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

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

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Rubi [A]  time = 0.04, antiderivative size = 80, 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 x-\frac {a^3 A}{x}+\frac {3}{2} a^2 B c x^2+a^3 B \log (x)+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 \end {gather*}

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]

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^2} \, dx &=\int \left (3 a^2 A c+\frac {a^3 A}{x^2}+\frac {a^3 B}{x}+3 a^2 B c x+3 a A c^2 x^2+3 a B c^2 x^3+A c^3 x^4+B c^3 x^5\right ) \, dx\\ &=-\frac {a^3 A}{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+a^3 B \log (x)\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 80, normalized size = 1.00 \begin {gather*} -\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 \end {gather*}

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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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^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.40, size = 79, normalized size = 0.99 \begin {gather*} \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 \relax (x) - 60 \, A a^{3}}{60 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^3/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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giac [A]  time = 0.18, size = 73, normalized size = 0.91 \begin {gather*} \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 ({\left | x \right |}\right ) - \frac {A a^{3}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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maple [A]  time = 0.05, size = 73, normalized size = 0.91 \begin {gather*} \frac {B \,c^{3} x^{6}}{6}+\frac {A \,c^{3} x^{5}}{5}+\frac {3 B a \,c^{2} x^{4}}{4}+A a \,c^{2} x^{3}+\frac {3 B \,a^{2} c \,x^{2}}{2}+3 A \,a^{2} c x +B \,a^{3} \ln \relax (x )-\frac {A \,a^{3}}{x} \end {gather*}

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*a^3/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+B*a^3*ln(x)

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maxima [A]  time = 0.55, size = 72, normalized size = 0.90 \begin {gather*} \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 \relax (x) - \frac {A a^{3}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^3/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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mupad [B]  time = 0.04, size = 72, normalized size = 0.90 \begin {gather*} \frac {A\,c^3\,x^5}{5}-\frac {A\,a^3}{x}+\frac {B\,c^3\,x^6}{6}+B\,a^3\,\ln \relax (x)+3\,A\,a^2\,c\,x+A\,a\,c^2\,x^3+\frac {3\,B\,a^2\,c\,x^2}{2}+\frac {3\,B\,a\,c^2\,x^4}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + c*x^2)^3*(A + B*x))/x^2,x)

[Out]

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

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sympy [A]  time = 0.20, size = 82, normalized size = 1.02 \begin {gather*} - \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 {\relax (x )} + \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} \end {gather*}

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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