3.175 \(\int \frac{x^3 (A+B x)}{a+b x} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0628807, 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} \[ \frac{a^2 x (A b-a B)}{b^4}-\frac{a^3 (A b-a B) \log (a+b x)}{b^5}+\frac{x^3 (A b-a B)}{3 b^2}-\frac{a x^2 (A b-a B)}{2 b^3}+\frac{B x^4}{4 b} \]

Antiderivative was successfully verified.

[In]

Int[(x^3*(A + B*x))/(a + b*x),x]

[Out]

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

Mathematica [A]  time = 0.0270113, size = 80, normalized size = 0.92 \[ \frac{b x \left (6 a^2 b (2 A+B x)-12 a^3 B-2 a b^2 x (3 A+2 B x)+b^3 x^2 (4 A+3 B x)\right )+12 a^3 (a B-A b) \log (a+b x)}{12 b^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^3*(A + B*x))/(a + b*x),x]

[Out]

(b*x*(-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)) + 12*a^3*(-(A*b) + a*
B)*Log[a + b*x])/(12*b^5)

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Maple [A]  time = 0.003, size = 100, normalized size = 1.2 \begin{align*}{\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}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(B*x+A)/(b*x+a),x)

[Out]

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

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Maxima [A]  time = 1.01558, size = 124, normalized size = 1.43 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(B*x+A)/(b*x+a),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.45249, size = 196, normalized size = 2.25 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(B*x+A)/(b*x+a),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.43022, size = 78, normalized size = 0.9 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(B*x+A)/(b*x+a),x)

[Out]

B*x**4/(4*b) + a**3*(-A*b + B*a)*log(a + b*x)/b**5 - x**3*(-A*b + B*a)/(3*b**2) + x**2*(-A*a*b + B*a**2)/(2*b*
*3) - x*(-A*a**2*b + B*a**3)/b**4

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Giac [A]  time = 1.36156, size = 127, normalized size = 1.46 \begin{align*} \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 )} \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*(B*x+A)/(b*x+a),x, algorithm="giac")

[Out]

1/12*(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)/b
^4 + (B*a^4 - A*a^3*b)*log(abs(b*x + a))/b^5