3.1.2 \(\int x^3 (A+B x) (b x+c x^2) \, dx\) [2]

Optimal. Leaf size=33 \[ \frac {1}{5} A b x^5+\frac {1}{6} (b B+A c) x^6+\frac {1}{7} B c x^7 \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 33, 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 = {779} \begin {gather*} \frac {1}{6} x^6 (A c+b B)+\frac {1}{5} A b x^5+\frac {1}{7} B c x^7 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3*(A + B*x)*(b*x + c*x^2),x]

[Out]

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

Rule 779

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

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Mathematica [A]
time = 0.00, size = 33, normalized size = 1.00 \begin {gather*} \frac {1}{5} A b x^5+\frac {1}{6} (b B+A c) x^6+\frac {1}{7} B c x^7 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.10, size = 28, normalized size = 0.85

method	result	size
gosper	\(\frac {x^{5} \left (30 B c \,x^{2}+35 A c x +35 b B x +42 A b \right )}{210}\)	\(28\)
default	\(\frac {A b \,x^{5}}{5}+\frac {\left (A c +B b \right ) x^{6}}{6}+\frac {B c \,x^{7}}{7}\)	\(28\)
norman	\(\frac {B c \,x^{7}}{7}+\left (\frac {A c}{6}+\frac {B b}{6}\right ) x^{6}+\frac {A b \,x^{5}}{5}\)	\(29\)
risch	\(\frac {1}{5} A b \,x^{5}+\frac {1}{6} A c \,x^{6}+\frac {1}{6} b B \,x^{6}+\frac {1}{7} B c \,x^{7}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(B*x+A)*(c*x^2+b*x),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.29, size = 27, normalized size = 0.82 \begin {gather*} \frac {1}{7} \, B c x^{7} + \frac {1}{5} \, A b x^{5} + \frac {1}{6} \, {\left (B b + A c\right )} x^{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 3.76, size = 27, normalized size = 0.82 \begin {gather*} \frac {1}{7} \, B c x^{7} + \frac {1}{5} \, A b x^{5} + \frac {1}{6} \, {\left (B b + A c\right )} x^{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.01, size = 29, normalized size = 0.88 \begin {gather*} \frac {A b x^{5}}{5} + \frac {B c x^{7}}{7} + x^{6} \left (\frac {A c}{6} + \frac {B b}{6}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 1.29, size = 29, normalized size = 0.88 \begin {gather*} \frac {1}{7} \, B c x^{7} + \frac {1}{6} \, B b x^{6} + \frac {1}{6} \, A c x^{6} + \frac {1}{5} \, A b x^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.04, size = 28, normalized size = 0.85 \begin {gather*} \frac {B\,c\,x^7}{7}+\left (\frac {A\,c}{6}+\frac {B\,b}{6}\right )\,x^6+\frac {A\,b\,x^5}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(b*x + c*x^2)*(A + B*x),x)

[Out]

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

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