3.1314 \(\int \frac{(c+d x)^{10}}{(a+b x)^3} \, dx\)

Optimal. Leaf size=262 \[ \frac{10 d^9 (a+b x)^7 (b c-a d)}{7 b^{11}}+\frac{15 d^8 (a+b x)^6 (b c-a d)^2}{2 b^{11}}+\frac{24 d^7 (a+b x)^5 (b c-a d)^3}{b^{11}}+\frac{105 d^6 (a+b x)^4 (b c-a d)^4}{2 b^{11}}+\frac{84 d^5 (a+b x)^3 (b c-a d)^5}{b^{11}}+\frac{105 d^4 (a+b x)^2 (b c-a d)^6}{b^{11}}+\frac{120 d^3 x (b c-a d)^7}{b^{10}}+\frac{45 d^2 (b c-a d)^8 \log (a+b x)}{b^{11}}-\frac{10 d (b c-a d)^9}{b^{11} (a+b x)}-\frac{(b c-a d)^{10}}{2 b^{11} (a+b x)^2}+\frac{d^{10} (a+b x)^8}{8 b^{11}} \]

[Out]

(120*d^3*(b*c - a*d)^7*x)/b^10 - (b*c - a*d)^10/(2*b^11*(a + b*x)^2) - (10*d*(b*c - a*d)^9)/(b^11*(a + b*x)) +
(105*d^4*(b*c - a*d)^6*(a + b*x)^2)/b^11 + (84*d^5*(b*c - a*d)^5*(a + b*x)^3)/b^11 + (105*d^6*(b*c - a*d)^4*(
a + b*x)^4)/(2*b^11) + (24*d^7*(b*c - a*d)^3*(a + b*x)^5)/b^11 + (15*d^8*(b*c - a*d)^2*(a + b*x)^6)/(2*b^11) +
(10*d^9*(b*c - a*d)*(a + b*x)^7)/(7*b^11) + (d^10*(a + b*x)^8)/(8*b^11) + (45*d^2*(b*c - a*d)^8*Log[a + b*x])
/b^11

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Rubi [A]  time = 0.444295, antiderivative size = 262, normalized size of antiderivative = 1., 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} \[ \frac{10 d^9 (a+b x)^7 (b c-a d)}{7 b^{11}}+\frac{15 d^8 (a+b x)^6 (b c-a d)^2}{2 b^{11}}+\frac{24 d^7 (a+b x)^5 (b c-a d)^3}{b^{11}}+\frac{105 d^6 (a+b x)^4 (b c-a d)^4}{2 b^{11}}+\frac{84 d^5 (a+b x)^3 (b c-a d)^5}{b^{11}}+\frac{105 d^4 (a+b x)^2 (b c-a d)^6}{b^{11}}+\frac{120 d^3 x (b c-a d)^7}{b^{10}}+\frac{45 d^2 (b c-a d)^8 \log (a+b x)}{b^{11}}-\frac{10 d (b c-a d)^9}{b^{11} (a+b x)}-\frac{(b c-a d)^{10}}{2 b^{11} (a+b x)^2}+\frac{d^{10} (a+b x)^8}{8 b^{11}} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^10/(a + b*x)^3,x]

[Out]

(120*d^3*(b*c - a*d)^7*x)/b^10 - (b*c - a*d)^10/(2*b^11*(a + b*x)^2) - (10*d*(b*c - a*d)^9)/(b^11*(a + b*x)) +
(105*d^4*(b*c - a*d)^6*(a + b*x)^2)/b^11 + (84*d^5*(b*c - a*d)^5*(a + b*x)^3)/b^11 + (105*d^6*(b*c - a*d)^4*(
a + b*x)^4)/(2*b^11) + (24*d^7*(b*c - a*d)^3*(a + b*x)^5)/b^11 + (15*d^8*(b*c - a*d)^2*(a + b*x)^6)/(2*b^11) +
(10*d^9*(b*c - a*d)*(a + b*x)^7)/(7*b^11) + (d^10*(a + b*x)^8)/(8*b^11) + (45*d^2*(b*c - a*d)^8*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)^3} \, dx &=\int \left (\frac{120 d^3 (b c-a d)^7}{b^{10}}+\frac{(b c-a d)^{10}}{b^{10} (a+b x)^3}+\frac{10 d (b c-a d)^9}{b^{10} (a+b x)^2}+\frac{45 d^2 (b c-a d)^8}{b^{10} (a+b x)}+\frac{210 d^4 (b c-a d)^6 (a+b x)}{b^{10}}+\frac{252 d^5 (b c-a d)^5 (a+b x)^2}{b^{10}}+\frac{210 d^6 (b c-a d)^4 (a+b x)^3}{b^{10}}+\frac{120 d^7 (b c-a d)^3 (a+b x)^4}{b^{10}}+\frac{45 d^8 (b c-a d)^2 (a+b x)^5}{b^{10}}+\frac{10 d^9 (b c-a d) (a+b x)^6}{b^{10}}+\frac{d^{10} (a+b x)^7}{b^{10}}\right ) \, dx\\ &=\frac{120 d^3 (b c-a d)^7 x}{b^{10}}-\frac{(b c-a d)^{10}}{2 b^{11} (a+b x)^2}-\frac{10 d (b c-a d)^9}{b^{11} (a+b x)}+\frac{105 d^4 (b c-a d)^6 (a+b x)^2}{b^{11}}+\frac{84 d^5 (b c-a d)^5 (a+b x)^3}{b^{11}}+\frac{105 d^6 (b c-a d)^4 (a+b x)^4}{2 b^{11}}+\frac{24 d^7 (b c-a d)^3 (a+b x)^5}{b^{11}}+\frac{15 d^8 (b c-a d)^2 (a+b x)^6}{2 b^{11}}+\frac{10 d^9 (b c-a d) (a+b x)^7}{7 b^{11}}+\frac{d^{10} (a+b x)^8}{8 b^{11}}+\frac{45 d^2 (b c-a d)^8 \log (a+b x)}{b^{11}}\\ \end{align*}

Mathematica [B]  time = 0.235511, size = 708, normalized size = 2.7 \[ \frac{3 a^2 b^8 d^2 \left (-21560 c^6 d^2 x^2+15680 c^5 d^3 x^3+4900 c^4 d^4 x^4+1568 c^3 d^5 x^5+392 c^2 d^6 x^6-4480 c^7 d x+1260 c^8+64 c d^7 x^7+5 d^8 x^8\right )-24 a^3 b^7 d^3 \left (-6174 c^5 d^2 x^2+2450 c^4 d^3 x^3+490 c^3 d^4 x^4+98 c^2 d^5 x^5-490 c^6 d x+700 c^7+14 c d^6 x^6+d^7 x^7\right )+42 a^4 b^6 d^4 \left (-4760 c^4 d^2 x^2+1120 c^3 d^3 x^3+140 c^2 d^4 x^4+336 c^5 d x+980 c^6+16 c d^5 x^5+d^6 x^6\right )-84 a^5 b^5 d^5 \left (-2000 c^3 d^2 x^2+280 c^2 d^3 x^3+560 c^4 d x+756 c^5+20 c d^4 x^4+d^5 x^5\right )+210 a^6 b^4 d^6 \left (-414 c^2 d^2 x^2+256 c^3 d x+308 c^4+32 c d^3 x^3+d^4 x^4\right )-280 a^7 b^3 d^7 \left (117 c^2 d x+156 c^3-91 c d^2 x^2+3 d^3 x^3\right )+28 a^8 b^2 d^8 \left (675 c^2+380 c d x-116 d^2 x^2\right )-56 a^9 b d^9 (85 c+26 d x)+532 a^{10} d^{10}-2 a b^9 d \left (-6720 c^7 d^2 x^2+11760 c^6 d^3 x^3+5880 c^5 d^4 x^4+2940 c^4 d^5 x^5+1176 c^3 d^6 x^6+336 c^2 d^7 x^7-2520 c^8 d x+140 c^9+60 c d^8 x^8+5 d^9 x^9\right )+2520 d^2 (a+b x)^2 (b c-a d)^8 \log (a+b x)+b^{10} \left (6720 c^7 d^3 x^3+5880 c^6 d^4 x^4+4704 c^5 d^5 x^5+2940 c^4 d^6 x^6+1344 c^3 d^7 x^7+420 c^2 d^8 x^8-560 c^9 d x-28 c^{10}+80 c d^9 x^9+7 d^{10} x^{10}\right )}{56 b^{11} (a+b x)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^10/(a + b*x)^3,x]

[Out]

(532*a^10*d^10 - 56*a^9*b*d^9*(85*c + 26*d*x) + 28*a^8*b^2*d^8*(675*c^2 + 380*c*d*x - 116*d^2*x^2) - 280*a^7*b
^3*d^7*(156*c^3 + 117*c^2*d*x - 91*c*d^2*x^2 + 3*d^3*x^3) + 210*a^6*b^4*d^6*(308*c^4 + 256*c^3*d*x - 414*c^2*d
^2*x^2 + 32*c*d^3*x^3 + d^4*x^4) - 84*a^5*b^5*d^5*(756*c^5 + 560*c^4*d*x - 2000*c^3*d^2*x^2 + 280*c^2*d^3*x^3
+ 20*c*d^4*x^4 + d^5*x^5) + 42*a^4*b^6*d^4*(980*c^6 + 336*c^5*d*x - 4760*c^4*d^2*x^2 + 1120*c^3*d^3*x^3 + 140*
c^2*d^4*x^4 + 16*c*d^5*x^5 + d^6*x^6) - 24*a^3*b^7*d^3*(700*c^7 - 490*c^6*d*x - 6174*c^5*d^2*x^2 + 2450*c^4*d^
3*x^3 + 490*c^3*d^4*x^4 + 98*c^2*d^5*x^5 + 14*c*d^6*x^6 + d^7*x^7) + 3*a^2*b^8*d^2*(1260*c^8 - 4480*c^7*d*x -
21560*c^6*d^2*x^2 + 15680*c^5*d^3*x^3 + 4900*c^4*d^4*x^4 + 1568*c^3*d^5*x^5 + 392*c^2*d^6*x^6 + 64*c*d^7*x^7 +
5*d^8*x^8) - 2*a*b^9*d*(140*c^9 - 2520*c^8*d*x - 6720*c^7*d^2*x^2 + 11760*c^6*d^3*x^3 + 5880*c^5*d^4*x^4 + 29
40*c^4*d^5*x^5 + 1176*c^3*d^6*x^6 + 336*c^2*d^7*x^7 + 60*c*d^8*x^8 + 5*d^9*x^9) + b^10*(-28*c^10 - 560*c^9*d*x
+ 6720*c^7*d^3*x^3 + 5880*c^6*d^4*x^4 + 4704*c^5*d^5*x^5 + 2940*c^4*d^6*x^6 + 1344*c^3*d^7*x^7 + 420*c^2*d^8*
x^8 + 80*c*d^9*x^9 + 7*d^10*x^10) + 2520*d^2*(b*c - a*d)^8*(a + b*x)^2*Log[a + b*x])/(56*b^11*(a + b*x)^2)

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Maple [B]  time = 0.016, size = 1105, normalized size = 4.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^10/(b*x+a)^3,x)

[Out]

-2520/b^8*d^7*ln(b*x+a)*a^5*c^3+3150/b^7*d^6*ln(b*x+a)*a^4*c^4-2520/b^6*d^5*ln(b*x+a)*a^3*c^5+1260/b^5*d^4*ln(
b*x+a)*a^2*c^6-360/b^4*d^3*ln(b*x+a)*a*c^7-45/2/b^9/(b*x+a)^2*a^8*c^2*d^8+60/b^8/(b*x+a)^2*a^7*c^3*d^7-105/b^7
/(b*x+a)^2*a^6*c^4*d^6+126/b^6/(b*x+a)^2*a^5*c^5*d^5-105/b^5/(b*x+a)^2*a^4*c^6*d^4+60/b^4/(b*x+a)^2*a^3*c^7*d^
3-45/2/b^3/(b*x+a)^2*a^2*c^8*d^2+5/b^2/(b*x+a)^2*a*c^9*d-1260/b^6*d^5/(b*x+a)*a^4*c^5+840/b^5*d^4/(b*x+a)*a^3*
c^6-360/b^4*d^3/(b*x+a)*a^2*c^7+90/b^3*d^2/(b*x+a)*a*c^8+5/b^10/(b*x+a)^2*a^9*c*d^9-630*d^4/b^4*a*c^6*x-150*d^
8/b^6*x^3*a^3*c^2+240*d^7/b^5*x^3*a^2*c^3-210*d^6/b^4*x^3*a*c^4-105*d^9/b^8*x^2*a^5*c+675/2*d^8/b^7*x^2*a^4*c^
2+1260/b^9*d^8*ln(b*x+a)*a^6*c^2-360/b^10*d^9*ln(b*x+a)*a^7*c-3/7*d^10/b^4*x^7*a+10/7*d^9/b^3*x^7*c+15/2*d^8/b
^3*x^6*c^2-2*d^10/b^6*x^5*a^3+120*d^3/b^3*c^7*x-36*d^10/b^10*a^7*x+d^10/b^5*x^6*a^2+10/b^11*d^10/(b*x+a)*a^9-1
0/b^2*d/(b*x+a)*c^9-1/2/b^11/(b*x+a)^2*a^10*d^10+45/b^11*d^10*ln(b*x+a)*a^8+45/b^3*d^2*ln(b*x+a)*c^8+24*d^7/b^
3*x^5*c^3+15/4*d^10/b^7*x^4*a^4+105/2*d^6/b^3*x^4*c^4-7*d^10/b^8*x^3*a^5+84*d^5/b^3*x^3*c^5+14*d^10/b^9*x^2*a^
6+105*d^4/b^3*x^2*c^6-1/2/b/(b*x+a)^2*c^10+1/8*d^10/b^3*x^8-600*d^7/b^6*x^2*a^3*c^3+630*d^6/b^5*x^2*a^2*c^4-37
8*d^5/b^4*x^2*a*c^5+280*d^9/b^9*a^6*c*x-90/b^10*d^9/(b*x+a)*a^8*c+360/b^9*d^8/(b*x+a)*a^7*c^2-840/b^8*d^7/(b*x
+a)*a^6*c^3+1260/b^7*d^6/(b*x+a)*a^5*c^4-27*d^8/b^4*x^5*a*c^2-25*d^9/b^6*x^4*a^3*c+135/2*d^8/b^5*x^4*a^2*c^2-9
0*d^7/b^4*x^4*a*c^3+50*d^9/b^7*x^3*a^4*c-945*d^8/b^8*a^5*c^2*x+1800*d^7/b^7*a^4*c^3*x-2100*d^6/b^6*a^3*c^4*x+1
512*d^5/b^5*a^2*c^5*x+12*d^9/b^5*x^5*a^2*c-5*d^9/b^4*x^6*a*c

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Maxima [B]  time = 1.02342, size = 1189, normalized size = 4.54 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^10/(b*x+a)^3,x, algorithm="maxima")

[Out]

-1/2*(b^10*c^10 + 10*a*b^9*c^9*d - 135*a^2*b^8*c^8*d^2 + 600*a^3*b^7*c^7*d^3 - 1470*a^4*b^6*c^6*d^4 + 2268*a^5
*b^5*c^5*d^5 - 2310*a^6*b^4*c^4*d^6 + 1560*a^7*b^3*c^3*d^7 - 675*a^8*b^2*c^2*d^8 + 170*a^9*b*c*d^9 - 19*a^10*d
^10 + 20*(b^10*c^9*d - 9*a*b^9*c^8*d^2 + 36*a^2*b^8*c^7*d^3 - 84*a^3*b^7*c^6*d^4 + 126*a^4*b^6*c^5*d^5 - 126*a
^5*b^5*c^4*d^6 + 84*a^6*b^4*c^3*d^7 - 36*a^7*b^3*c^2*d^8 + 9*a^8*b^2*c*d^9 - a^9*b*d^10)*x)/(b^13*x^2 + 2*a*b^
12*x + a^2*b^11) + 1/56*(7*b^7*d^10*x^8 + 8*(10*b^7*c*d^9 - 3*a*b^6*d^10)*x^7 + 28*(15*b^7*c^2*d^8 - 10*a*b^6*
c*d^9 + 2*a^2*b^5*d^10)*x^6 + 56*(24*b^7*c^3*d^7 - 27*a*b^6*c^2*d^8 + 12*a^2*b^5*c*d^9 - 2*a^3*b^4*d^10)*x^5 +
70*(42*b^7*c^4*d^6 - 72*a*b^6*c^3*d^7 + 54*a^2*b^5*c^2*d^8 - 20*a^3*b^4*c*d^9 + 3*a^4*b^3*d^10)*x^4 + 56*(84*
b^7*c^5*d^5 - 210*a*b^6*c^4*d^6 + 240*a^2*b^5*c^3*d^7 - 150*a^3*b^4*c^2*d^8 + 50*a^4*b^3*c*d^9 - 7*a^5*b^2*d^1
0)*x^3 + 28*(210*b^7*c^6*d^4 - 756*a*b^6*c^5*d^5 + 1260*a^2*b^5*c^4*d^6 - 1200*a^3*b^4*c^3*d^7 + 675*a^4*b^3*c
^2*d^8 - 210*a^5*b^2*c*d^9 + 28*a^6*b*d^10)*x^2 + 56*(120*b^7*c^7*d^3 - 630*a*b^6*c^6*d^4 + 1512*a^2*b^5*c^5*d
^5 - 2100*a^3*b^4*c^4*d^6 + 1800*a^4*b^3*c^3*d^7 - 945*a^5*b^2*c^2*d^8 + 280*a^6*b*c*d^9 - 36*a^7*d^10)*x)/b^1
0 + 45*(b^8*c^8*d^2 - 8*a*b^7*c^7*d^3 + 28*a^2*b^6*c^6*d^4 - 56*a^3*b^5*c^5*d^5 + 70*a^4*b^4*c^4*d^6 - 56*a^5*
b^3*c^3*d^7 + 28*a^6*b^2*c^2*d^8 - 8*a^7*b*c*d^9 + a^8*d^10)*log(b*x + a)/b^11

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Fricas [B]  time = 1.94856, size = 2612, normalized size = 9.97 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^10/(b*x+a)^3,x, algorithm="fricas")

[Out]

1/56*(7*b^10*d^10*x^10 - 28*b^10*c^10 - 280*a*b^9*c^9*d + 3780*a^2*b^8*c^8*d^2 - 16800*a^3*b^7*c^7*d^3 + 41160
*a^4*b^6*c^6*d^4 - 63504*a^5*b^5*c^5*d^5 + 64680*a^6*b^4*c^4*d^6 - 43680*a^7*b^3*c^3*d^7 + 18900*a^8*b^2*c^2*d
^8 - 4760*a^9*b*c*d^9 + 532*a^10*d^10 + 10*(8*b^10*c*d^9 - a*b^9*d^10)*x^9 + 15*(28*b^10*c^2*d^8 - 8*a*b^9*c*d
^9 + a^2*b^8*d^10)*x^8 + 24*(56*b^10*c^3*d^7 - 28*a*b^9*c^2*d^8 + 8*a^2*b^8*c*d^9 - a^3*b^7*d^10)*x^7 + 42*(70
*b^10*c^4*d^6 - 56*a*b^9*c^3*d^7 + 28*a^2*b^8*c^2*d^8 - 8*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^6 + 84*(56*b^10*c^5*
d^5 - 70*a*b^9*c^4*d^6 + 56*a^2*b^8*c^3*d^7 - 28*a^3*b^7*c^2*d^8 + 8*a^4*b^6*c*d^9 - a^5*b^5*d^10)*x^5 + 210*(
28*b^10*c^6*d^4 - 56*a*b^9*c^5*d^5 + 70*a^2*b^8*c^4*d^6 - 56*a^3*b^7*c^3*d^7 + 28*a^4*b^6*c^2*d^8 - 8*a^5*b^5*
c*d^9 + a^6*b^4*d^10)*x^4 + 840*(8*b^10*c^7*d^3 - 28*a*b^9*c^6*d^4 + 56*a^2*b^8*c^5*d^5 - 70*a^3*b^7*c^4*d^6 +
56*a^4*b^6*c^3*d^7 - 28*a^5*b^5*c^2*d^8 + 8*a^6*b^4*c*d^9 - a^7*b^3*d^10)*x^3 + 28*(480*a*b^9*c^7*d^3 - 2310*
a^2*b^8*c^6*d^4 + 5292*a^3*b^7*c^5*d^5 - 7140*a^4*b^6*c^4*d^6 + 6000*a^5*b^5*c^3*d^7 - 3105*a^6*b^4*c^2*d^8 +
910*a^7*b^3*c*d^9 - 116*a^8*b^2*d^10)*x^2 - 56*(10*b^10*c^9*d - 90*a*b^9*c^8*d^2 + 240*a^2*b^8*c^7*d^3 - 210*a
^3*b^7*c^6*d^4 - 252*a^4*b^6*c^5*d^5 + 840*a^5*b^5*c^4*d^6 - 960*a^6*b^4*c^3*d^7 + 585*a^7*b^3*c^2*d^8 - 190*a
^8*b^2*c*d^9 + 26*a^9*b*d^10)*x + 2520*(a^2*b^8*c^8*d^2 - 8*a^3*b^7*c^7*d^3 + 28*a^4*b^6*c^6*d^4 - 56*a^5*b^5*
c^5*d^5 + 70*a^6*b^4*c^4*d^6 - 56*a^7*b^3*c^3*d^7 + 28*a^8*b^2*c^2*d^8 - 8*a^9*b*c*d^9 + a^10*d^10 + (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 + 2
8*a^6*b^4*c^2*d^8 - 8*a^7*b^3*c*d^9 + a^8*b^2*d^10)*x^2 + 2*(a*b^9*c^8*d^2 - 8*a^2*b^8*c^7*d^3 + 28*a^3*b^7*c^
6*d^4 - 56*a^4*b^6*c^5*d^5 + 70*a^5*b^5*c^4*d^6 - 56*a^6*b^4*c^3*d^7 + 28*a^7*b^3*c^2*d^8 - 8*a^8*b^2*c*d^9 +
a^9*b*d^10)*x)*log(b*x + a))/(b^13*x^2 + 2*a*b^12*x + a^2*b^11)

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Sympy [B]  time = 6.57919, size = 828, normalized size = 3.16 \begin{align*} \frac{19 a^{10} d^{10} - 170 a^{9} b c d^{9} + 675 a^{8} b^{2} c^{2} d^{8} - 1560 a^{7} b^{3} c^{3} d^{7} + 2310 a^{6} b^{4} c^{4} d^{6} - 2268 a^{5} b^{5} c^{5} d^{5} + 1470 a^{4} b^{6} c^{6} d^{4} - 600 a^{3} b^{7} c^{7} d^{3} + 135 a^{2} b^{8} c^{8} d^{2} - 10 a b^{9} c^{9} d - b^{10} c^{10} + x \left (20 a^{9} b d^{10} - 180 a^{8} b^{2} c d^{9} + 720 a^{7} b^{3} c^{2} d^{8} - 1680 a^{6} b^{4} c^{3} d^{7} + 2520 a^{5} b^{5} c^{4} d^{6} - 2520 a^{4} b^{6} c^{5} d^{5} + 1680 a^{3} b^{7} c^{6} d^{4} - 720 a^{2} b^{8} c^{7} d^{3} + 180 a b^{9} c^{8} d^{2} - 20 b^{10} c^{9} d\right )}{2 a^{2} b^{11} + 4 a b^{12} x + 2 b^{13} x^{2}} + \frac{d^{10} x^{8}}{8 b^{3}} - \frac{x^{7} \left (3 a d^{10} - 10 b c d^{9}\right )}{7 b^{4}} + \frac{x^{6} \left (2 a^{2} d^{10} - 10 a b c d^{9} + 15 b^{2} c^{2} d^{8}\right )}{2 b^{5}} - \frac{x^{5} \left (2 a^{3} d^{10} - 12 a^{2} b c d^{9} + 27 a b^{2} c^{2} d^{8} - 24 b^{3} c^{3} d^{7}\right )}{b^{6}} + \frac{x^{4} \left (15 a^{4} d^{10} - 100 a^{3} b c d^{9} + 270 a^{2} b^{2} c^{2} d^{8} - 360 a b^{3} c^{3} d^{7} + 210 b^{4} c^{4} d^{6}\right )}{4 b^{7}} - \frac{x^{3} \left (7 a^{5} d^{10} - 50 a^{4} b c d^{9} + 150 a^{3} b^{2} c^{2} d^{8} - 240 a^{2} b^{3} c^{3} d^{7} + 210 a b^{4} c^{4} d^{6} - 84 b^{5} c^{5} d^{5}\right )}{b^{8}} + \frac{x^{2} \left (28 a^{6} d^{10} - 210 a^{5} b c d^{9} + 675 a^{4} b^{2} c^{2} d^{8} - 1200 a^{3} b^{3} c^{3} d^{7} + 1260 a^{2} b^{4} c^{4} d^{6} - 756 a b^{5} c^{5} d^{5} + 210 b^{6} c^{6} d^{4}\right )}{2 b^{9}} - \frac{x \left (36 a^{7} d^{10} - 280 a^{6} b c d^{9} + 945 a^{5} b^{2} c^{2} d^{8} - 1800 a^{4} b^{3} c^{3} d^{7} + 2100 a^{3} b^{4} c^{4} d^{6} - 1512 a^{2} b^{5} c^{5} d^{5} + 630 a b^{6} c^{6} d^{4} - 120 b^{7} c^{7} d^{3}\right )}{b^{10}} + \frac{45 d^{2} \left (a d - b c\right )^{8} \log{\left (a + b x \right )}}{b^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**10/(b*x+a)**3,x)

[Out]

(19*a**10*d**10 - 170*a**9*b*c*d**9 + 675*a**8*b**2*c**2*d**8 - 1560*a**7*b**3*c**3*d**7 + 2310*a**6*b**4*c**4
*d**6 - 2268*a**5*b**5*c**5*d**5 + 1470*a**4*b**6*c**6*d**4 - 600*a**3*b**7*c**7*d**3 + 135*a**2*b**8*c**8*d**
2 - 10*a*b**9*c**9*d - b**10*c**10 + x*(20*a**9*b*d**10 - 180*a**8*b**2*c*d**9 + 720*a**7*b**3*c**2*d**8 - 168
0*a**6*b**4*c**3*d**7 + 2520*a**5*b**5*c**4*d**6 - 2520*a**4*b**6*c**5*d**5 + 1680*a**3*b**7*c**6*d**4 - 720*a
**2*b**8*c**7*d**3 + 180*a*b**9*c**8*d**2 - 20*b**10*c**9*d))/(2*a**2*b**11 + 4*a*b**12*x + 2*b**13*x**2) + d*
*10*x**8/(8*b**3) - x**7*(3*a*d**10 - 10*b*c*d**9)/(7*b**4) + x**6*(2*a**2*d**10 - 10*a*b*c*d**9 + 15*b**2*c**
2*d**8)/(2*b**5) - x**5*(2*a**3*d**10 - 12*a**2*b*c*d**9 + 27*a*b**2*c**2*d**8 - 24*b**3*c**3*d**7)/b**6 + x**
4*(15*a**4*d**10 - 100*a**3*b*c*d**9 + 270*a**2*b**2*c**2*d**8 - 360*a*b**3*c**3*d**7 + 210*b**4*c**4*d**6)/(4
*b**7) - x**3*(7*a**5*d**10 - 50*a**4*b*c*d**9 + 150*a**3*b**2*c**2*d**8 - 240*a**2*b**3*c**3*d**7 + 210*a*b**
4*c**4*d**6 - 84*b**5*c**5*d**5)/b**8 + x**2*(28*a**6*d**10 - 210*a**5*b*c*d**9 + 675*a**4*b**2*c**2*d**8 - 12
00*a**3*b**3*c**3*d**7 + 1260*a**2*b**4*c**4*d**6 - 756*a*b**5*c**5*d**5 + 210*b**6*c**6*d**4)/(2*b**9) - x*(3
6*a**7*d**10 - 280*a**6*b*c*d**9 + 945*a**5*b**2*c**2*d**8 - 1800*a**4*b**3*c**3*d**7 + 2100*a**3*b**4*c**4*d*
*6 - 1512*a**2*b**5*c**5*d**5 + 630*a*b**6*c**6*d**4 - 120*b**7*c**7*d**3)/b**10 + 45*d**2*(a*d - b*c)**8*log(
a + b*x)/b**11

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Giac [B]  time = 1.04931, size = 1247, normalized size = 4.76 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^10/(b*x+a)^3,x, algorithm="giac")

[Out]

45*(b^8*c^8*d^2 - 8*a*b^7*c^7*d^3 + 28*a^2*b^6*c^6*d^4 - 56*a^3*b^5*c^5*d^5 + 70*a^4*b^4*c^4*d^6 - 56*a^5*b^3*
c^3*d^7 + 28*a^6*b^2*c^2*d^8 - 8*a^7*b*c*d^9 + a^8*d^10)*log(abs(b*x + a))/b^11 - 1/2*(b^10*c^10 + 10*a*b^9*c^
9*d - 135*a^2*b^8*c^8*d^2 + 600*a^3*b^7*c^7*d^3 - 1470*a^4*b^6*c^6*d^4 + 2268*a^5*b^5*c^5*d^5 - 2310*a^6*b^4*c
^4*d^6 + 1560*a^7*b^3*c^3*d^7 - 675*a^8*b^2*c^2*d^8 + 170*a^9*b*c*d^9 - 19*a^10*d^10 + 20*(b^10*c^9*d - 9*a*b^
9*c^8*d^2 + 36*a^2*b^8*c^7*d^3 - 84*a^3*b^7*c^6*d^4 + 126*a^4*b^6*c^5*d^5 - 126*a^5*b^5*c^4*d^6 + 84*a^6*b^4*c
^3*d^7 - 36*a^7*b^3*c^2*d^8 + 9*a^8*b^2*c*d^9 - a^9*b*d^10)*x)/((b*x + a)^2*b^11) + 1/56*(7*b^21*d^10*x^8 + 80
*b^21*c*d^9*x^7 - 24*a*b^20*d^10*x^7 + 420*b^21*c^2*d^8*x^6 - 280*a*b^20*c*d^9*x^6 + 56*a^2*b^19*d^10*x^6 + 13
44*b^21*c^3*d^7*x^5 - 1512*a*b^20*c^2*d^8*x^5 + 672*a^2*b^19*c*d^9*x^5 - 112*a^3*b^18*d^10*x^5 + 2940*b^21*c^4
*d^6*x^4 - 5040*a*b^20*c^3*d^7*x^4 + 3780*a^2*b^19*c^2*d^8*x^4 - 1400*a^3*b^18*c*d^9*x^4 + 210*a^4*b^17*d^10*x
^4 + 4704*b^21*c^5*d^5*x^3 - 11760*a*b^20*c^4*d^6*x^3 + 13440*a^2*b^19*c^3*d^7*x^3 - 8400*a^3*b^18*c^2*d^8*x^3
+ 2800*a^4*b^17*c*d^9*x^3 - 392*a^5*b^16*d^10*x^3 + 5880*b^21*c^6*d^4*x^2 - 21168*a*b^20*c^5*d^5*x^2 + 35280*
a^2*b^19*c^4*d^6*x^2 - 33600*a^3*b^18*c^3*d^7*x^2 + 18900*a^4*b^17*c^2*d^8*x^2 - 5880*a^5*b^16*c*d^9*x^2 + 784
*a^6*b^15*d^10*x^2 + 6720*b^21*c^7*d^3*x - 35280*a*b^20*c^6*d^4*x + 84672*a^2*b^19*c^5*d^5*x - 117600*a^3*b^18
*c^4*d^6*x + 100800*a^4*b^17*c^3*d^7*x - 52920*a^5*b^16*c^2*d^8*x + 15680*a^6*b^15*c*d^9*x - 2016*a^7*b^14*d^1
0*x)/b^24