3.1171 \(\int \frac{(b d+2 c d x)^5}{\left (a+b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=73 \[ 16 c^2 d^5 \log \left (a+b x+c x^2\right )-\frac{4 c d^5 (b+2 c x)^2}{a+b x+c x^2}-\frac{d^5 (b+2 c x)^4}{2 \left (a+b x+c x^2\right )^2} \]

[Out]

-(d^5*(b + 2*c*x)^4)/(2*(a + b*x + c*x^2)^2) - (4*c*d^5*(b + 2*c*x)^2)/(a + b*x
+ c*x^2) + 16*c^2*d^5*Log[a + b*x + c*x^2]

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Rubi [A]  time = 0.101806, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ 16 c^2 d^5 \log \left (a+b x+c x^2\right )-\frac{4 c d^5 (b+2 c x)^2}{a+b x+c x^2}-\frac{d^5 (b+2 c x)^4}{2 \left (a+b x+c x^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(b*d + 2*c*d*x)^5/(a + b*x + c*x^2)^3,x]

[Out]

-(d^5*(b + 2*c*x)^4)/(2*(a + b*x + c*x^2)^2) - (4*c*d^5*(b + 2*c*x)^2)/(a + b*x
+ c*x^2) + 16*c^2*d^5*Log[a + b*x + c*x^2]

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Rubi in Sympy [A]  time = 32.4715, size = 70, normalized size = 0.96 \[ 16 c^{2} d^{5} \log{\left (a + b x + c x^{2} \right )} - \frac{4 c d^{5} \left (b + 2 c x\right )^{2}}{a + b x + c x^{2}} - \frac{d^{5} \left (b + 2 c x\right )^{4}}{2 \left (a + b x + c x^{2}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*c*d*x+b*d)**5/(c*x**2+b*x+a)**3,x)

[Out]

16*c**2*d**5*log(a + b*x + c*x**2) - 4*c*d**5*(b + 2*c*x)**2/(a + b*x + c*x**2)
- d**5*(b + 2*c*x)**4/(2*(a + b*x + c*x**2)**2)

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Mathematica [A]  time = 0.0852064, size = 65, normalized size = 0.89 \[ d^5 \left (16 c^2 \log (a+x (b+c x))-\frac{\left (b^2-4 a c\right ) \left (4 c \left (3 a+4 c x^2\right )+b^2+16 b c x\right )}{2 (a+x (b+c x))^2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(b*d + 2*c*d*x)^5/(a + b*x + c*x^2)^3,x]

[Out]

d^5*(-((b^2 - 4*a*c)*(b^2 + 16*b*c*x + 4*c*(3*a + 4*c*x^2)))/(2*(a + x*(b + c*x)
)^2) + 16*c^2*Log[a + x*(b + c*x)])

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Maple [B]  time = 0.015, size = 181, normalized size = 2.5 \[ 32\,{\frac{{x}^{2}a{c}^{3}{d}^{5}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-8\,{\frac{{x}^{2}{b}^{2}{c}^{2}{d}^{5}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}+32\,{\frac{xab{c}^{2}{d}^{5}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-8\,{\frac{x{b}^{3}c{d}^{5}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}+24\,{\frac{{d}^{5}{a}^{2}{c}^{2}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-4\,{\frac{{d}^{5}ac{b}^{2}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-{\frac{{d}^{5}{b}^{4}}{2\, \left ( c{x}^{2}+bx+a \right ) ^{2}}}+16\,{c}^{2}{d}^{5}\ln \left ( c{x}^{2}+bx+a \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*c*d*x+b*d)^5/(c*x^2+b*x+a)^3,x)

[Out]

32*d^5/(c*x^2+b*x+a)^2*x^2*a*c^3-8*d^5/(c*x^2+b*x+a)^2*x^2*b^2*c^2+32*d^5/(c*x^2
+b*x+a)^2*x*a*b*c^2-8*d^5/(c*x^2+b*x+a)^2*b^3*c*x+24*d^5/(c*x^2+b*x+a)^2*a^2*c^2
-4*d^5/(c*x^2+b*x+a)^2*a*c*b^2-1/2*d^5/(c*x^2+b*x+a)^2*b^4+16*c^2*d^5*ln(c*x^2+b
*x+a)

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Maxima [A]  time = 0.68439, size = 167, normalized size = 2.29 \[ 16 \, c^{2} d^{5} \log \left (c x^{2} + b x + a\right ) - \frac{16 \,{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{5} x^{2} + 16 \,{\left (b^{3} c - 4 \, a b c^{2}\right )} d^{5} x +{\left (b^{4} + 8 \, a b^{2} c - 48 \, a^{2} c^{2}\right )} d^{5}}{2 \,{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^5/(c*x^2 + b*x + a)^3,x, algorithm="maxima")

[Out]

16*c^2*d^5*log(c*x^2 + b*x + a) - 1/2*(16*(b^2*c^2 - 4*a*c^3)*d^5*x^2 + 16*(b^3*
c - 4*a*b*c^2)*d^5*x + (b^4 + 8*a*b^2*c - 48*a^2*c^2)*d^5)/(c^2*x^4 + 2*b*c*x^3
+ 2*a*b*x + (b^2 + 2*a*c)*x^2 + a^2)

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Fricas [A]  time = 0.212664, size = 246, normalized size = 3.37 \[ -\frac{16 \,{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{5} x^{2} + 16 \,{\left (b^{3} c - 4 \, a b c^{2}\right )} d^{5} x +{\left (b^{4} + 8 \, a b^{2} c - 48 \, a^{2} c^{2}\right )} d^{5} - 32 \,{\left (c^{4} d^{5} x^{4} + 2 \, b c^{3} d^{5} x^{3} + 2 \, a b c^{2} d^{5} x + a^{2} c^{2} d^{5} +{\left (b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} x^{2}\right )} \log \left (c x^{2} + b x + a\right )}{2 \,{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^5/(c*x^2 + b*x + a)^3,x, algorithm="fricas")

[Out]

-1/2*(16*(b^2*c^2 - 4*a*c^3)*d^5*x^2 + 16*(b^3*c - 4*a*b*c^2)*d^5*x + (b^4 + 8*a
*b^2*c - 48*a^2*c^2)*d^5 - 32*(c^4*d^5*x^4 + 2*b*c^3*d^5*x^3 + 2*a*b*c^2*d^5*x +
a^2*c^2*d^5 + (b^2*c^2 + 2*a*c^3)*d^5*x^2)*log(c*x^2 + b*x + a))/(c^2*x^4 + 2*b
*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 + a^2)

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Sympy [A]  time = 16.9115, size = 141, normalized size = 1.93 \[ 16 c^{2} d^{5} \log{\left (a + b x + c x^{2} \right )} + \frac{48 a^{2} c^{2} d^{5} - 8 a b^{2} c d^{5} - b^{4} d^{5} + x^{2} \left (64 a c^{3} d^{5} - 16 b^{2} c^{2} d^{5}\right ) + x \left (64 a b c^{2} d^{5} - 16 b^{3} c d^{5}\right )}{2 a^{2} + 4 a b x + 4 b c x^{3} + 2 c^{2} x^{4} + x^{2} \left (4 a c + 2 b^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x+b*d)**5/(c*x**2+b*x+a)**3,x)

[Out]

16*c**2*d**5*log(a + b*x + c*x**2) + (48*a**2*c**2*d**5 - 8*a*b**2*c*d**5 - b**4
*d**5 + x**2*(64*a*c**3*d**5 - 16*b**2*c**2*d**5) + x*(64*a*b*c**2*d**5 - 16*b**
3*c*d**5))/(2*a**2 + 4*a*b*x + 4*b*c*x**3 + 2*c**2*x**4 + x**2*(4*a*c + 2*b**2))

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GIAC/XCAS [A]  time = 0.219431, size = 149, normalized size = 2.04 \[ 16 \, c^{2} d^{5}{\rm ln}\left (c x^{2} + b x + a\right ) - \frac{b^{4} d^{5} + 8 \, a b^{2} c d^{5} - 48 \, a^{2} c^{2} d^{5} + 16 \,{\left (b^{2} c^{2} d^{5} - 4 \, a c^{3} d^{5}\right )} x^{2} + 16 \,{\left (b^{3} c d^{5} - 4 \, a b c^{2} d^{5}\right )} x}{2 \,{\left (c x^{2} + b x + a\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^5/(c*x^2 + b*x + a)^3,x, algorithm="giac")

[Out]

16*c^2*d^5*ln(c*x^2 + b*x + a) - 1/2*(b^4*d^5 + 8*a*b^2*c*d^5 - 48*a^2*c^2*d^5 +
16*(b^2*c^2*d^5 - 4*a*c^3*d^5)*x^2 + 16*(b^3*c*d^5 - 4*a*b*c^2*d^5)*x)/(c*x^2 +
b*x + a)^2