3.1284 \(\int \frac{(c+d x)^7}{(a+b x)^2} \, dx\)

Optimal. Leaf size=187 \[ \frac{7 d^6 (a+b x)^5 (b c-a d)}{5 b^8}+\frac{21 d^5 (a+b x)^4 (b c-a d)^2}{4 b^8}+\frac{35 d^4 (a+b x)^3 (b c-a d)^3}{3 b^8}+\frac{35 d^3 (a+b x)^2 (b c-a d)^4}{2 b^8}-\frac{(b c-a d)^7}{b^8 (a+b x)}+\frac{7 d (b c-a d)^6 \log (a+b x)}{b^8}+\frac{d^7 (a+b x)^6}{6 b^8}+\frac{21 d^2 x (b c-a d)^5}{b^7} \]

[Out]

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

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Rubi [A]  time = 0.492174, antiderivative size = 187, 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 \[ \frac{7 d^6 (a+b x)^5 (b c-a d)}{5 b^8}+\frac{21 d^5 (a+b x)^4 (b c-a d)^2}{4 b^8}+\frac{35 d^4 (a+b x)^3 (b c-a d)^3}{3 b^8}+\frac{35 d^3 (a+b x)^2 (b c-a d)^4}{2 b^8}-\frac{(b c-a d)^7}{b^8 (a+b x)}+\frac{7 d (b c-a d)^6 \log (a+b x)}{b^8}+\frac{d^7 (a+b x)^6}{6 b^8}+\frac{21 d^2 x (b c-a d)^5}{b^7} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^7/(a + b*x)^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 77.3604, size = 172, normalized size = 0.92 \[ - \frac{21 d^{2} x \left (a d - b c\right )^{5}}{b^{7}} + \frac{d^{7} \left (a + b x\right )^{6}}{6 b^{8}} - \frac{7 d^{6} \left (a + b x\right )^{5} \left (a d - b c\right )}{5 b^{8}} + \frac{21 d^{5} \left (a + b x\right )^{4} \left (a d - b c\right )^{2}}{4 b^{8}} - \frac{35 d^{4} \left (a + b x\right )^{3} \left (a d - b c\right )^{3}}{3 b^{8}} + \frac{35 d^{3} \left (a + b x\right )^{2} \left (a d - b c\right )^{4}}{2 b^{8}} + \frac{7 d \left (a d - b c\right )^{6} \log{\left (a + b x \right )}}{b^{8}} + \frac{\left (a d - b c\right )^{7}}{b^{8} \left (a + b x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**7/(b*x+a)**2,x)

[Out]

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

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Mathematica [B]  time = 0.213495, size = 388, normalized size = 2.07 \[ \frac{60 a^7 d^7-60 a^6 b d^6 (7 c+6 d x)+210 a^5 b^2 d^5 \left (6 c^2+10 c d x-d^2 x^2\right )+70 a^4 b^3 d^4 \left (-30 c^3-72 c^2 d x+18 c d^2 x^2+d^3 x^3\right )-35 a^3 b^4 d^3 \left (-60 c^4-180 c^3 d x+90 c^2 d^2 x^2+12 c d^3 x^3+d^4 x^4\right )+21 a^2 b^5 d^2 \left (-60 c^5-200 c^4 d x+200 c^3 d^2 x^2+50 c^2 d^3 x^3+10 c d^4 x^4+d^5 x^5\right )-7 a b^6 d \left (-60 c^6-180 c^5 d x+450 c^4 d^2 x^2+200 c^3 d^3 x^3+75 c^2 d^4 x^4+18 c d^5 x^5+2 d^6 x^6\right )+420 d (a+b x) (b c-a d)^6 \log (a+b x)+b^7 \left (-60 c^7+1260 c^5 d^2 x^2+1050 c^4 d^3 x^3+700 c^3 d^4 x^4+315 c^2 d^5 x^5+84 c d^6 x^6+10 d^7 x^7\right )}{60 b^8 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^7/(a + b*x)^2,x]

[Out]

(60*a^7*d^7 - 60*a^6*b*d^6*(7*c + 6*d*x) + 210*a^5*b^2*d^5*(6*c^2 + 10*c*d*x - d
^2*x^2) + 70*a^4*b^3*d^4*(-30*c^3 - 72*c^2*d*x + 18*c*d^2*x^2 + d^3*x^3) - 35*a^
3*b^4*d^3*(-60*c^4 - 180*c^3*d*x + 90*c^2*d^2*x^2 + 12*c*d^3*x^3 + d^4*x^4) + 21
*a^2*b^5*d^2*(-60*c^5 - 200*c^4*d*x + 200*c^3*d^2*x^2 + 50*c^2*d^3*x^3 + 10*c*d^
4*x^4 + d^5*x^5) - 7*a*b^6*d*(-60*c^6 - 180*c^5*d*x + 450*c^4*d^2*x^2 + 200*c^3*
d^3*x^3 + 75*c^2*d^4*x^4 + 18*c*d^5*x^5 + 2*d^6*x^6) + b^7*(-60*c^7 + 1260*c^5*d
^2*x^2 + 1050*c^4*d^3*x^3 + 700*c^3*d^4*x^4 + 315*c^2*d^5*x^5 + 84*c*d^6*x^6 + 1
0*d^7*x^7) + 420*d*(b*c - a*d)^6*(a + b*x)*Log[a + b*x])/(60*b^8*(a + b*x))

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Maple [B]  time = 0.015, size = 571, normalized size = 3.1 \[{\frac{21\,{d}^{5}{x}^{4}{c}^{2}}{4\,{b}^{2}}}-{\frac{4\,{d}^{7}{x}^{3}{a}^{3}}{3\,{b}^{5}}}+{\frac{35\,{d}^{4}{x}^{3}{c}^{3}}{3\,{b}^{2}}}+{\frac{5\,{d}^{7}{x}^{2}{a}^{4}}{2\,{b}^{6}}}+{\frac{35\,{d}^{3}{x}^{2}{c}^{4}}{2\,{b}^{2}}}-6\,{\frac{{a}^{5}{d}^{7}x}{{b}^{7}}}+21\,{\frac{{c}^{5}{d}^{2}x}{{b}^{2}}}+7\,{\frac{{d}^{7}\ln \left ( bx+a \right ){a}^{6}}{{b}^{8}}}+7\,{\frac{d\ln \left ( bx+a \right ){c}^{6}}{{b}^{2}}}+{\frac{{a}^{7}{d}^{7}}{{b}^{8} \left ( bx+a \right ) }}-{\frac{2\,{d}^{7}{x}^{5}a}{5\,{b}^{3}}}+{\frac{7\,{d}^{6}{x}^{5}c}{5\,{b}^{2}}}+{\frac{3\,{d}^{7}{x}^{4}{a}^{2}}{4\,{b}^{4}}}-{\frac{{c}^{7}}{b \left ( bx+a \right ) }}+{\frac{{d}^{7}{x}^{6}}{6\,{b}^{2}}}-70\,{\frac{a{c}^{4}{d}^{3}x}{{b}^{3}}}-14\,{\frac{{d}^{5}{x}^{3}a{c}^{2}}{{b}^{3}}}-14\,{\frac{{d}^{6}{x}^{2}{a}^{3}c}{{b}^{5}}}+{\frac{63\,{d}^{5}{x}^{2}{a}^{2}{c}^{2}}{2\,{b}^{4}}}-35\,{\frac{{d}^{4}{x}^{2}a{c}^{3}}{{b}^{3}}}-{\frac{7\,{d}^{6}{x}^{4}ac}{2\,{b}^{3}}}+7\,{\frac{{d}^{6}{x}^{3}{a}^{2}c}{{b}^{4}}}+105\,{\frac{{a}^{2}{c}^{3}{d}^{4}x}{{b}^{4}}}+35\,{\frac{{a}^{4}c{d}^{6}x}{{b}^{6}}}-84\,{\frac{{a}^{3}{c}^{2}{d}^{5}x}{{b}^{5}}}-7\,{\frac{{a}^{6}c{d}^{6}}{{b}^{7} \left ( bx+a \right ) }}+21\,{\frac{{a}^{5}{c}^{2}{d}^{5}}{{b}^{6} \left ( bx+a \right ) }}-35\,{\frac{{a}^{4}{c}^{3}{d}^{4}}{{b}^{5} \left ( bx+a \right ) }}-42\,{\frac{{d}^{2}\ln \left ( bx+a \right ) a{c}^{5}}{{b}^{3}}}+35\,{\frac{{a}^{3}{c}^{4}{d}^{3}}{{b}^{4} \left ( bx+a \right ) }}-42\,{\frac{{d}^{6}\ln \left ( bx+a \right ){a}^{5}c}{{b}^{7}}}+105\,{\frac{{d}^{5}\ln \left ( bx+a \right ){a}^{4}{c}^{2}}{{b}^{6}}}-140\,{\frac{{d}^{4}\ln \left ( bx+a \right ){a}^{3}{c}^{3}}{{b}^{5}}}+105\,{\frac{{d}^{3}\ln \left ( bx+a \right ){a}^{2}{c}^{4}}{{b}^{4}}}-21\,{\frac{{a}^{2}{c}^{5}{d}^{2}}{{b}^{3} \left ( bx+a \right ) }}+7\,{\frac{a{c}^{6}d}{{b}^{2} \left ( bx+a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^7/(b*x+a)^2,x)

[Out]

21/4*d^5/b^2*x^4*c^2-4/3*d^7/b^5*x^3*a^3+35/3*d^4/b^2*x^3*c^3+5/2*d^7/b^6*x^2*a^
4+35/2*d^3/b^2*x^2*c^4-6*d^7/b^7*a^5*x+21*d^2/b^2*c^5*x+7/b^8*d^7*ln(b*x+a)*a^6+
7/b^2*d*ln(b*x+a)*c^6+1/b^8/(b*x+a)*a^7*d^7-2/5*d^7/b^3*x^5*a+7/5*d^6/b^2*x^5*c+
3/4*d^7/b^4*x^4*a^2-1/b/(b*x+a)*c^7+1/6*d^7/b^2*x^6-70*d^3/b^3*a*c^4*x-14*d^5/b^
3*x^3*a*c^2-14*d^6/b^5*x^2*a^3*c+63/2*d^5/b^4*x^2*a^2*c^2-35*d^4/b^3*x^2*a*c^3-7
/2*d^6/b^3*x^4*a*c+7*d^6/b^4*x^3*a^2*c+105*d^4/b^4*a^2*c^3*x+35*d^6/b^6*a^4*c*x-
84*d^5/b^5*a^3*c^2*x-7/b^7/(b*x+a)*a^6*c*d^6+21/b^6/(b*x+a)*a^5*c^2*d^5-35/b^5/(
b*x+a)*a^4*c^3*d^4-42/b^3*d^2*ln(b*x+a)*a*c^5+35/b^4/(b*x+a)*a^3*c^4*d^3-42/b^7*
d^6*ln(b*x+a)*a^5*c+105/b^6*d^5*ln(b*x+a)*a^4*c^2-140/b^5*d^4*ln(b*x+a)*a^3*c^3+
105/b^4*d^3*ln(b*x+a)*a^2*c^4-21/b^3/(b*x+a)*a^2*c^5*d^2+7/b^2/(b*x+a)*a*c^6*d

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Maxima [A]  time = 1.40486, size = 630, normalized size = 3.37 \[ -\frac{b^{7} c^{7} - 7 \, a b^{6} c^{6} d + 21 \, a^{2} b^{5} c^{5} d^{2} - 35 \, a^{3} b^{4} c^{4} d^{3} + 35 \, a^{4} b^{3} c^{3} d^{4} - 21 \, a^{5} b^{2} c^{2} d^{5} + 7 \, a^{6} b c d^{6} - a^{7} d^{7}}{b^{9} x + a b^{8}} + \frac{10 \, b^{5} d^{7} x^{6} + 12 \,{\left (7 \, b^{5} c d^{6} - 2 \, a b^{4} d^{7}\right )} x^{5} + 15 \,{\left (21 \, b^{5} c^{2} d^{5} - 14 \, a b^{4} c d^{6} + 3 \, a^{2} b^{3} d^{7}\right )} x^{4} + 20 \,{\left (35 \, b^{5} c^{3} d^{4} - 42 \, a b^{4} c^{2} d^{5} + 21 \, a^{2} b^{3} c d^{6} - 4 \, a^{3} b^{2} d^{7}\right )} x^{3} + 30 \,{\left (35 \, b^{5} c^{4} d^{3} - 70 \, a b^{4} c^{3} d^{4} + 63 \, a^{2} b^{3} c^{2} d^{5} - 28 \, a^{3} b^{2} c d^{6} + 5 \, a^{4} b d^{7}\right )} x^{2} + 60 \,{\left (21 \, b^{5} c^{5} d^{2} - 70 \, a b^{4} c^{4} d^{3} + 105 \, a^{2} b^{3} c^{3} d^{4} - 84 \, a^{3} b^{2} c^{2} d^{5} + 35 \, a^{4} b c d^{6} - 6 \, a^{5} d^{7}\right )} x}{60 \, b^{7}} + \frac{7 \,{\left (b^{6} c^{6} d - 6 \, a b^{5} c^{5} d^{2} + 15 \, a^{2} b^{4} c^{4} d^{3} - 20 \, a^{3} b^{3} c^{3} d^{4} + 15 \, a^{4} b^{2} c^{2} d^{5} - 6 \, a^{5} b c d^{6} + a^{6} d^{7}\right )} \log \left (b x + a\right )}{b^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^7/(b*x + a)^2,x, algorithm="maxima")

[Out]

-(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)/(b^9*x + a*b^8) + 1/60*
(10*b^5*d^7*x^6 + 12*(7*b^5*c*d^6 - 2*a*b^4*d^7)*x^5 + 15*(21*b^5*c^2*d^5 - 14*a
*b^4*c*d^6 + 3*a^2*b^3*d^7)*x^4 + 20*(35*b^5*c^3*d^4 - 42*a*b^4*c^2*d^5 + 21*a^2
*b^3*c*d^6 - 4*a^3*b^2*d^7)*x^3 + 30*(35*b^5*c^4*d^3 - 70*a*b^4*c^3*d^4 + 63*a^2
*b^3*c^2*d^5 - 28*a^3*b^2*c*d^6 + 5*a^4*b*d^7)*x^2 + 60*(21*b^5*c^5*d^2 - 70*a*b
^4*c^4*d^3 + 105*a^2*b^3*c^3*d^4 - 84*a^3*b^2*c^2*d^5 + 35*a^4*b*c*d^6 - 6*a^5*d
^7)*x)/b^7 + 7*(b^6*c^6*d - 6*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 + 15*a^4*b^2*c^2*d^5 - 6*a^5*b*c*d^6 + a^6*d^7)*log(b*x + a)/b^8

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Fricas [A]  time = 0.226378, size = 853, normalized size = 4.56 \[ \frac{10 \, b^{7} d^{7} x^{7} - 60 \, b^{7} c^{7} + 420 \, a b^{6} c^{6} d - 1260 \, a^{2} b^{5} c^{5} d^{2} + 2100 \, a^{3} b^{4} c^{4} d^{3} - 2100 \, a^{4} b^{3} c^{3} d^{4} + 1260 \, a^{5} b^{2} c^{2} d^{5} - 420 \, a^{6} b c d^{6} + 60 \, a^{7} d^{7} + 14 \,{\left (6 \, b^{7} c d^{6} - a b^{6} d^{7}\right )} x^{6} + 21 \,{\left (15 \, b^{7} c^{2} d^{5} - 6 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 35 \,{\left (20 \, b^{7} c^{3} d^{4} - 15 \, a b^{6} c^{2} d^{5} + 6 \, a^{2} b^{5} c d^{6} - a^{3} b^{4} d^{7}\right )} x^{4} + 70 \,{\left (15 \, b^{7} c^{4} d^{3} - 20 \, a b^{6} c^{3} d^{4} + 15 \, a^{2} b^{5} c^{2} d^{5} - 6 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 210 \,{\left (6 \, b^{7} c^{5} d^{2} - 15 \, a b^{6} c^{4} d^{3} + 20 \, a^{2} b^{5} c^{3} d^{4} - 15 \, a^{3} b^{4} c^{2} d^{5} + 6 \, a^{4} b^{3} c d^{6} - a^{5} b^{2} d^{7}\right )} x^{2} + 60 \,{\left (21 \, a b^{6} c^{5} d^{2} - 70 \, a^{2} b^{5} c^{4} d^{3} + 105 \, a^{3} b^{4} c^{3} d^{4} - 84 \, a^{4} b^{3} c^{2} d^{5} + 35 \, a^{5} b^{2} c d^{6} - 6 \, a^{6} b d^{7}\right )} x + 420 \,{\left (a b^{6} c^{6} d - 6 \, a^{2} b^{5} c^{5} d^{2} + 15 \, a^{3} b^{4} c^{4} d^{3} - 20 \, a^{4} b^{3} c^{3} d^{4} + 15 \, a^{5} b^{2} c^{2} d^{5} - 6 \, a^{6} b c d^{6} + a^{7} d^{7} +{\left (b^{7} c^{6} d - 6 \, a b^{6} c^{5} d^{2} + 15 \, a^{2} b^{5} c^{4} d^{3} - 20 \, a^{3} b^{4} c^{3} d^{4} + 15 \, a^{4} b^{3} c^{2} d^{5} - 6 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x\right )} \log \left (b x + a\right )}{60 \,{\left (b^{9} x + a b^{8}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^7/(b*x + a)^2,x, algorithm="fricas")

[Out]

1/60*(10*b^7*d^7*x^7 - 60*b^7*c^7 + 420*a*b^6*c^6*d - 1260*a^2*b^5*c^5*d^2 + 210
0*a^3*b^4*c^4*d^3 - 2100*a^4*b^3*c^3*d^4 + 1260*a^5*b^2*c^2*d^5 - 420*a^6*b*c*d^
6 + 60*a^7*d^7 + 14*(6*b^7*c*d^6 - a*b^6*d^7)*x^6 + 21*(15*b^7*c^2*d^5 - 6*a*b^6
*c*d^6 + a^2*b^5*d^7)*x^5 + 35*(20*b^7*c^3*d^4 - 15*a*b^6*c^2*d^5 + 6*a^2*b^5*c*
d^6 - a^3*b^4*d^7)*x^4 + 70*(15*b^7*c^4*d^3 - 20*a*b^6*c^3*d^4 + 15*a^2*b^5*c^2*
d^5 - 6*a^3*b^4*c*d^6 + a^4*b^3*d^7)*x^3 + 210*(6*b^7*c^5*d^2 - 15*a*b^6*c^4*d^3
+ 20*a^2*b^5*c^3*d^4 - 15*a^3*b^4*c^2*d^5 + 6*a^4*b^3*c*d^6 - a^5*b^2*d^7)*x^2
+ 60*(21*a*b^6*c^5*d^2 - 70*a^2*b^5*c^4*d^3 + 105*a^3*b^4*c^3*d^4 - 84*a^4*b^3*c
^2*d^5 + 35*a^5*b^2*c*d^6 - 6*a^6*b*d^7)*x + 420*(a*b^6*c^6*d - 6*a^2*b^5*c^5*d^
2 + 15*a^3*b^4*c^4*d^3 - 20*a^4*b^3*c^3*d^4 + 15*a^5*b^2*c^2*d^5 - 6*a^6*b*c*d^6
+ a^7*d^7 + (b^7*c^6*d - 6*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 + 15*a^4*b^3*c^2*d^5 - 6*a^5*b^2*c*d^6 + a^6*b*d^7)*x)*log(b*x + a))/(b^9*x
+ a*b^8)

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Sympy [A]  time = 6.29156, size = 410, normalized size = 2.19 \[ \frac{a^{7} d^{7} - 7 a^{6} b c d^{6} + 21 a^{5} b^{2} c^{2} d^{5} - 35 a^{4} b^{3} c^{3} d^{4} + 35 a^{3} b^{4} c^{4} d^{3} - 21 a^{2} b^{5} c^{5} d^{2} + 7 a b^{6} c^{6} d - b^{7} c^{7}}{a b^{8} + b^{9} x} + \frac{d^{7} x^{6}}{6 b^{2}} - \frac{x^{5} \left (2 a d^{7} - 7 b c d^{6}\right )}{5 b^{3}} + \frac{x^{4} \left (3 a^{2} d^{7} - 14 a b c d^{6} + 21 b^{2} c^{2} d^{5}\right )}{4 b^{4}} - \frac{x^{3} \left (4 a^{3} d^{7} - 21 a^{2} b c d^{6} + 42 a b^{2} c^{2} d^{5} - 35 b^{3} c^{3} d^{4}\right )}{3 b^{5}} + \frac{x^{2} \left (5 a^{4} d^{7} - 28 a^{3} b c d^{6} + 63 a^{2} b^{2} c^{2} d^{5} - 70 a b^{3} c^{3} d^{4} + 35 b^{4} c^{4} d^{3}\right )}{2 b^{6}} - \frac{x \left (6 a^{5} d^{7} - 35 a^{4} b c d^{6} + 84 a^{3} b^{2} c^{2} d^{5} - 105 a^{2} b^{3} c^{3} d^{4} + 70 a b^{4} c^{4} d^{3} - 21 b^{5} c^{5} d^{2}\right )}{b^{7}} + \frac{7 d \left (a d - b c\right )^{6} \log{\left (a + b x \right )}}{b^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x+c)**7/(b*x+a)**2,x)

[Out]

(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)/
(a*b**8 + b**9*x) + d**7*x**6/(6*b**2) - x**5*(2*a*d**7 - 7*b*c*d**6)/(5*b**3) +
x**4*(3*a**2*d**7 - 14*a*b*c*d**6 + 21*b**2*c**2*d**5)/(4*b**4) - x**3*(4*a**3*
d**7 - 21*a**2*b*c*d**6 + 42*a*b**2*c**2*d**5 - 35*b**3*c**3*d**4)/(3*b**5) + x*
*2*(5*a**4*d**7 - 28*a**3*b*c*d**6 + 63*a**2*b**2*c**2*d**5 - 70*a*b**3*c**3*d**
4 + 35*b**4*c**4*d**3)/(2*b**6) - x*(6*a**5*d**7 - 35*a**4*b*c*d**6 + 84*a**3*b*
*2*c**2*d**5 - 105*a**2*b**3*c**3*d**4 + 70*a*b**4*c**4*d**3 - 21*b**5*c**5*d**2
)/b**7 + 7*d*(a*d - b*c)**6*log(a + b*x)/b**8

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GIAC/XCAS [A]  time = 0.218383, size = 765, normalized size = 4.09 \[ \frac{{\left (10 \, d^{7} + \frac{84 \,{\left (b^{2} c d^{6} - a b d^{7}\right )}}{{\left (b x + a\right )} b} + \frac{315 \,{\left (b^{4} c^{2} d^{5} - 2 \, a b^{3} c d^{6} + a^{2} b^{2} d^{7}\right )}}{{\left (b x + a\right )}^{2} b^{2}} + \frac{700 \,{\left (b^{6} c^{3} d^{4} - 3 \, a b^{5} c^{2} d^{5} + 3 \, a^{2} b^{4} c d^{6} - a^{3} b^{3} d^{7}\right )}}{{\left (b x + a\right )}^{3} b^{3}} + \frac{1050 \,{\left (b^{8} c^{4} d^{3} - 4 \, a b^{7} c^{3} d^{4} + 6 \, a^{2} b^{6} c^{2} d^{5} - 4 \, a^{3} b^{5} c d^{6} + a^{4} b^{4} d^{7}\right )}}{{\left (b x + a\right )}^{4} b^{4}} + \frac{1260 \,{\left (b^{10} c^{5} d^{2} - 5 \, a b^{9} c^{4} d^{3} + 10 \, a^{2} b^{8} c^{3} d^{4} - 10 \, a^{3} b^{7} c^{2} d^{5} + 5 \, a^{4} b^{6} c d^{6} - a^{5} b^{5} d^{7}\right )}}{{\left (b x + a\right )}^{5} b^{5}}\right )}{\left (b x + a\right )}^{6}}{60 \, b^{8}} - \frac{7 \,{\left (b^{6} c^{6} d - 6 \, a b^{5} c^{5} d^{2} + 15 \, a^{2} b^{4} c^{4} d^{3} - 20 \, a^{3} b^{3} c^{3} d^{4} + 15 \, a^{4} b^{2} c^{2} d^{5} - 6 \, a^{5} b c d^{6} + a^{6} d^{7}\right )}{\rm ln}\left (\frac{{\left | b x + a \right |}}{{\left (b x + a\right )}^{2}{\left | b \right |}}\right )}{b^{8}} - \frac{\frac{b^{13} c^{7}}{b x + a} - \frac{7 \, a b^{12} c^{6} d}{b x + a} + \frac{21 \, a^{2} b^{11} c^{5} d^{2}}{b x + a} - \frac{35 \, a^{3} b^{10} c^{4} d^{3}}{b x + a} + \frac{35 \, a^{4} b^{9} c^{3} d^{4}}{b x + a} - \frac{21 \, a^{5} b^{8} c^{2} d^{5}}{b x + a} + \frac{7 \, a^{6} b^{7} c d^{6}}{b x + a} - \frac{a^{7} b^{6} d^{7}}{b x + a}}{b^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^7/(b*x + a)^2,x, algorithm="giac")

[Out]

1/60*(10*d^7 + 84*(b^2*c*d^6 - a*b*d^7)/((b*x + a)*b) + 315*(b^4*c^2*d^5 - 2*a*b
^3*c*d^6 + a^2*b^2*d^7)/((b*x + a)^2*b^2) + 700*(b^6*c^3*d^4 - 3*a*b^5*c^2*d^5 +
3*a^2*b^4*c*d^6 - a^3*b^3*d^7)/((b*x + a)^3*b^3) + 1050*(b^8*c^4*d^3 - 4*a*b^7*
c^3*d^4 + 6*a^2*b^6*c^2*d^5 - 4*a^3*b^5*c*d^6 + a^4*b^4*d^7)/((b*x + a)^4*b^4) +
1260*(b^10*c^5*d^2 - 5*a*b^9*c^4*d^3 + 10*a^2*b^8*c^3*d^4 - 10*a^3*b^7*c^2*d^5
+ 5*a^4*b^6*c*d^6 - a^5*b^5*d^7)/((b*x + a)^5*b^5))*(b*x + a)^6/b^8 - 7*(b^6*c^6
*d - 6*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 + 15*a^4*b^2*c^2*
d^5 - 6*a^5*b*c*d^6 + a^6*d^7)*ln(abs(b*x + a)/((b*x + a)^2*abs(b)))/b^8 - (b^13
*c^7/(b*x + a) - 7*a*b^12*c^6*d/(b*x + a) + 21*a^2*b^11*c^5*d^2/(b*x + a) - 35*a
^3*b^10*c^4*d^3/(b*x + a) + 35*a^4*b^9*c^3*d^4/(b*x + a) - 21*a^5*b^8*c^2*d^5/(b
*x + a) + 7*a^6*b^7*c*d^6/(b*x + a) - a^7*b^6*d^7/(b*x + a))/b^14