3.14.69 \(\int \frac {(a+b x)^2}{(c+d x)^8} \, dx\) [1369]

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

[Out]

-1/7*(-a*d+b*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

________________________________________________________________________________________

Rubi [A]
time = 0.03, antiderivative size = 65, normalized size of antiderivative = 1.00, 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 = {45} \begin {gather*} \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} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^2/(c + d*x)^8,x]

[Out]

-1/7*(b*c - a*d)^2/(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 45

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 = 0.02, size = 55, normalized size = 0.85 \begin {gather*} -\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} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^2/(c + d*x)^8,x]

[Out]

-1/105*(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))/(d^3*(c + d*x)^7)

________________________________________________________________________________________

Mathics [A]
time = 3.57, size = 125, normalized size = 1.92 \begin {gather*} \frac {-15 a^2 d^2-5 a b c d-b^2 c^2-7 b d x \left (5 a d+b c\right )-21 b^2 d^2 x^2}{105 d^3 \left (c^7+7 c^6 d x+21 c^5 d^2 x^2+35 c^4 d^3 x^3+35 c^3 d^4 x^4+21 c^2 d^5 x^5+7 c d^6 x^6+d^7 x^7\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.14, size = 71, normalized size = 1.09

method	result	size
gosper	\(-\frac {21 b^{2} x^{2} d^{2}+35 a b \,d^{2} x +7 b^{2} c d x +15 a^{2} d^{2}+5 a b c d +b^{2} c^{2}}{105 d^{3} \left (d x +c \right )^{7}}\)	\(62\)
risch	\(\frac {-\frac {b^{2} x^{2}}{5 d}-\frac {b \left (5 a d +b c \right ) x}{15 d^{2}}-\frac {15 a^{2} d^{2}+5 a b c d +b^{2} c^{2}}{105 d^{3}}}{\left (d x +c \right )^{7}}\)	\(63\)
default	\(-\frac {b \left (a d -b c \right )}{3 d^{3} \left (d x +c \right )^{6}}-\frac {b^{2}}{5 d^{3} \left (d x +c \right )^{5}}-\frac {a^{2} d^{2}-2 a b c d +b^{2} c^{2}}{7 d^{3} \left (d x +c \right )^{7}}\)	\(71\)
norman	\(\frac {-\frac {b^{2} x^{2}}{5 d}-\frac {\left (5 a b \,d^{5}+b^{2} c \,d^{4}\right ) x}{15 d^{6}}-\frac {15 a^{2} d^{6}+5 a b c \,d^{5}+b^{2} c^{2} d^{4}}{105 d^{7}}}{\left (d x +c \right )^{7}}\)	\(75\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 131 vs. \(2 (59) = 118\).
time = 0.28, size = 131, normalized size = 2.02 \begin {gather*} -\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 {gather*}

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)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 131 vs. \(2 (59) = 118\).
time = 0.29, size = 131, normalized size = 2.02 \begin {gather*} -\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 {gather*}

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)

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 139 vs. \(2 (54) = 108\).
time = 0.82, size = 139, normalized size = 2.14 \begin {gather*} \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 {gather*}

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 + 7
35*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)

________________________________________________________________________________________

Giac [A]
time = 0.00, size = 69, normalized size = 1.06 \begin {gather*} \frac {-21 x^{2} b^{2} d^{2}-7 x b^{2} d c-35 x b a d^{2}-b^{2} c^{2}-5 b a d c-15 a^{2} d^{2}}{105 d^{3} \left (x d+c\right )^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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)

________________________________________________________________________________________

Mupad [B]
time = 0.09, size = 129, normalized size = 1.98 \begin {gather*} -\frac {\frac {15\,a^2\,d^2+5\,a\,b\,c\,d+b^2\,c^2}{105\,d^3}+\frac {b^2\,x^2}{5\,d}+\frac {b\,x\,\left (5\,a\,d+b\,c\right )}{15\,d^2}}{c^7+7\,c^6\,d\,x+21\,c^5\,d^2\,x^2+35\,c^4\,d^3\,x^3+35\,c^3\,d^4\,x^4+21\,c^2\,d^5\,x^5+7\,c\,d^6\,x^6+d^7\,x^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^2/(c + d*x)^8,x)

[Out]

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

________________________________________________________________________________________