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

Optimal. Leaf size=138 \[ \frac {1}{3} a^3 B x^3+\frac {1}{4} a^3 C x^4+\frac {1}{5} a^2 x^5 (a D+3 b B)+\frac {1}{2} a^2 b C x^6+\frac {A \left (a+b x^2\right )^4}{8 b}+\frac {1}{9} b^2 x^9 (3 a D+b B)+\frac {3}{8} a b^2 C x^8+\frac {3}{7} a b x^7 (a D+b B)+\frac {1}{10} b^3 C x^{10}+\frac {1}{11} b^3 D x^{11} \]

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Rubi [A]  time = 0.09, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1582, 1810} \begin {gather*} \frac {1}{5} a^2 x^5 (a D+3 b B)+\frac {1}{2} a^2 b C x^6+\frac {1}{3} a^3 B x^3+\frac {1}{4} a^3 C x^4+\frac {A \left (a+b x^2\right )^4}{8 b}+\frac {1}{9} b^2 x^9 (3 a D+b B)+\frac {3}{8} a b^2 C x^8+\frac {3}{7} a b x^7 (a D+b B)+\frac {1}{10} b^3 C x^{10}+\frac {1}{11} b^3 D x^{11} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 1582

Int[(Px_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(Coeff[Px, x, n - 1]*(a + b*x^n)^(p + 1))/(b*n*(p +
1)), x] + Int[(Px - Coeff[Px, x, n - 1]*x^(n - 1))*(a + b*x^n)^p, x] /; FreeQ[{a, b}, x] && PolyQ[Px, x] && I
GtQ[p, 1] && IGtQ[n, 1] && NeQ[Coeff[Px, x, n - 1], 0] && NeQ[Px, Coeff[Px, x, n - 1]*x^(n - 1)] &&  !MatchQ[P
x, (Qx_.)*((c_) + (d_.)*x^(m_))^(q_) /; FreeQ[{c, d}, x] && PolyQ[Qx, x] && IGtQ[q, 1] && IGtQ[m, 1] && NeQ[Co
eff[Qx*(a + b*x^n)^p, x, m - 1], 0] && GtQ[m*q, n*p]]

Rule 1810

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

Rubi steps

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

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Mathematica [A]  time = 0.06, size = 124, normalized size = 0.90 \begin {gather*} \frac {462 a^3 x^2 (30 A+x (20 B+3 x (5 C+4 D x)))+198 a^2 b x^4 (105 A+2 x (42 B+5 x (7 C+6 D x)))+165 a b^2 x^6 (84 A+x (72 B+7 x (9 C+8 D x)))+7 b^3 x^8 \left (495 A+4 x \left (110 B+99 C x+90 D x^2\right )\right )}{27720} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(7*b^3*x^8*(495*A + 4*x*(110*B + 99*C*x + 90*D*x^2)) + 462*a^3*x^2*(30*A + x*(20*B + 3*x*(5*C + 4*D*x))) + 198
*a^2*b*x^4*(105*A + 2*x*(42*B + 5*x*(7*C + 6*D*x))) + 165*a*b^2*x^6*(84*A + x*(72*B + 7*x*(9*C + 8*D*x))))/277
20

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

[Out]

IntegrateAlgebraic[x*(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*(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 = 0.36, size = 153, normalized size = 1.11 \begin {gather*} \frac {1}{11} \, D b^{3} x^{11} + \frac {1}{10} \, C b^{3} x^{10} + \frac {1}{3} \, D a b^{2} x^{9} + \frac {1}{9} \, B b^{3} x^{9} + \frac {3}{8} \, C a b^{2} x^{8} + \frac {1}{8} \, A b^{3} x^{8} + \frac {3}{7} \, D a^{2} b x^{7} + \frac {3}{7} \, B a b^{2} x^{7} + \frac {1}{2} \, C a^{2} b x^{6} + \frac {1}{2} \, A a b^{2} x^{6} + \frac {1}{5} \, D a^{3} x^{5} + \frac {3}{5} \, B a^{2} b x^{5} + \frac {1}{4} \, C a^{3} x^{4} + \frac {3}{4} \, A a^{2} b x^{4} + \frac {1}{3} \, B a^{3} x^{3} + \frac {1}{2} \, A a^{3} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 150, normalized size = 1.09 \begin {gather*} \frac {D b^{3} x^{11}}{11}+\frac {C \,b^{3} x^{10}}{10}+\frac {\left (b^{3} B +3 a \,b^{2} D\right ) x^{9}}{9}+\frac {\left (A \,b^{3}+3 a \,b^{2} C \right ) x^{8}}{8}+\frac {B \,a^{3} x^{3}}{3}+\frac {\left (3 a \,b^{2} B +3 a^{2} b D\right ) x^{7}}{7}+\frac {A \,a^{3} x^{2}}{2}+\frac {\left (3 a \,b^{2} A +3 a^{2} b C \right ) x^{6}}{6}+\frac {\left (3 a^{2} b B +a^{3} D\right ) x^{5}}{5}+\frac {\left (3 A \,a^{2} b +a^{3} C \right ) x^{4}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.34, size = 145, normalized size = 1.05 \begin {gather*} \frac {1}{11} \, D b^{3} x^{11} + \frac {1}{10} \, C b^{3} x^{10} + \frac {1}{9} \, {\left (3 \, D a b^{2} + B b^{3}\right )} x^{9} + \frac {1}{8} \, {\left (3 \, C a b^{2} + A b^{3}\right )} x^{8} + \frac {3}{7} \, {\left (D a^{2} b + B a b^{2}\right )} x^{7} + \frac {1}{3} \, B a^{3} x^{3} + \frac {1}{2} \, {\left (C a^{2} b + A a b^{2}\right )} x^{6} + \frac {1}{2} \, A a^{3} x^{2} + \frac {1}{5} \, {\left (D a^{3} + 3 \, B a^{2} b\right )} x^{5} + \frac {1}{4} \, {\left (C a^{3} + 3 \, A a^{2} b\right )} x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.28, size = 153, normalized size = 1.11 \begin {gather*} \frac {A\,a^3\,x^2}{2}+\frac {B\,a^3\,x^3}{3}+\frac {A\,b^3\,x^8}{8}+\frac {C\,a^3\,x^4}{4}+\frac {B\,b^3\,x^9}{9}+\frac {C\,b^3\,x^{10}}{10}+\frac {a^3\,x^5\,D}{5}+\frac {b^3\,x^{11}\,D}{11}+\frac {3\,a^2\,b\,x^7\,D}{7}+\frac {a\,b^2\,x^9\,D}{3}+\frac {3\,A\,a^2\,b\,x^4}{4}+\frac {A\,a\,b^2\,x^6}{2}+\frac {3\,B\,a^2\,b\,x^5}{5}+\frac {3\,B\,a\,b^2\,x^7}{7}+\frac {C\,a^2\,b\,x^6}{2}+\frac {3\,C\,a\,b^2\,x^8}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.14, size = 163, normalized size = 1.18 \begin {gather*} \frac {A a^{3} x^{2}}{2} + \frac {B a^{3} x^{3}}{3} + \frac {C b^{3} x^{10}}{10} + \frac {D b^{3} x^{11}}{11} + x^{9} \left (\frac {B b^{3}}{9} + \frac {D a b^{2}}{3}\right ) + x^{8} \left (\frac {A b^{3}}{8} + \frac {3 C a b^{2}}{8}\right ) + x^{7} \left (\frac {3 B a b^{2}}{7} + \frac {3 D a^{2} b}{7}\right ) + x^{6} \left (\frac {A a b^{2}}{2} + \frac {C a^{2} b}{2}\right ) + x^{5} \left (\frac {3 B a^{2} b}{5} + \frac {D a^{3}}{5}\right ) + x^{4} \left (\frac {3 A a^{2} b}{4} + \frac {C a^{3}}{4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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