3.3.65 \(\int x^3 (A+B x) (a+c x^2)^3 \, dx\)

Optimal. Leaf size=93 \[ \frac {1}{4} a^3 A x^4+\frac {1}{5} a^3 B x^5+\frac {1}{2} a^2 A c x^6+\frac {3}{7} a^2 B c x^7+\frac {3}{8} a A c^2 x^8+\frac {1}{3} a B c^2 x^9+\frac {1}{10} A c^3 x^{10}+\frac {1}{11} B c^3 x^{11} \]

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Rubi [A]  time = 0.10, antiderivative size = 93, 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 {1}{2} a^2 A c x^6+\frac {1}{4} a^3 A x^4+\frac {3}{7} a^2 B c x^7+\frac {1}{5} a^3 B x^5+\frac {3}{8} a A c^2 x^8+\frac {1}{3} a B c^2 x^9+\frac {1}{10} A c^3 x^{10}+\frac {1}{11} B c^3 x^{11} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3*(A + B*x)*(a + c*x^2)^3,x]

[Out]

(a^3*A*x^4)/4 + (a^3*B*x^5)/5 + (a^2*A*c*x^6)/2 + (3*a^2*B*c*x^7)/7 + (3*a*A*c^2*x^8)/8 + (a*B*c^2*x^9)/3 + (A
*c^3*x^10)/10 + (B*c^3*x^11)/11

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 x^3 (A+B x) \left (a+c x^2\right )^3 \, dx &=\int \left (a^3 A x^3+a^3 B x^4+3 a^2 A c x^5+3 a^2 B c x^6+3 a A c^2 x^7+3 a B c^2 x^8+A c^3 x^9+B c^3 x^{10}\right ) \, dx\\ &=\frac {1}{4} a^3 A x^4+\frac {1}{5} a^3 B x^5+\frac {1}{2} a^2 A c x^6+\frac {3}{7} a^2 B c x^7+\frac {3}{8} a A c^2 x^8+\frac {1}{3} a B c^2 x^9+\frac {1}{10} A c^3 x^{10}+\frac {1}{11} B c^3 x^{11}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 93, normalized size = 1.00 \begin {gather*} \frac {1}{4} a^3 A x^4+\frac {1}{5} a^3 B x^5+\frac {1}{2} a^2 A c x^6+\frac {3}{7} a^2 B c x^7+\frac {3}{8} a A c^2 x^8+\frac {1}{3} a B c^2 x^9+\frac {1}{10} A c^3 x^{10}+\frac {1}{11} B c^3 x^{11} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*A*x^4)/4 + (a^3*B*x^5)/5 + (a^2*A*c*x^6)/2 + (3*a^2*B*c*x^7)/7 + (3*a*A*c^2*x^8)/8 + (a*B*c^2*x^9)/3 + (A
*c^3*x^10)/10 + (B*c^3*x^11)/11

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.35, size = 77, normalized size = 0.83 \begin {gather*} \frac {1}{11} x^{11} c^{3} B + \frac {1}{10} x^{10} c^{3} A + \frac {1}{3} x^{9} c^{2} a B + \frac {3}{8} x^{8} c^{2} a A + \frac {3}{7} x^{7} c a^{2} B + \frac {1}{2} x^{6} c a^{2} A + \frac {1}{5} x^{5} a^{3} B + \frac {1}{4} x^{4} a^{3} A \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*x^11*c^3*B + 1/10*x^10*c^3*A + 1/3*x^9*c^2*a*B + 3/8*x^8*c^2*a*A + 3/7*x^7*c*a^2*B + 1/2*x^6*c*a^2*A + 1/
5*x^5*a^3*B + 1/4*x^4*a^3*A

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giac [A]  time = 0.15, size = 77, normalized size = 0.83 \begin {gather*} \frac {1}{11} \, B c^{3} x^{11} + \frac {1}{10} \, A c^{3} x^{10} + \frac {1}{3} \, B a c^{2} x^{9} + \frac {3}{8} \, A a c^{2} x^{8} + \frac {3}{7} \, B a^{2} c x^{7} + \frac {1}{2} \, A a^{2} c x^{6} + \frac {1}{5} \, B a^{3} x^{5} + \frac {1}{4} \, A a^{3} x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*B*c^3*x^11 + 1/10*A*c^3*x^10 + 1/3*B*a*c^2*x^9 + 3/8*A*a*c^2*x^8 + 3/7*B*a^2*c*x^7 + 1/2*A*a^2*c*x^6 + 1/
5*B*a^3*x^5 + 1/4*A*a^3*x^4

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maple [A]  time = 0.04, size = 78, normalized size = 0.84 \begin {gather*} \frac {1}{11} B \,c^{3} x^{11}+\frac {1}{10} A \,c^{3} x^{10}+\frac {1}{3} B a \,c^{2} x^{9}+\frac {3}{8} A a \,c^{2} x^{8}+\frac {3}{7} B \,a^{2} c \,x^{7}+\frac {1}{2} A \,a^{2} c \,x^{6}+\frac {1}{5} B \,a^{3} x^{5}+\frac {1}{4} A \,a^{3} x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*A*a^3*x^4+1/5*B*a^3*x^5+1/2*a^2*A*c*x^6+3/7*a^2*B*c*x^7+3/8*a*A*c^2*x^8+1/3*a*B*c^2*x^9+1/10*A*c^3*x^10+1/
11*B*c^3*x^11

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maxima [A]  time = 0.57, size = 77, normalized size = 0.83 \begin {gather*} \frac {1}{11} \, B c^{3} x^{11} + \frac {1}{10} \, A c^{3} x^{10} + \frac {1}{3} \, B a c^{2} x^{9} + \frac {3}{8} \, A a c^{2} x^{8} + \frac {3}{7} \, B a^{2} c x^{7} + \frac {1}{2} \, A a^{2} c x^{6} + \frac {1}{5} \, B a^{3} x^{5} + \frac {1}{4} \, A a^{3} x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*B*c^3*x^11 + 1/10*A*c^3*x^10 + 1/3*B*a*c^2*x^9 + 3/8*A*a*c^2*x^8 + 3/7*B*a^2*c*x^7 + 1/2*A*a^2*c*x^6 + 1/
5*B*a^3*x^5 + 1/4*A*a^3*x^4

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mupad [B]  time = 0.03, size = 77, normalized size = 0.83 \begin {gather*} \frac {B\,a^3\,x^5}{5}+\frac {A\,a^3\,x^4}{4}+\frac {3\,B\,a^2\,c\,x^7}{7}+\frac {A\,a^2\,c\,x^6}{2}+\frac {B\,a\,c^2\,x^9}{3}+\frac {3\,A\,a\,c^2\,x^8}{8}+\frac {B\,c^3\,x^{11}}{11}+\frac {A\,c^3\,x^{10}}{10} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a + c*x^2)^3*(A + B*x),x)

[Out]

(A*a^3*x^4)/4 + (B*a^3*x^5)/5 + (A*c^3*x^10)/10 + (B*c^3*x^11)/11 + (A*a^2*c*x^6)/2 + (3*A*a*c^2*x^8)/8 + (3*B
*a^2*c*x^7)/7 + (B*a*c^2*x^9)/3

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sympy [A]  time = 0.08, size = 90, normalized size = 0.97 \begin {gather*} \frac {A a^{3} x^{4}}{4} + \frac {A a^{2} c x^{6}}{2} + \frac {3 A a c^{2} x^{8}}{8} + \frac {A c^{3} x^{10}}{10} + \frac {B a^{3} x^{5}}{5} + \frac {3 B a^{2} c x^{7}}{7} + \frac {B a c^{2} x^{9}}{3} + \frac {B c^{3} x^{11}}{11} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*a**3*x**4/4 + A*a**2*c*x**6/2 + 3*A*a*c**2*x**8/8 + A*c**3*x**10/10 + B*a**3*x**5/5 + 3*B*a**2*c*x**7/7 + B*
a*c**2*x**9/3 + B*c**3*x**11/11

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