3.255 $$\int \frac{(b x+c x^2)^3}{(d+e x)^7} \, dx$$

Optimal. Leaf size=228 $-\frac{3 c \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{2 e^7 (d+e x)^2}+\frac{(2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{3 e^7 (d+e x)^3}-\frac{3 d (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{4 e^7 (d+e x)^4}+\frac{3 c^2 (2 c d-b e)}{e^7 (d+e x)}+\frac{3 d^2 (c d-b e)^2 (2 c d-b e)}{5 e^7 (d+e x)^5}-\frac{d^3 (c d-b e)^3}{6 e^7 (d+e x)^6}+\frac{c^3 \log (d+e x)}{e^7}$

[Out]

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

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Rubi [A]  time = 0.174597, antiderivative size = 228, 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} $-\frac{3 c \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{2 e^7 (d+e x)^2}+\frac{(2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{3 e^7 (d+e x)^3}-\frac{3 d (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{4 e^7 (d+e x)^4}+\frac{3 c^2 (2 c d-b e)}{e^7 (d+e x)}+\frac{3 d^2 (c d-b e)^2 (2 c d-b e)}{5 e^7 (d+e x)^5}-\frac{d^3 (c d-b e)^3}{6 e^7 (d+e x)^6}+\frac{c^3 \log (d+e x)}{e^7}$

Antiderivative was successfully veriﬁed.

[In]

Int[(b*x + c*x^2)^3/(d + e*x)^7,x]

[Out]

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

Mathematica [A]  time = 0.0813611, size = 231, normalized size = 1.01 $\frac{-6 b^2 c e^2 \left (15 d^2 e^2 x^2+6 d^3 e x+d^4+20 d e^3 x^3+15 e^4 x^4\right )-b^3 e^3 \left (6 d^2 e x+d^3+15 d e^2 x^2+20 e^3 x^3\right )-30 b c^2 e \left (15 d^3 e^2 x^2+20 d^2 e^3 x^3+6 d^4 e x+d^5+15 d e^4 x^4+6 e^5 x^5\right )+c^3 d \left (1875 d^3 e^2 x^2+2200 d^2 e^3 x^3+822 d^4 e x+147 d^5+1350 d e^4 x^4+360 e^5 x^5\right )+60 c^3 (d+e x)^6 \log (d+e x)}{60 e^7 (d+e x)^6}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(b*x + c*x^2)^3/(d + e*x)^7,x]

[Out]

(-(b^3*e^3*(d^3 + 6*d^2*e*x + 15*d*e^2*x^2 + 20*e^3*x^3)) - 6*b^2*c*e^2*(d^4 + 6*d^3*e*x + 15*d^2*e^2*x^2 + 20
*d*e^3*x^3 + 15*e^4*x^4) - 30*b*c^2*e*(d^5 + 6*d^4*e*x + 15*d^3*e^2*x^2 + 20*d^2*e^3*x^3 + 15*d*e^4*x^4 + 6*e^
5*x^5) + c^3*d*(147*d^5 + 822*d^4*e*x + 1875*d^3*e^2*x^2 + 2200*d^2*e^3*x^3 + 1350*d*e^4*x^4 + 360*e^5*x^5) +
60*c^3*(d + e*x)^6*Log[d + e*x])/(60*e^7*(d + e*x)^6)

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Maple [A]  time = 0.05, size = 387, normalized size = 1.7 \begin{align*} -{\frac{3\,{b}^{2}c}{2\,{e}^{5} \left ( ex+d \right ) ^{2}}}+{\frac{15\,b{c}^{2}d}{2\,{e}^{6} \left ( ex+d \right ) ^{2}}}-{\frac{15\,{c}^{3}{d}^{2}}{2\,{e}^{7} \left ( ex+d \right ) ^{2}}}-{\frac{3\,{b}^{3}{d}^{2}}{5\,{e}^{4} \left ( ex+d \right ) ^{5}}}+{\frac{12\,{b}^{2}c{d}^{3}}{5\,{e}^{5} \left ( ex+d \right ) ^{5}}}-3\,{\frac{b{c}^{2}{d}^{4}}{{e}^{6} \left ( ex+d \right ) ^{5}}}+{\frac{6\,{c}^{3}{d}^{5}}{5\,{e}^{7} \left ( ex+d \right ) ^{5}}}+{\frac{{c}^{3}\ln \left ( ex+d \right ) }{{e}^{7}}}-{\frac{{b}^{3}}{3\,{e}^{4} \left ( ex+d \right ) ^{3}}}+4\,{\frac{{b}^{2}cd}{{e}^{5} \left ( ex+d \right ) ^{3}}}-10\,{\frac{b{c}^{2}{d}^{2}}{{e}^{6} \left ( ex+d \right ) ^{3}}}+{\frac{20\,{c}^{3}{d}^{3}}{3\,{e}^{7} \left ( ex+d \right ) ^{3}}}+{\frac{{b}^{3}{d}^{3}}{6\,{e}^{4} \left ( ex+d \right ) ^{6}}}-{\frac{{b}^{2}c{d}^{4}}{2\,{e}^{5} \left ( ex+d \right ) ^{6}}}+{\frac{b{c}^{2}{d}^{5}}{2\,{e}^{6} \left ( ex+d \right ) ^{6}}}-{\frac{{c}^{3}{d}^{6}}{6\,{e}^{7} \left ( ex+d \right ) ^{6}}}-3\,{\frac{b{c}^{2}}{{e}^{6} \left ( ex+d \right ) }}+6\,{\frac{{c}^{3}d}{{e}^{7} \left ( ex+d \right ) }}+{\frac{3\,{b}^{3}d}{4\,{e}^{4} \left ( ex+d \right ) ^{4}}}-{\frac{9\,{b}^{2}c{d}^{2}}{2\,{e}^{5} \left ( ex+d \right ) ^{4}}}+{\frac{15\,b{c}^{2}{d}^{3}}{2\,{e}^{6} \left ( ex+d \right ) ^{4}}}-{\frac{15\,{c}^{3}{d}^{4}}{4\,{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)^7,x)

[Out]

-3/2*c/e^5/(e*x+d)^2*b^2+15/2*c^2/e^6/(e*x+d)^2*b*d-15/2*c^3/e^7/(e*x+d)^2*d^2-3/5*d^2/e^4/(e*x+d)^5*b^3+12/5*
d^3/e^5/(e*x+d)^5*b^2*c-3*d^4/e^6/(e*x+d)^5*b*c^2+6/5*d^5/e^7/(e*x+d)^5*c^3+c^3*ln(e*x+d)/e^7-1/3/e^4/(e*x+d)^
3*b^3+4/e^5/(e*x+d)^3*b^2*c*d-10/e^6/(e*x+d)^3*b*c^2*d^2+20/3/e^7/(e*x+d)^3*c^3*d^3+1/6*d^3/e^4/(e*x+d)^6*b^3-
1/2*d^4/e^5/(e*x+d)^6*b^2*c+1/2*d^5/e^6/(e*x+d)^6*b*c^2-1/6*d^6/e^7/(e*x+d)^6*c^3-3*c^2/e^6/(e*x+d)*b+6*c^3*d/
e^7/(e*x+d)+3/4*d/e^4/(e*x+d)^4*b^3-9/2*d^2/e^5/(e*x+d)^4*b^2*c+15/2*d^3/e^6/(e*x+d)^4*b*c^2-15/4*d^4/e^7/(e*x
+d)^4*c^3

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Maxima [A]  time = 1.20406, size = 447, normalized size = 1.96 \begin{align*} \frac{147 \, c^{3} d^{6} - 30 \, b c^{2} d^{5} e - 6 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3} + 180 \,{\left (2 \, c^{3} d e^{5} - b c^{2} e^{6}\right )} x^{5} + 90 \,{\left (15 \, c^{3} d^{2} e^{4} - 5 \, b c^{2} d e^{5} - b^{2} c e^{6}\right )} x^{4} + 20 \,{\left (110 \, c^{3} d^{3} e^{3} - 30 \, b c^{2} d^{2} e^{4} - 6 \, b^{2} c d e^{5} - b^{3} e^{6}\right )} x^{3} + 15 \,{\left (125 \, c^{3} d^{4} e^{2} - 30 \, b c^{2} d^{3} e^{3} - 6 \, b^{2} c d^{2} e^{4} - b^{3} d e^{5}\right )} x^{2} + 6 \,{\left (137 \, c^{3} d^{5} e - 30 \, b c^{2} d^{4} e^{2} - 6 \, b^{2} c d^{3} e^{3} - b^{3} d^{2} e^{4}\right )} x}{60 \,{\left (e^{13} x^{6} + 6 \, d e^{12} x^{5} + 15 \, d^{2} e^{11} x^{4} + 20 \, d^{3} e^{10} x^{3} + 15 \, d^{4} e^{9} x^{2} + 6 \, d^{5} e^{8} x + d^{6} e^{7}\right )}} + \frac{c^{3} \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)^7,x, algorithm="maxima")

[Out]

1/60*(147*c^3*d^6 - 30*b*c^2*d^5*e - 6*b^2*c*d^4*e^2 - b^3*d^3*e^3 + 180*(2*c^3*d*e^5 - b*c^2*e^6)*x^5 + 90*(1
5*c^3*d^2*e^4 - 5*b*c^2*d*e^5 - b^2*c*e^6)*x^4 + 20*(110*c^3*d^3*e^3 - 30*b*c^2*d^2*e^4 - 6*b^2*c*d*e^5 - b^3*
e^6)*x^3 + 15*(125*c^3*d^4*e^2 - 30*b*c^2*d^3*e^3 - 6*b^2*c*d^2*e^4 - b^3*d*e^5)*x^2 + 6*(137*c^3*d^5*e - 30*b
*c^2*d^4*e^2 - 6*b^2*c*d^3*e^3 - b^3*d^2*e^4)*x)/(e^13*x^6 + 6*d*e^12*x^5 + 15*d^2*e^11*x^4 + 20*d^3*e^10*x^3
+ 15*d^4*e^9*x^2 + 6*d^5*e^8*x + d^6*e^7) + c^3*log(e*x + d)/e^7

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Fricas [A]  time = 1.66275, size = 840, normalized size = 3.68 \begin{align*} \frac{147 \, c^{3} d^{6} - 30 \, b c^{2} d^{5} e - 6 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3} + 180 \,{\left (2 \, c^{3} d e^{5} - b c^{2} e^{6}\right )} x^{5} + 90 \,{\left (15 \, c^{3} d^{2} e^{4} - 5 \, b c^{2} d e^{5} - b^{2} c e^{6}\right )} x^{4} + 20 \,{\left (110 \, c^{3} d^{3} e^{3} - 30 \, b c^{2} d^{2} e^{4} - 6 \, b^{2} c d e^{5} - b^{3} e^{6}\right )} x^{3} + 15 \,{\left (125 \, c^{3} d^{4} e^{2} - 30 \, b c^{2} d^{3} e^{3} - 6 \, b^{2} c d^{2} e^{4} - b^{3} d e^{5}\right )} x^{2} + 6 \,{\left (137 \, c^{3} d^{5} e - 30 \, b c^{2} d^{4} e^{2} - 6 \, b^{2} c d^{3} e^{3} - b^{3} d^{2} e^{4}\right )} x + 60 \,{\left (c^{3} e^{6} x^{6} + 6 \, c^{3} d e^{5} x^{5} + 15 \, c^{3} d^{2} e^{4} x^{4} + 20 \, c^{3} d^{3} e^{3} x^{3} + 15 \, c^{3} d^{4} e^{2} x^{2} + 6 \, c^{3} d^{5} e x + c^{3} d^{6}\right )} \log \left (e x + d\right )}{60 \,{\left (e^{13} x^{6} + 6 \, d e^{12} x^{5} + 15 \, d^{2} e^{11} x^{4} + 20 \, d^{3} e^{10} x^{3} + 15 \, d^{4} e^{9} x^{2} + 6 \, d^{5} e^{8} x + d^{6} 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)^7,x, algorithm="fricas")

[Out]

1/60*(147*c^3*d^6 - 30*b*c^2*d^5*e - 6*b^2*c*d^4*e^2 - b^3*d^3*e^3 + 180*(2*c^3*d*e^5 - b*c^2*e^6)*x^5 + 90*(1
5*c^3*d^2*e^4 - 5*b*c^2*d*e^5 - b^2*c*e^6)*x^4 + 20*(110*c^3*d^3*e^3 - 30*b*c^2*d^2*e^4 - 6*b^2*c*d*e^5 - b^3*
e^6)*x^3 + 15*(125*c^3*d^4*e^2 - 30*b*c^2*d^3*e^3 - 6*b^2*c*d^2*e^4 - b^3*d*e^5)*x^2 + 6*(137*c^3*d^5*e - 30*b
*c^2*d^4*e^2 - 6*b^2*c*d^3*e^3 - b^3*d^2*e^4)*x + 60*(c^3*e^6*x^6 + 6*c^3*d*e^5*x^5 + 15*c^3*d^2*e^4*x^4 + 20*
c^3*d^3*e^3*x^3 + 15*c^3*d^4*e^2*x^2 + 6*c^3*d^5*e*x + c^3*d^6)*log(e*x + d))/(e^13*x^6 + 6*d*e^12*x^5 + 15*d^
2*e^11*x^4 + 20*d^3*e^10*x^3 + 15*d^4*e^9*x^2 + 6*d^5*e^8*x + d^6*e^7)

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Sympy [A]  time = 103.83, size = 343, normalized size = 1.5 \begin{align*} \frac{c^{3} \log{\left (d + e x \right )}}{e^{7}} - \frac{b^{3} d^{3} e^{3} + 6 b^{2} c d^{4} e^{2} + 30 b c^{2} d^{5} e - 147 c^{3} d^{6} + x^{5} \left (180 b c^{2} e^{6} - 360 c^{3} d e^{5}\right ) + x^{4} \left (90 b^{2} c e^{6} + 450 b c^{2} d e^{5} - 1350 c^{3} d^{2} e^{4}\right ) + x^{3} \left (20 b^{3} e^{6} + 120 b^{2} c d e^{5} + 600 b c^{2} d^{2} e^{4} - 2200 c^{3} d^{3} e^{3}\right ) + x^{2} \left (15 b^{3} d e^{5} + 90 b^{2} c d^{2} e^{4} + 450 b c^{2} d^{3} e^{3} - 1875 c^{3} d^{4} e^{2}\right ) + x \left (6 b^{3} d^{2} e^{4} + 36 b^{2} c d^{3} e^{3} + 180 b c^{2} d^{4} e^{2} - 822 c^{3} d^{5} e\right )}{60 d^{6} e^{7} + 360 d^{5} e^{8} x + 900 d^{4} e^{9} x^{2} + 1200 d^{3} e^{10} x^{3} + 900 d^{2} e^{11} x^{4} + 360 d e^{12} x^{5} + 60 e^{13} x^{6}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x)**3/(e*x+d)**7,x)

[Out]

c**3*log(d + e*x)/e**7 - (b**3*d**3*e**3 + 6*b**2*c*d**4*e**2 + 30*b*c**2*d**5*e - 147*c**3*d**6 + x**5*(180*b
*c**2*e**6 - 360*c**3*d*e**5) + x**4*(90*b**2*c*e**6 + 450*b*c**2*d*e**5 - 1350*c**3*d**2*e**4) + x**3*(20*b**
3*e**6 + 120*b**2*c*d*e**5 + 600*b*c**2*d**2*e**4 - 2200*c**3*d**3*e**3) + x**2*(15*b**3*d*e**5 + 90*b**2*c*d*
*2*e**4 + 450*b*c**2*d**3*e**3 - 1875*c**3*d**4*e**2) + x*(6*b**3*d**2*e**4 + 36*b**2*c*d**3*e**3 + 180*b*c**2
*d**4*e**2 - 822*c**3*d**5*e))/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 9
00*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6)

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Giac [A]  time = 1.26178, size = 351, normalized size = 1.54 \begin{align*} c^{3} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{{\left (180 \,{\left (2 \, c^{3} d e^{4} - b c^{2} e^{5}\right )} x^{5} + 90 \,{\left (15 \, c^{3} d^{2} e^{3} - 5 \, b c^{2} d e^{4} - b^{2} c e^{5}\right )} x^{4} + 20 \,{\left (110 \, c^{3} d^{3} e^{2} - 30 \, b c^{2} d^{2} e^{3} - 6 \, b^{2} c d e^{4} - b^{3} e^{5}\right )} x^{3} + 15 \,{\left (125 \, c^{3} d^{4} e - 30 \, b c^{2} d^{3} e^{2} - 6 \, b^{2} c d^{2} e^{3} - b^{3} d e^{4}\right )} x^{2} + 6 \,{\left (137 \, c^{3} d^{5} - 30 \, b c^{2} d^{4} e - 6 \, b^{2} c d^{3} e^{2} - b^{3} d^{2} e^{3}\right )} x +{\left (147 \, c^{3} d^{6} - 30 \, b c^{2} d^{5} e - 6 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}\right )} e^{\left (-1\right )}\right )} e^{\left (-6\right )}}{60 \,{\left (x e + d\right )}^{6}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^3/(e*x+d)^7,x, algorithm="giac")

[Out]

c^3*e^(-7)*log(abs(x*e + d)) + 1/60*(180*(2*c^3*d*e^4 - b*c^2*e^5)*x^5 + 90*(15*c^3*d^2*e^3 - 5*b*c^2*d*e^4 -
b^2*c*e^5)*x^4 + 20*(110*c^3*d^3*e^2 - 30*b*c^2*d^2*e^3 - 6*b^2*c*d*e^4 - b^3*e^5)*x^3 + 15*(125*c^3*d^4*e - 3
0*b*c^2*d^3*e^2 - 6*b^2*c*d^2*e^3 - b^3*d*e^4)*x^2 + 6*(137*c^3*d^5 - 30*b*c^2*d^4*e - 6*b^2*c*d^3*e^2 - b^3*d
^2*e^3)*x + (147*c^3*d^6 - 30*b*c^2*d^5*e - 6*b^2*c*d^4*e^2 - b^3*d^3*e^3)*e^(-1))*e^(-6)/(x*e + d)^6