3.1283 \(\int \frac{(c+d x)^7}{a+b x} \, dx\)

Optimal. Leaf size=169 \[ \frac{d x (b c-a d)^6}{b^7}+\frac{(c+d x)^2 (b c-a d)^5}{2 b^6}+\frac{(c+d x)^3 (b c-a d)^4}{3 b^5}+\frac{(c+d x)^4 (b c-a d)^3}{4 b^4}+\frac{(c+d x)^5 (b c-a d)^2}{5 b^3}+\frac{(c+d x)^6 (b c-a d)}{6 b^2}+\frac{(b c-a d)^7 \log (a+b x)}{b^8}+\frac{(c+d x)^7}{7 b} \]

[Out]

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

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Rubi [A]  time = 0.0725765, antiderivative size = 169, 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} \[ \frac{d x (b c-a d)^6}{b^7}+\frac{(c+d x)^2 (b c-a d)^5}{2 b^6}+\frac{(c+d x)^3 (b c-a d)^4}{3 b^5}+\frac{(c+d x)^4 (b c-a d)^3}{4 b^4}+\frac{(c+d x)^5 (b c-a d)^2}{5 b^3}+\frac{(c+d x)^6 (b c-a d)}{6 b^2}+\frac{(b c-a d)^7 \log (a+b x)}{b^8}+\frac{(c+d x)^7}{7 b} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^7/(a + b*x),x]

[Out]

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

Mathematica [A]  time = 0.142045, size = 304, normalized size = 1.8 \[ \frac{d x \left (21 a^2 b^4 d^2 \left (140 c^2 d^2 x^2+350 c^3 d x+700 c^4+35 c d^3 x^3+4 d^4 x^4\right )-35 a^3 b^3 d^3 \left (126 c^2 d x+420 c^3+28 c d^2 x^2+3 d^3 x^3\right )+70 a^4 b^2 d^4 \left (126 c^2+21 c d x+2 d^2 x^2\right )-210 a^5 b d^5 (14 c+d x)+420 a^6 d^6-7 a b^5 d \left (700 c^3 d^2 x^2+315 c^2 d^3 x^3+1050 c^4 d x+1260 c^5+84 c d^4 x^4+10 d^5 x^5\right )+b^6 \left (4900 c^4 d^2 x^2+3675 c^3 d^3 x^3+1764 c^2 d^4 x^4+4410 c^5 d x+2940 c^6+490 c d^5 x^5+60 d^6 x^6\right )\right )}{420 b^7}+\frac{(b c-a d)^7 \log (a+b x)}{b^8} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^7/(a + b*x),x]

[Out]

(d*x*(420*a^6*d^6 - 210*a^5*b*d^5*(14*c + d*x) + 70*a^4*b^2*d^4*(126*c^2 + 21*c*d*x + 2*d^2*x^2) - 35*a^3*b^3*
d^3*(420*c^3 + 126*c^2*d*x + 28*c*d^2*x^2 + 3*d^3*x^3) + 21*a^2*b^4*d^2*(700*c^4 + 350*c^3*d*x + 140*c^2*d^2*x
^2 + 35*c*d^3*x^3 + 4*d^4*x^4) - 7*a*b^5*d*(1260*c^5 + 1050*c^4*d*x + 700*c^3*d^2*x^2 + 315*c^2*d^3*x^3 + 84*c
*d^4*x^4 + 10*d^5*x^5) + b^6*(2940*c^6 + 4410*c^5*d*x + 4900*c^4*d^2*x^2 + 3675*c^3*d^3*x^3 + 1764*c^2*d^4*x^4
+ 490*c*d^5*x^5 + 60*d^6*x^6)))/(420*b^7) + ((b*c - a*d)^7*Log[a + b*x])/b^8

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Maple [B]  time = 0.006, size = 539, normalized size = 3.2 \begin{align*}{\frac{{d}^{7}{x}^{3}{a}^{4}}{3\,{b}^{5}}}+{\frac{35\,{d}^{3}{x}^{3}{c}^{4}}{3\,b}}-{\frac{{d}^{7}{x}^{2}{a}^{5}}{2\,{b}^{6}}}+{\frac{21\,{d}^{2}{x}^{2}{c}^{5}}{2\,b}}-{\frac{{d}^{7}{x}^{6}a}{6\,{b}^{2}}}+{\frac{7\,{d}^{6}{x}^{6}c}{6\,b}}+{\frac{{d}^{7}{x}^{5}{a}^{2}}{5\,{b}^{3}}}+{\frac{21\,{d}^{5}{x}^{5}{c}^{2}}{5\,b}}-{\frac{\ln \left ( bx+a \right ){a}^{7}{d}^{7}}{{b}^{8}}}+7\,{\frac{d{c}^{6}x}{b}}+{\frac{{a}^{6}{d}^{7}x}{{b}^{7}}}-{\frac{{d}^{7}{x}^{4}{a}^{3}}{4\,{b}^{4}}}+{\frac{35\,{d}^{4}{x}^{4}{c}^{3}}{4\,b}}-{\frac{7\,{d}^{6}{x}^{3}{a}^{3}c}{3\,{b}^{4}}}+7\,{\frac{{d}^{5}{x}^{3}{a}^{2}{c}^{2}}{{b}^{3}}}-{\frac{35\,{d}^{4}{x}^{3}a{c}^{3}}{3\,{b}^{2}}}+{\frac{7\,{d}^{6}{x}^{4}{a}^{2}c}{4\,{b}^{3}}}-{\frac{21\,{d}^{5}{x}^{4}a{c}^{2}}{4\,{b}^{2}}}-{\frac{7\,{d}^{6}{x}^{5}ac}{5\,{b}^{2}}}-21\,{\frac{\ln \left ( bx+a \right ){a}^{5}{c}^{2}{d}^{5}}{{b}^{6}}}+35\,{\frac{{a}^{4}\ln \left ( bx+a \right ){c}^{3}{d}^{4}}{{b}^{5}}}-35\,{\frac{{a}^{3}\ln \left ( bx+a \right ){c}^{4}{d}^{3}}{{b}^{4}}}+21\,{\frac{{a}^{2}\ln \left ( bx+a \right ){c}^{5}{d}^{2}}{{b}^{3}}}+7\,{\frac{\ln \left ( bx+a \right ){a}^{6}c{d}^{6}}{{b}^{7}}}-{\frac{21\,{d}^{5}{x}^{2}{a}^{3}{c}^{2}}{2\,{b}^{4}}}-7\,{\frac{a\ln \left ( bx+a \right ){c}^{6}d}{{b}^{2}}}-7\,{\frac{{a}^{5}c{d}^{6}x}{{b}^{6}}}-21\,{\frac{a{c}^{5}{d}^{2}x}{{b}^{2}}}+{\frac{7\,{d}^{6}{x}^{2}{a}^{4}c}{2\,{b}^{5}}}+{\frac{35\,{d}^{4}{x}^{2}{a}^{2}{c}^{3}}{2\,{b}^{3}}}-{\frac{35\,{d}^{3}{x}^{2}a{c}^{4}}{2\,{b}^{2}}}+35\,{\frac{{a}^{2}{c}^{4}{d}^{3}x}{{b}^{3}}}+21\,{\frac{{d}^{5}{a}^{4}{c}^{2}x}{{b}^{5}}}-35\,{\frac{{a}^{3}{c}^{3}{d}^{4}x}{{b}^{4}}}+{\frac{{d}^{7}{x}^{7}}{7\,b}}+{\frac{\ln \left ( bx+a \right ){c}^{7}}{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 0.96687, size = 621, normalized size = 3.67 \begin{align*} \frac{60 \, b^{6} d^{7} x^{7} + 70 \,{\left (7 \, b^{6} c d^{6} - a b^{5} d^{7}\right )} x^{6} + 84 \,{\left (21 \, b^{6} c^{2} d^{5} - 7 \, a b^{5} c d^{6} + a^{2} b^{4} d^{7}\right )} x^{5} + 105 \,{\left (35 \, b^{6} c^{3} d^{4} - 21 \, a b^{5} c^{2} d^{5} + 7 \, a^{2} b^{4} c d^{6} - a^{3} b^{3} d^{7}\right )} x^{4} + 140 \,{\left (35 \, b^{6} c^{4} d^{3} - 35 \, a b^{5} c^{3} d^{4} + 21 \, a^{2} b^{4} c^{2} d^{5} - 7 \, a^{3} b^{3} c d^{6} + a^{4} b^{2} d^{7}\right )} x^{3} + 210 \,{\left (21 \, b^{6} c^{5} d^{2} - 35 \, a b^{5} c^{4} d^{3} + 35 \, a^{2} b^{4} c^{3} d^{4} - 21 \, a^{3} b^{3} c^{2} d^{5} + 7 \, a^{4} b^{2} c d^{6} - a^{5} b d^{7}\right )} x^{2} + 420 \,{\left (7 \, b^{6} c^{6} d - 21 \, a b^{5} c^{5} d^{2} + 35 \, a^{2} b^{4} c^{4} d^{3} - 35 \, a^{3} b^{3} c^{3} d^{4} + 21 \, a^{4} b^{2} c^{2} d^{5} - 7 \, a^{5} b c d^{6} + a^{6} d^{7}\right )} x}{420 \, b^{7}} + \frac{{\left (b^{7} c^{7} - 7 \, a b^{6} c^{6} d + 21 \, a^{2} b^{5} c^{5} d^{2} - 35 \, a^{3} b^{4} c^{4} d^{3} + 35 \, a^{4} b^{3} c^{3} d^{4} - 21 \, a^{5} b^{2} c^{2} d^{5} + 7 \, a^{6} b c d^{6} - a^{7} d^{7}\right )} \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),x, algorithm="maxima")

[Out]

1/420*(60*b^6*d^7*x^7 + 70*(7*b^6*c*d^6 - a*b^5*d^7)*x^6 + 84*(21*b^6*c^2*d^5 - 7*a*b^5*c*d^6 + a^2*b^4*d^7)*x
^5 + 105*(35*b^6*c^3*d^4 - 21*a*b^5*c^2*d^5 + 7*a^2*b^4*c*d^6 - a^3*b^3*d^7)*x^4 + 140*(35*b^6*c^4*d^3 - 35*a*
b^5*c^3*d^4 + 21*a^2*b^4*c^2*d^5 - 7*a^3*b^3*c*d^6 + a^4*b^2*d^7)*x^3 + 210*(21*b^6*c^5*d^2 - 35*a*b^5*c^4*d^3
+ 35*a^2*b^4*c^3*d^4 - 21*a^3*b^3*c^2*d^5 + 7*a^4*b^2*c*d^6 - a^5*b*d^7)*x^2 + 420*(7*b^6*c^6*d - 21*a*b^5*c^
5*d^2 + 35*a^2*b^4*c^4*d^3 - 35*a^3*b^3*c^3*d^4 + 21*a^4*b^2*c^2*d^5 - 7*a^5*b*c*d^6 + a^6*d^7)*x)/b^7 + (b^7*
c^7 - 7*a*b^6*c^6*d + 21*a^2*b^5*c^5*d^2 - 35*a^3*b^4*c^4*d^3 + 35*a^4*b^3*c^3*d^4 - 21*a^5*b^2*c^2*d^5 + 7*a^
6*b*c*d^6 - a^7*d^7)*log(b*x + a)/b^8

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Fricas [B]  time = 2.32019, size = 952, normalized size = 5.63 \begin{align*} \frac{60 \, b^{7} d^{7} x^{7} + 70 \,{\left (7 \, b^{7} c d^{6} - a b^{6} d^{7}\right )} x^{6} + 84 \,{\left (21 \, b^{7} c^{2} d^{5} - 7 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 105 \,{\left (35 \, b^{7} c^{3} d^{4} - 21 \, a b^{6} c^{2} d^{5} + 7 \, a^{2} b^{5} c d^{6} - a^{3} b^{4} d^{7}\right )} x^{4} + 140 \,{\left (35 \, b^{7} c^{4} d^{3} - 35 \, a b^{6} c^{3} d^{4} + 21 \, a^{2} b^{5} c^{2} d^{5} - 7 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 210 \,{\left (21 \, b^{7} c^{5} d^{2} - 35 \, a b^{6} c^{4} d^{3} + 35 \, a^{2} b^{5} c^{3} d^{4} - 21 \, a^{3} b^{4} c^{2} d^{5} + 7 \, a^{4} b^{3} c d^{6} - a^{5} b^{2} d^{7}\right )} x^{2} + 420 \,{\left (7 \, b^{7} c^{6} d - 21 \, a b^{6} c^{5} d^{2} + 35 \, a^{2} b^{5} c^{4} d^{3} - 35 \, a^{3} b^{4} c^{3} d^{4} + 21 \, a^{4} b^{3} c^{2} d^{5} - 7 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x + 420 \,{\left (b^{7} c^{7} - 7 \, a b^{6} c^{6} d + 21 \, a^{2} b^{5} c^{5} d^{2} - 35 \, a^{3} b^{4} c^{4} d^{3} + 35 \, a^{4} b^{3} c^{3} d^{4} - 21 \, a^{5} b^{2} c^{2} d^{5} + 7 \, a^{6} b c d^{6} - a^{7} d^{7}\right )} \log \left (b x + a\right )}{420 \, b^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/420*(60*b^7*d^7*x^7 + 70*(7*b^7*c*d^6 - a*b^6*d^7)*x^6 + 84*(21*b^7*c^2*d^5 - 7*a*b^6*c*d^6 + a^2*b^5*d^7)*x
^5 + 105*(35*b^7*c^3*d^4 - 21*a*b^6*c^2*d^5 + 7*a^2*b^5*c*d^6 - a^3*b^4*d^7)*x^4 + 140*(35*b^7*c^4*d^3 - 35*a*
b^6*c^3*d^4 + 21*a^2*b^5*c^2*d^5 - 7*a^3*b^4*c*d^6 + a^4*b^3*d^7)*x^3 + 210*(21*b^7*c^5*d^2 - 35*a*b^6*c^4*d^3
+ 35*a^2*b^5*c^3*d^4 - 21*a^3*b^4*c^2*d^5 + 7*a^4*b^3*c*d^6 - a^5*b^2*d^7)*x^2 + 420*(7*b^7*c^6*d - 21*a*b^6*
c^5*d^2 + 35*a^2*b^5*c^4*d^3 - 35*a^3*b^4*c^3*d^4 + 21*a^4*b^3*c^2*d^5 - 7*a^5*b^2*c*d^6 + a^6*b*d^7)*x + 420*
(b^7*c^7 - 7*a*b^6*c^6*d + 21*a^2*b^5*c^5*d^2 - 35*a^3*b^4*c^4*d^3 + 35*a^4*b^3*c^3*d^4 - 21*a^5*b^2*c^2*d^5 +
7*a^6*b*c*d^6 - a^7*d^7)*log(b*x + a))/b^8

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Sympy [B]  time = 0.968406, size = 384, normalized size = 2.27 \begin{align*} \frac{d^{7} x^{7}}{7 b} - \frac{x^{6} \left (a d^{7} - 7 b c d^{6}\right )}{6 b^{2}} + \frac{x^{5} \left (a^{2} d^{7} - 7 a b c d^{6} + 21 b^{2} c^{2} d^{5}\right )}{5 b^{3}} - \frac{x^{4} \left (a^{3} d^{7} - 7 a^{2} b c d^{6} + 21 a b^{2} c^{2} d^{5} - 35 b^{3} c^{3} d^{4}\right )}{4 b^{4}} + \frac{x^{3} \left (a^{4} d^{7} - 7 a^{3} b c d^{6} + 21 a^{2} b^{2} c^{2} d^{5} - 35 a b^{3} c^{3} d^{4} + 35 b^{4} c^{4} d^{3}\right )}{3 b^{5}} - \frac{x^{2} \left (a^{5} d^{7} - 7 a^{4} b c d^{6} + 21 a^{3} b^{2} c^{2} d^{5} - 35 a^{2} b^{3} c^{3} d^{4} + 35 a b^{4} c^{4} d^{3} - 21 b^{5} c^{5} d^{2}\right )}{2 b^{6}} + \frac{x \left (a^{6} d^{7} - 7 a^{5} b c d^{6} + 21 a^{4} b^{2} c^{2} d^{5} - 35 a^{3} b^{3} c^{3} d^{4} + 35 a^{2} b^{4} c^{4} d^{3} - 21 a b^{5} c^{5} d^{2} + 7 b^{6} c^{6} d\right )}{b^{7}} - \frac{\left (a d - b c\right )^{7} \log{\left (a + b x \right )}}{b^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.04438, size = 671, normalized size = 3.97 \begin{align*} \frac{60 \, b^{6} d^{7} x^{7} + 490 \, b^{6} c d^{6} x^{6} - 70 \, a b^{5} d^{7} x^{6} + 1764 \, b^{6} c^{2} d^{5} x^{5} - 588 \, a b^{5} c d^{6} x^{5} + 84 \, a^{2} b^{4} d^{7} x^{5} + 3675 \, b^{6} c^{3} d^{4} x^{4} - 2205 \, a b^{5} c^{2} d^{5} x^{4} + 735 \, a^{2} b^{4} c d^{6} x^{4} - 105 \, a^{3} b^{3} d^{7} x^{4} + 4900 \, b^{6} c^{4} d^{3} x^{3} - 4900 \, a b^{5} c^{3} d^{4} x^{3} + 2940 \, a^{2} b^{4} c^{2} d^{5} x^{3} - 980 \, a^{3} b^{3} c d^{6} x^{3} + 140 \, a^{4} b^{2} d^{7} x^{3} + 4410 \, b^{6} c^{5} d^{2} x^{2} - 7350 \, a b^{5} c^{4} d^{3} x^{2} + 7350 \, a^{2} b^{4} c^{3} d^{4} x^{2} - 4410 \, a^{3} b^{3} c^{2} d^{5} x^{2} + 1470 \, a^{4} b^{2} c d^{6} x^{2} - 210 \, a^{5} b d^{7} x^{2} + 2940 \, b^{6} c^{6} d x - 8820 \, a b^{5} c^{5} d^{2} x + 14700 \, a^{2} b^{4} c^{4} d^{3} x - 14700 \, a^{3} b^{3} c^{3} d^{4} x + 8820 \, a^{4} b^{2} c^{2} d^{5} x - 2940 \, a^{5} b c d^{6} x + 420 \, a^{6} d^{7} x}{420 \, b^{7}} + \frac{{\left (b^{7} c^{7} - 7 \, a b^{6} c^{6} d + 21 \, a^{2} b^{5} c^{5} d^{2} - 35 \, a^{3} b^{4} c^{4} d^{3} + 35 \, a^{4} b^{3} c^{3} d^{4} - 21 \, a^{5} b^{2} c^{2} d^{5} + 7 \, a^{6} b c d^{6} - a^{7} d^{7}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/420*(60*b^6*d^7*x^7 + 490*b^6*c*d^6*x^6 - 70*a*b^5*d^7*x^6 + 1764*b^6*c^2*d^5*x^5 - 588*a*b^5*c*d^6*x^5 + 84
*a^2*b^4*d^7*x^5 + 3675*b^6*c^3*d^4*x^4 - 2205*a*b^5*c^2*d^5*x^4 + 735*a^2*b^4*c*d^6*x^4 - 105*a^3*b^3*d^7*x^4
+ 4900*b^6*c^4*d^3*x^3 - 4900*a*b^5*c^3*d^4*x^3 + 2940*a^2*b^4*c^2*d^5*x^3 - 980*a^3*b^3*c*d^6*x^3 + 140*a^4*
b^2*d^7*x^3 + 4410*b^6*c^5*d^2*x^2 - 7350*a*b^5*c^4*d^3*x^2 + 7350*a^2*b^4*c^3*d^4*x^2 - 4410*a^3*b^3*c^2*d^5*
x^2 + 1470*a^4*b^2*c*d^6*x^2 - 210*a^5*b*d^7*x^2 + 2940*b^6*c^6*d*x - 8820*a*b^5*c^5*d^2*x + 14700*a^2*b^4*c^4
*d^3*x - 14700*a^3*b^3*c^3*d^4*x + 8820*a^4*b^2*c^2*d^5*x - 2940*a^5*b*c*d^6*x + 420*a^6*d^7*x)/b^7 + (b^7*c^7
- 7*a*b^6*c^6*d + 21*a^2*b^5*c^5*d^2 - 35*a^3*b^4*c^4*d^3 + 35*a^4*b^3*c^3*d^4 - 21*a^5*b^2*c^2*d^5 + 7*a^6*b
*c*d^6 - a^7*d^7)*log(abs(b*x + a))/b^8