Optimal. Leaf size=87 \[ -\frac{a^3 (A b-a B) \log (a+b x)}{b^5}+\frac{a^2 x (A b-a B)}{b^4}-\frac{a x^2 (A b-a B)}{2 b^3}+\frac{x^3 (A b-a B)}{3 b^2}+\frac{B x^4}{4 b} \]
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Rubi [A] time = 0.147805, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ -\frac{a^3 (A b-a B) \log (a+b x)}{b^5}+\frac{a^2 x (A b-a B)}{b^4}-\frac{a x^2 (A b-a B)}{2 b^3}+\frac{x^3 (A b-a B)}{3 b^2}+\frac{B x^4}{4 b} \]
Antiderivative was successfully verified.
[In] Int[(x^3*(A + B*x))/(a + b*x),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{B x^{4}}{4 b} - \frac{a^{3} \left (A b - B a\right ) \log{\left (a + b x \right )}}{b^{5}} - \frac{a \left (A b - B a\right ) \int x\, dx}{b^{3}} + \frac{x^{3} \left (A b - B a\right )}{3 b^{2}} + \frac{\left (A b - B a\right ) \int a^{2}\, dx}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3*(B*x+A)/(b*x+a),x)
[Out]
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Mathematica [A] time = 0.0478221, size = 80, normalized size = 0.92 \[ \frac{12 a^3 (a B-A b) \log (a+b x)+b x \left (-12 a^3 B+6 a^2 b (2 A+B x)-2 a b^2 x (3 A+2 B x)+b^3 x^2 (4 A+3 B x)\right )}{12 b^5} \]
Antiderivative was successfully verified.
[In] Integrate[(x^3*(A + B*x))/(a + b*x),x]
[Out]
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Maple [A] time = 0.004, size = 100, normalized size = 1.2 \[{\frac{B{x}^{4}}{4\,b}}+{\frac{A{x}^{3}}{3\,b}}-{\frac{B{x}^{3}a}{3\,{b}^{2}}}-{\frac{aA{x}^{2}}{2\,{b}^{2}}}+{\frac{B{x}^{2}{a}^{2}}{2\,{b}^{3}}}+{\frac{{a}^{2}Ax}{{b}^{3}}}-{\frac{{a}^{3}Bx}{{b}^{4}}}-{\frac{{a}^{3}\ln \left ( bx+a \right ) A}{{b}^{4}}}+{\frac{{a}^{4}\ln \left ( bx+a \right ) B}{{b}^{5}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3*(B*x+A)/(b*x+a),x)
[Out]
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Maxima [A] time = 1.34271, size = 124, normalized size = 1.43 \[ \frac{3 \, B b^{3} x^{4} - 4 \,{\left (B a b^{2} - A b^{3}\right )} x^{3} + 6 \,{\left (B a^{2} b - A a b^{2}\right )} x^{2} - 12 \,{\left (B a^{3} - A a^{2} b\right )} x}{12 \, b^{4}} + \frac{{\left (B a^{4} - A a^{3} b\right )} \log \left (b x + a\right )}{b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*x^3/(b*x + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.196737, size = 127, normalized size = 1.46 \[ \frac{3 \, B b^{4} x^{4} - 4 \,{\left (B a b^{3} - A b^{4}\right )} x^{3} + 6 \,{\left (B a^{2} b^{2} - A a b^{3}\right )} x^{2} - 12 \,{\left (B a^{3} b - A a^{2} b^{2}\right )} x + 12 \,{\left (B a^{4} - A a^{3} b\right )} \log \left (b x + a\right )}{12 \, b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*x^3/(b*x + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.44985, size = 78, normalized size = 0.9 \[ \frac{B x^{4}}{4 b} + \frac{a^{3} \left (- A b + B a\right ) \log{\left (a + b x \right )}}{b^{5}} - \frac{x^{3} \left (- A b + B a\right )}{3 b^{2}} + \frac{x^{2} \left (- A a b + B a^{2}\right )}{2 b^{3}} - \frac{x \left (- A a^{2} b + B a^{3}\right )}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3*(B*x+A)/(b*x+a),x)
[Out]
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GIAC/XCAS [A] time = 0.276005, size = 127, normalized size = 1.46 \[ \frac{3 \, B b^{3} x^{4} - 4 \, B a b^{2} x^{3} + 4 \, A b^{3} x^{3} + 6 \, B a^{2} b x^{2} - 6 \, A a b^{2} x^{2} - 12 \, B a^{3} x + 12 \, A a^{2} b x}{12 \, b^{4}} + \frac{{\left (B a^{4} - A a^{3} b\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*x^3/(b*x + a),x, algorithm="giac")
[Out]