3.1045 \(\int \frac{(a+b x)^3 (A+B x)}{(d+e x)^5} \, dx\)

Optimal. Leaf size=129 \[ -\frac{(a+b x)^4 (B d-A e)}{4 e (d+e x)^4 (b d-a e)}+\frac{3 b^2 B (b d-a e)}{e^5 (d+e x)}-\frac{3 b B (b d-a e)^2}{2 e^5 (d+e x)^2}+\frac{B (b d-a e)^3}{3 e^5 (d+e x)^3}+\frac{b^3 B \log (d+e x)}{e^5} \]

[Out]

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

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Rubi [A]  time = 0.11029, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {78, 43} \[ -\frac{(a+b x)^4 (B d-A e)}{4 e (d+e x)^4 (b d-a e)}+\frac{3 b^2 B (b d-a e)}{e^5 (d+e x)}-\frac{3 b B (b d-a e)^2}{2 e^5 (d+e x)^2}+\frac{B (b d-a e)^3}{3 e^5 (d+e x)^3}+\frac{b^3 B \log (d+e x)}{e^5} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^3*(A + B*x))/(d + e*x)^5,x]

[Out]

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

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.10322, size = 222, normalized size = 1.72 \[ \frac{-3 a^2 b e^2 \left (A e (d+4 e x)+B \left (d^2+4 d e x+6 e^2 x^2\right )\right )-a^3 e^3 (3 A e+B (d+4 e x))-3 a b^2 e \left (A e \left (d^2+4 d e x+6 e^2 x^2\right )+3 B \left (4 d^2 e x+d^3+6 d e^2 x^2+4 e^3 x^3\right )\right )+b^3 \left (B d \left (88 d^2 e x+25 d^3+108 d e^2 x^2+48 e^3 x^3\right )-3 A e \left (4 d^2 e x+d^3+6 d e^2 x^2+4 e^3 x^3\right )\right )+12 b^3 B (d+e x)^4 \log (d+e x)}{12 e^5 (d+e x)^4} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^3*(A + B*x))/(d + e*x)^5,x]

[Out]

(-(a^3*e^3*(3*A*e + B*(d + 4*e*x))) - 3*a^2*b*e^2*(A*e*(d + 4*e*x) + B*(d^2 + 4*d*e*x + 6*e^2*x^2)) - 3*a*b^2*
e*(A*e*(d^2 + 4*d*e*x + 6*e^2*x^2) + 3*B*(d^3 + 4*d^2*e*x + 6*d*e^2*x^2 + 4*e^3*x^3)) + b^3*(-3*A*e*(d^3 + 4*d
^2*e*x + 6*d*e^2*x^2 + 4*e^3*x^3) + B*d*(25*d^3 + 88*d^2*e*x + 108*d*e^2*x^2 + 48*e^3*x^3)) + 12*b^3*B*(d + e*
x)^4*Log[d + e*x])/(12*e^5*(d + e*x)^4)

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Maple [B]  time = 0.007, size = 430, normalized size = 3.3 \begin{align*} -{\frac{Ab{a}^{2}}{{e}^{2} \left ( ex+d \right ) ^{3}}}+2\,{\frac{Ada{b}^{2}}{{e}^{3} \left ( ex+d \right ) ^{3}}}-{\frac{A{d}^{2}{b}^{3}}{{e}^{4} \left ( ex+d \right ) ^{3}}}-{\frac{B{a}^{3}}{3\,{e}^{2} \left ( ex+d \right ) ^{3}}}+2\,{\frac{Bd{a}^{2}b}{{e}^{3} \left ( ex+d \right ) ^{3}}}-3\,{\frac{B{d}^{2}a{b}^{2}}{{e}^{4} \left ( ex+d \right ) ^{3}}}+{\frac{4\,{b}^{3}B{d}^{3}}{3\,{e}^{5} \left ( ex+d \right ) ^{3}}}-{\frac{3\,a{b}^{2}A}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}}+{\frac{3\,Ad{b}^{3}}{2\,{e}^{4} \left ( ex+d \right ) ^{2}}}-{\frac{3\,{a}^{2}bB}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}}+{\frac{9\,Bda{b}^{2}}{2\,{e}^{4} \left ( ex+d \right ) ^{2}}}-3\,{\frac{{b}^{3}B{d}^{2}}{{e}^{5} \left ( ex+d \right ) ^{2}}}-{\frac{{b}^{3}A}{{e}^{4} \left ( ex+d \right ) }}-3\,{\frac{a{b}^{2}B}{{e}^{4} \left ( ex+d \right ) }}+4\,{\frac{{b}^{3}Bd}{{e}^{5} \left ( ex+d \right ) }}+{\frac{B{b}^{3}\ln \left ( ex+d \right ) }{{e}^{5}}}-{\frac{A{a}^{3}}{4\,e \left ( ex+d \right ) ^{4}}}+{\frac{3\,Ad{a}^{2}b}{4\,{e}^{2} \left ( ex+d \right ) ^{4}}}-{\frac{3\,A{d}^{2}a{b}^{2}}{4\,{e}^{3} \left ( ex+d \right ) ^{4}}}+{\frac{A{d}^{3}{b}^{3}}{4\,{e}^{4} \left ( ex+d \right ) ^{4}}}+{\frac{Bd{a}^{3}}{4\,{e}^{2} \left ( ex+d \right ) ^{4}}}-{\frac{3\,B{d}^{2}{a}^{2}b}{4\,{e}^{3} \left ( ex+d \right ) ^{4}}}+{\frac{3\,B{d}^{3}a{b}^{2}}{4\,{e}^{4} \left ( ex+d \right ) ^{4}}}-{\frac{{b}^{3}B{d}^{4}}{4\,{e}^{5} \left ( ex+d \right ) ^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^3*(B*x+A)/(e*x+d)^5,x)

[Out]

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

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Maxima [B]  time = 1.22096, size = 408, normalized size = 3.16 \begin{align*} \frac{25 \, B b^{3} d^{4} - 3 \, A a^{3} e^{4} - 3 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e - 3 \,{\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} + 12 \,{\left (4 \, B b^{3} d e^{3} -{\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 18 \,{\left (6 \, 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} + 4 \,{\left (22 \, B b^{3} d^{3} e - 3 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} - 3 \,{\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}{12 \,{\left (e^{9} x^{4} + 4 \, d e^{8} x^{3} + 6 \, d^{2} e^{7} x^{2} + 4 \, d^{3} e^{6} x + d^{4} e^{5}\right )}} + \frac{B b^{3} \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)^5,x, algorithm="maxima")

[Out]

1/12*(25*B*b^3*d^4 - 3*A*a^3*e^4 - 3*(3*B*a*b^2 + A*b^3)*d^3*e - 3*(B*a^2*b + A*a*b^2)*d^2*e^2 - (B*a^3 + 3*A*
a^2*b)*d*e^3 + 12*(4*B*b^3*d*e^3 - (3*B*a*b^2 + A*b^3)*e^4)*x^3 + 18*(6*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 + 4*(22*B*b^3*d^3*e - 3*(3*B*a*b^2 + A*b^3)*d^2*e^2 - 3*(B*a^2*b + A*a*b^2)
*d*e^3 - (B*a^3 + 3*A*a^2*b)*e^4)*x)/(e^9*x^4 + 4*d*e^8*x^3 + 6*d^2*e^7*x^2 + 4*d^3*e^6*x + d^4*e^5) + B*b^3*l
og(e*x + d)/e^5

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Fricas [B]  time = 1.919, size = 726, normalized size = 5.63 \begin{align*} \frac{25 \, B b^{3} d^{4} - 3 \, A a^{3} e^{4} - 3 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e - 3 \,{\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} + 12 \,{\left (4 \, B b^{3} d e^{3} -{\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 18 \,{\left (6 \, 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} + 4 \,{\left (22 \, B b^{3} d^{3} e - 3 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} - 3 \,{\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 + 12 \,{\left (B b^{3} e^{4} x^{4} + 4 \, B b^{3} d e^{3} x^{3} + 6 \, B b^{3} d^{2} e^{2} x^{2} + 4 \, B b^{3} d^{3} e x + B b^{3} d^{4}\right )} \log \left (e x + d\right )}{12 \,{\left (e^{9} x^{4} + 4 \, d e^{8} x^{3} + 6 \, d^{2} e^{7} x^{2} + 4 \, d^{3} e^{6} x + d^{4} 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)^5,x, algorithm="fricas")

[Out]

1/12*(25*B*b^3*d^4 - 3*A*a^3*e^4 - 3*(3*B*a*b^2 + A*b^3)*d^3*e - 3*(B*a^2*b + A*a*b^2)*d^2*e^2 - (B*a^3 + 3*A*
a^2*b)*d*e^3 + 12*(4*B*b^3*d*e^3 - (3*B*a*b^2 + A*b^3)*e^4)*x^3 + 18*(6*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 + 4*(22*B*b^3*d^3*e - 3*(3*B*a*b^2 + A*b^3)*d^2*e^2 - 3*(B*a^2*b + A*a*b^2)
*d*e^3 - (B*a^3 + 3*A*a^2*b)*e^4)*x + 12*(B*b^3*e^4*x^4 + 4*B*b^3*d*e^3*x^3 + 6*B*b^3*d^2*e^2*x^2 + 4*B*b^3*d^
3*e*x + B*b^3*d^4)*log(e*x + d))/(e^9*x^4 + 4*d*e^8*x^3 + 6*d^2*e^7*x^2 + 4*d^3*e^6*x + d^4*e^5)

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Sympy [B]  time = 81.9577, size = 359, normalized size = 2.78 \begin{align*} \frac{B b^{3} \log{\left (d + e x \right )}}{e^{5}} - \frac{3 A a^{3} e^{4} + 3 A a^{2} b d e^{3} + 3 A a b^{2} d^{2} e^{2} + 3 A b^{3} d^{3} e + B a^{3} d e^{3} + 3 B a^{2} b d^{2} e^{2} + 9 B a b^{2} d^{3} e - 25 B b^{3} d^{4} + x^{3} \left (12 A b^{3} e^{4} + 36 B a b^{2} e^{4} - 48 B b^{3} d e^{3}\right ) + 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} - 108 B b^{3} d^{2} e^{2}\right ) + x \left (12 A a^{2} b e^{4} + 12 A a b^{2} d e^{3} + 12 A b^{3} d^{2} e^{2} + 4 B a^{3} e^{4} + 12 B a^{2} b d e^{3} + 36 B a b^{2} d^{2} e^{2} - 88 B b^{3} d^{3} e\right )}{12 d^{4} e^{5} + 48 d^{3} e^{6} x + 72 d^{2} e^{7} x^{2} + 48 d e^{8} x^{3} + 12 e^{9} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**3*(B*x+A)/(e*x+d)**5,x)

[Out]

B*b**3*log(d + e*x)/e**5 - (3*A*a**3*e**4 + 3*A*a**2*b*d*e**3 + 3*A*a*b**2*d**2*e**2 + 3*A*b**3*d**3*e + B*a**
3*d*e**3 + 3*B*a**2*b*d**2*e**2 + 9*B*a*b**2*d**3*e - 25*B*b**3*d**4 + x**3*(12*A*b**3*e**4 + 36*B*a*b**2*e**4
- 48*B*b**3*d*e**3) + 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 - 108
*B*b**3*d**2*e**2) + x*(12*A*a**2*b*e**4 + 12*A*a*b**2*d*e**3 + 12*A*b**3*d**2*e**2 + 4*B*a**3*e**4 + 12*B*a**
2*b*d*e**3 + 36*B*a*b**2*d**2*e**2 - 88*B*b**3*d**3*e))/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 4
8*d*e**8*x**3 + 12*e**9*x**4)

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Giac [B]  time = 1.23431, size = 603, normalized size = 4.67 \begin{align*} -B b^{3} e^{\left (-5\right )} \log \left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) + \frac{1}{12} \,{\left (\frac{48 \, B b^{3} d e^{15}}{x e + d} - \frac{36 \, B b^{3} d^{2} e^{15}}{{\left (x e + d\right )}^{2}} + \frac{16 \, B b^{3} d^{3} e^{15}}{{\left (x e + d\right )}^{3}} - \frac{3 \, B b^{3} d^{4} e^{15}}{{\left (x e + d\right )}^{4}} - \frac{36 \, B a b^{2} e^{16}}{x e + d} - \frac{12 \, A b^{3} e^{16}}{x e + d} + \frac{54 \, B a b^{2} d e^{16}}{{\left (x e + d\right )}^{2}} + \frac{18 \, A b^{3} d e^{16}}{{\left (x e + d\right )}^{2}} - \frac{36 \, B a b^{2} d^{2} e^{16}}{{\left (x e + d\right )}^{3}} - \frac{12 \, A b^{3} d^{2} e^{16}}{{\left (x e + d\right )}^{3}} + \frac{9 \, B a b^{2} d^{3} e^{16}}{{\left (x e + d\right )}^{4}} + \frac{3 \, A b^{3} d^{3} e^{16}}{{\left (x e + d\right )}^{4}} - \frac{18 \, B a^{2} b e^{17}}{{\left (x e + d\right )}^{2}} - \frac{18 \, A a b^{2} e^{17}}{{\left (x e + d\right )}^{2}} + \frac{24 \, B a^{2} b d e^{17}}{{\left (x e + d\right )}^{3}} + \frac{24 \, A a b^{2} d e^{17}}{{\left (x e + d\right )}^{3}} - \frac{9 \, B a^{2} b d^{2} e^{17}}{{\left (x e + d\right )}^{4}} - \frac{9 \, A a b^{2} d^{2} e^{17}}{{\left (x e + d\right )}^{4}} - \frac{4 \, B a^{3} e^{18}}{{\left (x e + d\right )}^{3}} - \frac{12 \, A a^{2} b e^{18}}{{\left (x e + d\right )}^{3}} + \frac{3 \, B a^{3} d e^{18}}{{\left (x e + d\right )}^{4}} + \frac{9 \, A a^{2} b d e^{18}}{{\left (x e + d\right )}^{4}} - \frac{3 \, A a^{3} e^{19}}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-20\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)^5,x, algorithm="giac")

[Out]

-B*b^3*e^(-5)*log(abs(x*e + d)*e^(-1)/(x*e + d)^2) + 1/12*(48*B*b^3*d*e^15/(x*e + d) - 36*B*b^3*d^2*e^15/(x*e
+ d)^2 + 16*B*b^3*d^3*e^15/(x*e + d)^3 - 3*B*b^3*d^4*e^15/(x*e + d)^4 - 36*B*a*b^2*e^16/(x*e + d) - 12*A*b^3*e
^16/(x*e + d) + 54*B*a*b^2*d*e^16/(x*e + d)^2 + 18*A*b^3*d*e^16/(x*e + d)^2 - 36*B*a*b^2*d^2*e^16/(x*e + d)^3
- 12*A*b^3*d^2*e^16/(x*e + d)^3 + 9*B*a*b^2*d^3*e^16/(x*e + d)^4 + 3*A*b^3*d^3*e^16/(x*e + d)^4 - 18*B*a^2*b*e
^17/(x*e + d)^2 - 18*A*a*b^2*e^17/(x*e + d)^2 + 24*B*a^2*b*d*e^17/(x*e + d)^3 + 24*A*a*b^2*d*e^17/(x*e + d)^3
- 9*B*a^2*b*d^2*e^17/(x*e + d)^4 - 9*A*a*b^2*d^2*e^17/(x*e + d)^4 - 4*B*a^3*e^18/(x*e + d)^3 - 12*A*a^2*b*e^18
/(x*e + d)^3 + 3*B*a^3*d*e^18/(x*e + d)^4 + 9*A*a^2*b*d*e^18/(x*e + d)^4 - 3*A*a^3*e^19/(x*e + d)^4)*e^(-20)