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

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

[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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Rubi [A]  time = 0.055733, antiderivative size = 107, 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, Rules used = {766} \[ 2 a^2 A c^2 x^3+4 a^3 A c x-\frac{a^4 A}{x}+\frac{3}{2} a^2 B c^2 x^4+2 a^3 B c x^2+a^4 B \log (x)+\frac{4}{5} a A c^3 x^5+\frac{2}{3} a B c^3 x^6+\frac{1}{7} A c^4 x^7+\frac{1}{8} B c^4 x^8 \]

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*}

Mathematica [A]  time = 0.0073365, size = 107, normalized size = 1. \[ 2 a^2 A c^2 x^3+4 a^3 A c x-\frac{a^4 A}{x}+\frac{3}{2} a^2 B c^2 x^4+2 a^3 B c x^2+a^4 B \log (x)+\frac{4}{5} a A c^3 x^5+\frac{2}{3} a B c^3 x^6+\frac{1}{7} A c^4 x^7+\frac{1}{8} B c^4 x^8 \]

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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Maple [A]  time = 0.006, size = 98, normalized size = 0.9 \begin{align*} -{\frac{A{a}^{4}}{x}}+4\,{a}^{3}Acx+2\,{a}^{3}Bc{x}^{2}+2\,{a}^{2}A{c}^{2}{x}^{3}+{\frac{3\,{a}^{2}B{c}^{2}{x}^{4}}{2}}+{\frac{4\,aA{c}^{3}{x}^{5}}{5}}+{\frac{2\,aB{c}^{3}{x}^{6}}{3}}+{\frac{A{c}^{4}{x}^{7}}{7}}+{\frac{B{c}^{4}{x}^{8}}{8}}+{a}^{4}B\ln \left ( x \right ) \end{align*}

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 = 1.04056, size = 131, normalized size = 1.22 \begin{align*} \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 (x\right ) - \frac{A a^{4}}{x} \end{align*}

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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Fricas [A]  time = 1.58013, size = 257, normalized size = 2.4 \begin{align*} \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 \left (x\right ) - 840 \, A a^{4}}{840 \, x} \end{align*}

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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Sympy [A]  time = 0.449545, size = 112, normalized size = 1.05 \begin{align*} - \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{\left (x \right )} + 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{align*}

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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Giac [A]  time = 1.13519, size = 132, normalized size = 1.23 \begin{align*} \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{align*}

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