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

Optimal. Leaf size=75 \[ \frac {1}{3} A b^3 x^3+\frac {1}{4} b^2 x^4 (3 A c+b B)+\frac {1}{6} c^2 x^6 (A c+3 b B)+\frac {3}{5} b c x^5 (A c+b B)+\frac {1}{7} B c^3 x^7 \]

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {765} \begin {gather*} \frac {1}{4} b^2 x^4 (3 A c+b B)+\frac {1}{3} A b^3 x^3+\frac {1}{6} c^2 x^6 (A c+3 b B)+\frac {3}{5} b c x^5 (A c+b B)+\frac {1}{7} B c^3 x^7 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(b*x + c*x^2)^3)/x,x]

[Out]

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

Rule 765

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand
Integrand[(e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, e, f, g, m}, x] && IntegerQ[p] && (
GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 75, normalized size = 1.00 \begin {gather*} \frac {1}{3} A b^3 x^3+\frac {1}{4} b^2 x^4 (3 A c+b B)+\frac {1}{6} c^2 x^6 (A c+3 b B)+\frac {3}{5} b c x^5 (A c+b B)+\frac {1}{7} B c^3 x^7 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.02, size = 93, normalized size = 1.24 \begin {gather*} \frac {1}{3} A b^3 x^3+\frac {3}{4} A b^2 c x^4+\frac {3}{5} A b c^2 x^5+\frac {1}{6} A c^3 x^6+\frac {1}{4} b^3 B x^4+\frac {3}{5} b^2 B c x^5+\frac {1}{2} b B c^2 x^6+\frac {1}{7} B c^3 x^7 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.38, size = 73, normalized size = 0.97 \begin {gather*} \frac {1}{7} \, B c^{3} x^{7} + \frac {1}{3} \, A b^{3} x^{3} + \frac {1}{6} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + \frac {3}{5} \, {\left (B b^{2} c + A b c^{2}\right )} x^{5} + \frac {1}{4} \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.15, size = 77, normalized size = 1.03 \begin {gather*} \frac {1}{7} \, B c^{3} x^{7} + \frac {1}{2} \, B b c^{2} x^{6} + \frac {1}{6} \, A c^{3} x^{6} + \frac {3}{5} \, B b^{2} c x^{5} + \frac {3}{5} \, A b c^{2} x^{5} + \frac {1}{4} \, B b^{3} x^{4} + \frac {3}{4} \, A b^{2} c x^{4} + \frac {1}{3} \, A b^{3} x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*B*c^3*x^7 + 1/2*B*b*c^2*x^6 + 1/6*A*c^3*x^6 + 3/5*B*b^2*c*x^5 + 3/5*A*b*c^2*x^5 + 1/4*B*b^3*x^4 + 3/4*A*b^
2*c*x^4 + 1/3*A*b^3*x^3

________________________________________________________________________________________

maple [A]  time = 0.04, size = 76, normalized size = 1.01 \begin {gather*} \frac {B \,c^{3} x^{7}}{7}+\frac {A \,b^{3} x^{3}}{3}+\frac {\left (A \,c^{3}+3 B b \,c^{2}\right ) x^{6}}{6}+\frac {\left (3 A b \,c^{2}+3 B \,b^{2} c \right ) x^{5}}{5}+\frac {\left (3 A \,b^{2} c +b^{3} B \right ) x^{4}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.94, size = 73, normalized size = 0.97 \begin {gather*} \frac {1}{7} \, B c^{3} x^{7} + \frac {1}{3} \, A b^{3} x^{3} + \frac {1}{6} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + \frac {3}{5} \, {\left (B b^{2} c + A b c^{2}\right )} x^{5} + \frac {1}{4} \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.03, size = 69, normalized size = 0.92 \begin {gather*} x^4\,\left (\frac {B\,b^3}{4}+\frac {3\,A\,c\,b^2}{4}\right )+x^6\,\left (\frac {A\,c^3}{6}+\frac {B\,b\,c^2}{2}\right )+\frac {A\,b^3\,x^3}{3}+\frac {B\,c^3\,x^7}{7}+\frac {3\,b\,c\,x^5\,\left (A\,c+B\,b\right )}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((b*x + c*x^2)^3*(A + B*x))/x,x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.08, size = 82, normalized size = 1.09 \begin {gather*} \frac {A b^{3} x^{3}}{3} + \frac {B c^{3} x^{7}}{7} + x^{6} \left (\frac {A c^{3}}{6} + \frac {B b c^{2}}{2}\right ) + x^{5} \left (\frac {3 A b c^{2}}{5} + \frac {3 B b^{2} c}{5}\right ) + x^{4} \left (\frac {3 A b^{2} c}{4} + \frac {B b^{3}}{4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________