+\frac {15 \sqrt {\left (b x +a \right )^{2}}\, e^{4} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \ln \left (b x +a \right )}{\left (b x +a \right ) b^{7}}\)	\(379\)
default	\(\frac {\left (168 x \,a^{5} b \,e^{6}-8 x \,b^{6} d^{5} e -12 x^{5} a \,b^{5} e^{6}+24 x^{5} b^{6} d \,e^{5}-68 x^{4} a^{2} b^{4} e^{6}-32 x^{3} a^{3} b^{3} e^{6}-80 x^{3} b^{6} d^{3} e^{3}+132 x^{2} a^{4} b^{2} e^{6}-120 \ln \left (b x +a \right ) a \,b^{5} d \,e^{5} x^{4}+240 \ln \left (b x +a \right ) a^{5} b \,e^{6} x +60 \ln \left (b x +a \right ) b^{6} d^{2} e^{4} x^{4}-120 \ln \left (b x +a \right ) a^{5} b d \,e^{5}+57 a^{6} e^{6}-b^{6} d^{6}-2 a \,b^{5} d^{5} e +125 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}-5 a^{2} b^{4} d^{4} e^{2}-154 a^{5} b d \,e^{5}+360 \ln \left (b x +a \right ) a^{4} b^{2} e^{6} x^{2}+60 \ln \left (b x +a \right ) a^{2} b^{4} e^{6} x^{4}+240 \ln \left (b x +a \right ) a^{3} b^{3} e^{6} x^{3}-480 \ln \left (b x +a \right ) a^{4} b^{2} d \,e^{5} x +240 \ln \left (b x +a \right ) a^{3} b^{3} d^{2} e^{4} x -480 \ln \left (b x +a \right ) a^{2} b^{4} d \,e^{5} x^{3}+240 \ln \left (b x +a \right ) a \,b^{5} d^{2} e^{4} x^{3}-720 \ln \left (b x +a \right ) a^{3} b^{3} d \,e^{5} x^{2}+360 \ln \left (b x +a \right ) a^{2} b^{4} d^{2} e^{4} x^{2}+60 \ln \left (b x +a \right ) a^{4} b^{2} d^{2} e^{4}+2 x^{6} b^{6} e^{6}+60 \ln \left (b x +a \right ) a^{6} e^{6}-30 x^{2} b^{6} d^{4} e^{2}+96 x^{4} a \,b^{5} d \,e^{5}-96 x^{3} a^{2} b^{4} d \,e^{5}+240 x^{3} a \,b^{5} d^{2} e^{4}-504 x^{2} a^{3} b^{3} d \,e^{5}+540 x^{2} a^{2} b^{4} d^{2} e^{4}-120 x^{2} a \,b^{5} d^{3} e^{3}-496 x \,a^{4} b^{2} d \,e^{5}+440 x \,a^{3} b^{3} d^{2} e^{4}-80 x \,a^{2} b^{4} d^{3} e^{3}-20 x a \,b^{5} d^{4} e^{2}\right ) \left (b x +a \right )}{4 b^{7} \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}\)	\(661\)

[In]

int((e*x+d)^6/(b^2*x^2+2*a*b*x+a^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-((b*x+a)^2)^(1/2)/(b*x+a)*e^5/b^6*(-1/2*b*e*x^2+5*a*e*x-6*b*d*x)+((b*x+a)^2)^(1/2)/(b*x+a)^5*((20*a^3*b^2*e^6
-60*a^2*b^3*d*e^5+60*a*b^4*d^2*e^4-20*b^5*d^3*e^3)*x^3+15/2*b*e^2*(7*a^4*e^4-20*a^3*b*d*e^3+18*a^2*b^2*d^2*e^2
-4*a*b^3*d^3*e-b^4*d^4)*x^2+e*(47*a^5*e^5-130*a^4*b*d*e^4+110*a^3*b^2*d^2*e^3-20*a^2*b^3*d^3*e^2-5*a*b^4*d^4*e
-2*b^5*d^5)*x+1/4*(57*a^6*e^6-154*a^5*b*d*e^5+125*a^4*b^2*d^2*e^4-20*a^3*b^3*d^3*e^3-5*a^2*b^4*d^4*e^2-2*a*b^5
*d^5*e-b^6*d^6)/b)/b^6+15*((b*x+a)^2)^(1/2)/(b*x+a)/b^7*e^4*(a^2*e^2-2*a*b*d*e+b^2*d^2)*ln(b*x+a)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 572 vs. \(2 (219) = 438\).

Time = 0.27 (sec) , antiderivative size = 572, normalized size of antiderivative = 1.89 \[ \int \frac {(d+e x)^6}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {2 \, b^{6} e^{6} x^{6} - b^{6} d^{6} - 2 \, a b^{5} d^{5} e - 5 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 125 \, a^{4} b^{2} d^{2} e^{4} - 154 \, a^{5} b d e^{5} + 57 \, a^{6} e^{6} + 12 \, {\left (2 \, b^{6} d e^{5} - a b^{5} e^{6}\right )} x^{5} + 4 \, {\left (24 \, a b^{5} d e^{5} - 17 \, a^{2} b^{4} e^{6}\right )} x^{4} - 16 \, {\left (5 \, b^{6} d^{3} e^{3} - 15 \, a b^{5} d^{2} e^{4} + 6 \, a^{2} b^{4} d e^{5} + 2 \, a^{3} b^{3} e^{6}\right )} x^{3} - 6 \, {\left (5 \, b^{6} d^{4} e^{2} + 20 \, a b^{5} d^{3} e^{3} - 90 \, a^{2} b^{4} d^{2} e^{4} + 84 \, a^{3} b^{3} d e^{5} - 22 \, a^{4} b^{2} e^{6}\right )} x^{2} - 4 \, {\left (2 \, b^{6} d^{5} e + 5 \, a b^{5} d^{4} e^{2} + 20 \, a^{2} b^{4} d^{3} e^{3} - 110 \, a^{3} b^{3} d^{2} e^{4} + 124 \, a^{4} b^{2} d e^{5} - 42 \, a^{5} b e^{6}\right )} x + 60 \, {\left (a^{4} b^{2} d^{2} e^{4} - 2 \, a^{5} b d e^{5} + a^{6} e^{6} + {\left (b^{6} d^{2} e^{4} - 2 \, a b^{5} d e^{5} + a^{2} b^{4} e^{6}\right )} x^{4} + 4 \, {\left (a b^{5} d^{2} e^{4} - 2 \, a^{2} b^{4} d e^{5} + a^{3} b^{3} e^{6}\right )} x^{3} + 6 \, {\left (a^{2} b^{4} d^{2} e^{4} - 2 \, a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{2} + 4 \, {\left (a^{3} b^{3} d^{2} e^{4} - 2 \, a^{4} b^{2} d e^{5} + a^{5} b e^{6}\right )} x\right )} \log \left (b x + a\right )}{4 \, {\left (b^{11} x^{4} + 4 \, a b^{10} x^{3} + 6 \, a^{2} b^{9} x^{2} + 4 \, a^{3} b^{8} x + a^{4} b^{7}\right )}} \]

[In]

integrate((e*x+d)^6/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")

[Out]

1/4*(2*b^6*e^6*x^6 - b^6*d^6 - 2*a*b^5*d^5*e - 5*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3 + 125*a^4*b^2*d^2*e^4 -
154*a^5*b*d*e^5 + 57*a^6*e^6 + 12*(2*b^6*d*e^5 - a*b^5*e^6)*x^5 + 4*(24*a*b^5*d*e^5 - 17*a^2*b^4*e^6)*x^4 - 16
*(5*b^6*d^3*e^3 - 15*a*b^5*d^2*e^4 + 6*a^2*b^4*d*e^5 + 2*a^3*b^3*e^6)*x^3 - 6*(5*b^6*d^4*e^2 + 20*a*b^5*d^3*e^
3 - 90*a^2*b^4*d^2*e^4 + 84*a^3*b^3*d*e^5 - 22*a^4*b^2*e^6)*x^2 - 4*(2*b^6*d^5*e + 5*a*b^5*d^4*e^2 + 20*a^2*b^
4*d^3*e^3 - 110*a^3*b^3*d^2*e^4 + 124*a^4*b^2*d*e^5 - 42*a^5*b*e^6)*x + 60*(a^4*b^2*d^2*e^4 - 2*a^5*b*d*e^5 +
a^6*e^6 + (b^6*d^2*e^4 - 2*a*b^5*d*e^5 + a^2*b^4*e^6)*x^4 + 4*(a*b^5*d^2*e^4 - 2*a^2*b^4*d*e^5 + a^3*b^3*e^6)*
x^3 + 6*(a^2*b^4*d^2*e^4 - 2*a^3*b^3*d*e^5 + a^4*b^2*e^6)*x^2 + 4*(a^3*b^3*d^2*e^4 - 2*a^4*b^2*d*e^5 + a^5*b*e
^6)*x)*log(b*x + a))/(b^11*x^4 + 4*a*b^10*x^3 + 6*a^2*b^9*x^2 + 4*a^3*b^8*x + a^4*b^7)

Sympy [F]

\[ \int \frac {(d+e x)^6}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int \frac {\left (d + e x\right )^{6}}{\left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}\, dx \]

[In]

integrate((e*x+d)**6/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Integral((d + e*x)**6/((a + b*x)**2)**(5/2), x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 576 vs. \(2 (219) = 438\).

Time = 0.25 (sec) , antiderivative size = 576, normalized size of antiderivative = 1.91 \[ \int \frac {(d+e x)^6}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {1}{4} \, e^{6} {\left (\frac {2 \, b^{6} x^{6} - 12 \, a b^{5} x^{5} - 68 \, a^{2} b^{4} x^{4} - 32 \, a^{3} b^{3} x^{3} + 132 \, a^{4} b^{2} x^{2} + 168 \, a^{5} b x + 57 \, a^{6}}{b^{11} x^{4} + 4 \, a b^{10} x^{3} + 6 \, a^{2} b^{9} x^{2} + 4 \, a^{3} b^{8} x + a^{4} b^{7}} + \frac {60 \, a^{2} \log \left (b x + a\right )}{b^{7}}\right )} + \frac {1}{2} \, d e^{5} {\left (\frac {12 \, b^{5} x^{5} + 48 \, a b^{4} x^{4} - 48 \, a^{2} b^{3} x^{3} - 252 \, a^{3} b^{2} x^{2} - 248 \, a^{4} b x - 77 \, a^{5}}{b^{10} x^{4} + 4 \, a b^{9} x^{3} + 6 \, a^{2} b^{8} x^{2} + 4 \, a^{3} b^{7} x + a^{4} b^{6}} - \frac {60 \, a \log \left (b x + a\right )}{b^{6}}\right )} + \frac {5}{4} \, d^{2} e^{4} {\left (\frac {48 \, a b^{3} x^{3} + 108 \, a^{2} b^{2} x^{2} + 88 \, a^{3} b x + 25 \, a^{4}}{b^{9} x^{4} + 4 \, a b^{8} x^{3} + 6 \, a^{2} b^{7} x^{2} + 4 \, a^{3} b^{6} x + a^{4} b^{5}} + \frac {12 \, \log \left (b x + a\right )}{b^{5}}\right )} - \frac {5}{3} \, d^{3} e^{3} {\left (\frac {12 \, x^{2}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b^{2}} + \frac {8 \, a^{2}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b^{4}} + \frac {6 \, a}{b^{6} {\left (x + \frac {a}{b}\right )}^{2}} - \frac {8 \, a^{2}}{b^{7} {\left (x + \frac {a}{b}\right )}^{3}} - \frac {3 \, a^{3}}{b^{8} {\left (x + \frac {a}{b}\right )}^{4}}\right )} - \frac {1}{2} \, d^{5} e {\left (\frac {4}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b^{2}} - \frac {3 \, a}{b^{6} {\left (x + \frac {a}{b}\right )}^{4}}\right )} - \frac {5}{4} \, d^{4} e^{2} {\left (\frac {6}{b^{5} {\left (x + \frac {a}{b}\right )}^{2}} - \frac {8 \, a}{b^{6} {\left (x + \frac {a}{b}\right )}^{3}} + \frac {3 \, a^{2}}{b^{7} {\left (x + \frac {a}{b}\right )}^{4}}\right )} - \frac {d^{6}}{4 \, b^{5} {\left (x + \frac {a}{b}\right )}^{4}} \]

[In]

integrate((e*x+d)^6/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="maxima")

[Out]

1/4*e^6*((2*b^6*x^6 - 12*a*b^5*x^5 - 68*a^2*b^4*x^4 - 32*a^3*b^3*x^3 + 132*a^4*b^2*x^2 + 168*a^5*b*x + 57*a^6)
/(b^11*x^4 + 4*a*b^10*x^3 + 6*a^2*b^9*x^2 + 4*a^3*b^8*x + a^4*b^7) + 60*a^2*log(b*x + a)/b^7) + 1/2*d*e^5*((12
*b^5*x^5 + 48*a*b^4*x^4 - 48*a^2*b^3*x^3 - 252*a^3*b^2*x^2 - 248*a^4*b*x - 77*a^5)/(b^10*x^4 + 4*a*b^9*x^3 + 6
*a^2*b^8*x^2 + 4*a^3*b^7*x + a^4*b^6) - 60*a*log(b*x + a)/b^6) + 5/4*d^2*e^4*((48*a*b^3*x^3 + 108*a^2*b^2*x^2
+ 88*a^3*b*x + 25*a^4)/(b^9*x^4 + 4*a*b^8*x^3 + 6*a^2*b^7*x^2 + 4*a^3*b^6*x + a^4*b^5) + 12*log(b*x + a)/b^5)
- 5/3*d^3*e^3*(12*x^2/((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*b^2) + 8*a^2/((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*b^4) + 6*
a/(b^6*(x + a/b)^2) - 8*a^2/(b^7*(x + a/b)^3) - 3*a^3/(b^8*(x + a/b)^4)) - 1/2*d^5*e*(4/((b^2*x^2 + 2*a*b*x +
a^2)^(3/2)*b^2) - 3*a/(b^6*(x + a/b)^4)) - 5/4*d^4*e^2*(6/(b^5*(x + a/b)^2) - 8*a/(b^6*(x + a/b)^3) + 3*a^2/(b
^7*(x + a/b)^4)) - 1/4*d^6/(b^5*(x + a/b)^4)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 389, normalized size of antiderivative = 1.29 \[ \int \frac {(d+e x)^6}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {15 \, {\left (b^{2} d^{2} e^{4} - 2 \, a b d e^{5} + a^{2} e^{6}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{7} \mathrm {sgn}\left (b x + a\right )} + \frac {b^{5} e^{6} x^{2} \mathrm {sgn}\left (b x + a\right ) + 12 \, b^{5} d e^{5} x \mathrm {sgn}\left (b x + a\right ) - 10 \, a b^{4} e^{6} x \mathrm {sgn}\left (b x + a\right )}{2 \, b^{10}} - \frac {b^{6} d^{6} + 2 \, a b^{5} d^{5} e + 5 \, a^{2} b^{4} d^{4} e^{2} + 20 \, a^{3} b^{3} d^{3} e^{3} - 125 \, a^{4} b^{2} d^{2} e^{4} + 154 \, a^{5} b d e^{5} - 57 \, a^{6} e^{6} + 80 \, {\left (b^{6} d^{3} e^{3} - 3 \, a b^{5} d^{2} e^{4} + 3 \, a^{2} b^{4} d e^{5} - a^{3} b^{3} e^{6}\right )} x^{3} + 30 \, {\left (b^{6} d^{4} e^{2} + 4 \, a b^{5} d^{3} e^{3} - 18 \, a^{2} b^{4} d^{2} e^{4} + 20 \, a^{3} b^{3} d e^{5} - 7 \, a^{4} b^{2} e^{6}\right )} x^{2} + 4 \, {\left (2 \, b^{6} d^{5} e + 5 \, a b^{5} d^{4} e^{2} + 20 \, a^{2} b^{4} d^{3} e^{3} - 110 \, a^{3} b^{3} d^{2} e^{4} + 130 \, a^{4} b^{2} d e^{5} - 47 \, a^{5} b e^{6}\right )} x}{4 \, {\left (b x + a\right )}^{4} b^{7} \mathrm {sgn}\left (b x + a\right )} \]

[In]

integrate((e*x+d)^6/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")

[Out]

15*(b^2*d^2*e^4 - 2*a*b*d*e^5 + a^2*e^6)*log(abs(b*x + a))/(b^7*sgn(b*x + a)) + 1/2*(b^5*e^6*x^2*sgn(b*x + a)
+ 12*b^5*d*e^5*x*sgn(b*x + a) - 10*a*b^4*e^6*x*sgn(b*x + a))/b^10 - 1/4*(b^6*d^6 + 2*a*b^5*d^5*e + 5*a^2*b^4*d
^4*e^2 + 20*a^3*b^3*d^3*e^3 - 125*a^4*b^2*d^2*e^4 + 154*a^5*b*d*e^5 - 57*a^6*e^6 + 80*(b^6*d^3*e^3 - 3*a*b^5*d
^2*e^4 + 3*a^2*b^4*d*e^5 - a^3*b^3*e^6)*x^3 + 30*(b^6*d^4*e^2 + 4*a*b^5*d^3*e^3 - 18*a^2*b^4*d^2*e^4 + 20*a^3*
b^3*d*e^5 - 7*a^4*b^2*e^6)*x^2 + 4*(2*b^6*d^5*e + 5*a*b^5*d^4*e^2 + 20*a^2*b^4*d^3*e^3 - 110*a^3*b^3*d^2*e^4 +
130*a^4*b^2*d*e^5 - 47*a^5*b*e^6)*x)/((b*x + a)^4*b^7*sgn(b*x + a))

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+e x)^6}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^6}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \]

[In]

int((d + e*x)^6/(a^2 + b^2*x^2 + 2*a*b*x)^(5/2),x)

[Out]

int((d + e*x)^6/(a^2 + b^2*x^2 + 2*a*b*x)^(5/2), x)