3.1158 \(\int \frac{(b d+2 c d x)^5}{\left (a+b x+c x^2\right )^2} \, dx\)

Optimal. Leaf size=65 \[ 8 c d^5 \left (b^2-4 a c\right ) \log \left (a+b x+c x^2\right )-\frac{d^5 (b+2 c x)^4}{a+b x+c x^2}+8 c d^5 (b+2 c x)^2 \]

[Out]

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

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Rubi [A]  time = 0.109052, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ 8 c d^5 \left (b^2-4 a c\right ) \log \left (a+b x+c x^2\right )-\frac{d^5 (b+2 c x)^4}{a+b x+c x^2}+8 c d^5 (b+2 c x)^2 \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 32.0886, size = 63, normalized size = 0.97 \[ 8 c d^{5} \left (b + 2 c x\right )^{2} + 8 c d^{5} \left (- 4 a c + b^{2}\right ) \log{\left (a + b x + c x^{2} \right )} - \frac{d^{5} \left (b + 2 c x\right )^{4}}{a + b x + c x^{2}} \]

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)**2,x)

[Out]

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

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Mathematica [A]  time = 0.0682047, size = 64, normalized size = 0.98 \[ d^5 \left (-\frac{\left (b^2-4 a c\right )^2}{a+x (b+c x)}+8 c \left (b^2-4 a c\right ) \log (a+x (b+c x))+16 b c^2 x+16 c^3 x^2\right ) \]

Antiderivative was successfully verified.

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

[Out]

d^5*(16*b*c^2*x + 16*c^3*x^2 - (b^2 - 4*a*c)^2/(a + x*(b + c*x)) + 8*c*(b^2 - 4*
a*c)*Log[a + x*(b + c*x)])

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Maple [A]  time = 0.013, size = 128, normalized size = 2. \[ 16\,{d}^{5}{c}^{3}{x}^{2}+16\,{d}^{5}b{c}^{2}x-16\,{\frac{{d}^{5}{a}^{2}{c}^{2}}{c{x}^{2}+bx+a}}+8\,{\frac{{d}^{5}ac{b}^{2}}{c{x}^{2}+bx+a}}-{\frac{{d}^{5}{b}^{4}}{c{x}^{2}+bx+a}}-32\,{d}^{5}\ln \left ( c{x}^{2}+bx+a \right ) a{c}^{2}+8\,{d}^{5}\ln \left ( c{x}^{2}+bx+a \right ){b}^{2}c \]

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)^2,x)

[Out]

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

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Maxima [A]  time = 0.674728, size = 116, normalized size = 1.78 \[ 16 \, c^{3} d^{5} x^{2} + 16 \, b c^{2} d^{5} x + 8 \,{\left (b^{2} c - 4 \, a c^{2}\right )} d^{5} \log \left (c x^{2} + b x + a\right ) - \frac{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} d^{5}}{c x^{2} + b x + a} \]

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)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.211222, size = 223, normalized size = 3.43 \[ \frac{16 \, c^{4} d^{5} x^{4} + 32 \, b c^{3} d^{5} x^{3} + 16 \, a b c^{2} d^{5} x + 16 \,{\left (b^{2} c^{2} + a c^{3}\right )} d^{5} x^{2} -{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} d^{5} + 8 \,{\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{5} x^{2} +{\left (b^{3} c - 4 \, a b c^{2}\right )} d^{5} x +{\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} d^{5}\right )} \log \left (c x^{2} + b x + a\right )}{c x^{2} + b x + a} \]

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)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 7.51919, size = 90, normalized size = 1.38 \[ 16 b c^{2} d^{5} x + 16 c^{3} d^{5} x^{2} - 8 c d^{5} \left (4 a c - b^{2}\right ) \log{\left (a + b x + c x^{2} \right )} - \frac{16 a^{2} c^{2} d^{5} - 8 a b^{2} c d^{5} + b^{4} d^{5}}{a + b x + c x^{2}} \]

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)**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.217054, size = 135, normalized size = 2.08 \[ 8 \,{\left (b^{2} c d^{5} - 4 \, a c^{2} d^{5}\right )}{\rm ln}\left (c x^{2} + b x + a\right ) - \frac{b^{4} d^{5} - 8 \, a b^{2} c d^{5} + 16 \, a^{2} c^{2} d^{5}}{c x^{2} + b x + a} + \frac{16 \,{\left (c^{7} d^{5} x^{2} + b c^{6} d^{5} x\right )}}{c^{4}} \]

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)^2,x, algorithm="giac")

[Out]

8*(b^2*c*d^5 - 4*a*c^2*d^5)*ln(c*x^2 + b*x + a) - (b^4*d^5 - 8*a*b^2*c*d^5 + 16*
a^2*c^2*d^5)/(c*x^2 + b*x + a) + 16*(c^7*d^5*x^2 + b*c^6*d^5*x)/c^4