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

Optimal. Leaf size=56 \[ -\frac {a^2 A}{2 x^2}-\frac {a^2 B}{x}+2 a A c \log (x)+2 a B c x+\frac {1}{2} A c^2 x^2+\frac {1}{3} B c^2 x^3 \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {766} \[ -\frac {a^2 A}{2 x^2}-\frac {a^2 B}{x}+2 a A c \log (x)+2 a B c x+\frac {1}{2} A c^2 x^2+\frac {1}{3} B c^2 x^3 \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + c*x^2)^2)/x^3,x]

[Out]

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

Rule 766

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(e*x
)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, e, f, g, m}, x] && IGtQ[p, 0]

Rubi steps

\begin {align*} \int \frac {(A+B x) \left (a+c x^2\right )^2}{x^3} \, dx &=\int \left (2 a B c+\frac {a^2 A}{x^3}+\frac {a^2 B}{x^2}+\frac {2 a A c}{x}+A c^2 x+B c^2 x^2\right ) \, dx\\ &=-\frac {a^2 A}{2 x^2}-\frac {a^2 B}{x}+2 a B c x+\frac {1}{2} A c^2 x^2+\frac {1}{3} B c^2 x^3+2 a A c \log (x)\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 56, normalized size = 1.00 \[ -\frac {a^2 A}{2 x^2}-\frac {a^2 B}{x}+2 a A c \log (x)+2 a B c x+\frac {1}{2} A c^2 x^2+\frac {1}{3} B c^2 x^3 \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(a^2*A)/x^2 - (a^2*B)/x + 2*a*B*c*x + (A*c^2*x^2)/2 + (B*c^2*x^3)/3 + 2*a*A*c*Log[x]

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fricas [A]  time = 0.90, size = 55, normalized size = 0.98 \[ \frac {2 \, B c^{2} x^{5} + 3 \, A c^{2} x^{4} + 12 \, B a c x^{3} + 12 \, A a c x^{2} \log \relax (x) - 6 \, B a^{2} x - 3 \, A a^{2}}{6 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.17, size = 51, normalized size = 0.91 \[ \frac {1}{3} \, B c^{2} x^{3} + \frac {1}{2} \, A c^{2} x^{2} + 2 \, B a c x + 2 \, A a c \log \left ({\left | x \right |}\right ) - \frac {2 \, B a^{2} x + A a^{2}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.06, size = 51, normalized size = 0.91 \[ \frac {B \,c^{2} x^{3}}{3}+\frac {A \,c^{2} x^{2}}{2}+2 A a c \ln \relax (x )+2 B a c x -\frac {B \,a^{2}}{x}-\frac {A \,a^{2}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+a)^2/x^3,x)

[Out]

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

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maxima [A]  time = 0.62, size = 50, normalized size = 0.89 \[ \frac {1}{3} \, B c^{2} x^{3} + \frac {1}{2} \, A c^{2} x^{2} + 2 \, B a c x + 2 \, A a c \log \relax (x) - \frac {2 \, B a^{2} x + A a^{2}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.03, size = 50, normalized size = 0.89 \[ \frac {A\,c^2\,x^2}{2}-\frac {\frac {A\,a^2}{2}+B\,a^2\,x}{x^2}+\frac {B\,c^2\,x^3}{3}+2\,A\,a\,c\,\ln \relax (x)+2\,B\,a\,c\,x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + c*x^2)^2*(A + B*x))/x^3,x)

[Out]

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

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sympy [A]  time = 0.25, size = 58, normalized size = 1.04 \[ 2 A a c \log {\relax (x )} + \frac {A c^{2} x^{2}}{2} + 2 B a c x + \frac {B c^{2} x^{3}}{3} + \frac {- A a^{2} - 2 B a^{2} x}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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