3.1137 \(\int \frac{(A+B x) (d+e x)^4}{b x+c x^2} \, dx\)

Optimal. Leaf size=207 \[ \frac{e^2 x^2 \left (A c e (4 c d-b e)+B \left (b^2 e^2-4 b c d e+6 c^2 d^2\right )\right )}{2 c^3}+\frac{e x \left (A c e \left (b^2 e^2-4 b c d e+6 c^2 d^2\right )+B \left (-b^3 e^3+4 b^2 c d e^2-6 b c^2 d^2 e+4 c^3 d^3\right )\right )}{c^4}+\frac{(b B-A c) (c d-b e)^4 \log (b+c x)}{b c^5}+\frac{e^3 x^3 (A c e-b B e+4 B c d)}{3 c^2}+\frac{A d^4 \log (x)}{b}+\frac{B e^4 x^4}{4 c} \]

[Out]

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

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Rubi [A]  time = 0.546526, antiderivative size = 207, 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 \[ \frac{e^2 x^2 \left (A c e (4 c d-b e)+B \left (b^2 e^2-4 b c d e+6 c^2 d^2\right )\right )}{2 c^3}+\frac{e x \left (A c e \left (b^2 e^2-4 b c d e+6 c^2 d^2\right )+B \left (-b^3 e^3+4 b^2 c d e^2-6 b c^2 d^2 e+4 c^3 d^3\right )\right )}{c^4}+\frac{(b B-A c) (c d-b e)^4 \log (b+c x)}{b c^5}+\frac{e^3 x^3 (A c e-b B e+4 B c d)}{3 c^2}+\frac{A d^4 \log (x)}{b}+\frac{B e^4 x^4}{4 c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{A d^{4} \log{\left (x \right )}}{b} + \frac{B e^{4} x^{4}}{4 c} + \frac{e^{3} x^{3} \left (A c e - B b e + 4 B c d\right )}{3 c^{2}} + \frac{e^{2} \left (- A b c e^{2} + 4 A c^{2} d e + B b^{2} e^{2} - 4 B b c d e + 6 B c^{2} d^{2}\right ) \int x\, dx}{c^{3}} - \frac{\left (- A b^{2} c e^{3} + 4 A b c^{2} d e^{2} - 6 A c^{3} d^{2} e + B b^{3} e^{3} - 4 B b^{2} c d e^{2} + 6 B b c^{2} d^{2} e - 4 B c^{3} d^{3}\right ) \int e\, dx}{c^{4}} - \frac{\left (A c - B b\right ) \left (b e - c d\right )^{4} \log{\left (b + c x \right )}}{b c^{5}} \]

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),x)

[Out]

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

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Mathematica [A]  time = 0.280792, size = 187, normalized size = 0.9 \[ \frac{e x \left (2 A c e \left (6 b^2 e^2-3 b c e (8 d+e x)+2 c^2 \left (18 d^2+6 d e x+e^2 x^2\right )\right )+B \left (-12 b^3 e^3+6 b^2 c e^2 (8 d+e x)-4 b c^2 e \left (18 d^2+6 d e x+e^2 x^2\right )+c^3 \left (48 d^3+36 d^2 e x+16 d e^2 x^2+3 e^3 x^3\right )\right )\right )}{12 c^4}+\frac{(b B-A c) (c d-b e)^4 \log (b+c x)}{b c^5}+\frac{A d^4 \log (x)}{b} \]

Antiderivative was successfully verified.

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

[Out]

(e*x*(2*A*c*e*(6*b^2*e^2 - 3*b*c*e*(8*d + e*x) + 2*c^2*(18*d^2 + 6*d*e*x + e^2*x
^2)) + B*(-12*b^3*e^3 + 6*b^2*c*e^2*(8*d + e*x) - 4*b*c^2*e*(18*d^2 + 6*d*e*x +
e^2*x^2) + c^3*(48*d^3 + 36*d^2*e*x + 16*d*e^2*x^2 + 3*e^3*x^3))))/(12*c^4) + (A
*d^4*Log[x])/b + ((b*B - A*c)*(c*d - b*e)^4*Log[b + c*x])/(b*c^5)

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Maple [A]  time = 0.014, size = 396, normalized size = 1.9 \[{\frac{A{d}^{4}\ln \left ( x \right ) }{b}}+{\frac{4\,{e}^{3}B{x}^{3}d}{3\,c}}-6\,{\frac{B{e}^{2}b{d}^{2}x}{{c}^{2}}}+4\,{\frac{{b}^{2}\ln \left ( cx+b \right ) Ad{e}^{3}}{{c}^{3}}}-6\,{\frac{b\ln \left ( cx+b \right ) A{d}^{2}{e}^{2}}{{c}^{2}}}-4\,{\frac{{b}^{3}\ln \left ( cx+b \right ) Bd{e}^{3}}{{c}^{4}}}+6\,{\frac{{b}^{2}\ln \left ( cx+b \right ) B{d}^{2}{e}^{2}}{{c}^{3}}}-4\,{\frac{b\ln \left ( cx+b \right ) B{d}^{3}e}{{c}^{2}}}-2\,{\frac{{e}^{3}B{x}^{2}bd}{{c}^{2}}}-4\,{\frac{A{e}^{3}bdx}{{c}^{2}}}+4\,{\frac{B{e}^{3}{b}^{2}dx}{{c}^{3}}}+{\frac{\ln \left ( cx+b \right ) B{d}^{4}}{c}}+{\frac{{e}^{4}A{b}^{2}x}{{c}^{3}}}+6\,{\frac{A{d}^{2}{e}^{2}x}{c}}-{\frac{B{e}^{4}{b}^{3}x}{{c}^{4}}}+4\,{\frac{B{d}^{3}ex}{c}}-{\frac{{e}^{4}A{x}^{2}b}{2\,{c}^{2}}}+2\,{\frac{A{e}^{3}{x}^{2}d}{c}}+{\frac{{e}^{4}A{x}^{3}}{3\,c}}-{\frac{B{e}^{4}{x}^{3}b}{3\,{c}^{2}}}+{\frac{B{e}^{4}{x}^{4}}{4\,c}}-{\frac{\ln \left ( cx+b \right ) A{d}^{4}}{b}}+{\frac{B{e}^{4}{x}^{2}{b}^{2}}{2\,{c}^{3}}}+3\,{\frac{{e}^{2}B{x}^{2}{d}^{2}}{c}}-{\frac{{b}^{3}\ln \left ( cx+b \right ) A{e}^{4}}{{c}^{4}}}+4\,{\frac{\ln \left ( cx+b \right ) A{d}^{3}e}{c}}+{\frac{{b}^{4}\ln \left ( cx+b \right ) B{e}^{4}}{{c}^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.723362, size = 417, normalized size = 2.01 \[ \frac{A d^{4} \log \left (x\right )}{b} + \frac{3 \, B c^{3} e^{4} x^{4} + 4 \,{\left (4 \, B c^{3} d e^{3} -{\left (B b c^{2} - A c^{3}\right )} e^{4}\right )} x^{3} + 6 \,{\left (6 \, B c^{3} d^{2} e^{2} - 4 \,{\left (B b c^{2} - A c^{3}\right )} d e^{3} +{\left (B b^{2} c - A b c^{2}\right )} e^{4}\right )} x^{2} + 12 \,{\left (4 \, B c^{3} d^{3} e - 6 \,{\left (B b c^{2} - A c^{3}\right )} d^{2} e^{2} + 4 \,{\left (B b^{2} c - A b c^{2}\right )} d e^{3} -{\left (B b^{3} - A b^{2} c\right )} e^{4}\right )} x}{12 \, c^{4}} + \frac{{\left ({\left (B b c^{4} - A c^{5}\right )} d^{4} - 4 \,{\left (B b^{2} c^{3} - A b c^{4}\right )} d^{3} e + 6 \,{\left (B b^{3} c^{2} - A b^{2} c^{3}\right )} d^{2} e^{2} - 4 \,{\left (B b^{4} c - A b^{3} c^{2}\right )} d e^{3} +{\left (B b^{5} - A b^{4} c\right )} e^{4}\right )} \log \left (c x + b\right )}{b c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.317118, size = 447, normalized size = 2.16 \[ \frac{3 \, B b c^{4} e^{4} x^{4} + 12 \, A c^{5} d^{4} \log \left (x\right ) + 4 \,{\left (4 \, B b c^{4} d e^{3} -{\left (B b^{2} c^{3} - A b c^{4}\right )} e^{4}\right )} x^{3} + 6 \,{\left (6 \, B b c^{4} d^{2} e^{2} - 4 \,{\left (B b^{2} c^{3} - A b c^{4}\right )} d e^{3} +{\left (B b^{3} c^{2} - A b^{2} c^{3}\right )} e^{4}\right )} x^{2} + 12 \,{\left (4 \, B b c^{4} d^{3} e - 6 \,{\left (B b^{2} c^{3} - A b c^{4}\right )} d^{2} e^{2} + 4 \,{\left (B b^{3} c^{2} - A b^{2} c^{3}\right )} d e^{3} -{\left (B b^{4} c - A b^{3} c^{2}\right )} e^{4}\right )} x + 12 \,{\left ({\left (B b c^{4} - A c^{5}\right )} d^{4} - 4 \,{\left (B b^{2} c^{3} - A b c^{4}\right )} d^{3} e + 6 \,{\left (B b^{3} c^{2} - A b^{2} c^{3}\right )} d^{2} e^{2} - 4 \,{\left (B b^{4} c - A b^{3} c^{2}\right )} d e^{3} +{\left (B b^{5} - A b^{4} c\right )} e^{4}\right )} \log \left (c x + b\right )}{12 \, b c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 25.8197, size = 391, normalized size = 1.89 \[ \frac{A d^{4} \log{\left (x \right )}}{b} + \frac{B e^{4} x^{4}}{4 c} - \frac{x^{3} \left (- A c e^{4} + B b e^{4} - 4 B c d e^{3}\right )}{3 c^{2}} + \frac{x^{2} \left (- A b c e^{4} + 4 A c^{2} d e^{3} + B b^{2} e^{4} - 4 B b c d e^{3} + 6 B c^{2} d^{2} e^{2}\right )}{2 c^{3}} - \frac{x \left (- A b^{2} c e^{4} + 4 A b c^{2} d e^{3} - 6 A c^{3} d^{2} e^{2} + B b^{3} e^{4} - 4 B b^{2} c d e^{3} + 6 B b c^{2} d^{2} e^{2} - 4 B c^{3} d^{3} e\right )}{c^{4}} + \frac{\left (- A c + B b\right ) \left (b e - c d\right )^{4} \log{\left (x + \frac{- A b c^{4} d^{4} + \frac{b \left (- A c + B b\right ) \left (b e - c d\right )^{4}}{c}}{- A b^{4} c e^{4} + 4 A b^{3} c^{2} d e^{3} - 6 A b^{2} c^{3} d^{2} e^{2} + 4 A b c^{4} d^{3} e - 2 A c^{5} d^{4} + B b^{5} e^{4} - 4 B b^{4} c d e^{3} + 6 B b^{3} c^{2} d^{2} e^{2} - 4 B b^{2} c^{3} d^{3} e + B b c^{4} d^{4}} \right )}}{b c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.277782, size = 446, normalized size = 2.15 \[ \frac{A d^{4}{\rm ln}\left ({\left | x \right |}\right )}{b} + \frac{3 \, B c^{3} x^{4} e^{4} + 16 \, B c^{3} d x^{3} e^{3} + 36 \, B c^{3} d^{2} x^{2} e^{2} + 48 \, B c^{3} d^{3} x e - 4 \, B b c^{2} x^{3} e^{4} + 4 \, A c^{3} x^{3} e^{4} - 24 \, B b c^{2} d x^{2} e^{3} + 24 \, A c^{3} d x^{2} e^{3} - 72 \, B b c^{2} d^{2} x e^{2} + 72 \, A c^{3} d^{2} x e^{2} + 6 \, B b^{2} c x^{2} e^{4} - 6 \, A b c^{2} x^{2} e^{4} + 48 \, B b^{2} c d x e^{3} - 48 \, A b c^{2} d x e^{3} - 12 \, B b^{3} x e^{4} + 12 \, A b^{2} c x e^{4}}{12 \, c^{4}} + \frac{{\left (B b c^{4} d^{4} - A c^{5} d^{4} - 4 \, B b^{2} c^{3} d^{3} e + 4 \, A b c^{4} d^{3} e + 6 \, B b^{3} c^{2} d^{2} e^{2} - 6 \, A b^{2} c^{3} d^{2} e^{2} - 4 \, B b^{4} c d e^{3} + 4 \, A b^{3} c^{2} d e^{3} + B b^{5} e^{4} - A b^{4} c e^{4}\right )}{\rm ln}\left ({\left | c x + b \right |}\right )}{b c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*d^4*ln(abs(x))/b + 1/12*(3*B*c^3*x^4*e^4 + 16*B*c^3*d*x^3*e^3 + 36*B*c^3*d^2*x
^2*e^2 + 48*B*c^3*d^3*x*e - 4*B*b*c^2*x^3*e^4 + 4*A*c^3*x^3*e^4 - 24*B*b*c^2*d*x
^2*e^3 + 24*A*c^3*d*x^2*e^3 - 72*B*b*c^2*d^2*x*e^2 + 72*A*c^3*d^2*x*e^2 + 6*B*b^
2*c*x^2*e^4 - 6*A*b*c^2*x^2*e^4 + 48*B*b^2*c*d*x*e^3 - 48*A*b*c^2*d*x*e^3 - 12*B
*b^3*x*e^4 + 12*A*b^2*c*x*e^4)/c^4 + (B*b*c^4*d^4 - A*c^5*d^4 - 4*B*b^2*c^3*d^3*
e + 4*A*b*c^4*d^3*e + 6*B*b^3*c^2*d^2*e^2 - 6*A*b^2*c^3*d^2*e^2 - 4*B*b^4*c*d*e^
3 + 4*A*b^3*c^2*d*e^3 + B*b^5*e^4 - A*b^4*c*e^4)*ln(abs(c*x + b))/(b*c^5)