3.3.26 \(\int (b x+c x^2)^3 \, dx\)

Optimal. Leaf size=43 \[ \frac {b^3 x^4}{4}+\frac {3}{5} b^2 c x^5+\frac {1}{2} b c^2 x^6+\frac {c^3 x^7}{7} \]

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Rubi [A]  time = 0.01, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {611} \begin {gather*} \frac {3}{5} b^2 c x^5+\frac {b^3 x^4}{4}+\frac {1}{2} b c^2 x^6+\frac {c^3 x^7}{7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*x + c*x^2)^3,x]

[Out]

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

Rule 611

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x + c*x^2)^p, x], x] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && (EqQ[a, 0] ||  !PerfectSquareQ[b^2 - 4*a*c])

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 43, normalized size = 1.00 \begin {gather*} \frac {b^3 x^4}{4}+\frac {3}{5} b^2 c x^5+\frac {1}{2} b c^2 x^6+\frac {c^3 x^7}{7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b*x + c*x^2)^3,x]

[Out]

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

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

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(b*x + c*x^2)^3,x]

[Out]

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

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fricas [A]  time = 0.34, size = 35, normalized size = 0.81 \begin {gather*} \frac {1}{7} x^{7} c^{3} + \frac {1}{2} x^{6} c^{2} b + \frac {3}{5} x^{5} c b^{2} + \frac {1}{4} x^{4} b^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.18, size = 35, normalized size = 0.81 \begin {gather*} \frac {1}{7} \, c^{3} x^{7} + \frac {1}{2} \, b c^{2} x^{6} + \frac {3}{5} \, b^{2} c x^{5} + \frac {1}{4} \, b^{3} x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.04, size = 36, normalized size = 0.84 \begin {gather*} \frac {1}{7} c^{3} x^{7}+\frac {1}{2} b \,c^{2} x^{6}+\frac {3}{5} b^{2} c \,x^{5}+\frac {1}{4} b^{3} x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.40, size = 35, normalized size = 0.81 \begin {gather*} \frac {1}{7} \, c^{3} x^{7} + \frac {1}{2} \, b c^{2} x^{6} + \frac {3}{5} \, b^{2} c x^{5} + \frac {1}{4} \, b^{3} x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.04, size = 35, normalized size = 0.81 \begin {gather*} \frac {b^3\,x^4}{4}+\frac {3\,b^2\,c\,x^5}{5}+\frac {b\,c^2\,x^6}{2}+\frac {c^3\,x^7}{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x + c*x^2)^3,x)

[Out]

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

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sympy [A]  time = 0.07, size = 37, normalized size = 0.86 \begin {gather*} \frac {b^{3} x^{4}}{4} + \frac {3 b^{2} c x^{5}}{5} + \frac {b c^{2} x^{6}}{2} + \frac {c^{3} x^{7}}{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b**3*x**4/4 + 3*b**2*c*x**5/5 + b*c**2*x**6/2 + c**3*x**7/7

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