3.1369 \(\int \frac{(a+b x)^2}{(c+d x)^8} \, dx\)

Optimal. Leaf size=65 \[ \frac{b (b c-a d)}{3 d^3 (c+d x)^6}-\frac{(b c-a d)^2}{7 d^3 (c+d x)^7}-\frac{b^2}{5 d^3 (c+d x)^5} \]

[Out]

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

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Rubi [A]  time = 0.0398526, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {43} \[ \frac{b (b c-a d)}{3 d^3 (c+d x)^6}-\frac{(b c-a d)^2}{7 d^3 (c+d x)^7}-\frac{b^2}{5 d^3 (c+d x)^5} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^2/(c + d*x)^8,x]

[Out]

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0236939, size = 55, normalized size = 0.85 \[ -\frac{15 a^2 d^2+5 a b d (c+7 d x)+b^2 \left (c^2+7 c d x+21 d^2 x^2\right )}{105 d^3 (c+d x)^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^2/(c + d*x)^8,x]

[Out]

-(15*a^2*d^2 + 5*a*b*d*(c + 7*d*x) + b^2*(c^2 + 7*c*d*x + 21*d^2*x^2))/(105*d^3*(c + d*x)^7)

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Maple [A]  time = 0.003, size = 71, normalized size = 1.1 \begin{align*} -{\frac{{a}^{2}{d}^{2}-2\,abcd+{c}^{2}{b}^{2}}{7\,{d}^{3} \left ( dx+c \right ) ^{7}}}-{\frac{b \left ( ad-bc \right ) }{3\,{d}^{3} \left ( dx+c \right ) ^{6}}}-{\frac{{b}^{2}}{5\,{d}^{3} \left ( dx+c \right ) ^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 0.982455, size = 177, normalized size = 2.72 \begin{align*} -\frac{21 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 5 \, a b c d + 15 \, a^{2} d^{2} + 7 \,{\left (b^{2} c d + 5 \, a b d^{2}\right )} x}{105 \,{\left (d^{10} x^{7} + 7 \, c d^{9} x^{6} + 21 \, c^{2} d^{8} x^{5} + 35 \, c^{3} d^{7} x^{4} + 35 \, c^{4} d^{6} x^{3} + 21 \, c^{5} d^{5} x^{2} + 7 \, c^{6} d^{4} x + c^{7} d^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/105*(21*b^2*d^2*x^2 + b^2*c^2 + 5*a*b*c*d + 15*a^2*d^2 + 7*(b^2*c*d + 5*a*b*d^2)*x)/(d^10*x^7 + 7*c*d^9*x^6
+ 21*c^2*d^8*x^5 + 35*c^3*d^7*x^4 + 35*c^4*d^6*x^3 + 21*c^5*d^5*x^2 + 7*c^6*d^4*x + c^7*d^3)

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Fricas [B]  time = 1.76736, size = 277, normalized size = 4.26 \begin{align*} -\frac{21 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 5 \, a b c d + 15 \, a^{2} d^{2} + 7 \,{\left (b^{2} c d + 5 \, a b d^{2}\right )} x}{105 \,{\left (d^{10} x^{7} + 7 \, c d^{9} x^{6} + 21 \, c^{2} d^{8} x^{5} + 35 \, c^{3} d^{7} x^{4} + 35 \, c^{4} d^{6} x^{3} + 21 \, c^{5} d^{5} x^{2} + 7 \, c^{6} d^{4} x + c^{7} d^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/105*(21*b^2*d^2*x^2 + b^2*c^2 + 5*a*b*c*d + 15*a^2*d^2 + 7*(b^2*c*d + 5*a*b*d^2)*x)/(d^10*x^7 + 7*c*d^9*x^6
+ 21*c^2*d^8*x^5 + 35*c^3*d^7*x^4 + 35*c^4*d^6*x^3 + 21*c^5*d^5*x^2 + 7*c^6*d^4*x + c^7*d^3)

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Sympy [B]  time = 1.51781, size = 139, normalized size = 2.14 \begin{align*} - \frac{15 a^{2} d^{2} + 5 a b c d + b^{2} c^{2} + 21 b^{2} d^{2} x^{2} + x \left (35 a b d^{2} + 7 b^{2} c d\right )}{105 c^{7} d^{3} + 735 c^{6} d^{4} x + 2205 c^{5} d^{5} x^{2} + 3675 c^{4} d^{6} x^{3} + 3675 c^{3} d^{7} x^{4} + 2205 c^{2} d^{8} x^{5} + 735 c d^{9} x^{6} + 105 d^{10} x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(15*a**2*d**2 + 5*a*b*c*d + b**2*c**2 + 21*b**2*d**2*x**2 + x*(35*a*b*d**2 + 7*b**2*c*d))/(105*c**7*d**3 + 73
5*c**6*d**4*x + 2205*c**5*d**5*x**2 + 3675*c**4*d**6*x**3 + 3675*c**3*d**7*x**4 + 2205*c**2*d**8*x**5 + 735*c*
d**9*x**6 + 105*d**10*x**7)

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Giac [A]  time = 1.05808, size = 82, normalized size = 1.26 \begin{align*} -\frac{21 \, b^{2} d^{2} x^{2} + 7 \, b^{2} c d x + 35 \, a b d^{2} x + b^{2} c^{2} + 5 \, a b c d + 15 \, a^{2} d^{2}}{105 \,{\left (d x + c\right )}^{7} d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/105*(21*b^2*d^2*x^2 + 7*b^2*c*d*x + 35*a*b*d^2*x + b^2*c^2 + 5*a*b*c*d + 15*a^2*d^2)/((d*x + c)^7*d^3)