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

Optimal. Leaf size=62 \[ -\frac {(b+c x)^5 (2 b B-A c)}{5 c^3}+\frac {b (b+c x)^4 (b B-A c)}{4 c^3}+\frac {B (b+c x)^6}{6 c^3} \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 62, 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} \[ -\frac {(b+c x)^5 (2 b B-A c)}{5 c^3}+\frac {b (b+c x)^4 (b B-A c)}{4 c^3}+\frac {B (b+c x)^6}{6 c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]  time = 0.01, size = 67, normalized size = 1.08 \[ \frac {1}{60} x^2 \left (30 A b^3+20 b^2 x (3 A c+b B)+12 c^2 x^3 (A c+3 b B)+45 b c x^2 (A c+b B)+10 B c^3 x^4\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^2*(30*A*b^3 + 20*b^2*(b*B + 3*A*c)*x + 45*b*c*(b*B + A*c)*x^2 + 12*c^2*(3*b*B + A*c)*x^3 + 10*B*c^3*x^4))/6
0

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fricas [A]  time = 0.72, size = 73, normalized size = 1.18 \[ \frac {1}{6} \, B c^{3} x^{6} + \frac {1}{2} \, A b^{3} x^{2} + \frac {1}{5} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{5} + \frac {3}{4} \, {\left (B b^{2} c + A b c^{2}\right )} x^{4} + \frac {1}{3} \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.15, size = 76, normalized size = 1.23 \[ \frac {1}{6} \, B c^{3} x^{6} + \frac {3}{5} \, B b c^{2} x^{5} + \frac {1}{5} \, A c^{3} x^{5} + \frac {3}{4} \, B b^{2} c x^{4} + \frac {3}{4} \, A b c^{2} x^{4} + \frac {1}{3} \, B b^{3} x^{3} + A b^{2} c x^{3} + \frac {1}{2} \, A b^{3} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.05, size = 76, normalized size = 1.23 \[ \frac {B \,c^{3} x^{6}}{6}+\frac {A \,b^{3} x^{2}}{2}+\frac {\left (A \,c^{3}+3 B b \,c^{2}\right ) x^{5}}{5}+\frac {\left (3 A b \,c^{2}+3 B \,b^{2} c \right ) x^{4}}{4}+\frac {\left (3 A \,b^{2} c +b^{3} B \right ) x^{3}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.82, size = 73, normalized size = 1.18 \[ \frac {1}{6} \, B c^{3} x^{6} + \frac {1}{2} \, A b^{3} x^{2} + \frac {1}{5} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{5} + \frac {3}{4} \, {\left (B b^{2} c + A b c^{2}\right )} x^{4} + \frac {1}{3} \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.03, size = 68, normalized size = 1.10 \[ x^3\,\left (\frac {B\,b^3}{3}+A\,c\,b^2\right )+x^5\,\left (\frac {A\,c^3}{5}+\frac {3\,B\,b\,c^2}{5}\right )+\frac {A\,b^3\,x^2}{2}+\frac {B\,c^3\,x^6}{6}+\frac {3\,b\,c\,x^4\,\left (A\,c+B\,b\right )}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.08, size = 80, normalized size = 1.29 \[ \frac {A b^{3} x^{2}}{2} + \frac {B c^{3} x^{6}}{6} + x^{5} \left (\frac {A c^{3}}{5} + \frac {3 B b c^{2}}{5}\right ) + x^{4} \left (\frac {3 A b c^{2}}{4} + \frac {3 B b^{2} c}{4}\right ) + x^{3} \left (A b^{2} c + \frac {B b^{3}}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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