3.1290 \(\int \frac{(c+d x)^7}{(a+b x)^8} \, dx\)

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

[Out]

-(b*c - a*d)^7/(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

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Rubi [A]  time = 0.155908, antiderivative size = 194, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {43} \[ -\frac{7 d^6 (b c-a d)}{b^8 (a+b x)}-\frac{21 d^5 (b c-a d)^2}{2 b^8 (a+b x)^2}-\frac{35 d^4 (b c-a d)^3}{3 b^8 (a+b x)^3}-\frac{35 d^3 (b c-a d)^4}{4 b^8 (a+b x)^4}-\frac{21 d^2 (b c-a d)^5}{5 b^8 (a+b x)^5}-\frac{7 d (b c-a d)^6}{6 b^8 (a+b x)^6}-\frac{(b c-a d)^7}{7 b^8 (a+b x)^7}+\frac{d^7 \log (a+b x)}{b^8} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^7/(a + b*x)^8,x]

[Out]

-(b*c - a*d)^7/(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 43

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*}

Mathematica [A]  time = 0.160317, size = 308, normalized size = 1.59 \[ \frac{d^7 \log (a+b x)}{b^8}-\frac{(b c-a d) \left (a^2 b^4 d^2 \left (6909 c^2 d^2 x^2+1813 c^3 d x+214 c^4+15925 c d^3 x^3+26950 d^4 x^4\right )+a^3 b^3 d^3 \left (2793 c^2 d x+319 c^3+11319 c d^2 x^2+30625 d^3 x^3\right )+3 a^4 b^2 d^4 \left (153 c^2+1421 c d x+6713 d^2 x^2\right )+3 a^5 b d^5 (223 c+2401 d x)+1089 a^6 d^6+a b^5 d \left (3969 c^3 d^2 x^2+8575 c^2 d^3 x^3+1078 c^4 d x+130 c^5+12250 c d^4 x^4+13230 d^5 x^5\right )+b^6 \left (1764 c^4 d^2 x^2+3675 c^3 d^3 x^3+4900 c^2 d^4 x^4+490 c^5 d x+60 c^6+4410 c d^5 x^5+2940 d^6 x^6\right )\right )}{420 b^8 (a+b x)^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^7/(a + b*x)^8,x]

[Out]

-((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 + 6713*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 + 1813*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)))/(420*b^8*(a + b*x)^7) + (d^7*Log[a + b*x])/
b^8

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Maple [B]  time = 0.01, size = 672, normalized size = 3.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

21*d^3/b^4/(b*x+a)^5*a*c^4+35*d^6/b^7/(b*x+a)^4*a^3*c-1/7/b/(b*x+a)^7*c^7-35/3*d^4/b^5/(b*x+a)^3*c^3+7/b^8*d^7
/(b*x+a)*a-7/b^7*d^6/(b*x+a)*c-7/6*d^7/b^8/(b*x+a)^6*a^6+21/5*d^7/b^8/(b*x+a)^5*a^5-21/5*d^2/b^3/(b*x+a)^5*c^5
-35/4*d^7/b^8/(b*x+a)^4*a^4-35/4*d^3/b^4/(b*x+a)^4*c^4+1/7/b^8/(b*x+a)^7*a^7*d^7-7/6*d/b^2/(b*x+a)^6*c^6-21/2*
d^7/b^8/(b*x+a)^2*a^2-21/2*d^5/b^6/(b*x+a)^2*c^2+35/3*d^7/b^8/(b*x+a)^3*a^3+1/b^2/(b*x+a)^7*a*c^6*d+7*d^6/b^7/
(b*x+a)^6*a^5*c-3/b^3/(b*x+a)^7*a^2*c^5*d^2-105/2*d^5/b^6/(b*x+a)^4*a^2*c^2+35*d^4/b^5/(b*x+a)^4*a*c^3-1/b^7/(
b*x+a)^7*a^6*c*d^6+3/b^6/(b*x+a)^7*a^5*c^2*d^5-5/b^5/(b*x+a)^7*c^3*d^4*a^4+5/b^4/(b*x+a)^7*a^3*c^4*d^3+35*d^5/
b^6/(b*x+a)^3*a*c^2-21*d^6/b^7/(b*x+a)^5*a^4*c+42*d^5/b^6/(b*x+a)^5*a^3*c^2-42*d^4/b^5/(b*x+a)^5*a^2*c^3-35*d^
6/b^7/(b*x+a)^3*a^2*c-35/2*d^5/b^6/(b*x+a)^6*a^4*c^2+70/3*d^4/b^5/(b*x+a)^6*a^3*c^3-35/2*d^3/b^4/(b*x+a)^6*a^2
*c^4+7*d^2/b^3/(b*x+a)^6*a*c^5+21*d^6/b^7/(b*x+a)^2*a*c+d^7*ln(b*x+a)/b^8

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Maxima [B]  time = 1.07051, size = 721, normalized size = 3.72 \begin{align*} -\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{align*}

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]  time = 2.15655, size = 1320, normalized size = 6.8 \begin{align*} -\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{align*}

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., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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]  time = 1.06038, size = 629, normalized size = 3.24 \begin{align*} \frac{d^{7} \log \left ({\left | b x + a \right |}\right )}{b^{8}} - \frac{2940 \,{\left (b^{6} c d^{6} - a b^{5} d^{7}\right )} x^{6} + 4410 \,{\left (b^{6} c^{2} d^{5} + 2 \, a b^{5} c d^{6} - 3 \, a^{2} b^{4} d^{7}\right )} x^{5} + 2450 \,{\left (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}\right )} x^{4} + 1225 \,{\left (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}\right )} x^{3} + 147 \,{\left (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}\right )} x^{2} + 49 \,{\left (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} - 147 \, a^{6} d^{7}\right )} x + \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}}{b}}{420 \,{\left (b x + a\right )}^{7} b^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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)