3.1155 \(\int \frac{(A+B x) (d+e x)^4}{(b x+c x^2)^3} \, dx\)

Optimal. Leaf size=235 \[ \frac{(c d-b e)^2 \log (b+c x) \left (-6 A c^3 d^2+2 b^2 B c d e+b^3 B e^2+3 b B c^2 d^2\right )}{b^5 c^3}+\frac{d^2 \log (x) \left (2 b^2 e (3 A e+2 B d)-3 b c d (4 A e+B d)+6 A c^2 d^2\right )}{b^5}-\frac{(c d-b e)^3 \left (-A b c e-3 A c^2 d+2 b^2 B e+2 b B c d\right )}{b^4 c^3 (b+c x)}-\frac{(b B-A c) (c d-b e)^4}{2 b^3 c^3 (b+c x)^2}-\frac{d^3 (4 A b e-3 A c d+b B d)}{b^4 x}-\frac{A d^4}{2 b^3 x^2} \]

[Out]

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

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Rubi [A]  time = 0.331914, antiderivative size = 235, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {771} \[ \frac{(c d-b e)^2 \log (b+c x) \left (-6 A c^3 d^2+2 b^2 B c d e+b^3 B e^2+3 b B c^2 d^2\right )}{b^5 c^3}+\frac{d^2 \log (x) \left (2 b^2 e (3 A e+2 B d)-3 b c d (4 A e+B d)+6 A c^2 d^2\right )}{b^5}-\frac{(c d-b e)^3 \left (-A b c e-3 A c^2 d+2 b^2 B e+2 b B c d\right )}{b^4 c^3 (b+c x)}-\frac{(b B-A c) (c d-b e)^4}{2 b^3 c^3 (b+c x)^2}-\frac{d^3 (4 A b e-3 A c d+b B d)}{b^4 x}-\frac{A d^4}{2 b^3 x^2} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(d + e*x)^4)/(b*x + c*x^2)^3,x]

[Out]

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

Rule 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

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

Mathematica [A]  time = 0.183113, size = 228, normalized size = 0.97 \[ -\frac{-\frac{2 (c d-b e)^2 \log (b+c x) \left (-6 A c^3 d^2+2 b^2 B c d e+b^3 B e^2+3 b B c^2 d^2\right )}{c^3}-2 d^2 \log (x) \left (2 b^2 e (3 A e+2 B d)-3 b c d (4 A e+B d)+6 A c^2 d^2\right )-\frac{2 b (b e-c d)^3 \left (b c (2 B d-A e)-3 A c^2 d+2 b^2 B e\right )}{c^3 (b+c x)}+\frac{b^2 (b B-A c) (c d-b e)^4}{c^3 (b+c x)^2}+\frac{A b^2 d^4}{x^2}+\frac{2 b d^3 (4 A b e-3 A c d+b B d)}{x}}{2 b^5} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(d + e*x)^4)/(b*x + c*x^2)^3,x]

[Out]

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

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Maple [B]  time = 0.016, size = 536, normalized size = 2.3 \begin{align*} -{\frac{A{d}^{4}}{2\,{b}^{3}{x}^{2}}}-12\,{\frac{{d}^{3}\ln \left ( x \right ) Ace}{{b}^{4}}}+2\,{\frac{Bbd{e}^{3}}{{c}^{2} \left ( cx+b \right ) ^{2}}}-8\,{\frac{A{d}^{3}ce}{{b}^{3} \left ( cx+b \right ) }}-2\,{\frac{A{d}^{3}ce}{{b}^{2} \left ( cx+b \right ) ^{2}}}+12\,{\frac{c\ln \left ( cx+b \right ) A{d}^{3}e}{{b}^{4}}}-{\frac{B{d}^{4}}{{b}^{3}x}}+{\frac{\ln \left ( cx+b \right ) B{e}^{4}}{{c}^{3}}}-{\frac{A{e}^{4}}{{c}^{2} \left ( cx+b \right ) }}+3\,{\frac{A{d}^{4}{c}^{2}}{{b}^{4} \left ( cx+b \right ) }}+2\,{\frac{B{e}^{4}b}{{c}^{3} \left ( cx+b \right ) }}-2\,{\frac{cB{d}^{4}}{{b}^{3} \left ( cx+b \right ) }}-6\,{\frac{\ln \left ( cx+b \right ) A{d}^{2}{e}^{2}}{{b}^{3}}}-6\,{\frac{{c}^{2}\ln \left ( cx+b \right ) A{d}^{4}}{{b}^{5}}}-4\,{\frac{\ln \left ( cx+b \right ) B{d}^{3}e}{{b}^{3}}}+3\,{\frac{c\ln \left ( cx+b \right ) B{d}^{4}}{{b}^{4}}}+{\frac{Ab{e}^{4}}{2\,{c}^{2} \left ( cx+b \right ) ^{2}}}+{\frac{A{d}^{4}{c}^{2}}{2\,{b}^{3} \left ( cx+b \right ) ^{2}}}-{\frac{{b}^{2}B{e}^{4}}{2\,{c}^{3} \left ( cx+b \right ) ^{2}}}-{\frac{cB{d}^{4}}{2\,{b}^{2} \left ( cx+b \right ) ^{2}}}+6\,{\frac{{d}^{2}\ln \left ( x \right ) A{e}^{2}}{{b}^{3}}}+6\,{\frac{{d}^{4}\ln \left ( x \right ) A{c}^{2}}{{b}^{5}}}+4\,{\frac{{d}^{3}\ln \left ( x \right ) Be}{{b}^{3}}}-3\,{\frac{{d}^{4}\ln \left ( x \right ) Bc}{{b}^{4}}}-4\,{\frac{A{d}^{3}e}{{b}^{3}x}}+3\,{\frac{A{d}^{4}c}{{b}^{4}x}}-2\,{\frac{Ad{e}^{3}}{c \left ( cx+b \right ) ^{2}}}+3\,{\frac{A{d}^{2}{e}^{2}}{b \left ( cx+b \right ) ^{2}}}+6\,{\frac{A{d}^{2}{e}^{2}}{{b}^{2} \left ( cx+b \right ) }}-4\,{\frac{Bd{e}^{3}}{{c}^{2} \left ( cx+b \right ) }}+4\,{\frac{B{d}^{3}e}{{b}^{2} \left ( cx+b \right ) }}-3\,{\frac{B{d}^{2}{e}^{2}}{c \left ( cx+b \right ) ^{2}}}+2\,{\frac{B{d}^{3}e}{b \left ( cx+b \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^4/(c*x^2+b*x)^3,x)

[Out]

-1/2*A*d^4/b^3/x^2-12*d^3/b^4*ln(x)*A*c*e+2/c^2*b/(c*x+b)^2*B*d*e^3-8*c/b^3/(c*x+b)*A*d^3*e-2*c/b^2/(c*x+b)^2*
A*d^3*e+12/b^4*c*ln(c*x+b)*A*d^3*e-d^4/b^3/x*B+1/c^3*ln(c*x+b)*B*e^4-1/c^2/(c*x+b)*A*e^4+3*c^2/b^4/(c*x+b)*A*d
^4+2/c^3*b/(c*x+b)*B*e^4-2*c/b^3/(c*x+b)*B*d^4-6/b^3*ln(c*x+b)*A*d^2*e^2-6/b^5*c^2*ln(c*x+b)*A*d^4-4/b^3*ln(c*
x+b)*B*d^3*e+3/b^4*c*ln(c*x+b)*B*d^4+1/2/c^2*b/(c*x+b)^2*A*e^4+1/2*c^2/b^3/(c*x+b)^2*A*d^4-1/2/c^3*b^2/(c*x+b)
^2*B*e^4-1/2*c/b^2/(c*x+b)^2*B*d^4+6*d^2/b^3*ln(x)*A*e^2+6*d^4/b^5*ln(x)*A*c^2+4*d^3/b^3*ln(x)*B*e-3*d^4/b^4*l
n(x)*B*c-4*d^3/b^3/x*A*e+3*d^4/b^4/x*A*c-2/c/(c*x+b)^2*A*d*e^3+3/b/(c*x+b)^2*A*d^2*e^2+6/b^2/(c*x+b)*A*d^2*e^2
-4/c^2/(c*x+b)*B*d*e^3+4/b^2/(c*x+b)*B*d^3*e-3/c/(c*x+b)^2*B*d^2*e^2+2/b/(c*x+b)^2*B*d^3*e

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Maxima [A]  time = 1.16044, size = 581, normalized size = 2.47 \begin{align*} -\frac{A b^{3} c^{3} d^{4} - 2 \,{\left (6 \, A b^{2} c^{4} d^{2} e^{2} - 4 \, B b^{4} c^{2} d e^{3} - 3 \,{\left (B b c^{5} - 2 \, A c^{6}\right )} d^{4} + 4 \,{\left (B b^{2} c^{4} - 3 \, A b c^{5}\right )} d^{3} e +{\left (2 \, B b^{5} c - A b^{4} c^{2}\right )} e^{4}\right )} x^{3} +{\left (9 \,{\left (B b^{2} c^{4} - 2 \, A b c^{5}\right )} d^{4} - 12 \,{\left (B b^{3} c^{3} - 3 \, A b^{2} c^{4}\right )} d^{3} e + 6 \,{\left (B b^{4} c^{2} - 3 \, A b^{3} c^{3}\right )} d^{2} e^{2} + 4 \,{\left (B b^{5} c + A b^{4} c^{2}\right )} d e^{3} -{\left (3 \, B b^{6} - A b^{5} c\right )} e^{4}\right )} x^{2} + 2 \,{\left (4 \, A b^{3} c^{3} d^{3} e +{\left (B b^{3} c^{3} - 2 \, A b^{2} c^{4}\right )} d^{4}\right )} x}{2 \,{\left (b^{4} c^{5} x^{4} + 2 \, b^{5} c^{4} x^{3} + b^{6} c^{3} x^{2}\right )}} + \frac{{\left (6 \, A b^{2} d^{2} e^{2} - 3 \,{\left (B b c - 2 \, A c^{2}\right )} d^{4} + 4 \,{\left (B b^{2} - 3 \, A b c\right )} d^{3} e\right )} \log \left (x\right )}{b^{5}} - \frac{{\left (6 \, A b^{2} c^{3} d^{2} e^{2} - B b^{5} e^{4} - 3 \,{\left (B b c^{4} - 2 \, A c^{5}\right )} d^{4} + 4 \,{\left (B b^{2} c^{3} - 3 \, A b c^{4}\right )} d^{3} e\right )} \log \left (c x + b\right )}{b^{5} c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^4/(c*x^2+b*x)^3,x, algorithm="maxima")

[Out]

-1/2*(A*b^3*c^3*d^4 - 2*(6*A*b^2*c^4*d^2*e^2 - 4*B*b^4*c^2*d*e^3 - 3*(B*b*c^5 - 2*A*c^6)*d^4 + 4*(B*b^2*c^4 -
3*A*b*c^5)*d^3*e + (2*B*b^5*c - A*b^4*c^2)*e^4)*x^3 + (9*(B*b^2*c^4 - 2*A*b*c^5)*d^4 - 12*(B*b^3*c^3 - 3*A*b^2
*c^4)*d^3*e + 6*(B*b^4*c^2 - 3*A*b^3*c^3)*d^2*e^2 + 4*(B*b^5*c + A*b^4*c^2)*d*e^3 - (3*B*b^6 - A*b^5*c)*e^4)*x
^2 + 2*(4*A*b^3*c^3*d^3*e + (B*b^3*c^3 - 2*A*b^2*c^4)*d^4)*x)/(b^4*c^5*x^4 + 2*b^5*c^4*x^3 + b^6*c^3*x^2) + (6
*A*b^2*d^2*e^2 - 3*(B*b*c - 2*A*c^2)*d^4 + 4*(B*b^2 - 3*A*b*c)*d^3*e)*log(x)/b^5 - (6*A*b^2*c^3*d^2*e^2 - B*b^
5*e^4 - 3*(B*b*c^4 - 2*A*c^5)*d^4 + 4*(B*b^2*c^3 - 3*A*b*c^4)*d^3*e)*log(c*x + b)/(b^5*c^3)

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Fricas [B]  time = 2.29296, size = 1467, normalized size = 6.24 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^4/(c*x^2+b*x)^3,x, algorithm="fricas")

[Out]

-1/2*(A*b^4*c^3*d^4 - 2*(6*A*b^3*c^4*d^2*e^2 - 4*B*b^5*c^2*d*e^3 - 3*(B*b^2*c^5 - 2*A*b*c^6)*d^4 + 4*(B*b^3*c^
4 - 3*A*b^2*c^5)*d^3*e + (2*B*b^6*c - A*b^5*c^2)*e^4)*x^3 + (9*(B*b^3*c^4 - 2*A*b^2*c^5)*d^4 - 12*(B*b^4*c^3 -
3*A*b^3*c^4)*d^3*e + 6*(B*b^5*c^2 - 3*A*b^4*c^3)*d^2*e^2 + 4*(B*b^6*c + A*b^5*c^2)*d*e^3 - (3*B*b^7 - A*b^6*c
)*e^4)*x^2 + 2*(4*A*b^4*c^3*d^3*e + (B*b^4*c^3 - 2*A*b^3*c^4)*d^4)*x + 2*((6*A*b^2*c^5*d^2*e^2 - B*b^5*c^2*e^4
- 3*(B*b*c^6 - 2*A*c^7)*d^4 + 4*(B*b^2*c^5 - 3*A*b*c^6)*d^3*e)*x^4 + 2*(6*A*b^3*c^4*d^2*e^2 - B*b^6*c*e^4 - 3
*(B*b^2*c^5 - 2*A*b*c^6)*d^4 + 4*(B*b^3*c^4 - 3*A*b^2*c^5)*d^3*e)*x^3 + (6*A*b^4*c^3*d^2*e^2 - B*b^7*e^4 - 3*(
B*b^3*c^4 - 2*A*b^2*c^5)*d^4 + 4*(B*b^4*c^3 - 3*A*b^3*c^4)*d^3*e)*x^2)*log(c*x + b) - 2*((6*A*b^2*c^5*d^2*e^2
- 3*(B*b*c^6 - 2*A*c^7)*d^4 + 4*(B*b^2*c^5 - 3*A*b*c^6)*d^3*e)*x^4 + 2*(6*A*b^3*c^4*d^2*e^2 - 3*(B*b^2*c^5 - 2
*A*b*c^6)*d^4 + 4*(B*b^3*c^4 - 3*A*b^2*c^5)*d^3*e)*x^3 + (6*A*b^4*c^3*d^2*e^2 - 3*(B*b^3*c^4 - 2*A*b^2*c^5)*d^
4 + 4*(B*b^4*c^3 - 3*A*b^3*c^4)*d^3*e)*x^2)*log(x))/(b^5*c^5*x^4 + 2*b^6*c^4*x^3 + b^7*c^3*x^2)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**4/(c*x**2+b*x)**3,x)

[Out]

Timed out

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Giac [A]  time = 1.31934, size = 578, normalized size = 2.46 \begin{align*} -\frac{{\left (3 \, B b c d^{4} - 6 \, A c^{2} d^{4} - 4 \, B b^{2} d^{3} e + 12 \, A b c d^{3} e - 6 \, A b^{2} d^{2} e^{2}\right )} \log \left ({\left | x \right |}\right )}{b^{5}} + \frac{{\left (3 \, B b c^{4} d^{4} - 6 \, A c^{5} d^{4} - 4 \, B b^{2} c^{3} d^{3} e + 12 \, A b c^{4} d^{3} e - 6 \, A b^{2} c^{3} d^{2} e^{2} + B b^{5} e^{4}\right )} \log \left ({\left | c x + b \right |}\right )}{b^{5} c^{3}} - \frac{A b^{3} c^{3} d^{4} + 2 \,{\left (3 \, B b c^{5} d^{4} - 6 \, A c^{6} d^{4} - 4 \, B b^{2} c^{4} d^{3} e + 12 \, A b c^{5} d^{3} e - 6 \, A b^{2} c^{4} d^{2} e^{2} + 4 \, B b^{4} c^{2} d e^{3} - 2 \, B b^{5} c e^{4} + A b^{4} c^{2} e^{4}\right )} x^{3} +{\left (9 \, B b^{2} c^{4} d^{4} - 18 \, A b c^{5} d^{4} - 12 \, B b^{3} c^{3} d^{3} e + 36 \, A b^{2} c^{4} d^{3} e + 6 \, B b^{4} c^{2} d^{2} e^{2} - 18 \, A b^{3} c^{3} d^{2} e^{2} + 4 \, B b^{5} c d e^{3} + 4 \, A b^{4} c^{2} d e^{3} - 3 \, B b^{6} e^{4} + A b^{5} c e^{4}\right )} x^{2} + 2 \,{\left (B b^{3} c^{3} d^{4} - 2 \, A b^{2} c^{4} d^{4} + 4 \, A b^{3} c^{3} d^{3} e\right )} x}{2 \,{\left (c x + b\right )}^{2} b^{4} c^{3} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^4/(c*x^2+b*x)^3,x, algorithm="giac")

[Out]

-(3*B*b*c*d^4 - 6*A*c^2*d^4 - 4*B*b^2*d^3*e + 12*A*b*c*d^3*e - 6*A*b^2*d^2*e^2)*log(abs(x))/b^5 + (3*B*b*c^4*d
^4 - 6*A*c^5*d^4 - 4*B*b^2*c^3*d^3*e + 12*A*b*c^4*d^3*e - 6*A*b^2*c^3*d^2*e^2 + B*b^5*e^4)*log(abs(c*x + b))/(
b^5*c^3) - 1/2*(A*b^3*c^3*d^4 + 2*(3*B*b*c^5*d^4 - 6*A*c^6*d^4 - 4*B*b^2*c^4*d^3*e + 12*A*b*c^5*d^3*e - 6*A*b^
2*c^4*d^2*e^2 + 4*B*b^4*c^2*d*e^3 - 2*B*b^5*c*e^4 + A*b^4*c^2*e^4)*x^3 + (9*B*b^2*c^4*d^4 - 18*A*b*c^5*d^4 - 1
2*B*b^3*c^3*d^3*e + 36*A*b^2*c^4*d^3*e + 6*B*b^4*c^2*d^2*e^2 - 18*A*b^3*c^3*d^2*e^2 + 4*B*b^5*c*d*e^3 + 4*A*b^
4*c^2*d*e^3 - 3*B*b^6*e^4 + A*b^5*c*e^4)*x^2 + 2*(B*b^3*c^3*d^4 - 2*A*b^2*c^4*d^4 + 4*A*b^3*c^3*d^3*e)*x)/((c*
x + b)^2*b^4*c^3*x^2)