3.64 \(\int x (a+b x^2) (A+B x+C x^2+D x^3) \, dx\)

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

[Out]

1/2*a*A*x^2+1/3*a*B*x^3+1/4*(A*b+C*a)*x^4+1/5*(B*b+D*a)*x^5+1/6*b*C*x^6+1/7*b*D*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 = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {1802} \[ \frac {1}{4} x^4 (a C+A b)+\frac {1}{2} a A x^2+\frac {1}{5} x^5 (a D+b B)+\frac {1}{3} a B x^3+\frac {1}{6} b C x^6+\frac {1}{7} b D x^7 \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x^2)*(A + B*x + C*x^2 + D*x^3),x]

[Out]

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

Rule 1802

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a + b*x
^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 65, normalized size = 1.00 \[ \frac {1}{4} x^4 (a C+A b)+\frac {1}{2} a A x^2+\frac {1}{5} x^5 (a D+b B)+\frac {1}{3} a B x^3+\frac {1}{6} b C x^6+\frac {1}{7} b D x^7 \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*x^2)*(A + B*x + C*x^2 + D*x^3),x]

[Out]

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

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> keys do not match self's parent

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giac [A]  time = 0.32, size = 57, normalized size = 0.88 \[ \frac {1}{7} \, D b x^{7} + \frac {1}{6} \, C b x^{6} + \frac {1}{5} \, D a x^{5} + \frac {1}{5} \, B b x^{5} + \frac {1}{4} \, C a x^{4} + \frac {1}{4} \, A b x^{4} + \frac {1}{3} \, B a x^{3} + \frac {1}{2} \, A a x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*D*b*x^7 + 1/6*C*b*x^6 + 1/5*D*a*x^5 + 1/5*B*b*x^5 + 1/4*C*a*x^4 + 1/4*A*b*x^4 + 1/3*B*a*x^3 + 1/2*A*a*x^2

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maple [A]  time = 0.00, size = 54, normalized size = 0.83 \[ \frac {D b \,x^{7}}{7}+\frac {C b \,x^{6}}{6}+\frac {B a \,x^{3}}{3}+\frac {\left (b B +a D\right ) x^{5}}{5}+\frac {A a \,x^{2}}{2}+\frac {\left (A b +a C \right ) x^{4}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(b*x^2+a)*(D*x^3+C*x^2+B*x+A),x)

[Out]

1/2*a*A*x^2+1/3*a*B*x^3+1/4*(A*b+C*a)*x^4+1/5*(B*b+D*a)*x^5+1/6*b*C*x^6+1/7*b*D*x^7

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maxima [A]  time = 1.31, size = 53, normalized size = 0.82 \[ \frac {1}{7} \, D b x^{7} + \frac {1}{6} \, C b x^{6} + \frac {1}{5} \, {\left (D a + B b\right )} x^{5} + \frac {1}{3} \, B a x^{3} + \frac {1}{4} \, {\left (C a + A b\right )} x^{4} + \frac {1}{2} \, A a x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*D*b*x^7 + 1/6*C*b*x^6 + 1/5*(D*a + B*b)*x^5 + 1/3*B*a*x^3 + 1/4*(C*a + A*b)*x^4 + 1/2*A*a*x^2

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mupad [B]  time = 1.19, size = 57, normalized size = 0.88 \[ \frac {a\,x^5\,D}{5}+\frac {b\,x^7\,D}{7}+\frac {A\,a\,x^2}{2}+\frac {B\,a\,x^3}{3}+\frac {A\,b\,x^4}{4}+\frac {C\,a\,x^4}{4}+\frac {B\,b\,x^5}{5}+\frac {C\,b\,x^6}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + b*x^2)*(A + B*x + C*x^2 + x^3*D),x)

[Out]

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

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sympy [A]  time = 0.14, size = 60, normalized size = 0.92 \[ \frac {A a x^{2}}{2} + \frac {B a x^{3}}{3} + \frac {C b x^{6}}{6} + \frac {D b x^{7}}{7} + x^{5} \left (\frac {B b}{5} + \frac {D a}{5}\right ) + x^{4} \left (\frac {A b}{4} + \frac {C a}{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x**2+a)*(D*x**3+C*x**2+B*x+A),x)

[Out]

A*a*x**2/2 + B*a*x**3/3 + C*b*x**6/6 + D*b*x**7/7 + x**5*(B*b/5 + D*a/5) + x**4*(A*b/4 + C*a/4)

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