3.1270 \(\int \frac{(c+d x)^3}{(a+b x)^7} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0504132, 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} \[ -\frac{3 d^2 (b c-a d)}{4 b^4 (a+b x)^4}-\frac{3 d (b c-a d)^2}{5 b^4 (a+b x)^5}-\frac{(b c-a d)^3}{6 b^4 (a+b x)^6}-\frac{d^3}{3 b^4 (a+b x)^3} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^3/(a + b*x)^7,x]

[Out]

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

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

Mathematica [A]  time = 0.0350424, size = 97, normalized size = 1.05 \[ -\frac{3 a^2 b d^2 (c+2 d x)+a^3 d^3+3 a b^2 d \left (2 c^2+6 c d x+5 d^2 x^2\right )+b^3 \left (36 c^2 d x+10 c^3+45 c d^2 x^2+20 d^3 x^3\right )}{60 b^4 (a+b x)^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^3/(a + b*x)^7,x]

[Out]

-(a^3*d^3 + 3*a^2*b*d^2*(c + 2*d*x) + 3*a*b^2*d*(2*c^2 + 6*c*d*x + 5*d^2*x^2) + b^3*(10*c^3 + 36*c^2*d*x + 45*
c*d^2*x^2 + 20*d^3*x^3))/(60*b^4*(a + b*x)^6)

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Maple [A]  time = 0.006, 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}}{6\,{b}^{4} \left ( bx+a \right ) ^{6}}}-{\frac{{d}^{3}}{3\,{b}^{4} \left ( bx+a \right ) ^{3}}}-{\frac{3\,d \left ({a}^{2}{d}^{2}-2\,abcd+{b}^{2}{c}^{2} \right ) }{5\,{b}^{4} \left ( bx+a \right ) ^{5}}}+{\frac{3\,{d}^{2} \left ( ad-bc \right ) }{4\,{b}^{4} \left ( bx+a \right ) ^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^3/(b*x+a)^7,x)

[Out]

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

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Maxima [B]  time = 0.970962, size = 231, normalized size = 2.51 \begin{align*} -\frac{20 \, b^{3} d^{3} x^{3} + 10 \, b^{3} c^{3} + 6 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + a^{3} d^{3} + 15 \,{\left (3 \, b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + 6 \,{\left (6 \, b^{3} c^{2} d + 3 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{60 \,{\left (b^{10} x^{6} + 6 \, a b^{9} x^{5} + 15 \, a^{2} b^{8} x^{4} + 20 \, a^{3} b^{7} x^{3} + 15 \, a^{4} b^{6} x^{2} + 6 \, a^{5} b^{5} x + a^{6} b^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3/(b*x+a)^7,x, algorithm="maxima")

[Out]

-1/60*(20*b^3*d^3*x^3 + 10*b^3*c^3 + 6*a*b^2*c^2*d + 3*a^2*b*c*d^2 + a^3*d^3 + 15*(3*b^3*c*d^2 + a*b^2*d^3)*x^
2 + 6*(6*b^3*c^2*d + 3*a*b^2*c*d^2 + a^2*b*d^3)*x)/(b^10*x^6 + 6*a*b^9*x^5 + 15*a^2*b^8*x^4 + 20*a^3*b^7*x^3 +
15*a^4*b^6*x^2 + 6*a^5*b^5*x + a^6*b^4)

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Fricas [B]  time = 2.23976, size = 354, normalized size = 3.85 \begin{align*} -\frac{20 \, b^{3} d^{3} x^{3} + 10 \, b^{3} c^{3} + 6 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + a^{3} d^{3} + 15 \,{\left (3 \, b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + 6 \,{\left (6 \, b^{3} c^{2} d + 3 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{60 \,{\left (b^{10} x^{6} + 6 \, a b^{9} x^{5} + 15 \, a^{2} b^{8} x^{4} + 20 \, a^{3} b^{7} x^{3} + 15 \, a^{4} b^{6} x^{2} + 6 \, a^{5} b^{5} x + a^{6} b^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3/(b*x+a)^7,x, algorithm="fricas")

[Out]

-1/60*(20*b^3*d^3*x^3 + 10*b^3*c^3 + 6*a*b^2*c^2*d + 3*a^2*b*c*d^2 + a^3*d^3 + 15*(3*b^3*c*d^2 + a*b^2*d^3)*x^
2 + 6*(6*b^3*c^2*d + 3*a*b^2*c*d^2 + a^2*b*d^3)*x)/(b^10*x^6 + 6*a*b^9*x^5 + 15*a^2*b^8*x^4 + 20*a^3*b^7*x^3 +
15*a^4*b^6*x^2 + 6*a^5*b^5*x + a^6*b^4)

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Sympy [B]  time = 3.05433, size = 182, normalized size = 1.98 \begin{align*} - \frac{a^{3} d^{3} + 3 a^{2} b c d^{2} + 6 a b^{2} c^{2} d + 10 b^{3} c^{3} + 20 b^{3} d^{3} x^{3} + x^{2} \left (15 a b^{2} d^{3} + 45 b^{3} c d^{2}\right ) + x \left (6 a^{2} b d^{3} + 18 a b^{2} c d^{2} + 36 b^{3} c^{2} d\right )}{60 a^{6} b^{4} + 360 a^{5} b^{5} x + 900 a^{4} b^{6} x^{2} + 1200 a^{3} b^{7} x^{3} + 900 a^{2} b^{8} x^{4} + 360 a b^{9} x^{5} + 60 b^{10} x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**3/(b*x+a)**7,x)

[Out]

-(a**3*d**3 + 3*a**2*b*c*d**2 + 6*a*b**2*c**2*d + 10*b**3*c**3 + 20*b**3*d**3*x**3 + x**2*(15*a*b**2*d**3 + 45
*b**3*c*d**2) + x*(6*a**2*b*d**3 + 18*a*b**2*c*d**2 + 36*b**3*c**2*d))/(60*a**6*b**4 + 360*a**5*b**5*x + 900*a
**4*b**6*x**2 + 1200*a**3*b**7*x**3 + 900*a**2*b**8*x**4 + 360*a*b**9*x**5 + 60*b**10*x**6)

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Giac [A]  time = 1.05761, size = 154, normalized size = 1.67 \begin{align*} -\frac{20 \, b^{3} d^{3} x^{3} + 45 \, b^{3} c d^{2} x^{2} + 15 \, a b^{2} d^{3} x^{2} + 36 \, b^{3} c^{2} d x + 18 \, a b^{2} c d^{2} x + 6 \, a^{2} b d^{3} x + 10 \, b^{3} c^{3} + 6 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + a^{3} d^{3}}{60 \,{\left (b x + a\right )}^{6} b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3/(b*x+a)^7,x, algorithm="giac")

[Out]

-1/60*(20*b^3*d^3*x^3 + 45*b^3*c*d^2*x^2 + 15*a*b^2*d^3*x^2 + 36*b^3*c^2*d*x + 18*a*b^2*c*d^2*x + 6*a^2*b*d^3*
x + 10*b^3*c^3 + 6*a*b^2*c^2*d + 3*a^2*b*c*d^2 + a^3*d^3)/((b*x + a)^6*b^4)