3.3.74 \(\int x (A+B x) (a+c x^2)^4 \, dx\)

Optimal. Leaf size=115 \[ \frac {1}{2} a^4 A x^2+\frac {1}{3} a^4 B x^3+a^3 A c x^4+\frac {4}{5} a^3 B c x^5+a^2 A c^2 x^6+\frac {6}{7} a^2 B c^2 x^7+\frac {1}{2} a A c^3 x^8+\frac {4}{9} a B c^3 x^9+\frac {1}{10} A c^4 x^{10}+\frac {1}{11} B c^4 x^{11} \]

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Rubi [A]  time = 0.12, antiderivative size = 115, 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} \begin {gather*} a^2 A c^2 x^6+a^3 A c x^4+\frac {1}{2} a^4 A x^2+\frac {6}{7} a^2 B c^2 x^7+\frac {4}{5} a^3 B c x^5+\frac {1}{3} a^4 B x^3+\frac {1}{2} a A c^3 x^8+\frac {4}{9} a B c^3 x^9+\frac {1}{10} A c^4 x^{10}+\frac {1}{11} B c^4 x^{11} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*(A + B*x)*(a + c*x^2)^4,x]

[Out]

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

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Mathematica [A]  time = 0.00, size = 115, normalized size = 1.00 \begin {gather*} \frac {1}{2} a^4 A x^2+\frac {1}{3} a^4 B x^3+a^3 A c x^4+\frac {4}{5} a^3 B c x^5+a^2 A c^2 x^6+\frac {6}{7} a^2 B c^2 x^7+\frac {1}{2} a A c^3 x^8+\frac {4}{9} a B c^3 x^9+\frac {1}{10} A c^4 x^{10}+\frac {1}{11} B c^4 x^{11} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*(A + B*x)*(a + c*x^2)^4,x]

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x (A+B x) \left (a+c x^2\right )^4 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x*(A + B*x)*(a + c*x^2)^4,x]

[Out]

IntegrateAlgebraic[x*(A + B*x)*(a + c*x^2)^4, x]

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fricas [A]  time = 0.36, size = 99, normalized size = 0.86 \begin {gather*} \frac {1}{11} x^{11} c^{4} B + \frac {1}{10} x^{10} c^{4} A + \frac {4}{9} x^{9} c^{3} a B + \frac {1}{2} x^{8} c^{3} a A + \frac {6}{7} x^{7} c^{2} a^{2} B + x^{6} c^{2} a^{2} A + \frac {4}{5} x^{5} c a^{3} B + x^{4} c a^{3} A + \frac {1}{3} x^{3} a^{4} B + \frac {1}{2} x^{2} a^{4} A \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(B*x+A)*(c*x^2+a)^4,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.19, size = 99, normalized size = 0.86 \begin {gather*} \frac {1}{11} \, B c^{4} x^{11} + \frac {1}{10} \, A c^{4} x^{10} + \frac {4}{9} \, B a c^{3} x^{9} + \frac {1}{2} \, A a c^{3} x^{8} + \frac {6}{7} \, B a^{2} c^{2} x^{7} + A a^{2} c^{2} x^{6} + \frac {4}{5} \, B a^{3} c x^{5} + A a^{3} c x^{4} + \frac {1}{3} \, B a^{4} x^{3} + \frac {1}{2} \, A a^{4} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(B*x+A)*(c*x^2+a)^4,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.05, size = 100, normalized size = 0.87 \begin {gather*} \frac {1}{11} B \,c^{4} x^{11}+\frac {1}{10} A \,c^{4} x^{10}+\frac {4}{9} B a \,c^{3} x^{9}+\frac {1}{2} A a \,c^{3} x^{8}+\frac {6}{7} B \,a^{2} c^{2} x^{7}+A \,a^{2} c^{2} x^{6}+\frac {4}{5} B \,a^{3} c \,x^{5}+A \,a^{3} c \,x^{4}+\frac {1}{3} B \,a^{4} x^{3}+\frac {1}{2} A \,a^{4} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(B*x+A)*(c*x^2+a)^4,x)

[Out]

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

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maxima [A]  time = 0.60, size = 99, normalized size = 0.86 \begin {gather*} \frac {1}{11} \, B c^{4} x^{11} + \frac {1}{10} \, A c^{4} x^{10} + \frac {4}{9} \, B a c^{3} x^{9} + \frac {1}{2} \, A a c^{3} x^{8} + \frac {6}{7} \, B a^{2} c^{2} x^{7} + A a^{2} c^{2} x^{6} + \frac {4}{5} \, B a^{3} c x^{5} + A a^{3} c x^{4} + \frac {1}{3} \, B a^{4} x^{3} + \frac {1}{2} \, A a^{4} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(B*x+A)*(c*x^2+a)^4,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.05, size = 99, normalized size = 0.86 \begin {gather*} \frac {B\,a^4\,x^3}{3}+\frac {A\,a^4\,x^2}{2}+\frac {4\,B\,a^3\,c\,x^5}{5}+A\,a^3\,c\,x^4+\frac {6\,B\,a^2\,c^2\,x^7}{7}+A\,a^2\,c^2\,x^6+\frac {4\,B\,a\,c^3\,x^9}{9}+\frac {A\,a\,c^3\,x^8}{2}+\frac {B\,c^4\,x^{11}}{11}+\frac {A\,c^4\,x^{10}}{10} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + c*x^2)^4*(A + B*x),x)

[Out]

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

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sympy [A]  time = 0.09, size = 116, normalized size = 1.01 \begin {gather*} \frac {A a^{4} x^{2}}{2} + A a^{3} c x^{4} + A a^{2} c^{2} x^{6} + \frac {A a c^{3} x^{8}}{2} + \frac {A c^{4} x^{10}}{10} + \frac {B a^{4} x^{3}}{3} + \frac {4 B a^{3} c x^{5}}{5} + \frac {6 B a^{2} c^{2} x^{7}}{7} + \frac {4 B a c^{3} x^{9}}{9} + \frac {B c^{4} x^{11}}{11} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(B*x+A)*(c*x**2+a)**4,x)

[Out]

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

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