3.1113 \(\int \frac{(A+B x) (d+e x)^3}{a+b x} \, dx\)

Optimal. Leaf size=123 \[ \frac{(d+e x)^3 (A b-a B)}{3 b^2}+\frac{(d+e x)^2 (A b-a B) (b d-a e)}{2 b^3}+\frac{e x (A b-a B) (b d-a e)^2}{b^4}+\frac{(A b-a B) (b d-a e)^3 \log (a+b x)}{b^5}+\frac{B (d+e x)^4}{4 b e} \]

[Out]

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

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Rubi [A]  time = 0.0756307, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {77} \[ \frac{(d+e x)^3 (A b-a B)}{3 b^2}+\frac{(d+e x)^2 (A b-a B) (b d-a e)}{2 b^3}+\frac{e x (A b-a B) (b d-a e)^2}{b^4}+\frac{(A b-a B) (b d-a e)^3 \log (a+b x)}{b^5}+\frac{B (d+e x)^4}{4 b e} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(d + e*x)^3)/(a + b*x),x]

[Out]

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

Mathematica [A]  time = 0.0862779, size = 169, normalized size = 1.37 \[ \frac{b x \left (6 a^2 b e^2 (2 A e+6 B d+B e x)-12 a^3 B e^3-2 a b^2 e \left (3 A e (6 d+e x)+B \left (18 d^2+9 d e x+2 e^2 x^2\right )\right )+b^3 \left (2 A e \left (18 d^2+9 d e x+2 e^2 x^2\right )+3 B \left (6 d^2 e x+4 d^3+4 d e^2 x^2+e^3 x^3\right )\right )\right )+12 (A b-a B) (b d-a e)^3 \log (a+b x)}{12 b^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*x*(-12*a^3*B*e^3 + 6*a^2*b*e^2*(6*B*d + 2*A*e + B*e*x) - 2*a*b^2*e*(3*A*e*(6*d + e*x) + B*(18*d^2 + 9*d*e*x
+ 2*e^2*x^2)) + b^3*(2*A*e*(18*d^2 + 9*d*e*x + 2*e^2*x^2) + 3*B*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3)))
+ 12*(A*b - a*B)*(b*d - a*e)^3*Log[a + b*x])/(12*b^5)

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Maple [B]  time = 0.003, size = 341, normalized size = 2.8 \begin{align*}{\frac{B{x}^{4}{e}^{3}}{4\,b}}+{\frac{A{x}^{3}{e}^{3}}{3\,b}}-{\frac{B{x}^{3}a{e}^{3}}{3\,{b}^{2}}}+{\frac{B{x}^{3}d{e}^{2}}{b}}-{\frac{aA{x}^{2}{e}^{3}}{2\,{b}^{2}}}+{\frac{3\,A{x}^{2}d{e}^{2}}{2\,b}}+{\frac{B{x}^{2}{a}^{2}{e}^{3}}{2\,{b}^{3}}}-{\frac{3\,B{x}^{2}ad{e}^{2}}{2\,{b}^{2}}}+{\frac{3\,B{x}^{2}{d}^{2}e}{2\,b}}+{\frac{{a}^{2}A{e}^{3}x}{{b}^{3}}}-3\,{\frac{aAd{e}^{2}x}{{b}^{2}}}+3\,{\frac{A{d}^{2}ex}{b}}-{\frac{B{a}^{3}{e}^{3}x}{{b}^{4}}}+3\,{\frac{B{a}^{2}d{e}^{2}x}{{b}^{3}}}-3\,{\frac{Ba{d}^{2}ex}{{b}^{2}}}+{\frac{B{d}^{3}x}{b}}-{\frac{\ln \left ( bx+a \right ) A{a}^{3}{e}^{3}}{{b}^{4}}}+3\,{\frac{\ln \left ( bx+a \right ) A{a}^{2}d{e}^{2}}{{b}^{3}}}-3\,{\frac{\ln \left ( bx+a \right ) Aa{d}^{2}e}{{b}^{2}}}+{\frac{\ln \left ( bx+a \right ) A{d}^{3}}{b}}+{\frac{\ln \left ( bx+a \right ) B{a}^{4}{e}^{3}}{{b}^{5}}}-3\,{\frac{\ln \left ( bx+a \right ) B{a}^{3}d{e}^{2}}{{b}^{4}}}+3\,{\frac{\ln \left ( bx+a \right ) B{a}^{2}{d}^{2}e}{{b}^{3}}}-{\frac{\ln \left ( bx+a \right ) Ba{d}^{3}}{{b}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.24789, size = 359, normalized size = 2.92 \begin{align*} \frac{3 \, B b^{3} e^{3} x^{4} + 4 \,{\left (3 \, B b^{3} d e^{2} -{\left (B a b^{2} - A b^{3}\right )} e^{3}\right )} x^{3} + 6 \,{\left (3 \, B b^{3} d^{2} e - 3 \,{\left (B a b^{2} - A b^{3}\right )} d e^{2} +{\left (B a^{2} b - A a b^{2}\right )} e^{3}\right )} x^{2} + 12 \,{\left (B b^{3} d^{3} - 3 \,{\left (B a b^{2} - A b^{3}\right )} d^{2} e + 3 \,{\left (B a^{2} b - A a b^{2}\right )} d e^{2} -{\left (B a^{3} - A a^{2} b\right )} e^{3}\right )} x}{12 \, b^{4}} - \frac{{\left ({\left (B a b^{3} - A b^{4}\right )} d^{3} - 3 \,{\left (B a^{2} b^{2} - A a b^{3}\right )} d^{2} e + 3 \,{\left (B a^{3} b - A a^{2} b^{2}\right )} d e^{2} -{\left (B a^{4} - A a^{3} b\right )} e^{3}\right )} \log \left (b x + a\right )}{b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.60546, size = 531, normalized size = 4.32 \begin{align*} \frac{3 \, B b^{4} e^{3} x^{4} + 4 \,{\left (3 \, B b^{4} d e^{2} -{\left (B a b^{3} - A b^{4}\right )} e^{3}\right )} x^{3} + 6 \,{\left (3 \, B b^{4} d^{2} e - 3 \,{\left (B a b^{3} - A b^{4}\right )} d e^{2} +{\left (B a^{2} b^{2} - A a b^{3}\right )} e^{3}\right )} x^{2} + 12 \,{\left (B b^{4} d^{3} - 3 \,{\left (B a b^{3} - A b^{4}\right )} d^{2} e + 3 \,{\left (B a^{2} b^{2} - A a b^{3}\right )} d e^{2} -{\left (B a^{3} b - A a^{2} b^{2}\right )} e^{3}\right )} x - 12 \,{\left ({\left (B a b^{3} - A b^{4}\right )} d^{3} - 3 \,{\left (B a^{2} b^{2} - A a b^{3}\right )} d^{2} e + 3 \,{\left (B a^{3} b - A a^{2} b^{2}\right )} d e^{2} -{\left (B a^{4} - A a^{3} b\right )} e^{3}\right )} \log \left (b x + a\right )}{12 \, b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.878373, size = 214, normalized size = 1.74 \begin{align*} \frac{B e^{3} x^{4}}{4 b} - \frac{x^{3} \left (- A b e^{3} + B a e^{3} - 3 B b d e^{2}\right )}{3 b^{2}} + \frac{x^{2} \left (- A a b e^{3} + 3 A b^{2} d e^{2} + B a^{2} e^{3} - 3 B a b d e^{2} + 3 B b^{2} d^{2} e\right )}{2 b^{3}} - \frac{x \left (- A a^{2} b e^{3} + 3 A a b^{2} d e^{2} - 3 A b^{3} d^{2} e + B a^{3} e^{3} - 3 B a^{2} b d e^{2} + 3 B a b^{2} d^{2} e - B b^{3} d^{3}\right )}{b^{4}} + \frac{\left (- A b + B a\right ) \left (a e - b d\right )^{3} \log{\left (a + b x \right )}}{b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 2.28732, size = 387, normalized size = 3.15 \begin{align*} \frac{3 \, B b^{3} x^{4} e^{3} + 12 \, B b^{3} d x^{3} e^{2} + 18 \, B b^{3} d^{2} x^{2} e + 12 \, B b^{3} d^{3} x - 4 \, B a b^{2} x^{3} e^{3} + 4 \, A b^{3} x^{3} e^{3} - 18 \, B a b^{2} d x^{2} e^{2} + 18 \, A b^{3} d x^{2} e^{2} - 36 \, B a b^{2} d^{2} x e + 36 \, A b^{3} d^{2} x e + 6 \, B a^{2} b x^{2} e^{3} - 6 \, A a b^{2} x^{2} e^{3} + 36 \, B a^{2} b d x e^{2} - 36 \, A a b^{2} d x e^{2} - 12 \, B a^{3} x e^{3} + 12 \, A a^{2} b x e^{3}}{12 \, b^{4}} - \frac{{\left (B a b^{3} d^{3} - A b^{4} d^{3} - 3 \, B a^{2} b^{2} d^{2} e + 3 \, A a b^{3} d^{2} e + 3 \, B a^{3} b d e^{2} - 3 \, A a^{2} b^{2} d e^{2} - B a^{4} e^{3} + A a^{3} b e^{3}\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((B*x+A)*(e*x+d)^3/(b*x+a),x, algorithm="giac")

[Out]

1/12*(3*B*b^3*x^4*e^3 + 12*B*b^3*d*x^3*e^2 + 18*B*b^3*d^2*x^2*e + 12*B*b^3*d^3*x - 4*B*a*b^2*x^3*e^3 + 4*A*b^3
*x^3*e^3 - 18*B*a*b^2*d*x^2*e^2 + 18*A*b^3*d*x^2*e^2 - 36*B*a*b^2*d^2*x*e + 36*A*b^3*d^2*x*e + 6*B*a^2*b*x^2*e
^3 - 6*A*a*b^2*x^2*e^3 + 36*B*a^2*b*d*x*e^2 - 36*A*a*b^2*d*x*e^2 - 12*B*a^3*x*e^3 + 12*A*a^2*b*x*e^3)/b^4 - (B
*a*b^3*d^3 - A*b^4*d^3 - 3*B*a^2*b^2*d^2*e + 3*A*a*b^3*d^2*e + 3*B*a^3*b*d*e^2 - 3*A*a^2*b^2*d*e^2 - B*a^4*e^3
+ A*a^3*b*e^3)*log(abs(b*x + a))/b^5