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

Optimal. Leaf size=262 \[ \frac {10 d^9 (a+b x)^3 (b c-a d)}{3 b^{11}}+\frac {45 d^8 (a+b x)^2 (b c-a d)^2}{2 b^{11}}+\frac {210 d^6 (b c-a d)^4 \log (a+b x)}{b^{11}}-\frac {252 d^5 (b c-a d)^5}{b^{11} (a+b x)}-\frac {105 d^4 (b c-a d)^6}{b^{11} (a+b x)^2}-\frac {40 d^3 (b c-a d)^7}{b^{11} (a+b x)^3}-\frac {45 d^2 (b c-a d)^8}{4 b^{11} (a+b x)^4}-\frac {2 d (b c-a d)^9}{b^{11} (a+b x)^5}-\frac {(b c-a d)^{10}}{6 b^{11} (a+b x)^6}+\frac {d^{10} (a+b x)^4}{4 b^{11}}+\frac {120 d^7 x (b c-a d)^3}{b^{10}} \]

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Rubi [A]  time = 0.39, antiderivative size = 262, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {10 d^9 (a+b x)^3 (b c-a d)}{3 b^{11}}+\frac {45 d^8 (a+b x)^2 (b c-a d)^2}{2 b^{11}}+\frac {120 d^7 x (b c-a d)^3}{b^{10}}-\frac {252 d^5 (b c-a d)^5}{b^{11} (a+b x)}-\frac {105 d^4 (b c-a d)^6}{b^{11} (a+b x)^2}-\frac {40 d^3 (b c-a d)^7}{b^{11} (a+b x)^3}-\frac {45 d^2 (b c-a d)^8}{4 b^{11} (a+b x)^4}+\frac {210 d^6 (b c-a d)^4 \log (a+b x)}{b^{11}}-\frac {2 d (b c-a d)^9}{b^{11} (a+b x)^5}-\frac {(b c-a d)^{10}}{6 b^{11} (a+b x)^6}+\frac {d^{10} (a+b x)^4}{4 b^{11}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^10/(a + b*x)^7,x]

[Out]

(120*d^7*(b*c - a*d)^3*x)/b^10 - (b*c - a*d)^10/(6*b^11*(a + b*x)^6) - (2*d*(b*c - a*d)^9)/(b^11*(a + b*x)^5)
- (45*d^2*(b*c - a*d)^8)/(4*b^11*(a + b*x)^4) - (40*d^3*(b*c - a*d)^7)/(b^11*(a + b*x)^3) - (105*d^4*(b*c - a*
d)^6)/(b^11*(a + b*x)^2) - (252*d^5*(b*c - a*d)^5)/(b^11*(a + b*x)) + (45*d^8*(b*c - a*d)^2*(a + b*x)^2)/(2*b^
11) + (10*d^9*(b*c - a*d)*(a + b*x)^3)/(3*b^11) + (d^10*(a + b*x)^4)/(4*b^11) + (210*d^6*(b*c - a*d)^4*Log[a +
b*x])/b^11

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)^{10}}{(a+b x)^7} \, dx &=\int \left (\frac {120 d^7 (b c-a d)^3}{b^{10}}+\frac {(b c-a d)^{10}}{b^{10} (a+b x)^7}+\frac {10 d (b c-a d)^9}{b^{10} (a+b x)^6}+\frac {45 d^2 (b c-a d)^8}{b^{10} (a+b x)^5}+\frac {120 d^3 (b c-a d)^7}{b^{10} (a+b x)^4}+\frac {210 d^4 (b c-a d)^6}{b^{10} (a+b x)^3}+\frac {252 d^5 (b c-a d)^5}{b^{10} (a+b x)^2}+\frac {210 d^6 (b c-a d)^4}{b^{10} (a+b x)}+\frac {45 d^8 (b c-a d)^2 (a+b x)}{b^{10}}+\frac {10 d^9 (b c-a d) (a+b x)^2}{b^{10}}+\frac {d^{10} (a+b x)^3}{b^{10}}\right ) \, dx\\ &=\frac {120 d^7 (b c-a d)^3 x}{b^{10}}-\frac {(b c-a d)^{10}}{6 b^{11} (a+b x)^6}-\frac {2 d (b c-a d)^9}{b^{11} (a+b x)^5}-\frac {45 d^2 (b c-a d)^8}{4 b^{11} (a+b x)^4}-\frac {40 d^3 (b c-a d)^7}{b^{11} (a+b x)^3}-\frac {105 d^4 (b c-a d)^6}{b^{11} (a+b x)^2}-\frac {252 d^5 (b c-a d)^5}{b^{11} (a+b x)}+\frac {45 d^8 (b c-a d)^2 (a+b x)^2}{2 b^{11}}+\frac {10 d^9 (b c-a d) (a+b x)^3}{3 b^{11}}+\frac {d^{10} (a+b x)^4}{4 b^{11}}+\frac {210 d^6 (b c-a d)^4 \log (a+b x)}{b^{11}}\\ \end {align*}

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Mathematica [A]  time = 0.22, size = 265, normalized size = 1.01 \begin {gather*} \frac {6 b^2 d^8 x^2 \left (28 a^2 d^2-70 a b c d+45 b^2 c^2\right )+12 b d^7 x \left (-84 a^3 d^3+280 a^2 b c d^2-315 a b^2 c^2 d+120 b^3 c^3\right )+4 b^3 d^9 x^3 (10 b c-7 a d)+2520 d^6 (b c-a d)^4 \log (a+b x)+\frac {3024 d^5 (a d-b c)^5}{a+b x}-\frac {1260 d^4 (b c-a d)^6}{(a+b x)^2}+\frac {480 d^3 (a d-b c)^7}{(a+b x)^3}-\frac {135 d^2 (b c-a d)^8}{(a+b x)^4}+\frac {24 d (a d-b c)^9}{(a+b x)^5}-\frac {2 (b c-a d)^{10}}{(a+b x)^6}+3 b^4 d^{10} x^4}{12 b^{11}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^10/(a + b*x)^7,x]

[Out]

(12*b*d^7*(120*b^3*c^3 - 315*a*b^2*c^2*d + 280*a^2*b*c*d^2 - 84*a^3*d^3)*x + 6*b^2*d^8*(45*b^2*c^2 - 70*a*b*c*
d + 28*a^2*d^2)*x^2 + 4*b^3*d^9*(10*b*c - 7*a*d)*x^3 + 3*b^4*d^10*x^4 - (2*(b*c - a*d)^10)/(a + b*x)^6 + (24*d
*(-(b*c) + a*d)^9)/(a + b*x)^5 - (135*d^2*(b*c - a*d)^8)/(a + b*x)^4 + (480*d^3*(-(b*c) + a*d)^7)/(a + b*x)^3
- (1260*d^4*(b*c - a*d)^6)/(a + b*x)^2 + (3024*d^5*(-(b*c) + a*d)^5)/(a + b*x) + 2520*d^6*(b*c - a*d)^4*Log[a
+ b*x])/(12*b^11)

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 1.23, size = 1386, normalized size = 5.29

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(3*b^10*d^10*x^10 - 2*b^10*c^10 - 4*a*b^9*c^9*d - 9*a^2*b^8*c^8*d^2 - 24*a^3*b^7*c^7*d^3 - 84*a^4*b^6*c^6
*d^4 - 504*a^5*b^5*c^5*d^5 + 6174*a^6*b^4*c^4*d^6 - 16056*a^7*b^3*c^3*d^7 + 18414*a^8*b^2*c^2*d^8 - 10036*a^9*
b*c*d^9 + 2131*a^10*d^10 + 10*(4*b^10*c*d^9 - a*b^9*d^10)*x^9 + 45*(6*b^10*c^2*d^8 - 4*a*b^9*c*d^9 + a^2*b^8*d
^10)*x^8 + 360*(4*b^10*c^3*d^7 - 6*a*b^9*c^2*d^8 + 4*a^2*b^8*c*d^9 - a^3*b^7*d^10)*x^7 + (8640*a*b^9*c^3*d^7 -
18630*a^2*b^8*c^2*d^8 + 14660*a^3*b^7*c*d^9 - 4043*a^4*b^6*d^10)*x^6 - 6*(504*b^10*c^5*d^5 - 2520*a*b^9*c^4*d
^6 + 1440*a^2*b^8*c^3*d^7 + 3510*a^3*b^7*c^2*d^8 - 4580*a^4*b^6*c*d^9 + 1523*a^5*b^5*d^10)*x^5 - 15*(84*b^10*c
^6*d^4 + 504*a*b^9*c^5*d^5 - 3780*a^2*b^8*c^4*d^6 + 6480*a^3*b^7*c^3*d^7 - 4050*a^4*b^6*c^2*d^8 + 460*a^5*b^5*
c*d^9 + 263*a^6*b^4*d^10)*x^4 - 20*(24*b^10*c^7*d^3 + 84*a*b^9*c^6*d^4 + 504*a^2*b^8*c^5*d^5 - 4620*a^3*b^7*c^
4*d^6 + 9840*a^4*b^6*c^3*d^7 - 9090*a^5*b^5*c^2*d^8 + 3820*a^6*b^4*c*d^9 - 577*a^7*b^3*d^10)*x^3 - 15*(9*b^10*
c^8*d^2 + 24*a*b^9*c^7*d^3 + 84*a^2*b^8*c^6*d^4 + 504*a^3*b^7*c^5*d^5 - 5250*a^4*b^6*c^4*d^6 + 12360*a^5*b^5*c
^3*d^7 - 12870*a^6*b^4*c^2*d^8 + 6340*a^7*b^3*c*d^9 - 1207*a^8*b^2*d^10)*x^2 - 6*(4*b^10*c^9*d + 9*a*b^9*c^8*d
^2 + 24*a^2*b^8*c^7*d^3 + 84*a^3*b^7*c^6*d^4 + 504*a^4*b^6*c^5*d^5 - 5754*a^5*b^5*c^4*d^6 + 14376*a^6*b^4*c^3*
d^7 - 15894*a^7*b^3*c^2*d^8 + 8356*a^8*b^2*c*d^9 - 1711*a^9*b*d^10)*x + 2520*(a^6*b^4*c^4*d^6 - 4*a^7*b^3*c^3*
d^7 + 6*a^8*b^2*c^2*d^8 - 4*a^9*b*c*d^9 + a^10*d^10 + (b^10*c^4*d^6 - 4*a*b^9*c^3*d^7 + 6*a^2*b^8*c^2*d^8 - 4*
a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^6 + 6*(a*b^9*c^4*d^6 - 4*a^2*b^8*c^3*d^7 + 6*a^3*b^7*c^2*d^8 - 4*a^4*b^6*c*d^9
+ a^5*b^5*d^10)*x^5 + 15*(a^2*b^8*c^4*d^6 - 4*a^3*b^7*c^3*d^7 + 6*a^4*b^6*c^2*d^8 - 4*a^5*b^5*c*d^9 + a^6*b^4
*d^10)*x^4 + 20*(a^3*b^7*c^4*d^6 - 4*a^4*b^6*c^3*d^7 + 6*a^5*b^5*c^2*d^8 - 4*a^6*b^4*c*d^9 + a^7*b^3*d^10)*x^3
+ 15*(a^4*b^6*c^4*d^6 - 4*a^5*b^5*c^3*d^7 + 6*a^6*b^4*c^2*d^8 - 4*a^7*b^3*c*d^9 + a^8*b^2*d^10)*x^2 + 6*(a^5*
b^5*c^4*d^6 - 4*a^6*b^4*c^3*d^7 + 6*a^7*b^3*c^2*d^8 - 4*a^8*b^2*c*d^9 + a^9*b*d^10)*x)*log(b*x + a))/(b^17*x^6
+ 6*a*b^16*x^5 + 15*a^2*b^15*x^4 + 20*a^3*b^14*x^3 + 15*a^4*b^13*x^2 + 6*a^5*b^12*x + a^6*b^11)

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giac [B]  time = 1.29, size = 878, normalized size = 3.35 \begin {gather*} \frac {210 \, {\left (b^{4} c^{4} d^{6} - 4 \, a b^{3} c^{3} d^{7} + 6 \, a^{2} b^{2} c^{2} d^{8} - 4 \, a^{3} b c d^{9} + a^{4} d^{10}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{11}} - \frac {2 \, b^{10} c^{10} + 4 \, a b^{9} c^{9} d + 9 \, a^{2} b^{8} c^{8} d^{2} + 24 \, a^{3} b^{7} c^{7} d^{3} + 84 \, a^{4} b^{6} c^{6} d^{4} + 504 \, a^{5} b^{5} c^{5} d^{5} - 6174 \, a^{6} b^{4} c^{4} d^{6} + 16056 \, a^{7} b^{3} c^{3} d^{7} - 18414 \, a^{8} b^{2} c^{2} d^{8} + 10036 \, a^{9} b c d^{9} - 2131 \, a^{10} d^{10} + 3024 \, {\left (b^{10} c^{5} d^{5} - 5 \, a b^{9} c^{4} d^{6} + 10 \, a^{2} b^{8} c^{3} d^{7} - 10 \, a^{3} b^{7} c^{2} d^{8} + 5 \, a^{4} b^{6} c d^{9} - a^{5} b^{5} d^{10}\right )} x^{5} + 1260 \, {\left (b^{10} c^{6} d^{4} + 6 \, a b^{9} c^{5} d^{5} - 45 \, a^{2} b^{8} c^{4} d^{6} + 100 \, a^{3} b^{7} c^{3} d^{7} - 105 \, a^{4} b^{6} c^{2} d^{8} + 54 \, a^{5} b^{5} c d^{9} - 11 \, a^{6} b^{4} d^{10}\right )} x^{4} + 240 \, {\left (2 \, b^{10} c^{7} d^{3} + 7 \, a b^{9} c^{6} d^{4} + 42 \, a^{2} b^{8} c^{5} d^{5} - 385 \, a^{3} b^{7} c^{4} d^{6} + 910 \, a^{4} b^{6} c^{3} d^{7} - 987 \, a^{5} b^{5} c^{2} d^{8} + 518 \, a^{6} b^{4} c d^{9} - 107 \, a^{7} b^{3} d^{10}\right )} x^{3} + 45 \, {\left (3 \, b^{10} c^{8} d^{2} + 8 \, a b^{9} c^{7} d^{3} + 28 \, a^{2} b^{8} c^{6} d^{4} + 168 \, a^{3} b^{7} c^{5} d^{5} - 1750 \, a^{4} b^{6} c^{4} d^{6} + 4312 \, a^{5} b^{5} c^{3} d^{7} - 4788 \, a^{6} b^{4} c^{2} d^{8} + 2552 \, a^{7} b^{3} c d^{9} - 533 \, a^{8} b^{2} d^{10}\right )} x^{2} + 6 \, {\left (4 \, b^{10} c^{9} d + 9 \, a b^{9} c^{8} d^{2} + 24 \, a^{2} b^{8} c^{7} d^{3} + 84 \, a^{3} b^{7} c^{6} d^{4} + 504 \, a^{4} b^{6} c^{5} d^{5} - 5754 \, a^{5} b^{5} c^{4} d^{6} + 14616 \, a^{6} b^{4} c^{3} d^{7} - 16524 \, a^{7} b^{3} c^{2} d^{8} + 8916 \, a^{8} b^{2} c d^{9} - 1879 \, a^{9} b d^{10}\right )} x}{12 \, {\left (b x + a\right )}^{6} b^{11}} + \frac {3 \, b^{21} d^{10} x^{4} + 40 \, b^{21} c d^{9} x^{3} - 28 \, a b^{20} d^{10} x^{3} + 270 \, b^{21} c^{2} d^{8} x^{2} - 420 \, a b^{20} c d^{9} x^{2} + 168 \, a^{2} b^{19} d^{10} x^{2} + 1440 \, b^{21} c^{3} d^{7} x - 3780 \, a b^{20} c^{2} d^{8} x + 3360 \, a^{2} b^{19} c d^{9} x - 1008 \, a^{3} b^{18} d^{10} x}{12 \, b^{28}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

210*(b^4*c^4*d^6 - 4*a*b^3*c^3*d^7 + 6*a^2*b^2*c^2*d^8 - 4*a^3*b*c*d^9 + a^4*d^10)*log(abs(b*x + a))/b^11 - 1/
12*(2*b^10*c^10 + 4*a*b^9*c^9*d + 9*a^2*b^8*c^8*d^2 + 24*a^3*b^7*c^7*d^3 + 84*a^4*b^6*c^6*d^4 + 504*a^5*b^5*c^
5*d^5 - 6174*a^6*b^4*c^4*d^6 + 16056*a^7*b^3*c^3*d^7 - 18414*a^8*b^2*c^2*d^8 + 10036*a^9*b*c*d^9 - 2131*a^10*d
^10 + 3024*(b^10*c^5*d^5 - 5*a*b^9*c^4*d^6 + 10*a^2*b^8*c^3*d^7 - 10*a^3*b^7*c^2*d^8 + 5*a^4*b^6*c*d^9 - a^5*b
^5*d^10)*x^5 + 1260*(b^10*c^6*d^4 + 6*a*b^9*c^5*d^5 - 45*a^2*b^8*c^4*d^6 + 100*a^3*b^7*c^3*d^7 - 105*a^4*b^6*c
^2*d^8 + 54*a^5*b^5*c*d^9 - 11*a^6*b^4*d^10)*x^4 + 240*(2*b^10*c^7*d^3 + 7*a*b^9*c^6*d^4 + 42*a^2*b^8*c^5*d^5
- 385*a^3*b^7*c^4*d^6 + 910*a^4*b^6*c^3*d^7 - 987*a^5*b^5*c^2*d^8 + 518*a^6*b^4*c*d^9 - 107*a^7*b^3*d^10)*x^3
+ 45*(3*b^10*c^8*d^2 + 8*a*b^9*c^7*d^3 + 28*a^2*b^8*c^6*d^4 + 168*a^3*b^7*c^5*d^5 - 1750*a^4*b^6*c^4*d^6 + 431
2*a^5*b^5*c^3*d^7 - 4788*a^6*b^4*c^2*d^8 + 2552*a^7*b^3*c*d^9 - 533*a^8*b^2*d^10)*x^2 + 6*(4*b^10*c^9*d + 9*a*
b^9*c^8*d^2 + 24*a^2*b^8*c^7*d^3 + 84*a^3*b^7*c^6*d^4 + 504*a^4*b^6*c^5*d^5 - 5754*a^5*b^5*c^4*d^6 + 14616*a^6
*b^4*c^3*d^7 - 16524*a^7*b^3*c^2*d^8 + 8916*a^8*b^2*c*d^9 - 1879*a^9*b*d^10)*x)/((b*x + a)^6*b^11) + 1/12*(3*b
^21*d^10*x^4 + 40*b^21*c*d^9*x^3 - 28*a*b^20*d^10*x^3 + 270*b^21*c^2*d^8*x^2 - 420*a*b^20*c*d^9*x^2 + 168*a^2*
b^19*d^10*x^2 + 1440*b^21*c^3*d^7*x - 3780*a*b^20*c^2*d^8*x + 3360*a^2*b^19*c*d^9*x - 1008*a^3*b^18*d^10*x)/b^
28

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maple [B]  time = 0.02, size = 1222, normalized size = 4.66

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^10/(b*x+a)^7,x)

[Out]

252/b^11*d^10/(b*x+a)*a^5-252/b^6*d^5/(b*x+a)*c^5-1/6/b^11/(b*x+a)^6*a^10*d^10+2/b^11*d^10/(b*x+a)^5*a^9-2/b^2
*d/(b*x+a)^5*c^9-105/b^11*d^10/(b*x+a)^2*a^6-105/b^5*d^4/(b*x+a)^2*c^6+210/b^11*d^10*ln(b*x+a)*a^4+210/b^7*d^6
*ln(b*x+a)*c^4-45/4/b^11*d^10/(b*x+a)^4*a^8-45/4/b^3*d^2/(b*x+a)^4*c^8+45/2*d^8/b^7*x^2*c^2-84*d^10/b^10*a^3*x
+120*d^7/b^7*c^3*x+40/b^11*d^10/(b*x+a)^3*a^7-40/b^4*d^3/(b*x+a)^3*c^7-7/3*d^10/b^8*x^3*a+10/3*d^9/b^7*x^3*c+1
4*d^10/b^9*x^2*a^2-315*d^8/b^8*a*c^2*x-280/b^10*d^9/(b*x+a)^3*a^6*c+2100/b^8*d^7/(b*x+a)^2*a^3*c^3-1575/b^7*d^
6/(b*x+a)^2*a^2*c^4+72/b^9*d^8/(b*x+a)^5*a^7*c^2-168/b^8*d^7/(b*x+a)^5*a^6*c^3+252/b^7*d^6/(b*x+a)^5*a^5*c^4-2
52/b^6*d^5/(b*x+a)^5*a^4*c^5+168/b^5*d^4/(b*x+a)^5*a^3*c^6-72/b^4*d^3/(b*x+a)^5*a^2*c^7+18/b^3*d^2/(b*x+a)^5*a
*c^8+840/b^9*d^8/(b*x+a)^3*a^5*c^2-1400/b^8*d^7/(b*x+a)^3*a^4*c^3+1400/b^7*d^6/(b*x+a)^3*a^3*c^4-840/b^6*d^5/(
b*x+a)^3*a^2*c^5+280/b^5*d^4/(b*x+a)^3*a*c^6+630/b^6*d^5/(b*x+a)^2*a*c^5-35*d^9/b^8*x^2*a*c+280*d^9/b^9*a^2*c*
x-18/b^10*d^9/(b*x+a)^5*a^8*c+1260/b^7*d^6/(b*x+a)*a*c^4+5/3/b^10/(b*x+a)^6*a^9*c*d^9-15/2/b^9/(b*x+a)^6*a^8*c
^2*d^8+20/b^8/(b*x+a)^6*a^7*c^3*d^7-35/b^7/(b*x+a)^6*a^6*c^4*d^6+42/b^6/(b*x+a)^6*a^5*c^5*d^5-35/b^5/(b*x+a)^6
*a^4*c^6*d^4+20/b^4/(b*x+a)^6*a^3*c^7*d^3-15/2/b^3/(b*x+a)^6*a^2*c^8*d^2+5/3/b^2/(b*x+a)^6*a*c^9*d+90/b^10*d^9
/(b*x+a)^4*a^7*c-315/b^9*d^8/(b*x+a)^4*a^6*c^2+630/b^8*d^7/(b*x+a)^4*a^5*c^3-1575/2/b^7*d^6/(b*x+a)^4*a^4*c^4+
630/b^6*d^5/(b*x+a)^4*a^3*c^5-315/b^5*d^4/(b*x+a)^4*a^2*c^6+90/b^4*d^3/(b*x+a)^4*a*c^7-1260/b^10*d^9/(b*x+a)*a
^4*c+2520/b^9*d^8/(b*x+a)*a^3*c^2-2520/b^8*d^7/(b*x+a)*a^2*c^3+1260/b^9*d^8*ln(b*x+a)*a^2*c^2-840/b^8*d^7*ln(b
*x+a)*a*c^3-840/b^10*d^9*ln(b*x+a)*a^3*c+630/b^10*d^9/(b*x+a)^2*a^5*c-1575/b^9*d^8/(b*x+a)^2*a^4*c^2+1/4*d^10/
b^7*x^4-1/6/b/(b*x+a)^6*c^10

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maxima [B]  time = 2.45, size = 925, normalized size = 3.53 \begin {gather*} -\frac {2 \, b^{10} c^{10} + 4 \, a b^{9} c^{9} d + 9 \, a^{2} b^{8} c^{8} d^{2} + 24 \, a^{3} b^{7} c^{7} d^{3} + 84 \, a^{4} b^{6} c^{6} d^{4} + 504 \, a^{5} b^{5} c^{5} d^{5} - 6174 \, a^{6} b^{4} c^{4} d^{6} + 16056 \, a^{7} b^{3} c^{3} d^{7} - 18414 \, a^{8} b^{2} c^{2} d^{8} + 10036 \, a^{9} b c d^{9} - 2131 \, a^{10} d^{10} + 3024 \, {\left (b^{10} c^{5} d^{5} - 5 \, a b^{9} c^{4} d^{6} + 10 \, a^{2} b^{8} c^{3} d^{7} - 10 \, a^{3} b^{7} c^{2} d^{8} + 5 \, a^{4} b^{6} c d^{9} - a^{5} b^{5} d^{10}\right )} x^{5} + 1260 \, {\left (b^{10} c^{6} d^{4} + 6 \, a b^{9} c^{5} d^{5} - 45 \, a^{2} b^{8} c^{4} d^{6} + 100 \, a^{3} b^{7} c^{3} d^{7} - 105 \, a^{4} b^{6} c^{2} d^{8} + 54 \, a^{5} b^{5} c d^{9} - 11 \, a^{6} b^{4} d^{10}\right )} x^{4} + 240 \, {\left (2 \, b^{10} c^{7} d^{3} + 7 \, a b^{9} c^{6} d^{4} + 42 \, a^{2} b^{8} c^{5} d^{5} - 385 \, a^{3} b^{7} c^{4} d^{6} + 910 \, a^{4} b^{6} c^{3} d^{7} - 987 \, a^{5} b^{5} c^{2} d^{8} + 518 \, a^{6} b^{4} c d^{9} - 107 \, a^{7} b^{3} d^{10}\right )} x^{3} + 45 \, {\left (3 \, b^{10} c^{8} d^{2} + 8 \, a b^{9} c^{7} d^{3} + 28 \, a^{2} b^{8} c^{6} d^{4} + 168 \, a^{3} b^{7} c^{5} d^{5} - 1750 \, a^{4} b^{6} c^{4} d^{6} + 4312 \, a^{5} b^{5} c^{3} d^{7} - 4788 \, a^{6} b^{4} c^{2} d^{8} + 2552 \, a^{7} b^{3} c d^{9} - 533 \, a^{8} b^{2} d^{10}\right )} x^{2} + 6 \, {\left (4 \, b^{10} c^{9} d + 9 \, a b^{9} c^{8} d^{2} + 24 \, a^{2} b^{8} c^{7} d^{3} + 84 \, a^{3} b^{7} c^{6} d^{4} + 504 \, a^{4} b^{6} c^{5} d^{5} - 5754 \, a^{5} b^{5} c^{4} d^{6} + 14616 \, a^{6} b^{4} c^{3} d^{7} - 16524 \, a^{7} b^{3} c^{2} d^{8} + 8916 \, a^{8} b^{2} c d^{9} - 1879 \, a^{9} b d^{10}\right )} x}{12 \, {\left (b^{17} x^{6} + 6 \, a b^{16} x^{5} + 15 \, a^{2} b^{15} x^{4} + 20 \, a^{3} b^{14} x^{3} + 15 \, a^{4} b^{13} x^{2} + 6 \, a^{5} b^{12} x + a^{6} b^{11}\right )}} + \frac {3 \, b^{3} d^{10} x^{4} + 4 \, {\left (10 \, b^{3} c d^{9} - 7 \, a b^{2} d^{10}\right )} x^{3} + 6 \, {\left (45 \, b^{3} c^{2} d^{8} - 70 \, a b^{2} c d^{9} + 28 \, a^{2} b d^{10}\right )} x^{2} + 12 \, {\left (120 \, b^{3} c^{3} d^{7} - 315 \, a b^{2} c^{2} d^{8} + 280 \, a^{2} b c d^{9} - 84 \, a^{3} d^{10}\right )} x}{12 \, b^{10}} + \frac {210 \, {\left (b^{4} c^{4} d^{6} - 4 \, a b^{3} c^{3} d^{7} + 6 \, a^{2} b^{2} c^{2} d^{8} - 4 \, a^{3} b c d^{9} + a^{4} d^{10}\right )} \log \left (b x + a\right )}{b^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(2*b^10*c^10 + 4*a*b^9*c^9*d + 9*a^2*b^8*c^8*d^2 + 24*a^3*b^7*c^7*d^3 + 84*a^4*b^6*c^6*d^4 + 504*a^5*b^5
*c^5*d^5 - 6174*a^6*b^4*c^4*d^6 + 16056*a^7*b^3*c^3*d^7 - 18414*a^8*b^2*c^2*d^8 + 10036*a^9*b*c*d^9 - 2131*a^1
0*d^10 + 3024*(b^10*c^5*d^5 - 5*a*b^9*c^4*d^6 + 10*a^2*b^8*c^3*d^7 - 10*a^3*b^7*c^2*d^8 + 5*a^4*b^6*c*d^9 - a^
5*b^5*d^10)*x^5 + 1260*(b^10*c^6*d^4 + 6*a*b^9*c^5*d^5 - 45*a^2*b^8*c^4*d^6 + 100*a^3*b^7*c^3*d^7 - 105*a^4*b^
6*c^2*d^8 + 54*a^5*b^5*c*d^9 - 11*a^6*b^4*d^10)*x^4 + 240*(2*b^10*c^7*d^3 + 7*a*b^9*c^6*d^4 + 42*a^2*b^8*c^5*d
^5 - 385*a^3*b^7*c^4*d^6 + 910*a^4*b^6*c^3*d^7 - 987*a^5*b^5*c^2*d^8 + 518*a^6*b^4*c*d^9 - 107*a^7*b^3*d^10)*x
^3 + 45*(3*b^10*c^8*d^2 + 8*a*b^9*c^7*d^3 + 28*a^2*b^8*c^6*d^4 + 168*a^3*b^7*c^5*d^5 - 1750*a^4*b^6*c^4*d^6 +
4312*a^5*b^5*c^3*d^7 - 4788*a^6*b^4*c^2*d^8 + 2552*a^7*b^3*c*d^9 - 533*a^8*b^2*d^10)*x^2 + 6*(4*b^10*c^9*d + 9
*a*b^9*c^8*d^2 + 24*a^2*b^8*c^7*d^3 + 84*a^3*b^7*c^6*d^4 + 504*a^4*b^6*c^5*d^5 - 5754*a^5*b^5*c^4*d^6 + 14616*
a^6*b^4*c^3*d^7 - 16524*a^7*b^3*c^2*d^8 + 8916*a^8*b^2*c*d^9 - 1879*a^9*b*d^10)*x)/(b^17*x^6 + 6*a*b^16*x^5 +
15*a^2*b^15*x^4 + 20*a^3*b^14*x^3 + 15*a^4*b^13*x^2 + 6*a^5*b^12*x + a^6*b^11) + 1/12*(3*b^3*d^10*x^4 + 4*(10*
b^3*c*d^9 - 7*a*b^2*d^10)*x^3 + 6*(45*b^3*c^2*d^8 - 70*a*b^2*c*d^9 + 28*a^2*b*d^10)*x^2 + 12*(120*b^3*c^3*d^7
- 315*a*b^2*c^2*d^8 + 280*a^2*b*c*d^9 - 84*a^3*d^10)*x)/b^10 + 210*(b^4*c^4*d^6 - 4*a*b^3*c^3*d^7 + 6*a^2*b^2*
c^2*d^8 - 4*a^3*b*c*d^9 + a^4*d^10)*log(b*x + a)/b^11

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mupad [B]  time = 0.42, size = 997, normalized size = 3.81 \begin {gather*} x^2\,\left (\frac {7\,a\,\left (\frac {7\,a\,d^{10}}{b^8}-\frac {10\,c\,d^9}{b^7}\right )}{2\,b}-\frac {21\,a^2\,d^{10}}{2\,b^9}+\frac {45\,c^2\,d^8}{2\,b^7}\right )-\frac {x^4\,\left (-1155\,a^6\,b^3\,d^{10}+5670\,a^5\,b^4\,c\,d^9-11025\,a^4\,b^5\,c^2\,d^8+10500\,a^3\,b^6\,c^3\,d^7-4725\,a^2\,b^7\,c^4\,d^6+630\,a\,b^8\,c^5\,d^5+105\,b^9\,c^6\,d^4\right )+\frac {-2131\,a^{10}\,d^{10}+10036\,a^9\,b\,c\,d^9-18414\,a^8\,b^2\,c^2\,d^8+16056\,a^7\,b^3\,c^3\,d^7-6174\,a^6\,b^4\,c^4\,d^6+504\,a^5\,b^5\,c^5\,d^5+84\,a^4\,b^6\,c^6\,d^4+24\,a^3\,b^7\,c^7\,d^3+9\,a^2\,b^8\,c^8\,d^2+4\,a\,b^9\,c^9\,d+2\,b^{10}\,c^{10}}{12\,b}+x\,\left (-\frac {1879\,a^9\,d^{10}}{2}+4458\,a^8\,b\,c\,d^9-8262\,a^7\,b^2\,c^2\,d^8+7308\,a^6\,b^3\,c^3\,d^7-2877\,a^5\,b^4\,c^4\,d^6+252\,a^4\,b^5\,c^5\,d^5+42\,a^3\,b^6\,c^6\,d^4+12\,a^2\,b^7\,c^7\,d^3+\frac {9\,a\,b^8\,c^8\,d^2}{2}+2\,b^9\,c^9\,d\right )+x^3\,\left (-2140\,a^7\,b^2\,d^{10}+10360\,a^6\,b^3\,c\,d^9-19740\,a^5\,b^4\,c^2\,d^8+18200\,a^4\,b^5\,c^3\,d^7-7700\,a^3\,b^6\,c^4\,d^6+840\,a^2\,b^7\,c^5\,d^5+140\,a\,b^8\,c^6\,d^4+40\,b^9\,c^7\,d^3\right )+x^2\,\left (-\frac {7995\,a^8\,b\,d^{10}}{4}+9570\,a^7\,b^2\,c\,d^9-17955\,a^6\,b^3\,c^2\,d^8+16170\,a^5\,b^4\,c^3\,d^7-\frac {13125\,a^4\,b^5\,c^4\,d^6}{2}+630\,a^3\,b^6\,c^5\,d^5+105\,a^2\,b^7\,c^6\,d^4+30\,a\,b^8\,c^7\,d^3+\frac {45\,b^9\,c^8\,d^2}{4}\right )-x^5\,\left (252\,a^5\,b^4\,d^{10}-1260\,a^4\,b^5\,c\,d^9+2520\,a^3\,b^6\,c^2\,d^8-2520\,a^2\,b^7\,c^3\,d^7+1260\,a\,b^8\,c^4\,d^6-252\,b^9\,c^5\,d^5\right )}{a^6\,b^{10}+6\,a^5\,b^{11}\,x+15\,a^4\,b^{12}\,x^2+20\,a^3\,b^{13}\,x^3+15\,a^2\,b^{14}\,x^4+6\,a\,b^{15}\,x^5+b^{16}\,x^6}-x^3\,\left (\frac {7\,a\,d^{10}}{3\,b^8}-\frac {10\,c\,d^9}{3\,b^7}\right )-x\,\left (\frac {7\,a\,\left (\frac {7\,a\,\left (\frac {7\,a\,d^{10}}{b^8}-\frac {10\,c\,d^9}{b^7}\right )}{b}-\frac {21\,a^2\,d^{10}}{b^9}+\frac {45\,c^2\,d^8}{b^7}\right )}{b}+\frac {35\,a^3\,d^{10}}{b^{10}}-\frac {120\,c^3\,d^7}{b^7}-\frac {21\,a^2\,\left (\frac {7\,a\,d^{10}}{b^8}-\frac {10\,c\,d^9}{b^7}\right )}{b^2}\right )+\frac {\ln \left (a+b\,x\right )\,\left (210\,a^4\,d^{10}-840\,a^3\,b\,c\,d^9+1260\,a^2\,b^2\,c^2\,d^8-840\,a\,b^3\,c^3\,d^7+210\,b^4\,c^4\,d^6\right )}{b^{11}}+\frac {d^{10}\,x^4}{4\,b^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^10/(a + b*x)^7,x)

[Out]

x^2*((7*a*((7*a*d^10)/b^8 - (10*c*d^9)/b^7))/(2*b) - (21*a^2*d^10)/(2*b^9) + (45*c^2*d^8)/(2*b^7)) - (x^4*(105
*b^9*c^6*d^4 - 1155*a^6*b^3*d^10 + 630*a*b^8*c^5*d^5 + 5670*a^5*b^4*c*d^9 - 4725*a^2*b^7*c^4*d^6 + 10500*a^3*b
^6*c^3*d^7 - 11025*a^4*b^5*c^2*d^8) + (2*b^10*c^10 - 2131*a^10*d^10 + 9*a^2*b^8*c^8*d^2 + 24*a^3*b^7*c^7*d^3 +
84*a^4*b^6*c^6*d^4 + 504*a^5*b^5*c^5*d^5 - 6174*a^6*b^4*c^4*d^6 + 16056*a^7*b^3*c^3*d^7 - 18414*a^8*b^2*c^2*d
^8 + 4*a*b^9*c^9*d + 10036*a^9*b*c*d^9)/(12*b) + x*(2*b^9*c^9*d - (1879*a^9*d^10)/2 + (9*a*b^8*c^8*d^2)/2 + 12
*a^2*b^7*c^7*d^3 + 42*a^3*b^6*c^6*d^4 + 252*a^4*b^5*c^5*d^5 - 2877*a^5*b^4*c^4*d^6 + 7308*a^6*b^3*c^3*d^7 - 82
62*a^7*b^2*c^2*d^8 + 4458*a^8*b*c*d^9) + x^3*(40*b^9*c^7*d^3 - 2140*a^7*b^2*d^10 + 140*a*b^8*c^6*d^4 + 10360*a
^6*b^3*c*d^9 + 840*a^2*b^7*c^5*d^5 - 7700*a^3*b^6*c^4*d^6 + 18200*a^4*b^5*c^3*d^7 - 19740*a^5*b^4*c^2*d^8) + x
^2*((45*b^9*c^8*d^2)/4 - (7995*a^8*b*d^10)/4 + 30*a*b^8*c^7*d^3 + 9570*a^7*b^2*c*d^9 + 105*a^2*b^7*c^6*d^4 + 6
30*a^3*b^6*c^5*d^5 - (13125*a^4*b^5*c^4*d^6)/2 + 16170*a^5*b^4*c^3*d^7 - 17955*a^6*b^3*c^2*d^8) - x^5*(252*a^5
*b^4*d^10 - 252*b^9*c^5*d^5 + 1260*a*b^8*c^4*d^6 - 1260*a^4*b^5*c*d^9 - 2520*a^2*b^7*c^3*d^7 + 2520*a^3*b^6*c^
2*d^8))/(a^6*b^10 + b^16*x^6 + 6*a^5*b^11*x + 6*a*b^15*x^5 + 15*a^4*b^12*x^2 + 20*a^3*b^13*x^3 + 15*a^2*b^14*x
^4) - x^3*((7*a*d^10)/(3*b^8) - (10*c*d^9)/(3*b^7)) - x*((7*a*((7*a*((7*a*d^10)/b^8 - (10*c*d^9)/b^7))/b - (21
*a^2*d^10)/b^9 + (45*c^2*d^8)/b^7))/b + (35*a^3*d^10)/b^10 - (120*c^3*d^7)/b^7 - (21*a^2*((7*a*d^10)/b^8 - (10
*c*d^9)/b^7))/b^2) + (log(a + b*x)*(210*a^4*d^10 + 210*b^4*c^4*d^6 - 840*a*b^3*c^3*d^7 + 1260*a^2*b^2*c^2*d^8
- 840*a^3*b*c*d^9))/b^11 + (d^10*x^4)/(4*b^7)

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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)**10/(b*x+a)**7,x)

[Out]

Timed out

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