### 3.1132 $$\int \frac{(a+b x+c x^2)^2}{(b d+2 c d x)^7} \, dx$$

Optimal. Leaf size=37 $\frac{\left (a+b x+c x^2\right )^3}{3 d^7 \left (b^2-4 a c\right ) (b+2 c x)^6}$

[Out]

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

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Rubi [A]  time = 0.0143671, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.042, Rules used = {682} $\frac{\left (a+b x+c x^2\right )^3}{3 d^7 \left (b^2-4 a c\right ) (b+2 c x)^6}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + b*x + c*x^2)^2/(b*d + 2*c*d*x)^7,x]

[Out]

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

Rule 682

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(2*c*(d + e*x)^(m +
1)*(a + b*x + c*x^2)^(p + 1))/(e*(p + 1)*(b^2 - 4*a*c)), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*
a*c, 0] && EqQ[2*c*d - b*e, 0] && EqQ[m + 2*p + 3, 0] && NeQ[p, -1]

Rubi steps

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

Mathematica [A]  time = 0.0290149, size = 65, normalized size = 1.76 $-\frac{16 a^2 c^2-3 \left (b^2-4 a c\right ) (b+2 c x)^2-8 a b^2 c+b^4+3 (b+2 c x)^4}{192 c^3 d^7 (b+2 c x)^6}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(b^4 - 8*a*b^2*c + 16*a^2*c^2 - 3*(b^2 - 4*a*c)*(b + 2*c*x)^2 + 3*(b + 2*c*x)^4)/(192*c^3*d^7*(b + 2*c*x)^6)

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Maple [B]  time = 0.044, size = 74, normalized size = 2. \begin{align*}{\frac{1}{{d}^{7}} \left ( -{\frac{16\,{a}^{2}{c}^{2}-8\,ac{b}^{2}+{b}^{4}}{192\,{c}^{3} \left ( 2\,cx+b \right ) ^{6}}}-{\frac{1}{64\,{c}^{3} \left ( 2\,cx+b \right ) ^{2}}}-{\frac{4\,ac-{b}^{2}}{64\,{c}^{3} \left ( 2\,cx+b \right ) ^{4}}} \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x+a)^2/(2*c*d*x+b*d)^7,x)

[Out]

1/d^7*(-1/192*(16*a^2*c^2-8*a*b^2*c+b^4)/c^3/(2*c*x+b)^6-1/64/c^3/(2*c*x+b)^2-1/64*(4*a*c-b^2)/c^3/(2*c*x+b)^4
)

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Maxima [B]  time = 1.18392, size = 221, normalized size = 5.97 \begin{align*} -\frac{48 \, c^{4} x^{4} + 96 \, b c^{3} x^{3} + b^{4} + 4 \, a b^{2} c + 16 \, a^{2} c^{2} + 12 \,{\left (5 \, b^{2} c^{2} + 4 \, a c^{3}\right )} x^{2} + 12 \,{\left (b^{3} c + 4 \, a b c^{2}\right )} x}{192 \,{\left (64 \, c^{9} d^{7} x^{6} + 192 \, b c^{8} d^{7} x^{5} + 240 \, b^{2} c^{7} d^{7} x^{4} + 160 \, b^{3} c^{6} d^{7} x^{3} + 60 \, b^{4} c^{5} d^{7} x^{2} + 12 \, b^{5} c^{4} d^{7} x + b^{6} c^{3} d^{7}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^2/(2*c*d*x+b*d)^7,x, algorithm="maxima")

[Out]

-1/192*(48*c^4*x^4 + 96*b*c^3*x^3 + b^4 + 4*a*b^2*c + 16*a^2*c^2 + 12*(5*b^2*c^2 + 4*a*c^3)*x^2 + 12*(b^3*c +
4*a*b*c^2)*x)/(64*c^9*d^7*x^6 + 192*b*c^8*d^7*x^5 + 240*b^2*c^7*d^7*x^4 + 160*b^3*c^6*d^7*x^3 + 60*b^4*c^5*d^7
*x^2 + 12*b^5*c^4*d^7*x + b^6*c^3*d^7)

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Fricas [B]  time = 1.88119, size = 351, normalized size = 9.49 \begin{align*} -\frac{48 \, c^{4} x^{4} + 96 \, b c^{3} x^{3} + b^{4} + 4 \, a b^{2} c + 16 \, a^{2} c^{2} + 12 \,{\left (5 \, b^{2} c^{2} + 4 \, a c^{3}\right )} x^{2} + 12 \,{\left (b^{3} c + 4 \, a b c^{2}\right )} x}{192 \,{\left (64 \, c^{9} d^{7} x^{6} + 192 \, b c^{8} d^{7} x^{5} + 240 \, b^{2} c^{7} d^{7} x^{4} + 160 \, b^{3} c^{6} d^{7} x^{3} + 60 \, b^{4} c^{5} d^{7} x^{2} + 12 \, b^{5} c^{4} d^{7} x + b^{6} c^{3} d^{7}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^2/(2*c*d*x+b*d)^7,x, algorithm="fricas")

[Out]

-1/192*(48*c^4*x^4 + 96*b*c^3*x^3 + b^4 + 4*a*b^2*c + 16*a^2*c^2 + 12*(5*b^2*c^2 + 4*a*c^3)*x^2 + 12*(b^3*c +
4*a*b*c^2)*x)/(64*c^9*d^7*x^6 + 192*b*c^8*d^7*x^5 + 240*b^2*c^7*d^7*x^4 + 160*b^3*c^6*d^7*x^3 + 60*b^4*c^5*d^7
*x^2 + 12*b^5*c^4*d^7*x + b^6*c^3*d^7)

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Sympy [B]  time = 3.89161, size = 172, normalized size = 4.65 \begin{align*} - \frac{16 a^{2} c^{2} + 4 a b^{2} c + b^{4} + 96 b c^{3} x^{3} + 48 c^{4} x^{4} + x^{2} \left (48 a c^{3} + 60 b^{2} c^{2}\right ) + x \left (48 a b c^{2} + 12 b^{3} c\right )}{192 b^{6} c^{3} d^{7} + 2304 b^{5} c^{4} d^{7} x + 11520 b^{4} c^{5} d^{7} x^{2} + 30720 b^{3} c^{6} d^{7} x^{3} + 46080 b^{2} c^{7} d^{7} x^{4} + 36864 b c^{8} d^{7} x^{5} + 12288 c^{9} d^{7} x^{6}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(16*a**2*c**2 + 4*a*b**2*c + b**4 + 96*b*c**3*x**3 + 48*c**4*x**4 + x**2*(48*a*c**3 + 60*b**2*c**2) + x*(48*a
*b*c**2 + 12*b**3*c))/(192*b**6*c**3*d**7 + 2304*b**5*c**4*d**7*x + 11520*b**4*c**5*d**7*x**2 + 30720*b**3*c**
6*d**7*x**3 + 46080*b**2*c**7*d**7*x**4 + 36864*b*c**8*d**7*x**5 + 12288*c**9*d**7*x**6)

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Giac [B]  time = 1.24655, size = 117, normalized size = 3.16 \begin{align*} -\frac{48 \, c^{4} x^{4} + 96 \, b c^{3} x^{3} + 60 \, b^{2} c^{2} x^{2} + 48 \, a c^{3} x^{2} + 12 \, b^{3} c x + 48 \, a b c^{2} x + b^{4} + 4 \, a b^{2} c + 16 \, a^{2} c^{2}}{192 \,{\left (2 \, c x + b\right )}^{6} c^{3} d^{7}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/192*(48*c^4*x^4 + 96*b*c^3*x^3 + 60*b^2*c^2*x^2 + 48*a*c^3*x^2 + 12*b^3*c*x + 48*a*b*c^2*x + b^4 + 4*a*b^2*
c + 16*a^2*c^2)/((2*c*x + b)^6*c^3*d^7)