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

Optimal. Leaf size=149 \[ \frac {1}{3} a^3 A x^3+\frac {1}{4} a^3 B x^4+\frac {1}{5} a^2 x^5 (a C+3 A b)+\frac {1}{6} a^2 x^6 (a D+3 b B)+\frac {1}{9} b^2 x^9 (3 a C+A b)+\frac {3}{7} a b x^7 (a C+A b)+\frac {1}{10} b^2 x^{10} (3 a D+b B)+\frac {3}{8} a b x^8 (a D+b B)+\frac {1}{11} b^3 C x^{11}+\frac {1}{12} b^3 D x^{12} \]

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Rubi [A]  time = 0.14, antiderivative size = 149, 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}{5} a^2 x^5 (a C+3 A b)+\frac {1}{3} a^3 A x^3+\frac {1}{6} a^2 x^6 (a D+3 b B)+\frac {1}{4} a^3 B x^4+\frac {1}{9} b^2 x^9 (3 a C+A b)+\frac {3}{7} a b x^7 (a C+A b)+\frac {1}{10} b^2 x^{10} (3 a D+b B)+\frac {3}{8} a b x^8 (a D+b B)+\frac {1}{11} b^3 C x^{11}+\frac {1}{12} b^3 D x^{12} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*A*x^3)/3 + (a^3*B*x^4)/4 + (a^2*(3*A*b + a*C)*x^5)/5 + (a^2*(3*b*B + a*D)*x^6)/6 + (3*a*b*(A*b + a*C)*x^7
)/7 + (3*a*b*(b*B + a*D)*x^8)/8 + (b^2*(A*b + 3*a*C)*x^9)/9 + (b^2*(b*B + 3*a*D)*x^10)/10 + (b^3*C*x^11)/11 +
(b^3*D*x^12)/12

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 )^3 \left (A+B x+C x^2+D x^3\right ) \, dx &=\int \left (a^3 A x^2+a^3 B x^3+a^2 (3 A b+a C) x^4+a^2 (3 b B+a D) x^5+3 a b (A b+a C) x^6+3 a b (b B+a D) x^7+b^2 (A b+3 a C) x^8+b^2 (b B+3 a D) x^9+b^3 C x^{10}+b^3 D x^{11}\right ) \, dx\\ &=\frac {1}{3} a^3 A x^3+\frac {1}{4} a^3 B x^4+\frac {1}{5} a^2 (3 A b+a C) x^5+\frac {1}{6} a^2 (3 b B+a D) x^6+\frac {3}{7} a b (A b+a C) x^7+\frac {3}{8} a b (b B+a D) x^8+\frac {1}{9} b^2 (A b+3 a C) x^9+\frac {1}{10} b^2 (b B+3 a D) x^{10}+\frac {1}{11} b^3 C x^{11}+\frac {1}{12} b^3 D x^{12}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 125, normalized size = 0.84 \begin {gather*} \frac {462 a^3 x^3 (20 A+x (15 B+2 x (6 C+5 D x)))+99 a^2 b x^5 (168 A+5 x (28 B+3 x (8 C+7 D x)))+33 a b^2 x^7 (360 A+7 x (45 B+4 x (10 C+9 D x)))+14 b^3 x^9 \left (220 A+3 x \left (66 B+60 C x+55 D x^2\right )\right )}{27720} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(14*b^3*x^9*(220*A + 3*x*(66*B + 60*C*x + 55*D*x^2)) + 462*a^3*x^3*(20*A + x*(15*B + 2*x*(6*C + 5*D*x))) + 99*
a^2*b*x^5*(168*A + 5*x*(28*B + 3*x*(8*C + 7*D*x))) + 33*a*b^2*x^7*(360*A + 7*x*(45*B + 4*x*(10*C + 9*D*x))))/2
7720

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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 )^3 \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)^3*(A + B*x + C*x^2 + D*x^3),x]

[Out]

IntegrateAlgebraic[x^2*(a + b*x^2)^3*(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)^3*(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 = 0.46, size = 153, normalized size = 1.03 \begin {gather*} \frac {1}{12} \, D b^{3} x^{12} + \frac {1}{11} \, C b^{3} x^{11} + \frac {3}{10} \, D a b^{2} x^{10} + \frac {1}{10} \, B b^{3} x^{10} + \frac {1}{3} \, C a b^{2} x^{9} + \frac {1}{9} \, A b^{3} x^{9} + \frac {3}{8} \, D a^{2} b x^{8} + \frac {3}{8} \, B a b^{2} x^{8} + \frac {3}{7} \, C a^{2} b x^{7} + \frac {3}{7} \, A a b^{2} x^{7} + \frac {1}{6} \, D a^{3} x^{6} + \frac {1}{2} \, B a^{2} b x^{6} + \frac {1}{5} \, C a^{3} x^{5} + \frac {3}{5} \, A a^{2} b x^{5} + \frac {1}{4} \, B a^{3} x^{4} + \frac {1}{3} \, A a^{3} x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*D*b^3*x^12 + 1/11*C*b^3*x^11 + 3/10*D*a*b^2*x^10 + 1/10*B*b^3*x^10 + 1/3*C*a*b^2*x^9 + 1/9*A*b^3*x^9 + 3/
8*D*a^2*b*x^8 + 3/8*B*a*b^2*x^8 + 3/7*C*a^2*b*x^7 + 3/7*A*a*b^2*x^7 + 1/6*D*a^3*x^6 + 1/2*B*a^2*b*x^6 + 1/5*C*
a^3*x^5 + 3/5*A*a^2*b*x^5 + 1/4*B*a^3*x^4 + 1/3*A*a^3*x^3

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maple [A]  time = 0.00, size = 150, normalized size = 1.01 \begin {gather*} \frac {D b^{3} x^{12}}{12}+\frac {C \,b^{3} x^{11}}{11}+\frac {\left (b^{3} B +3 a \,b^{2} D\right ) x^{10}}{10}+\frac {\left (A \,b^{3}+3 a \,b^{2} C \right ) x^{9}}{9}+\frac {B \,a^{3} x^{4}}{4}+\frac {\left (3 a \,b^{2} B +3 a^{2} b D\right ) x^{8}}{8}+\frac {A \,a^{3} x^{3}}{3}+\frac {\left (3 a \,b^{2} A +3 a^{2} b C \right ) x^{7}}{7}+\frac {\left (3 a^{2} b B +a^{3} D\right ) x^{6}}{6}+\frac {\left (3 A \,a^{2} b +a^{3} 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)^3*(D*x^3+C*x^2+B*x+A),x)

[Out]

1/12*b^3*D*x^12+1/11*b^3*C*x^11+1/10*(B*b^3+3*D*a*b^2)*x^10+1/9*(A*b^3+3*C*a*b^2)*x^9+1/8*(3*B*a*b^2+3*D*a^2*b
)*x^8+1/7*(3*A*a*b^2+3*C*a^2*b)*x^7+1/6*(3*B*a^2*b+D*a^3)*x^6+1/5*(3*A*a^2*b+C*a^3)*x^5+1/4*a^3*B*x^4+1/3*a^3*
A*x^3

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maxima [A]  time = 1.36, size = 145, normalized size = 0.97 \begin {gather*} \frac {1}{12} \, D b^{3} x^{12} + \frac {1}{11} \, C b^{3} x^{11} + \frac {1}{10} \, {\left (3 \, D a b^{2} + B b^{3}\right )} x^{10} + \frac {1}{9} \, {\left (3 \, C a b^{2} + A b^{3}\right )} x^{9} + \frac {3}{8} \, {\left (D a^{2} b + B a b^{2}\right )} x^{8} + \frac {1}{4} \, B a^{3} x^{4} + \frac {3}{7} \, {\left (C a^{2} b + A a b^{2}\right )} x^{7} + \frac {1}{3} \, A a^{3} x^{3} + \frac {1}{6} \, {\left (D a^{3} + 3 \, B a^{2} b\right )} x^{6} + \frac {1}{5} \, {\left (C a^{3} + 3 \, A a^{2} b\right )} x^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*D*b^3*x^12 + 1/11*C*b^3*x^11 + 1/10*(3*D*a*b^2 + B*b^3)*x^10 + 1/9*(3*C*a*b^2 + A*b^3)*x^9 + 3/8*(D*a^2*b
+ B*a*b^2)*x^8 + 1/4*B*a^3*x^4 + 3/7*(C*a^2*b + A*a*b^2)*x^7 + 1/3*A*a^3*x^3 + 1/6*(D*a^3 + 3*B*a^2*b)*x^6 +
1/5*(C*a^3 + 3*A*a^2*b)*x^5

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mupad [B]  time = 1.28, size = 153, normalized size = 1.03 \begin {gather*} \frac {A\,a^3\,x^3}{3}+\frac {B\,a^3\,x^4}{4}+\frac {A\,b^3\,x^9}{9}+\frac {C\,a^3\,x^5}{5}+\frac {B\,b^3\,x^{10}}{10}+\frac {C\,b^3\,x^{11}}{11}+\frac {a^3\,x^6\,D}{6}+\frac {b^3\,x^{12}\,D}{12}+\frac {3\,a^2\,b\,x^8\,D}{8}+\frac {3\,a\,b^2\,x^{10}\,D}{10}+\frac {3\,A\,a^2\,b\,x^5}{5}+\frac {3\,A\,a\,b^2\,x^7}{7}+\frac {B\,a^2\,b\,x^6}{2}+\frac {3\,B\,a\,b^2\,x^8}{8}+\frac {3\,C\,a^2\,b\,x^7}{7}+\frac {C\,a\,b^2\,x^9}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(A*a^3*x^3)/3 + (B*a^3*x^4)/4 + (A*b^3*x^9)/9 + (C*a^3*x^5)/5 + (B*b^3*x^10)/10 + (C*b^3*x^11)/11 + (a^3*x^6*D
)/6 + (b^3*x^12*D)/12 + (3*a^2*b*x^8*D)/8 + (3*a*b^2*x^10*D)/10 + (3*A*a^2*b*x^5)/5 + (3*A*a*b^2*x^7)/7 + (B*a
^2*b*x^6)/2 + (3*B*a*b^2*x^8)/8 + (3*C*a^2*b*x^7)/7 + (C*a*b^2*x^9)/3

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sympy [A]  time = 0.14, size = 165, normalized size = 1.11 \begin {gather*} \frac {A a^{3} x^{3}}{3} + \frac {B a^{3} x^{4}}{4} + \frac {C b^{3} x^{11}}{11} + \frac {D b^{3} x^{12}}{12} + x^{10} \left (\frac {B b^{3}}{10} + \frac {3 D a b^{2}}{10}\right ) + x^{9} \left (\frac {A b^{3}}{9} + \frac {C a b^{2}}{3}\right ) + x^{8} \left (\frac {3 B a b^{2}}{8} + \frac {3 D a^{2} b}{8}\right ) + x^{7} \left (\frac {3 A a b^{2}}{7} + \frac {3 C a^{2} b}{7}\right ) + x^{6} \left (\frac {B a^{2} b}{2} + \frac {D a^{3}}{6}\right ) + x^{5} \left (\frac {3 A a^{2} b}{5} + \frac {C a^{3}}{5}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*a**3*x**3/3 + B*a**3*x**4/4 + C*b**3*x**11/11 + D*b**3*x**12/12 + x**10*(B*b**3/10 + 3*D*a*b**2/10) + x**9*(
A*b**3/9 + C*a*b**2/3) + x**8*(3*B*a*b**2/8 + 3*D*a**2*b/8) + x**7*(3*A*a*b**2/7 + 3*C*a**2*b/7) + x**6*(B*a**
2*b/2 + D*a**3/6) + x**5*(3*A*a**2*b/5 + C*a**3/5)

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