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

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

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 81, 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*} \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 \end {gather*}

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 = 0.01, size = 81, normalized size = 1.00 \begin {gather*} a^3 A \log (x)+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 \end {gather*}

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]

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(A+B x) \left (a+c x^2\right )^3}{x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.39, size = 71, normalized size = 0.88 \begin {gather*} \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 \relax (x) \end {gather*}

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)

________________________________________________________________________________________

giac [A]  time = 0.15, size = 72, normalized size = 0.89 \begin {gather*} \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 {gather*}

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))

________________________________________________________________________________________

maple [A]  time = 0.05, size = 72, normalized size = 0.89 \begin {gather*} \frac {B \,c^{3} x^{7}}{7}+\frac {A \,c^{3} x^{6}}{6}+\frac {3 B a \,c^{2} x^{5}}{5}+\frac {3 A a \,c^{2} x^{4}}{4}+B \,a^{2} c \,x^{3}+\frac {3 A \,a^{2} c \,x^{2}}{2}+A \,a^{3} \ln \relax (x )+B \,a^{3} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

B*a^3*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*a^3*ln(x)

________________________________________________________________________________________

maxima [A]  time = 0.52, size = 71, normalized size = 0.88 \begin {gather*} \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 \relax (x) \end {gather*}

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)

________________________________________________________________________________________

mupad [B]  time = 0.04, size = 71, normalized size = 0.88 \begin {gather*} \frac {A\,c^3\,x^6}{6}+\frac {B\,c^3\,x^7}{7}+A\,a^3\,\ln \relax (x)+B\,a^3\,x+\frac {3\,A\,a^2\,c\,x^2}{2}+\frac {3\,A\,a\,c^2\,x^4}{4}+B\,a^2\,c\,x^3+\frac {3\,B\,a\,c^2\,x^5}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + c*x^2)^3*(A + B*x))/x,x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.18, size = 85, normalized size = 1.05 \begin {gather*} A a^{3} \log {\relax (x )} + \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 {gather*}

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

________________________________________________________________________________________