3.12.88 \(\int \frac {(c+d x)^7}{(a+b x)^{12}} \, dx\)

Optimal. Leaf size=120 \[ \frac {d^3 (c+d x)^8}{1320 (a+b x)^8 (b c-a d)^4}-\frac {d^2 (c+d x)^8}{165 (a+b x)^9 (b c-a d)^3}+\frac {3 d (c+d x)^8}{110 (a+b x)^{10} (b c-a d)^2}-\frac {(c+d x)^8}{11 (a+b x)^{11} (b c-a d)} \]

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Rubi [A]  time = 0.03, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 4, 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^3 (c+d x)^8}{1320 (a+b x)^8 (b c-a d)^4}-\frac {d^2 (c+d x)^8}{165 (a+b x)^9 (b c-a d)^3}+\frac {3 d (c+d x)^8}{110 (a+b x)^{10} (b c-a d)^2}-\frac {(c+d x)^8}{11 (a+b x)^{11} (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^7/(a + b*x)^12,x]

[Out]

-(c + d*x)^8/(11*(b*c - a*d)*(a + b*x)^11) + (3*d*(c + d*x)^8)/(110*(b*c - a*d)^2*(a + b*x)^10) - (d^2*(c + d*
x)^8)/(165*(b*c - a*d)^3*(a + b*x)^9) + (d^3*(c + d*x)^8)/(1320*(b*c - a*d)^4*(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)^{12}} \, dx &=-\frac {(c+d x)^8}{11 (b c-a d) (a+b x)^{11}}-\frac {(3 d) \int \frac {(c+d x)^7}{(a+b x)^{11}} \, dx}{11 (b c-a d)}\\ &=-\frac {(c+d x)^8}{11 (b c-a d) (a+b x)^{11}}+\frac {3 d (c+d x)^8}{110 (b c-a d)^2 (a+b x)^{10}}+\frac {\left (3 d^2\right ) \int \frac {(c+d x)^7}{(a+b x)^{10}} \, dx}{55 (b c-a d)^2}\\ &=-\frac {(c+d x)^8}{11 (b c-a d) (a+b x)^{11}}+\frac {3 d (c+d x)^8}{110 (b c-a d)^2 (a+b x)^{10}}-\frac {d^2 (c+d x)^8}{165 (b c-a d)^3 (a+b x)^9}-\frac {d^3 \int \frac {(c+d x)^7}{(a+b x)^9} \, dx}{165 (b c-a d)^3}\\ &=-\frac {(c+d x)^8}{11 (b c-a d) (a+b x)^{11}}+\frac {3 d (c+d x)^8}{110 (b c-a d)^2 (a+b x)^{10}}-\frac {d^2 (c+d x)^8}{165 (b c-a d)^3 (a+b x)^9}+\frac {d^3 (c+d x)^8}{1320 (b c-a d)^4 (a+b x)^8}\\ \end {align*}

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Mathematica [B]  time = 0.12, size = 369, normalized size = 3.08 \begin {gather*} -\frac {a^7 d^7+a^6 b d^6 (4 c+11 d x)+a^5 b^2 d^5 \left (10 c^2+44 c d x+55 d^2 x^2\right )+5 a^4 b^3 d^4 \left (4 c^3+22 c^2 d x+44 c d^2 x^2+33 d^3 x^3\right )+5 a^3 b^4 d^3 \left (7 c^4+44 c^3 d x+110 c^2 d^2 x^2+132 c d^3 x^3+66 d^4 x^4\right )+a^2 b^5 d^2 \left (56 c^5+385 c^4 d x+1100 c^3 d^2 x^2+1650 c^2 d^3 x^3+1320 c d^4 x^4+462 d^5 x^5\right )+a b^6 d \left (84 c^6+616 c^5 d x+1925 c^4 d^2 x^2+3300 c^3 d^3 x^3+3300 c^2 d^4 x^4+1848 c d^5 x^5+462 d^6 x^6\right )+b^7 \left (120 c^7+924 c^6 d x+3080 c^5 d^2 x^2+5775 c^4 d^3 x^3+6600 c^3 d^4 x^4+4620 c^2 d^5 x^5+1848 c d^6 x^6+330 d^7 x^7\right )}{1320 b^8 (a+b x)^{11}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^7/(a + b*x)^12,x]

[Out]

-1/1320*(a^7*d^7 + a^6*b*d^6*(4*c + 11*d*x) + a^5*b^2*d^5*(10*c^2 + 44*c*d*x + 55*d^2*x^2) + 5*a^4*b^3*d^4*(4*
c^3 + 22*c^2*d*x + 44*c*d^2*x^2 + 33*d^3*x^3) + 5*a^3*b^4*d^3*(7*c^4 + 44*c^3*d*x + 110*c^2*d^2*x^2 + 132*c*d^
3*x^3 + 66*d^4*x^4) + a^2*b^5*d^2*(56*c^5 + 385*c^4*d*x + 1100*c^3*d^2*x^2 + 1650*c^2*d^3*x^3 + 1320*c*d^4*x^4
+ 462*d^5*x^5) + a*b^6*d*(84*c^6 + 616*c^5*d*x + 1925*c^4*d^2*x^2 + 3300*c^3*d^3*x^3 + 3300*c^2*d^4*x^4 + 184
8*c*d^5*x^5 + 462*d^6*x^6) + b^7*(120*c^7 + 924*c^6*d*x + 3080*c^5*d^2*x^2 + 5775*c^4*d^3*x^3 + 6600*c^3*d^4*x
^4 + 4620*c^2*d^5*x^5 + 1848*c*d^6*x^6 + 330*d^7*x^7))/(b^8*(a + b*x)^11)

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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)^{12}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(c + d*x)^7/(a + b*x)^12,x]

[Out]

IntegrateAlgebraic[(c + d*x)^7/(a + b*x)^12, x]

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fricas [B]  time = 0.71, size = 570, normalized size = 4.75 \begin {gather*} -\frac {330 \, b^{7} d^{7} x^{7} + 120 \, b^{7} c^{7} + 84 \, a b^{6} c^{6} d + 56 \, a^{2} b^{5} c^{5} d^{2} + 35 \, a^{3} b^{4} c^{4} d^{3} + 20 \, a^{4} b^{3} c^{3} d^{4} + 10 \, a^{5} b^{2} c^{2} d^{5} + 4 \, a^{6} b c d^{6} + a^{7} d^{7} + 462 \, {\left (4 \, b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{6} + 462 \, {\left (10 \, b^{7} c^{2} d^{5} + 4 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 330 \, {\left (20 \, b^{7} c^{3} d^{4} + 10 \, a b^{6} c^{2} d^{5} + 4 \, a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 165 \, {\left (35 \, b^{7} c^{4} d^{3} + 20 \, a b^{6} c^{3} d^{4} + 10 \, a^{2} b^{5} c^{2} d^{5} + 4 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 55 \, {\left (56 \, b^{7} c^{5} d^{2} + 35 \, a b^{6} c^{4} d^{3} + 20 \, a^{2} b^{5} c^{3} d^{4} + 10 \, a^{3} b^{4} c^{2} d^{5} + 4 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 11 \, {\left (84 \, b^{7} c^{6} d + 56 \, a b^{6} c^{5} d^{2} + 35 \, a^{2} b^{5} c^{4} d^{3} + 20 \, a^{3} b^{4} c^{3} d^{4} + 10 \, a^{4} b^{3} c^{2} d^{5} + 4 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x}{1320 \, {\left (b^{19} x^{11} + 11 \, a b^{18} x^{10} + 55 \, a^{2} b^{17} x^{9} + 165 \, a^{3} b^{16} x^{8} + 330 \, a^{4} b^{15} x^{7} + 462 \, a^{5} b^{14} x^{6} + 462 \, a^{6} b^{13} x^{5} + 330 \, a^{7} b^{12} x^{4} + 165 \, a^{8} b^{11} x^{3} + 55 \, a^{9} b^{10} x^{2} + 11 \, a^{10} b^{9} x + a^{11} b^{8}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^7/(b*x+a)^12,x, algorithm="fricas")

[Out]

-1/1320*(330*b^7*d^7*x^7 + 120*b^7*c^7 + 84*a*b^6*c^6*d + 56*a^2*b^5*c^5*d^2 + 35*a^3*b^4*c^4*d^3 + 20*a^4*b^3
*c^3*d^4 + 10*a^5*b^2*c^2*d^5 + 4*a^6*b*c*d^6 + a^7*d^7 + 462*(4*b^7*c*d^6 + a*b^6*d^7)*x^6 + 462*(10*b^7*c^2*
d^5 + 4*a*b^6*c*d^6 + a^2*b^5*d^7)*x^5 + 330*(20*b^7*c^3*d^4 + 10*a*b^6*c^2*d^5 + 4*a^2*b^5*c*d^6 + a^3*b^4*d^
7)*x^4 + 165*(35*b^7*c^4*d^3 + 20*a*b^6*c^3*d^4 + 10*a^2*b^5*c^2*d^5 + 4*a^3*b^4*c*d^6 + a^4*b^3*d^7)*x^3 + 55
*(56*b^7*c^5*d^2 + 35*a*b^6*c^4*d^3 + 20*a^2*b^5*c^3*d^4 + 10*a^3*b^4*c^2*d^5 + 4*a^4*b^3*c*d^6 + a^5*b^2*d^7)
*x^2 + 11*(84*b^7*c^6*d + 56*a*b^6*c^5*d^2 + 35*a^2*b^5*c^4*d^3 + 20*a^3*b^4*c^3*d^4 + 10*a^4*b^3*c^2*d^5 + 4*
a^5*b^2*c*d^6 + a^6*b*d^7)*x)/(b^19*x^11 + 11*a*b^18*x^10 + 55*a^2*b^17*x^9 + 165*a^3*b^16*x^8 + 330*a^4*b^15*
x^7 + 462*a^5*b^14*x^6 + 462*a^6*b^13*x^5 + 330*a^7*b^12*x^4 + 165*a^8*b^11*x^3 + 55*a^9*b^10*x^2 + 11*a^10*b^
9*x + a^11*b^8)

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giac [B]  time = 1.30, size = 496, normalized size = 4.13 \begin {gather*} -\frac {330 \, b^{7} d^{7} x^{7} + 1848 \, b^{7} c d^{6} x^{6} + 462 \, a b^{6} d^{7} x^{6} + 4620 \, b^{7} c^{2} d^{5} x^{5} + 1848 \, a b^{6} c d^{6} x^{5} + 462 \, a^{2} b^{5} d^{7} x^{5} + 6600 \, b^{7} c^{3} d^{4} x^{4} + 3300 \, a b^{6} c^{2} d^{5} x^{4} + 1320 \, a^{2} b^{5} c d^{6} x^{4} + 330 \, a^{3} b^{4} d^{7} x^{4} + 5775 \, b^{7} c^{4} d^{3} x^{3} + 3300 \, a b^{6} c^{3} d^{4} x^{3} + 1650 \, a^{2} b^{5} c^{2} d^{5} x^{3} + 660 \, a^{3} b^{4} c d^{6} x^{3} + 165 \, a^{4} b^{3} d^{7} x^{3} + 3080 \, b^{7} c^{5} d^{2} x^{2} + 1925 \, a b^{6} c^{4} d^{3} x^{2} + 1100 \, a^{2} b^{5} c^{3} d^{4} x^{2} + 550 \, a^{3} b^{4} c^{2} d^{5} x^{2} + 220 \, a^{4} b^{3} c d^{6} x^{2} + 55 \, a^{5} b^{2} d^{7} x^{2} + 924 \, b^{7} c^{6} d x + 616 \, a b^{6} c^{5} d^{2} x + 385 \, a^{2} b^{5} c^{4} d^{3} x + 220 \, a^{3} b^{4} c^{3} d^{4} x + 110 \, a^{4} b^{3} c^{2} d^{5} x + 44 \, a^{5} b^{2} c d^{6} x + 11 \, a^{6} b d^{7} x + 120 \, b^{7} c^{7} + 84 \, a b^{6} c^{6} d + 56 \, a^{2} b^{5} c^{5} d^{2} + 35 \, a^{3} b^{4} c^{4} d^{3} + 20 \, a^{4} b^{3} c^{3} d^{4} + 10 \, a^{5} b^{2} c^{2} d^{5} + 4 \, a^{6} b c d^{6} + a^{7} d^{7}}{1320 \, {\left (b x + a\right )}^{11} b^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^7/(b*x+a)^12,x, algorithm="giac")

[Out]

-1/1320*(330*b^7*d^7*x^7 + 1848*b^7*c*d^6*x^6 + 462*a*b^6*d^7*x^6 + 4620*b^7*c^2*d^5*x^5 + 1848*a*b^6*c*d^6*x^
5 + 462*a^2*b^5*d^7*x^5 + 6600*b^7*c^3*d^4*x^4 + 3300*a*b^6*c^2*d^5*x^4 + 1320*a^2*b^5*c*d^6*x^4 + 330*a^3*b^4
*d^7*x^4 + 5775*b^7*c^4*d^3*x^3 + 3300*a*b^6*c^3*d^4*x^3 + 1650*a^2*b^5*c^2*d^5*x^3 + 660*a^3*b^4*c*d^6*x^3 +
165*a^4*b^3*d^7*x^3 + 3080*b^7*c^5*d^2*x^2 + 1925*a*b^6*c^4*d^3*x^2 + 1100*a^2*b^5*c^3*d^4*x^2 + 550*a^3*b^4*c
^2*d^5*x^2 + 220*a^4*b^3*c*d^6*x^2 + 55*a^5*b^2*d^7*x^2 + 924*b^7*c^6*d*x + 616*a*b^6*c^5*d^2*x + 385*a^2*b^5*
c^4*d^3*x + 220*a^3*b^4*c^3*d^4*x + 110*a^4*b^3*c^2*d^5*x + 44*a^5*b^2*c*d^6*x + 11*a^6*b*d^7*x + 120*b^7*c^7
+ 84*a*b^6*c^6*d + 56*a^2*b^5*c^5*d^2 + 35*a^3*b^4*c^4*d^3 + 20*a^4*b^3*c^3*d^4 + 10*a^5*b^2*c^2*d^5 + 4*a^6*b
*c*d^6 + a^7*d^7)/((b*x + a)^11*b^8)

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maple [B]  time = 0.00, size = 464, normalized size = 3.87 \begin {gather*} -\frac {d^{7}}{4 \left (b x +a \right )^{4} b^{8}}+\frac {7 \left (a d -b c \right ) d^{6}}{5 \left (b x +a \right )^{5} b^{8}}-\frac {7 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) d^{5}}{2 \left (b x +a \right )^{6} b^{8}}+\frac {5 \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 )^{7} 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}}{8 \left (b x +a \right )^{8} b^{8}}+\frac {7 \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}}{3 \left (b x +a \right )^{9} 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}{10 \left (b x +a \right )^{10} 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}}{11 \left (b x +a \right )^{11} b^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^7/(b*x+a)^12,x)

[Out]

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

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maxima [B]  time = 1.81, size = 570, normalized size = 4.75 \begin {gather*} -\frac {330 \, b^{7} d^{7} x^{7} + 120 \, b^{7} c^{7} + 84 \, a b^{6} c^{6} d + 56 \, a^{2} b^{5} c^{5} d^{2} + 35 \, a^{3} b^{4} c^{4} d^{3} + 20 \, a^{4} b^{3} c^{3} d^{4} + 10 \, a^{5} b^{2} c^{2} d^{5} + 4 \, a^{6} b c d^{6} + a^{7} d^{7} + 462 \, {\left (4 \, b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{6} + 462 \, {\left (10 \, b^{7} c^{2} d^{5} + 4 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 330 \, {\left (20 \, b^{7} c^{3} d^{4} + 10 \, a b^{6} c^{2} d^{5} + 4 \, a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 165 \, {\left (35 \, b^{7} c^{4} d^{3} + 20 \, a b^{6} c^{3} d^{4} + 10 \, a^{2} b^{5} c^{2} d^{5} + 4 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 55 \, {\left (56 \, b^{7} c^{5} d^{2} + 35 \, a b^{6} c^{4} d^{3} + 20 \, a^{2} b^{5} c^{3} d^{4} + 10 \, a^{3} b^{4} c^{2} d^{5} + 4 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 11 \, {\left (84 \, b^{7} c^{6} d + 56 \, a b^{6} c^{5} d^{2} + 35 \, a^{2} b^{5} c^{4} d^{3} + 20 \, a^{3} b^{4} c^{3} d^{4} + 10 \, a^{4} b^{3} c^{2} d^{5} + 4 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x}{1320 \, {\left (b^{19} x^{11} + 11 \, a b^{18} x^{10} + 55 \, a^{2} b^{17} x^{9} + 165 \, a^{3} b^{16} x^{8} + 330 \, a^{4} b^{15} x^{7} + 462 \, a^{5} b^{14} x^{6} + 462 \, a^{6} b^{13} x^{5} + 330 \, a^{7} b^{12} x^{4} + 165 \, a^{8} b^{11} x^{3} + 55 \, a^{9} b^{10} x^{2} + 11 \, a^{10} b^{9} x + a^{11} b^{8}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^7/(b*x+a)^12,x, algorithm="maxima")

[Out]

-1/1320*(330*b^7*d^7*x^7 + 120*b^7*c^7 + 84*a*b^6*c^6*d + 56*a^2*b^5*c^5*d^2 + 35*a^3*b^4*c^4*d^3 + 20*a^4*b^3
*c^3*d^4 + 10*a^5*b^2*c^2*d^5 + 4*a^6*b*c*d^6 + a^7*d^7 + 462*(4*b^7*c*d^6 + a*b^6*d^7)*x^6 + 462*(10*b^7*c^2*
d^5 + 4*a*b^6*c*d^6 + a^2*b^5*d^7)*x^5 + 330*(20*b^7*c^3*d^4 + 10*a*b^6*c^2*d^5 + 4*a^2*b^5*c*d^6 + a^3*b^4*d^
7)*x^4 + 165*(35*b^7*c^4*d^3 + 20*a*b^6*c^3*d^4 + 10*a^2*b^5*c^2*d^5 + 4*a^3*b^4*c*d^6 + a^4*b^3*d^7)*x^3 + 55
*(56*b^7*c^5*d^2 + 35*a*b^6*c^4*d^3 + 20*a^2*b^5*c^3*d^4 + 10*a^3*b^4*c^2*d^5 + 4*a^4*b^3*c*d^6 + a^5*b^2*d^7)
*x^2 + 11*(84*b^7*c^6*d + 56*a*b^6*c^5*d^2 + 35*a^2*b^5*c^4*d^3 + 20*a^3*b^4*c^3*d^4 + 10*a^4*b^3*c^2*d^5 + 4*
a^5*b^2*c*d^6 + a^6*b*d^7)*x)/(b^19*x^11 + 11*a*b^18*x^10 + 55*a^2*b^17*x^9 + 165*a^3*b^16*x^8 + 330*a^4*b^15*
x^7 + 462*a^5*b^14*x^6 + 462*a^6*b^13*x^5 + 330*a^7*b^12*x^4 + 165*a^8*b^11*x^3 + 55*a^9*b^10*x^2 + 11*a^10*b^
9*x + a^11*b^8)

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mupad [B]  time = 0.52, size = 548, normalized size = 4.57 \begin {gather*} -\frac {\frac {a^7\,d^7+4\,a^6\,b\,c\,d^6+10\,a^5\,b^2\,c^2\,d^5+20\,a^4\,b^3\,c^3\,d^4+35\,a^3\,b^4\,c^4\,d^3+56\,a^2\,b^5\,c^5\,d^2+84\,a\,b^6\,c^6\,d+120\,b^7\,c^7}{1320\,b^8}+\frac {d^7\,x^7}{4\,b}+\frac {d^2\,x^2\,\left (a^5\,d^5+4\,a^4\,b\,c\,d^4+10\,a^3\,b^2\,c^2\,d^3+20\,a^2\,b^3\,c^3\,d^2+35\,a\,b^4\,c^4\,d+56\,b^5\,c^5\right )}{24\,b^6}+\frac {d^4\,x^4\,\left (a^3\,d^3+4\,a^2\,b\,c\,d^2+10\,a\,b^2\,c^2\,d+20\,b^3\,c^3\right )}{4\,b^4}+\frac {7\,d^6\,x^6\,\left (a\,d+4\,b\,c\right )}{20\,b^2}+\frac {d^3\,x^3\,\left (a^4\,d^4+4\,a^3\,b\,c\,d^3+10\,a^2\,b^2\,c^2\,d^2+20\,a\,b^3\,c^3\,d+35\,b^4\,c^4\right )}{8\,b^5}+\frac {d\,x\,\left (a^6\,d^6+4\,a^5\,b\,c\,d^5+10\,a^4\,b^2\,c^2\,d^4+20\,a^3\,b^3\,c^3\,d^3+35\,a^2\,b^4\,c^4\,d^2+56\,a\,b^5\,c^5\,d+84\,b^6\,c^6\right )}{120\,b^7}+\frac {7\,d^5\,x^5\,\left (a^2\,d^2+4\,a\,b\,c\,d+10\,b^2\,c^2\right )}{20\,b^3}}{a^{11}+11\,a^{10}\,b\,x+55\,a^9\,b^2\,x^2+165\,a^8\,b^3\,x^3+330\,a^7\,b^4\,x^4+462\,a^6\,b^5\,x^5+462\,a^5\,b^6\,x^6+330\,a^4\,b^7\,x^7+165\,a^3\,b^8\,x^8+55\,a^2\,b^9\,x^9+11\,a\,b^{10}\,x^{10}+b^{11}\,x^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^7/(a + b*x)^12,x)

[Out]

-((a^7*d^7 + 120*b^7*c^7 + 56*a^2*b^5*c^5*d^2 + 35*a^3*b^4*c^4*d^3 + 20*a^4*b^3*c^3*d^4 + 10*a^5*b^2*c^2*d^5 +
84*a*b^6*c^6*d + 4*a^6*b*c*d^6)/(1320*b^8) + (d^7*x^7)/(4*b) + (d^2*x^2*(a^5*d^5 + 56*b^5*c^5 + 20*a^2*b^3*c^
3*d^2 + 10*a^3*b^2*c^2*d^3 + 35*a*b^4*c^4*d + 4*a^4*b*c*d^4))/(24*b^6) + (d^4*x^4*(a^3*d^3 + 20*b^3*c^3 + 10*a
*b^2*c^2*d + 4*a^2*b*c*d^2))/(4*b^4) + (7*d^6*x^6*(a*d + 4*b*c))/(20*b^2) + (d^3*x^3*(a^4*d^4 + 35*b^4*c^4 + 1
0*a^2*b^2*c^2*d^2 + 20*a*b^3*c^3*d + 4*a^3*b*c*d^3))/(8*b^5) + (d*x*(a^6*d^6 + 84*b^6*c^6 + 35*a^2*b^4*c^4*d^2
+ 20*a^3*b^3*c^3*d^3 + 10*a^4*b^2*c^2*d^4 + 56*a*b^5*c^5*d + 4*a^5*b*c*d^5))/(120*b^7) + (7*d^5*x^5*(a^2*d^2
+ 10*b^2*c^2 + 4*a*b*c*d))/(20*b^3))/(a^11 + b^11*x^11 + 11*a*b^10*x^10 + 55*a^9*b^2*x^2 + 165*a^8*b^3*x^3 + 3
30*a^7*b^4*x^4 + 462*a^6*b^5*x^5 + 462*a^5*b^6*x^6 + 330*a^4*b^7*x^7 + 165*a^3*b^8*x^8 + 55*a^2*b^9*x^9 + 11*a
^10*b*x)

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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)**12,x)

[Out]

Timed out

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