3.80 \(\int \frac{(a+b x)^3 (A+B x)}{x^3} \, dx\)

Optimal. Leaf size=65 \[ -\frac{a^3 A}{2 x^2}-\frac{a^2 (a B+3 A b)}{x}+b^2 x (3 a B+A b)+3 a b \log (x) (a B+A b)+\frac{1}{2} b^3 B x^2 \]

[Out]

-(a^3*A)/(2*x^2) - (a^2*(3*A*b + a*B))/x + b^2*(A*b + 3*a*B)*x + (b^3*B*x^2)/2 +
3*a*b*(A*b + a*B)*Log[x]

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Rubi [A]  time = 0.11127, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ -\frac{a^3 A}{2 x^2}-\frac{a^2 (a B+3 A b)}{x}+b^2 x (3 a B+A b)+3 a b \log (x) (a B+A b)+\frac{1}{2} b^3 B x^2 \]

Antiderivative was successfully verified.

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

[Out]

-(a^3*A)/(2*x^2) - (a^2*(3*A*b + a*B))/x + b^2*(A*b + 3*a*B)*x + (b^3*B*x^2)/2 +
3*a*b*(A*b + a*B)*Log[x]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**3*(B*x+A)/x**3,x)

[Out]

-A*a**3/(2*x**2) + B*b**3*Integral(x, x) - a**2*(3*A*b + B*a)/x + 3*a*b*(A*b + B
*a)*log(x) + b**2*(A*b + 3*B*a)*Integral(A, x)/A

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Mathematica [A]  time = 0.043502, size = 62, normalized size = 0.95 \[ \frac{1}{2} \left (-\frac{a^3 (A+2 B x)}{x^2}-\frac{6 a^2 A b}{x}+6 a b \log (x) (a B+A b)+6 a b^2 B x+b^3 x (2 A+B x)\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 71, normalized size = 1.1 \[{\frac{{b}^{3}B{x}^{2}}{2}}+Ax{b}^{3}+3\,Bxa{b}^{2}+3\,A\ln \left ( x \right ) a{b}^{2}+3\,B\ln \left ( x \right ){a}^{2}b-{\frac{A{a}^{3}}{2\,{x}^{2}}}-3\,{\frac{{a}^{2}bA}{x}}-{\frac{{a}^{3}B}{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^3*(B*x+A)/x^3,x)

[Out]

1/2*b^3*B*x^2+A*x*b^3+3*B*x*a*b^2+3*A*ln(x)*a*b^2+3*B*ln(x)*a^2*b-1/2*a^3*A/x^2-
3*a^2/x*A*b-a^3/x*B

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Maxima [A]  time = 1.35414, size = 93, normalized size = 1.43 \[ \frac{1}{2} \, B b^{3} x^{2} +{\left (3 \, B a b^{2} + A b^{3}\right )} x + 3 \,{\left (B a^{2} b + A a b^{2}\right )} \log \left (x\right ) - \frac{A a^{3} + 2 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^3/x^3,x, algorithm="maxima")

[Out]

1/2*B*b^3*x^2 + (3*B*a*b^2 + A*b^3)*x + 3*(B*a^2*b + A*a*b^2)*log(x) - 1/2*(A*a^
3 + 2*(B*a^3 + 3*A*a^2*b)*x)/x^2

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Fricas [A]  time = 0.204833, size = 100, normalized size = 1.54 \[ \frac{B b^{3} x^{4} - A a^{3} + 2 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 6 \,{\left (B a^{2} b + A a b^{2}\right )} x^{2} \log \left (x\right ) - 2 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^3/x^3,x, algorithm="fricas")

[Out]

1/2*(B*b^3*x^4 - A*a^3 + 2*(3*B*a*b^2 + A*b^3)*x^3 + 6*(B*a^2*b + A*a*b^2)*x^2*l
og(x) - 2*(B*a^3 + 3*A*a^2*b)*x)/x^2

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Sympy [A]  time = 2.14464, size = 66, normalized size = 1.02 \[ \frac{B b^{3} x^{2}}{2} + 3 a b \left (A b + B a\right ) \log{\left (x \right )} + x \left (A b^{3} + 3 B a b^{2}\right ) - \frac{A a^{3} + x \left (6 A a^{2} b + 2 B a^{3}\right )}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**3*(B*x+A)/x**3,x)

[Out]

B*b**3*x**2/2 + 3*a*b*(A*b + B*a)*log(x) + x*(A*b**3 + 3*B*a*b**2) - (A*a**3 + x
*(6*A*a**2*b + 2*B*a**3))/(2*x**2)

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GIAC/XCAS [A]  time = 0.235512, size = 93, normalized size = 1.43 \[ \frac{1}{2} \, B b^{3} x^{2} + 3 \, B a b^{2} x + A b^{3} x + 3 \,{\left (B a^{2} b + A a b^{2}\right )}{\rm ln}\left ({\left | x \right |}\right ) - \frac{A a^{3} + 2 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^3/x^3,x, algorithm="giac")

[Out]

1/2*B*b^3*x^2 + 3*B*a*b^2*x + A*b^3*x + 3*(B*a^2*b + A*a*b^2)*ln(abs(x)) - 1/2*(
A*a^3 + 2*(B*a^3 + 3*A*a^2*b)*x)/x^2