\(\int \frac {(b x+c x^2)^3}{(d+e x)^6} \, dx\) [254]

Optimal result

Integrand size = 19, antiderivative size = 218 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^6} \, dx=\frac {c^3 x}{e^6}-\frac {d^3 (c d-b e)^3}{5 e^7 (d+e x)^5}+\frac {3 d^2 (c d-b e)^2 (2 c d-b e)}{4 e^7 (d+e x)^4}-\frac {d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )}{e^7 (d+e x)^3}+\frac {(2 c d-b e) \left (10 c^2 d^2-10 b c d e+b^2 e^2\right )}{2 e^7 (d+e x)^2}-\frac {3 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )}{e^7 (d+e x)}-\frac {3 c^2 (2 c d-b e) \log (d+e x)}{e^7} \]

[Out]

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

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {712} \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^6} \, dx=-\frac {3 c \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^7 (d+e x)}+\frac {(2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{2 e^7 (d+e x)^2}-\frac {d (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^7 (d+e x)^3}-\frac {3 c^2 (2 c d-b e) \log (d+e x)}{e^7}-\frac {d^3 (c d-b e)^3}{5 e^7 (d+e x)^5}+\frac {3 d^2 (c d-b e)^2 (2 c d-b e)}{4 e^7 (d+e x)^4}+\frac {c^3 x}{e^6} \]

[In]

Int[(b*x + c*x^2)^3/(d + e*x)^6,x]

[Out]

(c^3*x)/e^6 - (d^3*(c*d - b*e)^3)/(5*e^7*(d + e*x)^5) + (3*d^2*(c*d - b*e)^2*(2*c*d - b*e))/(4*e^7*(d + e*x)^4
) - (d*(c*d - b*e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2))/(e^7*(d + e*x)^3) + ((2*c*d - b*e)*(10*c^2*d^2 - 10*b*c*
d*e + b^2*e^2))/(2*e^7*(d + e*x)^2) - (3*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2))/(e^7*(d + e*x)) - (3*c^2*(2*c*d
- b*e)*Log[d + e*x])/e^7

Rule 712

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {c^3}{e^6}+\frac {d^3 (c d-b e)^3}{e^6 (d+e x)^6}-\frac {3 d^2 (c d-b e)^2 (2 c d-b e)}{e^6 (d+e x)^5}+\frac {3 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )}{e^6 (d+e x)^4}+\frac {(2 c d-b e) \left (-10 c^2 d^2+10 b c d e-b^2 e^2\right )}{e^6 (d+e x)^3}+\frac {3 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )}{e^6 (d+e x)^2}-\frac {3 c^2 (2 c d-b e)}{e^6 (d+e x)}\right ) \, dx \\ & = \frac {c^3 x}{e^6}-\frac {d^3 (c d-b e)^3}{5 e^7 (d+e x)^5}+\frac {3 d^2 (c d-b e)^2 (2 c d-b e)}{4 e^7 (d+e x)^4}-\frac {d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )}{e^7 (d+e x)^3}+\frac {(2 c d-b e) \left (10 c^2 d^2-10 b c d e+b^2 e^2\right )}{2 e^7 (d+e x)^2}-\frac {3 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )}{e^7 (d+e x)}-\frac {3 c^2 (2 c d-b e) \log (d+e x)}{e^7} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.11 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^6} \, dx=-\frac {b^3 e^3 \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )+12 b^2 c e^2 \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )-b c^2 d e \left (137 d^4+625 d^3 e x+1100 d^2 e^2 x^2+900 d e^3 x^3+300 e^4 x^4\right )+2 c^3 \left (87 d^6+375 d^5 e x+600 d^4 e^2 x^2+400 d^3 e^3 x^3+50 d^2 e^4 x^4-50 d e^5 x^5-10 e^6 x^6\right )+60 c^2 (2 c d-b e) (d+e x)^5 \log (d+e x)}{20 e^7 (d+e x)^5} \]

[In]

Integrate[(b*x + c*x^2)^3/(d + e*x)^6,x]

[Out]

-1/20*(b^3*e^3*(d^3 + 5*d^2*e*x + 10*d*e^2*x^2 + 10*e^3*x^3) + 12*b^2*c*e^2*(d^4 + 5*d^3*e*x + 10*d^2*e^2*x^2
+ 10*d*e^3*x^3 + 5*e^4*x^4) - b*c^2*d*e*(137*d^4 + 625*d^3*e*x + 1100*d^2*e^2*x^2 + 900*d*e^3*x^3 + 300*e^4*x^
4) + 2*c^3*(87*d^6 + 375*d^5*e*x + 600*d^4*e^2*x^2 + 400*d^3*e^3*x^3 + 50*d^2*e^4*x^4 - 50*d*e^5*x^5 - 10*e^6*
x^6) + 60*c^2*(2*c*d - b*e)*(d + e*x)^5*Log[d + e*x])/(e^7*(d + e*x)^5)

Maple [A] (verified)

Time = 1.96 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.17

method	result	size
norman	\(\frac {\frac {c^{3} x^{6}}{e}-\frac {d^{3} \left (b^{3} e^{3}+12 b^{2} d \,e^{2} c -137 b \,c^{2} d^{2} e +274 c^{3} d^{3}\right )}{20 e^{7}}-\frac {\left (3 b^{2} e^{2} c -15 d e b \,c^{2}+30 c^{3} d^{2}\right ) x^{4}}{e^{3}}-\frac {\left (b^{3} e^{3}+12 b^{2} d \,e^{2} c -90 b \,c^{2} d^{2} e +180 c^{3} d^{3}\right ) x^{3}}{2 e^{4}}-\frac {d \left (b^{3} e^{3}+12 b^{2} d \,e^{2} c -110 b \,c^{2} d^{2} e +220 c^{3} d^{3}\right ) x^{2}}{2 e^{5}}-\frac {d^{2} \left (b^{3} e^{3}+12 b^{2} d \,e^{2} c -125 b \,c^{2} d^{2} e +250 c^{3} d^{3}\right ) x}{4 e^{6}}}{\left (e x +d \right )^{5}}+\frac {3 c^{2} \left (b e -2 c d \right ) \ln \left (e x +d \right )}{e^{7}}\)	\(255\)
risch	\(\frac {c^{3} x}{e^{6}}+\frac {\left (-3 b^{2} c \,e^{5}+15 b \,c^{2} d \,e^{4}-15 c^{3} d^{2} e^{3}\right ) x^{4}-\frac {e^{2} \left (b^{3} e^{3}+12 b^{2} d \,e^{2} c -90 b \,c^{2} d^{2} e +100 c^{3} d^{3}\right ) x^{3}}{2}-\frac {d e \left (b^{3} e^{3}+12 b^{2} d \,e^{2} c -110 b \,c^{2} d^{2} e +130 c^{3} d^{3}\right ) x^{2}}{2}-\frac {d^{2} \left (b^{3} e^{3}+12 b^{2} d \,e^{2} c -125 b \,c^{2} d^{2} e +154 c^{3} d^{3}\right ) x}{4}-\frac {d^{3} \left (b^{3} e^{3}+12 b^{2} d \,e^{2} c -137 b \,c^{2} d^{2} e +174 c^{3} d^{3}\right )}{20 e}}{e^{6} \left (e x +d \right )^{5}}+\frac {3 c^{2} \ln \left (e x +d \right ) b}{e^{6}}-\frac {6 c^{3} d \ln \left (e x +d \right )}{e^{7}}\)	\(260\)
default	\(\frac {c^{3} x}{e^{6}}-\frac {3 c \left (b^{2} e^{2}-5 b c d e +5 c^{2} d^{2}\right )}{e^{7} \left (e x +d \right )}+\frac {d \left (b^{3} e^{3}-6 b^{2} d \,e^{2} c +10 b \,c^{2} d^{2} e -5 c^{3} d^{3}\right )}{e^{7} \left (e x +d \right )^{3}}-\frac {3 d^{2} \left (b^{3} e^{3}-4 b^{2} d \,e^{2} c +5 b \,c^{2} d^{2} e -2 c^{3} d^{3}\right )}{4 e^{7} \left (e x +d \right )^{4}}+\frac {d^{3} \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}{5 e^{7} \left (e x +d \right )^{5}}-\frac {b^{3} e^{3}-12 b^{2} d \,e^{2} c +30 b \,c^{2} d^{2} e -20 c^{3} d^{3}}{2 e^{7} \left (e x +d \right )^{2}}+\frac {3 c^{2} \left (b e -2 c d \right ) \ln \left (e x +d \right )}{e^{7}}\)	\(265\)
parallelrisch	\(\frac {300 \ln \left (e x +d \right ) x b \,c^{2} d^{4} e^{2}-b^{3} d^{3} e^{3}-60 x^{4} b^{2} c \,e^{6}-600 x^{4} c^{3} d^{2} e^{4}-1800 x^{3} c^{3} d^{3} e^{3}-10 x^{2} b^{3} d \,e^{5}-2200 x^{2} c^{3} d^{4} e^{2}-5 x \,b^{3} d^{2} e^{4}-1250 x \,c^{3} d^{5} e -274 c^{3} d^{6}+600 \ln \left (e x +d \right ) x^{2} b \,c^{2} d^{3} e^{3}+600 \ln \left (e x +d \right ) x^{3} b \,c^{2} d^{2} e^{4}-600 \ln \left (e x +d \right ) x \,c^{3} d^{5} e +20 x^{6} c^{3} e^{6}+300 \ln \left (e x +d \right ) x^{4} b \,c^{2} d \,e^{5}+300 x^{4} b \,c^{2} d \,e^{5}-120 x^{3} b^{2} c d \,e^{5}-1200 \ln \left (e x +d \right ) x^{2} c^{3} d^{4} e^{2}-1200 \ln \left (e x +d \right ) x^{3} c^{3} d^{3} e^{3}-12 b^{2} c \,d^{4} e^{2}+137 b \,c^{2} d^{5} e -600 \ln \left (e x +d \right ) x^{4} c^{3} d^{2} e^{4}+900 x^{3} b \,c^{2} d^{2} e^{4}-120 x^{2} b^{2} c \,d^{2} e^{4}+1100 x^{2} b \,c^{2} d^{3} e^{3}+60 \ln \left (e x +d \right ) b \,c^{2} d^{5} e -60 x \,b^{2} c \,d^{3} e^{3}+60 \ln \left (e x +d \right ) x^{5} b \,c^{2} e^{6}-120 \ln \left (e x +d \right ) x^{5} c^{3} d \,e^{5}-10 x^{3} b^{3} e^{6}-120 \ln \left (e x +d \right ) c^{3} d^{6}+625 x b \,c^{2} d^{4} e^{2}}{20 e^{7} \left (e x +d \right )^{5}}\)	\(484\)

[In]

int((c*x^2+b*x)^3/(e*x+d)^6,x,method=_RETURNVERBOSE)

[Out]

(c^3*x^6/e-1/20*d^3*(b^3*e^3+12*b^2*c*d*e^2-137*b*c^2*d^2*e+274*c^3*d^3)/e^7-(3*b^2*c*e^2-15*b*c^2*d*e+30*c^3*
d^2)/e^3*x^4-1/2*(b^3*e^3+12*b^2*c*d*e^2-90*b*c^2*d^2*e+180*c^3*d^3)/e^4*x^3-1/2*d*(b^3*e^3+12*b^2*c*d*e^2-110
*b*c^2*d^2*e+220*c^3*d^3)/e^5*x^2-1/4*d^2*(b^3*e^3+12*b^2*c*d*e^2-125*b*c^2*d^2*e+250*c^3*d^3)/e^6*x)/(e*x+d)^
5+3*c^2/e^7*(b*e-2*c*d)*ln(e*x+d)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 462 vs. \(2 (212) = 424\).

Time = 0.26 (sec) , antiderivative size = 462, normalized size of antiderivative = 2.12 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^6} \, dx=\frac {20 \, c^{3} e^{6} x^{6} + 100 \, c^{3} d e^{5} x^{5} - 174 \, c^{3} d^{6} + 137 \, b c^{2} d^{5} e - 12 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3} - 20 \, {\left (5 \, c^{3} d^{2} e^{4} - 15 \, b c^{2} d e^{5} + 3 \, b^{2} c e^{6}\right )} x^{4} - 10 \, {\left (80 \, c^{3} d^{3} e^{3} - 90 \, b c^{2} d^{2} e^{4} + 12 \, b^{2} c d e^{5} + b^{3} e^{6}\right )} x^{3} - 10 \, {\left (120 \, c^{3} d^{4} e^{2} - 110 \, b c^{2} d^{3} e^{3} + 12 \, b^{2} c d^{2} e^{4} + b^{3} d e^{5}\right )} x^{2} - 5 \, {\left (150 \, c^{3} d^{5} e - 125 \, b c^{2} d^{4} e^{2} + 12 \, b^{2} c d^{3} e^{3} + b^{3} d^{2} e^{4}\right )} x - 60 \, {\left (2 \, c^{3} d^{6} - b c^{2} d^{5} e + {\left (2 \, c^{3} d e^{5} - b c^{2} e^{6}\right )} x^{5} + 5 \, {\left (2 \, c^{3} d^{2} e^{4} - b c^{2} d e^{5}\right )} x^{4} + 10 \, {\left (2 \, c^{3} d^{3} e^{3} - b c^{2} d^{2} e^{4}\right )} x^{3} + 10 \, {\left (2 \, c^{3} d^{4} e^{2} - b c^{2} d^{3} e^{3}\right )} x^{2} + 5 \, {\left (2 \, c^{3} d^{5} e - b c^{2} d^{4} e^{2}\right )} x\right )} \log \left (e x + d\right )}{20 \, {\left (e^{12} x^{5} + 5 \, d e^{11} x^{4} + 10 \, d^{2} e^{10} x^{3} + 10 \, d^{3} e^{9} x^{2} + 5 \, d^{4} e^{8} x + d^{5} e^{7}\right )}} \]

[In]

integrate((c*x^2+b*x)^3/(e*x+d)^6,x, algorithm="fricas")

[Out]

1/20*(20*c^3*e^6*x^6 + 100*c^3*d*e^5*x^5 - 174*c^3*d^6 + 137*b*c^2*d^5*e - 12*b^2*c*d^4*e^2 - b^3*d^3*e^3 - 20
*(5*c^3*d^2*e^4 - 15*b*c^2*d*e^5 + 3*b^2*c*e^6)*x^4 - 10*(80*c^3*d^3*e^3 - 90*b*c^2*d^2*e^4 + 12*b^2*c*d*e^5 +
b^3*e^6)*x^3 - 10*(120*c^3*d^4*e^2 - 110*b*c^2*d^3*e^3 + 12*b^2*c*d^2*e^4 + b^3*d*e^5)*x^2 - 5*(150*c^3*d^5*e
- 125*b*c^2*d^4*e^2 + 12*b^2*c*d^3*e^3 + b^3*d^2*e^4)*x - 60*(2*c^3*d^6 - b*c^2*d^5*e + (2*c^3*d*e^5 - b*c^2*
e^6)*x^5 + 5*(2*c^3*d^2*e^4 - b*c^2*d*e^5)*x^4 + 10*(2*c^3*d^3*e^3 - b*c^2*d^2*e^4)*x^3 + 10*(2*c^3*d^4*e^2 -
b*c^2*d^3*e^3)*x^2 + 5*(2*c^3*d^5*e - b*c^2*d^4*e^2)*x)*log(e*x + d))/(e^12*x^5 + 5*d*e^11*x^4 + 10*d^2*e^10*x
^3 + 10*d^3*e^9*x^2 + 5*d^4*e^8*x + d^5*e^7)

Sympy [A] (verification not implemented)

Time = 145.84 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.50 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^6} \, dx=\frac {c^{3} x}{e^{6}} + \frac {3 c^{2} \left (b e - 2 c d\right ) \log {\left (d + e x \right )}}{e^{7}} + \frac {- b^{3} d^{3} e^{3} - 12 b^{2} c d^{4} e^{2} + 137 b c^{2} d^{5} e - 174 c^{3} d^{6} + x^{4} \left (- 60 b^{2} c e^{6} + 300 b c^{2} d e^{5} - 300 c^{3} d^{2} e^{4}\right ) + x^{3} \left (- 10 b^{3} e^{6} - 120 b^{2} c d e^{5} + 900 b c^{2} d^{2} e^{4} - 1000 c^{3} d^{3} e^{3}\right ) + x^{2} \left (- 10 b^{3} d e^{5} - 120 b^{2} c d^{2} e^{4} + 1100 b c^{2} d^{3} e^{3} - 1300 c^{3} d^{4} e^{2}\right ) + x \left (- 5 b^{3} d^{2} e^{4} - 60 b^{2} c d^{3} e^{3} + 625 b c^{2} d^{4} e^{2} - 770 c^{3} d^{5} e\right )}{20 d^{5} e^{7} + 100 d^{4} e^{8} x + 200 d^{3} e^{9} x^{2} + 200 d^{2} e^{10} x^{3} + 100 d e^{11} x^{4} + 20 e^{12} x^{5}} \]

[In]

integrate((c*x**2+b*x)**3/(e*x+d)**6,x)

[Out]

c**3*x/e**6 + 3*c**2*(b*e - 2*c*d)*log(d + e*x)/e**7 + (-b**3*d**3*e**3 - 12*b**2*c*d**4*e**2 + 137*b*c**2*d**
5*e - 174*c**3*d**6 + x**4*(-60*b**2*c*e**6 + 300*b*c**2*d*e**5 - 300*c**3*d**2*e**4) + x**3*(-10*b**3*e**6 -
120*b**2*c*d*e**5 + 900*b*c**2*d**2*e**4 - 1000*c**3*d**3*e**3) + x**2*(-10*b**3*d*e**5 - 120*b**2*c*d**2*e**4
+ 1100*b*c**2*d**3*e**3 - 1300*c**3*d**4*e**2) + x*(-5*b**3*d**2*e**4 - 60*b**2*c*d**3*e**3 + 625*b*c**2*d**4
*e**2 - 770*c**3*d**5*e))/(20*d**5*e**7 + 100*d**4*e**8*x + 200*d**3*e**9*x**2 + 200*d**2*e**10*x**3 + 100*d*e
**11*x**4 + 20*e**12*x**5)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.43 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^6} \, dx=-\frac {174 \, c^{3} d^{6} - 137 \, b c^{2} d^{5} e + 12 \, b^{2} c d^{4} e^{2} + b^{3} d^{3} e^{3} + 60 \, {\left (5 \, c^{3} d^{2} e^{4} - 5 \, b c^{2} d e^{5} + b^{2} c e^{6}\right )} x^{4} + 10 \, {\left (100 \, c^{3} d^{3} e^{3} - 90 \, b c^{2} d^{2} e^{4} + 12 \, b^{2} c d e^{5} + b^{3} e^{6}\right )} x^{3} + 10 \, {\left (130 \, c^{3} d^{4} e^{2} - 110 \, b c^{2} d^{3} e^{3} + 12 \, b^{2} c d^{2} e^{4} + b^{3} d e^{5}\right )} x^{2} + 5 \, {\left (154 \, c^{3} d^{5} e - 125 \, b c^{2} d^{4} e^{2} + 12 \, b^{2} c d^{3} e^{3} + b^{3} d^{2} e^{4}\right )} x}{20 \, {\left (e^{12} x^{5} + 5 \, d e^{11} x^{4} + 10 \, d^{2} e^{10} x^{3} + 10 \, d^{3} e^{9} x^{2} + 5 \, d^{4} e^{8} x + d^{5} e^{7}\right )}} + \frac {c^{3} x}{e^{6}} - \frac {3 \, {\left (2 \, c^{3} d - b c^{2} e\right )} \log \left (e x + d\right )}{e^{7}} \]

[In]

integrate((c*x^2+b*x)^3/(e*x+d)^6,x, algorithm="maxima")

[Out]

-1/20*(174*c^3*d^6 - 137*b*c^2*d^5*e + 12*b^2*c*d^4*e^2 + b^3*d^3*e^3 + 60*(5*c^3*d^2*e^4 - 5*b*c^2*d*e^5 + b^
2*c*e^6)*x^4 + 10*(100*c^3*d^3*e^3 - 90*b*c^2*d^2*e^4 + 12*b^2*c*d*e^5 + b^3*e^6)*x^3 + 10*(130*c^3*d^4*e^2 -
110*b*c^2*d^3*e^3 + 12*b^2*c*d^2*e^4 + b^3*d*e^5)*x^2 + 5*(154*c^3*d^5*e - 125*b*c^2*d^4*e^2 + 12*b^2*c*d^3*e^
3 + b^3*d^2*e^4)*x)/(e^12*x^5 + 5*d*e^11*x^4 + 10*d^2*e^10*x^3 + 10*d^3*e^9*x^2 + 5*d^4*e^8*x + d^5*e^7) + c^3
*x/e^6 - 3*(2*c^3*d - b*c^2*e)*log(e*x + d)/e^7

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.22 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^6} \, dx=\frac {c^{3} x}{e^{6}} - \frac {3 \, {\left (2 \, c^{3} d - b c^{2} e\right )} \log \left ({\left | e x + d \right |}\right )}{e^{7}} - \frac {174 \, c^{3} d^{6} - 137 \, b c^{2} d^{5} e + 12 \, b^{2} c d^{4} e^{2} + b^{3} d^{3} e^{3} + 60 \, {\left (5 \, c^{3} d^{2} e^{4} - 5 \, b c^{2} d e^{5} + b^{2} c e^{6}\right )} x^{4} + 10 \, {\left (100 \, c^{3} d^{3} e^{3} - 90 \, b c^{2} d^{2} e^{4} + 12 \, b^{2} c d e^{5} + b^{3} e^{6}\right )} x^{3} + 10 \, {\left (130 \, c^{3} d^{4} e^{2} - 110 \, b c^{2} d^{3} e^{3} + 12 \, b^{2} c d^{2} e^{4} + b^{3} d e^{5}\right )} x^{2} + 5 \, {\left (154 \, c^{3} d^{5} e - 125 \, b c^{2} d^{4} e^{2} + 12 \, b^{2} c d^{3} e^{3} + b^{3} d^{2} e^{4}\right )} x}{20 \, {\left (e x + d\right )}^{5} e^{7}} \]

[In]

integrate((c*x^2+b*x)^3/(e*x+d)^6,x, algorithm="giac")

[Out]

c^3*x/e^6 - 3*(2*c^3*d - b*c^2*e)*log(abs(e*x + d))/e^7 - 1/20*(174*c^3*d^6 - 137*b*c^2*d^5*e + 12*b^2*c*d^4*e
^2 + b^3*d^3*e^3 + 60*(5*c^3*d^2*e^4 - 5*b*c^2*d*e^5 + b^2*c*e^6)*x^4 + 10*(100*c^3*d^3*e^3 - 90*b*c^2*d^2*e^4
+ 12*b^2*c*d*e^5 + b^3*e^6)*x^3 + 10*(130*c^3*d^4*e^2 - 110*b*c^2*d^3*e^3 + 12*b^2*c*d^2*e^4 + b^3*d*e^5)*x^2
+ 5*(154*c^3*d^5*e - 125*b*c^2*d^4*e^2 + 12*b^2*c*d^3*e^3 + b^3*d^2*e^4)*x)/((e*x + d)^5*e^7)

Mupad [B] (verification not implemented)

Time = 9.64 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.43 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^6} \, dx=\frac {c^3\,x}{e^6}-\frac {x^4\,\left (3\,b^2\,c\,e^5-15\,b\,c^2\,d\,e^4+15\,c^3\,d^2\,e^3\right )+x^2\,\left (\frac {b^3\,d\,e^4}{2}+6\,b^2\,c\,d^2\,e^3-55\,b\,c^2\,d^3\,e^2+65\,c^3\,d^4\,e\right )+x\,\left (\frac {b^3\,d^2\,e^3}{4}+3\,b^2\,c\,d^3\,e^2-\frac {125\,b\,c^2\,d^4\,e}{4}+\frac {77\,c^3\,d^5}{2}\right )+\frac {b^3\,d^3\,e^3+12\,b^2\,c\,d^4\,e^2-137\,b\,c^2\,d^5\,e+174\,c^3\,d^6}{20\,e}+x^3\,\left (\frac {b^3\,e^5}{2}+6\,b^2\,c\,d\,e^4-45\,b\,c^2\,d^2\,e^3+50\,c^3\,d^3\,e^2\right )}{d^5\,e^6+5\,d^4\,e^7\,x+10\,d^3\,e^8\,x^2+10\,d^2\,e^9\,x^3+5\,d\,e^{10}\,x^4+e^{11}\,x^5}-\frac {\ln \left (d+e\,x\right )\,\left (6\,c^3\,d-3\,b\,c^2\,e\right )}{e^7} \]

[In]

int((b*x + c*x^2)^3/(d + e*x)^6,x)

[Out]

(c^3*x)/e^6 - (x^4*(3*b^2*c*e^5 + 15*c^3*d^2*e^3 - 15*b*c^2*d*e^4) + x^2*((b^3*d*e^4)/2 + 65*c^3*d^4*e - 55*b*
c^2*d^3*e^2 + 6*b^2*c*d^2*e^3) + x*((77*c^3*d^5)/2 + (b^3*d^2*e^3)/4 + 3*b^2*c*d^3*e^2 - (125*b*c^2*d^4*e)/4)
+ (174*c^3*d^6 + b^3*d^3*e^3 + 12*b^2*c*d^4*e^2 - 137*b*c^2*d^5*e)/(20*e) + x^3*((b^3*e^5)/2 + 50*c^3*d^3*e^2
- 45*b*c^2*d^2*e^3 + 6*b^2*c*d*e^4))/(d^5*e^6 + e^11*x^5 + 5*d^4*e^7*x + 5*d*e^10*x^4 + 10*d^3*e^8*x^2 + 10*d^
2*e^9*x^3) - (log(d + e*x)*(6*c^3*d - 3*b*c^2*e))/e^7