3.1292 \(\int \frac{(c+d x)^7}{(a+b x)^{10}} \, dx\)

Optimal. Leaf size=58 \[ \frac{d (c+d x)^8}{72 (a+b x)^8 (b c-a d)^2}-\frac{(c+d x)^8}{9 (a+b x)^9 (b c-a d)} \]

[Out]

-(c + d*x)^8/(9*(b*c - a*d)*(a + b*x)^9) + (d*(c + d*x)^8)/(72*(b*c - a*d)^2*(a + b*x)^8)

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Rubi [A]  time = 0.009459, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {45, 37} \[ \frac{d (c+d x)^8}{72 (a+b x)^8 (b c-a d)^2}-\frac{(c+d x)^8}{9 (a+b x)^9 (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^7/(a + b*x)^10,x]

[Out]

-(c + d*x)^8/(9*(b*c - a*d)*(a + b*x)^9) + (d*(c + d*x)^8)/(72*(b*c - a*d)^2*(a + b*x)^8)

Rule 45

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] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

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)^{10}} \, dx &=-\frac{(c+d x)^8}{9 (b c-a d) (a+b x)^9}-\frac{d \int \frac{(c+d x)^7}{(a+b x)^9} \, dx}{9 (b c-a d)}\\ &=-\frac{(c+d x)^8}{9 (b c-a d) (a+b x)^9}+\frac{d (c+d x)^8}{72 (b c-a d)^2 (a+b x)^8}\\ \end{align*}

Mathematica [B]  time = 0.118556, size = 367, normalized size = 6.33 \[ -\frac{3 a^2 b^5 d^2 \left (48 c^3 d^2 x^2+84 c^2 d^3 x^3+15 c^4 d x+2 c^5+84 c d^4 x^4+42 d^5 x^5\right )+a^3 b^4 d^3 \left (108 c^2 d^2 x^2+36 c^3 d x+5 c^4+168 c d^3 x^3+126 d^4 x^4\right )+a^4 b^3 d^4 \left (27 c^2 d x+4 c^3+72 c d^2 x^2+84 d^3 x^3\right )+3 a^5 b^2 d^5 \left (c^2+6 c d x+12 d^2 x^2\right )+a^6 b d^6 (2 c+9 d x)+a^7 d^7+a b^6 d \left (180 c^4 d^2 x^2+336 c^3 d^3 x^3+378 c^2 d^4 x^4+54 c^5 d x+7 c^6+252 c d^5 x^5+84 d^6 x^6\right )+b^7 \left (216 c^5 d^2 x^2+420 c^4 d^3 x^3+504 c^3 d^4 x^4+378 c^2 d^5 x^5+63 c^6 d x+8 c^7+168 c d^6 x^6+36 d^7 x^7\right )}{72 b^8 (a+b x)^9} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^7/(a + b*x)^10,x]

[Out]

-(a^7*d^7 + a^6*b*d^6*(2*c + 9*d*x) + 3*a^5*b^2*d^5*(c^2 + 6*c*d*x + 12*d^2*x^2) + a^4*b^3*d^4*(4*c^3 + 27*c^2
*d*x + 72*c*d^2*x^2 + 84*d^3*x^3) + a^3*b^4*d^3*(5*c^4 + 36*c^3*d*x + 108*c^2*d^2*x^2 + 168*c*d^3*x^3 + 126*d^
4*x^4) + 3*a^2*b^5*d^2*(2*c^5 + 15*c^4*d*x + 48*c^3*d^2*x^2 + 84*c^2*d^3*x^3 + 84*c*d^4*x^4 + 42*d^5*x^5) + a*
b^6*d*(7*c^6 + 54*c^5*d*x + 180*c^4*d^2*x^2 + 336*c^3*d^3*x^3 + 378*c^2*d^4*x^4 + 252*c*d^5*x^5 + 84*d^6*x^6)
+ b^7*(8*c^7 + 63*c^6*d*x + 216*c^5*d^2*x^2 + 420*c^4*d^3*x^3 + 504*c^3*d^4*x^4 + 378*c^2*d^5*x^5 + 168*c*d^6*
x^6 + 36*d^7*x^7))/(72*b^8*(a + b*x)^9)

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Maple [B]  time = 0.008, size = 464, normalized size = 8. \begin{align*} -{\frac{7\,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 ) }{8\,{b}^{8} \left ( bx+a \right ) ^{8}}}-{\frac{35\,{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 ) }{6\,{b}^{8} \left ( bx+a \right ) ^{6}}}-{\frac{{d}^{7}}{2\,{b}^{8} \left ( bx+a \right ) ^{2}}}+{\frac{7\,{d}^{6} \left ( ad-bc \right ) }{3\,{b}^{8} \left ( bx+a \right ) ^{3}}}+7\,{\frac{{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 ) }{{b}^{8} \left ( bx+a \right ) ^{5}}}-{\frac{21\,{d}^{5} \left ({a}^{2}{d}^{2}-2\,abcd+{b}^{2}{c}^{2} \right ) }{4\,{b}^{8} \left ( bx+a \right ) ^{4}}}-{\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}}{9\,{b}^{8} \left ( bx+a \right ) ^{9}}}+3\,{\frac{{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 ) }{{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)^10,x)

[Out]

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

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Maxima [B]  time = 1.0716, size = 740, normalized size = 12.76 \begin{align*} -\frac{36 \, b^{7} d^{7} x^{7} + 8 \, b^{7} c^{7} + 7 \, a b^{6} c^{6} d + 6 \, a^{2} b^{5} c^{5} d^{2} + 5 \, a^{3} b^{4} c^{4} d^{3} + 4 \, a^{4} b^{3} c^{3} d^{4} + 3 \, a^{5} b^{2} c^{2} d^{5} + 2 \, a^{6} b c d^{6} + a^{7} d^{7} + 84 \,{\left (2 \, b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{6} + 126 \,{\left (3 \, b^{7} c^{2} d^{5} + 2 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 126 \,{\left (4 \, b^{7} c^{3} d^{4} + 3 \, a b^{6} c^{2} d^{5} + 2 \, a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 84 \,{\left (5 \, b^{7} c^{4} d^{3} + 4 \, a b^{6} c^{3} d^{4} + 3 \, a^{2} b^{5} c^{2} d^{5} + 2 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 36 \,{\left (6 \, b^{7} c^{5} d^{2} + 5 \, a b^{6} c^{4} d^{3} + 4 \, a^{2} b^{5} c^{3} d^{4} + 3 \, a^{3} b^{4} c^{2} d^{5} + 2 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 9 \,{\left (7 \, b^{7} c^{6} d + 6 \, a b^{6} c^{5} d^{2} + 5 \, a^{2} b^{5} c^{4} d^{3} + 4 \, a^{3} b^{4} c^{3} d^{4} + 3 \, a^{4} b^{3} c^{2} d^{5} + 2 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x}{72 \,{\left (b^{17} x^{9} + 9 \, a b^{16} x^{8} + 36 \, a^{2} b^{15} x^{7} + 84 \, a^{3} b^{14} x^{6} + 126 \, a^{4} b^{13} x^{5} + 126 \, a^{5} b^{12} x^{4} + 84 \, a^{6} b^{11} x^{3} + 36 \, a^{7} b^{10} x^{2} + 9 \, a^{8} b^{9} x + a^{9} b^{8}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^7/(b*x+a)^10,x, algorithm="maxima")

[Out]

-1/72*(36*b^7*d^7*x^7 + 8*b^7*c^7 + 7*a*b^6*c^6*d + 6*a^2*b^5*c^5*d^2 + 5*a^3*b^4*c^4*d^3 + 4*a^4*b^3*c^3*d^4
+ 3*a^5*b^2*c^2*d^5 + 2*a^6*b*c*d^6 + a^7*d^7 + 84*(2*b^7*c*d^6 + a*b^6*d^7)*x^6 + 126*(3*b^7*c^2*d^5 + 2*a*b^
6*c*d^6 + a^2*b^5*d^7)*x^5 + 126*(4*b^7*c^3*d^4 + 3*a*b^6*c^2*d^5 + 2*a^2*b^5*c*d^6 + a^3*b^4*d^7)*x^4 + 84*(5
*b^7*c^4*d^3 + 4*a*b^6*c^3*d^4 + 3*a^2*b^5*c^2*d^5 + 2*a^3*b^4*c*d^6 + a^4*b^3*d^7)*x^3 + 36*(6*b^7*c^5*d^2 +
5*a*b^6*c^4*d^3 + 4*a^2*b^5*c^3*d^4 + 3*a^3*b^4*c^2*d^5 + 2*a^4*b^3*c*d^6 + a^5*b^2*d^7)*x^2 + 9*(7*b^7*c^6*d
+ 6*a*b^6*c^5*d^2 + 5*a^2*b^5*c^4*d^3 + 4*a^3*b^4*c^3*d^4 + 3*a^4*b^3*c^2*d^5 + 2*a^5*b^2*c*d^6 + a^6*b*d^7)*x
)/(b^17*x^9 + 9*a*b^16*x^8 + 36*a^2*b^15*x^7 + 84*a^3*b^14*x^6 + 126*a^4*b^13*x^5 + 126*a^5*b^12*x^4 + 84*a^6*
b^11*x^3 + 36*a^7*b^10*x^2 + 9*a^8*b^9*x + a^9*b^8)

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Fricas [B]  time = 2.03853, size = 1111, normalized size = 19.16 \begin{align*} -\frac{36 \, b^{7} d^{7} x^{7} + 8 \, b^{7} c^{7} + 7 \, a b^{6} c^{6} d + 6 \, a^{2} b^{5} c^{5} d^{2} + 5 \, a^{3} b^{4} c^{4} d^{3} + 4 \, a^{4} b^{3} c^{3} d^{4} + 3 \, a^{5} b^{2} c^{2} d^{5} + 2 \, a^{6} b c d^{6} + a^{7} d^{7} + 84 \,{\left (2 \, b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{6} + 126 \,{\left (3 \, b^{7} c^{2} d^{5} + 2 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 126 \,{\left (4 \, b^{7} c^{3} d^{4} + 3 \, a b^{6} c^{2} d^{5} + 2 \, a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 84 \,{\left (5 \, b^{7} c^{4} d^{3} + 4 \, a b^{6} c^{3} d^{4} + 3 \, a^{2} b^{5} c^{2} d^{5} + 2 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 36 \,{\left (6 \, b^{7} c^{5} d^{2} + 5 \, a b^{6} c^{4} d^{3} + 4 \, a^{2} b^{5} c^{3} d^{4} + 3 \, a^{3} b^{4} c^{2} d^{5} + 2 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 9 \,{\left (7 \, b^{7} c^{6} d + 6 \, a b^{6} c^{5} d^{2} + 5 \, a^{2} b^{5} c^{4} d^{3} + 4 \, a^{3} b^{4} c^{3} d^{4} + 3 \, a^{4} b^{3} c^{2} d^{5} + 2 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x}{72 \,{\left (b^{17} x^{9} + 9 \, a b^{16} x^{8} + 36 \, a^{2} b^{15} x^{7} + 84 \, a^{3} b^{14} x^{6} + 126 \, a^{4} b^{13} x^{5} + 126 \, a^{5} b^{12} x^{4} + 84 \, a^{6} b^{11} x^{3} + 36 \, a^{7} b^{10} x^{2} + 9 \, a^{8} b^{9} x + a^{9} b^{8}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^7/(b*x+a)^10,x, algorithm="fricas")

[Out]

-1/72*(36*b^7*d^7*x^7 + 8*b^7*c^7 + 7*a*b^6*c^6*d + 6*a^2*b^5*c^5*d^2 + 5*a^3*b^4*c^4*d^3 + 4*a^4*b^3*c^3*d^4
+ 3*a^5*b^2*c^2*d^5 + 2*a^6*b*c*d^6 + a^7*d^7 + 84*(2*b^7*c*d^6 + a*b^6*d^7)*x^6 + 126*(3*b^7*c^2*d^5 + 2*a*b^
6*c*d^6 + a^2*b^5*d^7)*x^5 + 126*(4*b^7*c^3*d^4 + 3*a*b^6*c^2*d^5 + 2*a^2*b^5*c*d^6 + a^3*b^4*d^7)*x^4 + 84*(5
*b^7*c^4*d^3 + 4*a*b^6*c^3*d^4 + 3*a^2*b^5*c^2*d^5 + 2*a^3*b^4*c*d^6 + a^4*b^3*d^7)*x^3 + 36*(6*b^7*c^5*d^2 +
5*a*b^6*c^4*d^3 + 4*a^2*b^5*c^3*d^4 + 3*a^3*b^4*c^2*d^5 + 2*a^4*b^3*c*d^6 + a^5*b^2*d^7)*x^2 + 9*(7*b^7*c^6*d
+ 6*a*b^6*c^5*d^2 + 5*a^2*b^5*c^4*d^3 + 4*a^3*b^4*c^3*d^4 + 3*a^4*b^3*c^2*d^5 + 2*a^5*b^2*c*d^6 + a^6*b*d^7)*x
)/(b^17*x^9 + 9*a*b^16*x^8 + 36*a^2*b^15*x^7 + 84*a^3*b^14*x^6 + 126*a^4*b^13*x^5 + 126*a^5*b^12*x^4 + 84*a^6*
b^11*x^3 + 36*a^7*b^10*x^2 + 9*a^8*b^9*x + a^9*b^8)

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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)**10,x)

[Out]

Timed out

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Giac [B]  time = 1.06109, size = 670, normalized size = 11.55 \begin{align*} -\frac{36 \, b^{7} d^{7} x^{7} + 168 \, b^{7} c d^{6} x^{6} + 84 \, a b^{6} d^{7} x^{6} + 378 \, b^{7} c^{2} d^{5} x^{5} + 252 \, a b^{6} c d^{6} x^{5} + 126 \, a^{2} b^{5} d^{7} x^{5} + 504 \, b^{7} c^{3} d^{4} x^{4} + 378 \, a b^{6} c^{2} d^{5} x^{4} + 252 \, a^{2} b^{5} c d^{6} x^{4} + 126 \, a^{3} b^{4} d^{7} x^{4} + 420 \, b^{7} c^{4} d^{3} x^{3} + 336 \, a b^{6} c^{3} d^{4} x^{3} + 252 \, a^{2} b^{5} c^{2} d^{5} x^{3} + 168 \, a^{3} b^{4} c d^{6} x^{3} + 84 \, a^{4} b^{3} d^{7} x^{3} + 216 \, b^{7} c^{5} d^{2} x^{2} + 180 \, a b^{6} c^{4} d^{3} x^{2} + 144 \, a^{2} b^{5} c^{3} d^{4} x^{2} + 108 \, a^{3} b^{4} c^{2} d^{5} x^{2} + 72 \, a^{4} b^{3} c d^{6} x^{2} + 36 \, a^{5} b^{2} d^{7} x^{2} + 63 \, b^{7} c^{6} d x + 54 \, a b^{6} c^{5} d^{2} x + 45 \, a^{2} b^{5} c^{4} d^{3} x + 36 \, a^{3} b^{4} c^{3} d^{4} x + 27 \, a^{4} b^{3} c^{2} d^{5} x + 18 \, a^{5} b^{2} c d^{6} x + 9 \, a^{6} b d^{7} x + 8 \, b^{7} c^{7} + 7 \, a b^{6} c^{6} d + 6 \, a^{2} b^{5} c^{5} d^{2} + 5 \, a^{3} b^{4} c^{4} d^{3} + 4 \, a^{4} b^{3} c^{3} d^{4} + 3 \, a^{5} b^{2} c^{2} d^{5} + 2 \, a^{6} b c d^{6} + a^{7} d^{7}}{72 \,{\left (b x + a\right )}^{9} b^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^7/(b*x+a)^10,x, algorithm="giac")

[Out]

-1/72*(36*b^7*d^7*x^7 + 168*b^7*c*d^6*x^6 + 84*a*b^6*d^7*x^6 + 378*b^7*c^2*d^5*x^5 + 252*a*b^6*c*d^6*x^5 + 126
*a^2*b^5*d^7*x^5 + 504*b^7*c^3*d^4*x^4 + 378*a*b^6*c^2*d^5*x^4 + 252*a^2*b^5*c*d^6*x^4 + 126*a^3*b^4*d^7*x^4 +
420*b^7*c^4*d^3*x^3 + 336*a*b^6*c^3*d^4*x^3 + 252*a^2*b^5*c^2*d^5*x^3 + 168*a^3*b^4*c*d^6*x^3 + 84*a^4*b^3*d^
7*x^3 + 216*b^7*c^5*d^2*x^2 + 180*a*b^6*c^4*d^3*x^2 + 144*a^2*b^5*c^3*d^4*x^2 + 108*a^3*b^4*c^2*d^5*x^2 + 72*a
^4*b^3*c*d^6*x^2 + 36*a^5*b^2*d^7*x^2 + 63*b^7*c^6*d*x + 54*a*b^6*c^5*d^2*x + 45*a^2*b^5*c^4*d^3*x + 36*a^3*b^
4*c^3*d^4*x + 27*a^4*b^3*c^2*d^5*x + 18*a^5*b^2*c*d^6*x + 9*a^6*b*d^7*x + 8*b^7*c^7 + 7*a*b^6*c^6*d + 6*a^2*b^
5*c^5*d^2 + 5*a^3*b^4*c^4*d^3 + 4*a^4*b^3*c^3*d^4 + 3*a^5*b^2*c^2*d^5 + 2*a^6*b*c*d^6 + a^7*d^7)/((b*x + a)^9*
b^8)