3.1353 \(\int \frac{(a+b x)^5}{(c+d x)^3} \, dx\)

Optimal. Leaf size=133 \[ -\frac{5 b^4 (c+d x)^2 (b c-a d)}{2 d^6}+\frac{10 b^3 x (b c-a d)^2}{d^5}-\frac{10 b^2 (b c-a d)^3 \log (c+d x)}{d^6}-\frac{5 b (b c-a d)^4}{d^6 (c+d x)}+\frac{(b c-a d)^5}{2 d^6 (c+d x)^2}+\frac{b^5 (c+d x)^3}{3 d^6} \]

[Out]

(10*b^3*(b*c - a*d)^2*x)/d^5 + (b*c - a*d)^5/(2*d^6*(c + d*x)^2) - (5*b*(b*c - a
*d)^4)/(d^6*(c + d*x)) - (5*b^4*(b*c - a*d)*(c + d*x)^2)/(2*d^6) + (b^5*(c + d*x
)^3)/(3*d^6) - (10*b^2*(b*c - a*d)^3*Log[c + d*x])/d^6

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Rubi [A]  time = 0.263985, antiderivative size = 133, 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 \[ -\frac{5 b^4 (c+d x)^2 (b c-a d)}{2 d^6}+\frac{10 b^3 x (b c-a d)^2}{d^5}-\frac{10 b^2 (b c-a d)^3 \log (c+d x)}{d^6}-\frac{5 b (b c-a d)^4}{d^6 (c+d x)}+\frac{(b c-a d)^5}{2 d^6 (c+d x)^2}+\frac{b^5 (c+d x)^3}{3 d^6} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^5/(c + d*x)^3,x]

[Out]

(10*b^3*(b*c - a*d)^2*x)/d^5 + (b*c - a*d)^5/(2*d^6*(c + d*x)^2) - (5*b*(b*c - a
*d)^4)/(d^6*(c + d*x)) - (5*b^4*(b*c - a*d)*(c + d*x)^2)/(2*d^6) + (b^5*(c + d*x
)^3)/(3*d^6) - (10*b^2*(b*c - a*d)^3*Log[c + d*x])/d^6

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Rubi in Sympy [A]  time = 40.5298, size = 121, normalized size = 0.91 \[ \frac{b^{5} \left (c + d x\right )^{3}}{3 d^{6}} + \frac{5 b^{4} \left (c + d x\right )^{2} \left (a d - b c\right )}{2 d^{6}} + \frac{10 b^{3} x \left (a d - b c\right )^{2}}{d^{5}} + \frac{10 b^{2} \left (a d - b c\right )^{3} \log{\left (c + d x \right )}}{d^{6}} - \frac{5 b \left (a d - b c\right )^{4}}{d^{6} \left (c + d x\right )} - \frac{\left (a d - b c\right )^{5}}{2 d^{6} \left (c + d x\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**5/(d*x+c)**3,x)

[Out]

b**5*(c + d*x)**3/(3*d**6) + 5*b**4*(c + d*x)**2*(a*d - b*c)/(2*d**6) + 10*b**3*
x*(a*d - b*c)**2/d**5 + 10*b**2*(a*d - b*c)**3*log(c + d*x)/d**6 - 5*b*(a*d - b*
c)**4/(d**6*(c + d*x)) - (a*d - b*c)**5/(2*d**6*(c + d*x)**2)

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Mathematica [A]  time = 0.136562, size = 230, normalized size = 1.73 \[ \frac{-3 a^5 d^5-15 a^4 b d^4 (c+2 d x)+30 a^3 b^2 c d^3 (3 c+4 d x)+30 a^2 b^3 d^2 \left (-5 c^3-4 c^2 d x+4 c d^2 x^2+2 d^3 x^3\right )+15 a b^4 d \left (7 c^4+2 c^3 d x-11 c^2 d^2 x^2-4 c d^3 x^3+d^4 x^4\right )-60 b^2 (c+d x)^2 (b c-a d)^3 \log (c+d x)+b^5 \left (-27 c^5+6 c^4 d x+63 c^3 d^2 x^2+20 c^2 d^3 x^3-5 c d^4 x^4+2 d^5 x^5\right )}{6 d^6 (c+d x)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^5/(c + d*x)^3,x]

[Out]

(-3*a^5*d^5 - 15*a^4*b*d^4*(c + 2*d*x) + 30*a^3*b^2*c*d^3*(3*c + 4*d*x) + 30*a^2
*b^3*d^2*(-5*c^3 - 4*c^2*d*x + 4*c*d^2*x^2 + 2*d^3*x^3) + 15*a*b^4*d*(7*c^4 + 2*
c^3*d*x - 11*c^2*d^2*x^2 - 4*c*d^3*x^3 + d^4*x^4) + b^5*(-27*c^5 + 6*c^4*d*x + 6
3*c^3*d^2*x^2 + 20*c^2*d^3*x^3 - 5*c*d^4*x^4 + 2*d^5*x^5) - 60*b^2*(b*c - a*d)^3
*(c + d*x)^2*Log[c + d*x])/(6*d^6*(c + d*x)^2)

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Maple [B]  time = 0.013, size = 346, normalized size = 2.6 \[{\frac{{b}^{5}{x}^{3}}{3\,{d}^{3}}}+{\frac{5\,a{b}^{4}{x}^{2}}{2\,{d}^{3}}}-{\frac{3\,{b}^{5}{x}^{2}c}{2\,{d}^{4}}}+10\,{\frac{{a}^{2}{b}^{3}x}{{d}^{3}}}-15\,{\frac{a{b}^{4}cx}{{d}^{4}}}+6\,{\frac{{b}^{5}{c}^{2}x}{{d}^{5}}}+10\,{\frac{{b}^{2}\ln \left ( dx+c \right ){a}^{3}}{{d}^{3}}}-30\,{\frac{{b}^{3}\ln \left ( dx+c \right ){a}^{2}c}{{d}^{4}}}+30\,{\frac{{b}^{4}\ln \left ( dx+c \right ) a{c}^{2}}{{d}^{5}}}-10\,{\frac{{b}^{5}\ln \left ( dx+c \right ){c}^{3}}{{d}^{6}}}-{\frac{{a}^{5}}{2\,d \left ( dx+c \right ) ^{2}}}+{\frac{5\,{a}^{4}bc}{2\,{d}^{2} \left ( dx+c \right ) ^{2}}}-5\,{\frac{{a}^{3}{b}^{2}{c}^{2}}{{d}^{3} \left ( dx+c \right ) ^{2}}}+5\,{\frac{{a}^{2}{b}^{3}{c}^{3}}{{d}^{4} \left ( dx+c \right ) ^{2}}}-{\frac{5\,a{b}^{4}{c}^{4}}{2\,{d}^{5} \left ( dx+c \right ) ^{2}}}+{\frac{{b}^{5}{c}^{5}}{2\,{d}^{6} \left ( dx+c \right ) ^{2}}}-5\,{\frac{{a}^{4}b}{{d}^{2} \left ( dx+c \right ) }}+20\,{\frac{{a}^{3}{b}^{2}c}{{d}^{3} \left ( dx+c \right ) }}-30\,{\frac{{a}^{2}{b}^{3}{c}^{2}}{{d}^{4} \left ( dx+c \right ) }}+20\,{\frac{a{b}^{4}{c}^{3}}{{d}^{5} \left ( dx+c \right ) }}-5\,{\frac{{b}^{5}{c}^{4}}{{d}^{6} \left ( dx+c \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^5/(d*x+c)^3,x)

[Out]

1/3*b^5/d^3*x^3+5/2*b^4/d^3*x^2*a-3/2*b^5/d^4*x^2*c+10*b^3/d^3*a^2*x-15*b^4/d^4*
a*c*x+6*b^5/d^5*c^2*x+10*b^2/d^3*ln(d*x+c)*a^3-30*b^3/d^4*ln(d*x+c)*a^2*c+30*b^4
/d^5*ln(d*x+c)*a*c^2-10*b^5/d^6*ln(d*x+c)*c^3-1/2/d/(d*x+c)^2*a^5+5/2/d^2/(d*x+c
)^2*a^4*b*c-5/d^3/(d*x+c)^2*a^3*b^2*c^2+5/d^4/(d*x+c)^2*a^2*b^3*c^3-5/2/d^5/(d*x
+c)^2*a*b^4*c^4+1/2/d^6/(d*x+c)^2*b^5*c^5-5*b/d^2/(d*x+c)*a^4+20*b^2/d^3/(d*x+c)
*a^3*c-30*b^3/d^4/(d*x+c)*a^2*c^2+20*b^4/d^5/(d*x+c)*a*c^3-5*b^5/d^6/(d*x+c)*c^4

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Maxima [A]  time = 1.38576, size = 366, normalized size = 2.75 \[ -\frac{9 \, b^{5} c^{5} - 35 \, a b^{4} c^{4} d + 50 \, a^{2} b^{3} c^{3} d^{2} - 30 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} + a^{5} d^{5} + 10 \,{\left (b^{5} c^{4} d - 4 \, a b^{4} c^{3} d^{2} + 6 \, a^{2} b^{3} c^{2} d^{3} - 4 \, a^{3} b^{2} c d^{4} + a^{4} b d^{5}\right )} x}{2 \,{\left (d^{8} x^{2} + 2 \, c d^{7} x + c^{2} d^{6}\right )}} + \frac{2 \, b^{5} d^{2} x^{3} - 3 \,{\left (3 \, b^{5} c d - 5 \, a b^{4} d^{2}\right )} x^{2} + 6 \,{\left (6 \, b^{5} c^{2} - 15 \, a b^{4} c d + 10 \, a^{2} b^{3} d^{2}\right )} x}{6 \, d^{5}} - \frac{10 \,{\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} \log \left (d x + c\right )}{d^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5/(d*x + c)^3,x, algorithm="maxima")

[Out]

-1/2*(9*b^5*c^5 - 35*a*b^4*c^4*d + 50*a^2*b^3*c^3*d^2 - 30*a^3*b^2*c^2*d^3 + 5*a
^4*b*c*d^4 + a^5*d^5 + 10*(b^5*c^4*d - 4*a*b^4*c^3*d^2 + 6*a^2*b^3*c^2*d^3 - 4*a
^3*b^2*c*d^4 + a^4*b*d^5)*x)/(d^8*x^2 + 2*c*d^7*x + c^2*d^6) + 1/6*(2*b^5*d^2*x^
3 - 3*(3*b^5*c*d - 5*a*b^4*d^2)*x^2 + 6*(6*b^5*c^2 - 15*a*b^4*c*d + 10*a^2*b^3*d
^2)*x)/d^5 - 10*(b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*log(d*
x + c)/d^6

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Fricas [A]  time = 0.209958, size = 562, normalized size = 4.23 \[ \frac{2 \, b^{5} d^{5} x^{5} - 27 \, b^{5} c^{5} + 105 \, a b^{4} c^{4} d - 150 \, a^{2} b^{3} c^{3} d^{2} + 90 \, a^{3} b^{2} c^{2} d^{3} - 15 \, a^{4} b c d^{4} - 3 \, a^{5} d^{5} - 5 \,{\left (b^{5} c d^{4} - 3 \, a b^{4} d^{5}\right )} x^{4} + 20 \,{\left (b^{5} c^{2} d^{3} - 3 \, a b^{4} c d^{4} + 3 \, a^{2} b^{3} d^{5}\right )} x^{3} + 3 \,{\left (21 \, b^{5} c^{3} d^{2} - 55 \, a b^{4} c^{2} d^{3} + 40 \, a^{2} b^{3} c d^{4}\right )} x^{2} + 6 \,{\left (b^{5} c^{4} d + 5 \, a b^{4} c^{3} d^{2} - 20 \, a^{2} b^{3} c^{2} d^{3} + 20 \, a^{3} b^{2} c d^{4} - 5 \, a^{4} b d^{5}\right )} x - 60 \,{\left (b^{5} c^{5} - 3 \, a b^{4} c^{4} d + 3 \, a^{2} b^{3} c^{3} d^{2} - a^{3} b^{2} c^{2} d^{3} +{\left (b^{5} c^{3} d^{2} - 3 \, a b^{4} c^{2} d^{3} + 3 \, a^{2} b^{3} c d^{4} - a^{3} b^{2} d^{5}\right )} x^{2} + 2 \,{\left (b^{5} c^{4} d - 3 \, a b^{4} c^{3} d^{2} + 3 \, a^{2} b^{3} c^{2} d^{3} - a^{3} b^{2} c d^{4}\right )} x\right )} \log \left (d x + c\right )}{6 \,{\left (d^{8} x^{2} + 2 \, c d^{7} x + c^{2} d^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5/(d*x + c)^3,x, algorithm="fricas")

[Out]

1/6*(2*b^5*d^5*x^5 - 27*b^5*c^5 + 105*a*b^4*c^4*d - 150*a^2*b^3*c^3*d^2 + 90*a^3
*b^2*c^2*d^3 - 15*a^4*b*c*d^4 - 3*a^5*d^5 - 5*(b^5*c*d^4 - 3*a*b^4*d^5)*x^4 + 20
*(b^5*c^2*d^3 - 3*a*b^4*c*d^4 + 3*a^2*b^3*d^5)*x^3 + 3*(21*b^5*c^3*d^2 - 55*a*b^
4*c^2*d^3 + 40*a^2*b^3*c*d^4)*x^2 + 6*(b^5*c^4*d + 5*a*b^4*c^3*d^2 - 20*a^2*b^3*
c^2*d^3 + 20*a^3*b^2*c*d^4 - 5*a^4*b*d^5)*x - 60*(b^5*c^5 - 3*a*b^4*c^4*d + 3*a^
2*b^3*c^3*d^2 - a^3*b^2*c^2*d^3 + (b^5*c^3*d^2 - 3*a*b^4*c^2*d^3 + 3*a^2*b^3*c*d
^4 - a^3*b^2*d^5)*x^2 + 2*(b^5*c^4*d - 3*a*b^4*c^3*d^2 + 3*a^2*b^3*c^2*d^3 - a^3
*b^2*c*d^4)*x)*log(d*x + c))/(d^8*x^2 + 2*c*d^7*x + c^2*d^6)

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Sympy [A]  time = 3.63093, size = 253, normalized size = 1.9 \[ \frac{b^{5} x^{3}}{3 d^{3}} + \frac{10 b^{2} \left (a d - b c\right )^{3} \log{\left (c + d x \right )}}{d^{6}} - \frac{a^{5} d^{5} + 5 a^{4} b c d^{4} - 30 a^{3} b^{2} c^{2} d^{3} + 50 a^{2} b^{3} c^{3} d^{2} - 35 a b^{4} c^{4} d + 9 b^{5} c^{5} + x \left (10 a^{4} b d^{5} - 40 a^{3} b^{2} c d^{4} + 60 a^{2} b^{3} c^{2} d^{3} - 40 a b^{4} c^{3} d^{2} + 10 b^{5} c^{4} d\right )}{2 c^{2} d^{6} + 4 c d^{7} x + 2 d^{8} x^{2}} + \frac{x^{2} \left (5 a b^{4} d - 3 b^{5} c\right )}{2 d^{4}} + \frac{x \left (10 a^{2} b^{3} d^{2} - 15 a b^{4} c d + 6 b^{5} c^{2}\right )}{d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**5/(d*x+c)**3,x)

[Out]

b**5*x**3/(3*d**3) + 10*b**2*(a*d - b*c)**3*log(c + d*x)/d**6 - (a**5*d**5 + 5*a
**4*b*c*d**4 - 30*a**3*b**2*c**2*d**3 + 50*a**2*b**3*c**3*d**2 - 35*a*b**4*c**4*
d + 9*b**5*c**5 + x*(10*a**4*b*d**5 - 40*a**3*b**2*c*d**4 + 60*a**2*b**3*c**2*d*
*3 - 40*a*b**4*c**3*d**2 + 10*b**5*c**4*d))/(2*c**2*d**6 + 4*c*d**7*x + 2*d**8*x
**2) + x**2*(5*a*b**4*d - 3*b**5*c)/(2*d**4) + x*(10*a**2*b**3*d**2 - 15*a*b**4*
c*d + 6*b**5*c**2)/d**5

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GIAC/XCAS [A]  time = 0.224181, size = 356, normalized size = 2.68 \[ -\frac{10 \,{\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )}{\rm ln}\left ({\left | d x + c \right |}\right )}{d^{6}} - \frac{9 \, b^{5} c^{5} - 35 \, a b^{4} c^{4} d + 50 \, a^{2} b^{3} c^{3} d^{2} - 30 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} + a^{5} d^{5} + 10 \,{\left (b^{5} c^{4} d - 4 \, a b^{4} c^{3} d^{2} + 6 \, a^{2} b^{3} c^{2} d^{3} - 4 \, a^{3} b^{2} c d^{4} + a^{4} b d^{5}\right )} x}{2 \,{\left (d x + c\right )}^{2} d^{6}} + \frac{2 \, b^{5} d^{6} x^{3} - 9 \, b^{5} c d^{5} x^{2} + 15 \, a b^{4} d^{6} x^{2} + 36 \, b^{5} c^{2} d^{4} x - 90 \, a b^{4} c d^{5} x + 60 \, a^{2} b^{3} d^{6} x}{6 \, d^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5/(d*x + c)^3,x, algorithm="giac")

[Out]

-10*(b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*ln(abs(d*x + c))/d
^6 - 1/2*(9*b^5*c^5 - 35*a*b^4*c^4*d + 50*a^2*b^3*c^3*d^2 - 30*a^3*b^2*c^2*d^3 +
5*a^4*b*c*d^4 + a^5*d^5 + 10*(b^5*c^4*d - 4*a*b^4*c^3*d^2 + 6*a^2*b^3*c^2*d^3 -
4*a^3*b^2*c*d^4 + a^4*b*d^5)*x)/((d*x + c)^2*d^6) + 1/6*(2*b^5*d^6*x^3 - 9*b^5*
c*d^5*x^2 + 15*a*b^4*d^6*x^2 + 36*b^5*c^2*d^4*x - 90*a*b^4*c*d^5*x + 60*a^2*b^3*
d^6*x)/d^9