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

Optimal. Leaf size=109 \[ \frac {1}{3} a^2 A x^3+\frac {1}{4} a^2 B x^4+\frac {1}{7} b x^7 (2 a C+A b)+\frac {1}{5} a x^5 (a C+2 A b)+\frac {1}{8} b x^8 (2 a D+b B)+\frac {1}{6} a x^6 (a D+2 b B)+\frac {1}{9} b^2 C x^9+\frac {1}{10} b^2 D x^{10} \]

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Rubi [A]  time = 0.11, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {1802} \begin {gather*} \frac {1}{3} a^2 A x^3+\frac {1}{4} a^2 B x^4+\frac {1}{7} b x^7 (2 a C+A b)+\frac {1}{5} a x^5 (a C+2 A b)+\frac {1}{8} b x^8 (2 a D+b B)+\frac {1}{6} a x^6 (a D+2 b B)+\frac {1}{9} b^2 C x^9+\frac {1}{10} b^2 D x^{10} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*A*x^3)/3 + (a^2*B*x^4)/4 + (a*(2*A*b + a*C)*x^5)/5 + (a*(2*b*B + a*D)*x^6)/6 + (b*(A*b + 2*a*C)*x^7)/7 +
(b*(b*B + 2*a*D)*x^8)/8 + (b^2*C*x^9)/9 + (b^2*D*x^10)/10

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

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Mathematica [A]  time = 0.07, size = 92, normalized size = 0.84 \begin {gather*} \frac {42 a^2 x^3 (20 A+x (15 B+2 x (6 C+5 D x)))+6 a b x^5 (168 A+5 x (28 B+3 x (8 C+7 D x)))+b^2 x^7 (360 A+7 x (45 B+4 x (10 C+9 D x)))}{2520} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(42*a^2*x^3*(20*A + x*(15*B + 2*x*(6*C + 5*D*x))) + 6*a*b*x^5*(168*A + 5*x*(28*B + 3*x*(8*C + 7*D*x))) + b^2*x
^7*(360*A + 7*x*(45*B + 4*x*(10*C + 9*D*x))))/2520

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x^2+a)^2*(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 = 0.37, size = 105, normalized size = 0.96 \begin {gather*} \frac {1}{10} \, D b^{2} x^{10} + \frac {1}{9} \, C b^{2} x^{9} + \frac {1}{4} \, D a b x^{8} + \frac {1}{8} \, B b^{2} x^{8} + \frac {2}{7} \, C a b x^{7} + \frac {1}{7} \, A b^{2} x^{7} + \frac {1}{6} \, D a^{2} x^{6} + \frac {1}{3} \, B a b x^{6} + \frac {1}{5} \, C a^{2} x^{5} + \frac {2}{5} \, A a b x^{5} + \frac {1}{4} \, B a^{2} x^{4} + \frac {1}{3} \, A a^{2} x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*D*b^2*x^10 + 1/9*C*b^2*x^9 + 1/4*D*a*b*x^8 + 1/8*B*b^2*x^8 + 2/7*C*a*b*x^7 + 1/7*A*b^2*x^7 + 1/6*D*a^2*x^
6 + 1/3*B*a*b*x^6 + 1/5*C*a^2*x^5 + 2/5*A*a*b*x^5 + 1/4*B*a^2*x^4 + 1/3*A*a^2*x^3

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maple [A]  time = 0.00, size = 102, normalized size = 0.94 \begin {gather*} \frac {D b^{2} x^{10}}{10}+\frac {C \,b^{2} x^{9}}{9}+\frac {\left (b^{2} B +2 a b D\right ) x^{8}}{8}+\frac {B \,a^{2} x^{4}}{4}+\frac {\left (b^{2} A +2 a b C \right ) x^{7}}{7}+\frac {A \,a^{2} x^{3}}{3}+\frac {\left (2 a b B +a^{2} D\right ) x^{6}}{6}+\frac {\left (2 A a b +a^{2} C \right ) x^{5}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*b^2*D*x^10+1/9*b^2*C*x^9+1/8*(B*b^2+2*D*a*b)*x^8+1/7*(A*b^2+2*C*a*b)*x^7+1/6*(2*B*a*b+D*a^2)*x^6+1/5*(2*A
*a*b+C*a^2)*x^5+1/4*a^2*B*x^4+1/3*A*a^2*x^3

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maxima [A]  time = 1.30, size = 101, normalized size = 0.93 \begin {gather*} \frac {1}{10} \, D b^{2} x^{10} + \frac {1}{9} \, C b^{2} x^{9} + \frac {1}{8} \, {\left (2 \, D a b + B b^{2}\right )} x^{8} + \frac {1}{7} \, {\left (2 \, C a b + A b^{2}\right )} x^{7} + \frac {1}{4} \, B a^{2} x^{4} + \frac {1}{6} \, {\left (D a^{2} + 2 \, B a b\right )} x^{6} + \frac {1}{3} \, A a^{2} x^{3} + \frac {1}{5} \, {\left (C a^{2} + 2 \, A a b\right )} x^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*D*b^2*x^10 + 1/9*C*b^2*x^9 + 1/8*(2*D*a*b + B*b^2)*x^8 + 1/7*(2*C*a*b + A*b^2)*x^7 + 1/4*B*a^2*x^4 + 1/6*
(D*a^2 + 2*B*a*b)*x^6 + 1/3*A*a^2*x^3 + 1/5*(C*a^2 + 2*A*a*b)*x^5

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mupad [B]  time = 1.11, size = 108, normalized size = 0.99 \begin {gather*} \frac {a^2\,x^6\,D}{6}+\frac {b^2\,x^{10}\,D}{10}+\frac {A\,x^3\,\left (35\,a^2+42\,a\,b\,x^2+15\,b^2\,x^4\right )}{105}+\frac {B\,x^4\,\left (6\,a^2+8\,a\,b\,x^2+3\,b^2\,x^4\right )}{24}+\frac {C\,x^5\,\left (63\,a^2+90\,a\,b\,x^2+35\,b^2\,x^4\right )}{315}+\frac {a\,b\,x^8\,D}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a^2*x^6*D)/6 + (b^2*x^10*D)/10 + (A*x^3*(35*a^2 + 15*b^2*x^4 + 42*a*b*x^2))/105 + (B*x^4*(6*a^2 + 3*b^2*x^4 +
8*a*b*x^2))/24 + (C*x^5*(63*a^2 + 35*b^2*x^4 + 90*a*b*x^2))/315 + (a*b*x^8*D)/4

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sympy [A]  time = 0.14, size = 110, normalized size = 1.01 \begin {gather*} \frac {A a^{2} x^{3}}{3} + \frac {B a^{2} x^{4}}{4} + \frac {C b^{2} x^{9}}{9} + \frac {D b^{2} x^{10}}{10} + x^{8} \left (\frac {B b^{2}}{8} + \frac {D a b}{4}\right ) + x^{7} \left (\frac {A b^{2}}{7} + \frac {2 C a b}{7}\right ) + x^{6} \left (\frac {B a b}{3} + \frac {D a^{2}}{6}\right ) + x^{5} \left (\frac {2 A a b}{5} + \frac {C a^{2}}{5}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*a**2*x**3/3 + B*a**2*x**4/4 + C*b**2*x**9/9 + D*b**2*x**10/10 + x**8*(B*b**2/8 + D*a*b/4) + x**7*(A*b**2/7 +
2*C*a*b/7) + x**6*(B*a*b/3 + D*a**2/6) + x**5*(2*A*a*b/5 + C*a**2/5)

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