3.1095 \(\int \frac{(A+B x) (d+e x)^4}{a+b x} \, dx\)

Optimal. Leaf size=155 \[ \frac{(A b-a B) (b d-a e)^4 \log (a+b x)}{b^6}+\frac{e x (A b-a B) (b d-a e)^3}{b^5}+\frac{(d+e x)^2 (A b-a B) (b d-a e)^2}{2 b^4}+\frac{(d+e x)^3 (A b-a B) (b d-a e)}{3 b^3}+\frac{(d+e x)^4 (A b-a B)}{4 b^2}+\frac{B (d+e x)^5}{5 b e} \]

[Out]

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

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Rubi [A]  time = 0.21558, antiderivative size = 155, 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 \[ \frac{(A b-a B) (b d-a e)^4 \log (a+b x)}{b^6}+\frac{e x (A b-a B) (b d-a e)^3}{b^5}+\frac{(d+e x)^2 (A b-a B) (b d-a e)^2}{2 b^4}+\frac{(d+e x)^3 (A b-a B) (b d-a e)}{3 b^3}+\frac{(d+e x)^4 (A b-a B)}{4 b^2}+\frac{B (d+e x)^5}{5 b e} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.246446, size = 257, normalized size = 1.66 \[ \frac{b x \left (60 a^4 B e^4-30 a^3 b e^3 (2 A e+8 B d+B e x)+10 a^2 b^2 e^2 \left (3 A e (8 d+e x)+2 B \left (18 d^2+6 d e x+e^2 x^2\right )\right )-5 a b^3 e \left (4 A e \left (18 d^2+6 d e x+e^2 x^2\right )+B \left (48 d^3+36 d^2 e x+16 d e^2 x^2+3 e^3 x^3\right )\right )+b^4 \left (5 A e \left (48 d^3+36 d^2 e x+16 d e^2 x^2+3 e^3 x^3\right )+12 B \left (5 d^4+10 d^3 e x+10 d^2 e^2 x^2+5 d e^3 x^3+e^4 x^4\right )\right )\right )+60 (A b-a B) (b d-a e)^4 \log (a+b x)}{60 b^6} \]

Antiderivative was successfully verified.

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

[Out]

(b*x*(60*a^4*B*e^4 - 30*a^3*b*e^3*(8*B*d + 2*A*e + B*e*x) + 10*a^2*b^2*e^2*(3*A*
e*(8*d + e*x) + 2*B*(18*d^2 + 6*d*e*x + e^2*x^2)) - 5*a*b^3*e*(4*A*e*(18*d^2 + 6
*d*e*x + e^2*x^2) + B*(48*d^3 + 36*d^2*e*x + 16*d*e^2*x^2 + 3*e^3*x^3)) + b^4*(5
*A*e*(48*d^3 + 36*d^2*e*x + 16*d*e^2*x^2 + 3*e^3*x^3) + 12*B*(5*d^4 + 10*d^3*e*x
+ 10*d^2*e^2*x^2 + 5*d*e^3*x^3 + e^4*x^4))) + 60*(A*b - a*B)*(b*d - a*e)^4*Log[
a + b*x])/(60*b^6)

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Maple [B]  time = 0.008, size = 521, normalized size = 3.4 \[ -4\,{\frac{B{a}^{3}d{e}^{3}x}{{b}^{4}}}+6\,{\frac{B{a}^{2}{d}^{2}{e}^{2}x}{{b}^{3}}}-4\,{\frac{\ln \left ( bx+a \right ) A{a}^{3}d{e}^{3}}{{b}^{4}}}+6\,{\frac{\ln \left ( bx+a \right ) A{a}^{2}{d}^{2}{e}^{2}}{{b}^{3}}}+{\frac{A{x}^{4}{e}^{4}}{4\,b}}+{\frac{\ln \left ( bx+a \right ) A{d}^{4}}{b}}+{\frac{B{d}^{4}x}{b}}+{\frac{B{x}^{5}{e}^{4}}{5\,b}}-4\,{\frac{Ba{d}^{3}ex}{{b}^{2}}}-{\frac{4\,B{x}^{3}ad{e}^{3}}{3\,{b}^{2}}}-3\,{\frac{B{x}^{2}a{d}^{2}{e}^{2}}{{b}^{2}}}+4\,{\frac{{a}^{2}Ad{e}^{3}x}{{b}^{3}}}-6\,{\frac{Aa{d}^{2}{e}^{2}x}{{b}^{2}}}-{\frac{aA{x}^{3}{e}^{4}}{3\,{b}^{2}}}-{\frac{{a}^{3}A{e}^{4}x}{{b}^{4}}}-{\frac{B{x}^{2}{a}^{3}{e}^{4}}{2\,{b}^{4}}}+2\,{\frac{B{x}^{2}{d}^{3}e}{b}}+{\frac{B{x}^{3}{a}^{2}{e}^{4}}{3\,{b}^{3}}}+2\,{\frac{B{x}^{3}{d}^{2}{e}^{2}}{b}}+{\frac{{a}^{2}A{x}^{2}{e}^{4}}{2\,{b}^{3}}}+4\,{\frac{A{d}^{3}ex}{b}}+{\frac{B{a}^{4}{e}^{4}x}{{b}^{5}}}+{\frac{B{x}^{4}d{e}^{3}}{b}}+{\frac{4\,A{x}^{3}d{e}^{3}}{3\,b}}+3\,{\frac{A{x}^{2}{d}^{2}{e}^{2}}{b}}-{\frac{B{x}^{4}a{e}^{4}}{4\,{b}^{2}}}-{\frac{\ln \left ( bx+a \right ) Ba{d}^{4}}{{b}^{2}}}+{\frac{\ln \left ( bx+a \right ) A{a}^{4}{e}^{4}}{{b}^{5}}}-{\frac{\ln \left ( bx+a \right ) B{a}^{5}{e}^{4}}{{b}^{6}}}-4\,{\frac{\ln \left ( bx+a \right ) Aa{d}^{3}e}{{b}^{2}}}+4\,{\frac{\ln \left ( bx+a \right ) B{a}^{4}d{e}^{3}}{{b}^{5}}}-6\,{\frac{\ln \left ( bx+a \right ) B{a}^{3}{d}^{2}{e}^{2}}{{b}^{4}}}+4\,{\frac{\ln \left ( bx+a \right ) B{a}^{2}{d}^{3}e}{{b}^{3}}}-2\,{\frac{aA{x}^{2}d{e}^{3}}{{b}^{2}}}+2\,{\frac{B{x}^{2}{a}^{2}d{e}^{3}}{{b}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.34963, size = 540, normalized size = 3.48 \[ \frac{12 \, B b^{4} e^{4} x^{5} + 15 \,{\left (4 \, B b^{4} d e^{3} -{\left (B a b^{3} - A b^{4}\right )} e^{4}\right )} x^{4} + 20 \,{\left (6 \, B b^{4} d^{2} e^{2} - 4 \,{\left (B a b^{3} - A b^{4}\right )} d e^{3} +{\left (B a^{2} b^{2} - A a b^{3}\right )} e^{4}\right )} x^{3} + 30 \,{\left (4 \, B b^{4} d^{3} e - 6 \,{\left (B a b^{3} - A b^{4}\right )} d^{2} e^{2} + 4 \,{\left (B a^{2} b^{2} - A a b^{3}\right )} d e^{3} -{\left (B a^{3} b - A a^{2} b^{2}\right )} e^{4}\right )} x^{2} + 60 \,{\left (B b^{4} d^{4} - 4 \,{\left (B a b^{3} - A b^{4}\right )} d^{3} e + 6 \,{\left (B a^{2} b^{2} - A a b^{3}\right )} d^{2} e^{2} - 4 \,{\left (B a^{3} b - A a^{2} b^{2}\right )} d e^{3} +{\left (B a^{4} - A a^{3} b\right )} e^{4}\right )} x}{60 \, b^{5}} - \frac{{\left ({\left (B a b^{4} - A b^{5}\right )} d^{4} - 4 \,{\left (B a^{2} b^{3} - A a b^{4}\right )} d^{3} e + 6 \,{\left (B a^{3} b^{2} - A a^{2} b^{3}\right )} d^{2} e^{2} - 4 \,{\left (B a^{4} b - A a^{3} b^{2}\right )} d e^{3} +{\left (B a^{5} - A a^{4} b\right )} e^{4}\right )} \log \left (b x + a\right )}{b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/60*(12*B*b^4*e^4*x^5 + 15*(4*B*b^4*d*e^3 - (B*a*b^3 - A*b^4)*e^4)*x^4 + 20*(6*
B*b^4*d^2*e^2 - 4*(B*a*b^3 - A*b^4)*d*e^3 + (B*a^2*b^2 - A*a*b^3)*e^4)*x^3 + 30*
(4*B*b^4*d^3*e - 6*(B*a*b^3 - A*b^4)*d^2*e^2 + 4*(B*a^2*b^2 - A*a*b^3)*d*e^3 - (
B*a^3*b - A*a^2*b^2)*e^4)*x^2 + 60*(B*b^4*d^4 - 4*(B*a*b^3 - A*b^4)*d^3*e + 6*(B
*a^2*b^2 - A*a*b^3)*d^2*e^2 - 4*(B*a^3*b - A*a^2*b^2)*d*e^3 + (B*a^4 - A*a^3*b)*
e^4)*x)/b^5 - ((B*a*b^4 - A*b^5)*d^4 - 4*(B*a^2*b^3 - A*a*b^4)*d^3*e + 6*(B*a^3*
b^2 - A*a^2*b^3)*d^2*e^2 - 4*(B*a^4*b - A*a^3*b^2)*d*e^3 + (B*a^5 - A*a^4*b)*e^4
)*log(b*x + a)/b^6

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Fricas [A]  time = 0.213599, size = 544, normalized size = 3.51 \[ \frac{12 \, B b^{5} e^{4} x^{5} + 15 \,{\left (4 \, B b^{5} d e^{3} -{\left (B a b^{4} - A b^{5}\right )} e^{4}\right )} x^{4} + 20 \,{\left (6 \, B b^{5} d^{2} e^{2} - 4 \,{\left (B a b^{4} - A b^{5}\right )} d e^{3} +{\left (B a^{2} b^{3} - A a b^{4}\right )} e^{4}\right )} x^{3} + 30 \,{\left (4 \, B b^{5} d^{3} e - 6 \,{\left (B a b^{4} - A b^{5}\right )} d^{2} e^{2} + 4 \,{\left (B a^{2} b^{3} - A a b^{4}\right )} d e^{3} -{\left (B a^{3} b^{2} - A a^{2} b^{3}\right )} e^{4}\right )} x^{2} + 60 \,{\left (B b^{5} d^{4} - 4 \,{\left (B a b^{4} - A b^{5}\right )} d^{3} e + 6 \,{\left (B a^{2} b^{3} - A a b^{4}\right )} d^{2} e^{2} - 4 \,{\left (B a^{3} b^{2} - A a^{2} b^{3}\right )} d e^{3} +{\left (B a^{4} b - A a^{3} b^{2}\right )} e^{4}\right )} x - 60 \,{\left ({\left (B a b^{4} - A b^{5}\right )} d^{4} - 4 \,{\left (B a^{2} b^{3} - A a b^{4}\right )} d^{3} e + 6 \,{\left (B a^{3} b^{2} - A a^{2} b^{3}\right )} d^{2} e^{2} - 4 \,{\left (B a^{4} b - A a^{3} b^{2}\right )} d e^{3} +{\left (B a^{5} - A a^{4} b\right )} e^{4}\right )} \log \left (b x + a\right )}{60 \, b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/60*(12*B*b^5*e^4*x^5 + 15*(4*B*b^5*d*e^3 - (B*a*b^4 - A*b^5)*e^4)*x^4 + 20*(6*
B*b^5*d^2*e^2 - 4*(B*a*b^4 - A*b^5)*d*e^3 + (B*a^2*b^3 - A*a*b^4)*e^4)*x^3 + 30*
(4*B*b^5*d^3*e - 6*(B*a*b^4 - A*b^5)*d^2*e^2 + 4*(B*a^2*b^3 - A*a*b^4)*d*e^3 - (
B*a^3*b^2 - A*a^2*b^3)*e^4)*x^2 + 60*(B*b^5*d^4 - 4*(B*a*b^4 - A*b^5)*d^3*e + 6*
(B*a^2*b^3 - A*a*b^4)*d^2*e^2 - 4*(B*a^3*b^2 - A*a^2*b^3)*d*e^3 + (B*a^4*b - A*a
^3*b^2)*e^4)*x - 60*((B*a*b^4 - A*b^5)*d^4 - 4*(B*a^2*b^3 - A*a*b^4)*d^3*e + 6*(
B*a^3*b^2 - A*a^2*b^3)*d^2*e^2 - 4*(B*a^4*b - A*a^3*b^2)*d*e^3 + (B*a^5 - A*a^4*
b)*e^4)*log(b*x + a))/b^6

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Sympy [A]  time = 4.93179, size = 345, normalized size = 2.23 \[ \frac{B e^{4} x^{5}}{5 b} - \frac{x^{4} \left (- A b e^{4} + B a e^{4} - 4 B b d e^{3}\right )}{4 b^{2}} + \frac{x^{3} \left (- A a b e^{4} + 4 A b^{2} d e^{3} + B a^{2} e^{4} - 4 B a b d e^{3} + 6 B b^{2} d^{2} e^{2}\right )}{3 b^{3}} - \frac{x^{2} \left (- A a^{2} b e^{4} + 4 A a b^{2} d e^{3} - 6 A b^{3} d^{2} e^{2} + B a^{3} e^{4} - 4 B a^{2} b d e^{3} + 6 B a b^{2} d^{2} e^{2} - 4 B b^{3} d^{3} e\right )}{2 b^{4}} + \frac{x \left (- A a^{3} b e^{4} + 4 A a^{2} b^{2} d e^{3} - 6 A a b^{3} d^{2} e^{2} + 4 A b^{4} d^{3} e + B a^{4} e^{4} - 4 B a^{3} b d e^{3} + 6 B a^{2} b^{2} d^{2} e^{2} - 4 B a b^{3} d^{3} e + B b^{4} d^{4}\right )}{b^{5}} - \frac{\left (- A b + B a\right ) \left (a e - b d\right )^{4} \log{\left (a + b x \right )}}{b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.216763, size = 599, normalized size = 3.86 \[ \frac{12 \, B b^{4} x^{5} e^{4} + 60 \, B b^{4} d x^{4} e^{3} + 120 \, B b^{4} d^{2} x^{3} e^{2} + 120 \, B b^{4} d^{3} x^{2} e + 60 \, B b^{4} d^{4} x - 15 \, B a b^{3} x^{4} e^{4} + 15 \, A b^{4} x^{4} e^{4} - 80 \, B a b^{3} d x^{3} e^{3} + 80 \, A b^{4} d x^{3} e^{3} - 180 \, B a b^{3} d^{2} x^{2} e^{2} + 180 \, A b^{4} d^{2} x^{2} e^{2} - 240 \, B a b^{3} d^{3} x e + 240 \, A b^{4} d^{3} x e + 20 \, B a^{2} b^{2} x^{3} e^{4} - 20 \, A a b^{3} x^{3} e^{4} + 120 \, B a^{2} b^{2} d x^{2} e^{3} - 120 \, A a b^{3} d x^{2} e^{3} + 360 \, B a^{2} b^{2} d^{2} x e^{2} - 360 \, A a b^{3} d^{2} x e^{2} - 30 \, B a^{3} b x^{2} e^{4} + 30 \, A a^{2} b^{2} x^{2} e^{4} - 240 \, B a^{3} b d x e^{3} + 240 \, A a^{2} b^{2} d x e^{3} + 60 \, B a^{4} x e^{4} - 60 \, A a^{3} b x e^{4}}{60 \, b^{5}} - \frac{{\left (B a b^{4} d^{4} - A b^{5} d^{4} - 4 \, B a^{2} b^{3} d^{3} e + 4 \, A a b^{4} d^{3} e + 6 \, B a^{3} b^{2} d^{2} e^{2} - 6 \, A a^{2} b^{3} d^{2} e^{2} - 4 \, B a^{4} b d e^{3} + 4 \, A a^{3} b^{2} d e^{3} + B a^{5} e^{4} - A a^{4} b e^{4}\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/60*(12*B*b^4*x^5*e^4 + 60*B*b^4*d*x^4*e^3 + 120*B*b^4*d^2*x^3*e^2 + 120*B*b^4*
d^3*x^2*e + 60*B*b^4*d^4*x - 15*B*a*b^3*x^4*e^4 + 15*A*b^4*x^4*e^4 - 80*B*a*b^3*
d*x^3*e^3 + 80*A*b^4*d*x^3*e^3 - 180*B*a*b^3*d^2*x^2*e^2 + 180*A*b^4*d^2*x^2*e^2
- 240*B*a*b^3*d^3*x*e + 240*A*b^4*d^3*x*e + 20*B*a^2*b^2*x^3*e^4 - 20*A*a*b^3*x
^3*e^4 + 120*B*a^2*b^2*d*x^2*e^3 - 120*A*a*b^3*d*x^2*e^3 + 360*B*a^2*b^2*d^2*x*e
^2 - 360*A*a*b^3*d^2*x*e^2 - 30*B*a^3*b*x^2*e^4 + 30*A*a^2*b^2*x^2*e^4 - 240*B*a
^3*b*d*x*e^3 + 240*A*a^2*b^2*d*x*e^3 + 60*B*a^4*x*e^4 - 60*A*a^3*b*x*e^4)/b^5 -
(B*a*b^4*d^4 - A*b^5*d^4 - 4*B*a^2*b^3*d^3*e + 4*A*a*b^4*d^3*e + 6*B*a^3*b^2*d^2
*e^2 - 6*A*a^2*b^3*d^2*e^2 - 4*B*a^4*b*d*e^3 + 4*A*a^3*b^2*d*e^3 + B*a^5*e^4 - A
*a^4*b*e^4)*ln(abs(b*x + a))/b^6