3.13.90 \(\int \frac {(c+d x)^7}{(a+b x)^8} \, dx\) [1290]

Optimal. Leaf size=194 \[ -\frac {(b c-a d)^7}{7 b^8 (a+b x)^7}-\frac {7 d (b c-a d)^6}{6 b^8 (a+b x)^6}-\frac {21 d^2 (b c-a d)^5}{5 b^8 (a+b x)^5}-\frac {35 d^3 (b c-a d)^4}{4 b^8 (a+b x)^4}-\frac {35 d^4 (b c-a d)^3}{3 b^8 (a+b x)^3}-\frac {21 d^5 (b c-a d)^2}{2 b^8 (a+b x)^2}-\frac {7 d^6 (b c-a d)}{b^8 (a+b x)}+\frac {d^7 \log (a+b x)}{b^8} \]

[Out]

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

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Rubi [A]
time = 0.11, antiderivative size = 194, 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 {7 d^6 (b c-a d)}{b^8 (a+b x)}-\frac {21 d^5 (b c-a d)^2}{2 b^8 (a+b x)^2}-\frac {35 d^4 (b c-a d)^3}{3 b^8 (a+b x)^3}-\frac {35 d^3 (b c-a d)^4}{4 b^8 (a+b x)^4}-\frac {21 d^2 (b c-a d)^5}{5 b^8 (a+b x)^5}-\frac {7 d (b c-a d)^6}{6 b^8 (a+b x)^6}-\frac {(b c-a d)^7}{7 b^8 (a+b x)^7}+\frac {d^7 \log (a+b x)}{b^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^7/(a + b*x)^8,x]

[Out]

-1/7*(b*c - a*d)^7/(b^8*(a + b*x)^7) - (7*d*(b*c - a*d)^6)/(6*b^8*(a + b*x)^6) - (21*d^2*(b*c - a*d)^5)/(5*b^8
*(a + b*x)^5) - (35*d^3*(b*c - a*d)^4)/(4*b^8*(a + b*x)^4) - (35*d^4*(b*c - a*d)^3)/(3*b^8*(a + b*x)^3) - (21*
d^5*(b*c - a*d)^2)/(2*b^8*(a + b*x)^2) - (7*d^6*(b*c - a*d))/(b^8*(a + b*x)) + (d^7*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)^8} \, dx &=\int \left (\frac {(b c-a d)^7}{b^7 (a+b x)^8}+\frac {7 d (b c-a d)^6}{b^7 (a+b x)^7}+\frac {21 d^2 (b c-a d)^5}{b^7 (a+b x)^6}+\frac {35 d^3 (b c-a d)^4}{b^7 (a+b x)^5}+\frac {35 d^4 (b c-a d)^3}{b^7 (a+b x)^4}+\frac {21 d^5 (b c-a d)^2}{b^7 (a+b x)^3}+\frac {7 d^6 (b c-a d)}{b^7 (a+b x)^2}+\frac {d^7}{b^7 (a+b x)}\right ) \, dx\\ &=-\frac {(b c-a d)^7}{7 b^8 (a+b x)^7}-\frac {7 d (b c-a d)^6}{6 b^8 (a+b x)^6}-\frac {21 d^2 (b c-a d)^5}{5 b^8 (a+b x)^5}-\frac {35 d^3 (b c-a d)^4}{4 b^8 (a+b x)^4}-\frac {35 d^4 (b c-a d)^3}{3 b^8 (a+b x)^3}-\frac {21 d^5 (b c-a d)^2}{2 b^8 (a+b x)^2}-\frac {7 d^6 (b c-a d)}{b^8 (a+b x)}+\frac {d^7 \log (a+b x)}{b^8}\\ \end {align*}

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Mathematica [A]
time = 0.10, size = 308, normalized size = 1.59 \begin {gather*} -\frac {(b c-a d) \left (1089 a^6 d^6+3 a^5 b d^5 (223 c+2401 d x)+3 a^4 b^2 d^4 \left (153 c^2+1421 c d x+6713 d^2 x^2\right )+a^3 b^3 d^3 \left (319 c^3+2793 c^2 d x+11319 c d^2 x^2+30625 d^3 x^3\right )+a^2 b^4 d^2 \left (214 c^4+1813 c^3 d x+6909 c^2 d^2 x^2+15925 c d^3 x^3+26950 d^4 x^4\right )+a b^5 d \left (130 c^5+1078 c^4 d x+3969 c^3 d^2 x^2+8575 c^2 d^3 x^3+12250 c d^4 x^4+13230 d^5 x^5\right )+b^6 \left (60 c^6+490 c^5 d x+1764 c^4 d^2 x^2+3675 c^3 d^3 x^3+4900 c^2 d^4 x^4+4410 c d^5 x^5+2940 d^6 x^6\right )\right )}{420 b^8 (a+b x)^7}+\frac {d^7 \log (a+b x)}{b^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^7/(a + b*x)^8,x]

[Out]

-1/420*((b*c - a*d)*(1089*a^6*d^6 + 3*a^5*b*d^5*(223*c + 2401*d*x) + 3*a^4*b^2*d^4*(153*c^2 + 1421*c*d*x + 671
3*d^2*x^2) + a^3*b^3*d^3*(319*c^3 + 2793*c^2*d*x + 11319*c*d^2*x^2 + 30625*d^3*x^3) + a^2*b^4*d^2*(214*c^4 + 1
813*c^3*d*x + 6909*c^2*d^2*x^2 + 15925*c*d^3*x^3 + 26950*d^4*x^4) + a*b^5*d*(130*c^5 + 1078*c^4*d*x + 3969*c^3
*d^2*x^2 + 8575*c^2*d^3*x^3 + 12250*c*d^4*x^4 + 13230*d^5*x^5) + b^6*(60*c^6 + 490*c^5*d*x + 1764*c^4*d^2*x^2
+ 3675*c^3*d^3*x^3 + 4900*c^2*d^4*x^4 + 4410*c*d^5*x^5 + 2940*d^6*x^6)))/(b^8*(a + b*x)^7) + (d^7*Log[a + b*x]
)/b^8

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Mathics [F(-1)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[(c + d*x)^7/(a + b*x)^8,x]')

[Out]

Timed out

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(461\) vs. \(2(182)=364\).
time = 0.15, size = 462, normalized size = 2.38

method	result	size
risch	\(\frac {\frac {7 d^{6} \left (a d -b c \right ) x^{6}}{b^{2}}+\frac {21 d^{5} \left (3 a^{2} d^{2}-2 a b c d -b^{2} c^{2}\right ) x^{5}}{2 b^{3}}+\frac {35 d^{4} \left (11 a^{3} d^{3}-6 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d -2 b^{3} c^{3}\right ) x^{4}}{6 b^{4}}+\frac {35 d^{3} \left (25 a^{4} d^{4}-12 a^{3} b c \,d^{3}-6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d -3 b^{4} c^{4}\right ) x^{3}}{12 b^{5}}+\frac {7 d^{2} \left (137 a^{5} d^{5}-60 a^{4} b c \,d^{4}-30 a^{3} b^{2} c^{2} d^{3}-20 a^{2} b^{3} c^{3} d^{2}-15 a \,b^{4} c^{4} d -12 b^{5} c^{5}\right ) x^{2}}{20 b^{6}}+\frac {7 d \left (147 a^{6} d^{6}-60 a^{5} b c \,d^{5}-30 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}-15 a^{2} b^{4} c^{4} d^{2}-12 a \,b^{5} c^{5} d -10 b^{6} c^{6}\right ) x}{60 b^{7}}+\frac {1089 a^{7} d^{7}-420 a^{6} b c \,d^{6}-210 a^{5} b^{2} c^{2} d^{5}-140 a^{4} b^{3} c^{3} d^{4}-105 a^{3} b^{4} c^{4} d^{3}-84 a^{2} b^{5} c^{5} d^{2}-70 a \,b^{6} c^{6} d -60 b^{7} c^{7}}{420 b^{8}}}{\left (b x +a \right )^{7}}+\frac {d^{7} \ln \left (b x +a \right )}{b^{8}}\)	\(447\)
norman	\(\frac {\frac {1089 a^{7} d^{7}-420 a^{6} b c \,d^{6}-210 a^{5} b^{2} c^{2} d^{5}-140 a^{4} b^{3} c^{3} d^{4}-105 a^{3} b^{4} c^{4} d^{3}-84 a^{2} b^{5} c^{5} d^{2}-70 a \,b^{6} c^{6} d -60 b^{7} c^{7}}{420 b^{8}}+\frac {7 \left (a \,d^{7}-b c \,d^{6}\right ) x^{6}}{b^{2}}+\frac {21 \left (3 a^{2} d^{7}-2 a b c \,d^{6}-b^{2} c^{2} d^{5}\right ) x^{5}}{2 b^{3}}+\frac {35 \left (11 a^{3} d^{7}-6 a^{2} b c \,d^{6}-3 a \,b^{2} c^{2} d^{5}-2 b^{3} c^{3} d^{4}\right ) x^{4}}{6 b^{4}}+\frac {35 \left (25 a^{4} d^{7}-12 a^{3} b c \,d^{6}-6 b^{2} a^{2} c^{2} d^{5}-4 a \,b^{3} c^{3} d^{4}-3 b^{4} c^{4} d^{3}\right ) x^{3}}{12 b^{5}}+\frac {7 \left (137 a^{5} d^{7}-60 a^{4} b c \,d^{6}-30 a^{3} b^{2} c^{2} d^{5}-20 a^{2} b^{3} c^{3} d^{4}-15 a \,b^{4} c^{4} d^{3}-12 b^{5} c^{5} d^{2}\right ) x^{2}}{20 b^{6}}+\frac {7 \left (147 a^{6} d^{7}-60 a^{5} b c \,d^{6}-30 a^{4} b^{2} c^{2} d^{5}-20 a^{3} b^{3} c^{3} d^{4}-15 a^{2} b^{4} c^{4} d^{3}-12 a \,b^{5} c^{5} d^{2}-10 b^{6} c^{6} d \right ) x}{60 b^{7}}}{\left (b x +a \right )^{7}}+\frac {d^{7} \ln \left (b x +a \right )}{b^{8}}\)	\(459\)
default	\(\frac {7 d^{6} \left (a d -b c \right )}{b^{8} \left (b x +a \right )}-\frac {35 d^{3} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}{4 b^{8} \left (b x +a \right )^{4}}+\frac {21 d^{2} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right )}{5 b^{8} \left (b x +a \right )^{5}}-\frac {-a^{7} d^{7}+7 a^{6} b c \,d^{6}-21 a^{5} b^{2} c^{2} d^{5}+35 a^{4} b^{3} c^{3} d^{4}-35 a^{3} b^{4} c^{4} d^{3}+21 a^{2} b^{5} c^{5} d^{2}-7 a \,b^{6} c^{6} d +b^{7} c^{7}}{7 b^{8} \left (b x +a \right )^{7}}-\frac {21 d^{5} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{2 b^{8} \left (b x +a \right )^{2}}-\frac {7 d \left (a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +b^{6} c^{6}\right )}{6 b^{8} \left (b x +a \right )^{6}}+\frac {d^{7} \ln \left (b x +a \right )}{b^{8}}+\frac {35 d^{4} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{3 b^{8} \left (b x +a \right )^{3}}\)	\(462\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^7/(b*x+a)^8,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 534 vs. \(2 (182) = 364\).
time = 0.28, size = 534, normalized size = 2.75 \begin {gather*} -\frac {60 \, b^{7} c^{7} + 70 \, a b^{6} c^{6} d + 84 \, a^{2} b^{5} c^{5} d^{2} + 105 \, a^{3} b^{4} c^{4} d^{3} + 140 \, a^{4} b^{3} c^{3} d^{4} + 210 \, a^{5} b^{2} c^{2} d^{5} + 420 \, a^{6} b c d^{6} - 1089 \, a^{7} d^{7} + 2940 \, {\left (b^{7} c d^{6} - a b^{6} d^{7}\right )} x^{6} + 4410 \, {\left (b^{7} c^{2} d^{5} + 2 \, a b^{6} c d^{6} - 3 \, a^{2} b^{5} d^{7}\right )} x^{5} + 2450 \, {\left (2 \, b^{7} c^{3} d^{4} + 3 \, a b^{6} c^{2} d^{5} + 6 \, a^{2} b^{5} c d^{6} - 11 \, a^{3} b^{4} d^{7}\right )} x^{4} + 1225 \, {\left (3 \, b^{7} c^{4} d^{3} + 4 \, a b^{6} c^{3} d^{4} + 6 \, a^{2} b^{5} c^{2} d^{5} + 12 \, a^{3} b^{4} c d^{6} - 25 \, a^{4} b^{3} d^{7}\right )} x^{3} + 147 \, {\left (12 \, b^{7} c^{5} d^{2} + 15 \, a b^{6} c^{4} d^{3} + 20 \, a^{2} b^{5} c^{3} d^{4} + 30 \, a^{3} b^{4} c^{2} d^{5} + 60 \, a^{4} b^{3} c d^{6} - 137 \, a^{5} b^{2} d^{7}\right )} x^{2} + 49 \, {\left (10 \, b^{7} c^{6} d + 12 \, a b^{6} c^{5} d^{2} + 15 \, a^{2} b^{5} c^{4} d^{3} + 20 \, a^{3} b^{4} c^{3} d^{4} + 30 \, a^{4} b^{3} c^{2} d^{5} + 60 \, a^{5} b^{2} c d^{6} - 147 \, a^{6} b d^{7}\right )} x}{420 \, {\left (b^{15} x^{7} + 7 \, a b^{14} x^{6} + 21 \, a^{2} b^{13} x^{5} + 35 \, a^{3} b^{12} x^{4} + 35 \, a^{4} b^{11} x^{3} + 21 \, a^{5} b^{10} x^{2} + 7 \, a^{6} b^{9} x + a^{7} b^{8}\right )}} + \frac {d^{7} \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)^8,x, algorithm="maxima")

[Out]

-1/420*(60*b^7*c^7 + 70*a*b^6*c^6*d + 84*a^2*b^5*c^5*d^2 + 105*a^3*b^4*c^4*d^3 + 140*a^4*b^3*c^3*d^4 + 210*a^5
*b^2*c^2*d^5 + 420*a^6*b*c*d^6 - 1089*a^7*d^7 + 2940*(b^7*c*d^6 - a*b^6*d^7)*x^6 + 4410*(b^7*c^2*d^5 + 2*a*b^6
*c*d^6 - 3*a^2*b^5*d^7)*x^5 + 2450*(2*b^7*c^3*d^4 + 3*a*b^6*c^2*d^5 + 6*a^2*b^5*c*d^6 - 11*a^3*b^4*d^7)*x^4 +
1225*(3*b^7*c^4*d^3 + 4*a*b^6*c^3*d^4 + 6*a^2*b^5*c^2*d^5 + 12*a^3*b^4*c*d^6 - 25*a^4*b^3*d^7)*x^3 + 147*(12*b
^7*c^5*d^2 + 15*a*b^6*c^4*d^3 + 20*a^2*b^5*c^3*d^4 + 30*a^3*b^4*c^2*d^5 + 60*a^4*b^3*c*d^6 - 137*a^5*b^2*d^7)*
x^2 + 49*(10*b^7*c^6*d + 12*a*b^6*c^5*d^2 + 15*a^2*b^5*c^4*d^3 + 20*a^3*b^4*c^3*d^4 + 30*a^4*b^3*c^2*d^5 + 60*
a^5*b^2*c*d^6 - 147*a^6*b*d^7)*x)/(b^15*x^7 + 7*a*b^14*x^6 + 21*a^2*b^13*x^5 + 35*a^3*b^12*x^4 + 35*a^4*b^11*x
^3 + 21*a^5*b^10*x^2 + 7*a^6*b^9*x + a^7*b^8) + 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. 624 vs. \(2 (182) = 364\).
time = 0.30, size = 624, normalized size = 3.22 \begin {gather*} -\frac {60 \, b^{7} c^{7} + 70 \, a b^{6} c^{6} d + 84 \, a^{2} b^{5} c^{5} d^{2} + 105 \, a^{3} b^{4} c^{4} d^{3} + 140 \, a^{4} b^{3} c^{3} d^{4} + 210 \, a^{5} b^{2} c^{2} d^{5} + 420 \, a^{6} b c d^{6} - 1089 \, a^{7} d^{7} + 2940 \, {\left (b^{7} c d^{6} - a b^{6} d^{7}\right )} x^{6} + 4410 \, {\left (b^{7} c^{2} d^{5} + 2 \, a b^{6} c d^{6} - 3 \, a^{2} b^{5} d^{7}\right )} x^{5} + 2450 \, {\left (2 \, b^{7} c^{3} d^{4} + 3 \, a b^{6} c^{2} d^{5} + 6 \, a^{2} b^{5} c d^{6} - 11 \, a^{3} b^{4} d^{7}\right )} x^{4} + 1225 \, {\left (3 \, b^{7} c^{4} d^{3} + 4 \, a b^{6} c^{3} d^{4} + 6 \, a^{2} b^{5} c^{2} d^{5} + 12 \, a^{3} b^{4} c d^{6} - 25 \, a^{4} b^{3} d^{7}\right )} x^{3} + 147 \, {\left (12 \, b^{7} c^{5} d^{2} + 15 \, a b^{6} c^{4} d^{3} + 20 \, a^{2} b^{5} c^{3} d^{4} + 30 \, a^{3} b^{4} c^{2} d^{5} + 60 \, a^{4} b^{3} c d^{6} - 137 \, a^{5} b^{2} d^{7}\right )} x^{2} + 49 \, {\left (10 \, b^{7} c^{6} d + 12 \, a b^{6} c^{5} d^{2} + 15 \, a^{2} b^{5} c^{4} d^{3} + 20 \, a^{3} b^{4} c^{3} d^{4} + 30 \, a^{4} b^{3} c^{2} d^{5} + 60 \, a^{5} b^{2} c d^{6} - 147 \, a^{6} b d^{7}\right )} x - 420 \, {\left (b^{7} d^{7} x^{7} + 7 \, a b^{6} d^{7} x^{6} + 21 \, a^{2} b^{5} d^{7} x^{5} + 35 \, a^{3} b^{4} d^{7} x^{4} + 35 \, a^{4} b^{3} d^{7} x^{3} + 21 \, a^{5} b^{2} d^{7} x^{2} + 7 \, a^{6} b d^{7} x + a^{7} d^{7}\right )} \log \left (b x + a\right )}{420 \, {\left (b^{15} x^{7} + 7 \, a b^{14} x^{6} + 21 \, a^{2} b^{13} x^{5} + 35 \, a^{3} b^{12} x^{4} + 35 \, a^{4} b^{11} x^{3} + 21 \, a^{5} b^{10} x^{2} + 7 \, a^{6} b^{9} x + a^{7} b^{8}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^7/(b*x+a)^8,x, algorithm="fricas")

[Out]

-1/420*(60*b^7*c^7 + 70*a*b^6*c^6*d + 84*a^2*b^5*c^5*d^2 + 105*a^3*b^4*c^4*d^3 + 140*a^4*b^3*c^3*d^4 + 210*a^5
*b^2*c^2*d^5 + 420*a^6*b*c*d^6 - 1089*a^7*d^7 + 2940*(b^7*c*d^6 - a*b^6*d^7)*x^6 + 4410*(b^7*c^2*d^5 + 2*a*b^6
*c*d^6 - 3*a^2*b^5*d^7)*x^5 + 2450*(2*b^7*c^3*d^4 + 3*a*b^6*c^2*d^5 + 6*a^2*b^5*c*d^6 - 11*a^3*b^4*d^7)*x^4 +
1225*(3*b^7*c^4*d^3 + 4*a*b^6*c^3*d^4 + 6*a^2*b^5*c^2*d^5 + 12*a^3*b^4*c*d^6 - 25*a^4*b^3*d^7)*x^3 + 147*(12*b
^7*c^5*d^2 + 15*a*b^6*c^4*d^3 + 20*a^2*b^5*c^3*d^4 + 30*a^3*b^4*c^2*d^5 + 60*a^4*b^3*c*d^6 - 137*a^5*b^2*d^7)*
x^2 + 49*(10*b^7*c^6*d + 12*a*b^6*c^5*d^2 + 15*a^2*b^5*c^4*d^3 + 20*a^3*b^4*c^3*d^4 + 30*a^4*b^3*c^2*d^5 + 60*
a^5*b^2*c*d^6 - 147*a^6*b*d^7)*x - 420*(b^7*d^7*x^7 + 7*a*b^6*d^7*x^6 + 21*a^2*b^5*d^7*x^5 + 35*a^3*b^4*d^7*x^
4 + 35*a^4*b^3*d^7*x^3 + 21*a^5*b^2*d^7*x^2 + 7*a^6*b*d^7*x + a^7*d^7)*log(b*x + a))/(b^15*x^7 + 7*a*b^14*x^6
+ 21*a^2*b^13*x^5 + 35*a^3*b^12*x^4 + 35*a^4*b^11*x^3 + 21*a^5*b^10*x^2 + 7*a^6*b^9*x + a^7*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)**8,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 466 vs. \(2 (182) = 364\).
time = 0.00, size = 494, normalized size = 2.55 \begin {gather*} \frac {\frac {1}{420} \left (\left (2940 d^{7} b^{5} a-2940 d^{6} b^{6} c\right ) x^{6}+\left (13230 d^{7} b^{4} a^{2}-8820 d^{6} b^{5} a c-4410 d^{5} b^{6} c^{2}\right ) x^{5}+\left (26950 d^{7} b^{3} a^{3}-14700 d^{6} b^{4} a^{2} c-7350 d^{5} b^{5} a c^{2}-4900 d^{4} b^{6} c^{3}\right ) x^{4}+\left (30625 d^{7} b^{2} a^{4}-14700 d^{6} b^{3} a^{3} c-7350 d^{5} b^{4} a^{2} c^{2}-4900 d^{4} b^{5} a c^{3}-3675 d^{3} b^{6} c^{4}\right ) x^{3}+\left (20139 d^{7} b a^{5}-8820 d^{6} b^{2} a^{4} c-4410 d^{5} b^{3} a^{3} c^{2}-2940 d^{4} b^{4} a^{2} c^{3}-2205 d^{3} b^{5} a c^{4}-1764 d^{2} b^{6} c^{5}\right ) x^{2}+\left (7203 d^{7} a^{6}-2940 d^{6} b a^{5} c-1470 d^{5} b^{2} a^{4} c^{2}-980 d^{4} b^{3} a^{3} c^{3}-735 d^{3} b^{4} a^{2} c^{4}-588 d^{2} b^{5} a c^{5}-490 d b^{6} c^{6}\right ) x+\frac {1089 d^{7} a^{7}-420 d^{6} b a^{6} c-210 d^{5} b^{2} a^{5} c^{2}-140 d^{4} b^{3} a^{4} c^{3}-105 d^{3} b^{4} a^{3} c^{4}-84 d^{2} b^{5} a^{2} c^{5}-70 d b^{6} a c^{6}-60 b^{7} c^{7}}{b}\right )}{b^{7} \left (x b+a\right )^{7}}+\frac {d^{7} \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)^8,x)

[Out]

d^7*log(abs(b*x + a))/b^8 - 1/420*(2940*(b^6*c*d^6 - a*b^5*d^7)*x^6 + 4410*(b^6*c^2*d^5 + 2*a*b^5*c*d^6 - 3*a^
2*b^4*d^7)*x^5 + 2450*(2*b^6*c^3*d^4 + 3*a*b^5*c^2*d^5 + 6*a^2*b^4*c*d^6 - 11*a^3*b^3*d^7)*x^4 + 1225*(3*b^6*c
^4*d^3 + 4*a*b^5*c^3*d^4 + 6*a^2*b^4*c^2*d^5 + 12*a^3*b^3*c*d^6 - 25*a^4*b^2*d^7)*x^3 + 147*(12*b^6*c^5*d^2 +
15*a*b^5*c^4*d^3 + 20*a^2*b^4*c^3*d^4 + 30*a^3*b^3*c^2*d^5 + 60*a^4*b^2*c*d^6 - 137*a^5*b*d^7)*x^2 + 49*(10*b^
6*c^6*d + 12*a*b^5*c^5*d^2 + 15*a^2*b^4*c^4*d^3 + 20*a^3*b^3*c^3*d^4 + 30*a^4*b^2*c^2*d^5 + 60*a^5*b*c*d^6 - 1
47*a^6*d^7)*x + (60*b^7*c^7 + 70*a*b^6*c^6*d + 84*a^2*b^5*c^5*d^2 + 105*a^3*b^4*c^4*d^3 + 140*a^4*b^3*c^3*d^4
+ 210*a^5*b^2*c^2*d^5 + 420*a^6*b*c*d^6 - 1089*a^7*d^7)/b)/((b*x + a)^7*b^7)

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Mupad [B]
time = 0.35, size = 461, normalized size = 2.38 \begin {gather*} \frac {d^7\,\ln \left (a+b\,x\right )}{b^8}-\frac {x\,\left (-\frac {343\,a^6\,b\,d^7}{20}+7\,a^5\,b^2\,c\,d^6+\frac {7\,a^4\,b^3\,c^2\,d^5}{2}+\frac {7\,a^3\,b^4\,c^3\,d^4}{3}+\frac {7\,a^2\,b^5\,c^4\,d^3}{4}+\frac {7\,a\,b^6\,c^5\,d^2}{5}+\frac {7\,b^7\,c^6\,d}{6}\right )-x^6\,\left (7\,a\,b^6\,d^7-7\,b^7\,c\,d^6\right )+x^3\,\left (-\frac {875\,a^4\,b^3\,d^7}{12}+35\,a^3\,b^4\,c\,d^6+\frac {35\,a^2\,b^5\,c^2\,d^5}{2}+\frac {35\,a\,b^6\,c^3\,d^4}{3}+\frac {35\,b^7\,c^4\,d^3}{4}\right )+x^5\,\left (-\frac {63\,a^2\,b^5\,d^7}{2}+21\,a\,b^6\,c\,d^6+\frac {21\,b^7\,c^2\,d^5}{2}\right )+x^2\,\left (-\frac {959\,a^5\,b^2\,d^7}{20}+21\,a^4\,b^3\,c\,d^6+\frac {21\,a^3\,b^4\,c^2\,d^5}{2}+7\,a^2\,b^5\,c^3\,d^4+\frac {21\,a\,b^6\,c^4\,d^3}{4}+\frac {21\,b^7\,c^5\,d^2}{5}\right )-\frac {363\,a^7\,d^7}{140}+\frac {b^7\,c^7}{7}+x^4\,\left (-\frac {385\,a^3\,b^4\,d^7}{6}+35\,a^2\,b^5\,c\,d^6+\frac {35\,a\,b^6\,c^2\,d^5}{2}+\frac {35\,b^7\,c^3\,d^4}{3}\right )+\frac {a^2\,b^5\,c^5\,d^2}{5}+\frac {a^3\,b^4\,c^4\,d^3}{4}+\frac {a^4\,b^3\,c^3\,d^4}{3}+\frac {a^5\,b^2\,c^2\,d^5}{2}+\frac {a\,b^6\,c^6\,d}{6}+a^6\,b\,c\,d^6}{b^8\,{\left (a+b\,x\right )}^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^7/(a + b*x)^8,x)

[Out]

(d^7*log(a + b*x))/b^8 - (x*((7*b^7*c^6*d)/6 - (343*a^6*b*d^7)/20 + (7*a*b^6*c^5*d^2)/5 + 7*a^5*b^2*c*d^6 + (7
*a^2*b^5*c^4*d^3)/4 + (7*a^3*b^4*c^3*d^4)/3 + (7*a^4*b^3*c^2*d^5)/2) - x^6*(7*a*b^6*d^7 - 7*b^7*c*d^6) + x^3*(
(35*b^7*c^4*d^3)/4 - (875*a^4*b^3*d^7)/12 + (35*a*b^6*c^3*d^4)/3 + 35*a^3*b^4*c*d^6 + (35*a^2*b^5*c^2*d^5)/2)
+ x^5*((21*b^7*c^2*d^5)/2 - (63*a^2*b^5*d^7)/2 + 21*a*b^6*c*d^6) + x^2*((21*b^7*c^5*d^2)/5 - (959*a^5*b^2*d^7)
/20 + (21*a*b^6*c^4*d^3)/4 + 21*a^4*b^3*c*d^6 + 7*a^2*b^5*c^3*d^4 + (21*a^3*b^4*c^2*d^5)/2) - (363*a^7*d^7)/14
0 + (b^7*c^7)/7 + x^4*((35*b^7*c^3*d^4)/3 - (385*a^3*b^4*d^7)/6 + (35*a*b^6*c^2*d^5)/2 + 35*a^2*b^5*c*d^6) + (
a^2*b^5*c^5*d^2)/5 + (a^3*b^4*c^4*d^3)/4 + (a^4*b^3*c^3*d^4)/3 + (a^5*b^2*c^2*d^5)/2 + (a*b^6*c^6*d)/6 + a^6*b
*c*d^6)/(b^8*(a + b*x)^7)

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