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

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

[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 = 0.0757904, 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, Rules used = {77} \[ \frac{a^2 x (b c-a d)}{b^4}-\frac{a^3 (b c-a d) \log (a+b x)}{b^5}+\frac{x^3 (b c-a d)}{3 b^2}-\frac{a x^2 (b c-a d)}{2 b^3}+\frac{d x^4}{4 b} \]

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

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int \frac{x^3 (c+d x)}{a+b x} \, dx &=\int \left (-\frac{a^2 (-b c+a d)}{b^4}+\frac{a (-b c+a d) x}{b^3}+\frac{(b c-a d) x^2}{b^2}+\frac{d x^3}{b}+\frac{a^3 (-b c+a d)}{b^4 (a+b x)}\right ) \, dx\\ &=\frac{a^2 (b c-a d) x}{b^4}-\frac{a (b c-a d) x^2}{2 b^3}+\frac{(b c-a d) x^3}{3 b^2}+\frac{d x^4}{4 b}-\frac{a^3 (b c-a d) \log (a+b x)}{b^5}\\ \end{align*}

Mathematica [A]  time = 0.0287512, size = 80, normalized size = 0.92 \[ \frac{b x \left (6 a^2 b (2 c+d x)-12 a^3 d-2 a b^2 x (3 c+2 d x)+b^3 x^2 (4 c+3 d x)\right )+12 a^3 (a d-b c) \log (a+b x)}{12 b^5} \]

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 = 0.004, size = 100, normalized size = 1.2 \begin{align*}{\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{a{x}^{2}c}{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}}} \end{align*}

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 = 2.17067, size = 126, normalized size = 1.45 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(d*x+c)/(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 = 1.90481, size = 196, normalized size = 2.25 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(d*x+c)/(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 = 0.441104, size = 78, normalized size = 0.9 \begin{align*} \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}} \end{align*}

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 [A]  time = 1.19723, size = 128, normalized size = 1.47 \begin{align*} \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 )} \log \left ({\left | b x + a \right |}\right )}{b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(d*x+c)/(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)*log(abs(b*x + a))/b^5