\(\int \frac {(b x+c x^2)^3}{(d+e x)^3} \, dx\) [251]

Optimal result

Integrand size = 19, antiderivative size = 200 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^3} \, dx=-\frac {(c d-b e) \left (10 c^2 d^2-8 b c d e+b^2 e^2\right ) x}{e^6}+\frac {3 c (c d-b e) (2 c d-b e) x^2}{2 e^5}-\frac {c^2 (c d-b e) x^3}{e^4}+\frac {c^3 x^4}{4 e^3}-\frac {d^3 (c d-b e)^3}{2 e^7 (d+e x)^2}+\frac {3 d^2 (c d-b e)^2 (2 c d-b e)}{e^7 (d+e x)}+\frac {3 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) \log (d+e x)}{e^7} \]

[Out]

-(-b*e+c*d)*(b^2*e^2-8*b*c*d*e+10*c^2*d^2)*x/e^6+3/2*c*(-b*e+c*d)*(-b*e+2*c*d)*x^2/e^5-c^2*(-b*e+c*d)*x^3/e^4+
1/4*c^3*x^4/e^3-1/2*d^3*(-b*e+c*d)^3/e^7/(e*x+d)^2+3*d^2*(-b*e+c*d)^2*(-b*e+2*c*d)/e^7/(e*x+d)+3*d*(-b*e+c*d)*
(b^2*e^2-5*b*c*d*e+5*c^2*d^2)*ln(e*x+d)/e^7

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 200, 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} \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^3} \, dx=\frac {3 d (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right ) \log (d+e x)}{e^7}-\frac {x (c d-b e) \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )}{e^6}-\frac {c^2 x^3 (c d-b e)}{e^4}-\frac {d^3 (c d-b e)^3}{2 e^7 (d+e x)^2}+\frac {3 d^2 (c d-b e)^2 (2 c d-b e)}{e^7 (d+e x)}+\frac {3 c x^2 (c d-b e) (2 c d-b e)}{2 e^5}+\frac {c^3 x^4}{4 e^3} \]

[In]

Int[(b*x + c*x^2)^3/(d + e*x)^3,x]

[Out]

-(((c*d - b*e)*(10*c^2*d^2 - 8*b*c*d*e + b^2*e^2)*x)/e^6) + (3*c*(c*d - b*e)*(2*c*d - b*e)*x^2)/(2*e^5) - (c^2
*(c*d - b*e)*x^3)/e^4 + (c^3*x^4)/(4*e^3) - (d^3*(c*d - b*e)^3)/(2*e^7*(d + e*x)^2) + (3*d^2*(c*d - b*e)^2*(2*
c*d - b*e))/(e^7*(d + e*x)) + (3*d*(c*d - b*e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*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 d-b e) \left (-10 c^2 d^2+8 b c d e-b^2 e^2\right )}{e^6}+\frac {3 c (c d-b e) (2 c d-b e) x}{e^5}-\frac {3 c^2 (c d-b e) x^2}{e^4}+\frac {c^3 x^3}{e^3}+\frac {d^3 (c d-b e)^3}{e^6 (d+e x)^3}-\frac {3 d^2 (c d-b e)^2 (2 c d-b e)}{e^6 (d+e x)^2}+\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)}\right ) \, dx \\ & = -\frac {(c d-b e) \left (10 c^2 d^2-8 b c d e+b^2 e^2\right ) x}{e^6}+\frac {3 c (c d-b e) (2 c d-b e) x^2}{2 e^5}-\frac {c^2 (c d-b e) x^3}{e^4}+\frac {c^3 x^4}{4 e^3}-\frac {d^3 (c d-b e)^3}{2 e^7 (d+e x)^2}+\frac {3 d^2 (c d-b e)^2 (2 c d-b e)}{e^7 (d+e x)}+\frac {3 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) \log (d+e x)}{e^7} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.04 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^3} \, dx=\frac {4 e \left (-10 c^3 d^3+18 b c^2 d^2 e-9 b^2 c d e^2+b^3 e^3\right ) x+6 c e^2 \left (2 c^2 d^2-3 b c d e+b^2 e^2\right ) x^2-4 c^2 e^3 (c d-b e) x^3+c^3 e^4 x^4-\frac {2 d^3 (c d-b e)^3}{(d+e x)^2}+\frac {12 d^2 (c d-b e)^2 (2 c d-b e)}{d+e x}+12 d \left (5 c^3 d^3-10 b c^2 d^2 e+6 b^2 c d e^2-b^3 e^3\right ) \log (d+e x)}{4 e^7} \]

[In]

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

[Out]

(4*e*(-10*c^3*d^3 + 18*b*c^2*d^2*e - 9*b^2*c*d*e^2 + b^3*e^3)*x + 6*c*e^2*(2*c^2*d^2 - 3*b*c*d*e + b^2*e^2)*x^
2 - 4*c^2*e^3*(c*d - b*e)*x^3 + c^3*e^4*x^4 - (2*d^3*(c*d - b*e)^3)/(d + e*x)^2 + (12*d^2*(c*d - b*e)^2*(2*c*d
- b*e))/(d + e*x) + 12*d*(5*c^3*d^3 - 10*b*c^2*d^2*e + 6*b^2*c*d*e^2 - b^3*e^3)*Log[d + e*x])/(4*e^7)

Maple [A] (verified)

Time = 2.03 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.30

method	result	size
norman	\(\frac {\frac {\left (b^{3} e^{3}-6 b^{2} d \,e^{2} c +10 b \,c^{2} d^{2} e -5 c^{3} d^{3}\right ) x^{3}}{e^{4}}+\frac {c^{3} x^{6}}{4 e}-\frac {d^{2} \left (9 b^{3} d \,e^{3}-54 d^{2} e^{2} b^{2} c +90 d^{3} e b \,c^{2}-45 d^{4} c^{3}\right )}{2 e^{7}}+\frac {c \left (6 b^{2} e^{2}-10 b c d e +5 c^{2} d^{2}\right ) x^{4}}{4 e^{3}}+\frac {c^{2} \left (2 b e -c d \right ) x^{5}}{2 e^{2}}-\frac {2 d \left (3 b^{3} d \,e^{3}-18 d^{2} e^{2} b^{2} c +30 d^{3} e b \,c^{2}-15 d^{4} c^{3}\right ) x}{e^{6}}}{\left (e x +d \right )^{2}}-\frac {3 d \left (b^{3} e^{3}-6 b^{2} d \,e^{2} c +10 b \,c^{2} d^{2} e -5 c^{3} d^{3}\right ) \ln \left (e x +d \right )}{e^{7}}\)	\(260\)
default	\(\frac {\frac {1}{4} c^{3} x^{4} e^{3}+b \,c^{2} e^{3} x^{3}-c^{3} d \,e^{2} x^{3}+\frac {3}{2} b^{2} c \,e^{3} x^{2}-\frac {9}{2} b \,c^{2} d \,e^{2} x^{2}+3 c^{3} d^{2} e \,x^{2}+b^{3} e^{3} x -9 b^{2} d \,e^{2} c x +18 b \,c^{2} d^{2} e x -10 c^{3} d^{3} x}{e^{6}}-\frac {3 d^{2} \left (b^{3} e^{3}-4 b^{2} d \,e^{2} c +5 b \,c^{2} d^{2} e -2 c^{3} d^{3}\right )}{e^{7} \left (e x +d \right )}+\frac {d^{3} \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}{2 e^{7} \left (e x +d \right )^{2}}-\frac {3 d \left (b^{3} e^{3}-6 b^{2} d \,e^{2} c +10 b \,c^{2} d^{2} e -5 c^{3} d^{3}\right ) \ln \left (e x +d \right )}{e^{7}}\)	\(267\)
risch	\(\frac {c^{3} x^{4}}{4 e^{3}}+\frac {b \,c^{2} x^{3}}{e^{3}}-\frac {c^{3} d \,x^{3}}{e^{4}}+\frac {3 b^{2} c \,x^{2}}{2 e^{3}}-\frac {9 b \,c^{2} d \,x^{2}}{2 e^{4}}+\frac {3 c^{3} d^{2} x^{2}}{e^{5}}+\frac {b^{3} x}{e^{3}}-\frac {9 b^{2} d c x}{e^{4}}+\frac {18 b \,c^{2} d^{2} x}{e^{5}}-\frac {10 c^{3} d^{3} x}{e^{6}}+\frac {\left (-3 b^{3} d^{2} e^{3}+12 b^{2} c \,d^{3} e^{2}-15 b \,c^{2} d^{4} e +6 c^{3} d^{5}\right ) x -\frac {d^{3} \left (5 b^{3} e^{3}-21 b^{2} d \,e^{2} c +27 b \,c^{2} d^{2} e -11 c^{3} d^{3}\right )}{2 e}}{e^{6} \left (e x +d \right )^{2}}-\frac {3 d \ln \left (e x +d \right ) b^{3}}{e^{4}}+\frac {18 d^{2} \ln \left (e x +d \right ) b^{2} c}{e^{5}}-\frac {30 d^{3} \ln \left (e x +d \right ) b \,c^{2}}{e^{6}}+\frac {15 d^{4} \ln \left (e x +d \right ) c^{3}}{e^{7}}\)	\(288\)
parallelrisch	\(-\frac {240 \ln \left (e x +d \right ) x b \,c^{2} d^{4} e^{2}+18 b^{3} d^{3} e^{3}-144 \ln \left (e x +d \right ) x \,b^{2} c \,d^{3} e^{3}+12 \ln \left (e x +d \right ) x^{2} b^{3} d \,e^{5}-4 x^{5} b \,c^{2} e^{6}-6 x^{4} b^{2} c \,e^{6}-5 x^{4} c^{3} d^{2} e^{4}+20 x^{3} c^{3} d^{3} e^{3}+12 \ln \left (e x +d \right ) b^{3} d^{3} e^{3}+24 x \,b^{3} d^{2} e^{4}-120 x \,c^{3} d^{5} e -90 c^{3} d^{6}+120 \ln \left (e x +d \right ) x^{2} b \,c^{2} d^{3} e^{3}+24 \ln \left (e x +d \right ) x \,b^{3} d^{2} e^{4}-120 \ln \left (e x +d \right ) x \,c^{3} d^{5} e -x^{6} c^{3} e^{6}-72 \ln \left (e x +d \right ) x^{2} b^{2} c \,d^{2} e^{4}+10 x^{4} b \,c^{2} d \,e^{5}+24 x^{3} b^{2} c d \,e^{5}-60 \ln \left (e x +d \right ) x^{2} c^{3} d^{4} e^{2}-108 b^{2} c \,d^{4} e^{2}+180 b \,c^{2} d^{5} e +2 x^{5} c^{3} d \,e^{5}-40 x^{3} b \,c^{2} d^{2} e^{4}-72 \ln \left (e x +d \right ) b^{2} c \,d^{4} e^{2}+120 \ln \left (e x +d \right ) b \,c^{2} d^{5} e -144 x \,b^{2} c \,d^{3} e^{3}-4 x^{3} b^{3} e^{6}-60 \ln \left (e x +d \right ) c^{3} d^{6}+240 x b \,c^{2} d^{4} e^{2}}{4 e^{7} \left (e x +d \right )^{2}}\)	\(447\)

[In]

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

[Out]

(1/e^4*(b^3*e^3-6*b^2*c*d*e^2+10*b*c^2*d^2*e-5*c^3*d^3)*x^3+1/4*c^3*x^6/e-1/2*d^2*(9*b^3*d*e^3-54*b^2*c*d^2*e^
2+90*b*c^2*d^3*e-45*c^3*d^4)/e^7+1/4*c*(6*b^2*e^2-10*b*c*d*e+5*c^2*d^2)/e^3*x^4+1/2*c^2*(2*b*e-c*d)/e^2*x^5-2*
d*(3*b^3*d*e^3-18*b^2*c*d^2*e^2+30*b*c^2*d^3*e-15*c^3*d^4)/e^6*x)/(e*x+d)^2-3*d/e^7*(b^3*e^3-6*b^2*c*d*e^2+10*
b*c^2*d^2*e-5*c^3*d^3)*ln(e*x+d)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 428 vs. \(2 (194) = 388\).

Time = 0.27 (sec) , antiderivative size = 428, normalized size of antiderivative = 2.14 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^3} \, dx=\frac {c^{3} e^{6} x^{6} + 22 \, c^{3} d^{6} - 54 \, b c^{2} d^{5} e + 42 \, b^{2} c d^{4} e^{2} - 10 \, b^{3} d^{3} e^{3} - 2 \, {\left (c^{3} d e^{5} - 2 \, b c^{2} e^{6}\right )} x^{5} + {\left (5 \, c^{3} d^{2} e^{4} - 10 \, b c^{2} d e^{5} + 6 \, b^{2} c e^{6}\right )} x^{4} - 4 \, {\left (5 \, c^{3} d^{3} e^{3} - 10 \, b c^{2} d^{2} e^{4} + 6 \, b^{2} c d e^{5} - b^{3} e^{6}\right )} x^{3} - 2 \, {\left (34 \, c^{3} d^{4} e^{2} - 63 \, b c^{2} d^{3} e^{3} + 33 \, b^{2} c d^{2} e^{4} - 4 \, b^{3} d e^{5}\right )} x^{2} - 4 \, {\left (4 \, c^{3} d^{5} e - 3 \, b c^{2} d^{4} e^{2} - 3 \, b^{2} c d^{3} e^{3} + 2 \, b^{3} d^{2} e^{4}\right )} x + 12 \, {\left (5 \, c^{3} d^{6} - 10 \, b c^{2} d^{5} e + 6 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3} + {\left (5 \, c^{3} d^{4} e^{2} - 10 \, b c^{2} d^{3} e^{3} + 6 \, b^{2} c d^{2} e^{4} - b^{3} d e^{5}\right )} x^{2} + 2 \, {\left (5 \, c^{3} d^{5} e - 10 \, b c^{2} d^{4} e^{2} + 6 \, b^{2} c d^{3} e^{3} - b^{3} d^{2} e^{4}\right )} x\right )} \log \left (e x + d\right )}{4 \, {\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} e^{7}\right )}} \]

[In]

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

[Out]

1/4*(c^3*e^6*x^6 + 22*c^3*d^6 - 54*b*c^2*d^5*e + 42*b^2*c*d^4*e^2 - 10*b^3*d^3*e^3 - 2*(c^3*d*e^5 - 2*b*c^2*e^
6)*x^5 + (5*c^3*d^2*e^4 - 10*b*c^2*d*e^5 + 6*b^2*c*e^6)*x^4 - 4*(5*c^3*d^3*e^3 - 10*b*c^2*d^2*e^4 + 6*b^2*c*d*
e^5 - b^3*e^6)*x^3 - 2*(34*c^3*d^4*e^2 - 63*b*c^2*d^3*e^3 + 33*b^2*c*d^2*e^4 - 4*b^3*d*e^5)*x^2 - 4*(4*c^3*d^5
*e - 3*b*c^2*d^4*e^2 - 3*b^2*c*d^3*e^3 + 2*b^3*d^2*e^4)*x + 12*(5*c^3*d^6 - 10*b*c^2*d^5*e + 6*b^2*c*d^4*e^2 -
b^3*d^3*e^3 + (5*c^3*d^4*e^2 - 10*b*c^2*d^3*e^3 + 6*b^2*c*d^2*e^4 - b^3*d*e^5)*x^2 + 2*(5*c^3*d^5*e - 10*b*c^
2*d^4*e^2 + 6*b^2*c*d^3*e^3 - b^3*d^2*e^4)*x)*log(e*x + d))/(e^9*x^2 + 2*d*e^8*x + d^2*e^7)

Sympy [A] (verification not implemented)

Time = 0.83 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.42 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^3} \, dx=\frac {c^{3} x^{4}}{4 e^{3}} - \frac {3 d \left (b e - c d\right ) \left (b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right ) \log {\left (d + e x \right )}}{e^{7}} + x^{3} \left (\frac {b c^{2}}{e^{3}} - \frac {c^{3} d}{e^{4}}\right ) + x^{2} \cdot \left (\frac {3 b^{2} c}{2 e^{3}} - \frac {9 b c^{2} d}{2 e^{4}} + \frac {3 c^{3} d^{2}}{e^{5}}\right ) + x \left (\frac {b^{3}}{e^{3}} - \frac {9 b^{2} c d}{e^{4}} + \frac {18 b c^{2} d^{2}}{e^{5}} - \frac {10 c^{3} d^{3}}{e^{6}}\right ) + \frac {- 5 b^{3} d^{3} e^{3} + 21 b^{2} c d^{4} e^{2} - 27 b c^{2} d^{5} e + 11 c^{3} d^{6} + x \left (- 6 b^{3} d^{2} e^{4} + 24 b^{2} c d^{3} e^{3} - 30 b c^{2} d^{4} e^{2} + 12 c^{3} d^{5} e\right )}{2 d^{2} e^{7} + 4 d e^{8} x + 2 e^{9} x^{2}} \]

[In]

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

[Out]

c**3*x**4/(4*e**3) - 3*d*(b*e - c*d)*(b**2*e**2 - 5*b*c*d*e + 5*c**2*d**2)*log(d + e*x)/e**7 + x**3*(b*c**2/e*
*3 - c**3*d/e**4) + x**2*(3*b**2*c/(2*e**3) - 9*b*c**2*d/(2*e**4) + 3*c**3*d**2/e**5) + x*(b**3/e**3 - 9*b**2*
c*d/e**4 + 18*b*c**2*d**2/e**5 - 10*c**3*d**3/e**6) + (-5*b**3*d**3*e**3 + 21*b**2*c*d**4*e**2 - 27*b*c**2*d**
5*e + 11*c**3*d**6 + x*(-6*b**3*d**2*e**4 + 24*b**2*c*d**3*e**3 - 30*b*c**2*d**4*e**2 + 12*c**3*d**5*e))/(2*d*
*2*e**7 + 4*d*e**8*x + 2*e**9*x**2)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.40 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^3} \, dx=\frac {11 \, c^{3} d^{6} - 27 \, b c^{2} d^{5} e + 21 \, b^{2} c d^{4} e^{2} - 5 \, b^{3} d^{3} e^{3} + 6 \, {\left (2 \, c^{3} d^{5} e - 5 \, b c^{2} d^{4} e^{2} + 4 \, b^{2} c d^{3} e^{3} - b^{3} d^{2} e^{4}\right )} x}{2 \, {\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} e^{7}\right )}} + \frac {c^{3} e^{3} x^{4} - 4 \, {\left (c^{3} d e^{2} - b c^{2} e^{3}\right )} x^{3} + 6 \, {\left (2 \, c^{3} d^{2} e - 3 \, b c^{2} d e^{2} + b^{2} c e^{3}\right )} x^{2} - 4 \, {\left (10 \, c^{3} d^{3} - 18 \, b c^{2} d^{2} e + 9 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} x}{4 \, e^{6}} + \frac {3 \, {\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \, b^{2} c d^{2} e^{2} - b^{3} d e^{3}\right )} \log \left (e x + d\right )}{e^{7}} \]

[In]

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

[Out]

1/2*(11*c^3*d^6 - 27*b*c^2*d^5*e + 21*b^2*c*d^4*e^2 - 5*b^3*d^3*e^3 + 6*(2*c^3*d^5*e - 5*b*c^2*d^4*e^2 + 4*b^2
*c*d^3*e^3 - b^3*d^2*e^4)*x)/(e^9*x^2 + 2*d*e^8*x + d^2*e^7) + 1/4*(c^3*e^3*x^4 - 4*(c^3*d*e^2 - b*c^2*e^3)*x^
3 + 6*(2*c^3*d^2*e - 3*b*c^2*d*e^2 + b^2*c*e^3)*x^2 - 4*(10*c^3*d^3 - 18*b*c^2*d^2*e + 9*b^2*c*d*e^2 - b^3*e^3
)*x)/e^6 + 3*(5*c^3*d^4 - 10*b*c^2*d^3*e + 6*b^2*c*d^2*e^2 - b^3*d*e^3)*log(e*x + d)/e^7

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.40 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^3} \, dx=\frac {3 \, {\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \, b^{2} c d^{2} e^{2} - b^{3} d e^{3}\right )} \log \left ({\left | e x + d \right |}\right )}{e^{7}} + \frac {11 \, c^{3} d^{6} - 27 \, b c^{2} d^{5} e + 21 \, b^{2} c d^{4} e^{2} - 5 \, b^{3} d^{3} e^{3} + 6 \, {\left (2 \, c^{3} d^{5} e - 5 \, b c^{2} d^{4} e^{2} + 4 \, b^{2} c d^{3} e^{3} - b^{3} d^{2} e^{4}\right )} x}{2 \, {\left (e x + d\right )}^{2} e^{7}} + \frac {c^{3} e^{9} x^{4} - 4 \, c^{3} d e^{8} x^{3} + 4 \, b c^{2} e^{9} x^{3} + 12 \, c^{3} d^{2} e^{7} x^{2} - 18 \, b c^{2} d e^{8} x^{2} + 6 \, b^{2} c e^{9} x^{2} - 40 \, c^{3} d^{3} e^{6} x + 72 \, b c^{2} d^{2} e^{7} x - 36 \, b^{2} c d e^{8} x + 4 \, b^{3} e^{9} x}{4 \, e^{12}} \]

[In]

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

[Out]

3*(5*c^3*d^4 - 10*b*c^2*d^3*e + 6*b^2*c*d^2*e^2 - b^3*d*e^3)*log(abs(e*x + d))/e^7 + 1/2*(11*c^3*d^6 - 27*b*c^
2*d^5*e + 21*b^2*c*d^4*e^2 - 5*b^3*d^3*e^3 + 6*(2*c^3*d^5*e - 5*b*c^2*d^4*e^2 + 4*b^2*c*d^3*e^3 - b^3*d^2*e^4)
*x)/((e*x + d)^2*e^7) + 1/4*(c^3*e^9*x^4 - 4*c^3*d*e^8*x^3 + 4*b*c^2*e^9*x^3 + 12*c^3*d^2*e^7*x^2 - 18*b*c^2*d
*e^8*x^2 + 6*b^2*c*e^9*x^2 - 40*c^3*d^3*e^6*x + 72*b*c^2*d^2*e^7*x - 36*b^2*c*d*e^8*x + 4*b^3*e^9*x)/e^12

Mupad [B] (verification not implemented)

Time = 9.55 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.76 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^3} \, dx=x^3\,\left (\frac {b\,c^2}{e^3}-\frac {c^3\,d}{e^4}\right )-x^2\,\left (\frac {3\,d\,\left (\frac {3\,b\,c^2}{e^3}-\frac {3\,c^3\,d}{e^4}\right )}{2\,e}-\frac {3\,b^2\,c}{2\,e^3}+\frac {3\,c^3\,d^2}{2\,e^5}\right )+\frac {x\,\left (-3\,b^3\,d^2\,e^3+12\,b^2\,c\,d^3\,e^2-15\,b\,c^2\,d^4\,e+6\,c^3\,d^5\right )+\frac {-5\,b^3\,d^3\,e^3+21\,b^2\,c\,d^4\,e^2-27\,b\,c^2\,d^5\,e+11\,c^3\,d^6}{2\,e}}{d^2\,e^6+2\,d\,e^7\,x+e^8\,x^2}+x\,\left (\frac {b^3}{e^3}+\frac {3\,d\,\left (\frac {3\,d\,\left (\frac {3\,b\,c^2}{e^3}-\frac {3\,c^3\,d}{e^4}\right )}{e}-\frac {3\,b^2\,c}{e^3}+\frac {3\,c^3\,d^2}{e^5}\right )}{e}-\frac {c^3\,d^3}{e^6}-\frac {3\,d^2\,\left (\frac {3\,b\,c^2}{e^3}-\frac {3\,c^3\,d}{e^4}\right )}{e^2}\right )+\frac {c^3\,x^4}{4\,e^3}+\frac {\ln \left (d+e\,x\right )\,\left (-3\,b^3\,d\,e^3+18\,b^2\,c\,d^2\,e^2-30\,b\,c^2\,d^3\,e+15\,c^3\,d^4\right )}{e^7} \]

[In]

int((b*x + c*x^2)^3/(d + e*x)^3,x)

[Out]

x^3*((b*c^2)/e^3 - (c^3*d)/e^4) - x^2*((3*d*((3*b*c^2)/e^3 - (3*c^3*d)/e^4))/(2*e) - (3*b^2*c)/(2*e^3) + (3*c^
3*d^2)/(2*e^5)) + (x*(6*c^3*d^5 - 3*b^3*d^2*e^3 + 12*b^2*c*d^3*e^2 - 15*b*c^2*d^4*e) + (11*c^3*d^6 - 5*b^3*d^3
*e^3 + 21*b^2*c*d^4*e^2 - 27*b*c^2*d^5*e)/(2*e))/(d^2*e^6 + e^8*x^2 + 2*d*e^7*x) + x*(b^3/e^3 + (3*d*((3*d*((3
*b*c^2)/e^3 - (3*c^3*d)/e^4))/e - (3*b^2*c)/e^3 + (3*c^3*d^2)/e^5))/e - (c^3*d^3)/e^6 - (3*d^2*((3*b*c^2)/e^3
- (3*c^3*d)/e^4))/e^2) + (c^3*x^4)/(4*e^3) + (log(d + e*x)*(15*c^3*d^4 - 3*b^3*d*e^3 + 18*b^2*c*d^2*e^2 - 30*b
*c^2*d^3*e))/e^7