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

Optimal. Leaf size=110 \[ \frac{3}{2} a^2 A c^2 x^4+2 a^3 A c x^2+a^4 A \log (x)+\frac{6}{5} a^2 B c^2 x^5+\frac{4}{3} a^3 B c x^3+a^4 B x+\frac{2}{3} a A c^3 x^6+\frac{4}{7} a B c^3 x^7+\frac{1}{8} A c^4 x^8+\frac{1}{9} B c^4 x^9 \]

[Out]

a^4*B*x + 2*a^3*A*c*x^2 + (4*a^3*B*c*x^3)/3 + (3*a^2*A*c^2*x^4)/2 + (6*a^2*B*c^2*x^5)/5 + (2*a*A*c^3*x^6)/3 +
(4*a*B*c^3*x^7)/7 + (A*c^4*x^8)/8 + (B*c^4*x^9)/9 + a^4*A*Log[x]

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Rubi [A]  time = 0.052258, antiderivative size = 110, 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} \[ \frac{3}{2} a^2 A c^2 x^4+2 a^3 A c x^2+a^4 A \log (x)+\frac{6}{5} a^2 B c^2 x^5+\frac{4}{3} a^3 B c x^3+a^4 B x+\frac{2}{3} a A c^3 x^6+\frac{4}{7} a B c^3 x^7+\frac{1}{8} A c^4 x^8+\frac{1}{9} B c^4 x^9 \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + c*x^2)^4)/x,x]

[Out]

a^4*B*x + 2*a^3*A*c*x^2 + (4*a^3*B*c*x^3)/3 + (3*a^2*A*c^2*x^4)/2 + (6*a^2*B*c^2*x^5)/5 + (2*a*A*c^3*x^6)/3 +
(4*a*B*c^3*x^7)/7 + (A*c^4*x^8)/8 + (B*c^4*x^9)/9 + a^4*A*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} \, dx &=\int \left (a^4 B+\frac{a^4 A}{x}+4 a^3 A c x+4 a^3 B c x^2+6 a^2 A c^2 x^3+6 a^2 B c^2 x^4+4 a A c^3 x^5+4 a B c^3 x^6+A c^4 x^7+B c^4 x^8\right ) \, dx\\ &=a^4 B x+2 a^3 A c x^2+\frac{4}{3} a^3 B c x^3+\frac{3}{2} a^2 A c^2 x^4+\frac{6}{5} a^2 B c^2 x^5+\frac{2}{3} a A c^3 x^6+\frac{4}{7} a B c^3 x^7+\frac{1}{8} A c^4 x^8+\frac{1}{9} B c^4 x^9+a^4 A \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0071966, size = 110, normalized size = 1. \[ \frac{3}{2} a^2 A c^2 x^4+2 a^3 A c x^2+a^4 A \log (x)+\frac{6}{5} a^2 B c^2 x^5+\frac{4}{3} a^3 B c x^3+a^4 B x+\frac{2}{3} a A c^3 x^6+\frac{4}{7} a B c^3 x^7+\frac{1}{8} A c^4 x^8+\frac{1}{9} B c^4 x^9 \]

Antiderivative was successfully verified.

[In]

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

[Out]

a^4*B*x + 2*a^3*A*c*x^2 + (4*a^3*B*c*x^3)/3 + (3*a^2*A*c^2*x^4)/2 + (6*a^2*B*c^2*x^5)/5 + (2*a*A*c^3*x^6)/3 +
(4*a*B*c^3*x^7)/7 + (A*c^4*x^8)/8 + (B*c^4*x^9)/9 + a^4*A*Log[x]

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Maple [A]  time = 0.003, size = 97, normalized size = 0.9 \begin{align*}{a}^{4}Bx+2\,{a}^{3}Ac{x}^{2}+{\frac{4\,{a}^{3}Bc{x}^{3}}{3}}+{\frac{3\,{a}^{2}A{c}^{2}{x}^{4}}{2}}+{\frac{6\,{a}^{2}B{c}^{2}{x}^{5}}{5}}+{\frac{2\,aA{c}^{3}{x}^{6}}{3}}+{\frac{4\,aB{c}^{3}{x}^{7}}{7}}+{\frac{A{c}^{4}{x}^{8}}{8}}+{\frac{B{c}^{4}{x}^{9}}{9}}+{a}^{4}A\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,x)

[Out]

a^4*B*x+2*a^3*A*c*x^2+4/3*a^3*B*c*x^3+3/2*a^2*A*c^2*x^4+6/5*a^2*B*c^2*x^5+2/3*a*A*c^3*x^6+4/7*a*B*c^3*x^7+1/8*
A*c^4*x^8+1/9*B*c^4*x^9+a^4*A*ln(x)

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Maxima [A]  time = 1.08801, size = 130, normalized size = 1.18 \begin{align*} \frac{1}{9} \, B c^{4} x^{9} + \frac{1}{8} \, A c^{4} x^{8} + \frac{4}{7} \, B a c^{3} x^{7} + \frac{2}{3} \, A a c^{3} x^{6} + \frac{6}{5} \, B a^{2} c^{2} x^{5} + \frac{3}{2} \, A a^{2} c^{2} x^{4} + \frac{4}{3} \, B a^{3} c x^{3} + 2 \, A a^{3} c x^{2} + B a^{4} x + A a^{4} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*B*c^4*x^9 + 1/8*A*c^4*x^8 + 4/7*B*a*c^3*x^7 + 2/3*A*a*c^3*x^6 + 6/5*B*a^2*c^2*x^5 + 3/2*A*a^2*c^2*x^4 + 4/
3*B*a^3*c*x^3 + 2*A*a^3*c*x^2 + B*a^4*x + A*a^4*log(x)

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Fricas [A]  time = 1.55654, size = 224, normalized size = 2.04 \begin{align*} \frac{1}{9} \, B c^{4} x^{9} + \frac{1}{8} \, A c^{4} x^{8} + \frac{4}{7} \, B a c^{3} x^{7} + \frac{2}{3} \, A a c^{3} x^{6} + \frac{6}{5} \, B a^{2} c^{2} x^{5} + \frac{3}{2} \, A a^{2} c^{2} x^{4} + \frac{4}{3} \, B a^{3} c x^{3} + 2 \, A a^{3} c x^{2} + B a^{4} x + A a^{4} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*B*c^4*x^9 + 1/8*A*c^4*x^8 + 4/7*B*a*c^3*x^7 + 2/3*A*a*c^3*x^6 + 6/5*B*a^2*c^2*x^5 + 3/2*A*a^2*c^2*x^4 + 4/
3*B*a^3*c*x^3 + 2*A*a^3*c*x^2 + B*a^4*x + A*a^4*log(x)

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Sympy [A]  time = 0.423524, size = 117, normalized size = 1.06 \begin{align*} A a^{4} \log{\left (x \right )} + 2 A a^{3} c x^{2} + \frac{3 A a^{2} c^{2} x^{4}}{2} + \frac{2 A a c^{3} x^{6}}{3} + \frac{A c^{4} x^{8}}{8} + B a^{4} x + \frac{4 B a^{3} c x^{3}}{3} + \frac{6 B a^{2} c^{2} x^{5}}{5} + \frac{4 B a c^{3} x^{7}}{7} + \frac{B c^{4} x^{9}}{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*a**4*log(x) + 2*A*a**3*c*x**2 + 3*A*a**2*c**2*x**4/2 + 2*A*a*c**3*x**6/3 + A*c**4*x**8/8 + B*a**4*x + 4*B*a*
*3*c*x**3/3 + 6*B*a**2*c**2*x**5/5 + 4*B*a*c**3*x**7/7 + B*c**4*x**9/9

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Giac [A]  time = 1.18686, size = 131, normalized size = 1.19 \begin{align*} \frac{1}{9} \, B c^{4} x^{9} + \frac{1}{8} \, A c^{4} x^{8} + \frac{4}{7} \, B a c^{3} x^{7} + \frac{2}{3} \, A a c^{3} x^{6} + \frac{6}{5} \, B a^{2} c^{2} x^{5} + \frac{3}{2} \, A a^{2} c^{2} x^{4} + \frac{4}{3} \, B a^{3} c x^{3} + 2 \, A a^{3} c x^{2} + B a^{4} x + A a^{4} \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*B*c^4*x^9 + 1/8*A*c^4*x^8 + 4/7*B*a*c^3*x^7 + 2/3*A*a*c^3*x^6 + 6/5*B*a^2*c^2*x^5 + 3/2*A*a^2*c^2*x^4 + 4/
3*B*a^3*c*x^3 + 2*A*a^3*c*x^2 + B*a^4*x + A*a^4*log(abs(x))