3.999 \(\int \frac{(a+b x) (A+B x)}{d+e x} \, dx\)

Optimal. Leaf size=60 \[ \frac{(b d-a e) (B d-A e) \log (d+e x)}{e^3}+\frac{B (a+b x)^2}{2 b e}-\frac{b x (B d-A e)}{e^2} \]

[Out]

-((b*(B*d - A*e)*x)/e^2) + (B*(a + b*x)^2)/(2*b*e) + ((b*d - a*e)*(B*d - A*e)*Lo
g[d + e*x])/e^3

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Rubi [A]  time = 0.10138, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ \frac{(b d-a e) (B d-A e) \log (d+e x)}{e^3}+\frac{B (a+b x)^2}{2 b e}-\frac{b x (B d-A e)}{e^2} \]

Antiderivative was successfully verified.

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

[Out]

-((b*(B*d - A*e)*x)/e^2) + (B*(a + b*x)^2)/(2*b*e) + ((b*d - a*e)*(B*d - A*e)*Lo
g[d + e*x])/e^3

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0416093, size = 56, normalized size = 0.93 \[ \frac{e x (2 a B e+b (2 A e-2 B d+B e x))+2 (b d-a e) (B d-A e) \log (d+e x)}{2 e^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 90, normalized size = 1.5 \[{\frac{bB{x}^{2}}{2\,e}}+{\frac{Abx}{e}}+{\frac{Bax}{e}}-{\frac{Bbdx}{{e}^{2}}}+{\frac{\ln \left ( ex+d \right ) aA}{e}}-{\frac{\ln \left ( ex+d \right ) Abd}{{e}^{2}}}-{\frac{\ln \left ( ex+d \right ) Bad}{{e}^{2}}}+{\frac{\ln \left ( ex+d \right ) bB{d}^{2}}{{e}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(B*x+A)/(e*x+d),x)

[Out]

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

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Maxima [A]  time = 1.33978, size = 89, normalized size = 1.48 \[ \frac{B b e x^{2} - 2 \,{\left (B b d -{\left (B a + A b\right )} e\right )} x}{2 \, e^{2}} + \frac{{\left (B b d^{2} + A a e^{2} -{\left (B a + A b\right )} d e\right )} \log \left (e x + d\right )}{e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.202655, size = 92, normalized size = 1.53 \[ \frac{B b e^{2} x^{2} - 2 \,{\left (B b d e -{\left (B a + A b\right )} e^{2}\right )} x + 2 \,{\left (B b d^{2} + A a e^{2} -{\left (B a + A b\right )} d e\right )} \log \left (e x + d\right )}{2 \, e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.81423, size = 53, normalized size = 0.88 \[ \frac{B b x^{2}}{2 e} + \frac{x \left (A b e + B a e - B b d\right )}{e^{2}} - \frac{\left (- A e + B d\right ) \left (a e - b d\right ) \log{\left (d + e x \right )}}{e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.218211, size = 96, normalized size = 1.6 \[{\left (B b d^{2} - B a d e - A b d e + A a e^{2}\right )} e^{\left (-3\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + \frac{1}{2} \,{\left (B b x^{2} e - 2 \, B b d x + 2 \, B a x e + 2 \, A b x e\right )} e^{\left (-2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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