3.1291 \(\int \frac{(c+d x)^7}{(a+b x)^9} \, dx\)

Optimal. Leaf size=28 \[ -\frac{(c+d x)^8}{8 (a+b x)^8 (b c-a d)} \]

[Out]

-(c + d*x)^8/(8*(b*c - a*d)*(a + b*x)^8)

________________________________________________________________________________________

Rubi [A]  time = 0.0034644, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {37} \[ -\frac{(c+d x)^8}{8 (a+b x)^8 (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^7/(a + b*x)^9,x]

[Out]

-(c + d*x)^8/(8*(b*c - a*d)*(a + b*x)^8)

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{(c+d x)^7}{(a+b x)^9} \, dx &=-\frac{(c+d x)^8}{8 (b c-a d) (a+b x)^8}\\ \end{align*}

Mathematica [B]  time = 0.119213, size = 353, normalized size = 12.61 \[ -\frac{a^2 b^5 d^2 \left (28 c^3 d^2 x^2+56 c^2 d^3 x^3+8 c^4 d x+c^5+70 c d^4 x^4+56 d^5 x^5\right )+a^3 b^4 d^3 \left (28 c^2 d^2 x^2+8 c^3 d x+c^4+56 c d^3 x^3+70 d^4 x^4\right )+a^4 b^3 d^4 \left (8 c^2 d x+c^3+28 c d^2 x^2+56 d^3 x^3\right )+a^5 b^2 d^5 \left (c^2+8 c d x+28 d^2 x^2\right )+a^6 b d^6 (c+8 d x)+a^7 d^7+a b^6 d \left (28 c^4 d^2 x^2+56 c^3 d^3 x^3+70 c^2 d^4 x^4+8 c^5 d x+c^6+56 c d^5 x^5+28 d^6 x^6\right )+b^7 \left (28 c^5 d^2 x^2+56 c^4 d^3 x^3+70 c^3 d^4 x^4+56 c^2 d^5 x^5+8 c^6 d x+c^7+28 c d^6 x^6+8 d^7 x^7\right )}{8 b^8 (a+b x)^8} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^7/(a + b*x)^9,x]

[Out]

-(a^7*d^7 + a^6*b*d^6*(c + 8*d*x) + a^5*b^2*d^5*(c^2 + 8*c*d*x + 28*d^2*x^2) + a^4*b^3*d^4*(c^3 + 8*c^2*d*x +
28*c*d^2*x^2 + 56*d^3*x^3) + a^3*b^4*d^3*(c^4 + 8*c^3*d*x + 28*c^2*d^2*x^2 + 56*c*d^3*x^3 + 70*d^4*x^4) + a^2*
b^5*d^2*(c^5 + 8*c^4*d*x + 28*c^3*d^2*x^2 + 56*c^2*d^3*x^3 + 70*c*d^4*x^4 + 56*d^5*x^5) + a*b^6*d*(c^6 + 8*c^5
*d*x + 28*c^4*d^2*x^2 + 56*c^3*d^3*x^3 + 70*c^2*d^4*x^4 + 56*c*d^5*x^5 + 28*d^6*x^6) + b^7*(c^7 + 8*c^6*d*x +
28*c^5*d^2*x^2 + 56*c^4*d^3*x^3 + 70*c^3*d^4*x^4 + 56*c^2*d^5*x^5 + 28*c*d^6*x^6 + 8*d^7*x^7))/(8*b^8*(a + b*x
)^8)

________________________________________________________________________________________

Maple [B]  time = 0.007, size = 464, normalized size = 16.6 \begin{align*} -{\frac{{d}^{7}}{{b}^{8} \left ( bx+a \right ) }}-{\frac{-{a}^{7}{d}^{7}+7\,{a}^{6}c{d}^{6}b-21\,{a}^{5}{b}^{2}{c}^{2}{d}^{5}+35\,{c}^{3}{d}^{4}{a}^{4}{b}^{3}-35\,{a}^{3}{b}^{4}{c}^{4}{d}^{3}+21\,{a}^{2}{c}^{5}{d}^{2}{b}^{5}-7\,a{c}^{6}d{b}^{6}+{b}^{7}{c}^{7}}{8\,{b}^{8} \left ( bx+a \right ) ^{8}}}+{\frac{7\,{d}^{2} \left ({a}^{5}{d}^{5}-5\,{a}^{4}bc{d}^{4}+10\,{a}^{3}{b}^{2}{c}^{2}{d}^{3}-10\,{a}^{2}{b}^{3}{c}^{3}{d}^{2}+5\,a{b}^{4}{c}^{4}d-{b}^{5}{c}^{5} \right ) }{2\,{b}^{8} \left ( bx+a \right ) ^{6}}}+{\frac{7\,{d}^{6} \left ( ad-bc \right ) }{2\,{b}^{8} \left ( bx+a \right ) ^{2}}}-7\,{\frac{{d}^{5} \left ({a}^{2}{d}^{2}-2\,abcd+{b}^{2}{c}^{2} \right ) }{{b}^{8} \left ( bx+a \right ) ^{3}}}-7\,{\frac{{d}^{3} \left ({a}^{4}{d}^{4}-4\,{a}^{3}bc{d}^{3}+6\,{a}^{2}{b}^{2}{c}^{2}{d}^{2}-4\,a{b}^{3}{c}^{3}d+{b}^{4}{c}^{4} \right ) }{{b}^{8} \left ( bx+a \right ) ^{5}}}+{\frac{35\,{d}^{4} \left ({a}^{3}{d}^{3}-3\,{a}^{2}bc{d}^{2}+3\,a{b}^{2}{c}^{2}d-{b}^{3}{c}^{3} \right ) }{4\,{b}^{8} \left ( bx+a \right ) ^{4}}}-{\frac{d \left ({a}^{6}{d}^{6}-6\,{a}^{5}bc{d}^{5}+15\,{a}^{4}{b}^{2}{c}^{2}{d}^{4}-20\,{a}^{3}{b}^{3}{c}^{3}{d}^{3}+15\,{a}^{2}{b}^{4}{c}^{4}{d}^{2}-6\,a{b}^{5}{c}^{5}d+{b}^{6}{c}^{6} \right ) }{{b}^{8} \left ( bx+a \right ) ^{7}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^7/(b*x+a)^9,x)

[Out]

-d^7/b^8/(b*x+a)-1/8*(-a^7*d^7+7*a^6*b*c*d^6-21*a^5*b^2*c^2*d^5+35*a^4*b^3*c^3*d^4-35*a^3*b^4*c^4*d^3+21*a^2*b
^5*c^5*d^2-7*a*b^6*c^6*d+b^7*c^7)/b^8/(b*x+a)^8+7/2*d^2*(a^5*d^5-5*a^4*b*c*d^4+10*a^3*b^2*c^2*d^3-10*a^2*b^3*c
^3*d^2+5*a*b^4*c^4*d-b^5*c^5)/b^8/(b*x+a)^6+7/2*d^6*(a*d-b*c)/b^8/(b*x+a)^2-7*d^5*(a^2*d^2-2*a*b*c*d+b^2*c^2)/
b^8/(b*x+a)^3-7*d^3*(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)/b^8/(b*x+a)^5+35/4*d^4*(a^
3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/b^8/(b*x+a)^4-d*(a^6*d^6-6*a^5*b*c*d^5+15*a^4*b^2*c^2*d^4-20*a^3*b^
3*c^3*d^3+15*a^2*b^4*c^4*d^2-6*a*b^5*c^5*d+b^6*c^6)/b^8/(b*x+a)^7

________________________________________________________________________________________

Maxima [B]  time = 1.05627, size = 687, normalized size = 24.54 \begin{align*} -\frac{8 \, b^{7} d^{7} x^{7} + b^{7} c^{7} + a b^{6} c^{6} d + a^{2} b^{5} c^{5} d^{2} + a^{3} b^{4} c^{4} d^{3} + a^{4} b^{3} c^{3} d^{4} + a^{5} b^{2} c^{2} d^{5} + a^{6} b c d^{6} + a^{7} d^{7} + 28 \,{\left (b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{6} + 56 \,{\left (b^{7} c^{2} d^{5} + a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 70 \,{\left (b^{7} c^{3} d^{4} + a b^{6} c^{2} d^{5} + a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 56 \,{\left (b^{7} c^{4} d^{3} + a b^{6} c^{3} d^{4} + a^{2} b^{5} c^{2} d^{5} + a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 28 \,{\left (b^{7} c^{5} d^{2} + a b^{6} c^{4} d^{3} + a^{2} b^{5} c^{3} d^{4} + a^{3} b^{4} c^{2} d^{5} + a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 8 \,{\left (b^{7} c^{6} d + a b^{6} c^{5} d^{2} + a^{2} b^{5} c^{4} d^{3} + a^{3} b^{4} c^{3} d^{4} + a^{4} b^{3} c^{2} d^{5} + a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x}{8 \,{\left (b^{16} x^{8} + 8 \, a b^{15} x^{7} + 28 \, a^{2} b^{14} x^{6} + 56 \, a^{3} b^{13} x^{5} + 70 \, a^{4} b^{12} x^{4} + 56 \, a^{5} b^{11} x^{3} + 28 \, a^{6} b^{10} x^{2} + 8 \, a^{7} b^{9} x + a^{8} b^{8}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^7/(b*x+a)^9,x, algorithm="maxima")

[Out]

-1/8*(8*b^7*d^7*x^7 + b^7*c^7 + a*b^6*c^6*d + a^2*b^5*c^5*d^2 + a^3*b^4*c^4*d^3 + a^4*b^3*c^3*d^4 + a^5*b^2*c^
2*d^5 + a^6*b*c*d^6 + a^7*d^7 + 28*(b^7*c*d^6 + a*b^6*d^7)*x^6 + 56*(b^7*c^2*d^5 + a*b^6*c*d^6 + a^2*b^5*d^7)*
x^5 + 70*(b^7*c^3*d^4 + a*b^6*c^2*d^5 + a^2*b^5*c*d^6 + a^3*b^4*d^7)*x^4 + 56*(b^7*c^4*d^3 + a*b^6*c^3*d^4 + a
^2*b^5*c^2*d^5 + a^3*b^4*c*d^6 + a^4*b^3*d^7)*x^3 + 28*(b^7*c^5*d^2 + a*b^6*c^4*d^3 + a^2*b^5*c^3*d^4 + a^3*b^
4*c^2*d^5 + a^4*b^3*c*d^6 + a^5*b^2*d^7)*x^2 + 8*(b^7*c^6*d + a*b^6*c^5*d^2 + a^2*b^5*c^4*d^3 + a^3*b^4*c^3*d^
4 + a^4*b^3*c^2*d^5 + a^5*b^2*c*d^6 + a^6*b*d^7)*x)/(b^16*x^8 + 8*a*b^15*x^7 + 28*a^2*b^14*x^6 + 56*a^3*b^13*x
^5 + 70*a^4*b^12*x^4 + 56*a^5*b^11*x^3 + 28*a^6*b^10*x^2 + 8*a^7*b^9*x + a^8*b^8)

________________________________________________________________________________________

Fricas [B]  time = 2.10975, size = 1003, normalized size = 35.82 \begin{align*} -\frac{8 \, b^{7} d^{7} x^{7} + b^{7} c^{7} + a b^{6} c^{6} d + a^{2} b^{5} c^{5} d^{2} + a^{3} b^{4} c^{4} d^{3} + a^{4} b^{3} c^{3} d^{4} + a^{5} b^{2} c^{2} d^{5} + a^{6} b c d^{6} + a^{7} d^{7} + 28 \,{\left (b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{6} + 56 \,{\left (b^{7} c^{2} d^{5} + a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 70 \,{\left (b^{7} c^{3} d^{4} + a b^{6} c^{2} d^{5} + a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 56 \,{\left (b^{7} c^{4} d^{3} + a b^{6} c^{3} d^{4} + a^{2} b^{5} c^{2} d^{5} + a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 28 \,{\left (b^{7} c^{5} d^{2} + a b^{6} c^{4} d^{3} + a^{2} b^{5} c^{3} d^{4} + a^{3} b^{4} c^{2} d^{5} + a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 8 \,{\left (b^{7} c^{6} d + a b^{6} c^{5} d^{2} + a^{2} b^{5} c^{4} d^{3} + a^{3} b^{4} c^{3} d^{4} + a^{4} b^{3} c^{2} d^{5} + a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x}{8 \,{\left (b^{16} x^{8} + 8 \, a b^{15} x^{7} + 28 \, a^{2} b^{14} x^{6} + 56 \, a^{3} b^{13} x^{5} + 70 \, a^{4} b^{12} x^{4} + 56 \, a^{5} b^{11} x^{3} + 28 \, a^{6} b^{10} x^{2} + 8 \, a^{7} b^{9} x + a^{8} b^{8}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^7/(b*x+a)^9,x, algorithm="fricas")

[Out]

-1/8*(8*b^7*d^7*x^7 + b^7*c^7 + a*b^6*c^6*d + a^2*b^5*c^5*d^2 + a^3*b^4*c^4*d^3 + a^4*b^3*c^3*d^4 + a^5*b^2*c^
2*d^5 + a^6*b*c*d^6 + a^7*d^7 + 28*(b^7*c*d^6 + a*b^6*d^7)*x^6 + 56*(b^7*c^2*d^5 + a*b^6*c*d^6 + a^2*b^5*d^7)*
x^5 + 70*(b^7*c^3*d^4 + a*b^6*c^2*d^5 + a^2*b^5*c*d^6 + a^3*b^4*d^7)*x^4 + 56*(b^7*c^4*d^3 + a*b^6*c^3*d^4 + a
^2*b^5*c^2*d^5 + a^3*b^4*c*d^6 + a^4*b^3*d^7)*x^3 + 28*(b^7*c^5*d^2 + a*b^6*c^4*d^3 + a^2*b^5*c^3*d^4 + a^3*b^
4*c^2*d^5 + a^4*b^3*c*d^6 + a^5*b^2*d^7)*x^2 + 8*(b^7*c^6*d + a*b^6*c^5*d^2 + a^2*b^5*c^4*d^3 + a^3*b^4*c^3*d^
4 + a^4*b^3*c^2*d^5 + a^5*b^2*c*d^6 + a^6*b*d^7)*x)/(b^16*x^8 + 8*a*b^15*x^7 + 28*a^2*b^14*x^6 + 56*a^3*b^13*x
^5 + 70*a^4*b^12*x^4 + 56*a^5*b^11*x^3 + 28*a^6*b^10*x^2 + 8*a^7*b^9*x + a^8*b^8)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**7/(b*x+a)**9,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.06372, size = 660, normalized size = 23.57 \begin{align*} -\frac{8 \, b^{7} d^{7} x^{7} + 28 \, b^{7} c d^{6} x^{6} + 28 \, a b^{6} d^{7} x^{6} + 56 \, b^{7} c^{2} d^{5} x^{5} + 56 \, a b^{6} c d^{6} x^{5} + 56 \, a^{2} b^{5} d^{7} x^{5} + 70 \, b^{7} c^{3} d^{4} x^{4} + 70 \, a b^{6} c^{2} d^{5} x^{4} + 70 \, a^{2} b^{5} c d^{6} x^{4} + 70 \, a^{3} b^{4} d^{7} x^{4} + 56 \, b^{7} c^{4} d^{3} x^{3} + 56 \, a b^{6} c^{3} d^{4} x^{3} + 56 \, a^{2} b^{5} c^{2} d^{5} x^{3} + 56 \, a^{3} b^{4} c d^{6} x^{3} + 56 \, a^{4} b^{3} d^{7} x^{3} + 28 \, b^{7} c^{5} d^{2} x^{2} + 28 \, a b^{6} c^{4} d^{3} x^{2} + 28 \, a^{2} b^{5} c^{3} d^{4} x^{2} + 28 \, a^{3} b^{4} c^{2} d^{5} x^{2} + 28 \, a^{4} b^{3} c d^{6} x^{2} + 28 \, a^{5} b^{2} d^{7} x^{2} + 8 \, b^{7} c^{6} d x + 8 \, a b^{6} c^{5} d^{2} x + 8 \, a^{2} b^{5} c^{4} d^{3} x + 8 \, a^{3} b^{4} c^{3} d^{4} x + 8 \, a^{4} b^{3} c^{2} d^{5} x + 8 \, a^{5} b^{2} c d^{6} x + 8 \, a^{6} b d^{7} x + b^{7} c^{7} + a b^{6} c^{6} d + a^{2} b^{5} c^{5} d^{2} + a^{3} b^{4} c^{4} d^{3} + a^{4} b^{3} c^{3} d^{4} + a^{5} b^{2} c^{2} d^{5} + a^{6} b c d^{6} + a^{7} d^{7}}{8 \,{\left (b x + a\right )}^{8} b^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^7/(b*x+a)^9,x, algorithm="giac")

[Out]

-1/8*(8*b^7*d^7*x^7 + 28*b^7*c*d^6*x^6 + 28*a*b^6*d^7*x^6 + 56*b^7*c^2*d^5*x^5 + 56*a*b^6*c*d^6*x^5 + 56*a^2*b
^5*d^7*x^5 + 70*b^7*c^3*d^4*x^4 + 70*a*b^6*c^2*d^5*x^4 + 70*a^2*b^5*c*d^6*x^4 + 70*a^3*b^4*d^7*x^4 + 56*b^7*c^
4*d^3*x^3 + 56*a*b^6*c^3*d^4*x^3 + 56*a^2*b^5*c^2*d^5*x^3 + 56*a^3*b^4*c*d^6*x^3 + 56*a^4*b^3*d^7*x^3 + 28*b^7
*c^5*d^2*x^2 + 28*a*b^6*c^4*d^3*x^2 + 28*a^2*b^5*c^3*d^4*x^2 + 28*a^3*b^4*c^2*d^5*x^2 + 28*a^4*b^3*c*d^6*x^2 +
28*a^5*b^2*d^7*x^2 + 8*b^7*c^6*d*x + 8*a*b^6*c^5*d^2*x + 8*a^2*b^5*c^4*d^3*x + 8*a^3*b^4*c^3*d^4*x + 8*a^4*b^
3*c^2*d^5*x + 8*a^5*b^2*c*d^6*x + 8*a^6*b*d^7*x + b^7*c^7 + a*b^6*c^6*d + a^2*b^5*c^5*d^2 + a^3*b^4*c^4*d^3 +
a^4*b^3*c^3*d^4 + a^5*b^2*c^2*d^5 + a^6*b*c*d^6 + a^7*d^7)/((b*x + a)^8*b^8)