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

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

[Out]

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

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Rubi [A]  time = 0.08, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {766} \[ \frac {3}{4} a^2 A c x^4+\frac {1}{2} a^3 A x^2+\frac {3}{5} a^2 B c x^5+\frac {1}{3} a^3 B x^3+\frac {1}{2} a A c^2 x^6+\frac {3}{7} a B c^2 x^7+\frac {1}{8} A c^3 x^8+\frac {1}{9} B c^3 x^9 \]

Antiderivative was successfully verified.

[In]

Int[x*(A + B*x)*(a + c*x^2)^3,x]

[Out]

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

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

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

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.52, size = 77, normalized size = 0.83 \[ \frac {1}{9} x^{9} c^{3} B + \frac {1}{8} x^{8} c^{3} A + \frac {3}{7} x^{7} c^{2} a B + \frac {1}{2} x^{6} c^{2} a A + \frac {3}{5} x^{5} c a^{2} B + \frac {3}{4} x^{4} c a^{2} A + \frac {1}{3} x^{3} a^{3} B + \frac {1}{2} x^{2} a^{3} A \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.05, size = 78, normalized size = 0.84 \[ \frac {1}{9} B \,c^{3} x^{9}+\frac {1}{8} A \,c^{3} x^{8}+\frac {3}{7} B a \,c^{2} x^{7}+\frac {1}{2} A a \,c^{2} x^{6}+\frac {3}{5} B \,a^{2} c \,x^{5}+\frac {3}{4} A \,a^{2} c \,x^{4}+\frac {1}{3} B \,a^{3} x^{3}+\frac {1}{2} A \,a^{3} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.56, size = 77, normalized size = 0.83 \[ \frac {1}{9} \, B c^{3} x^{9} + \frac {1}{8} \, A c^{3} x^{8} + \frac {3}{7} \, B a c^{2} x^{7} + \frac {1}{2} \, A a c^{2} x^{6} + \frac {3}{5} \, B a^{2} c x^{5} + \frac {3}{4} \, A a^{2} c x^{4} + \frac {1}{3} \, B a^{3} x^{3} + \frac {1}{2} \, A a^{3} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + c*x^2)^3*(A + B*x),x)

[Out]

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

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sympy [A]  time = 0.08, size = 92, normalized size = 0.99 \[ \frac {A a^{3} x^{2}}{2} + \frac {3 A a^{2} c x^{4}}{4} + \frac {A a c^{2} x^{6}}{2} + \frac {A c^{3} x^{8}}{8} + \frac {B a^{3} x^{3}}{3} + \frac {3 B a^{2} c x^{5}}{5} + \frac {3 B a c^{2} x^{7}}{7} + \frac {B c^{3} x^{9}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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