3.13.9 \(\int \frac {(c+d x)^{10}}{(a+b x)^4} \, dx\)

Optimal. Leaf size=258 \[ \frac {5 d^9 (a+b x)^6 (b c-a d)}{3 b^{11}}+\frac {9 d^8 (a+b x)^5 (b c-a d)^2}{b^{11}}+\frac {30 d^7 (a+b x)^4 (b c-a d)^3}{b^{11}}+\frac {70 d^6 (a+b x)^3 (b c-a d)^4}{b^{11}}+\frac {126 d^5 (a+b x)^2 (b c-a d)^5}{b^{11}}+\frac {120 d^3 (b c-a d)^7 \log (a+b x)}{b^{11}}-\frac {45 d^2 (b c-a d)^8}{b^{11} (a+b x)}-\frac {5 d (b c-a d)^9}{b^{11} (a+b x)^2}-\frac {(b c-a d)^{10}}{3 b^{11} (a+b x)^3}+\frac {d^{10} (a+b x)^7}{7 b^{11}}+\frac {210 d^4 x (b c-a d)^6}{b^{10}} \]

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Rubi [A]  time = 0.44, antiderivative size = 258, 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 {5 d^9 (a+b x)^6 (b c-a d)}{3 b^{11}}+\frac {9 d^8 (a+b x)^5 (b c-a d)^2}{b^{11}}+\frac {30 d^7 (a+b x)^4 (b c-a d)^3}{b^{11}}+\frac {70 d^6 (a+b x)^3 (b c-a d)^4}{b^{11}}+\frac {126 d^5 (a+b x)^2 (b c-a d)^5}{b^{11}}+\frac {210 d^4 x (b c-a d)^6}{b^{10}}-\frac {45 d^2 (b c-a d)^8}{b^{11} (a+b x)}+\frac {120 d^3 (b c-a d)^7 \log (a+b x)}{b^{11}}-\frac {5 d (b c-a d)^9}{b^{11} (a+b x)^2}-\frac {(b c-a d)^{10}}{3 b^{11} (a+b x)^3}+\frac {d^{10} (a+b x)^7}{7 b^{11}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^10/(a + b*x)^4,x]

[Out]

(210*d^4*(b*c - a*d)^6*x)/b^10 - (b*c - a*d)^10/(3*b^11*(a + b*x)^3) - (5*d*(b*c - a*d)^9)/(b^11*(a + b*x)^2)
- (45*d^2*(b*c - a*d)^8)/(b^11*(a + b*x)) + (126*d^5*(b*c - a*d)^5*(a + b*x)^2)/b^11 + (70*d^6*(b*c - a*d)^4*(
a + b*x)^3)/b^11 + (30*d^7*(b*c - a*d)^3*(a + b*x)^4)/b^11 + (9*d^8*(b*c - a*d)^2*(a + b*x)^5)/b^11 + (5*d^9*(
b*c - a*d)*(a + b*x)^6)/(3*b^11) + (d^10*(a + b*x)^7)/(7*b^11) + (120*d^3*(b*c - a*d)^7*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)^4} \, dx &=\int \left (\frac {210 d^4 (b c-a d)^6}{b^{10}}+\frac {(b c-a d)^{10}}{b^{10} (a+b x)^4}+\frac {10 d (b c-a d)^9}{b^{10} (a+b x)^3}+\frac {45 d^2 (b c-a d)^8}{b^{10} (a+b x)^2}+\frac {120 d^3 (b c-a d)^7}{b^{10} (a+b x)}+\frac {252 d^5 (b c-a d)^5 (a+b x)}{b^{10}}+\frac {210 d^6 (b c-a d)^4 (a+b x)^2}{b^{10}}+\frac {120 d^7 (b c-a d)^3 (a+b x)^3}{b^{10}}+\frac {45 d^8 (b c-a d)^2 (a+b x)^4}{b^{10}}+\frac {10 d^9 (b c-a d) (a+b x)^5}{b^{10}}+\frac {d^{10} (a+b x)^6}{b^{10}}\right ) \, dx\\ &=\frac {210 d^4 (b c-a d)^6 x}{b^{10}}-\frac {(b c-a d)^{10}}{3 b^{11} (a+b x)^3}-\frac {5 d (b c-a d)^9}{b^{11} (a+b x)^2}-\frac {45 d^2 (b c-a d)^8}{b^{11} (a+b x)}+\frac {126 d^5 (b c-a d)^5 (a+b x)^2}{b^{11}}+\frac {70 d^6 (b c-a d)^4 (a+b x)^3}{b^{11}}+\frac {30 d^7 (b c-a d)^3 (a+b x)^4}{b^{11}}+\frac {9 d^8 (b c-a d)^2 (a+b x)^5}{b^{11}}+\frac {5 d^9 (b c-a d) (a+b x)^6}{3 b^{11}}+\frac {d^{10} (a+b x)^7}{7 b^{11}}+\frac {120 d^3 (b c-a d)^7 \log (a+b x)}{b^{11}}\\ \end {align*}

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Mathematica [A]  time = 0.18, size = 427, normalized size = 1.66 \begin {gather*} \frac {21 b^5 d^8 x^5 \left (2 a^2 d^2-8 a b c d+9 b^2 c^2\right )+105 b^4 d^7 x^4 \left (-a^3 d^3+5 a^2 b c d^2-9 a b^2 c^2 d+6 b^3 c^3\right )+35 b^3 d^6 x^3 \left (7 a^4 d^4-40 a^3 b c d^3+90 a^2 b^2 c^2 d^2-96 a b^3 c^3 d+42 b^4 c^4\right )+21 b^2 d^5 x^2 \left (-28 a^5 d^5+175 a^4 b c d^4-450 a^3 b^2 c^2 d^3+600 a^2 b^3 c^3 d^2-420 a b^4 c^4 d+126 b^5 c^5\right )+21 b d^4 x \left (84 a^6 d^6-560 a^5 b c d^5+1575 a^4 b^2 c^2 d^4-2400 a^3 b^3 c^3 d^3+2100 a^2 b^4 c^4 d^2-1008 a b^5 c^5 d+210 b^6 c^6\right )+7 b^6 d^9 x^6 (5 b c-2 a d)+2520 d^3 (b c-a d)^7 \log (a+b x)-\frac {945 d^2 (b c-a d)^8}{a+b x}+\frac {105 d (a d-b c)^9}{(a+b x)^2}-\frac {7 (b c-a d)^{10}}{(a+b x)^3}+3 b^7 d^{10} x^7}{21 b^{11}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^10/(a + b*x)^4,x]

[Out]

(21*b*d^4*(210*b^6*c^6 - 1008*a*b^5*c^5*d + 2100*a^2*b^4*c^4*d^2 - 2400*a^3*b^3*c^3*d^3 + 1575*a^4*b^2*c^2*d^4
- 560*a^5*b*c*d^5 + 84*a^6*d^6)*x + 21*b^2*d^5*(126*b^5*c^5 - 420*a*b^4*c^4*d + 600*a^2*b^3*c^3*d^2 - 450*a^3
*b^2*c^2*d^3 + 175*a^4*b*c*d^4 - 28*a^5*d^5)*x^2 + 35*b^3*d^6*(42*b^4*c^4 - 96*a*b^3*c^3*d + 90*a^2*b^2*c^2*d^
2 - 40*a^3*b*c*d^3 + 7*a^4*d^4)*x^3 + 105*b^4*d^7*(6*b^3*c^3 - 9*a*b^2*c^2*d + 5*a^2*b*c*d^2 - a^3*d^3)*x^4 +
21*b^5*d^8*(9*b^2*c^2 - 8*a*b*c*d + 2*a^2*d^2)*x^5 + 7*b^6*d^9*(5*b*c - 2*a*d)*x^6 + 3*b^7*d^10*x^7 - (7*(b*c
- a*d)^10)/(a + b*x)^3 + (105*d*(-(b*c) + a*d)^9)/(a + b*x)^2 - (945*d^2*(b*c - a*d)^8)/(a + b*x) + 2520*d^3*(
b*c - a*d)^7*Log[a + b*x])/(21*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)^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(c + d*x)^10/(a + b*x)^4,x]

[Out]

IntegrateAlgebraic[(c + d*x)^10/(a + b*x)^4, x]

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fricas [B]  time = 1.25, size = 1316, normalized size = 5.10

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^10/(b*x+a)^4,x, algorithm="fricas")

[Out]

1/21*(3*b^10*d^10*x^10 - 7*b^10*c^10 - 35*a*b^9*c^9*d - 315*a^2*b^8*c^8*d^2 + 4620*a^3*b^7*c^7*d^3 - 19110*a^4
*b^6*c^6*d^4 + 41454*a^5*b^5*c^5*d^5 - 54390*a^6*b^4*c^4*d^6 + 44940*a^7*b^3*c^3*d^7 - 22995*a^8*b^2*c^2*d^8 +
6685*a^9*b*c*d^9 - 847*a^10*d^10 + 5*(7*b^10*c*d^9 - a*b^9*d^10)*x^9 + 9*(21*b^10*c^2*d^8 - 7*a*b^9*c*d^9 + a
^2*b^8*d^10)*x^8 + 18*(35*b^10*c^3*d^7 - 21*a*b^9*c^2*d^8 + 7*a^2*b^8*c*d^9 - a^3*b^7*d^10)*x^7 + 42*(35*b^10*
c^4*d^6 - 35*a*b^9*c^3*d^7 + 21*a^2*b^8*c^2*d^8 - 7*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^6 + 126*(21*b^10*c^5*d^5 -
35*a*b^9*c^4*d^6 + 35*a^2*b^8*c^3*d^7 - 21*a^3*b^7*c^2*d^8 + 7*a^4*b^6*c*d^9 - a^5*b^5*d^10)*x^5 + 630*(7*b^1
0*c^6*d^4 - 21*a*b^9*c^5*d^5 + 35*a^2*b^8*c^4*d^6 - 35*a^3*b^7*c^3*d^7 + 21*a^4*b^6*c^2*d^8 - 7*a^5*b^5*c*d^9
+ a^6*b^4*d^10)*x^4 + 7*(1890*a*b^9*c^6*d^4 - 7938*a^2*b^8*c^5*d^5 + 15330*a^3*b^7*c^4*d^6 - 16680*a^4*b^6*c^3
*d^7 + 10575*a^5*b^5*c^2*d^8 - 3665*a^6*b^4*c*d^9 + 539*a^7*b^3*d^10)*x^3 - 21*(45*b^10*c^8*d^2 - 360*a*b^9*c^
7*d^3 + 630*a^2*b^8*c^6*d^4 + 378*a^3*b^7*c^5*d^5 - 2730*a^4*b^6*c^4*d^6 + 4080*a^5*b^5*c^3*d^7 - 3015*a^6*b^4
*c^2*d^8 + 1145*a^7*b^3*c*d^9 - 179*a^8*b^2*d^10)*x^2 - 21*(5*b^10*c^9*d + 45*a*b^9*c^8*d^2 - 540*a^2*b^8*c^7*
d^3 + 1890*a^3*b^7*c^6*d^4 - 3402*a^4*b^6*c^5*d^5 + 3570*a^5*b^5*c^4*d^6 - 2220*a^6*b^4*c^3*d^7 + 765*a^7*b^3*
c^2*d^8 - 115*a^8*b^2*c*d^9 + a^9*b*d^10)*x + 2520*(a^3*b^7*c^7*d^3 - 7*a^4*b^6*c^6*d^4 + 21*a^5*b^5*c^5*d^5 -
35*a^6*b^4*c^4*d^6 + 35*a^7*b^3*c^3*d^7 - 21*a^8*b^2*c^2*d^8 + 7*a^9*b*c*d^9 - a^10*d^10 + (b^10*c^7*d^3 - 7*
a*b^9*c^6*d^4 + 21*a^2*b^8*c^5*d^5 - 35*a^3*b^7*c^4*d^6 + 35*a^4*b^6*c^3*d^7 - 21*a^5*b^5*c^2*d^8 + 7*a^6*b^4*
c*d^9 - a^7*b^3*d^10)*x^3 + 3*(a*b^9*c^7*d^3 - 7*a^2*b^8*c^6*d^4 + 21*a^3*b^7*c^5*d^5 - 35*a^4*b^6*c^4*d^6 + 3
5*a^5*b^5*c^3*d^7 - 21*a^6*b^4*c^2*d^8 + 7*a^7*b^3*c*d^9 - a^8*b^2*d^10)*x^2 + 3*(a^2*b^8*c^7*d^3 - 7*a^3*b^7*
c^6*d^4 + 21*a^4*b^6*c^5*d^5 - 35*a^5*b^5*c^4*d^6 + 35*a^6*b^4*c^3*d^7 - 21*a^7*b^3*c^2*d^8 + 7*a^8*b^2*c*d^9
- a^9*b*d^10)*x)*log(b*x + a))/(b^14*x^3 + 3*a*b^13*x^2 + 3*a^2*b^12*x + a^3*b^11)

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giac [B]  time = 1.26, size = 907, normalized size = 3.52 \begin {gather*} \frac {120 \, {\left (b^{7} c^{7} d^{3} - 7 \, a b^{6} c^{6} d^{4} + 21 \, a^{2} b^{5} c^{5} d^{5} - 35 \, a^{3} b^{4} c^{4} d^{6} + 35 \, a^{4} b^{3} c^{3} d^{7} - 21 \, a^{5} b^{2} c^{2} d^{8} + 7 \, a^{6} b c d^{9} - a^{7} d^{10}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{11}} - \frac {b^{10} c^{10} + 5 \, a b^{9} c^{9} d + 45 \, a^{2} b^{8} c^{8} d^{2} - 660 \, a^{3} b^{7} c^{7} d^{3} + 2730 \, a^{4} b^{6} c^{6} d^{4} - 5922 \, a^{5} b^{5} c^{5} d^{5} + 7770 \, a^{6} b^{4} c^{4} d^{6} - 6420 \, a^{7} b^{3} c^{3} d^{7} + 3285 \, a^{8} b^{2} c^{2} d^{8} - 955 \, a^{9} b c d^{9} + 121 \, a^{10} d^{10} + 135 \, {\left (b^{10} c^{8} d^{2} - 8 \, a b^{9} c^{7} d^{3} + 28 \, a^{2} b^{8} c^{6} d^{4} - 56 \, a^{3} b^{7} c^{5} d^{5} + 70 \, a^{4} b^{6} c^{4} d^{6} - 56 \, a^{5} b^{5} c^{3} d^{7} + 28 \, a^{6} b^{4} c^{2} d^{8} - 8 \, a^{7} b^{3} c d^{9} + a^{8} b^{2} d^{10}\right )} x^{2} + 15 \, {\left (b^{10} c^{9} d + 9 \, a b^{9} c^{8} d^{2} - 108 \, a^{2} b^{8} c^{7} d^{3} + 420 \, a^{3} b^{7} c^{6} d^{4} - 882 \, a^{4} b^{6} c^{5} d^{5} + 1134 \, a^{5} b^{5} c^{4} d^{6} - 924 \, a^{6} b^{4} c^{3} d^{7} + 468 \, a^{7} b^{3} c^{2} d^{8} - 135 \, a^{8} b^{2} c d^{9} + 17 \, a^{9} b d^{10}\right )} x}{3 \, {\left (b x + a\right )}^{3} b^{11}} + \frac {3 \, b^{24} d^{10} x^{7} + 35 \, b^{24} c d^{9} x^{6} - 14 \, a b^{23} d^{10} x^{6} + 189 \, b^{24} c^{2} d^{8} x^{5} - 168 \, a b^{23} c d^{9} x^{5} + 42 \, a^{2} b^{22} d^{10} x^{5} + 630 \, b^{24} c^{3} d^{7} x^{4} - 945 \, a b^{23} c^{2} d^{8} x^{4} + 525 \, a^{2} b^{22} c d^{9} x^{4} - 105 \, a^{3} b^{21} d^{10} x^{4} + 1470 \, b^{24} c^{4} d^{6} x^{3} - 3360 \, a b^{23} c^{3} d^{7} x^{3} + 3150 \, a^{2} b^{22} c^{2} d^{8} x^{3} - 1400 \, a^{3} b^{21} c d^{9} x^{3} + 245 \, a^{4} b^{20} d^{10} x^{3} + 2646 \, b^{24} c^{5} d^{5} x^{2} - 8820 \, a b^{23} c^{4} d^{6} x^{2} + 12600 \, a^{2} b^{22} c^{3} d^{7} x^{2} - 9450 \, a^{3} b^{21} c^{2} d^{8} x^{2} + 3675 \, a^{4} b^{20} c d^{9} x^{2} - 588 \, a^{5} b^{19} d^{10} x^{2} + 4410 \, b^{24} c^{6} d^{4} x - 21168 \, a b^{23} c^{5} d^{5} x + 44100 \, a^{2} b^{22} c^{4} d^{6} x - 50400 \, a^{3} b^{21} c^{3} d^{7} x + 33075 \, a^{4} b^{20} c^{2} d^{8} x - 11760 \, a^{5} b^{19} c d^{9} x + 1764 \, a^{6} b^{18} d^{10} x}{21 \, b^{28}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^10/(b*x+a)^4,x, algorithm="giac")

[Out]

120*(b^7*c^7*d^3 - 7*a*b^6*c^6*d^4 + 21*a^2*b^5*c^5*d^5 - 35*a^3*b^4*c^4*d^6 + 35*a^4*b^3*c^3*d^7 - 21*a^5*b^2
*c^2*d^8 + 7*a^6*b*c*d^9 - a^7*d^10)*log(abs(b*x + a))/b^11 - 1/3*(b^10*c^10 + 5*a*b^9*c^9*d + 45*a^2*b^8*c^8*
d^2 - 660*a^3*b^7*c^7*d^3 + 2730*a^4*b^6*c^6*d^4 - 5922*a^5*b^5*c^5*d^5 + 7770*a^6*b^4*c^4*d^6 - 6420*a^7*b^3*
c^3*d^7 + 3285*a^8*b^2*c^2*d^8 - 955*a^9*b*c*d^9 + 121*a^10*d^10 + 135*(b^10*c^8*d^2 - 8*a*b^9*c^7*d^3 + 28*a^
2*b^8*c^6*d^4 - 56*a^3*b^7*c^5*d^5 + 70*a^4*b^6*c^4*d^6 - 56*a^5*b^5*c^3*d^7 + 28*a^6*b^4*c^2*d^8 - 8*a^7*b^3*
c*d^9 + a^8*b^2*d^10)*x^2 + 15*(b^10*c^9*d + 9*a*b^9*c^8*d^2 - 108*a^2*b^8*c^7*d^3 + 420*a^3*b^7*c^6*d^4 - 882
*a^4*b^6*c^5*d^5 + 1134*a^5*b^5*c^4*d^6 - 924*a^6*b^4*c^3*d^7 + 468*a^7*b^3*c^2*d^8 - 135*a^8*b^2*c*d^9 + 17*a
^9*b*d^10)*x)/((b*x + a)^3*b^11) + 1/21*(3*b^24*d^10*x^7 + 35*b^24*c*d^9*x^6 - 14*a*b^23*d^10*x^6 + 189*b^24*c
^2*d^8*x^5 - 168*a*b^23*c*d^9*x^5 + 42*a^2*b^22*d^10*x^5 + 630*b^24*c^3*d^7*x^4 - 945*a*b^23*c^2*d^8*x^4 + 525
*a^2*b^22*c*d^9*x^4 - 105*a^3*b^21*d^10*x^4 + 1470*b^24*c^4*d^6*x^3 - 3360*a*b^23*c^3*d^7*x^3 + 3150*a^2*b^22*
c^2*d^8*x^3 - 1400*a^3*b^21*c*d^9*x^3 + 245*a^4*b^20*d^10*x^3 + 2646*b^24*c^5*d^5*x^2 - 8820*a*b^23*c^4*d^6*x^
2 + 12600*a^2*b^22*c^3*d^7*x^2 - 9450*a^3*b^21*c^2*d^8*x^2 + 3675*a^4*b^20*c*d^9*x^2 - 588*a^5*b^19*d^10*x^2 +
4410*b^24*c^6*d^4*x - 21168*a*b^23*c^5*d^5*x + 44100*a^2*b^22*c^4*d^6*x - 50400*a^3*b^21*c^3*d^7*x + 33075*a^
4*b^20*c^2*d^8*x - 11760*a^5*b^19*c*d^9*x + 1764*a^6*b^18*d^10*x)/b^28

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maple [B]  time = 0.02, size = 1141, normalized size = 4.42

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^10/(b*x+a)^4,x)

[Out]

-1/3/b^11/(b*x+a)^3*a^10*d^10+5/b^11*d^10/(b*x+a)^2*a^9-5/b^2*d/(b*x+a)^2*c^9-120/b^11*d^10*ln(b*x+a)*a^7+120/
b^4*d^3*ln(b*x+a)*c^7-45/b^11*d^10/(b*x+a)*a^8-45/b^3*d^2/(b*x+a)*c^8+9*d^8/b^4*x^5*c^2-5*d^10/b^7*x^4*a^3+30*
d^7/b^4*x^4*c^3+35/3*d^10/b^8*x^3*a^4+70*d^6/b^4*x^3*c^4-28*d^10/b^9*x^2*a^5+126*d^5/b^4*x^2*c^5-2/3*d^10/b^5*
x^6*a+5/3*d^9/b^4*x^6*c+2*d^10/b^6*x^5*a^2+210*d^4/b^4*c^6*x+84*d^10/b^10*a^6*x+1/7*d^10/b^4*x^7-1/3/b/(b*x+a)
^3*c^10-70/b^5/(b*x+a)^3*a^4*c^6*d^4+40/b^4/(b*x+a)^3*a^3*c^7*d^3-15/b^3/(b*x+a)^3*a^2*c^8*d^2+10/3/b^2/(b*x+a
)^3*a*c^9*d-45/b^10*d^9/(b*x+a)^2*a^8*c+180/b^9*d^8/(b*x+a)^2*a^7*c^2-420/b^8*d^7/(b*x+a)^2*a^6*c^3+630/b^7*d^
6/(b*x+a)^2*a^5*c^4-630/b^6*d^5/(b*x+a)^2*a^4*c^5+420/b^5*d^4/(b*x+a)^2*a^3*c^6-180/b^4*d^3/(b*x+a)^2*a^2*c^7+
45/b^3*d^2/(b*x+a)^2*a*c^8+840/b^10*d^9*ln(b*x+a)*a^6*c-2520/b^9*d^8*ln(b*x+a)*a^5*c^2+4200/b^8*d^7*ln(b*x+a)*
a^4*c^3-4200/b^7*d^6*ln(b*x+a)*a^3*c^4+2520/b^6*d^5*ln(b*x+a)*a^2*c^5-840/b^5*d^4*ln(b*x+a)*a*c^6+360/b^10*d^9
/(b*x+a)*a^7*c-1260/b^9*d^8/(b*x+a)*a^6*c^2+2520/b^8*d^7/(b*x+a)*a^5*c^3-3150/b^7*d^6/(b*x+a)*a^4*c^4+2520/b^6
*d^5/(b*x+a)*a^3*c^5-1260/b^5*d^4/(b*x+a)*a^2*c^6+360/b^4*d^3/(b*x+a)*a*c^7-200/3*d^9/b^7*x^3*a^3*c-8*d^9/b^5*
x^5*a*c-560*d^9/b^9*a^5*c*x+1575*d^8/b^8*a^4*c^2*x-2400*d^7/b^7*a^3*c^3*x+2100*d^6/b^6*a^2*c^4*x-1008*d^5/b^5*
a*c^5*x-450*d^8/b^7*x^2*a^3*c^2+600*d^7/b^6*x^2*a^2*c^3-420*d^6/b^5*x^2*a*c^4+175*d^9/b^8*x^2*a^4*c+150*d^8/b^
6*x^3*a^2*c^2-160*d^7/b^5*x^3*a*c^3+10/3/b^10/(b*x+a)^3*a^9*c*d^9-15/b^9/(b*x+a)^3*a^8*c^2*d^8+40/b^8/(b*x+a)^
3*a^7*c^3*d^7+25*d^9/b^6*x^4*a^2*c-45*d^8/b^5*x^4*a*c^2-70/b^7/(b*x+a)^3*a^6*c^4*d^6+84/b^6/(b*x+a)^3*a^5*c^5*
d^5

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maxima [B]  time = 1.73, size = 891, normalized size = 3.45 \begin {gather*} -\frac {b^{10} c^{10} + 5 \, a b^{9} c^{9} d + 45 \, a^{2} b^{8} c^{8} d^{2} - 660 \, a^{3} b^{7} c^{7} d^{3} + 2730 \, a^{4} b^{6} c^{6} d^{4} - 5922 \, a^{5} b^{5} c^{5} d^{5} + 7770 \, a^{6} b^{4} c^{4} d^{6} - 6420 \, a^{7} b^{3} c^{3} d^{7} + 3285 \, a^{8} b^{2} c^{2} d^{8} - 955 \, a^{9} b c d^{9} + 121 \, a^{10} d^{10} + 135 \, {\left (b^{10} c^{8} d^{2} - 8 \, a b^{9} c^{7} d^{3} + 28 \, a^{2} b^{8} c^{6} d^{4} - 56 \, a^{3} b^{7} c^{5} d^{5} + 70 \, a^{4} b^{6} c^{4} d^{6} - 56 \, a^{5} b^{5} c^{3} d^{7} + 28 \, a^{6} b^{4} c^{2} d^{8} - 8 \, a^{7} b^{3} c d^{9} + a^{8} b^{2} d^{10}\right )} x^{2} + 15 \, {\left (b^{10} c^{9} d + 9 \, a b^{9} c^{8} d^{2} - 108 \, a^{2} b^{8} c^{7} d^{3} + 420 \, a^{3} b^{7} c^{6} d^{4} - 882 \, a^{4} b^{6} c^{5} d^{5} + 1134 \, a^{5} b^{5} c^{4} d^{6} - 924 \, a^{6} b^{4} c^{3} d^{7} + 468 \, a^{7} b^{3} c^{2} d^{8} - 135 \, a^{8} b^{2} c d^{9} + 17 \, a^{9} b d^{10}\right )} x}{3 \, {\left (b^{14} x^{3} + 3 \, a b^{13} x^{2} + 3 \, a^{2} b^{12} x + a^{3} b^{11}\right )}} + \frac {3 \, b^{6} d^{10} x^{7} + 7 \, {\left (5 \, b^{6} c d^{9} - 2 \, a b^{5} d^{10}\right )} x^{6} + 21 \, {\left (9 \, b^{6} c^{2} d^{8} - 8 \, a b^{5} c d^{9} + 2 \, a^{2} b^{4} d^{10}\right )} x^{5} + 105 \, {\left (6 \, b^{6} c^{3} d^{7} - 9 \, a b^{5} c^{2} d^{8} + 5 \, a^{2} b^{4} c d^{9} - a^{3} b^{3} d^{10}\right )} x^{4} + 35 \, {\left (42 \, b^{6} c^{4} d^{6} - 96 \, a b^{5} c^{3} d^{7} + 90 \, a^{2} b^{4} c^{2} d^{8} - 40 \, a^{3} b^{3} c d^{9} + 7 \, a^{4} b^{2} d^{10}\right )} x^{3} + 21 \, {\left (126 \, b^{6} c^{5} d^{5} - 420 \, a b^{5} c^{4} d^{6} + 600 \, a^{2} b^{4} c^{3} d^{7} - 450 \, a^{3} b^{3} c^{2} d^{8} + 175 \, a^{4} b^{2} c d^{9} - 28 \, a^{5} b d^{10}\right )} x^{2} + 21 \, {\left (210 \, b^{6} c^{6} d^{4} - 1008 \, a b^{5} c^{5} d^{5} + 2100 \, a^{2} b^{4} c^{4} d^{6} - 2400 \, a^{3} b^{3} c^{3} d^{7} + 1575 \, a^{4} b^{2} c^{2} d^{8} - 560 \, a^{5} b c d^{9} + 84 \, a^{6} d^{10}\right )} x}{21 \, b^{10}} + \frac {120 \, {\left (b^{7} c^{7} d^{3} - 7 \, a b^{6} c^{6} d^{4} + 21 \, a^{2} b^{5} c^{5} d^{5} - 35 \, a^{3} b^{4} c^{4} d^{6} + 35 \, a^{4} b^{3} c^{3} d^{7} - 21 \, a^{5} b^{2} c^{2} d^{8} + 7 \, a^{6} b c d^{9} - a^{7} 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)^4,x, algorithm="maxima")

[Out]

-1/3*(b^10*c^10 + 5*a*b^9*c^9*d + 45*a^2*b^8*c^8*d^2 - 660*a^3*b^7*c^7*d^3 + 2730*a^4*b^6*c^6*d^4 - 5922*a^5*b
^5*c^5*d^5 + 7770*a^6*b^4*c^4*d^6 - 6420*a^7*b^3*c^3*d^7 + 3285*a^8*b^2*c^2*d^8 - 955*a^9*b*c*d^9 + 121*a^10*d
^10 + 135*(b^10*c^8*d^2 - 8*a*b^9*c^7*d^3 + 28*a^2*b^8*c^6*d^4 - 56*a^3*b^7*c^5*d^5 + 70*a^4*b^6*c^4*d^6 - 56*
a^5*b^5*c^3*d^7 + 28*a^6*b^4*c^2*d^8 - 8*a^7*b^3*c*d^9 + a^8*b^2*d^10)*x^2 + 15*(b^10*c^9*d + 9*a*b^9*c^8*d^2
- 108*a^2*b^8*c^7*d^3 + 420*a^3*b^7*c^6*d^4 - 882*a^4*b^6*c^5*d^5 + 1134*a^5*b^5*c^4*d^6 - 924*a^6*b^4*c^3*d^7
+ 468*a^7*b^3*c^2*d^8 - 135*a^8*b^2*c*d^9 + 17*a^9*b*d^10)*x)/(b^14*x^3 + 3*a*b^13*x^2 + 3*a^2*b^12*x + a^3*b
^11) + 1/21*(3*b^6*d^10*x^7 + 7*(5*b^6*c*d^9 - 2*a*b^5*d^10)*x^6 + 21*(9*b^6*c^2*d^8 - 8*a*b^5*c*d^9 + 2*a^2*b
^4*d^10)*x^5 + 105*(6*b^6*c^3*d^7 - 9*a*b^5*c^2*d^8 + 5*a^2*b^4*c*d^9 - a^3*b^3*d^10)*x^4 + 35*(42*b^6*c^4*d^6
- 96*a*b^5*c^3*d^7 + 90*a^2*b^4*c^2*d^8 - 40*a^3*b^3*c*d^9 + 7*a^4*b^2*d^10)*x^3 + 21*(126*b^6*c^5*d^5 - 420*
a*b^5*c^4*d^6 + 600*a^2*b^4*c^3*d^7 - 450*a^3*b^3*c^2*d^8 + 175*a^4*b^2*c*d^9 - 28*a^5*b*d^10)*x^2 + 21*(210*b
^6*c^6*d^4 - 1008*a*b^5*c^5*d^5 + 2100*a^2*b^4*c^4*d^6 - 2400*a^3*b^3*c^3*d^7 + 1575*a^4*b^2*c^2*d^8 - 560*a^5
*b*c*d^9 + 84*a^6*d^10)*x)/b^10 + 120*(b^7*c^7*d^3 - 7*a*b^6*c^6*d^4 + 21*a^2*b^5*c^5*d^5 - 35*a^3*b^4*c^4*d^6
+ 35*a^4*b^3*c^3*d^7 - 21*a^5*b^2*c^2*d^8 + 7*a^6*b*c*d^9 - a^7*d^10)*log(b*x + a)/b^11

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mupad [B]  time = 0.39, size = 2219, normalized size = 8.60

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^10/(a + b*x)^4,x)

[Out]

x^3*((4*a*((4*a*((4*a*((4*a*d^10)/b^5 - (10*c*d^9)/b^4))/b - (6*a^2*d^10)/b^6 + (45*c^2*d^8)/b^4))/b + (4*a^3*
d^10)/b^7 - (120*c^3*d^7)/b^4 - (6*a^2*((4*a*d^10)/b^5 - (10*c*d^9)/b^4))/b^2))/(3*b) - (a^4*d^10)/(3*b^8) + (
70*c^4*d^6)/b^4 + (4*a^3*((4*a*d^10)/b^5 - (10*c*d^9)/b^4))/(3*b^3) - (2*a^2*((4*a*((4*a*d^10)/b^5 - (10*c*d^9
)/b^4))/b - (6*a^2*d^10)/b^6 + (45*c^2*d^8)/b^4))/b^2) - x^6*((2*a*d^10)/(3*b^5) - (5*c*d^9)/(3*b^4)) - x^4*((
a*((4*a*((4*a*d^10)/b^5 - (10*c*d^9)/b^4))/b - (6*a^2*d^10)/b^6 + (45*c^2*d^8)/b^4))/b + (a^3*d^10)/b^7 - (30*
c^3*d^7)/b^4 - (3*a^2*((4*a*d^10)/b^5 - (10*c*d^9)/b^4))/(2*b^2)) + x^5*((4*a*((4*a*d^10)/b^5 - (10*c*d^9)/b^4
))/(5*b) - (6*a^2*d^10)/(5*b^6) + (9*c^2*d^8)/b^4) - x*((4*a*((252*c^5*d^5)/b^4 - (4*a*((4*a*((4*a*((4*a*((4*a
*d^10)/b^5 - (10*c*d^9)/b^4))/b - (6*a^2*d^10)/b^6 + (45*c^2*d^8)/b^4))/b + (4*a^3*d^10)/b^7 - (120*c^3*d^7)/b
^4 - (6*a^2*((4*a*d^10)/b^5 - (10*c*d^9)/b^4))/b^2))/b - (a^4*d^10)/b^8 + (210*c^4*d^6)/b^4 + (4*a^3*((4*a*d^1
0)/b^5 - (10*c*d^9)/b^4))/b^3 - (6*a^2*((4*a*((4*a*d^10)/b^5 - (10*c*d^9)/b^4))/b - (6*a^2*d^10)/b^6 + (45*c^2
*d^8)/b^4))/b^2))/b + (a^4*((4*a*d^10)/b^5 - (10*c*d^9)/b^4))/b^4 + (6*a^2*((4*a*((4*a*((4*a*d^10)/b^5 - (10*c
*d^9)/b^4))/b - (6*a^2*d^10)/b^6 + (45*c^2*d^8)/b^4))/b + (4*a^3*d^10)/b^7 - (120*c^3*d^7)/b^4 - (6*a^2*((4*a*
d^10)/b^5 - (10*c*d^9)/b^4))/b^2))/b^2 - (4*a^3*((4*a*((4*a*d^10)/b^5 - (10*c*d^9)/b^4))/b - (6*a^2*d^10)/b^6
+ (45*c^2*d^8)/b^4))/b^3))/b - (210*c^6*d^4)/b^4 + (6*a^2*((4*a*((4*a*((4*a*((4*a*d^10)/b^5 - (10*c*d^9)/b^4))
/b - (6*a^2*d^10)/b^6 + (45*c^2*d^8)/b^4))/b + (4*a^3*d^10)/b^7 - (120*c^3*d^7)/b^4 - (6*a^2*((4*a*d^10)/b^5 -
(10*c*d^9)/b^4))/b^2))/b - (a^4*d^10)/b^8 + (210*c^4*d^6)/b^4 + (4*a^3*((4*a*d^10)/b^5 - (10*c*d^9)/b^4))/b^3
- (6*a^2*((4*a*((4*a*d^10)/b^5 - (10*c*d^9)/b^4))/b - (6*a^2*d^10)/b^6 + (45*c^2*d^8)/b^4))/b^2))/b^2 - (4*a^
3*((4*a*((4*a*((4*a*d^10)/b^5 - (10*c*d^9)/b^4))/b - (6*a^2*d^10)/b^6 + (45*c^2*d^8)/b^4))/b + (4*a^3*d^10)/b^
7 - (120*c^3*d^7)/b^4 - (6*a^2*((4*a*d^10)/b^5 - (10*c*d^9)/b^4))/b^2))/b^3 + (a^4*((4*a*((4*a*d^10)/b^5 - (10
*c*d^9)/b^4))/b - (6*a^2*d^10)/b^6 + (45*c^2*d^8)/b^4))/b^4) + x^2*((126*c^5*d^5)/b^4 - (2*a*((4*a*((4*a*((4*a
*((4*a*d^10)/b^5 - (10*c*d^9)/b^4))/b - (6*a^2*d^10)/b^6 + (45*c^2*d^8)/b^4))/b + (4*a^3*d^10)/b^7 - (120*c^3*
d^7)/b^4 - (6*a^2*((4*a*d^10)/b^5 - (10*c*d^9)/b^4))/b^2))/b - (a^4*d^10)/b^8 + (210*c^4*d^6)/b^4 + (4*a^3*((4
*a*d^10)/b^5 - (10*c*d^9)/b^4))/b^3 - (6*a^2*((4*a*((4*a*d^10)/b^5 - (10*c*d^9)/b^4))/b - (6*a^2*d^10)/b^6 + (
45*c^2*d^8)/b^4))/b^2))/b + (a^4*((4*a*d^10)/b^5 - (10*c*d^9)/b^4))/(2*b^4) + (3*a^2*((4*a*((4*a*((4*a*d^10)/b
^5 - (10*c*d^9)/b^4))/b - (6*a^2*d^10)/b^6 + (45*c^2*d^8)/b^4))/b + (4*a^3*d^10)/b^7 - (120*c^3*d^7)/b^4 - (6*
a^2*((4*a*d^10)/b^5 - (10*c*d^9)/b^4))/b^2))/b^2 - (2*a^3*((4*a*((4*a*d^10)/b^5 - (10*c*d^9)/b^4))/b - (6*a^2*
d^10)/b^6 + (45*c^2*d^8)/b^4))/b^3) - ((121*a^10*d^10 + b^10*c^10 + 45*a^2*b^8*c^8*d^2 - 660*a^3*b^7*c^7*d^3 +
2730*a^4*b^6*c^6*d^4 - 5922*a^5*b^5*c^5*d^5 + 7770*a^6*b^4*c^4*d^6 - 6420*a^7*b^3*c^3*d^7 + 3285*a^8*b^2*c^2*
d^8 + 5*a*b^9*c^9*d - 955*a^9*b*c*d^9)/(3*b) + x*(85*a^9*d^10 + 5*b^9*c^9*d + 45*a*b^8*c^8*d^2 - 540*a^2*b^7*c
^7*d^3 + 2100*a^3*b^6*c^6*d^4 - 4410*a^4*b^5*c^5*d^5 + 5670*a^5*b^4*c^4*d^6 - 4620*a^6*b^3*c^3*d^7 + 2340*a^7*
b^2*c^2*d^8 - 675*a^8*b*c*d^9) + x^2*(45*a^8*b*d^10 + 45*b^9*c^8*d^2 - 360*a*b^8*c^7*d^3 - 360*a^7*b^2*c*d^9 +
1260*a^2*b^7*c^6*d^4 - 2520*a^3*b^6*c^5*d^5 + 3150*a^4*b^5*c^4*d^6 - 2520*a^5*b^4*c^3*d^7 + 1260*a^6*b^3*c^2*
d^8))/(a^3*b^10 + b^13*x^3 + 3*a^2*b^11*x + 3*a*b^12*x^2) + (d^10*x^7)/(7*b^4) - (log(a + b*x)*(120*a^7*d^10 -
120*b^7*c^7*d^3 + 840*a*b^6*c^6*d^4 - 2520*a^2*b^5*c^5*d^5 + 4200*a^3*b^4*c^4*d^6 - 4200*a^4*b^3*c^3*d^7 + 25
20*a^5*b^2*c^2*d^8 - 840*a^6*b*c*d^9))/b^11

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sympy [B]  time = 32.53, size = 867, normalized size = 3.36 \begin {gather*} x^{6} \left (- \frac {2 a d^{10}}{3 b^{5}} + \frac {5 c d^{9}}{3 b^{4}}\right ) + x^{5} \left (\frac {2 a^{2} d^{10}}{b^{6}} - \frac {8 a c d^{9}}{b^{5}} + \frac {9 c^{2} d^{8}}{b^{4}}\right ) + x^{4} \left (- \frac {5 a^{3} d^{10}}{b^{7}} + \frac {25 a^{2} c d^{9}}{b^{6}} - \frac {45 a c^{2} d^{8}}{b^{5}} + \frac {30 c^{3} d^{7}}{b^{4}}\right ) + x^{3} \left (\frac {35 a^{4} d^{10}}{3 b^{8}} - \frac {200 a^{3} c d^{9}}{3 b^{7}} + \frac {150 a^{2} c^{2} d^{8}}{b^{6}} - \frac {160 a c^{3} d^{7}}{b^{5}} + \frac {70 c^{4} d^{6}}{b^{4}}\right ) + x^{2} \left (- \frac {28 a^{5} d^{10}}{b^{9}} + \frac {175 a^{4} c d^{9}}{b^{8}} - \frac {450 a^{3} c^{2} d^{8}}{b^{7}} + \frac {600 a^{2} c^{3} d^{7}}{b^{6}} - \frac {420 a c^{4} d^{6}}{b^{5}} + \frac {126 c^{5} d^{5}}{b^{4}}\right ) + x \left (\frac {84 a^{6} d^{10}}{b^{10}} - \frac {560 a^{5} c d^{9}}{b^{9}} + \frac {1575 a^{4} c^{2} d^{8}}{b^{8}} - \frac {2400 a^{3} c^{3} d^{7}}{b^{7}} + \frac {2100 a^{2} c^{4} d^{6}}{b^{6}} - \frac {1008 a c^{5} d^{5}}{b^{5}} + \frac {210 c^{6} d^{4}}{b^{4}}\right ) + \frac {- 121 a^{10} d^{10} + 955 a^{9} b c d^{9} - 3285 a^{8} b^{2} c^{2} d^{8} + 6420 a^{7} b^{3} c^{3} d^{7} - 7770 a^{6} b^{4} c^{4} d^{6} + 5922 a^{5} b^{5} c^{5} d^{5} - 2730 a^{4} b^{6} c^{6} d^{4} + 660 a^{3} b^{7} c^{7} d^{3} - 45 a^{2} b^{8} c^{8} d^{2} - 5 a b^{9} c^{9} d - b^{10} c^{10} + x^{2} \left (- 135 a^{8} b^{2} d^{10} + 1080 a^{7} b^{3} c d^{9} - 3780 a^{6} b^{4} c^{2} d^{8} + 7560 a^{5} b^{5} c^{3} d^{7} - 9450 a^{4} b^{6} c^{4} d^{6} + 7560 a^{3} b^{7} c^{5} d^{5} - 3780 a^{2} b^{8} c^{6} d^{4} + 1080 a b^{9} c^{7} d^{3} - 135 b^{10} c^{8} d^{2}\right ) + x \left (- 255 a^{9} b d^{10} + 2025 a^{8} b^{2} c d^{9} - 7020 a^{7} b^{3} c^{2} d^{8} + 13860 a^{6} b^{4} c^{3} d^{7} - 17010 a^{5} b^{5} c^{4} d^{6} + 13230 a^{4} b^{6} c^{5} d^{5} - 6300 a^{3} b^{7} c^{6} d^{4} + 1620 a^{2} b^{8} c^{7} d^{3} - 135 a b^{9} c^{8} d^{2} - 15 b^{10} c^{9} d\right )}{3 a^{3} b^{11} + 9 a^{2} b^{12} x + 9 a b^{13} x^{2} + 3 b^{14} x^{3}} + \frac {d^{10} x^{7}}{7 b^{4}} - \frac {120 d^{3} \left (a d - b c\right )^{7} \log {\left (a + b x \right )}}{b^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**10/(b*x+a)**4,x)

[Out]

x**6*(-2*a*d**10/(3*b**5) + 5*c*d**9/(3*b**4)) + x**5*(2*a**2*d**10/b**6 - 8*a*c*d**9/b**5 + 9*c**2*d**8/b**4)
+ x**4*(-5*a**3*d**10/b**7 + 25*a**2*c*d**9/b**6 - 45*a*c**2*d**8/b**5 + 30*c**3*d**7/b**4) + x**3*(35*a**4*d
**10/(3*b**8) - 200*a**3*c*d**9/(3*b**7) + 150*a**2*c**2*d**8/b**6 - 160*a*c**3*d**7/b**5 + 70*c**4*d**6/b**4)
+ x**2*(-28*a**5*d**10/b**9 + 175*a**4*c*d**9/b**8 - 450*a**3*c**2*d**8/b**7 + 600*a**2*c**3*d**7/b**6 - 420*
a*c**4*d**6/b**5 + 126*c**5*d**5/b**4) + x*(84*a**6*d**10/b**10 - 560*a**5*c*d**9/b**9 + 1575*a**4*c**2*d**8/b
**8 - 2400*a**3*c**3*d**7/b**7 + 2100*a**2*c**4*d**6/b**6 - 1008*a*c**5*d**5/b**5 + 210*c**6*d**4/b**4) + (-12
1*a**10*d**10 + 955*a**9*b*c*d**9 - 3285*a**8*b**2*c**2*d**8 + 6420*a**7*b**3*c**3*d**7 - 7770*a**6*b**4*c**4*
d**6 + 5922*a**5*b**5*c**5*d**5 - 2730*a**4*b**6*c**6*d**4 + 660*a**3*b**7*c**7*d**3 - 45*a**2*b**8*c**8*d**2
- 5*a*b**9*c**9*d - b**10*c**10 + x**2*(-135*a**8*b**2*d**10 + 1080*a**7*b**3*c*d**9 - 3780*a**6*b**4*c**2*d**
8 + 7560*a**5*b**5*c**3*d**7 - 9450*a**4*b**6*c**4*d**6 + 7560*a**3*b**7*c**5*d**5 - 3780*a**2*b**8*c**6*d**4
+ 1080*a*b**9*c**7*d**3 - 135*b**10*c**8*d**2) + x*(-255*a**9*b*d**10 + 2025*a**8*b**2*c*d**9 - 7020*a**7*b**3
*c**2*d**8 + 13860*a**6*b**4*c**3*d**7 - 17010*a**5*b**5*c**4*d**6 + 13230*a**4*b**6*c**5*d**5 - 6300*a**3*b**
7*c**6*d**4 + 1620*a**2*b**8*c**7*d**3 - 135*a*b**9*c**8*d**2 - 15*b**10*c**9*d))/(3*a**3*b**11 + 9*a**2*b**12
*x + 9*a*b**13*x**2 + 3*b**14*x**3) + d**10*x**7/(7*b**4) - 120*d**3*(a*d - b*c)**7*log(a + b*x)/b**11

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