\(\int \frac {(a+b x)^2}{(a c-b c x)^7} \, dx\) [1048]

Optimal result

Integrand size = 19, antiderivative size = 59 \[ \int \frac {(a+b x)^2}{(a c-b c x)^7} \, dx=\frac {2 a^2}{3 b c^7 (a-b x)^6}-\frac {4 a}{5 b c^7 (a-b x)^5}+\frac {1}{4 b c^7 (a-b x)^4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {45} \[ \int \frac {(a+b x)^2}{(a c-b c x)^7} \, dx=\frac {2 a^2}{3 b c^7 (a-b x)^6}-\frac {4 a}{5 b c^7 (a-b x)^5}+\frac {1}{4 b c^7 (a-b x)^4} \]

[In]

Int[(a + b*x)^2/(a*c - b*c*x)^7,x]

[Out]

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

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*} \text {integral}& = \int \left (\frac {4 a^2}{c^7 (a-b x)^7}-\frac {4 a}{c^7 (a-b x)^6}+\frac {1}{c^7 (a-b x)^5}\right ) \, dx \\ & = \frac {2 a^2}{3 b c^7 (a-b x)^6}-\frac {4 a}{5 b c^7 (a-b x)^5}+\frac {1}{4 b c^7 (a-b x)^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.63 \[ \int \frac {(a+b x)^2}{(a c-b c x)^7} \, dx=\frac {7 a^2+18 a b x+15 b^2 x^2}{60 b c^7 (a-b x)^6} \]

[In]

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

[Out]

(7*a^2 + 18*a*b*x + 15*b^2*x^2)/(60*b*c^7*(a - b*x)^6)

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.54

method	result	size
risch	\(\frac {\frac {b \,x^{2}}{4}+\frac {3 a x}{10}+\frac {7 a^{2}}{60 b}}{c^{7} \left (-b x +a \right )^{6}}\)	\(32\)
gosper	\(\frac {15 b^{2} x^{2}+18 a b x +7 a^{2}}{60 \left (-b x +a \right )^{6} c^{7} b}\)	\(36\)
norman	\(\frac {\frac {7 a^{2}}{60 b c}+\frac {b \,x^{2}}{4 c}+\frac {3 a x}{10 c}}{c^{6} \left (-b x +a \right )^{6}}\)	\(41\)
parallelrisch	\(\frac {15 b^{7} x^{2}+18 a \,b^{6} x +7 a^{2} b^{5}}{60 b^{6} c^{7} \left (b x -a \right )^{6}}\)	\(42\)
default	\(\frac {\frac {1}{4 b \left (-b x +a \right )^{4}}+\frac {2 a^{2}}{3 b \left (-b x +a \right )^{6}}-\frac {4 a}{5 b \left (-b x +a \right )^{5}}}{c^{7}}\)	\(49\)

[In]

int((b*x+a)^2/(-b*c*x+a*c)^7,x,method=_RETURNVERBOSE)

[Out]

(1/4*b*x^2+3/10*a*x+7/60*a^2/b)/c^7/(-b*x+a)^6

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.83 \[ \int \frac {(a+b x)^2}{(a c-b c x)^7} \, dx=\frac {15 \, b^{2} x^{2} + 18 \, a b x + 7 \, a^{2}}{60 \, {\left (b^{7} c^{7} x^{6} - 6 \, a b^{6} c^{7} x^{5} + 15 \, a^{2} b^{5} c^{7} x^{4} - 20 \, a^{3} b^{4} c^{7} x^{3} + 15 \, a^{4} b^{3} c^{7} x^{2} - 6 \, a^{5} b^{2} c^{7} x + a^{6} b c^{7}\right )}} \]

[In]

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

[Out]

1/60*(15*b^2*x^2 + 18*a*b*x + 7*a^2)/(b^7*c^7*x^6 - 6*a*b^6*c^7*x^5 + 15*a^2*b^5*c^7*x^4 - 20*a^3*b^4*c^7*x^3
+ 15*a^4*b^3*c^7*x^2 - 6*a^5*b^2*c^7*x + a^6*b*c^7)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 117 vs. \(2 (49) = 98\).

Time = 0.38 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.98 \[ \int \frac {(a+b x)^2}{(a c-b c x)^7} \, dx=- \frac {- 7 a^{2} - 18 a b x - 15 b^{2} x^{2}}{60 a^{6} b c^{7} - 360 a^{5} b^{2} c^{7} x + 900 a^{4} b^{3} c^{7} x^{2} - 1200 a^{3} b^{4} c^{7} x^{3} + 900 a^{2} b^{5} c^{7} x^{4} - 360 a b^{6} c^{7} x^{5} + 60 b^{7} c^{7} x^{6}} \]

[In]

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

[Out]

-(-7*a**2 - 18*a*b*x - 15*b**2*x**2)/(60*a**6*b*c**7 - 360*a**5*b**2*c**7*x + 900*a**4*b**3*c**7*x**2 - 1200*a
**3*b**4*c**7*x**3 + 900*a**2*b**5*c**7*x**4 - 360*a*b**6*c**7*x**5 + 60*b**7*c**7*x**6)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.83 \[ \int \frac {(a+b x)^2}{(a c-b c x)^7} \, dx=\frac {15 \, b^{2} x^{2} + 18 \, a b x + 7 \, a^{2}}{60 \, {\left (b^{7} c^{7} x^{6} - 6 \, a b^{6} c^{7} x^{5} + 15 \, a^{2} b^{5} c^{7} x^{4} - 20 \, a^{3} b^{4} c^{7} x^{3} + 15 \, a^{4} b^{3} c^{7} x^{2} - 6 \, a^{5} b^{2} c^{7} x + a^{6} b c^{7}\right )}} \]

[In]

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

[Out]

1/60*(15*b^2*x^2 + 18*a*b*x + 7*a^2)/(b^7*c^7*x^6 - 6*a*b^6*c^7*x^5 + 15*a^2*b^5*c^7*x^4 - 20*a^3*b^4*c^7*x^3
+ 15*a^4*b^3*c^7*x^2 - 6*a^5*b^2*c^7*x + a^6*b*c^7)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.61 \[ \int \frac {(a+b x)^2}{(a c-b c x)^7} \, dx=\frac {15 \, b^{2} x^{2} + 18 \, a b x + 7 \, a^{2}}{60 \, {\left (b x - a\right )}^{6} b c^{7}} \]

[In]

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

[Out]

1/60*(15*b^2*x^2 + 18*a*b*x + 7*a^2)/((b*x - a)^6*b*c^7)

Mupad [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.76 \[ \int \frac {(a+b x)^2}{(a c-b c x)^7} \, dx=\frac {\frac {3\,a\,x}{10}+\frac {b\,x^2}{4}+\frac {7\,a^2}{60\,b}}{a^6\,c^7-6\,a^5\,b\,c^7\,x+15\,a^4\,b^2\,c^7\,x^2-20\,a^3\,b^3\,c^7\,x^3+15\,a^2\,b^4\,c^7\,x^4-6\,a\,b^5\,c^7\,x^5+b^6\,c^7\,x^6} \]

[In]

int((a + b*x)^2/(a*c - b*c*x)^7,x)

[Out]

((3*a*x)/10 + (b*x^2)/4 + (7*a^2)/(60*b))/(a^6*c^7 + b^6*c^7*x^6 - 6*a*b^5*c^7*x^5 + 15*a^4*b^2*c^7*x^2 - 20*a
^3*b^3*c^7*x^3 + 15*a^2*b^4*c^7*x^4 - 6*a^5*b*c^7*x)