3.1315 \(\int \frac{(c+d x)^{10}}{(a+b x)^4} \, dx\)

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

[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

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Rubi [A]  time = 0.959487, antiderivative size = 258, 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 \[ \frac{5 d^9 (a+b x)^6 (b c-a d)}{3 b^{11}}+\frac{9 d^8 (a+b x)^5 (b c-a d)^2}{b^{11}}+\frac{30 d^7 (a+b x)^4 (b c-a d)^3}{b^{11}}+\frac{70 d^6 (a+b x)^3 (b c-a d)^4}{b^{11}}+\frac{126 d^5 (a+b x)^2 (b c-a d)^5}{b^{11}}+\frac{120 d^3 (b c-a d)^7 \log (a+b x)}{b^{11}}-\frac{45 d^2 (b c-a d)^8}{b^{11} (a+b x)}-\frac{5 d (b c-a d)^9}{b^{11} (a+b x)^2}-\frac{(b c-a d)^{10}}{3 b^{11} (a+b x)^3}+\frac{d^{10} (a+b x)^7}{7 b^{11}}+\frac{210 d^4 x (b c-a d)^6}{b^{10}} \]

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

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Rubi in Sympy [A]  time = 143.012, size = 240, normalized size = 0.93 \[ \frac{210 d^{4} x \left (a d - b c\right )^{6}}{b^{10}} + \frac{d^{10} \left (a + b x\right )^{7}}{7 b^{11}} - \frac{5 d^{9} \left (a + b x\right )^{6} \left (a d - b c\right )}{3 b^{11}} + \frac{9 d^{8} \left (a + b x\right )^{5} \left (a d - b c\right )^{2}}{b^{11}} - \frac{30 d^{7} \left (a + b x\right )^{4} \left (a d - b c\right )^{3}}{b^{11}} + \frac{70 d^{6} \left (a + b x\right )^{3} \left (a d - b c\right )^{4}}{b^{11}} - \frac{126 d^{5} \left (a + b x\right )^{2} \left (a d - b c\right )^{5}}{b^{11}} - \frac{120 d^{3} \left (a d - b c\right )^{7} \log{\left (a + b x \right )}}{b^{11}} - \frac{45 d^{2} \left (a d - b c\right )^{8}}{b^{11} \left (a + b x\right )} + \frac{5 d \left (a d - b c\right )^{9}}{b^{11} \left (a + b x\right )^{2}} - \frac{\left (a d - b c\right )^{10}}{3 b^{11} \left (a + b x\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**10/(b*x+a)**4,x)

[Out]

210*d**4*x*(a*d - b*c)**6/b**10 + d**10*(a + b*x)**7/(7*b**11) - 5*d**9*(a + b*x
)**6*(a*d - b*c)/(3*b**11) + 9*d**8*(a + b*x)**5*(a*d - b*c)**2/b**11 - 30*d**7*
(a + b*x)**4*(a*d - b*c)**3/b**11 + 70*d**6*(a + b*x)**3*(a*d - b*c)**4/b**11 -
126*d**5*(a + b*x)**2*(a*d - b*c)**5/b**11 - 120*d**3*(a*d - b*c)**7*log(a + b*x
)/b**11 - 45*d**2*(a*d - b*c)**8/(b**11*(a + b*x)) + 5*d*(a*d - b*c)**9/(b**11*(
a + b*x)**2) - (a*d - b*c)**10/(3*b**11*(a + b*x)**3)

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Mathematica [A]  time = 0.296313, size = 427, normalized size = 1.66 \[ \frac{21 b^5 d^8 x^5 \left (2 a^2 d^2-8 a b c d+9 b^2 c^2\right )+105 b^4 d^7 x^4 \left (-a^3 d^3+5 a^2 b c d^2-9 a b^2 c^2 d+6 b^3 c^3\right )+35 b^3 d^6 x^3 \left (7 a^4 d^4-40 a^3 b c d^3+90 a^2 b^2 c^2 d^2-96 a b^3 c^3 d+42 b^4 c^4\right )+21 b^2 d^5 x^2 \left (-28 a^5 d^5+175 a^4 b c d^4-450 a^3 b^2 c^2 d^3+600 a^2 b^3 c^3 d^2-420 a b^4 c^4 d+126 b^5 c^5\right )+21 b d^4 x \left (84 a^6 d^6-560 a^5 b c d^5+1575 a^4 b^2 c^2 d^4-2400 a^3 b^3 c^3 d^3+2100 a^2 b^4 c^4 d^2-1008 a b^5 c^5 d+210 b^6 c^6\right )+7 b^6 d^9 x^6 (5 b c-2 a d)+2520 d^3 (b c-a d)^7 \log (a+b x)-\frac{945 d^2 (b c-a d)^8}{a+b x}+\frac{105 d (a d-b c)^9}{(a+b x)^2}-\frac{7 (b c-a d)^{10}}{(a+b x)^3}+3 b^7 d^{10} x^7}{21 b^{11}} \]

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*(1
26*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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Maple [B]  time = 0.027, size = 1141, normalized size = 4.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^10/(b*x+a)^4,x)

[Out]

-120/b^11*d^10*ln(b*x+a)*a^7+120/b^4*d^3*ln(b*x+a)*c^7-1/3/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^9-45/b^11*d^10/(b*x+a)*a^8-45/
b^3*d^2/(b*x+a)*c^8-28*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+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+210*d^4/b^4*c^6*x+84*d^10/b
^10*a^6*x-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+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+4
0/b^8/(b*x+a)^3*a^7*c^3*d^7-70/b^7/(b*x+a)^3*a^6*c^4*d^6+84/b^6/(b*x+a)^3*a^5*c^
5*d^5-70/b^5/(b*x+a)^3*a^4*c^6*d^4-2400*d^7/b^7*a^3*c^3*x+2100*d^6/b^6*a^2*c^4*x
-8*d^9/b^5*x^5*a*c-1008*d^5/b^5*a*c^5*x-200/3*d^9/b^7*x^3*a^3*c+150*d^8/b^6*x^3*
a^2*c^2-560*d^9/b^9*a^5*c*x+1575*d^8/b^8*a^4*c^2*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^4-160*d^7/b^5*x^3*a*c^3+175*d^9/b^8*x^2*a
^4*c-45*d^8/b^5*x^4*a*c^2+25*d^9/b^6*x^4*a^2*c+10/3/b^2/(b*x+a)^3*a*c^9*d-45/b^1
0*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+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+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+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+1/7*d^10/b^4*x^7-1/3/b/(b*x+a)^3*c^10

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Maxima [A]  time = 1.41311, size = 1203, normalized size = 4.66 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^10/(b*x + a)^4,x, algorithm="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 + 273
0*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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Fricas [A]  time = 0.208774, size = 1777, normalized size = 6.89 \[ \text{result too large to display} \]

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 + 46
20*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^10*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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Sympy [A]  time = 89.0359, size = 853, normalized size = 3.31 \[ - \frac{121 a^{10} d^{10} - 955 a^{9} b c d^{9} + 3285 a^{8} b^{2} c^{2} d^{8} - 6420 a^{7} b^{3} c^{3} d^{7} + 7770 a^{6} b^{4} c^{4} d^{6} - 5922 a^{5} b^{5} c^{5} d^{5} + 2730 a^{4} b^{6} c^{6} d^{4} - 660 a^{3} b^{7} c^{7} d^{3} + 45 a^{2} b^{8} c^{8} d^{2} + 5 a b^{9} c^{9} d + b^{10} c^{10} + x^{2} \left (135 a^{8} b^{2} d^{10} - 1080 a^{7} b^{3} c d^{9} + 3780 a^{6} b^{4} c^{2} d^{8} - 7560 a^{5} b^{5} c^{3} d^{7} + 9450 a^{4} b^{6} c^{4} d^{6} - 7560 a^{3} b^{7} c^{5} d^{5} + 3780 a^{2} b^{8} c^{6} d^{4} - 1080 a b^{9} c^{7} d^{3} + 135 b^{10} c^{8} d^{2}\right ) + x \left (255 a^{9} b d^{10} - 2025 a^{8} b^{2} c d^{9} + 7020 a^{7} b^{3} c^{2} d^{8} - 13860 a^{6} b^{4} c^{3} d^{7} + 17010 a^{5} b^{5} c^{4} d^{6} - 13230 a^{4} b^{6} c^{5} d^{5} + 6300 a^{3} b^{7} c^{6} d^{4} - 1620 a^{2} b^{8} c^{7} d^{3} + 135 a b^{9} c^{8} d^{2} + 15 b^{10} c^{9} d\right )}{3 a^{3} b^{11} + 9 a^{2} b^{12} x + 9 a b^{13} x^{2} + 3 b^{14} x^{3}} + \frac{d^{10} x^{7}}{7 b^{4}} - \frac{x^{6} \left (2 a d^{10} - 5 b c d^{9}\right )}{3 b^{5}} + \frac{x^{5} \left (2 a^{2} d^{10} - 8 a b c d^{9} + 9 b^{2} c^{2} d^{8}\right )}{b^{6}} - \frac{x^{4} \left (5 a^{3} d^{10} - 25 a^{2} b c d^{9} + 45 a b^{2} c^{2} d^{8} - 30 b^{3} c^{3} d^{7}\right )}{b^{7}} + \frac{x^{3} \left (35 a^{4} d^{10} - 200 a^{3} b c d^{9} + 450 a^{2} b^{2} c^{2} d^{8} - 480 a b^{3} c^{3} d^{7} + 210 b^{4} c^{4} d^{6}\right )}{3 b^{8}} - \frac{x^{2} \left (28 a^{5} d^{10} - 175 a^{4} b c d^{9} + 450 a^{3} b^{2} c^{2} d^{8} - 600 a^{2} b^{3} c^{3} d^{7} + 420 a b^{4} c^{4} d^{6} - 126 b^{5} c^{5} d^{5}\right )}{b^{9}} + \frac{x \left (84 a^{6} d^{10} - 560 a^{5} b c d^{9} + 1575 a^{4} b^{2} c^{2} d^{8} - 2400 a^{3} b^{3} c^{3} d^{7} + 2100 a^{2} b^{4} c^{4} d^{6} - 1008 a b^{5} c^{5} d^{5} + 210 b^{6} c^{6} d^{4}\right )}{b^{10}} - \frac{120 d^{3} \left (a d - b c\right )^{7} \log{\left (a + b x \right )}}{b^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(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*(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**1
4*x**3) + d**10*x**7/(7*b**4) - x**6*(2*a*d**10 - 5*b*c*d**9)/(3*b**5) + x**5*(2
*a**2*d**10 - 8*a*b*c*d**9 + 9*b**2*c**2*d**8)/b**6 - x**4*(5*a**3*d**10 - 25*a*
*2*b*c*d**9 + 45*a*b**2*c**2*d**8 - 30*b**3*c**3*d**7)/b**7 + x**3*(35*a**4*d**1
0 - 200*a**3*b*c*d**9 + 450*a**2*b**2*c**2*d**8 - 480*a*b**3*c**3*d**7 + 210*b**
4*c**4*d**6)/(3*b**8) - x**2*(28*a**5*d**10 - 175*a**4*b*c*d**9 + 450*a**3*b**2*
c**2*d**8 - 600*a**2*b**3*c**3*d**7 + 420*a*b**4*c**4*d**6 - 126*b**5*c**5*d**5)
/b**9 + x*(84*a**6*d**10 - 560*a**5*b*c*d**9 + 1575*a**4*b**2*c**2*d**8 - 2400*a
**3*b**3*c**3*d**7 + 2100*a**2*b**4*c**4*d**6 - 1008*a*b**5*c**5*d**5 + 210*b**6
*c**6*d**4)/b**10 - 120*d**3*(a*d - b*c)**7*log(a + b*x)/b**11

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GIAC/XCAS [A]  time = 0.225648, size = 1224, normalized size = 4.74 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^10/(b*x + a)^4,x, algorithm="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 + 3
5*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)*ln(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