3.175 \(\int \frac{x^2 (c+d x)}{a+b x} \, dx\)

Optimal. Leaf size=66 \[ \frac{a^2 (b c-a d) \log (a+b x)}{b^4}-\frac{a x (b c-a d)}{b^3}+\frac{x^2 (b c-a d)}{2 b^2}+\frac{d x^3}{3 b} \]

[Out]

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

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Rubi [A]  time = 0.114618, antiderivative size = 66, 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 \[ \frac{a^2 (b c-a d) \log (a+b x)}{b^4}-\frac{a x (b c-a d)}{b^3}+\frac{x^2 (b c-a d)}{2 b^2}+\frac{d x^3}{3 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(d*x+c)/(b*x+a),x)

[Out]

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

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Mathematica [A]  time = 0.0348711, size = 61, normalized size = 0.92 \[ \frac{b x \left (6 a^2 d-3 a b (2 c+d x)+b^2 x (3 c+2 d x)\right )+6 a^2 (b c-a d) \log (a+b x)}{6 b^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.004, size = 76, normalized size = 1.2 \[{\frac{d{x}^{3}}{3\,b}}-{\frac{{x}^{2}ad}{2\,{b}^{2}}}+{\frac{c{x}^{2}}{2\,b}}+{\frac{{a}^{2}dx}{{b}^{3}}}-{\frac{acx}{{b}^{2}}}-{\frac{{a}^{3}\ln \left ( bx+a \right ) d}{{b}^{4}}}+{\frac{{a}^{2}\ln \left ( bx+a \right ) c}{{b}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(d*x+c)/(b*x+a),x)

[Out]

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

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Maxima [A]  time = 1.34616, size = 93, normalized size = 1.41 \[ \frac{2 \, b^{2} d x^{3} + 3 \,{\left (b^{2} c - a b d\right )} x^{2} - 6 \,{\left (a b c - a^{2} d\right )} x}{6 \, b^{3}} + \frac{{\left (a^{2} b c - a^{3} d\right )} \log \left (b x + a\right )}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.203537, size = 96, normalized size = 1.45 \[ \frac{2 \, b^{3} d x^{3} + 3 \,{\left (b^{3} c - a b^{2} d\right )} x^{2} - 6 \,{\left (a b^{2} c - a^{2} b d\right )} x + 6 \,{\left (a^{2} b c - a^{3} d\right )} \log \left (b x + a\right )}{6 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.28825, size = 58, normalized size = 0.88 \[ - \frac{a^{2} \left (a d - b c\right ) \log{\left (a + b x \right )}}{b^{4}} + \frac{d x^{3}}{3 b} - \frac{x^{2} \left (a d - b c\right )}{2 b^{2}} + \frac{x \left (a^{2} d - a b c\right )}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(d*x+c)/(b*x+a),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.245121, size = 95, normalized size = 1.44 \[ \frac{2 \, b^{2} d x^{3} + 3 \, b^{2} c x^{2} - 3 \, a b d x^{2} - 6 \, a b c x + 6 \, a^{2} d x}{6 \, b^{3}} + \frac{{\left (a^{2} b c - a^{3} d\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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