3.14.15 \(\int \frac {(c+d x)^{10}}{(a+b x)^4} \, dx\) [1315]

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

[Out]

210*d^4*(-a*d+b*c)^6*x/b^10-1/3*(-a*d+b*c)^10/b^11/(b*x+a)^3-5*d*(-a*d+b*c)^9/b^11/(b*x+a)^2-45*d^2*(-a*d+b*c)
^8/b^11/(b*x+a)+126*d^5*(-a*d+b*c)^5*(b*x+a)^2/b^11+70*d^6*(-a*d+b*c)^4*(b*x+a)^3/b^11+30*d^7*(-a*d+b*c)^3*(b*
x+a)^4/b^11+9*d^8*(-a*d+b*c)^2*(b*x+a)^5/b^11+5/3*d^9*(-a*d+b*c)*(b*x+a)^6/b^11+1/7*d^10*(b*x+a)^7/b^11+120*d^
3*(-a*d+b*c)^7*ln(b*x+a)/b^11

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Rubi [A]
time = 0.30, 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 = {45} \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 {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}} \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 45

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 = 0.11, size = 427, normalized size = 1.66 \begin {gather*} \frac {21 b d^4 \left (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\right ) x+21 b^2 d^5 \left (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\right ) x^2+35 b^3 d^6 \left (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\right ) x^3+105 b^4 d^7 \left (6 b^3 c^3-9 a b^2 c^2 d+5 a^2 b c d^2-a^3 d^3\right ) x^4+21 b^5 d^8 \left (9 b^2 c^2-8 a b c d+2 a^2 d^2\right ) x^5+7 b^6 d^9 (5 b c-2 a d) x^6+3 b^7 d^{10} x^7-\frac {7 (b c-a d)^{10}}{(a+b x)^3}+\frac {105 d (-b c+a d)^9}{(a+b x)^2}-\frac {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}} \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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Mathics [B] Leaf count is larger than twice the leaf count of optimal. \(987\) vs. \(2(258)=516\).
time = 28.97, size = 985, normalized size = 3.82 \begin {gather*} \frac {-2520 d^3 \text {Log}\left [a+b x\right ] \left (a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3\right ) \left (a d-b c\right )^7-847 a^{10} d^{10}+6685 a^9 b c d^9-22995 a^8 b^2 c^2 d^8+44940 a^7 b^3 c^3 d^7-54390 a^6 b^4 c^4 d^6+41454 a^5 b^5 c^5 d^5-19110 a^4 b^6 c^6 d^4+4620 a^3 b^7 c^7 d^3-315 a^2 b^8 c^8 d^2-35 a b^9 c^9 d-7 b^{10} c^{10}-105 b d x \left (17 a^9 d^9-135 a^8 b c d^8+468 a^7 b^2 c^2 d^7-924 a^6 b^3 c^3 d^6+1134 a^5 b^4 c^4 d^5-882 a^4 b^5 c^5 d^4+420 a^3 b^6 c^6 d^3-108 a^2 b^7 c^7 d^2+9 a b^8 c^8 d+b^9 c^9\right )+945 b^2 d^2 x^2 \left (-a^8 d^8+8 a^7 b c d^7-28 a^6 b^2 c^2 d^6+56 a^5 b^3 c^3 d^5-70 a^4 b^4 c^4 d^4+56 a^3 b^5 c^5 d^3-28 a^2 b^6 c^6 d^2+8 a b^7 c^7 d-b^8 c^8\right )+21 b d^4 x \left (a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3\right ) \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 )-21 b^2 d^5 x^2 \left (a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3\right ) \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 )+35 b^3 d^6 x^3 \left (a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3\right ) \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 )-105 b^4 d^7 x^4 \left (a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3\right ) \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 )+21 b^5 d^8 x^5 \left (a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3\right ) \left (2 a^2 d^2-8 a b c d+9 b^2 c^2\right )-7 b^6 d^9 x^6 \left (a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3\right ) \left (2 a d-5 b c\right )+3 b^7 d^{10} x^7 \left (a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3\right )}{21 b^{11} \left (a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2520 d ^ 3 Log[a + b x] (a ^ 3 + 3 a ^ 2 b x + 3 a b ^ 2 x ^ 2 + b ^ 3 x ^ 3) (a d - b c) ^ 7 - 847 a ^ 10 d
^ 10 + 6685 a ^ 9 b c d ^ 9 - 22995 a ^ 8 b ^ 2 c ^ 2 d ^ 8 + 44940 a ^ 7 b ^ 3 c ^ 3 d ^ 7 - 54390 a ^ 6 b ^
4 c ^ 4 d ^ 6 + 41454 a ^ 5 b ^ 5 c ^ 5 d ^ 5 - 19110 a ^ 4 b ^ 6 c ^ 6 d ^ 4 + 4620 a ^ 3 b ^ 7 c ^ 7 d ^ 3
- 315 a ^ 2 b ^ 8 c ^ 8 d ^ 2 - 35 a b ^ 9 c ^ 9 d - 7 b ^ 10 c ^ 10 - 105 b d x (17 a ^ 9 d ^ 9 - 135 a ^ 8 b
c d ^ 8 + 468 a ^ 7 b ^ 2 c ^ 2 d ^ 7 - 924 a ^ 6 b ^ 3 c ^ 3 d ^ 6 + 1134 a ^ 5 b ^ 4 c ^ 4 d ^ 5 - 882 a ^
4 b ^ 5 c ^ 5 d ^ 4 + 420 a ^ 3 b ^ 6 c ^ 6 d ^ 3 - 108 a ^ 2 b ^ 7 c ^ 7 d ^ 2 + 9 a b ^ 8 c ^ 8 d + b ^ 9 c
^ 9) + 945 b ^ 2 d ^ 2 x ^ 2 (-a ^ 8 d ^ 8 + 8 a ^ 7 b c d ^ 7 - 28 a ^ 6 b ^ 2 c ^ 2 d ^ 6 + 56 a ^ 5 b ^ 3 c
^ 3 d ^ 5 - 70 a ^ 4 b ^ 4 c ^ 4 d ^ 4 + 56 a ^ 3 b ^ 5 c ^ 5 d ^ 3 - 28 a ^ 2 b ^ 6 c ^ 6 d ^ 2 + 8 a b ^ 7
c ^ 7 d - b ^ 8 c ^ 8) + 21 b d ^ 4 x (a ^ 3 + 3 a ^ 2 b x + 3 a b ^ 2 x ^ 2 + b ^ 3 x ^ 3) (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) - 21 b ^ 2 d ^ 5 x ^ 2 (a ^ 3 + 3 a ^ 2 b x + 3 a b ^ 2 x ^ 2 + b
^ 3 x ^ 3) (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) + 35 b ^ 3 d ^ 6 x ^ 3 (a ^ 3 + 3 a ^ 2 b x + 3 a b ^ 2 x ^ 2 + b ^ 3
x ^ 3) (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
) - 105 b ^ 4 d ^ 7 x ^ 4 (a ^ 3 + 3 a ^ 2 b x + 3 a b ^ 2 x ^ 2 + b ^ 3 x ^ 3) (a ^ 3 d ^ 3 - 5 a ^ 2 b c d ^
2 + 9 a b ^ 2 c ^ 2 d - 6 b ^ 3 c ^ 3) + 21 b ^ 5 d ^ 8 x ^ 5 (a ^ 3 + 3 a ^ 2 b x + 3 a b ^ 2 x ^ 2 + b ^ 3
x ^ 3) (2 a ^ 2 d ^ 2 - 8 a b c d + 9 b ^ 2 c ^ 2) - 7 b ^ 6 d ^ 9 x ^ 6 (a ^ 3 + 3 a ^ 2 b x + 3 a b ^ 2 x ^
2 + b ^ 3 x ^ 3) (2 a d - 5 b c) + 3 b ^ 7 d ^ 10 x ^ 7 (a ^ 3 + 3 a ^ 2 b x + 3 a b ^ 2 x ^ 2 + b ^ 3 x ^ 3))
/ (21 b ^ 11 (a ^ 3 + 3 a ^ 2 b x + 3 a b ^ 2 x ^ 2 + b ^ 3 x ^ 3))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(895\) vs. \(2(252)=504\).
time = 0.17, size = 896, normalized size = 3.47

method	result	size
norman	\(\frac {-\frac {660 a^{10} d^{10}-4620 a^{9} b c \,d^{9}+13860 a^{8} b^{2} c^{2} d^{8}-23100 a^{7} b^{3} c^{3} d^{7}+23100 a^{6} b^{4} c^{4} d^{6}-13860 a^{5} b^{5} c^{5} d^{5}+4620 a^{4} b^{6} c^{6} d^{4}-660 a^{3} b^{7} c^{7} d^{3}+45 a^{2} b^{8} c^{8} d^{2}+5 a \,b^{9} c^{9} d +b^{10} c^{10}}{3 b^{11}}+\frac {d^{10} x^{10}}{7 b}-\frac {3 \left (120 a^{8} d^{10}-840 a^{7} b c \,d^{9}+2520 a^{6} b^{2} c^{2} d^{8}-4200 a^{5} b^{3} c^{3} d^{7}+4200 a^{4} b^{4} c^{4} d^{6}-2520 a^{3} b^{5} c^{5} d^{5}+840 b^{6} a^{2} c^{6} d^{4}-120 a \,b^{7} c^{7} d^{3}+15 b^{8} c^{8} d^{2}\right ) x^{2}}{b^{9}}-\frac {\left (540 a^{9} d^{10}-3780 a^{8} b c \,d^{9}+11340 a^{7} b^{2} c^{2} d^{8}-18900 a^{6} b^{3} c^{3} d^{7}+18900 a^{5} b^{4} c^{4} d^{6}-11340 a^{4} b^{5} c^{5} d^{5}+3780 a^{3} b^{6} c^{6} d^{4}-540 a^{2} b^{7} c^{7} d^{3}+45 a \,b^{8} c^{8} d^{2}+5 b^{9} c^{9} d \right ) x}{b^{10}}+\frac {30 d^{4} \left (a^{6} d^{6}-7 a^{5} b c \,d^{5}+21 a^{4} b^{2} c^{2} d^{4}-35 a^{3} b^{3} c^{3} d^{3}+35 a^{2} b^{4} c^{4} d^{2}-21 a \,b^{5} c^{5} d +7 b^{6} c^{6}\right ) x^{4}}{b^{7}}-\frac {6 d^{5} \left (a^{5} d^{5}-7 a^{4} b c \,d^{4}+21 a^{3} b^{2} c^{2} d^{3}-35 a^{2} b^{3} c^{3} d^{2}+35 a \,b^{4} c^{4} d -21 b^{5} c^{5}\right ) x^{5}}{b^{6}}+\frac {2 d^{6} \left (a^{4} d^{4}-7 a^{3} b c \,d^{3}+21 a^{2} b^{2} c^{2} d^{2}-35 a \,b^{3} c^{3} d +35 b^{4} c^{4}\right ) x^{6}}{b^{5}}-\frac {6 d^{7} \left (a^{3} d^{3}-7 a^{2} b c \,d^{2}+21 a \,b^{2} c^{2} d -35 b^{3} c^{3}\right ) x^{7}}{7 b^{4}}+\frac {3 d^{8} \left (a^{2} d^{2}-7 a b c d +21 b^{2} c^{2}\right ) x^{8}}{7 b^{3}}-\frac {5 d^{9} \left (a d -7 b c \right ) x^{9}}{21 b^{2}}}{\left (b x +a \right )^{3}}-\frac {120 d^{3} \left (a^{7} d^{7}-7 a^{6} b c \,d^{6}+21 a^{5} b^{2} c^{2} d^{5}-35 a^{4} b^{3} c^{3} d^{4}+35 a^{3} b^{4} c^{4} d^{3}-21 a^{2} b^{5} c^{5} d^{2}+7 a \,b^{6} c^{6} d -b^{7} c^{7}\right ) \ln \left (b x +a \right )}{b^{11}}\)	\(841\)
default	\(\frac {d^{4} \left (-\frac {2}{3} a \,b^{5} d^{6} x^{6}+\frac {5}{3} b^{6} c \,d^{5} x^{6}+2 a^{2} b^{4} d^{6} x^{5}+9 b^{6} c^{2} d^{4} x^{5}-5 a^{3} b^{3} d^{6} x^{4}+30 b^{6} c^{3} d^{3} x^{4}+\frac {35}{3} a^{4} b^{2} d^{6} x^{3}+70 b^{6} c^{4} d^{2} x^{3}-28 a^{5} b \,d^{6} x^{2}+\frac {1}{7} d^{6} x^{7} b^{6}+84 a^{6} d^{6} x +210 b^{6} c^{6} x -45 a \,b^{5} c^{2} d^{4} x^{4}-\frac {200}{3} a^{3} b^{3} c \,d^{5} x^{3}+150 a^{2} b^{4} c^{2} d^{4} x^{3}-160 a \,b^{5} c^{3} d^{3} x^{3}+175 a^{4} b^{2} c \,d^{5} x^{2}-450 a^{3} b^{3} c^{2} d^{4} x^{2}+600 a^{2} b^{4} c^{3} d^{3} x^{2}-420 a \,b^{5} c^{4} d^{2} x^{2}-560 a^{5} b c \,d^{5} x +1575 a^{4} b^{2} c^{2} d^{4} x -2400 a^{3} b^{3} c^{3} d^{3} x +2100 a^{2} b^{4} c^{4} d^{2} x -1008 a \,b^{5} c^{5} d x +126 b^{6} c^{5} d \,x^{2}-8 a \,b^{5} c \,d^{5} x^{5}+25 a^{2} b^{4} c \,d^{5} x^{4}\right )}{b^{10}}-\frac {45 d^{2} \left (a^{8} d^{8}-8 a^{7} b c \,d^{7}+28 a^{6} b^{2} c^{2} d^{6}-56 a^{5} b^{3} c^{3} d^{5}+70 a^{4} b^{4} c^{4} d^{4}-56 a^{3} b^{5} c^{5} d^{3}+28 a^{2} b^{6} c^{6} d^{2}-8 a \,b^{7} c^{7} d +b^{8} c^{8}\right )}{b^{11} \left (b x +a \right )}+\frac {5 d \left (a^{9} d^{9}-9 a^{8} b c \,d^{8}+36 a^{7} b^{2} c^{2} d^{7}-84 a^{6} b^{3} c^{3} d^{6}+126 a^{5} b^{4} c^{4} d^{5}-126 a^{4} b^{5} c^{5} d^{4}+84 a^{3} b^{6} c^{6} d^{3}-36 a^{2} b^{7} c^{7} d^{2}+9 a \,b^{8} c^{8} d -b^{9} c^{9}\right )}{b^{11} \left (b x +a \right )^{2}}-\frac {120 d^{3} \left (a^{7} d^{7}-7 a^{6} b c \,d^{6}+21 a^{5} b^{2} c^{2} d^{5}-35 a^{4} b^{3} c^{3} d^{4}+35 a^{3} b^{4} c^{4} d^{3}-21 a^{2} b^{5} c^{5} d^{2}+7 a \,b^{6} c^{6} d -b^{7} c^{7}\right ) \ln \left (b x +a \right )}{b^{11}}-\frac {a^{10} d^{10}-10 a^{9} b c \,d^{9}+45 a^{8} b^{2} c^{2} d^{8}-120 a^{7} b^{3} c^{3} d^{7}+210 a^{6} b^{4} c^{4} d^{6}-252 a^{5} b^{5} c^{5} d^{5}+210 a^{4} b^{6} c^{6} d^{4}-120 a^{3} b^{7} c^{7} d^{3}+45 a^{2} b^{8} c^{8} d^{2}-10 a \,b^{9} c^{9} d +b^{10} c^{10}}{3 b^{11} \left (b x +a \right )^{3}}\)	\(896\)
risch	\(-\frac {420 d^{6} a \,c^{4} x^{2}}{b^{5}}-\frac {560 d^{9} a^{5} c x}{b^{9}}+\frac {1575 d^{8} a^{4} c^{2} x}{b^{8}}-\frac {2400 d^{7} a^{3} c^{3} x}{b^{7}}+\frac {2100 d^{6} a^{2} c^{4} x}{b^{6}}-\frac {1008 d^{5} a \,c^{5} x}{b^{5}}-\frac {8 d^{9} a c \,x^{5}}{b^{5}}+\frac {25 d^{9} a^{2} c \,x^{4}}{b^{6}}+\frac {150 d^{8} a^{2} c^{2} x^{3}}{b^{6}}-\frac {160 d^{7} a \,c^{3} x^{3}}{b^{5}}+\frac {175 d^{9} a^{4} c \,x^{2}}{b^{8}}-\frac {450 d^{8} a^{3} c^{2} x^{2}}{b^{7}}+\frac {600 d^{7} a^{2} c^{3} x^{2}}{b^{6}}-\frac {45 d^{8} a \,c^{2} x^{4}}{b^{5}}-\frac {200 d^{9} a^{3} c \,x^{3}}{3 b^{7}}+\frac {\left (-45 a^{8} b \,d^{10}+360 a^{7} b^{2} c \,d^{9}-1260 a^{6} b^{3} c^{2} d^{8}+2520 a^{5} b^{4} c^{3} d^{7}-3150 a^{4} b^{5} c^{4} d^{6}+2520 a^{3} b^{6} c^{5} d^{5}-1260 a^{2} b^{7} c^{6} d^{4}+360 a \,b^{8} c^{7} d^{3}-45 b^{9} c^{8} d^{2}\right ) x^{2}-5 d \left (17 a^{9} d^{9}-135 a^{8} b c \,d^{8}+468 a^{7} b^{2} c^{2} d^{7}-924 a^{6} b^{3} c^{3} d^{6}+1134 a^{5} b^{4} c^{4} d^{5}-882 a^{4} b^{5} c^{5} d^{4}+420 a^{3} b^{6} c^{6} d^{3}-108 a^{2} b^{7} c^{7} d^{2}+9 a \,b^{8} c^{8} d +b^{9} c^{9}\right ) x -\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}}{3 b}}{b^{10} \left (b x +a \right )^{3}}+\frac {840 d^{9} \ln \left (b x +a \right ) a^{6} c}{b^{10}}-\frac {2520 d^{8} \ln \left (b x +a \right ) a^{5} c^{2}}{b^{9}}+\frac {4200 d^{7} \ln \left (b x +a \right ) a^{4} c^{3}}{b^{8}}-\frac {4200 d^{6} \ln \left (b x +a \right ) a^{3} c^{4}}{b^{7}}+\frac {2520 d^{5} \ln \left (b x +a \right ) a^{2} c^{5}}{b^{6}}-\frac {840 d^{4} \ln \left (b x +a \right ) a \,c^{6}}{b^{5}}+\frac {d^{10} x^{7}}{7 b^{4}}-\frac {2 d^{10} a \,x^{6}}{3 b^{5}}+\frac {5 d^{9} c \,x^{6}}{3 b^{4}}+\frac {2 d^{10} a^{2} x^{5}}{b^{6}}+\frac {9 d^{8} c^{2} x^{5}}{b^{4}}-\frac {5 d^{10} a^{3} x^{4}}{b^{7}}+\frac {30 d^{7} c^{3} x^{4}}{b^{4}}+\frac {35 d^{10} a^{4} x^{3}}{3 b^{8}}+\frac {70 d^{6} c^{4} x^{3}}{b^{4}}-\frac {28 d^{10} a^{5} x^{2}}{b^{9}}+\frac {84 d^{10} a^{6} x}{b^{10}}+\frac {210 d^{4} c^{6} x}{b^{4}}+\frac {126 d^{5} c^{5} x^{2}}{b^{4}}-\frac {120 d^{10} \ln \left (b x +a \right ) a^{7}}{b^{11}}+\frac {120 d^{3} \ln \left (b x +a \right ) c^{7}}{b^{4}}\)	\(942\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d^4/b^10*(-2/3*a*b^5*d^6*x^6+5/3*b^6*c*d^5*x^6+2*a^2*b^4*d^6*x^5+9*b^6*c^2*d^4*x^5-5*a^3*b^3*d^6*x^4+30*b^6*c^
3*d^3*x^4+35/3*a^4*b^2*d^6*x^3+70*b^6*c^4*d^2*x^3-28*a^5*b*d^6*x^2+1/7*d^6*x^7*b^6+84*a^6*d^6*x+210*b^6*c^6*x-
45*a*b^5*c^2*d^4*x^4-200/3*a^3*b^3*c*d^5*x^3+150*a^2*b^4*c^2*d^4*x^3-160*a*b^5*c^3*d^3*x^3+175*a^4*b^2*c*d^5*x
^2-450*a^3*b^3*c^2*d^4*x^2+600*a^2*b^4*c^3*d^3*x^2-420*a*b^5*c^4*d^2*x^2-560*a^5*b*c*d^5*x+1575*a^4*b^2*c^2*d^
4*x-2400*a^3*b^3*c^3*d^3*x+2100*a^2*b^4*c^4*d^2*x-1008*a*b^5*c^5*d*x+126*b^6*c^5*d*x^2-8*a*b^5*c*d^5*x^5+25*a^
2*b^4*c*d^5*x^4)-45/b^11*d^2*(a^8*d^8-8*a^7*b*c*d^7+28*a^6*b^2*c^2*d^6-56*a^5*b^3*c^3*d^5+70*a^4*b^4*c^4*d^4-5
6*a^3*b^5*c^5*d^3+28*a^2*b^6*c^6*d^2-8*a*b^7*c^7*d+b^8*c^8)/(b*x+a)+5/b^11*d*(a^9*d^9-9*a^8*b*c*d^8+36*a^7*b^2
*c^2*d^7-84*a^6*b^3*c^3*d^6+126*a^5*b^4*c^4*d^5-126*a^4*b^5*c^5*d^4+84*a^3*b^6*c^6*d^3-36*a^2*b^7*c^7*d^2+9*a*
b^8*c^8*d-b^9*c^9)/(b*x+a)^2-120/b^11*d^3*(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)*ln(b*x+a)-1/3*(a^10*d^10-10*a^9*b*c*d^9+45*a^8*b^2*c^2*d
^8-120*a^7*b^3*c^3*d^7+210*a^6*b^4*c^4*d^6-252*a^5*b^5*c^5*d^5+210*a^4*b^6*c^6*d^4-120*a^3*b^7*c^7*d^3+45*a^2*
b^8*c^8*d^2-10*a*b^9*c^9*d+b^10*c^10)/b^11/(b*x+a)^3

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 891 vs. \(2 (252) = 504\).
time = 0.30, 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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1316 vs. \(2 (252) = 504\).
time = 0.31, 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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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)**4,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 907 vs. \(2 (252) = 504\).
time = 0.00, size = 957, normalized size = 3.71 \begin {gather*} \frac {\frac {1}{7} x^{7} d^{10} b^{24}-\frac {2}{3} x^{6} d^{10} b^{23} a+\frac {5}{3} x^{6} d^{9} c b^{24}+2 x^{5} d^{10} b^{22} a^{2}-8 x^{5} d^{9} c b^{23} a+9 x^{5} d^{8} c^{2} b^{24}-5 x^{4} d^{10} b^{21} a^{3}+25 x^{4} d^{9} c b^{22} a^{2}-45 x^{4} d^{8} c^{2} b^{23} a+30 x^{4} d^{7} c^{3} b^{24}+\frac {35}{3} x^{3} d^{10} b^{20} a^{4}-\frac {200}{3} x^{3} d^{9} c b^{21} a^{3}+150 x^{3} d^{8} c^{2} b^{22} a^{2}-160 x^{3} d^{7} c^{3} b^{23} a+70 x^{3} d^{6} c^{4} b^{24}-28 x^{2} d^{10} b^{19} a^{5}+175 x^{2} d^{9} c b^{20} a^{4}-450 x^{2} d^{8} c^{2} b^{21} a^{3}+600 x^{2} d^{7} c^{3} b^{22} a^{2}-420 x^{2} d^{6} c^{4} b^{23} a+126 x^{2} d^{5} c^{5} b^{24}+84 x d^{10} b^{18} a^{6}-560 x d^{9} c b^{19} a^{5}+1575 x d^{8} c^{2} b^{20} a^{4}-2400 x d^{7} c^{3} b^{21} a^{3}+2100 x d^{6} c^{4} b^{22} a^{2}-1008 x d^{5} c^{5} b^{23} a+210 x d^{4} c^{6} b^{24}}{b^{28}}+\frac {\frac {1}{3} \left (\left (-135 d^{10} b^{2} a^{8}+1080 d^{9} b^{3} a^{7} c-3780 d^{8} b^{4} a^{6} c^{2}+7560 d^{7} b^{5} a^{5} c^{3}-9450 d^{6} b^{6} a^{4} c^{4}+7560 d^{5} b^{7} a^{3} c^{5}-3780 d^{4} b^{8} a^{2} c^{6}+1080 d^{3} b^{9} a c^{7}-135 d^{2} b^{10} c^{8}\right ) x^{2}+\left (-255 d^{10} b a^{9}+2025 d^{9} b^{2} a^{8} c-7020 d^{8} b^{3} a^{7} c^{2}+13860 d^{7} b^{4} a^{6} c^{3}-17010 d^{6} b^{5} a^{5} c^{4}+13230 d^{5} b^{6} a^{4} c^{5}-6300 d^{4} b^{7} a^{3} c^{6}+1620 d^{3} b^{8} a^{2} c^{7}-135 d^{2} b^{9} a c^{8}-15 d b^{10} c^{9}\right ) x-121 d^{10} a^{10}+955 d^{9} b a^{9} c-3285 d^{8} b^{2} a^{8} c^{2}+6420 d^{7} b^{3} a^{7} c^{3}-7770 d^{6} b^{4} a^{6} c^{4}+5922 d^{5} b^{5} a^{5} c^{5}-2730 d^{4} b^{6} a^{4} c^{6}+660 d^{3} b^{7} a^{3} c^{7}-45 d^{2} b^{8} a^{2} c^{8}-5 d b^{9} a c^{9}-b^{10} c^{10}\right )}{b^{11} \left (x b+a\right )^{3}}+\frac {\left (-120 d^{10} a^{7}+840 d^{9} c b a^{6}-2520 d^{8} c^{2} b^{2} a^{5}+4200 d^{7} c^{3} b^{3} a^{4}-4200 d^{6} c^{4} b^{4} a^{3}+2520 d^{5} c^{5} b^{5} a^{2}-840 d^{4} c^{6} b^{6} a+120 d^{3} c^{7} b^{7}\right ) \ln \left |x b+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)

[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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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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