3.3.77 \(\int \frac {(A+B x) (a+c x^2)^4}{x^2} \, dx\)

Optimal. Leaf size=107 \[ -\frac {a^4 A}{x}+a^4 B \log (x)+4 a^3 A c x+2 a^3 B c x^2+2 a^2 A c^2 x^3+\frac {3}{2} a^2 B c^2 x^4+\frac {4}{5} a A c^3 x^5+\frac {2}{3} a B c^3 x^6+\frac {1}{7} A c^4 x^7+\frac {1}{8} B c^4 x^8 \]

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Rubi [A]  time = 0.06, antiderivative size = 107, 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*} 2 a^2 A c^2 x^3+4 a^3 A c x-\frac {a^4 A}{x}+\frac {3}{2} a^2 B c^2 x^4+2 a^3 B c x^2+a^4 B \log (x)+\frac {4}{5} a A c^3 x^5+\frac {2}{3} a B c^3 x^6+\frac {1}{7} A c^4 x^7+\frac {1}{8} B c^4 x^8 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + c*x^2)^4)/x^2,x]

[Out]

-((a^4*A)/x) + 4*a^3*A*c*x + 2*a^3*B*c*x^2 + 2*a^2*A*c^2*x^3 + (3*a^2*B*c^2*x^4)/2 + (4*a*A*c^3*x^5)/5 + (2*a*
B*c^3*x^6)/3 + (A*c^4*x^7)/7 + (B*c^4*x^8)/8 + a^4*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 )^4}{x^2} \, dx &=\int \left (4 a^3 A c+\frac {a^4 A}{x^2}+\frac {a^4 B}{x}+4 a^3 B c x+6 a^2 A c^2 x^2+6 a^2 B c^2 x^3+4 a A c^3 x^4+4 a B c^3 x^5+A c^4 x^6+B c^4 x^7\right ) \, dx\\ &=-\frac {a^4 A}{x}+4 a^3 A c x+2 a^3 B c x^2+2 a^2 A c^2 x^3+\frac {3}{2} a^2 B c^2 x^4+\frac {4}{5} a A c^3 x^5+\frac {2}{3} a B c^3 x^6+\frac {1}{7} A c^4 x^7+\frac {1}{8} B c^4 x^8+a^4 B \log (x)\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 107, normalized size = 1.00 \begin {gather*} -\frac {a^4 A}{x}+a^4 B \log (x)+4 a^3 A c x+2 a^3 B c x^2+2 a^2 A c^2 x^3+\frac {3}{2} a^2 B c^2 x^4+\frac {4}{5} a A c^3 x^5+\frac {2}{3} a B c^3 x^6+\frac {1}{7} A c^4 x^7+\frac {1}{8} B c^4 x^8 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + c*x^2)^4)/x^2,x]

[Out]

-((a^4*A)/x) + 4*a^3*A*c*x + 2*a^3*B*c*x^2 + 2*a^2*A*c^2*x^3 + (3*a^2*B*c^2*x^4)/2 + (4*a*A*c^3*x^5)/5 + (2*a*
B*c^3*x^6)/3 + (A*c^4*x^7)/7 + (B*c^4*x^8)/8 + a^4*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 )^4}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((A + B*x)*(a + c*x^2)^4)/x^2,x]

[Out]

IntegrateAlgebraic[((A + B*x)*(a + c*x^2)^4)/x^2, x]

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fricas [A]  time = 0.40, size = 103, normalized size = 0.96 \begin {gather*} \frac {105 \, B c^{4} x^{9} + 120 \, A c^{4} x^{8} + 560 \, B a c^{3} x^{7} + 672 \, A a c^{3} x^{6} + 1260 \, B a^{2} c^{2} x^{5} + 1680 \, A a^{2} c^{2} x^{4} + 1680 \, B a^{3} c x^{3} + 3360 \, A a^{3} c x^{2} + 840 \, B a^{4} x \log \relax (x) - 840 \, A a^{4}}{840 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^4/x^2,x, algorithm="fricas")

[Out]

1/840*(105*B*c^4*x^9 + 120*A*c^4*x^8 + 560*B*a*c^3*x^7 + 672*A*a*c^3*x^6 + 1260*B*a^2*c^2*x^5 + 1680*A*a^2*c^2
*x^4 + 1680*B*a^3*c*x^3 + 3360*A*a^3*c*x^2 + 840*B*a^4*x*log(x) - 840*A*a^4)/x

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giac [A]  time = 0.17, size = 98, normalized size = 0.92 \begin {gather*} \frac {1}{8} \, B c^{4} x^{8} + \frac {1}{7} \, A c^{4} x^{7} + \frac {2}{3} \, B a c^{3} x^{6} + \frac {4}{5} \, A a c^{3} x^{5} + \frac {3}{2} \, B a^{2} c^{2} x^{4} + 2 \, A a^{2} c^{2} x^{3} + 2 \, B a^{3} c x^{2} + 4 \, A a^{3} c x + B a^{4} \log \left ({\left | x \right |}\right ) - \frac {A a^{4}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^4/x^2,x, algorithm="giac")

[Out]

1/8*B*c^4*x^8 + 1/7*A*c^4*x^7 + 2/3*B*a*c^3*x^6 + 4/5*A*a*c^3*x^5 + 3/2*B*a^2*c^2*x^4 + 2*A*a^2*c^2*x^3 + 2*B*
a^3*c*x^2 + 4*A*a^3*c*x + B*a^4*log(abs(x)) - A*a^4/x

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maple [A]  time = 0.05, size = 98, normalized size = 0.92 \begin {gather*} \frac {B \,c^{4} x^{8}}{8}+\frac {A \,c^{4} x^{7}}{7}+\frac {2 B a \,c^{3} x^{6}}{3}+\frac {4 A a \,c^{3} x^{5}}{5}+\frac {3 B \,a^{2} c^{2} x^{4}}{2}+2 A \,a^{2} c^{2} x^{3}+2 B \,a^{3} c \,x^{2}+4 A \,a^{3} c x +B \,a^{4} \ln \relax (x )-\frac {A \,a^{4}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+a)^4/x^2,x)

[Out]

-a^4*A/x+4*a^3*A*c*x+2*a^3*B*c*x^2+2*a^2*A*c^2*x^3+3/2*a^2*B*c^2*x^4+4/5*a*A*c^3*x^5+2/3*a*B*c^3*x^6+1/7*A*c^4
*x^7+1/8*B*c^4*x^8+a^4*B*ln(x)

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maxima [A]  time = 0.52, size = 97, normalized size = 0.91 \begin {gather*} \frac {1}{8} \, B c^{4} x^{8} + \frac {1}{7} \, A c^{4} x^{7} + \frac {2}{3} \, B a c^{3} x^{6} + \frac {4}{5} \, A a c^{3} x^{5} + \frac {3}{2} \, B a^{2} c^{2} x^{4} + 2 \, A a^{2} c^{2} x^{3} + 2 \, B a^{3} c x^{2} + 4 \, A a^{3} c x + B a^{4} \log \relax (x) - \frac {A a^{4}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^4/x^2,x, algorithm="maxima")

[Out]

1/8*B*c^4*x^8 + 1/7*A*c^4*x^7 + 2/3*B*a*c^3*x^6 + 4/5*A*a*c^3*x^5 + 3/2*B*a^2*c^2*x^4 + 2*A*a^2*c^2*x^3 + 2*B*
a^3*c*x^2 + 4*A*a^3*c*x + B*a^4*log(x) - A*a^4/x

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mupad [B]  time = 0.05, size = 97, normalized size = 0.91 \begin {gather*} \frac {A\,c^4\,x^7}{7}-\frac {A\,a^4}{x}+\frac {B\,c^4\,x^8}{8}+B\,a^4\,\ln \relax (x)+4\,A\,a^3\,c\,x+\frac {4\,A\,a\,c^3\,x^5}{5}+2\,B\,a^3\,c\,x^2+\frac {2\,B\,a\,c^3\,x^6}{3}+2\,A\,a^2\,c^2\,x^3+\frac {3\,B\,a^2\,c^2\,x^4}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + c*x^2)^4*(A + B*x))/x^2,x)

[Out]

(A*c^4*x^7)/7 - (A*a^4)/x + (B*c^4*x^8)/8 + B*a^4*log(x) + 4*A*a^3*c*x + (4*A*a*c^3*x^5)/5 + 2*B*a^3*c*x^2 + (
2*B*a*c^3*x^6)/3 + 2*A*a^2*c^2*x^3 + (3*B*a^2*c^2*x^4)/2

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sympy [A]  time = 0.25, size = 112, normalized size = 1.05 \begin {gather*} - \frac {A a^{4}}{x} + 4 A a^{3} c x + 2 A a^{2} c^{2} x^{3} + \frac {4 A a c^{3} x^{5}}{5} + \frac {A c^{4} x^{7}}{7} + B a^{4} \log {\relax (x )} + 2 B a^{3} c x^{2} + \frac {3 B a^{2} c^{2} x^{4}}{2} + \frac {2 B a c^{3} x^{6}}{3} + \frac {B c^{4} x^{8}}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+a)**4/x**2,x)

[Out]

-A*a**4/x + 4*A*a**3*c*x + 2*A*a**2*c**2*x**3 + 4*A*a*c**3*x**5/5 + A*c**4*x**7/7 + B*a**4*log(x) + 2*B*a**3*c
*x**2 + 3*B*a**2*c**2*x**4/2 + 2*B*a*c**3*x**6/3 + B*c**4*x**8/8

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