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

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

[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 = 0.689864, antiderivative size = 235, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{d^3 (4 A b e-3 A c d+b B d)}{b^4 x}-\frac{(b B-A c) (c d-b e)^4}{2 b^3 c^3 (b+c x)^2}-\frac{A d^4}{2 b^3 x^2}+\frac{d^2 \log (x) \left (2 b^2 e (3 A e+2 B d)-3 b c d (4 A e+B d)+6 A c^2 d^2\right )}{b^5}+\frac{(c d-b e)^3 \left (-b c (2 B d-A e)+3 A c^2 d-2 b^2 B e\right )}{b^4 c^3 (b+c x)}+\frac{(c d-b e)^2 \log (b+c x) \left (-6 A c^3 d^2+b^3 B e^2+2 b^2 B c d e+3 b B c^2 d^2\right )}{b^5 c^3} \]

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*(3*A*c^2*d - 2*b^2*B*e -
b*c*(2*B*d - A*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 in Sympy [A]  time = 107.96, size = 248, normalized size = 1.06 \[ - \frac{A d^{4}}{2 b^{3} x^{2}} + \frac{\left (A c - B b\right ) \left (b e - c d\right )^{4}}{2 b^{3} c^{3} \left (b + c x\right )^{2}} - \frac{d^{3} \left (4 A b e - 3 A c d + B b d\right )}{b^{4} x} - \frac{\left (b e - c d\right )^{3} \left (A b c e + 3 A c^{2} d - 2 B b^{2} e - 2 B b c d\right )}{b^{4} c^{3} \left (b + c x\right )} + \frac{d^{2} \left (6 A b^{2} e^{2} - 12 A b c d e + 6 A c^{2} d^{2} + 4 B b^{2} d e - 3 B b c d^{2}\right ) \log{\left (x \right )}}{b^{5}} - \frac{\left (b e - c d\right )^{2} \left (6 A c^{3} d^{2} - B b^{3} e^{2} - 2 B b^{2} c d e - 3 B b c^{2} d^{2}\right ) \log{\left (b + c x \right )}}{b^{5} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.296241, size = 228, normalized size = 0.97 \[ -\frac{\frac{b^2 (b B-A c) (c d-b e)^4}{c^3 (b+c x)^2}-2 d^2 \log (x) \left (2 b^2 e (3 A e+2 B d)-3 b c d (4 A e+B d)+6 A c^2 d^2\right )-\frac{2 b (b e-c d)^3 \left (b c (2 B d-A e)-3 A c^2 d+2 b^2 B e\right )}{c^3 (b+c x)}+\frac{A b^2 d^4}{x^2}-\frac{2 (c d-b e)^2 \log (b+c x) \left (-6 A c^3 d^2+b^3 B e^2+2 b^2 B c d e+3 b B c^2 d^2\right )}{c^3}+\frac{2 b d^3 (4 A b e-3 A c d+b B d)}{x}}{2 b^5} \]

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 = 0.024, size = 536, normalized size = 2.3 \[ -12\,{\frac{{d}^{3}\ln \left ( x \right ) Ace}{{b}^{4}}}-8\,{\frac{A{d}^{3}ce}{{b}^{3} \left ( cx+b \right ) }}+6\,{\frac{A{d}^{2}{e}^{2}}{{b}^{2} \left ( cx+b \right ) }}+3\,{\frac{A{d}^{4}{c}^{2}}{{b}^{4} \left ( cx+b \right ) }}+2\,{\frac{B{e}^{4}b}{{c}^{3} \left ( cx+b \right ) }}-2\,{\frac{cB{d}^{4}}{{b}^{3} \left ( cx+b \right ) }}-6\,{\frac{\ln \left ( cx+b \right ) A{d}^{2}{e}^{2}}{{b}^{3}}}-6\,{\frac{{c}^{2}\ln \left ( cx+b \right ) A{d}^{4}}{{b}^{5}}}-4\,{\frac{\ln \left ( cx+b \right ) B{d}^{3}e}{{b}^{3}}}+3\,{\frac{c\ln \left ( cx+b \right ) B{d}^{4}}{{b}^{4}}}+{\frac{Ab{e}^{4}}{2\,{c}^{2} \left ( cx+b \right ) ^{2}}}+{\frac{A{d}^{4}{c}^{2}}{2\,{b}^{3} \left ( cx+b \right ) ^{2}}}-{\frac{{b}^{2}B{e}^{4}}{2\,{c}^{3} \left ( cx+b \right ) ^{2}}}-{\frac{cB{d}^{4}}{2\,{b}^{2} \left ( cx+b \right ) ^{2}}}+6\,{\frac{{d}^{2}\ln \left ( x \right ) A{e}^{2}}{{b}^{3}}}+6\,{\frac{{d}^{4}\ln \left ( x \right ) A{c}^{2}}{{b}^{5}}}+4\,{\frac{{d}^{3}\ln \left ( x \right ) Be}{{b}^{3}}}-3\,{\frac{{d}^{4}\ln \left ( x \right ) Bc}{{b}^{4}}}-4\,{\frac{A{d}^{3}e}{{b}^{3}x}}+3\,{\frac{A{d}^{4}c}{{b}^{4}x}}-2\,{\frac{dA{e}^{3}}{c \left ( cx+b \right ) ^{2}}}+3\,{\frac{A{d}^{2}{e}^{2}}{b \left ( cx+b \right ) ^{2}}}-3\,{\frac{B{d}^{2}{e}^{2}}{c \left ( cx+b \right ) ^{2}}}+2\,{\frac{B{d}^{3}e}{b \left ( cx+b \right ) ^{2}}}-4\,{\frac{Bd{e}^{3}}{{c}^{2} \left ( cx+b \right ) }}+4\,{\frac{B{d}^{3}e}{{b}^{2} \left ( cx+b \right ) }}-{\frac{A{d}^{4}}{2\,{b}^{3}{x}^{2}}}+12\,{\frac{c\ln \left ( cx+b \right ) A{d}^{3}e}{{b}^{4}}}-2\,{\frac{A{d}^{3}ce}{{b}^{2} \left ( cx+b \right ) ^{2}}}+2\,{\frac{Bbd{e}^{3}}{{c}^{2} \left ( cx+b \right ) ^{2}}}-{\frac{B{d}^{4}}{{b}^{3}x}}-{\frac{A{e}^{4}}{{c}^{2} \left ( cx+b \right ) }}+{\frac{\ln \left ( cx+b \right ) B{e}^{4}}{{c}^{3}}} \]

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]

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

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Maxima [A]  time = 0.703563, size = 581, normalized size = 2.47 \[ -\frac{A b^{3} c^{3} d^{4} - 2 \,{\left (6 \, A b^{2} c^{4} d^{2} e^{2} - 4 \, B b^{4} c^{2} d e^{3} - 3 \,{\left (B b c^{5} - 2 \, A c^{6}\right )} d^{4} + 4 \,{\left (B b^{2} c^{4} - 3 \, A b c^{5}\right )} d^{3} e +{\left (2 \, B b^{5} c - A b^{4} c^{2}\right )} e^{4}\right )} x^{3} +{\left (9 \,{\left (B b^{2} c^{4} - 2 \, A b c^{5}\right )} d^{4} - 12 \,{\left (B b^{3} c^{3} - 3 \, A b^{2} c^{4}\right )} d^{3} e + 6 \,{\left (B b^{4} c^{2} - 3 \, A b^{3} c^{3}\right )} d^{2} e^{2} + 4 \,{\left (B b^{5} c + A b^{4} c^{2}\right )} d e^{3} -{\left (3 \, B b^{6} - A b^{5} c\right )} e^{4}\right )} x^{2} + 2 \,{\left (4 \, A b^{3} c^{3} d^{3} e +{\left (B b^{3} c^{3} - 2 \, A b^{2} c^{4}\right )} d^{4}\right )} x}{2 \,{\left (b^{4} c^{5} x^{4} + 2 \, b^{5} c^{4} x^{3} + b^{6} c^{3} x^{2}\right )}} + \frac{{\left (6 \, A b^{2} d^{2} e^{2} - 3 \,{\left (B b c - 2 \, A c^{2}\right )} d^{4} + 4 \,{\left (B b^{2} - 3 \, A b c\right )} d^{3} e\right )} \log \left (x\right )}{b^{5}} - \frac{{\left (6 \, A b^{2} c^{3} d^{2} e^{2} - B b^{5} e^{4} - 3 \,{\left (B b c^{4} - 2 \, A c^{5}\right )} d^{4} + 4 \,{\left (B b^{2} c^{3} - 3 \, A b c^{4}\right )} d^{3} e\right )} \log \left (c x + b\right )}{b^{5} c^{3}} \]

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 [A]  time = 0.321085, size = 994, normalized size = 4.23 \[ \text{result too large to display} \]

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. \[ \text{Timed out} \]

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/XCAS [A]  time = 0.279403, size = 578, normalized size = 2.46 \[ -\frac{{\left (3 \, B b c d^{4} - 6 \, A c^{2} d^{4} - 4 \, B b^{2} d^{3} e + 12 \, A b c d^{3} e - 6 \, A b^{2} d^{2} e^{2}\right )}{\rm ln}\left ({\left | x \right |}\right )}{b^{5}} + \frac{{\left (3 \, B b c^{4} d^{4} - 6 \, A c^{5} d^{4} - 4 \, B b^{2} c^{3} d^{3} e + 12 \, A b c^{4} d^{3} e - 6 \, A b^{2} c^{3} d^{2} e^{2} + B b^{5} e^{4}\right )}{\rm ln}\left ({\left | c x + b \right |}\right )}{b^{5} c^{3}} - \frac{A b^{3} c^{3} d^{4} + 2 \,{\left (3 \, B b c^{5} d^{4} - 6 \, A c^{6} d^{4} - 4 \, B b^{2} c^{4} d^{3} e + 12 \, A b c^{5} d^{3} e - 6 \, A b^{2} c^{4} d^{2} e^{2} + 4 \, B b^{4} c^{2} d e^{3} - 2 \, B b^{5} c e^{4} + A b^{4} c^{2} e^{4}\right )} x^{3} +{\left (9 \, B b^{2} c^{4} d^{4} - 18 \, A b c^{5} d^{4} - 12 \, B b^{3} c^{3} d^{3} e + 36 \, A b^{2} c^{4} d^{3} e + 6 \, B b^{4} c^{2} d^{2} e^{2} - 18 \, A b^{3} c^{3} d^{2} e^{2} + 4 \, B b^{5} c d e^{3} + 4 \, A b^{4} c^{2} d e^{3} - 3 \, B b^{6} e^{4} + A b^{5} c e^{4}\right )} x^{2} + 2 \,{\left (B b^{3} c^{3} d^{4} - 2 \, A b^{2} c^{4} d^{4} + 4 \, A b^{3} c^{3} d^{3} e\right )} x}{2 \,{\left (c x + b\right )}^{2} b^{4} c^{3} x^{2}} \]

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)*
ln(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)*ln(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 - 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)*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)