3.174 \(\int \frac{x^3 (c+d x)}{a+b x} \, dx\)

Optimal. Leaf size=87 \[ -\frac{a^3 (b c-a d) \log (a+b x)}{b^5}+\frac{a^2 x (b c-a d)}{b^4}-\frac{a x^2 (b c-a d)}{2 b^3}+\frac{x^3 (b c-a d)}{3 b^2}+\frac{d x^4}{4 b} \]

[Out]

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

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Rubi [A]  time = 0.166751, 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 \[ -\frac{a^3 (b c-a d) \log (a+b x)}{b^5}+\frac{a^2 x (b c-a d)}{b^4}-\frac{a x^2 (b c-a d)}{2 b^3}+\frac{x^3 (b c-a d)}{3 b^2}+\frac{d x^4}{4 b} \]

Antiderivative was successfully verified.

[In]  Int[(x^3*(c + d*x))/(a + b*x),x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{a^{3} \left (a d - b c\right ) \log{\left (a + b x \right )}}{b^{5}} + \frac{a \left (a d - b c\right ) \int x\, dx}{b^{3}} + \frac{d x^{4}}{4 b} - \frac{x^{3} \left (a d - b c\right )}{3 b^{2}} - \frac{\left (a d - b c\right ) \int a^{2}\, dx}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(d*x+c)/(b*x+a),x)

[Out]

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

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Mathematica [A]  time = 0.0470087, size = 80, normalized size = 0.92 \[ \frac{12 a^3 (a d-b c) \log (a+b x)+b x \left (-12 a^3 d+6 a^2 b (2 c+d x)-2 a b^2 x (3 c+2 d x)+b^3 x^2 (4 c+3 d x)\right )}{12 b^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^3*(c + d*x))/(a + b*x),x]

[Out]

(b*x*(-12*a^3*d + 6*a^2*b*(2*c + d*x) - 2*a*b^2*x*(3*c + 2*d*x) + b^3*x^2*(4*c +
3*d*x)) + 12*a^3*(-(b*c) + a*d)*Log[a + b*x])/(12*b^5)

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Maple [A]  time = 0.005, size = 100, normalized size = 1.2 \[{\frac{d{x}^{4}}{4\,b}}-{\frac{{x}^{3}ad}{3\,{b}^{2}}}+{\frac{c{x}^{3}}{3\,b}}+{\frac{{a}^{2}{x}^{2}d}{2\,{b}^{3}}}-{\frac{{x}^{2}ac}{2\,{b}^{2}}}-{\frac{{a}^{3}dx}{{b}^{4}}}+{\frac{{a}^{2}cx}{{b}^{3}}}+{\frac{{a}^{4}\ln \left ( bx+a \right ) d}{{b}^{5}}}-{\frac{{a}^{3}\ln \left ( bx+a \right ) c}{{b}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(d*x+c)/(b*x+a),x)

[Out]

1/4*d*x^4/b-1/3/b^2*x^3*a*d+1/3/b*x^3*c+1/2/b^3*x^2*a^2*d-1/2/b^2*x^2*a*c-1/b^4*
a^3*d*x+1/b^3*a^2*c*x+a^4/b^5*ln(b*x+a)*d-a^3/b^4*ln(b*x+a)*c

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Maxima [A]  time = 1.34564, size = 126, normalized size = 1.45 \[ \frac{3 \, b^{3} d x^{4} + 4 \,{\left (b^{3} c - a b^{2} d\right )} x^{3} - 6 \,{\left (a b^{2} c - a^{2} b d\right )} x^{2} + 12 \,{\left (a^{2} b c - a^{3} d\right )} x}{12 \, b^{4}} - \frac{{\left (a^{3} b c - a^{4} d\right )} \log \left (b x + a\right )}{b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)*x^3/(b*x + a),x, algorithm="maxima")

[Out]

1/12*(3*b^3*d*x^4 + 4*(b^3*c - a*b^2*d)*x^3 - 6*(a*b^2*c - a^2*b*d)*x^2 + 12*(a^
2*b*c - a^3*d)*x)/b^4 - (a^3*b*c - a^4*d)*log(b*x + a)/b^5

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Fricas [A]  time = 0.198067, size = 127, normalized size = 1.46 \[ \frac{3 \, b^{4} d x^{4} + 4 \,{\left (b^{4} c - a b^{3} d\right )} x^{3} - 6 \,{\left (a b^{3} c - a^{2} b^{2} d\right )} x^{2} + 12 \,{\left (a^{2} b^{2} c - a^{3} b d\right )} x - 12 \,{\left (a^{3} b c - a^{4} d\right )} \log \left (b x + a\right )}{12 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)*x^3/(b*x + a),x, algorithm="fricas")

[Out]

1/12*(3*b^4*d*x^4 + 4*(b^4*c - a*b^3*d)*x^3 - 6*(a*b^3*c - a^2*b^2*d)*x^2 + 12*(
a^2*b^2*c - a^3*b*d)*x - 12*(a^3*b*c - a^4*d)*log(b*x + a))/b^5

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Sympy [A]  time = 2.34299, size = 78, normalized size = 0.9 \[ \frac{a^{3} \left (a d - b c\right ) \log{\left (a + b x \right )}}{b^{5}} + \frac{d x^{4}}{4 b} - \frac{x^{3} \left (a d - b c\right )}{3 b^{2}} + \frac{x^{2} \left (a^{2} d - a b c\right )}{2 b^{3}} - \frac{x \left (a^{3} d - a^{2} b c\right )}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(d*x+c)/(b*x+a),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.262375, size = 128, normalized size = 1.47 \[ \frac{3 \, b^{3} d x^{4} + 4 \, b^{3} c x^{3} - 4 \, a b^{2} d x^{3} - 6 \, a b^{2} c x^{2} + 6 \, a^{2} b d x^{2} + 12 \, a^{2} b c x - 12 \, a^{3} d x}{12 \, b^{4}} - \frac{{\left (a^{3} b c - a^{4} d\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)*x^3/(b*x + a),x, algorithm="giac")

[Out]

1/12*(3*b^3*d*x^4 + 4*b^3*c*x^3 - 4*a*b^2*d*x^3 - 6*a*b^2*c*x^2 + 6*a^2*b*d*x^2
+ 12*a^2*b*c*x - 12*a^3*d*x)/b^4 - (a^3*b*c - a^4*d)*ln(abs(b*x + a))/b^5