3.271 \(\int \frac{(A+B x) \left (a+c x^2\right )^3}{x^3} \, dx\)

Optimal. Leaf size=81 \[ -\frac{a^3 A}{2 x^2}-\frac{a^3 B}{x}+3 a^2 A c \log (x)+3 a^2 B c x+\frac{3}{2} a A c^2 x^2+a B c^2 x^3+\frac{1}{4} A c^3 x^4+\frac{1}{5} B c^3 x^5 \]

[Out]

-(a^3*A)/(2*x^2) - (a^3*B)/x + 3*a^2*B*c*x + (3*a*A*c^2*x^2)/2 + a*B*c^2*x^3 + (
A*c^3*x^4)/4 + (B*c^3*x^5)/5 + 3*a^2*A*c*Log[x]

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Rubi [A]  time = 0.100936, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ -\frac{a^3 A}{2 x^2}-\frac{a^3 B}{x}+3 a^2 A c \log (x)+3 a^2 B c x+\frac{3}{2} a A c^2 x^2+a B c^2 x^3+\frac{1}{4} A c^3 x^4+\frac{1}{5} B c^3 x^5 \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a + c*x^2)^3)/x^3,x]

[Out]

-(a^3*A)/(2*x^2) - (a^3*B)/x + 3*a^2*B*c*x + (3*a*A*c^2*x^2)/2 + a*B*c^2*x^3 + (
A*c^3*x^4)/4 + (B*c^3*x^5)/5 + 3*a^2*A*c*Log[x]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+a)**3/x**3,x)

[Out]

-A*a**3/(2*x**2) + 3*A*a**2*c*log(x) + 3*A*a*c**2*Integral(x, x) + A*c**3*x**4/4
- B*a**3/x + 3*B*a**2*c*x + B*a*c**2*x**3 + B*c**3*x**5/5

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Mathematica [A]  time = 0.0130655, size = 81, normalized size = 1. \[ -\frac{a^3 A}{2 x^2}-\frac{a^3 B}{x}+3 a^2 A c \log (x)+3 a^2 B c x+\frac{3}{2} a A c^2 x^2+a B c^2 x^3+\frac{1}{4} A c^3 x^4+\frac{1}{5} B c^3 x^5 \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a + c*x^2)^3)/x^3,x]

[Out]

-(a^3*A)/(2*x^2) - (a^3*B)/x + 3*a^2*B*c*x + (3*a*A*c^2*x^2)/2 + a*B*c^2*x^3 + (
A*c^3*x^4)/4 + (B*c^3*x^5)/5 + 3*a^2*A*c*Log[x]

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Maple [A]  time = 0.008, size = 74, normalized size = 0.9 \[ -{\frac{A{a}^{3}}{2\,{x}^{2}}}-{\frac{B{a}^{3}}{x}}+3\,{a}^{2}Bcx+{\frac{3\,aA{c}^{2}{x}^{2}}{2}}+aB{c}^{2}{x}^{3}+{\frac{A{c}^{3}{x}^{4}}{4}}+{\frac{B{c}^{3}{x}^{5}}{5}}+3\,{a}^{2}Ac\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+a)^3/x^3,x)

[Out]

-1/2*a^3*A/x^2-a^3*B/x+3*a^2*B*c*x+3/2*a*A*c^2*x^2+a*B*c^2*x^3+1/4*A*c^3*x^4+1/5
*B*c^3*x^5+3*a^2*A*c*ln(x)

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Maxima [A]  time = 0.692395, size = 99, normalized size = 1.22 \[ \frac{1}{5} \, B c^{3} x^{5} + \frac{1}{4} \, A c^{3} x^{4} + B a c^{2} x^{3} + \frac{3}{2} \, A a c^{2} x^{2} + 3 \, B a^{2} c x + 3 \, A a^{2} c \log \left (x\right ) - \frac{2 \, B a^{3} x + A a^{3}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^3*(B*x + A)/x^3,x, algorithm="maxima")

[Out]

1/5*B*c^3*x^5 + 1/4*A*c^3*x^4 + B*a*c^2*x^3 + 3/2*A*a*c^2*x^2 + 3*B*a^2*c*x + 3*
A*a^2*c*log(x) - 1/2*(2*B*a^3*x + A*a^3)/x^2

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Fricas [A]  time = 0.283225, size = 107, normalized size = 1.32 \[ \frac{4 \, B c^{3} x^{7} + 5 \, A c^{3} x^{6} + 20 \, B a c^{2} x^{5} + 30 \, A a c^{2} x^{4} + 60 \, B a^{2} c x^{3} + 60 \, A a^{2} c x^{2} \log \left (x\right ) - 20 \, B a^{3} x - 10 \, A a^{3}}{20 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^3*(B*x + A)/x^3,x, algorithm="fricas")

[Out]

1/20*(4*B*c^3*x^7 + 5*A*c^3*x^6 + 20*B*a*c^2*x^5 + 30*A*a*c^2*x^4 + 60*B*a^2*c*x
^3 + 60*A*a^2*c*x^2*log(x) - 20*B*a^3*x - 10*A*a^3)/x^2

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Sympy [A]  time = 1.65139, size = 83, normalized size = 1.02 \[ 3 A a^{2} c \log{\left (x \right )} + \frac{3 A a c^{2} x^{2}}{2} + \frac{A c^{3} x^{4}}{4} + 3 B a^{2} c x + B a c^{2} x^{3} + \frac{B c^{3} x^{5}}{5} - \frac{A a^{3} + 2 B a^{3} x}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+a)**3/x**3,x)

[Out]

3*A*a**2*c*log(x) + 3*A*a*c**2*x**2/2 + A*c**3*x**4/4 + 3*B*a**2*c*x + B*a*c**2*
x**3 + B*c**3*x**5/5 - (A*a**3 + 2*B*a**3*x)/(2*x**2)

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GIAC/XCAS [A]  time = 0.268167, size = 100, normalized size = 1.23 \[ \frac{1}{5} \, B c^{3} x^{5} + \frac{1}{4} \, A c^{3} x^{4} + B a c^{2} x^{3} + \frac{3}{2} \, A a c^{2} x^{2} + 3 \, B a^{2} c x + 3 \, A a^{2} c{\rm ln}\left ({\left | x \right |}\right ) - \frac{2 \, B a^{3} x + A a^{3}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^3*(B*x + A)/x^3,x, algorithm="giac")

[Out]

1/5*B*c^3*x^5 + 1/4*A*c^3*x^4 + B*a*c^2*x^3 + 3/2*A*a*c^2*x^2 + 3*B*a^2*c*x + 3*
A*a^2*c*ln(abs(x)) - 1/2*(2*B*a^3*x + A*a^3)/x^2