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 = 0.12, 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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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 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,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 = 0.33, 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 = 0.88, size = 1316, normalized size = 5.10 \begin {gather*} \frac {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 \, {\left (7 \, b^{10} c d^{9} - a b^{9} d^{10}\right )} x^{9} + 9 \, {\left (21 \, b^{10} c^{2} d^{8} - 7 \, a b^{9} c d^{9} + a^{2} b^{8} d^{10}\right )} x^{8} + 18 \, {\left (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}\right )} x^{7} + 42 \, {\left (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}\right )} x^{6} + 126 \, {\left (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}\right )} x^{5} + 630 \, {\left (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}\right )} x^{4} + 7 \, {\left (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}\right )} x^{3} - 21 \, {\left (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}\right )} x^{2} - 21 \, {\left (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}\right )} x + 2520 \, {\left (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} + {\left (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}\right )} x^{3} + 3 \, {\left (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} + 35 \, 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}\right )} x^{2} + 3 \, {\left (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}\right )} x\right )} \log \left (b x + a\right )}{21 \, {\left (b^{14} x^{3} + 3 \, a b^{13} x^{2} + 3 \, a^{2} b^{12} x + a^{3} b^{11}\right )}} \end {gather*}

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)] Timed out
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 = 0.66, 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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Mupad [B]
time = 0.39, size = 2219, normalized size = 8.60 \begin {gather*} \text {Too large to display} \end {gather*}

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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