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

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

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Rubi [A]  time = 0.01, antiderivative size = 58, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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)} \end {gather*}

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 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]

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])

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*}

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Mathematica [B]  time = 0.13, size = 367, normalized size = 6.33 \begin {gather*} -\frac {a^7 d^7+a^6 b d^6 (2 c+9 d x)+3 a^5 b^2 d^5 \left (c^2+6 c d x+12 d^2 x^2\right )+a^4 b^3 d^4 \left (4 c^3+27 c^2 d x+72 c d^2 x^2+84 d^3 x^3\right )+a^3 b^4 d^3 \left (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\right )+3 a^2 b^5 d^2 \left (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\right )+a b^6 d \left (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\right )+b^7 \left (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\right )}{72 b^8 (a+b x)^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^7/(a + b*x)^10,x]

[Out]

-1/72*(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 + 2
7*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 + 1
26*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))/(b^8*(a + b*x)^9)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(c+d x)^7}{(a+b x)^{10}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(c + d*x)^7/(a + b*x)^10,x]

[Out]

IntegrateAlgebraic[(c + d*x)^7/(a + b*x)^10, x]

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fricas [B]  time = 1.41, size = 548, normalized size = 9.45 \begin {gather*} -\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 {gather*}

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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giac [B]  time = 1.27, size = 496, normalized size = 8.55 \begin {gather*} -\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 {gather*}

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)

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maple [B]  time = 0.01, size = 464, normalized size = 8.00 \begin {gather*} -\frac {d^{7}}{2 \left (b x +a \right )^{2} b^{8}}+\frac {7 \left (a d -b c \right ) d^{6}}{3 \left (b x +a \right )^{3} b^{8}}-\frac {21 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) d^{5}}{4 \left (b x +a \right )^{4} b^{8}}+\frac {7 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) d^{4}}{\left (b x +a \right )^{5} b^{8}}-\frac {35 \left (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}\right ) d^{3}}{6 \left (b x +a \right )^{6} b^{8}}+\frac {3 \left (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}\right ) d^{2}}{\left (b x +a \right )^{7} b^{8}}-\frac {7 \left (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}\right ) d}{8 \left (b x +a \right )^{8} b^{8}}-\frac {-a^{7} d^{7}+7 a^{6} b c \,d^{6}-21 a^{5} b^{2} c^{2} d^{5}+35 a^{4} c^{3} d^{4} b^{3}-35 a^{3} b^{4} c^{4} d^{3}+21 a^{2} c^{5} d^{2} b^{5}-7 a \,b^{6} c^{6} d +b^{7} c^{7}}{9 \left (b x +a \right )^{9} b^{8}} \end {gather*}

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

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maxima [B]  time = 1.71, size = 548, normalized size = 9.45 \begin {gather*} -\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 {gather*}

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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mupad [B]  time = 0.15, size = 39, normalized size = 0.67 \begin {gather*} \frac {{\left (c+d\,x\right )}^8\,\left (9\,a\,d-8\,b\,c+b\,d\,x\right )}{72\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^7/(a + b*x)^10,x)

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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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