3.1139 \(\int \frac{(A+B x) (d+e x)^2}{b x+c x^2} \, dx\)

Optimal. Leaf size=77 \[ \frac{(b B-A c) (c d-b e)^2 \log (b+c x)}{b c^3}+\frac{e x (A c e-b B e+2 B c d)}{c^2}+\frac{A d^2 \log (x)}{b}+\frac{B e^2 x^2}{2 c} \]

[Out]

(e*(2*B*c*d - b*B*e + A*c*e)*x)/c^2 + (B*e^2*x^2)/(2*c) + (A*d^2*Log[x])/b + ((b
*B - A*c)*(c*d - b*e)^2*Log[b + c*x])/(b*c^3)

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Rubi [A]  time = 0.188192, antiderivative size = 77, 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 \[ \frac{(b B-A c) (c d-b e)^2 \log (b+c x)}{b c^3}+\frac{e x (A c e-b B e+2 B c d)}{c^2}+\frac{A d^2 \log (x)}{b}+\frac{B e^2 x^2}{2 c} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(d + e*x)^2)/(b*x + c*x^2),x]

[Out]

(e*(2*B*c*d - b*B*e + A*c*e)*x)/c^2 + (B*e^2*x^2)/(2*c) + (A*d^2*Log[x])/b + ((b
*B - A*c)*(c*d - b*e)^2*Log[b + c*x])/(b*c^3)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{A d^{2} \log{\left (x \right )}}{b} + \frac{B e^{2} \int x\, dx}{c} + \frac{\left (A c e - B b e + 2 B c d\right ) \int e\, dx}{c^{2}} - \frac{\left (A c - B b\right ) \left (b e - c d\right )^{2} \log{\left (b + c x \right )}}{b c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(e*x+d)**2/(c*x**2+b*x),x)

[Out]

A*d**2*log(x)/b + B*e**2*Integral(x, x)/c + (A*c*e - B*b*e + 2*B*c*d)*Integral(e
, x)/c**2 - (A*c - B*b)*(b*e - c*d)**2*log(b + c*x)/(b*c**3)

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Mathematica [A]  time = 0.0806757, size = 74, normalized size = 0.96 \[ \frac{b c e x (2 A c e+B (-2 b e+4 c d+c e x))+2 (b B-A c) (c d-b e)^2 \log (b+c x)+2 A c^3 d^2 \log (x)}{2 b c^3} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(d + e*x)^2)/(b*x + c*x^2),x]

[Out]

(b*c*e*x*(2*A*c*e + B*(4*c*d - 2*b*e + c*e*x)) + 2*A*c^3*d^2*Log[x] + 2*(b*B - A
*c)*(c*d - b*e)^2*Log[b + c*x])/(2*b*c^3)

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Maple [A]  time = 0.01, size = 144, normalized size = 1.9 \[{\frac{B{e}^{2}{x}^{2}}{2\,c}}+{\frac{{e}^{2}Ax}{c}}-{\frac{B{e}^{2}bx}{{c}^{2}}}+2\,{\frac{Bdex}{c}}+{\frac{A{d}^{2}\ln \left ( x \right ) }{b}}-{\frac{b\ln \left ( cx+b \right ) A{e}^{2}}{{c}^{2}}}+2\,{\frac{\ln \left ( cx+b \right ) Ade}{c}}-{\frac{\ln \left ( cx+b \right ) A{d}^{2}}{b}}+{\frac{{b}^{2}\ln \left ( cx+b \right ) B{e}^{2}}{{c}^{3}}}-2\,{\frac{b\ln \left ( cx+b \right ) Bde}{{c}^{2}}}+{\frac{\ln \left ( cx+b \right ) B{d}^{2}}{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(e*x+d)^2/(c*x^2+b*x),x)

[Out]

1/2*B*e^2*x^2/c+1/c*e^2*A*x-1/c^2*e^2*b*B*x+2/c*e*B*d*x+A*d^2*ln(x)/b-b/c^2*ln(c
*x+b)*A*e^2+2/c*ln(c*x+b)*A*d*e-1/b*ln(c*x+b)*A*d^2+b^2/c^3*ln(c*x+b)*B*e^2-2*b/
c^2*ln(c*x+b)*B*d*e+1/c*ln(c*x+b)*B*d^2

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Maxima [A]  time = 0.712104, size = 155, normalized size = 2.01 \[ \frac{A d^{2} \log \left (x\right )}{b} + \frac{B c e^{2} x^{2} + 2 \,{\left (2 \, B c d e -{\left (B b - A c\right )} e^{2}\right )} x}{2 \, c^{2}} + \frac{{\left ({\left (B b c^{2} - A c^{3}\right )} d^{2} - 2 \,{\left (B b^{2} c - A b c^{2}\right )} d e +{\left (B b^{3} - A b^{2} c\right )} e^{2}\right )} \log \left (c x + b\right )}{b c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*d^2*log(x)/b + 1/2*(B*c*e^2*x^2 + 2*(2*B*c*d*e - (B*b - A*c)*e^2)*x)/c^2 + ((B
*b*c^2 - A*c^3)*d^2 - 2*(B*b^2*c - A*b*c^2)*d*e + (B*b^3 - A*b^2*c)*e^2)*log(c*x
+ b)/(b*c^3)

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Fricas [A]  time = 0.284021, size = 169, normalized size = 2.19 \[ \frac{B b c^{2} e^{2} x^{2} + 2 \, A c^{3} d^{2} \log \left (x\right ) + 2 \,{\left (2 \, B b c^{2} d e -{\left (B b^{2} c - A b c^{2}\right )} e^{2}\right )} x + 2 \,{\left ({\left (B b c^{2} - A c^{3}\right )} d^{2} - 2 \,{\left (B b^{2} c - A b c^{2}\right )} d e +{\left (B b^{3} - A b^{2} c\right )} e^{2}\right )} \log \left (c x + b\right )}{2 \, b c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(B*b*c^2*e^2*x^2 + 2*A*c^3*d^2*log(x) + 2*(2*B*b*c^2*d*e - (B*b^2*c - A*b*c^
2)*e^2)*x + 2*((B*b*c^2 - A*c^3)*d^2 - 2*(B*b^2*c - A*b*c^2)*d*e + (B*b^3 - A*b^
2*c)*e^2)*log(c*x + b))/(b*c^3)

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Sympy [A]  time = 11.6098, size = 163, normalized size = 2.12 \[ \frac{A d^{2} \log{\left (x \right )}}{b} + \frac{B e^{2} x^{2}}{2 c} - \frac{x \left (- A c e^{2} + B b e^{2} - 2 B c d e\right )}{c^{2}} + \frac{\left (- A c + B b\right ) \left (b e - c d\right )^{2} \log{\left (x + \frac{- A b c^{2} d^{2} + \frac{b \left (- A c + B b\right ) \left (b e - c d\right )^{2}}{c}}{- A b^{2} c e^{2} + 2 A b c^{2} d e - 2 A c^{3} d^{2} + B b^{3} e^{2} - 2 B b^{2} c d e + B b c^{2} d^{2}} \right )}}{b c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(e*x+d)**2/(c*x**2+b*x),x)

[Out]

A*d**2*log(x)/b + B*e**2*x**2/(2*c) - x*(-A*c*e**2 + B*b*e**2 - 2*B*c*d*e)/c**2
+ (-A*c + B*b)*(b*e - c*d)**2*log(x + (-A*b*c**2*d**2 + b*(-A*c + B*b)*(b*e - c*
d)**2/c)/(-A*b**2*c*e**2 + 2*A*b*c**2*d*e - 2*A*c**3*d**2 + B*b**3*e**2 - 2*B*b*
*2*c*d*e + B*b*c**2*d**2))/(b*c**3)

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GIAC/XCAS [A]  time = 0.275954, size = 158, normalized size = 2.05 \[ \frac{A d^{2}{\rm ln}\left ({\left | x \right |}\right )}{b} + \frac{B c x^{2} e^{2} + 4 \, B c d x e - 2 \, B b x e^{2} + 2 \, A c x e^{2}}{2 \, c^{2}} + \frac{{\left (B b c^{2} d^{2} - A c^{3} d^{2} - 2 \, B b^{2} c d e + 2 \, A b c^{2} d e + B b^{3} e^{2} - A b^{2} c e^{2}\right )}{\rm ln}\left ({\left | c x + b \right |}\right )}{b c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*d^2*ln(abs(x))/b + 1/2*(B*c*x^2*e^2 + 4*B*c*d*x*e - 2*B*b*x*e^2 + 2*A*c*x*e^2)
/c^2 + (B*b*c^2*d^2 - A*c^3*d^2 - 2*B*b^2*c*d*e + 2*A*b*c^2*d*e + B*b^3*e^2 - A*
b^2*c*e^2)*ln(abs(c*x + b))/(b*c^3)