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

Optimal. Leaf size=218 \[ -\frac{3 c \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^7 (d+e x)}+\frac{(2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{2 e^7 (d+e x)^2}-\frac{d (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^7 (d+e x)^3}-\frac{3 c^2 (2 c d-b e) \log (d+e x)}{e^7}+\frac{3 d^2 (c d-b e)^2 (2 c d-b e)}{4 e^7 (d+e x)^4}-\frac{d^3 (c d-b e)^3}{5 e^7 (d+e x)^5}+\frac{c^3 x}{e^6} \]

[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

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Rubi [A]  time = 0.183574, antiderivative size = 218, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {698} \[ -\frac{3 c \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^7 (d+e x)}+\frac{(2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{2 e^7 (d+e x)^2}-\frac{d (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^7 (d+e x)^3}-\frac{3 c^2 (2 c d-b e) \log (d+e x)}{e^7}+\frac{3 d^2 (c d-b e)^2 (2 c d-b e)}{4 e^7 (d+e x)^4}-\frac{d^3 (c d-b e)^3}{5 e^7 (d+e x)^5}+\frac{c^3 x}{e^6} \]

Antiderivative was successfully verified.

[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 698

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*} \int \frac{\left (b x+c x^2\right )^3}{(d+e x)^6} \, dx &=\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]  time = 0.108065, size = 242, normalized size = 1.11 \[ -\frac{12 b^2 c e^2 \left (10 d^2 e^2 x^2+5 d^3 e x+d^4+10 d e^3 x^3+5 e^4 x^4\right )+b^3 e^3 \left (5 d^2 e x+d^3+10 d e^2 x^2+10 e^3 x^3\right )-b c^2 d e \left (1100 d^2 e^2 x^2+625 d^3 e x+137 d^4+900 d e^3 x^3+300 e^4 x^4\right )+60 c^2 (d+e x)^5 (2 c d-b e) \log (d+e x)+2 c^3 \left (600 d^4 e^2 x^2+400 d^3 e^3 x^3+50 d^2 e^4 x^4+375 d^5 e x+87 d^6-50 d e^5 x^5-10 e^6 x^6\right )}{20 e^7 (d+e x)^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(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])/(20*e^7*(d + e*x)^5)

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Maple [A]  time = 0.053, size = 379, normalized size = 1.7 \begin{align*}{\frac{{c}^{3}x}{{e}^{6}}}-{\frac{{b}^{3}}{2\,{e}^{4} \left ( ex+d \right ) ^{2}}}+6\,{\frac{{b}^{2}cd}{{e}^{5} \left ( ex+d \right ) ^{2}}}-15\,{\frac{b{c}^{2}{d}^{2}}{{e}^{6} \left ( ex+d \right ) ^{2}}}+10\,{\frac{{c}^{3}{d}^{3}}{{e}^{7} \left ( ex+d \right ) ^{2}}}+{\frac{{d}^{3}{b}^{3}}{5\,{e}^{4} \left ( ex+d \right ) ^{5}}}-{\frac{3\,{d}^{4}{b}^{2}c}{5\,{e}^{5} \left ( ex+d \right ) ^{5}}}+{\frac{3\,{d}^{5}b{c}^{2}}{5\,{e}^{6} \left ( ex+d \right ) ^{5}}}-{\frac{{c}^{3}{d}^{6}}{5\,{e}^{7} \left ( ex+d \right ) ^{5}}}+3\,{\frac{{c}^{2}\ln \left ( ex+d \right ) b}{{e}^{6}}}-6\,{\frac{{c}^{3}d\ln \left ( ex+d \right ) }{{e}^{7}}}+{\frac{d{b}^{3}}{{e}^{4} \left ( ex+d \right ) ^{3}}}-6\,{\frac{{b}^{2}{d}^{2}c}{{e}^{5} \left ( ex+d \right ) ^{3}}}+10\,{\frac{{d}^{3}b{c}^{2}}{{e}^{6} \left ( ex+d \right ) ^{3}}}-5\,{\frac{{c}^{3}{d}^{4}}{{e}^{7} \left ( ex+d \right ) ^{3}}}-3\,{\frac{{b}^{2}c}{{e}^{5} \left ( ex+d \right ) }}+15\,{\frac{b{c}^{2}d}{{e}^{6} \left ( ex+d \right ) }}-15\,{\frac{{c}^{3}{d}^{2}}{{e}^{7} \left ( ex+d \right ) }}-{\frac{3\,{d}^{2}{b}^{3}}{4\,{e}^{4} \left ( ex+d \right ) ^{4}}}+3\,{\frac{{d}^{3}{b}^{2}c}{{e}^{5} \left ( ex+d \right ) ^{4}}}-{\frac{15\,b{d}^{4}{c}^{2}}{4\,{e}^{6} \left ( ex+d \right ) ^{4}}}+{\frac{3\,{c}^{3}{d}^{5}}{2\,{e}^{7} \left ( ex+d \right ) ^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.18628, size = 420, normalized size = 1.93 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [B]  time = 1.7414, size = 953, normalized size = 4.37 \begin{align*} \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]  time = 38.9897, size = 326, normalized size = 1.5 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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 - 1
100*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)

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Giac [A]  time = 1.26617, size = 339, normalized size = 1.56 \begin{align*} c^{3} x e^{\left (-6\right )} - 3 \,{\left (2 \, c^{3} d - b c^{2} e\right )} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) - \frac{{\left (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\right )} e^{\left (-7\right )}}{20 \,{\left (x e + d\right )}^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)*e^(-7)*log(abs(x*e + d)) - 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^(-7)/(x*e + d)^5