3.3.60 \(\int x (A+B x) (a+c x^2)^2 \, dx\) [260]

Optimal. Leaf size=65 \[ \frac {1}{2} a^2 A x^2+\frac {1}{3} a^2 B x^3+\frac {1}{2} a A c x^4+\frac {2}{5} a B c x^5+\frac {1}{6} A c^2 x^6+\frac {1}{7} B c^2 x^7 \]

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 65, 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 = {780} \begin {gather*} \frac {1}{2} a^2 A x^2+\frac {1}{3} a^2 B x^3+\frac {1}{2} a A c x^4+\frac {2}{5} a B c x^5+\frac {1}{6} A c^2 x^6+\frac {1}{7} B c^2 x^7 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*(A + B*x)*(a + c*x^2)^2,x]

[Out]

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

Rule 780

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

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Mathematica [A]
time = 0.00, size = 65, normalized size = 1.00 \begin {gather*} \frac {1}{2} a^2 A x^2+\frac {1}{3} a^2 B x^3+\frac {1}{2} a A c x^4+\frac {2}{5} a B c x^5+\frac {1}{6} A c^2 x^6+\frac {1}{7} B c^2 x^7 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.57, size = 54, normalized size = 0.83

method	result	size
gosper	\(\frac {1}{2} a^{2} A \,x^{2}+\frac {1}{3} a^{2} B \,x^{3}+\frac {1}{2} a A c \,x^{4}+\frac {2}{5} a B c \,x^{5}+\frac {1}{6} A \,c^{2} x^{6}+\frac {1}{7} B \,c^{2} x^{7}\)	\(54\)
default	\(\frac {1}{2} a^{2} A \,x^{2}+\frac {1}{3} a^{2} B \,x^{3}+\frac {1}{2} a A c \,x^{4}+\frac {2}{5} a B c \,x^{5}+\frac {1}{6} A \,c^{2} x^{6}+\frac {1}{7} B \,c^{2} x^{7}\)	\(54\)
norman	\(\frac {1}{2} a^{2} A \,x^{2}+\frac {1}{3} a^{2} B \,x^{3}+\frac {1}{2} a A c \,x^{4}+\frac {2}{5} a B c \,x^{5}+\frac {1}{6} A \,c^{2} x^{6}+\frac {1}{7} B \,c^{2} x^{7}\)	\(54\)
risch	\(\frac {1}{2} a^{2} A \,x^{2}+\frac {1}{3} a^{2} B \,x^{3}+\frac {1}{2} a A c \,x^{4}+\frac {2}{5} a B c \,x^{5}+\frac {1}{6} A \,c^{2} x^{6}+\frac {1}{7} B \,c^{2} x^{7}\)	\(54\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 53, normalized size = 0.82 \begin {gather*} \frac {1}{7} \, B c^{2} x^{7} + \frac {1}{6} \, A c^{2} x^{6} + \frac {2}{5} \, B a c x^{5} + \frac {1}{2} \, A a c x^{4} + \frac {1}{3} \, B a^{2} x^{3} + \frac {1}{2} \, A a^{2} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 1.88, size = 53, normalized size = 0.82 \begin {gather*} \frac {1}{7} \, B c^{2} x^{7} + \frac {1}{6} \, A c^{2} x^{6} + \frac {2}{5} \, B a c x^{5} + \frac {1}{2} \, A a c x^{4} + \frac {1}{3} \, B a^{2} x^{3} + \frac {1}{2} \, A a^{2} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.02, size = 61, normalized size = 0.94 \begin {gather*} \frac {A a^{2} x^{2}}{2} + \frac {A a c x^{4}}{2} + \frac {A c^{2} x^{6}}{6} + \frac {B a^{2} x^{3}}{3} + \frac {2 B a c x^{5}}{5} + \frac {B c^{2} x^{7}}{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.84, size = 53, normalized size = 0.82 \begin {gather*} \frac {1}{7} \, B c^{2} x^{7} + \frac {1}{6} \, A c^{2} x^{6} + \frac {2}{5} \, B a c x^{5} + \frac {1}{2} \, A a c x^{4} + \frac {1}{3} \, B a^{2} x^{3} + \frac {1}{2} \, A a^{2} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.02, size = 53, normalized size = 0.82 \begin {gather*} \frac {B\,a^2\,x^3}{3}+\frac {A\,a^2\,x^2}{2}+\frac {2\,B\,a\,c\,x^5}{5}+\frac {A\,a\,c\,x^4}{2}+\frac {B\,c^2\,x^7}{7}+\frac {A\,c^2\,x^6}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + c*x^2)^2*(A + B*x),x)

[Out]

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

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