3.1134 \(\int \frac{(A+B x) (d+e x)^4}{(a+b x)^3} \, dx\)

Optimal. Leaf size=191 \[ \frac{e^3 (a+b x)^2 (-5 a B e+A b e+4 b B d)}{2 b^6}+\frac{2 e^2 x (b d-a e) (-5 a B e+2 A b e+3 b B d)}{b^5}-\frac{(b d-a e)^3 (-5 a B e+4 A b e+b B d)}{b^6 (a+b x)}-\frac{(A b-a B) (b d-a e)^4}{2 b^6 (a+b x)^2}+\frac{2 e (b d-a e)^2 \log (a+b x) (-5 a B e+3 A b e+2 b B d)}{b^6}+\frac{B e^4 (a+b x)^3}{3 b^6} \]

[Out]

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

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Rubi [A]  time = 0.253351, antiderivative size = 191, 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} \[ \frac{e^3 (a+b x)^2 (-5 a B e+A b e+4 b B d)}{2 b^6}+\frac{2 e^2 x (b d-a e) (-5 a B e+2 A b e+3 b B d)}{b^5}-\frac{(b d-a e)^3 (-5 a B e+4 A b e+b B d)}{b^6 (a+b x)}-\frac{(A b-a B) (b d-a e)^4}{2 b^6 (a+b x)^2}+\frac{2 e (b d-a e)^2 \log (a+b x) (-5 a B e+3 A b e+2 b B d)}{b^6}+\frac{B e^4 (a+b x)^3}{3 b^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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)^4}{(a+b x)^3} \, dx &=\int \left (\frac{2 e^2 (b d-a e) (3 b B d+2 A b e-5 a B e)}{b^5}+\frac{(A b-a B) (b d-a e)^4}{b^5 (a+b x)^3}+\frac{(b d-a e)^3 (b B d+4 A b e-5 a B e)}{b^5 (a+b x)^2}+\frac{2 e (b d-a e)^2 (2 b B d+3 A b e-5 a B e)}{b^5 (a+b x)}+\frac{e^3 (4 b B d+A b e-5 a B e) (a+b x)}{b^5}+\frac{B e^4 (a+b x)^2}{b^5}\right ) \, dx\\ &=\frac{2 e^2 (b d-a e) (3 b B d+2 A b e-5 a B e) x}{b^5}-\frac{(A b-a B) (b d-a e)^4}{2 b^6 (a+b x)^2}-\frac{(b d-a e)^3 (b B d+4 A b e-5 a B e)}{b^6 (a+b x)}+\frac{e^3 (4 b B d+A b e-5 a B e) (a+b x)^2}{2 b^6}+\frac{B e^4 (a+b x)^3}{3 b^6}+\frac{2 e (b d-a e)^2 (2 b B d+3 A b e-5 a B e) \log (a+b x)}{b^6}\\ \end{align*}

Mathematica [A]  time = 0.111683, size = 187, normalized size = 0.98 \[ \frac{6 b e^2 x \left (6 a^2 B e^2-3 a b e (A e+4 B d)+2 b^2 d (2 A e+3 B d)\right )+3 b^2 e^3 x^2 (-3 a B e+A b e+4 b B d)-\frac{6 (b d-a e)^3 (-5 a B e+4 A b e+b B d)}{a+b x}-\frac{3 (A b-a B) (b d-a e)^4}{(a+b x)^2}+12 e (b d-a e)^2 \log (a+b x) (-5 a B e+3 A b e+2 b B d)+2 b^3 B e^4 x^3}{6 b^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.011, size = 601, normalized size = 3.2 \begin{align*} 2\,{\frac{aA{d}^{3}e}{{b}^{2} \left ( bx+a \right ) ^{2}}}-2\,{\frac{B{a}^{4}d{e}^{3}}{{b}^{5} \left ( bx+a \right ) ^{2}}}+12\,{\frac{aA{d}^{2}{e}^{2}}{{b}^{3} \left ( bx+a \right ) }}-12\,{\frac{{e}^{3}\ln \left ( bx+a \right ) Aad}{{b}^{4}}}+2\,{\frac{{e}^{3}B{x}^{2}d}{{b}^{3}}}-3\,{\frac{aA{e}^{4}x}{{b}^{4}}}+4\,{\frac{{e}^{3}Adx}{{b}^{3}}}+6\,{\frac{B{a}^{2}{e}^{4}x}{{b}^{5}}}+6\,{\frac{{e}^{2}B{d}^{2}x}{{b}^{3}}}+6\,{\frac{{e}^{4}\ln \left ( bx+a \right ) A{a}^{2}}{{b}^{5}}}+6\,{\frac{{e}^{2}\ln \left ( bx+a \right ) A{d}^{2}}{{b}^{3}}}-10\,{\frac{{e}^{4}\ln \left ( bx+a \right ) B{a}^{3}}{{b}^{6}}}+4\,{\frac{e\ln \left ( bx+a \right ) B{d}^{3}}{{b}^{3}}}+4\,{\frac{{a}^{3}A{e}^{4}}{{b}^{5} \left ( bx+a \right ) }}-4\,{\frac{A{d}^{3}e}{{b}^{2} \left ( bx+a \right ) }}-5\,{\frac{B{a}^{4}{e}^{4}}{{b}^{6} \left ( bx+a \right ) }}-{\frac{A{a}^{4}{e}^{4}}{2\,{b}^{5} \left ( bx+a \right ) ^{2}}}-18\,{\frac{B{a}^{2}{d}^{2}{e}^{2}}{{b}^{4} \left ( bx+a \right ) }}+8\,{\frac{Ba{d}^{3}e}{{b}^{3} \left ( bx+a \right ) }}+24\,{\frac{{e}^{3}\ln \left ( bx+a \right ) B{a}^{2}d}{{b}^{5}}}-18\,{\frac{{e}^{2}\ln \left ( bx+a \right ) Ba{d}^{2}}{{b}^{4}}}-12\,{\frac{{a}^{2}Ad{e}^{3}}{{b}^{4} \left ( bx+a \right ) }}+{\frac{B{a}^{5}{e}^{4}}{2\,{b}^{6} \left ( bx+a \right ) ^{2}}}+{\frac{Ba{d}^{4}}{2\,{b}^{2} \left ( bx+a \right ) ^{2}}}+{\frac{B{e}^{4}{x}^{3}}{3\,{b}^{3}}}+{\frac{{e}^{4}A{x}^{2}}{2\,{b}^{3}}}-{\frac{B{d}^{4}}{{b}^{2} \left ( bx+a \right ) }}-{\frac{A{d}^{4}}{2\,b \left ( bx+a \right ) ^{2}}}+3\,{\frac{B{a}^{3}{d}^{2}{e}^{2}}{{b}^{4} \left ( bx+a \right ) ^{2}}}-2\,{\frac{B{a}^{2}{d}^{3}e}{{b}^{3} \left ( bx+a \right ) ^{2}}}+16\,{\frac{B{a}^{3}d{e}^{3}}{{b}^{5} \left ( bx+a \right ) }}-12\,{\frac{{e}^{3}Badx}{{b}^{4}}}+2\,{\frac{{a}^{3}Ad{e}^{3}}{{b}^{4} \left ( bx+a \right ) ^{2}}}-3\,{\frac{{a}^{2}A{d}^{2}{e}^{2}}{{b}^{3} \left ( bx+a \right ) ^{2}}}-{\frac{3\,B{e}^{4}{x}^{2}a}{2\,{b}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/b^2/(b*x+a)^2*A*a*d^3*e-2/b^5/(b*x+a)^2*B*a^4*d*e^3+12/b^3/(b*x+a)*A*a*d^2*e^2-12/b^4*e^3*ln(b*x+a)*A*a*d+2*
e^3/b^3*B*x^2*d-3*e^4/b^4*A*a*x+4*e^3/b^3*A*d*x+6*e^4/b^5*a^2*B*x+6*e^2/b^3*B*d^2*x+6/b^5*e^4*ln(b*x+a)*A*a^2+
6/b^3*e^2*ln(b*x+a)*A*d^2-10/b^6*e^4*ln(b*x+a)*B*a^3+4/b^3*e*ln(b*x+a)*B*d^3+4/b^5/(b*x+a)*A*a^3*e^4-4/b^2/(b*
x+a)*A*d^3*e-5/b^6/(b*x+a)*B*a^4*e^4-1/2/b^5/(b*x+a)^2*A*a^4*e^4-18/b^4/(b*x+a)*B*a^2*d^2*e^2+8/b^3/(b*x+a)*B*
a*d^3*e+24/b^5*e^3*ln(b*x+a)*B*a^2*d-18/b^4*e^2*ln(b*x+a)*B*a*d^2-12/b^4/(b*x+a)*A*a^2*d*e^3+1/2/b^6/(b*x+a)^2
*B*a^5*e^4+1/2/b^2/(b*x+a)^2*B*a*d^4+1/3*e^4/b^3*B*x^3+1/2*e^4/b^3*A*x^2-1/b^2/(b*x+a)*B*d^4-1/2/b/(b*x+a)^2*A
*d^4+3/b^4/(b*x+a)^2*B*a^3*d^2*e^2-2/b^3/(b*x+a)^2*B*a^2*d^3*e+16/b^5/(b*x+a)*B*a^3*d*e^3-12*e^3/b^4*B*a*d*x+2
/b^4/(b*x+a)^2*A*a^3*d*e^3-3/b^3/(b*x+a)^2*A*a^2*d^2*e^2-3/2*e^4/b^4*B*x^2*a

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Maxima [B]  time = 1.29102, size = 572, normalized size = 2.99 \begin{align*} -\frac{{\left (B a b^{4} + A b^{5}\right )} d^{4} - 4 \,{\left (3 \, B a^{2} b^{3} - A a b^{4}\right )} d^{3} e + 6 \,{\left (5 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} d^{2} e^{2} - 4 \,{\left (7 \, B a^{4} b - 5 \, A a^{3} b^{2}\right )} d e^{3} +{\left (9 \, B a^{5} - 7 \, A a^{4} b\right )} e^{4} + 2 \,{\left (B b^{5} d^{4} - 4 \,{\left (2 \, B a b^{4} - A b^{5}\right )} d^{3} e + 6 \,{\left (3 \, B a^{2} b^{3} - 2 \, A a b^{4}\right )} d^{2} e^{2} - 4 \,{\left (4 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} d e^{3} +{\left (5 \, B a^{4} b - 4 \, A a^{3} b^{2}\right )} e^{4}\right )} x}{2 \,{\left (b^{8} x^{2} + 2 \, a b^{7} x + a^{2} b^{6}\right )}} + \frac{2 \, B b^{2} e^{4} x^{3} + 3 \,{\left (4 \, B b^{2} d e^{3} -{\left (3 \, B a b - A b^{2}\right )} e^{4}\right )} x^{2} + 6 \,{\left (6 \, B b^{2} d^{2} e^{2} - 4 \,{\left (3 \, B a b - A b^{2}\right )} d e^{3} + 3 \,{\left (2 \, B a^{2} - A a b\right )} e^{4}\right )} x}{6 \, b^{5}} + \frac{2 \,{\left (2 \, B b^{3} d^{3} e - 3 \,{\left (3 \, B a b^{2} - A b^{3}\right )} d^{2} e^{2} + 6 \,{\left (2 \, B a^{2} b - A a b^{2}\right )} d e^{3} -{\left (5 \, B a^{3} - 3 \, A a^{2} b\right )} e^{4}\right )} \log \left (b x + a\right )}{b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.64519, size = 1350, normalized size = 7.07 \begin{align*} \frac{2 \, B b^{5} e^{4} x^{5} - 3 \,{\left (B a b^{4} + A b^{5}\right )} d^{4} + 12 \,{\left (3 \, B a^{2} b^{3} - A a b^{4}\right )} d^{3} e - 18 \,{\left (5 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} d^{2} e^{2} + 12 \,{\left (7 \, B a^{4} b - 5 \, A a^{3} b^{2}\right )} d e^{3} - 3 \,{\left (9 \, B a^{5} - 7 \, A a^{4} b\right )} e^{4} +{\left (12 \, B b^{5} d e^{3} -{\left (5 \, B a b^{4} - 3 \, A b^{5}\right )} e^{4}\right )} x^{4} + 4 \,{\left (9 \, B b^{5} d^{2} e^{2} - 6 \,{\left (2 \, B a b^{4} - A b^{5}\right )} d e^{3} +{\left (5 \, B a^{2} b^{3} - 3 \, A a b^{4}\right )} e^{4}\right )} x^{3} + 3 \,{\left (24 \, B a b^{4} d^{2} e^{2} - 4 \,{\left (11 \, B a^{2} b^{3} - 4 \, A a b^{4}\right )} d e^{3} +{\left (21 \, B a^{3} b^{2} - 11 \, A a^{2} b^{3}\right )} e^{4}\right )} x^{2} - 6 \,{\left (B b^{5} d^{4} - 4 \,{\left (2 \, B a b^{4} - A b^{5}\right )} d^{3} e + 12 \,{\left (B a^{2} b^{3} - A a b^{4}\right )} d^{2} e^{2} - 4 \,{\left (B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )} d e^{3} -{\left (B a^{4} b + A a^{3} b^{2}\right )} e^{4}\right )} x + 12 \,{\left (2 \, B a^{2} b^{3} d^{3} e - 3 \,{\left (3 \, B a^{3} b^{2} - A a^{2} b^{3}\right )} d^{2} e^{2} + 6 \,{\left (2 \, B a^{4} b - A a^{3} b^{2}\right )} d e^{3} -{\left (5 \, B a^{5} - 3 \, A a^{4} b\right )} e^{4} +{\left (2 \, B b^{5} d^{3} e - 3 \,{\left (3 \, B a b^{4} - A b^{5}\right )} d^{2} e^{2} + 6 \,{\left (2 \, B a^{2} b^{3} - A a b^{4}\right )} d e^{3} -{\left (5 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} e^{4}\right )} x^{2} + 2 \,{\left (2 \, B a b^{4} d^{3} e - 3 \,{\left (3 \, B a^{2} b^{3} - A a b^{4}\right )} d^{2} e^{2} + 6 \,{\left (2 \, B a^{3} b^{2} - A a^{2} b^{3}\right )} d e^{3} -{\left (5 \, B a^{4} b - 3 \, A a^{3} b^{2}\right )} e^{4}\right )} x\right )} \log \left (b x + a\right )}{6 \,{\left (b^{8} x^{2} + 2 \, a b^{7} x + a^{2} b^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(2*B*b^5*e^4*x^5 - 3*(B*a*b^4 + A*b^5)*d^4 + 12*(3*B*a^2*b^3 - A*a*b^4)*d^3*e - 18*(5*B*a^3*b^2 - 3*A*a^2*
b^3)*d^2*e^2 + 12*(7*B*a^4*b - 5*A*a^3*b^2)*d*e^3 - 3*(9*B*a^5 - 7*A*a^4*b)*e^4 + (12*B*b^5*d*e^3 - (5*B*a*b^4
- 3*A*b^5)*e^4)*x^4 + 4*(9*B*b^5*d^2*e^2 - 6*(2*B*a*b^4 - A*b^5)*d*e^3 + (5*B*a^2*b^3 - 3*A*a*b^4)*e^4)*x^3 +
3*(24*B*a*b^4*d^2*e^2 - 4*(11*B*a^2*b^3 - 4*A*a*b^4)*d*e^3 + (21*B*a^3*b^2 - 11*A*a^2*b^3)*e^4)*x^2 - 6*(B*b^
5*d^4 - 4*(2*B*a*b^4 - A*b^5)*d^3*e + 12*(B*a^2*b^3 - A*a*b^4)*d^2*e^2 - 4*(B*a^3*b^2 - 2*A*a^2*b^3)*d*e^3 - (
B*a^4*b + A*a^3*b^2)*e^4)*x + 12*(2*B*a^2*b^3*d^3*e - 3*(3*B*a^3*b^2 - A*a^2*b^3)*d^2*e^2 + 6*(2*B*a^4*b - A*a
^3*b^2)*d*e^3 - (5*B*a^5 - 3*A*a^4*b)*e^4 + (2*B*b^5*d^3*e - 3*(3*B*a*b^4 - A*b^5)*d^2*e^2 + 6*(2*B*a^2*b^3 -
A*a*b^4)*d*e^3 - (5*B*a^3*b^2 - 3*A*a^2*b^3)*e^4)*x^2 + 2*(2*B*a*b^4*d^3*e - 3*(3*B*a^2*b^3 - A*a*b^4)*d^2*e^2
+ 6*(2*B*a^3*b^2 - A*a^2*b^3)*d*e^3 - (5*B*a^4*b - 3*A*a^3*b^2)*e^4)*x)*log(b*x + a))/(b^8*x^2 + 2*a*b^7*x +
a^2*b^6)

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Sympy [B]  time = 9.1656, size = 435, normalized size = 2.28 \begin{align*} \frac{B e^{4} x^{3}}{3 b^{3}} - \frac{- 7 A a^{4} b e^{4} + 20 A a^{3} b^{2} d e^{3} - 18 A a^{2} b^{3} d^{2} e^{2} + 4 A a b^{4} d^{3} e + A b^{5} d^{4} + 9 B a^{5} e^{4} - 28 B a^{4} b d e^{3} + 30 B a^{3} b^{2} d^{2} e^{2} - 12 B a^{2} b^{3} d^{3} e + B a b^{4} d^{4} + x \left (- 8 A a^{3} b^{2} e^{4} + 24 A a^{2} b^{3} d e^{3} - 24 A a b^{4} d^{2} e^{2} + 8 A b^{5} d^{3} e + 10 B a^{4} b e^{4} - 32 B a^{3} b^{2} d e^{3} + 36 B a^{2} b^{3} d^{2} e^{2} - 16 B a b^{4} d^{3} e + 2 B b^{5} d^{4}\right )}{2 a^{2} b^{6} + 4 a b^{7} x + 2 b^{8} x^{2}} - \frac{x^{2} \left (- A b e^{4} + 3 B a e^{4} - 4 B b d e^{3}\right )}{2 b^{4}} + \frac{x \left (- 3 A a b e^{4} + 4 A b^{2} d e^{3} + 6 B a^{2} e^{4} - 12 B a b d e^{3} + 6 B b^{2} d^{2} e^{2}\right )}{b^{5}} - \frac{2 e \left (a e - b d\right )^{2} \left (- 3 A b e + 5 B a e - 2 B b d\right ) \log{\left (a + b x \right )}}{b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

B*e**4*x**3/(3*b**3) - (-7*A*a**4*b*e**4 + 20*A*a**3*b**2*d*e**3 - 18*A*a**2*b**3*d**2*e**2 + 4*A*a*b**4*d**3*
e + A*b**5*d**4 + 9*B*a**5*e**4 - 28*B*a**4*b*d*e**3 + 30*B*a**3*b**2*d**2*e**2 - 12*B*a**2*b**3*d**3*e + B*a*
b**4*d**4 + x*(-8*A*a**3*b**2*e**4 + 24*A*a**2*b**3*d*e**3 - 24*A*a*b**4*d**2*e**2 + 8*A*b**5*d**3*e + 10*B*a*
*4*b*e**4 - 32*B*a**3*b**2*d*e**3 + 36*B*a**2*b**3*d**2*e**2 - 16*B*a*b**4*d**3*e + 2*B*b**5*d**4))/(2*a**2*b*
*6 + 4*a*b**7*x + 2*b**8*x**2) - x**2*(-A*b*e**4 + 3*B*a*e**4 - 4*B*b*d*e**3)/(2*b**4) + x*(-3*A*a*b*e**4 + 4*
A*b**2*d*e**3 + 6*B*a**2*e**4 - 12*B*a*b*d*e**3 + 6*B*b**2*d**2*e**2)/b**5 - 2*e*(a*e - b*d)**2*(-3*A*b*e + 5*
B*a*e - 2*B*b*d)*log(a + b*x)/b**6

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Giac [B]  time = 1.85248, size = 567, normalized size = 2.97 \begin{align*} \frac{2 \,{\left (2 \, B b^{3} d^{3} e - 9 \, B a b^{2} d^{2} e^{2} + 3 \, A b^{3} d^{2} e^{2} + 12 \, B a^{2} b d e^{3} - 6 \, A a b^{2} d e^{3} - 5 \, B a^{3} e^{4} + 3 \, A a^{2} b e^{4}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{6}} - \frac{B a b^{4} d^{4} + A b^{5} d^{4} - 12 \, B a^{2} b^{3} d^{3} e + 4 \, A a b^{4} d^{3} e + 30 \, B a^{3} b^{2} d^{2} e^{2} - 18 \, A a^{2} b^{3} d^{2} e^{2} - 28 \, B a^{4} b d e^{3} + 20 \, A a^{3} b^{2} d e^{3} + 9 \, B a^{5} e^{4} - 7 \, A a^{4} b e^{4} + 2 \,{\left (B b^{5} d^{4} - 8 \, B a b^{4} d^{3} e + 4 \, A b^{5} d^{3} e + 18 \, B a^{2} b^{3} d^{2} e^{2} - 12 \, A a b^{4} d^{2} e^{2} - 16 \, B a^{3} b^{2} d e^{3} + 12 \, A a^{2} b^{3} d e^{3} + 5 \, B a^{4} b e^{4} - 4 \, A a^{3} b^{2} e^{4}\right )} x}{2 \,{\left (b x + a\right )}^{2} b^{6}} + \frac{2 \, B b^{6} x^{3} e^{4} + 12 \, B b^{6} d x^{2} e^{3} + 36 \, B b^{6} d^{2} x e^{2} - 9 \, B a b^{5} x^{2} e^{4} + 3 \, A b^{6} x^{2} e^{4} - 72 \, B a b^{5} d x e^{3} + 24 \, A b^{6} d x e^{3} + 36 \, B a^{2} b^{4} x e^{4} - 18 \, A a b^{5} x e^{4}}{6 \, b^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(2*B*b^3*d^3*e - 9*B*a*b^2*d^2*e^2 + 3*A*b^3*d^2*e^2 + 12*B*a^2*b*d*e^3 - 6*A*a*b^2*d*e^3 - 5*B*a^3*e^4 + 3*
A*a^2*b*e^4)*log(abs(b*x + a))/b^6 - 1/2*(B*a*b^4*d^4 + A*b^5*d^4 - 12*B*a^2*b^3*d^3*e + 4*A*a*b^4*d^3*e + 30*
B*a^3*b^2*d^2*e^2 - 18*A*a^2*b^3*d^2*e^2 - 28*B*a^4*b*d*e^3 + 20*A*a^3*b^2*d*e^3 + 9*B*a^5*e^4 - 7*A*a^4*b*e^4
+ 2*(B*b^5*d^4 - 8*B*a*b^4*d^3*e + 4*A*b^5*d^3*e + 18*B*a^2*b^3*d^2*e^2 - 12*A*a*b^4*d^2*e^2 - 16*B*a^3*b^2*d
*e^3 + 12*A*a^2*b^3*d*e^3 + 5*B*a^4*b*e^4 - 4*A*a^3*b^2*e^4)*x)/((b*x + a)^2*b^6) + 1/6*(2*B*b^6*x^3*e^4 + 12*
B*b^6*d*x^2*e^3 + 36*B*b^6*d^2*x*e^2 - 9*B*a*b^5*x^2*e^4 + 3*A*b^6*x^2*e^4 - 72*B*a*b^5*d*x*e^3 + 24*A*b^6*d*x
*e^3 + 36*B*a^2*b^4*x*e^4 - 18*A*a*b^5*x*e^4)/b^9