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

Optimal. Leaf size=81 \[ \frac{3}{2} a^2 A c x^2+a^3 A \log (x)+a^2 B c x^3+a^3 B x+\frac{3}{4} a A c^2 x^4+\frac{3}{5} a B c^2 x^5+\frac{1}{6} A c^3 x^6+\frac{1}{7} B c^3 x^7 \]

[Out]

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

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Rubi [A]  time = 0.0361952, antiderivative size = 81, 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} \[ \frac{3}{2} a^2 A c x^2+a^3 A \log (x)+a^2 B c x^3+a^3 B x+\frac{3}{4} a A c^2 x^4+\frac{3}{5} a B c^2 x^5+\frac{1}{6} A c^3 x^6+\frac{1}{7} B c^3 x^7 \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + c*x^2)^3)/x,x]

[Out]

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

Mathematica [A]  time = 0.0066101, size = 81, normalized size = 1. \[ \frac{3}{2} a^2 A c x^2+a^3 A \log (x)+a^2 B c x^3+a^3 B x+\frac{3}{4} a A c^2 x^4+\frac{3}{5} a B c^2 x^5+\frac{1}{6} A c^3 x^6+\frac{1}{7} B c^3 x^7 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.002, size = 72, normalized size = 0.9 \begin{align*}{a}^{3}Bx+{\frac{3\,{a}^{2}Ac{x}^{2}}{2}}+{a}^{2}Bc{x}^{3}+{\frac{3\,aA{c}^{2}{x}^{4}}{4}}+{\frac{3\,aB{c}^{2}{x}^{5}}{5}}+{\frac{A{c}^{3}{x}^{6}}{6}}+{\frac{B{c}^{3}{x}^{7}}{7}}+{a}^{3}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)^3/x,x)

[Out]

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

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Maxima [A]  time = 1.03982, size = 96, normalized size = 1.19 \begin{align*} \frac{1}{7} \, B c^{3} x^{7} + \frac{1}{6} \, A c^{3} x^{6} + \frac{3}{5} \, B a c^{2} x^{5} + \frac{3}{4} \, A a c^{2} x^{4} + B a^{2} c x^{3} + \frac{3}{2} \, A a^{2} c x^{2} + B a^{3} x + A a^{3} \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)^3/x,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.5257, size = 167, normalized size = 2.06 \begin{align*} \frac{1}{7} \, B c^{3} x^{7} + \frac{1}{6} \, A c^{3} x^{6} + \frac{3}{5} \, B a c^{2} x^{5} + \frac{3}{4} \, A a c^{2} x^{4} + B a^{2} c x^{3} + \frac{3}{2} \, A a^{2} c x^{2} + B a^{3} x + A a^{3} \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)^3/x,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.385519, size = 85, normalized size = 1.05 \begin{align*} A a^{3} \log{\left (x \right )} + \frac{3 A a^{2} c x^{2}}{2} + \frac{3 A a c^{2} x^{4}}{4} + \frac{A c^{3} x^{6}}{6} + B a^{3} x + B a^{2} c x^{3} + \frac{3 B a c^{2} x^{5}}{5} + \frac{B c^{3} x^{7}}{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+a)**3/x,x)

[Out]

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

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Giac [A]  time = 1.12998, size = 97, normalized size = 1.2 \begin{align*} \frac{1}{7} \, B c^{3} x^{7} + \frac{1}{6} \, A c^{3} x^{6} + \frac{3}{5} \, B a c^{2} x^{5} + \frac{3}{4} \, A a c^{2} x^{4} + B a^{2} c x^{3} + \frac{3}{2} \, A a^{2} c x^{2} + B a^{3} x + A a^{3} \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)^3/x,x, algorithm="giac")

[Out]

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