3.1098 \(\int \frac{(A+B x) (d+e x)}{a+b x} \, dx\)

Optimal. Leaf size=59 \[ \frac{(A b-a B) (b d-a e) \log (a+b x)}{b^3}+\frac{B x (b d-a e)}{b^2}+\frac{e (A+B x)^2}{2 b B} \]

[Out]

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

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Rubi [A]  time = 0.0948833, antiderivative size = 59, 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 \[ \frac{(A b-a B) (b d-a e) \log (a+b x)}{b^3}+\frac{B x (b d-a e)}{b^2}+\frac{e (A+B x)^2}{2 b B} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0385519, size = 56, normalized size = 0.95 \[ \frac{b x (b (2 A e+2 B d+B e x)-2 a B e)+2 (A b-a B) (b d-a e) \log (a+b x)}{2 b^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.004, size = 90, normalized size = 1.5 \[{\frac{B{x}^{2}e}{2\,b}}+{\frac{Aex}{b}}-{\frac{Baex}{{b}^{2}}}+{\frac{Bdx}{b}}-{\frac{\ln \left ( bx+a \right ) Aae}{{b}^{2}}}+{\frac{\ln \left ( bx+a \right ) Ad}{b}}+{\frac{\ln \left ( bx+a \right ) B{a}^{2}e}{{b}^{3}}}-{\frac{\ln \left ( bx+a \right ) Bad}{{b}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(e*x+d)/(b*x+a),x)

[Out]

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

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Maxima [A]  time = 1.35463, size = 97, normalized size = 1.64 \[ \frac{B b e x^{2} + 2 \,{\left (B b d -{\left (B a - A b\right )} e\right )} x}{2 \, b^{2}} - \frac{{\left ({\left (B a b - A b^{2}\right )} d -{\left (B a^{2} - A a b\right )} e\right )} \log \left (b x + a\right )}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.222031, size = 101, normalized size = 1.71 \[ \frac{B b^{2} e x^{2} + 2 \,{\left (B b^{2} d -{\left (B a b - A b^{2}\right )} e\right )} x - 2 \,{\left ({\left (B a b - A b^{2}\right )} d -{\left (B a^{2} - A a b\right )} e\right )} \log \left (b x + a\right )}{2 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.80958, size = 53, normalized size = 0.9 \[ \frac{B e x^{2}}{2 b} - \frac{x \left (- A b e + B a e - B b d\right )}{b^{2}} + \frac{\left (- A b + B a\right ) \left (a e - b d\right ) \log{\left (a + b x \right )}}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.214571, size = 100, normalized size = 1.69 \[ \frac{B b x^{2} e + 2 \, B b d x - 2 \, B a x e + 2 \, A b x e}{2 \, b^{2}} - \frac{{\left (B a b d - A b^{2} d - B a^{2} e + A a b e\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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