3.278 \(\int \frac{(A+B x) (a+c x^2)^4}{x^3} \, dx\)

Optimal. Leaf size=105 \[ 3 a^2 A c^2 x^2+4 a^3 A c \log (x)-\frac{a^4 A}{2 x^2}+2 a^2 B c^2 x^3+4 a^3 B c x-\frac{a^4 B}{x}+a A c^3 x^4+\frac{4}{5} a B c^3 x^5+\frac{1}{6} A c^4 x^6+\frac{1}{7} B c^4 x^7 \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0548459, antiderivative size = 105, 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} \[ 3 a^2 A c^2 x^2+4 a^3 A c \log (x)-\frac{a^4 A}{2 x^2}+2 a^2 B c^2 x^3+4 a^3 B c x-\frac{a^4 B}{x}+a A c^3 x^4+\frac{4}{5} a B c^3 x^5+\frac{1}{6} A c^4 x^6+\frac{1}{7} B c^4 x^7 \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + c*x^2)^4)/x^3,x]

[Out]

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

Mathematica [A]  time = 0.0071459, size = 105, normalized size = 1. \[ 3 a^2 A c^2 x^2+4 a^3 A c \log (x)-\frac{a^4 A}{2 x^2}+2 a^2 B c^2 x^3+4 a^3 B c x-\frac{a^4 B}{x}+a A c^3 x^4+\frac{4}{5} a B c^3 x^5+\frac{1}{6} A c^4 x^6+\frac{1}{7} B c^4 x^7 \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + c*x^2)^4)/x^3,x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 98, normalized size = 0.9 \begin{align*} -{\frac{A{a}^{4}}{2\,{x}^{2}}}-{\frac{B{a}^{4}}{x}}+4\,{a}^{3}Bcx+3\,{a}^{2}A{c}^{2}{x}^{2}+2\,{a}^{2}B{c}^{2}{x}^{3}+aA{c}^{3}{x}^{4}+{\frac{4\,aB{c}^{3}{x}^{5}}{5}}+{\frac{A{c}^{4}{x}^{6}}{6}}+{\frac{B{c}^{4}{x}^{7}}{7}}+4\,{a}^{3}Ac\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^3,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.03021, size = 131, normalized size = 1.25 \begin{align*} \frac{1}{7} \, B c^{4} x^{7} + \frac{1}{6} \, A c^{4} x^{6} + \frac{4}{5} \, B a c^{3} x^{5} + A a c^{3} x^{4} + 2 \, B a^{2} c^{2} x^{3} + 3 \, A a^{2} c^{2} x^{2} + 4 \, B a^{3} c x + 4 \, A a^{3} c \log \left (x\right ) - \frac{2 \, B a^{4} x + A a^{4}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^4/x^3,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.53623, size = 251, normalized size = 2.39 \begin{align*} \frac{30 \, B c^{4} x^{9} + 35 \, A c^{4} x^{8} + 168 \, B a c^{3} x^{7} + 210 \, A a c^{3} x^{6} + 420 \, B a^{2} c^{2} x^{5} + 630 \, A a^{2} c^{2} x^{4} + 840 \, B a^{3} c x^{3} + 840 \, A a^{3} c x^{2} \log \left (x\right ) - 210 \, B a^{4} x - 105 \, A a^{4}}{210 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^4/x^3,x, algorithm="fricas")

[Out]

1/210*(30*B*c^4*x^9 + 35*A*c^4*x^8 + 168*B*a*c^3*x^7 + 210*A*a*c^3*x^6 + 420*B*a^2*c^2*x^5 + 630*A*a^2*c^2*x^4
+ 840*B*a^3*c*x^3 + 840*A*a^3*c*x^2*log(x) - 210*B*a^4*x - 105*A*a^4)/x^2

________________________________________________________________________________________

Sympy [A]  time = 0.578763, size = 110, normalized size = 1.05 \begin{align*} 4 A a^{3} c \log{\left (x \right )} + 3 A a^{2} c^{2} x^{2} + A a c^{3} x^{4} + \frac{A c^{4} x^{6}}{6} + 4 B a^{3} c x + 2 B a^{2} c^{2} x^{3} + \frac{4 B a c^{3} x^{5}}{5} + \frac{B c^{4} x^{7}}{7} - \frac{A a^{4} + 2 B a^{4} x}{2 x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+a)**4/x**3,x)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.12201, size = 132, normalized size = 1.26 \begin{align*} \frac{1}{7} \, B c^{4} x^{7} + \frac{1}{6} \, A c^{4} x^{6} + \frac{4}{5} \, B a c^{3} x^{5} + A a c^{3} x^{4} + 2 \, B a^{2} c^{2} x^{3} + 3 \, A a^{2} c^{2} x^{2} + 4 \, B a^{3} c x + 4 \, A a^{3} c \log \left ({\left | x \right |}\right ) - \frac{2 \, B a^{4} x + A a^{4}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^4/x^3,x, algorithm="giac")

[Out]

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