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

Optimal. Leaf size=80 \[ -\frac{a^3 A}{x}+a^3 B \log (x)+3 a^2 A c x+\frac{3}{2} a^2 B c x^2+a A c^2 x^3+\frac{3}{4} a B c^2 x^4+\frac{1}{5} A c^3 x^5+\frac{1}{6} B c^3 x^6 \]

[Out]

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

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Rubi [A]  time = 0.103942, antiderivative size = 80, 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 \[ -\frac{a^3 A}{x}+a^3 B \log (x)+3 a^2 A c x+\frac{3}{2} a^2 B c x^2+a A c^2 x^3+\frac{3}{4} a B c^2 x^4+\frac{1}{5} A c^3 x^5+\frac{1}{6} B c^3 x^6 \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0131599, size = 80, normalized size = 1. \[ -\frac{a^3 A}{x}+a^3 B \log (x)+3 a^2 A c x+\frac{3}{2} a^2 B c x^2+a A c^2 x^3+\frac{3}{4} a B c^2 x^4+\frac{1}{5} A c^3 x^5+\frac{1}{6} B c^3 x^6 \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.009, size = 73, normalized size = 0.9 \[ -{\frac{A{a}^{3}}{x}}+3\,{a}^{2}Acx+{\frac{3\,{a}^{2}Bc{x}^{2}}{2}}+aA{c}^{2}{x}^{3}+{\frac{3\,aB{c}^{2}{x}^{4}}{4}}+{\frac{A{c}^{3}{x}^{5}}{5}}+{\frac{B{c}^{3}{x}^{6}}{6}}+{a}^{3}B\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+a)^3/x^2,x)

[Out]

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

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Maxima [A]  time = 0.681495, size = 97, normalized size = 1.21 \[ \frac{1}{6} \, B c^{3} x^{6} + \frac{1}{5} \, A c^{3} x^{5} + \frac{3}{4} \, B a c^{2} x^{4} + A a c^{2} x^{3} + \frac{3}{2} \, B a^{2} c x^{2} + 3 \, A a^{2} c x + B a^{3} \log \left (x\right ) - \frac{A a^{3}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*B*c^3*x^6 + 1/5*A*c^3*x^5 + 3/4*B*a*c^2*x^4 + A*a*c^2*x^3 + 3/2*B*a^2*c*x^2
+ 3*A*a^2*c*x + B*a^3*log(x) - A*a^3/x

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Fricas [A]  time = 0.275294, size = 107, normalized size = 1.34 \[ \frac{10 \, B c^{3} x^{7} + 12 \, A c^{3} x^{6} + 45 \, B a c^{2} x^{5} + 60 \, A a c^{2} x^{4} + 90 \, B a^{2} c x^{3} + 180 \, A a^{2} c x^{2} + 60 \, B a^{3} x \log \left (x\right ) - 60 \, A a^{3}}{60 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/60*(10*B*c^3*x^7 + 12*A*c^3*x^6 + 45*B*a*c^2*x^5 + 60*A*a*c^2*x^4 + 90*B*a^2*c
*x^3 + 180*A*a^2*c*x^2 + 60*B*a^3*x*log(x) - 60*A*a^3)/x

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Sympy [A]  time = 1.34368, size = 82, normalized size = 1.02 \[ - \frac{A a^{3}}{x} + 3 A a^{2} c x + A a c^{2} x^{3} + \frac{A c^{3} x^{5}}{5} + B a^{3} \log{\left (x \right )} + \frac{3 B a^{2} c x^{2}}{2} + \frac{3 B a c^{2} x^{4}}{4} + \frac{B c^{3} x^{6}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.271563, size = 99, normalized size = 1.24 \[ \frac{1}{6} \, B c^{3} x^{6} + \frac{1}{5} \, A c^{3} x^{5} + \frac{3}{4} \, B a c^{2} x^{4} + A a c^{2} x^{3} + \frac{3}{2} \, B a^{2} c x^{2} + 3 \, A a^{2} c x + B a^{3}{\rm ln}\left ({\left | x \right |}\right ) - \frac{A a^{3}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*B*c^3*x^6 + 1/5*A*c^3*x^5 + 3/4*B*a*c^2*x^4 + A*a*c^2*x^3 + 3/2*B*a^2*c*x^2
+ 3*A*a^2*c*x + B*a^3*ln(abs(x)) - A*a^3/x