3.1368 \(\int \frac{(a+b x)^3}{(c+d x)^8} \, dx\)

Optimal. Leaf size=92 \[ \frac{3 b^2 (b c-a d)}{5 d^4 (c+d x)^5}-\frac{b (b c-a d)^2}{2 d^4 (c+d x)^6}+\frac{(b c-a d)^3}{7 d^4 (c+d x)^7}-\frac{b^3}{4 d^4 (c+d x)^4} \]

[Out]

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

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Rubi [A]  time = 0.057465, antiderivative size = 92, 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} \[ \frac{3 b^2 (b c-a d)}{5 d^4 (c+d x)^5}-\frac{b (b c-a d)^2}{2 d^4 (c+d x)^6}+\frac{(b c-a d)^3}{7 d^4 (c+d x)^7}-\frac{b^3}{4 d^4 (c+d x)^4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^3/(c + d*x)^8,x]

[Out]

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

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

Mathematica [A]  time = 0.0292335, size = 94, normalized size = 1.02 \[ -\frac{10 a^2 b d^2 (c+7 d x)+20 a^3 d^3+4 a b^2 d \left (c^2+7 c d x+21 d^2 x^2\right )+b^3 \left (7 c^2 d x+c^3+21 c d^2 x^2+35 d^3 x^3\right )}{140 d^4 (c+d x)^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^3/(c + d*x)^8,x]

[Out]

-(20*a^3*d^3 + 10*a^2*b*d^2*(c + 7*d*x) + 4*a*b^2*d*(c^2 + 7*c*d*x + 21*d^2*x^2) + b^3*(c^3 + 7*c^2*d*x + 21*c
*d^2*x^2 + 35*d^3*x^3))/(140*d^4*(c + d*x)^7)

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Maple [A]  time = 0.005, size = 122, normalized size = 1.3 \begin{align*} -{\frac{{a}^{3}{d}^{3}-3\,{a}^{2}bc{d}^{2}+3\,a{b}^{2}{c}^{2}d-{b}^{3}{c}^{3}}{7\,{d}^{4} \left ( dx+c \right ) ^{7}}}-{\frac{b \left ({a}^{2}{d}^{2}-2\,abcd+{b}^{2}{c}^{2} \right ) }{2\,{d}^{4} \left ( dx+c \right ) ^{6}}}-{\frac{3\,{b}^{2} \left ( ad-bc \right ) }{5\,{d}^{4} \left ( dx+c \right ) ^{5}}}-{\frac{{b}^{3}}{4\,{d}^{4} \left ( dx+c \right ) ^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^3/(d*x+c)^8,x)

[Out]

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

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Maxima [B]  time = 1.00277, size = 246, normalized size = 2.67 \begin{align*} -\frac{35 \, b^{3} d^{3} x^{3} + b^{3} c^{3} + 4 \, a b^{2} c^{2} d + 10 \, a^{2} b c d^{2} + 20 \, a^{3} d^{3} + 21 \,{\left (b^{3} c d^{2} + 4 \, a b^{2} d^{3}\right )} x^{2} + 7 \,{\left (b^{3} c^{2} d + 4 \, a b^{2} c d^{2} + 10 \, a^{2} b d^{3}\right )} x}{140 \,{\left (d^{11} x^{7} + 7 \, c d^{10} x^{6} + 21 \, c^{2} d^{9} x^{5} + 35 \, c^{3} d^{8} x^{4} + 35 \, c^{4} d^{7} x^{3} + 21 \, c^{5} d^{6} x^{2} + 7 \, c^{6} d^{5} x + c^{7} d^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3/(d*x+c)^8,x, algorithm="maxima")

[Out]

-1/140*(35*b^3*d^3*x^3 + b^3*c^3 + 4*a*b^2*c^2*d + 10*a^2*b*c*d^2 + 20*a^3*d^3 + 21*(b^3*c*d^2 + 4*a*b^2*d^3)*
x^2 + 7*(b^3*c^2*d + 4*a*b^2*c*d^2 + 10*a^2*b*d^3)*x)/(d^11*x^7 + 7*c*d^10*x^6 + 21*c^2*d^9*x^5 + 35*c^3*d^8*x
^4 + 35*c^4*d^7*x^3 + 21*c^5*d^6*x^2 + 7*c^6*d^5*x + c^7*d^4)

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Fricas [B]  time = 1.76952, size = 382, normalized size = 4.15 \begin{align*} -\frac{35 \, b^{3} d^{3} x^{3} + b^{3} c^{3} + 4 \, a b^{2} c^{2} d + 10 \, a^{2} b c d^{2} + 20 \, a^{3} d^{3} + 21 \,{\left (b^{3} c d^{2} + 4 \, a b^{2} d^{3}\right )} x^{2} + 7 \,{\left (b^{3} c^{2} d + 4 \, a b^{2} c d^{2} + 10 \, a^{2} b d^{3}\right )} x}{140 \,{\left (d^{11} x^{7} + 7 \, c d^{10} x^{6} + 21 \, c^{2} d^{9} x^{5} + 35 \, c^{3} d^{8} x^{4} + 35 \, c^{4} d^{7} x^{3} + 21 \, c^{5} d^{6} x^{2} + 7 \, c^{6} d^{5} x + c^{7} d^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3/(d*x+c)^8,x, algorithm="fricas")

[Out]

-1/140*(35*b^3*d^3*x^3 + b^3*c^3 + 4*a*b^2*c^2*d + 10*a^2*b*c*d^2 + 20*a^3*d^3 + 21*(b^3*c*d^2 + 4*a*b^2*d^3)*
x^2 + 7*(b^3*c^2*d + 4*a*b^2*c*d^2 + 10*a^2*b*d^3)*x)/(d^11*x^7 + 7*c*d^10*x^6 + 21*c^2*d^9*x^5 + 35*c^3*d^8*x
^4 + 35*c^4*d^7*x^3 + 21*c^5*d^6*x^2 + 7*c^6*d^5*x + c^7*d^4)

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Sympy [B]  time = 3.48897, size = 194, normalized size = 2.11 \begin{align*} - \frac{20 a^{3} d^{3} + 10 a^{2} b c d^{2} + 4 a b^{2} c^{2} d + b^{3} c^{3} + 35 b^{3} d^{3} x^{3} + x^{2} \left (84 a b^{2} d^{3} + 21 b^{3} c d^{2}\right ) + x \left (70 a^{2} b d^{3} + 28 a b^{2} c d^{2} + 7 b^{3} c^{2} d\right )}{140 c^{7} d^{4} + 980 c^{6} d^{5} x + 2940 c^{5} d^{6} x^{2} + 4900 c^{4} d^{7} x^{3} + 4900 c^{3} d^{8} x^{4} + 2940 c^{2} d^{9} x^{5} + 980 c d^{10} x^{6} + 140 d^{11} x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**3/(d*x+c)**8,x)

[Out]

-(20*a**3*d**3 + 10*a**2*b*c*d**2 + 4*a*b**2*c**2*d + b**3*c**3 + 35*b**3*d**3*x**3 + x**2*(84*a*b**2*d**3 + 2
1*b**3*c*d**2) + x*(70*a**2*b*d**3 + 28*a*b**2*c*d**2 + 7*b**3*c**2*d))/(140*c**7*d**4 + 980*c**6*d**5*x + 294
0*c**5*d**6*x**2 + 4900*c**4*d**7*x**3 + 4900*c**3*d**8*x**4 + 2940*c**2*d**9*x**5 + 980*c*d**10*x**6 + 140*d*
*11*x**7)

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Giac [A]  time = 1.07896, size = 154, normalized size = 1.67 \begin{align*} -\frac{35 \, b^{3} d^{3} x^{3} + 21 \, b^{3} c d^{2} x^{2} + 84 \, a b^{2} d^{3} x^{2} + 7 \, b^{3} c^{2} d x + 28 \, a b^{2} c d^{2} x + 70 \, a^{2} b d^{3} x + b^{3} c^{3} + 4 \, a b^{2} c^{2} d + 10 \, a^{2} b c d^{2} + 20 \, a^{3} d^{3}}{140 \,{\left (d x + c\right )}^{7} d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3/(d*x+c)^8,x, algorithm="giac")

[Out]

-1/140*(35*b^3*d^3*x^3 + 21*b^3*c*d^2*x^2 + 84*a*b^2*d^3*x^2 + 7*b^3*c^2*d*x + 28*a*b^2*c*d^2*x + 70*a^2*b*d^3
*x + b^3*c^3 + 4*a*b^2*c^2*d + 10*a^2*b*c*d^2 + 20*a^3*d^3)/((d*x + c)^7*d^4)