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

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

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Rubi [A]  time = 0.04, antiderivative size = 81, 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} \begin {gather*} 3 a^2 A c \log (x)-\frac {a^3 A}{2 x^2}+3 a^2 B c x-\frac {a^3 B}{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 \end {gather*}

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]

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 )^3}{x^3} \, dx &=\int \left (3 a^2 B c+\frac {a^3 A}{x^3}+\frac {a^3 B}{x^2}+\frac {3 a^2 A c}{x}+3 a A c^2 x+3 a B c^2 x^2+A c^3 x^3+B c^3 x^4\right ) \, dx\\ &=-\frac {a^3 A}{2 x^2}-\frac {a^3 B}{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+3 a^2 A c \log (x)\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 81, normalized size = 1.00 \begin {gather*} -\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 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(a^3*A)/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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(A+B x) \left (a+c x^2\right )^3}{x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.39, size = 79, normalized size = 0.98 \begin {gather*} \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 \relax (x) - 20 \, B a^{3} x - 10 \, A a^{3}}{20 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^3/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) - 2
0*B*a^3*x - 10*A*a^3)/x^2

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giac [A]  time = 0.16, size = 74, normalized size = 0.91 \begin {gather*} \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 ({\left | x \right |}\right ) - \frac {2 \, B a^{3} x + A a^{3}}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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maple [A]  time = 0.05, size = 74, normalized size = 0.91 \begin {gather*} \frac {B \,c^{3} x^{5}}{5}+\frac {A \,c^{3} x^{4}}{4}+B a \,c^{2} x^{3}+\frac {3 A a \,c^{2} x^{2}}{2}+3 A \,a^{2} c \ln \relax (x )+3 B \,a^{2} c x -\frac {B \,a^{3}}{x}-\frac {A \,a^{3}}{2 x^{2}} \end {gather*}

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*a^3/x^2-B*a^3/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 = 0.62, size = 73, normalized size = 0.90 \begin {gather*} \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 \relax (x) - \frac {2 \, B a^{3} x + A a^{3}}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^3/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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mupad [B]  time = 0.03, size = 73, normalized size = 0.90 \begin {gather*} \frac {A\,c^3\,x^4}{4}-\frac {\frac {A\,a^3}{2}+B\,a^3\,x}{x^2}+\frac {B\,c^3\,x^5}{5}+3\,B\,a^2\,c\,x+\frac {3\,A\,a\,c^2\,x^2}{2}+B\,a\,c^2\,x^3+3\,A\,a^2\,c\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + c*x^2)^3*(A + B*x))/x^3,x)

[Out]

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

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sympy [A]  time = 0.29, size = 85, normalized size = 1.05 \begin {gather*} 3 A a^{2} c \log {\relax (x )} + \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}} \end {gather*}

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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