3.1044 \(\int \frac{(a+b x)^3 (A+B x)}{(d+e x)^4} \, dx\)

Optimal. Leaf size=149 \[ -\frac{b^2 \log (d+e x) (-3 a B e-A b e+4 b B d)}{e^5}-\frac{3 b (b d-a e) (-a B e-A b e+2 b B d)}{e^5 (d+e x)}+\frac{(b d-a e)^2 (-a B e-3 A b e+4 b B d)}{2 e^5 (d+e x)^2}-\frac{(b d-a e)^3 (B d-A e)}{3 e^5 (d+e x)^3}+\frac{b^3 B x}{e^4} \]

[Out]

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

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Rubi [A]  time = 0.138566, antiderivative size = 149, 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} \[ -\frac{b^2 \log (d+e x) (-3 a B e-A b e+4 b B d)}{e^5}-\frac{3 b (b d-a e) (-a B e-A b e+2 b B d)}{e^5 (d+e x)}+\frac{(b d-a e)^2 (-a B e-3 A b e+4 b B d)}{2 e^5 (d+e x)^2}-\frac{(b d-a e)^3 (B d-A e)}{3 e^5 (d+e x)^3}+\frac{b^3 B x}{e^4} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^3*(A + B*x))/(d + e*x)^4,x]

[Out]

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

Mathematica [A]  time = 0.111807, size = 232, normalized size = 1.56 \[ \frac{-3 a^2 b e^2 \left (A e (d+3 e x)+2 B \left (d^2+3 d e x+3 e^2 x^2\right )\right )-a^3 e^3 (2 A e+B (d+3 e x))+3 a b^2 e \left (B d \left (11 d^2+27 d e x+18 e^2 x^2\right )-2 A e \left (d^2+3 d e x+3 e^2 x^2\right )\right )-6 b^2 (d+e x)^3 \log (d+e x) (-3 a B e-A b e+4 b B d)+b^3 \left (A d e \left (11 d^2+27 d e x+18 e^2 x^2\right )-2 B \left (9 d^2 e^2 x^2+27 d^3 e x+13 d^4-9 d e^3 x^3-3 e^4 x^4\right )\right )}{6 e^5 (d+e x)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^3*(A + B*x))/(d + e*x)^4,x]

[Out]

(-(a^3*e^3*(2*A*e + B*(d + 3*e*x))) - 3*a^2*b*e^2*(A*e*(d + 3*e*x) + 2*B*(d^2 + 3*d*e*x + 3*e^2*x^2)) + 3*a*b^
2*e*(-2*A*e*(d^2 + 3*d*e*x + 3*e^2*x^2) + B*d*(11*d^2 + 27*d*e*x + 18*e^2*x^2)) + b^3*(A*d*e*(11*d^2 + 27*d*e*
x + 18*e^2*x^2) - 2*B*(13*d^4 + 27*d^3*e*x + 9*d^2*e^2*x^2 - 9*d*e^3*x^3 - 3*e^4*x^4)) - 6*b^2*(4*b*B*d - A*b*
e - 3*a*B*e)*(d + e*x)^3*Log[d + e*x])/(6*e^5*(d + e*x)^3)

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Maple [B]  time = 0.009, size = 419, normalized size = 2.8 \begin{align*}{\frac{{b}^{3}Bx}{{e}^{4}}}-{\frac{{a}^{3}A}{3\,e \left ( ex+d \right ) ^{3}}}+{\frac{Ad{a}^{2}b}{{e}^{2} \left ( ex+d \right ) ^{3}}}-{\frac{A{d}^{2}a{b}^{2}}{{e}^{3} \left ( ex+d \right ) ^{3}}}+{\frac{A{d}^{3}{b}^{3}}{3\,{e}^{4} \left ( ex+d \right ) ^{3}}}+{\frac{Bd{a}^{3}}{3\,{e}^{2} \left ( ex+d \right ) ^{3}}}-{\frac{B{d}^{2}{a}^{2}b}{{e}^{3} \left ( ex+d \right ) ^{3}}}+{\frac{B{d}^{3}a{b}^{2}}{{e}^{4} \left ( ex+d \right ) ^{3}}}-{\frac{{b}^{3}B{d}^{4}}{3\,{e}^{5} \left ( ex+d \right ) ^{3}}}-{\frac{3\,Ab{a}^{2}}{2\,{e}^{2} \left ( ex+d \right ) ^{2}}}+3\,{\frac{a{b}^{2}Ad}{{e}^{3} \left ( ex+d \right ) ^{2}}}-{\frac{3\,{b}^{3}A{d}^{2}}{2\,{e}^{4} \left ( ex+d \right ) ^{2}}}-{\frac{B{a}^{3}}{2\,{e}^{2} \left ( ex+d \right ) ^{2}}}+3\,{\frac{{a}^{2}bBd}{{e}^{3} \left ( ex+d \right ) ^{2}}}-{\frac{9\,a{b}^{2}B{d}^{2}}{2\,{e}^{4} \left ( ex+d \right ) ^{2}}}+2\,{\frac{{b}^{3}B{d}^{3}}{{e}^{5} \left ( ex+d \right ) ^{2}}}-3\,{\frac{a{b}^{2}A}{{e}^{3} \left ( ex+d \right ) }}+3\,{\frac{{b}^{3}Ad}{{e}^{4} \left ( ex+d \right ) }}-3\,{\frac{{a}^{2}bB}{{e}^{3} \left ( ex+d \right ) }}+9\,{\frac{{b}^{2}Bda}{{e}^{4} \left ( ex+d \right ) }}-6\,{\frac{{b}^{3}B{d}^{2}}{{e}^{5} \left ( ex+d \right ) }}+{\frac{{b}^{3}\ln \left ( ex+d \right ) A}{{e}^{4}}}+3\,{\frac{{b}^{2}\ln \left ( ex+d \right ) Ba}{{e}^{4}}}-4\,{\frac{{b}^{3}\ln \left ( ex+d \right ) Bd}{{e}^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^3*(B*x+A)/(e*x+d)^4,x)

[Out]

b^3*B*x/e^4-1/3/e/(e*x+d)^3*a^3*A+1/e^2/(e*x+d)^3*A*d*a^2*b-1/e^3/(e*x+d)^3*A*d^2*a*b^2+1/3/e^4/(e*x+d)^3*A*d^
3*b^3+1/3/e^2/(e*x+d)^3*B*d*a^3-1/e^3/(e*x+d)^3*B*d^2*a^2*b+1/e^4/(e*x+d)^3*B*d^3*a*b^2-1/3/e^5/(e*x+d)^3*b^3*
B*d^4-3/2/e^2/(e*x+d)^2*A*a^2*b+3/e^3/(e*x+d)^2*A*a*b^2*d-3/2/e^4/(e*x+d)^2*A*b^3*d^2-1/2/e^2/(e*x+d)^2*B*a^3+
3/e^3/(e*x+d)^2*B*a^2*b*d-9/2/e^4/(e*x+d)^2*B*a*b^2*d^2+2/e^5/(e*x+d)^2*b^3*B*d^3-3*b^2/e^3/(e*x+d)*A*a+3*b^3/
e^4/(e*x+d)*A*d-3*b/e^3/(e*x+d)*B*a^2+9*b^2/e^4/(e*x+d)*B*d*a-6*b^3/e^5/(e*x+d)*B*d^2+b^3/e^4*ln(e*x+d)*A+3*b^
2/e^4*ln(e*x+d)*B*a-4*b^3/e^5*ln(e*x+d)*B*d

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Maxima [A]  time = 1.13037, size = 383, normalized size = 2.57 \begin{align*} \frac{B b^{3} x}{e^{4}} - \frac{26 \, B b^{3} d^{4} + 2 \, A a^{3} e^{4} - 11 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 6 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} +{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} + 18 \,{\left (2 \, B b^{3} d^{2} e^{2} -{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} +{\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} + 3 \,{\left (20 \, B b^{3} d^{3} e - 9 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 6 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{3} +{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x}{6 \,{\left (e^{8} x^{3} + 3 \, d e^{7} x^{2} + 3 \, d^{2} e^{6} x + d^{3} e^{5}\right )}} - \frac{{\left (4 \, B b^{3} d -{\left (3 \, B a b^{2} + A b^{3}\right )} e\right )} \log \left (e x + d\right )}{e^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)/(e*x+d)^4,x, algorithm="maxima")

[Out]

B*b^3*x/e^4 - 1/6*(26*B*b^3*d^4 + 2*A*a^3*e^4 - 11*(3*B*a*b^2 + A*b^3)*d^3*e + 6*(B*a^2*b + A*a*b^2)*d^2*e^2 +
(B*a^3 + 3*A*a^2*b)*d*e^3 + 18*(2*B*b^3*d^2*e^2 - (3*B*a*b^2 + A*b^3)*d*e^3 + (B*a^2*b + A*a*b^2)*e^4)*x^2 +
3*(20*B*b^3*d^3*e - 9*(3*B*a*b^2 + A*b^3)*d^2*e^2 + 6*(B*a^2*b + A*a*b^2)*d*e^3 + (B*a^3 + 3*A*a^2*b)*e^4)*x)/
(e^8*x^3 + 3*d*e^7*x^2 + 3*d^2*e^6*x + d^3*e^5) - (4*B*b^3*d - (3*B*a*b^2 + A*b^3)*e)*log(e*x + d)/e^5

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Fricas [B]  time = 1.94901, size = 830, normalized size = 5.57 \begin{align*} \frac{6 \, B b^{3} e^{4} x^{4} + 18 \, B b^{3} d e^{3} x^{3} - 26 \, B b^{3} d^{4} - 2 \, A a^{3} e^{4} + 11 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e - 6 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} -{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} - 18 \,{\left (B b^{3} d^{2} e^{2} -{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} +{\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} - 3 \,{\left (18 \, B b^{3} d^{3} e - 9 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 6 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{3} +{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x - 6 \,{\left (4 \, B b^{3} d^{4} -{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e +{\left (4 \, B b^{3} d e^{3} -{\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 3 \,{\left (4 \, B b^{3} d^{2} e^{2} -{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3}\right )} x^{2} + 3 \,{\left (4 \, B b^{3} d^{3} e -{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2}\right )} x\right )} \log \left (e x + d\right )}{6 \,{\left (e^{8} x^{3} + 3 \, d e^{7} x^{2} + 3 \, d^{2} e^{6} x + d^{3} e^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)/(e*x+d)^4,x, algorithm="fricas")

[Out]

1/6*(6*B*b^3*e^4*x^4 + 18*B*b^3*d*e^3*x^3 - 26*B*b^3*d^4 - 2*A*a^3*e^4 + 11*(3*B*a*b^2 + A*b^3)*d^3*e - 6*(B*a
^2*b + A*a*b^2)*d^2*e^2 - (B*a^3 + 3*A*a^2*b)*d*e^3 - 18*(B*b^3*d^2*e^2 - (3*B*a*b^2 + A*b^3)*d*e^3 + (B*a^2*b
+ A*a*b^2)*e^4)*x^2 - 3*(18*B*b^3*d^3*e - 9*(3*B*a*b^2 + A*b^3)*d^2*e^2 + 6*(B*a^2*b + A*a*b^2)*d*e^3 + (B*a^
3 + 3*A*a^2*b)*e^4)*x - 6*(4*B*b^3*d^4 - (3*B*a*b^2 + A*b^3)*d^3*e + (4*B*b^3*d*e^3 - (3*B*a*b^2 + A*b^3)*e^4)
*x^3 + 3*(4*B*b^3*d^2*e^2 - (3*B*a*b^2 + A*b^3)*d*e^3)*x^2 + 3*(4*B*b^3*d^3*e - (3*B*a*b^2 + A*b^3)*d^2*e^2)*x
)*log(e*x + d))/(e^8*x^3 + 3*d*e^7*x^2 + 3*d^2*e^6*x + d^3*e^5)

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Sympy [B]  time = 22.1649, size = 337, normalized size = 2.26 \begin{align*} \frac{B b^{3} x}{e^{4}} + \frac{b^{2} \left (A b e + 3 B a e - 4 B b d\right ) \log{\left (d + e x \right )}}{e^{5}} - \frac{2 A a^{3} e^{4} + 3 A a^{2} b d e^{3} + 6 A a b^{2} d^{2} e^{2} - 11 A b^{3} d^{3} e + B a^{3} d e^{3} + 6 B a^{2} b d^{2} e^{2} - 33 B a b^{2} d^{3} e + 26 B b^{3} d^{4} + x^{2} \left (18 A a b^{2} e^{4} - 18 A b^{3} d e^{3} + 18 B a^{2} b e^{4} - 54 B a b^{2} d e^{3} + 36 B b^{3} d^{2} e^{2}\right ) + x \left (9 A a^{2} b e^{4} + 18 A a b^{2} d e^{3} - 27 A b^{3} d^{2} e^{2} + 3 B a^{3} e^{4} + 18 B a^{2} b d e^{3} - 81 B a b^{2} d^{2} e^{2} + 60 B b^{3} d^{3} e\right )}{6 d^{3} e^{5} + 18 d^{2} e^{6} x + 18 d e^{7} x^{2} + 6 e^{8} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**3*(B*x+A)/(e*x+d)**4,x)

[Out]

B*b**3*x/e**4 + b**2*(A*b*e + 3*B*a*e - 4*B*b*d)*log(d + e*x)/e**5 - (2*A*a**3*e**4 + 3*A*a**2*b*d*e**3 + 6*A*
a*b**2*d**2*e**2 - 11*A*b**3*d**3*e + B*a**3*d*e**3 + 6*B*a**2*b*d**2*e**2 - 33*B*a*b**2*d**3*e + 26*B*b**3*d*
*4 + x**2*(18*A*a*b**2*e**4 - 18*A*b**3*d*e**3 + 18*B*a**2*b*e**4 - 54*B*a*b**2*d*e**3 + 36*B*b**3*d**2*e**2)
+ x*(9*A*a**2*b*e**4 + 18*A*a*b**2*d*e**3 - 27*A*b**3*d**2*e**2 + 3*B*a**3*e**4 + 18*B*a**2*b*d*e**3 - 81*B*a*
b**2*d**2*e**2 + 60*B*b**3*d**3*e))/(6*d**3*e**5 + 18*d**2*e**6*x + 18*d*e**7*x**2 + 6*e**8*x**3)

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Giac [A]  time = 2.81399, size = 360, normalized size = 2.42 \begin{align*} B b^{3} x e^{\left (-4\right )} -{\left (4 \, B b^{3} d - 3 \, B a b^{2} e - A b^{3} e\right )} e^{\left (-5\right )} \log \left ({\left | x e + d \right |}\right ) - \frac{{\left (26 \, B b^{3} d^{4} - 33 \, B a b^{2} d^{3} e - 11 \, A b^{3} d^{3} e + 6 \, B a^{2} b d^{2} e^{2} + 6 \, A a b^{2} d^{2} e^{2} + B a^{3} d e^{3} + 3 \, A a^{2} b d e^{3} + 2 \, A a^{3} e^{4} + 18 \,{\left (2 \, B b^{3} d^{2} e^{2} - 3 \, B a b^{2} d e^{3} - A b^{3} d e^{3} + B a^{2} b e^{4} + A a b^{2} e^{4}\right )} x^{2} + 3 \,{\left (20 \, B b^{3} d^{3} e - 27 \, B a b^{2} d^{2} e^{2} - 9 \, A b^{3} d^{2} e^{2} + 6 \, B a^{2} b d e^{3} + 6 \, A a b^{2} d e^{3} + B a^{3} e^{4} + 3 \, A a^{2} b e^{4}\right )} x\right )} e^{\left (-5\right )}}{6 \,{\left (x e + d\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)/(e*x+d)^4,x, algorithm="giac")

[Out]

B*b^3*x*e^(-4) - (4*B*b^3*d - 3*B*a*b^2*e - A*b^3*e)*e^(-5)*log(abs(x*e + d)) - 1/6*(26*B*b^3*d^4 - 33*B*a*b^2
*d^3*e - 11*A*b^3*d^3*e + 6*B*a^2*b*d^2*e^2 + 6*A*a*b^2*d^2*e^2 + B*a^3*d*e^3 + 3*A*a^2*b*d*e^3 + 2*A*a^3*e^4
+ 18*(2*B*b^3*d^2*e^2 - 3*B*a*b^2*d*e^3 - A*b^3*d*e^3 + B*a^2*b*e^4 + A*a*b^2*e^4)*x^2 + 3*(20*B*b^3*d^3*e - 2
7*B*a*b^2*d^2*e^2 - 9*A*b^3*d^2*e^2 + 6*B*a^2*b*d*e^3 + 6*A*a*b^2*d*e^3 + B*a^3*e^4 + 3*A*a^2*b*e^4)*x)*e^(-5)
/(x*e + d)^3