### 3.243 $$\int \frac{(b x+c x^2)^2}{(d+e x)^8} \, dx$$

Optimal. Leaf size=137 $-\frac{b^2 e^2-6 b c d e+6 c^2 d^2}{5 e^5 (d+e x)^5}-\frac{d^2 (c d-b e)^2}{7 e^5 (d+e x)^7}+\frac{c (2 c d-b e)}{2 e^5 (d+e x)^4}+\frac{d (c d-b e) (2 c d-b e)}{3 e^5 (d+e x)^6}-\frac{c^2}{3 e^5 (d+e x)^3}$

[Out]

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

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Rubi [A]  time = 0.0880271, antiderivative size = 137, 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} $-\frac{b^2 e^2-6 b c d e+6 c^2 d^2}{5 e^5 (d+e x)^5}-\frac{d^2 (c d-b e)^2}{7 e^5 (d+e x)^7}+\frac{c (2 c d-b e)}{2 e^5 (d+e x)^4}+\frac{d (c d-b e) (2 c d-b e)}{3 e^5 (d+e x)^6}-\frac{c^2}{3 e^5 (d+e x)^3}$

Antiderivative was successfully veriﬁed.

[In]

Int[(b*x + c*x^2)^2/(d + e*x)^8,x]

[Out]

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

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

Mathematica [A]  time = 0.0477294, size = 117, normalized size = 0.85 $-\frac{2 b^2 e^2 \left (d^2+7 d e x+21 e^2 x^2\right )+3 b c e \left (7 d^2 e x+d^3+21 d e^2 x^2+35 e^3 x^3\right )+2 c^2 \left (21 d^2 e^2 x^2+7 d^3 e x+d^4+35 d e^3 x^3+35 e^4 x^4\right )}{210 e^5 (d+e x)^7}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(b*x + c*x^2)^2/(d + e*x)^8,x]

[Out]

-(2*b^2*e^2*(d^2 + 7*d*e*x + 21*e^2*x^2) + 3*b*c*e*(d^3 + 7*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3) + 2*c^2*(d^4
+ 7*d^3*e*x + 21*d^2*e^2*x^2 + 35*d*e^3*x^3 + 35*e^4*x^4))/(210*e^5*(d + e*x)^7)

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Maple [A]  time = 0.056, size = 143, normalized size = 1. \begin{align*} -{\frac{{d}^{2} \left ({b}^{2}{e}^{2}-2\,bcde+{c}^{2}{d}^{2} \right ) }{7\,{e}^{5} \left ( ex+d \right ) ^{7}}}-{\frac{{b}^{2}{e}^{2}-6\,bcde+6\,{c}^{2}{d}^{2}}{5\,{e}^{5} \left ( ex+d \right ) ^{5}}}-{\frac{{c}^{2}}{3\,{e}^{5} \left ( ex+d \right ) ^{3}}}+{\frac{d \left ({b}^{2}{e}^{2}-3\,bcde+2\,{c}^{2}{d}^{2} \right ) }{3\,{e}^{5} \left ( ex+d \right ) ^{6}}}-{\frac{c \left ( be-2\,cd \right ) }{2\,{e}^{5} \left ( ex+d \right ) ^{4}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x)^2/(e*x+d)^8,x)

[Out]

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

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Maxima [A]  time = 1.16599, size = 279, normalized size = 2.04 \begin{align*} -\frac{70 \, c^{2} e^{4} x^{4} + 2 \, c^{2} d^{4} + 3 \, b c d^{3} e + 2 \, b^{2} d^{2} e^{2} + 35 \,{\left (2 \, c^{2} d e^{3} + 3 \, b c e^{4}\right )} x^{3} + 21 \,{\left (2 \, c^{2} d^{2} e^{2} + 3 \, b c d e^{3} + 2 \, b^{2} e^{4}\right )} x^{2} + 7 \,{\left (2 \, c^{2} d^{3} e + 3 \, b c d^{2} e^{2} + 2 \, b^{2} d e^{3}\right )} x}{210 \,{\left (e^{12} x^{7} + 7 \, d e^{11} x^{6} + 21 \, d^{2} e^{10} x^{5} + 35 \, d^{3} e^{9} x^{4} + 35 \, d^{4} e^{8} x^{3} + 21 \, d^{5} e^{7} x^{2} + 7 \, d^{6} e^{6} x + d^{7} e^{5}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^2/(e*x+d)^8,x, algorithm="maxima")

[Out]

-1/210*(70*c^2*e^4*x^4 + 2*c^2*d^4 + 3*b*c*d^3*e + 2*b^2*d^2*e^2 + 35*(2*c^2*d*e^3 + 3*b*c*e^4)*x^3 + 21*(2*c^
2*d^2*e^2 + 3*b*c*d*e^3 + 2*b^2*e^4)*x^2 + 7*(2*c^2*d^3*e + 3*b*c*d^2*e^2 + 2*b^2*d*e^3)*x)/(e^12*x^7 + 7*d*e^
11*x^6 + 21*d^2*e^10*x^5 + 35*d^3*e^9*x^4 + 35*d^4*e^8*x^3 + 21*d^5*e^7*x^2 + 7*d^6*e^6*x + d^7*e^5)

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Fricas [A]  time = 1.56862, size = 435, normalized size = 3.18 \begin{align*} -\frac{70 \, c^{2} e^{4} x^{4} + 2 \, c^{2} d^{4} + 3 \, b c d^{3} e + 2 \, b^{2} d^{2} e^{2} + 35 \,{\left (2 \, c^{2} d e^{3} + 3 \, b c e^{4}\right )} x^{3} + 21 \,{\left (2 \, c^{2} d^{2} e^{2} + 3 \, b c d e^{3} + 2 \, b^{2} e^{4}\right )} x^{2} + 7 \,{\left (2 \, c^{2} d^{3} e + 3 \, b c d^{2} e^{2} + 2 \, b^{2} d e^{3}\right )} x}{210 \,{\left (e^{12} x^{7} + 7 \, d e^{11} x^{6} + 21 \, d^{2} e^{10} x^{5} + 35 \, d^{3} e^{9} x^{4} + 35 \, d^{4} e^{8} x^{3} + 21 \, d^{5} e^{7} x^{2} + 7 \, d^{6} e^{6} x + d^{7} e^{5}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^2/(e*x+d)^8,x, algorithm="fricas")

[Out]

-1/210*(70*c^2*e^4*x^4 + 2*c^2*d^4 + 3*b*c*d^3*e + 2*b^2*d^2*e^2 + 35*(2*c^2*d*e^3 + 3*b*c*e^4)*x^3 + 21*(2*c^
2*d^2*e^2 + 3*b*c*d*e^3 + 2*b^2*e^4)*x^2 + 7*(2*c^2*d^3*e + 3*b*c*d^2*e^2 + 2*b^2*d*e^3)*x)/(e^12*x^7 + 7*d*e^
11*x^6 + 21*d^2*e^10*x^5 + 35*d^3*e^9*x^4 + 35*d^4*e^8*x^3 + 21*d^5*e^7*x^2 + 7*d^6*e^6*x + d^7*e^5)

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Sympy [A]  time = 22.3192, size = 218, normalized size = 1.59 \begin{align*} - \frac{2 b^{2} d^{2} e^{2} + 3 b c d^{3} e + 2 c^{2} d^{4} + 70 c^{2} e^{4} x^{4} + x^{3} \left (105 b c e^{4} + 70 c^{2} d e^{3}\right ) + x^{2} \left (42 b^{2} e^{4} + 63 b c d e^{3} + 42 c^{2} d^{2} e^{2}\right ) + x \left (14 b^{2} d e^{3} + 21 b c d^{2} e^{2} + 14 c^{2} d^{3} e\right )}{210 d^{7} e^{5} + 1470 d^{6} e^{6} x + 4410 d^{5} e^{7} x^{2} + 7350 d^{4} e^{8} x^{3} + 7350 d^{3} e^{9} x^{4} + 4410 d^{2} e^{10} x^{5} + 1470 d e^{11} x^{6} + 210 e^{12} x^{7}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x)**2/(e*x+d)**8,x)

[Out]

-(2*b**2*d**2*e**2 + 3*b*c*d**3*e + 2*c**2*d**4 + 70*c**2*e**4*x**4 + x**3*(105*b*c*e**4 + 70*c**2*d*e**3) + x
**2*(42*b**2*e**4 + 63*b*c*d*e**3 + 42*c**2*d**2*e**2) + x*(14*b**2*d*e**3 + 21*b*c*d**2*e**2 + 14*c**2*d**3*e
))/(210*d**7*e**5 + 1470*d**6*e**6*x + 4410*d**5*e**7*x**2 + 7350*d**4*e**8*x**3 + 7350*d**3*e**9*x**4 + 4410*
d**2*e**10*x**5 + 1470*d*e**11*x**6 + 210*e**12*x**7)

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Giac [A]  time = 1.25388, size = 180, normalized size = 1.31 \begin{align*} -\frac{{\left (70 \, c^{2} x^{4} e^{4} + 70 \, c^{2} d x^{3} e^{3} + 42 \, c^{2} d^{2} x^{2} e^{2} + 14 \, c^{2} d^{3} x e + 2 \, c^{2} d^{4} + 105 \, b c x^{3} e^{4} + 63 \, b c d x^{2} e^{3} + 21 \, b c d^{2} x e^{2} + 3 \, b c d^{3} e + 42 \, b^{2} x^{2} e^{4} + 14 \, b^{2} d x e^{3} + 2 \, b^{2} d^{2} e^{2}\right )} e^{\left (-5\right )}}{210 \,{\left (x e + d\right )}^{7}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^2/(e*x+d)^8,x, algorithm="giac")

[Out]

-1/210*(70*c^2*x^4*e^4 + 70*c^2*d*x^3*e^3 + 42*c^2*d^2*x^2*e^2 + 14*c^2*d^3*x*e + 2*c^2*d^4 + 105*b*c*x^3*e^4
+ 63*b*c*d*x^2*e^3 + 21*b*c*d^2*x*e^2 + 3*b*c*d^3*e + 42*b^2*x^2*e^4 + 14*b^2*d*x*e^3 + 2*b^2*d^2*e^2)*e^(-5)/
(x*e + d)^7