3.13.88 \(\int \frac {(c+d x)^7}{(a+b x)^6} \, dx\) [1288]

Optimal. Leaf size=181 \[ \frac {d^6 (7 b c-6 a d) x}{b^7}+\frac {d^7 x^2}{2 b^6}-\frac {(b c-a d)^7}{5 b^8 (a+b x)^5}-\frac {7 d (b c-a d)^6}{4 b^8 (a+b x)^4}-\frac {7 d^2 (b c-a d)^5}{b^8 (a+b x)^3}-\frac {35 d^3 (b c-a d)^4}{2 b^8 (a+b x)^2}-\frac {35 d^4 (b c-a d)^3}{b^8 (a+b x)}+\frac {21 d^5 (b c-a d)^2 \log (a+b x)}{b^8} \]

[Out]

d^6*(-6*a*d+7*b*c)*x/b^7+1/2*d^7*x^2/b^6-1/5*(-a*d+b*c)^7/b^8/(b*x+a)^5-7/4*d*(-a*d+b*c)^6/b^8/(b*x+a)^4-7*d^2
*(-a*d+b*c)^5/b^8/(b*x+a)^3-35/2*d^3*(-a*d+b*c)^4/b^8/(b*x+a)^2-35*d^4*(-a*d+b*c)^3/b^8/(b*x+a)+21*d^5*(-a*d+b
*c)^2*ln(b*x+a)/b^8

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Rubi [A]
time = 0.13, antiderivative size = 181, 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 {21 d^5 (b c-a d)^2 \log (a+b x)}{b^8}-\frac {35 d^4 (b c-a d)^3}{b^8 (a+b x)}-\frac {35 d^3 (b c-a d)^4}{2 b^8 (a+b x)^2}-\frac {7 d^2 (b c-a d)^5}{b^8 (a+b x)^3}-\frac {7 d (b c-a d)^6}{4 b^8 (a+b x)^4}-\frac {(b c-a d)^7}{5 b^8 (a+b x)^5}+\frac {d^6 x (7 b c-6 a d)}{b^7}+\frac {d^7 x^2}{2 b^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^7/(a + b*x)^6,x]

[Out]

(d^6*(7*b*c - 6*a*d)*x)/b^7 + (d^7*x^2)/(2*b^6) - (b*c - a*d)^7/(5*b^8*(a + b*x)^5) - (7*d*(b*c - a*d)^6)/(4*b
^8*(a + b*x)^4) - (7*d^2*(b*c - a*d)^5)/(b^8*(a + b*x)^3) - (35*d^3*(b*c - a*d)^4)/(2*b^8*(a + b*x)^2) - (35*d
^4*(b*c - a*d)^3)/(b^8*(a + b*x)) + (21*d^5*(b*c - a*d)^2*Log[a + b*x])/b^8

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)^7}{(a+b x)^6} \, dx &=\int \left (\frac {d^6 (7 b c-6 a d)}{b^7}+\frac {d^7 x}{b^6}+\frac {(b c-a d)^7}{b^7 (a+b x)^6}+\frac {7 d (b c-a d)^6}{b^7 (a+b x)^5}+\frac {21 d^2 (b c-a d)^5}{b^7 (a+b x)^4}+\frac {35 d^3 (b c-a d)^4}{b^7 (a+b x)^3}+\frac {35 d^4 (b c-a d)^3}{b^7 (a+b x)^2}+\frac {21 d^5 (b c-a d)^2}{b^7 (a+b x)}\right ) \, dx\\ &=\frac {d^6 (7 b c-6 a d) x}{b^7}+\frac {d^7 x^2}{2 b^6}-\frac {(b c-a d)^7}{5 b^8 (a+b x)^5}-\frac {7 d (b c-a d)^6}{4 b^8 (a+b x)^4}-\frac {7 d^2 (b c-a d)^5}{b^8 (a+b x)^3}-\frac {35 d^3 (b c-a d)^4}{2 b^8 (a+b x)^2}-\frac {35 d^4 (b c-a d)^3}{b^8 (a+b x)}+\frac {21 d^5 (b c-a d)^2 \log (a+b x)}{b^8}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(389\) vs. \(2(181)=362\).
time = 0.09, size = 389, normalized size = 2.15 \begin {gather*} \frac {459 a^7 d^7+3 a^6 b d^6 (-406 c+625 d x)+a^5 b^2 d^5 \left (959 c^2-5250 c d x+2700 d^2 x^2\right )+5 a^4 b^3 d^4 \left (-28 c^3+875 c^2 d x-1680 c d^2 x^2+260 d^3 x^3\right )-5 a^3 b^4 d^3 \left (7 c^4+140 c^3 d x-1540 c^2 d^2 x^2+1120 c d^3 x^3+80 d^4 x^4\right )-a^2 b^5 d^2 \left (14 c^5+175 c^4 d x+1400 c^3 d^2 x^2-6300 c^2 d^3 x^3+700 c d^4 x^4+500 d^5 x^5\right )-7 a b^6 d \left (c^6+10 c^5 d x+50 c^4 d^2 x^2+200 c^3 d^3 x^3-300 c^2 d^4 x^4-100 c d^5 x^5+10 d^6 x^6\right )-b^7 \left (4 c^7+35 c^6 d x+140 c^5 d^2 x^2+350 c^4 d^3 x^3+700 c^3 d^4 x^4-140 c d^6 x^6-10 d^7 x^7\right )+420 d^5 (b c-a d)^2 (a+b x)^5 \log (a+b x)}{20 b^8 (a+b x)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^7/(a + b*x)^6,x]

[Out]

(459*a^7*d^7 + 3*a^6*b*d^6*(-406*c + 625*d*x) + a^5*b^2*d^5*(959*c^2 - 5250*c*d*x + 2700*d^2*x^2) + 5*a^4*b^3*
d^4*(-28*c^3 + 875*c^2*d*x - 1680*c*d^2*x^2 + 260*d^3*x^3) - 5*a^3*b^4*d^3*(7*c^4 + 140*c^3*d*x - 1540*c^2*d^2
*x^2 + 1120*c*d^3*x^3 + 80*d^4*x^4) - a^2*b^5*d^2*(14*c^5 + 175*c^4*d*x + 1400*c^3*d^2*x^2 - 6300*c^2*d^3*x^3
+ 700*c*d^4*x^4 + 500*d^5*x^5) - 7*a*b^6*d*(c^6 + 10*c^5*d*x + 50*c^4*d^2*x^2 + 200*c^3*d^3*x^3 - 300*c^2*d^4*
x^4 - 100*c*d^5*x^5 + 10*d^6*x^6) - b^7*(4*c^7 + 35*c^6*d*x + 140*c^5*d^2*x^2 + 350*c^4*d^3*x^3 + 700*c^3*d^4*
x^4 - 140*c*d^6*x^6 - 10*d^7*x^7) + 420*d^5*(b*c - a*d)^2*(a + b*x)^5*Log[a + b*x])/(20*b^8*(a + b*x)^5)

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Mathics [B] Leaf count is larger than twice the leaf count of optimal. \(614\) vs. \(2(181)=362\).
time = 66.68, size = 612, normalized size = 3.38 \begin {gather*} \frac {459 a^7 d^7-1218 a^6 b c d^6+959 a^5 b^2 c^2 d^5-140 a^4 b^3 c^3 d^4-35 a^3 b^4 c^4 d^3-14 a^2 b^5 c^5 d^2-7 a b^6 c^6 d-4 b^7 c^7-35 b d x \left (-57 a^6 d^6+154 a^5 b c d^5-125 a^4 b^2 c^2 d^4+20 a^3 b^3 c^3 d^3+5 a^2 b^4 c^4 d^2+2 a b^5 c^5 d+b^6 c^6\right )+70 b^2 d^2 x^2 \left (47 a^5 d^5-130 a^4 b c d^4+110 a^3 b^2 c^2 d^3-20 a^2 b^3 c^3 d^2-5 a b^4 c^4 d-2 b^5 c^5\right )+350 b^3 d^3 x^3 \left (7 a^4 d^4-20 a^3 b c d^3+18 a^2 b^2 c^2 d^2-4 a b^3 c^3 d-b^4 c^4\right )+700 b^4 d^4 x^4 \left (a^3 d^3-3 a^2 b c d^2+3 a b^2 c^2 d-b^3 c^3\right )+420 d^5 \text {Log}\left [a+b x\right ] \left (a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5\right ) \left (a d-b c\right )^2-20 b d^6 x \left (a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5\right ) \left (6 a d-7 b c\right )+10 b^2 d^7 x^2 \left (a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5\right )}{20 b^8 \left (a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[(c + d*x)^7/(a + b*x)^6,x]')

[Out]

(459 a ^ 7 d ^ 7 - 1218 a ^ 6 b c d ^ 6 + 959 a ^ 5 b ^ 2 c ^ 2 d ^ 5 - 140 a ^ 4 b ^ 3 c ^ 3 d ^ 4 - 35 a ^ 3
b ^ 4 c ^ 4 d ^ 3 - 14 a ^ 2 b ^ 5 c ^ 5 d ^ 2 - 7 a b ^ 6 c ^ 6 d - 4 b ^ 7 c ^ 7 - 35 b d x (-57 a ^ 6 d ^
6 + 154 a ^ 5 b c d ^ 5 - 125 a ^ 4 b ^ 2 c ^ 2 d ^ 4 + 20 a ^ 3 b ^ 3 c ^ 3 d ^ 3 + 5 a ^ 2 b ^ 4 c ^ 4 d ^ 2
+ 2 a b ^ 5 c ^ 5 d + b ^ 6 c ^ 6) + 70 b ^ 2 d ^ 2 x ^ 2 (47 a ^ 5 d ^ 5 - 130 a ^ 4 b c d ^ 4 + 110 a ^ 3 b
^ 2 c ^ 2 d ^ 3 - 20 a ^ 2 b ^ 3 c ^ 3 d ^ 2 - 5 a b ^ 4 c ^ 4 d - 2 b ^ 5 c ^ 5) + 350 b ^ 3 d ^ 3 x ^ 3 (7
a ^ 4 d ^ 4 - 20 a ^ 3 b c d ^ 3 + 18 a ^ 2 b ^ 2 c ^ 2 d ^ 2 - 4 a b ^ 3 c ^ 3 d - b ^ 4 c ^ 4) + 700 b ^ 4 d
^ 4 x ^ 4 (a ^ 3 d ^ 3 - 3 a ^ 2 b c d ^ 2 + 3 a b ^ 2 c ^ 2 d - b ^ 3 c ^ 3) + 420 d ^ 5 Log[a + b x] (a ^ 5
+ 5 a ^ 4 b x + 10 a ^ 3 b ^ 2 x ^ 2 + 10 a ^ 2 b ^ 3 x ^ 3 + 5 a b ^ 4 x ^ 4 + b ^ 5 x ^ 5) (a d - b c) ^ 2
- 20 b d ^ 6 x (a ^ 5 + 5 a ^ 4 b x + 10 a ^ 3 b ^ 2 x ^ 2 + 10 a ^ 2 b ^ 3 x ^ 3 + 5 a b ^ 4 x ^ 4 + b ^ 5 x
^ 5) (6 a d - 7 b c) + 10 b ^ 2 d ^ 7 x ^ 2 (a ^ 5 + 5 a ^ 4 b x + 10 a ^ 3 b ^ 2 x ^ 2 + 10 a ^ 2 b ^ 3 x ^ 3
+ 5 a b ^ 4 x ^ 4 + b ^ 5 x ^ 5)) / (20 b ^ 8 (a ^ 5 + 5 a ^ 4 b x + 10 a ^ 3 b ^ 2 x ^ 2 + 10 a ^ 2 b ^ 3 x
^ 3 + 5 a b ^ 4 x ^ 4 + b ^ 5 x ^ 5))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(450\) vs. \(2(173)=346\).
time = 0.18, size = 451, normalized size = 2.49

method	result	size
default	\(-\frac {d^{6} \left (-\frac {1}{2} b d \,x^{2}+6 a d x -7 b c x \right )}{b^{7}}+\frac {35 d^{4} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{b^{8} \left (b x +a \right )}-\frac {7 d \left (a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +b^{6} c^{6}\right )}{4 b^{8} \left (b x +a \right )^{4}}-\frac {-a^{7} d^{7}+7 a^{6} b c \,d^{6}-21 a^{5} b^{2} c^{2} d^{5}+35 a^{4} b^{3} c^{3} d^{4}-35 a^{3} b^{4} c^{4} d^{3}+21 a^{2} b^{5} c^{5} d^{2}-7 a \,b^{6} c^{6} d +b^{7} c^{7}}{5 b^{8} \left (b x +a \right )^{5}}-\frac {35 d^{3} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}{2 b^{8} \left (b x +a \right )^{2}}+\frac {21 d^{5} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (b x +a \right )}{b^{8}}+\frac {7 d^{2} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right )}{b^{8} \left (b x +a \right )^{3}}\)	\(451\)
norman	\(\frac {\frac {959 a^{7} d^{7}-1918 a^{6} b c \,d^{6}+959 a^{5} b^{2} c^{2} d^{5}-140 a^{4} b^{3} c^{3} d^{4}-35 a^{3} b^{4} c^{4} d^{3}-14 a^{2} b^{5} c^{5} d^{2}-7 a \,b^{6} c^{6} d -4 b^{7} c^{7}}{20 b^{8}}+\frac {d^{7} x^{7}}{2 b}+\frac {5 \left (21 a^{3} d^{7}-42 a^{2} b c \,d^{6}+21 a \,b^{2} c^{2} d^{5}-7 b^{3} c^{3} d^{4}\right ) x^{4}}{b^{4}}+\frac {5 \left (126 a^{4} d^{7}-252 a^{3} b c \,d^{6}+126 b^{2} a^{2} c^{2} d^{5}-28 a \,b^{3} c^{3} d^{4}-7 b^{4} c^{4} d^{3}\right ) x^{3}}{2 b^{5}}+\frac {\left (770 a^{5} d^{7}-1540 a^{4} b c \,d^{6}+770 a^{3} b^{2} c^{2} d^{5}-140 a^{2} b^{3} c^{3} d^{4}-35 a \,b^{4} c^{4} d^{3}-14 b^{5} c^{5} d^{2}\right ) x^{2}}{2 b^{6}}+\frac {\left (875 a^{6} d^{7}-1750 a^{5} b c \,d^{6}+875 a^{4} b^{2} c^{2} d^{5}-140 a^{3} b^{3} c^{3} d^{4}-35 a^{2} b^{4} c^{4} d^{3}-14 a \,b^{5} c^{5} d^{2}-7 b^{6} c^{6} d \right ) x}{4 b^{7}}-\frac {7 d^{6} \left (a d -2 b c \right ) x^{6}}{2 b^{2}}}{\left (b x +a \right )^{5}}+\frac {21 d^{5} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (b x +a \right )}{b^{8}}\)	\(454\)
risch	\(\frac {d^{7} x^{2}}{2 b^{6}}-\frac {6 d^{7} a x}{b^{7}}+\frac {7 d^{6} c x}{b^{6}}+\frac {\left (35 a^{3} b^{3} d^{7}-105 a^{2} b^{4} c \,d^{6}+105 a \,b^{5} c^{2} d^{5}-35 b^{6} c^{3} d^{4}\right ) x^{4}+\frac {35 b^{2} d^{3} \left (7 a^{4} d^{4}-20 a^{3} b c \,d^{3}+18 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d -b^{4} c^{4}\right ) x^{3}}{2}+\frac {7 b \,d^{2} \left (47 a^{5} d^{5}-130 a^{4} b c \,d^{4}+110 a^{3} b^{2} c^{2} d^{3}-20 a^{2} b^{3} c^{3} d^{2}-5 a \,b^{4} c^{4} d -2 b^{5} c^{5}\right ) x^{2}}{2}+\frac {7 d \left (57 a^{6} d^{6}-154 a^{5} b c \,d^{5}+125 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}-5 a^{2} b^{4} c^{4} d^{2}-2 a \,b^{5} c^{5} d -b^{6} c^{6}\right ) x}{4}+\frac {459 a^{7} d^{7}-1218 a^{6} b c \,d^{6}+959 a^{5} b^{2} c^{2} d^{5}-140 a^{4} b^{3} c^{3} d^{4}-35 a^{3} b^{4} c^{4} d^{3}-14 a^{2} b^{5} c^{5} d^{2}-7 a \,b^{6} c^{6} d -4 b^{7} c^{7}}{20 b}}{b^{7} \left (b x +a \right )^{5}}+\frac {21 d^{7} \ln \left (b x +a \right ) a^{2}}{b^{8}}-\frac {42 d^{6} \ln \left (b x +a \right ) a c}{b^{7}}+\frac {21 d^{5} \ln \left (b x +a \right ) c^{2}}{b^{6}}\)	\(463\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^7/(b*x+a)^6,x,method=_RETURNVERBOSE)

[Out]

-d^6/b^7*(-1/2*b*d*x^2+6*a*d*x-7*b*c*x)+35/b^8*d^4*(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/(b*x+a)-7/4/b
^8*d*(a^6*d^6-6*a^5*b*c*d^5+15*a^4*b^2*c^2*d^4-20*a^3*b^3*c^3*d^3+15*a^2*b^4*c^4*d^2-6*a*b^5*c^5*d+b^6*c^6)/(b
*x+a)^4-1/5/b^8*(-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)/(b*x+a)^5-35/2/b^8*d^3*(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4
*c^4)/(b*x+a)^2+21/b^8*d^5*(a^2*d^2-2*a*b*c*d+b^2*c^2)*ln(b*x+a)+7/b^8*d^2*(a^5*d^5-5*a^4*b*c*d^4+10*a^3*b^2*c
^2*d^3-10*a^2*b^3*c^3*d^2+5*a*b^4*c^4*d-b^5*c^5)/(b*x+a)^3

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 504 vs. \(2 (173) = 346\).
time = 0.31, size = 504, normalized size = 2.78 \begin {gather*} -\frac {4 \, b^{7} c^{7} + 7 \, a b^{6} c^{6} d + 14 \, a^{2} b^{5} c^{5} d^{2} + 35 \, a^{3} b^{4} c^{4} d^{3} + 140 \, a^{4} b^{3} c^{3} d^{4} - 959 \, a^{5} b^{2} c^{2} d^{5} + 1218 \, a^{6} b c d^{6} - 459 \, a^{7} d^{7} + 700 \, {\left (b^{7} c^{3} d^{4} - 3 \, a b^{6} c^{2} d^{5} + 3 \, a^{2} b^{5} c d^{6} - a^{3} b^{4} d^{7}\right )} x^{4} + 350 \, {\left (b^{7} c^{4} d^{3} + 4 \, a b^{6} c^{3} d^{4} - 18 \, a^{2} b^{5} c^{2} d^{5} + 20 \, a^{3} b^{4} c d^{6} - 7 \, a^{4} b^{3} d^{7}\right )} x^{3} + 70 \, {\left (2 \, b^{7} c^{5} d^{2} + 5 \, a b^{6} c^{4} d^{3} + 20 \, a^{2} b^{5} c^{3} d^{4} - 110 \, a^{3} b^{4} c^{2} d^{5} + 130 \, a^{4} b^{3} c d^{6} - 47 \, a^{5} b^{2} d^{7}\right )} x^{2} + 35 \, {\left (b^{7} c^{6} d + 2 \, a b^{6} c^{5} d^{2} + 5 \, a^{2} b^{5} c^{4} d^{3} + 20 \, a^{3} b^{4} c^{3} d^{4} - 125 \, a^{4} b^{3} c^{2} d^{5} + 154 \, a^{5} b^{2} c d^{6} - 57 \, a^{6} b d^{7}\right )} x}{20 \, {\left (b^{13} x^{5} + 5 \, a b^{12} x^{4} + 10 \, a^{2} b^{11} x^{3} + 10 \, a^{3} b^{10} x^{2} + 5 \, a^{4} b^{9} x + a^{5} b^{8}\right )}} + \frac {b d^{7} x^{2} + 2 \, {\left (7 \, b c d^{6} - 6 \, a d^{7}\right )} x}{2 \, b^{7}} + \frac {21 \, {\left (b^{2} c^{2} d^{5} - 2 \, a b c d^{6} + a^{2} d^{7}\right )} \log \left (b x + a\right )}{b^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^7/(b*x+a)^6,x, algorithm="maxima")

[Out]

-1/20*(4*b^7*c^7 + 7*a*b^6*c^6*d + 14*a^2*b^5*c^5*d^2 + 35*a^3*b^4*c^4*d^3 + 140*a^4*b^3*c^3*d^4 - 959*a^5*b^2
*c^2*d^5 + 1218*a^6*b*c*d^6 - 459*a^7*d^7 + 700*(b^7*c^3*d^4 - 3*a*b^6*c^2*d^5 + 3*a^2*b^5*c*d^6 - a^3*b^4*d^7
)*x^4 + 350*(b^7*c^4*d^3 + 4*a*b^6*c^3*d^4 - 18*a^2*b^5*c^2*d^5 + 20*a^3*b^4*c*d^6 - 7*a^4*b^3*d^7)*x^3 + 70*(
2*b^7*c^5*d^2 + 5*a*b^6*c^4*d^3 + 20*a^2*b^5*c^3*d^4 - 110*a^3*b^4*c^2*d^5 + 130*a^4*b^3*c*d^6 - 47*a^5*b^2*d^
7)*x^2 + 35*(b^7*c^6*d + 2*a*b^6*c^5*d^2 + 5*a^2*b^5*c^4*d^3 + 20*a^3*b^4*c^3*d^4 - 125*a^4*b^3*c^2*d^5 + 154*
a^5*b^2*c*d^6 - 57*a^6*b*d^7)*x)/(b^13*x^5 + 5*a*b^12*x^4 + 10*a^2*b^11*x^3 + 10*a^3*b^10*x^2 + 5*a^4*b^9*x +
a^5*b^8) + 1/2*(b*d^7*x^2 + 2*(7*b*c*d^6 - 6*a*d^7)*x)/b^7 + 21*(b^2*c^2*d^5 - 2*a*b*c*d^6 + a^2*d^7)*log(b*x
+ a)/b^8

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 732 vs. \(2 (173) = 346\).
time = 0.29, size = 732, normalized size = 4.04 \begin {gather*} \frac {10 \, b^{7} d^{7} x^{7} - 4 \, b^{7} c^{7} - 7 \, a b^{6} c^{6} d - 14 \, a^{2} b^{5} c^{5} d^{2} - 35 \, a^{3} b^{4} c^{4} d^{3} - 140 \, a^{4} b^{3} c^{3} d^{4} + 959 \, a^{5} b^{2} c^{2} d^{5} - 1218 \, a^{6} b c d^{6} + 459 \, a^{7} d^{7} + 70 \, {\left (2 \, b^{7} c d^{6} - a b^{6} d^{7}\right )} x^{6} + 100 \, {\left (7 \, a b^{6} c d^{6} - 5 \, a^{2} b^{5} d^{7}\right )} x^{5} - 100 \, {\left (7 \, b^{7} c^{3} d^{4} - 21 \, a b^{6} c^{2} d^{5} + 7 \, a^{2} b^{5} c d^{6} + 4 \, a^{3} b^{4} d^{7}\right )} x^{4} - 50 \, {\left (7 \, b^{7} c^{4} d^{3} + 28 \, a b^{6} c^{3} d^{4} - 126 \, a^{2} b^{5} c^{2} d^{5} + 112 \, a^{3} b^{4} c d^{6} - 26 \, a^{4} b^{3} d^{7}\right )} x^{3} - 10 \, {\left (14 \, b^{7} c^{5} d^{2} + 35 \, a b^{6} c^{4} d^{3} + 140 \, a^{2} b^{5} c^{3} d^{4} - 770 \, a^{3} b^{4} c^{2} d^{5} + 840 \, a^{4} b^{3} c d^{6} - 270 \, a^{5} b^{2} d^{7}\right )} x^{2} - 5 \, {\left (7 \, b^{7} c^{6} d + 14 \, a b^{6} c^{5} d^{2} + 35 \, a^{2} b^{5} c^{4} d^{3} + 140 \, a^{3} b^{4} c^{3} d^{4} - 875 \, a^{4} b^{3} c^{2} d^{5} + 1050 \, a^{5} b^{2} c d^{6} - 375 \, a^{6} b d^{7}\right )} x + 420 \, {\left (a^{5} b^{2} c^{2} d^{5} - 2 \, a^{6} b c d^{6} + a^{7} d^{7} + {\left (b^{7} c^{2} d^{5} - 2 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 5 \, {\left (a b^{6} c^{2} d^{5} - 2 \, a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 10 \, {\left (a^{2} b^{5} c^{2} d^{5} - 2 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 10 \, {\left (a^{3} b^{4} c^{2} d^{5} - 2 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 5 \, {\left (a^{4} b^{3} c^{2} d^{5} - 2 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x\right )} \log \left (b x + a\right )}{20 \, {\left (b^{13} x^{5} + 5 \, a b^{12} x^{4} + 10 \, a^{2} b^{11} x^{3} + 10 \, a^{3} b^{10} x^{2} + 5 \, a^{4} b^{9} x + a^{5} b^{8}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^7/(b*x+a)^6,x, algorithm="fricas")

[Out]

1/20*(10*b^7*d^7*x^7 - 4*b^7*c^7 - 7*a*b^6*c^6*d - 14*a^2*b^5*c^5*d^2 - 35*a^3*b^4*c^4*d^3 - 140*a^4*b^3*c^3*d
^4 + 959*a^5*b^2*c^2*d^5 - 1218*a^6*b*c*d^6 + 459*a^7*d^7 + 70*(2*b^7*c*d^6 - a*b^6*d^7)*x^6 + 100*(7*a*b^6*c*
d^6 - 5*a^2*b^5*d^7)*x^5 - 100*(7*b^7*c^3*d^4 - 21*a*b^6*c^2*d^5 + 7*a^2*b^5*c*d^6 + 4*a^3*b^4*d^7)*x^4 - 50*(
7*b^7*c^4*d^3 + 28*a*b^6*c^3*d^4 - 126*a^2*b^5*c^2*d^5 + 112*a^3*b^4*c*d^6 - 26*a^4*b^3*d^7)*x^3 - 10*(14*b^7*
c^5*d^2 + 35*a*b^6*c^4*d^3 + 140*a^2*b^5*c^3*d^4 - 770*a^3*b^4*c^2*d^5 + 840*a^4*b^3*c*d^6 - 270*a^5*b^2*d^7)*
x^2 - 5*(7*b^7*c^6*d + 14*a*b^6*c^5*d^2 + 35*a^2*b^5*c^4*d^3 + 140*a^3*b^4*c^3*d^4 - 875*a^4*b^3*c^2*d^5 + 105
0*a^5*b^2*c*d^6 - 375*a^6*b*d^7)*x + 420*(a^5*b^2*c^2*d^5 - 2*a^6*b*c*d^6 + a^7*d^7 + (b^7*c^2*d^5 - 2*a*b^6*c
*d^6 + a^2*b^5*d^7)*x^5 + 5*(a*b^6*c^2*d^5 - 2*a^2*b^5*c*d^6 + a^3*b^4*d^7)*x^4 + 10*(a^2*b^5*c^2*d^5 - 2*a^3*
b^4*c*d^6 + a^4*b^3*d^7)*x^3 + 10*(a^3*b^4*c^2*d^5 - 2*a^4*b^3*c*d^6 + a^5*b^2*d^7)*x^2 + 5*(a^4*b^3*c^2*d^5 -
2*a^5*b^2*c*d^6 + a^6*b*d^7)*x)*log(b*x + a))/(b^13*x^5 + 5*a*b^12*x^4 + 10*a^2*b^11*x^3 + 10*a^3*b^10*x^2 +
5*a^4*b^9*x + a^5*b^8)

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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)**7/(b*x+a)**6,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 463 vs. \(2 (173) = 346\).
time = 0.00, size = 493, normalized size = 2.72 \begin {gather*} \frac {\frac {1}{2} x^{2} d^{7} b^{6}-6 x d^{7} b^{5} a+7 x d^{6} c b^{6}}{b^{12}}+\frac {\frac {1}{20} \left (\left (700 d^{7} b^{4} a^{3}-2100 d^{6} b^{5} a^{2} c+2100 d^{5} b^{6} a c^{2}-700 d^{4} b^{7} c^{3}\right ) x^{4}+\left (2450 d^{7} b^{3} a^{4}-7000 d^{6} b^{4} a^{3} c+6300 d^{5} b^{5} a^{2} c^{2}-1400 d^{4} b^{6} a c^{3}-350 d^{3} b^{7} c^{4}\right ) x^{3}+\left (3290 d^{7} b^{2} a^{5}-9100 d^{6} b^{3} a^{4} c+7700 d^{5} b^{4} a^{3} c^{2}-1400 d^{4} b^{5} a^{2} c^{3}-350 d^{3} b^{6} a c^{4}-140 d^{2} b^{7} c^{5}\right ) x^{2}+\left (1995 d^{7} b a^{6}-5390 d^{6} b^{2} a^{5} c+4375 d^{5} b^{3} a^{4} c^{2}-700 d^{4} b^{4} a^{3} c^{3}-175 d^{3} b^{5} a^{2} c^{4}-70 d^{2} b^{6} a c^{5}-35 d b^{7} c^{6}\right ) x+459 d^{7} a^{7}-1218 d^{6} b a^{6} c+959 d^{5} b^{2} a^{5} c^{2}-140 d^{4} b^{3} a^{4} c^{3}-35 d^{3} b^{4} a^{3} c^{4}-14 d^{2} b^{5} a^{2} c^{5}-7 d b^{6} a c^{6}-4 b^{7} c^{7}\right )}{b^{8} \left (x b+a\right )^{5}}+\frac {\left (21 d^{7} a^{2}-42 d^{6} c b a+21 d^{5} c^{2} b^{2}\right ) \ln \left |x b+a\right |}{b^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^7/(b*x+a)^6,x)

[Out]

21*(b^2*c^2*d^5 - 2*a*b*c*d^6 + a^2*d^7)*log(abs(b*x + a))/b^8 + 1/2*(b^6*d^7*x^2 + 14*b^6*c*d^6*x - 12*a*b^5*
d^7*x)/b^12 - 1/20*(4*b^7*c^7 + 7*a*b^6*c^6*d + 14*a^2*b^5*c^5*d^2 + 35*a^3*b^4*c^4*d^3 + 140*a^4*b^3*c^3*d^4
- 959*a^5*b^2*c^2*d^5 + 1218*a^6*b*c*d^6 - 459*a^7*d^7 + 700*(b^7*c^3*d^4 - 3*a*b^6*c^2*d^5 + 3*a^2*b^5*c*d^6
- a^3*b^4*d^7)*x^4 + 350*(b^7*c^4*d^3 + 4*a*b^6*c^3*d^4 - 18*a^2*b^5*c^2*d^5 + 20*a^3*b^4*c*d^6 - 7*a^4*b^3*d^
7)*x^3 + 70*(2*b^7*c^5*d^2 + 5*a*b^6*c^4*d^3 + 20*a^2*b^5*c^3*d^4 - 110*a^3*b^4*c^2*d^5 + 130*a^4*b^3*c*d^6 -
47*a^5*b^2*d^7)*x^2 + 35*(b^7*c^6*d + 2*a*b^6*c^5*d^2 + 5*a^2*b^5*c^4*d^3 + 20*a^3*b^4*c^3*d^4 - 125*a^4*b^3*c
^2*d^5 + 154*a^5*b^2*c*d^6 - 57*a^6*b*d^7)*x)/((b*x + a)^5*b^8)

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Mupad [B]
time = 0.34, size = 508, normalized size = 2.81 \begin {gather*} \frac {\ln \left (a+b\,x\right )\,\left (21\,a^2\,d^7-42\,a\,b\,c\,d^6+21\,b^2\,c^2\,d^5\right )}{b^8}-x\,\left (\frac {6\,a\,d^7}{b^7}-\frac {7\,c\,d^6}{b^6}\right )-\frac {\frac {-459\,a^7\,d^7+1218\,a^6\,b\,c\,d^6-959\,a^5\,b^2\,c^2\,d^5+140\,a^4\,b^3\,c^3\,d^4+35\,a^3\,b^4\,c^4\,d^3+14\,a^2\,b^5\,c^5\,d^2+7\,a\,b^6\,c^6\,d+4\,b^7\,c^7}{20\,b}+x\,\left (-\frac {399\,a^6\,d^7}{4}+\frac {539\,a^5\,b\,c\,d^6}{2}-\frac {875\,a^4\,b^2\,c^2\,d^5}{4}+35\,a^3\,b^3\,c^3\,d^4+\frac {35\,a^2\,b^4\,c^4\,d^3}{4}+\frac {7\,a\,b^5\,c^5\,d^2}{2}+\frac {7\,b^6\,c^6\,d}{4}\right )+x^3\,\left (-\frac {245\,a^4\,b^2\,d^7}{2}+350\,a^3\,b^3\,c\,d^6-315\,a^2\,b^4\,c^2\,d^5+70\,a\,b^5\,c^3\,d^4+\frac {35\,b^6\,c^4\,d^3}{2}\right )+x^2\,\left (-\frac {329\,a^5\,b\,d^7}{2}+455\,a^4\,b^2\,c\,d^6-385\,a^3\,b^3\,c^2\,d^5+70\,a^2\,b^4\,c^3\,d^4+\frac {35\,a\,b^5\,c^4\,d^3}{2}+7\,b^6\,c^5\,d^2\right )-x^4\,\left (35\,a^3\,b^3\,d^7-105\,a^2\,b^4\,c\,d^6+105\,a\,b^5\,c^2\,d^5-35\,b^6\,c^3\,d^4\right )}{a^5\,b^7+5\,a^4\,b^8\,x+10\,a^3\,b^9\,x^2+10\,a^2\,b^{10}\,x^3+5\,a\,b^{11}\,x^4+b^{12}\,x^5}+\frac {d^7\,x^2}{2\,b^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^7/(a + b*x)^6,x)

[Out]

(log(a + b*x)*(21*a^2*d^7 + 21*b^2*c^2*d^5 - 42*a*b*c*d^6))/b^8 - x*((6*a*d^7)/b^7 - (7*c*d^6)/b^6) - ((4*b^7*
c^7 - 459*a^7*d^7 + 14*a^2*b^5*c^5*d^2 + 35*a^3*b^4*c^4*d^3 + 140*a^4*b^3*c^3*d^4 - 959*a^5*b^2*c^2*d^5 + 7*a*
b^6*c^6*d + 1218*a^6*b*c*d^6)/(20*b) + x*((7*b^6*c^6*d)/4 - (399*a^6*d^7)/4 + (7*a*b^5*c^5*d^2)/2 + (35*a^2*b^
4*c^4*d^3)/4 + 35*a^3*b^3*c^3*d^4 - (875*a^4*b^2*c^2*d^5)/4 + (539*a^5*b*c*d^6)/2) + x^3*((35*b^6*c^4*d^3)/2 -
(245*a^4*b^2*d^7)/2 + 70*a*b^5*c^3*d^4 + 350*a^3*b^3*c*d^6 - 315*a^2*b^4*c^2*d^5) + x^2*(7*b^6*c^5*d^2 - (329
*a^5*b*d^7)/2 + (35*a*b^5*c^4*d^3)/2 + 455*a^4*b^2*c*d^6 + 70*a^2*b^4*c^3*d^4 - 385*a^3*b^3*c^2*d^5) - x^4*(35
*a^3*b^3*d^7 - 35*b^6*c^3*d^4 + 105*a*b^5*c^2*d^5 - 105*a^2*b^4*c*d^6))/(a^5*b^7 + b^12*x^5 + 5*a^4*b^8*x + 5*
a*b^11*x^4 + 10*a^3*b^9*x^2 + 10*a^2*b^10*x^3) + (d^7*x^2)/(2*b^6)

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