3.1.52 \(\int \frac {x^2 (A+B x+C x^2)}{(a+b x^2)^{9/2}} \, dx\)

Optimal. Leaf size=139 \[ \frac {2 x (3 a C+4 A b)}{105 a^3 b^2 \sqrt {a+b x^2}}+\frac {x (3 a C+4 A b)}{105 a^2 b^2 \left (a+b x^2\right )^{3/2}}-\frac {x (3 a C+4 A b)+2 a B}{35 a b^2 \left (a+b x^2\right )^{5/2}}-\frac {x^2 (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}} \]

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Rubi [A]  time = 0.13, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {1804, 778, 192, 191} \begin {gather*} \frac {2 x (3 a C+4 A b)}{105 a^3 b^2 \sqrt {a+b x^2}}+\frac {x (3 a C+4 A b)}{105 a^2 b^2 \left (a+b x^2\right )^{3/2}}-\frac {x (3 a C+4 A b)+2 a B}{35 a b^2 \left (a+b x^2\right )^{5/2}}-\frac {x^2 (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^2*(A + B*x + C*x^2))/(a + b*x^2)^(9/2),x]

[Out]

-(x^2*(a*B - (A*b - a*C)*x))/(7*a*b*(a + b*x^2)^(7/2)) - (2*a*B + (4*A*b + 3*a*C)*x)/(35*a*b^2*(a + b*x^2)^(5/
2)) + ((4*A*b + 3*a*C)*x)/(105*a^2*b^2*(a + b*x^2)^(3/2)) + (2*(4*A*b + 3*a*C)*x)/(105*a^3*b^2*Sqrt[a + b*x^2]
)

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 192

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, n, p}, x] && ILtQ[Simplify[1/n + p + 1
], 0] && NeQ[p, -1]

Rule 778

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((a*(e*f + d*g) -
(c*d*f - a*e*g)*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1)), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(2*a*c*(p + 1)),
Int[(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && LtQ[p, -1]

Rule 1804

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x
^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x
], x, 1]}, Simp[((c*x)^m*(a + b*x^2)^(p + 1)*(a*g - b*f*x))/(2*a*b*(p + 1)), x] + Dist[c/(2*a*b*(p + 1)), Int[
(c*x)^(m - 1)*(a + b*x^2)^(p + 1)*ExpandToSum[2*a*b*(p + 1)*x*Q - a*g*m + b*f*(m + 2*p + 3)*x, x], x], x]] /;
FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && GtQ[m, 0]

Rubi steps

\begin {align*} \int \frac {x^2 \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx &=-\frac {x^2 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac {\int \frac {x (-2 a B-(4 A b+3 a C) x)}{\left (a+b x^2\right )^{7/2}} \, dx}{7 a b}\\ &=-\frac {x^2 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac {2 a B+(4 A b+3 a C) x}{35 a b^2 \left (a+b x^2\right )^{5/2}}+\frac {(4 A b+3 a C) \int \frac {1}{\left (a+b x^2\right )^{5/2}} \, dx}{35 a b^2}\\ &=-\frac {x^2 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac {2 a B+(4 A b+3 a C) x}{35 a b^2 \left (a+b x^2\right )^{5/2}}+\frac {(4 A b+3 a C) x}{105 a^2 b^2 \left (a+b x^2\right )^{3/2}}+\frac {(2 (4 A b+3 a C)) \int \frac {1}{\left (a+b x^2\right )^{3/2}} \, dx}{105 a^2 b^2}\\ &=-\frac {x^2 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac {2 a B+(4 A b+3 a C) x}{35 a b^2 \left (a+b x^2\right )^{5/2}}+\frac {(4 A b+3 a C) x}{105 a^2 b^2 \left (a+b x^2\right )^{3/2}}+\frac {2 (4 A b+3 a C) x}{105 a^3 b^2 \sqrt {a+b x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.14, size = 87, normalized size = 0.63 \begin {gather*} \frac {-6 a^4 B-21 a^3 b B x^2+7 a^2 b^2 x^3 \left (5 A+3 C x^2\right )+2 a b^3 x^5 \left (14 A+3 C x^2\right )+8 A b^4 x^7}{105 a^3 b^2 \left (a+b x^2\right )^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^2*(A + B*x + C*x^2))/(a + b*x^2)^(9/2),x]

[Out]

(-6*a^4*B - 21*a^3*b*B*x^2 + 8*A*b^4*x^7 + 7*a^2*b^2*x^3*(5*A + 3*C*x^2) + 2*a*b^3*x^5*(14*A + 3*C*x^2))/(105*
a^3*b^2*(a + b*x^2)^(7/2))

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IntegrateAlgebraic [A]  time = 0.85, size = 91, normalized size = 0.65 \begin {gather*} \frac {-6 a^4 B-21 a^3 b B x^2+35 a^2 A b^2 x^3+21 a^2 b^2 C x^5+28 a A b^3 x^5+6 a b^3 C x^7+8 A b^4 x^7}{105 a^3 b^2 \left (a+b x^2\right )^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(x^2*(A + B*x + C*x^2))/(a + b*x^2)^(9/2),x]

[Out]

(-6*a^4*B - 21*a^3*b*B*x^2 + 35*a^2*A*b^2*x^3 + 28*a*A*b^3*x^5 + 21*a^2*b^2*C*x^5 + 8*A*b^4*x^7 + 6*a*b^3*C*x^
7)/(105*a^3*b^2*(a + b*x^2)^(7/2))

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fricas [A]  time = 0.99, size = 134, normalized size = 0.96 \begin {gather*} \frac {{\left (35 \, A a^{2} b^{2} x^{3} + 2 \, {\left (3 \, C a b^{3} + 4 \, A b^{4}\right )} x^{7} - 21 \, B a^{3} b x^{2} + 7 \, {\left (3 \, C a^{2} b^{2} + 4 \, A a b^{3}\right )} x^{5} - 6 \, B a^{4}\right )} \sqrt {b x^{2} + a}}{105 \, {\left (a^{3} b^{6} x^{8} + 4 \, a^{4} b^{5} x^{6} + 6 \, a^{5} b^{4} x^{4} + 4 \, a^{6} b^{3} x^{2} + a^{7} b^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(C*x^2+B*x+A)/(b*x^2+a)^(9/2),x, algorithm="fricas")

[Out]

1/105*(35*A*a^2*b^2*x^3 + 2*(3*C*a*b^3 + 4*A*b^4)*x^7 - 21*B*a^3*b*x^2 + 7*(3*C*a^2*b^2 + 4*A*a*b^3)*x^5 - 6*B
*a^4)*sqrt(b*x^2 + a)/(a^3*b^6*x^8 + 4*a^4*b^5*x^6 + 6*a^5*b^4*x^4 + 4*a^6*b^3*x^2 + a^7*b^2)

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giac [A]  time = 0.52, size = 94, normalized size = 0.68 \begin {gather*} \frac {{\left ({\left (x^{2} {\left (\frac {2 \, {\left (3 \, C a b^{4} + 4 \, A b^{5}\right )} x^{2}}{a^{3} b^{3}} + \frac {7 \, {\left (3 \, C a^{2} b^{3} + 4 \, A a b^{4}\right )}}{a^{3} b^{3}}\right )} + \frac {35 \, A}{a}\right )} x - \frac {21 \, B}{b}\right )} x^{2} - \frac {6 \, B a}{b^{2}}}{105 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(C*x^2+B*x+A)/(b*x^2+a)^(9/2),x, algorithm="giac")

[Out]

1/105*(((x^2*(2*(3*C*a*b^4 + 4*A*b^5)*x^2/(a^3*b^3) + 7*(3*C*a^2*b^3 + 4*A*a*b^4)/(a^3*b^3)) + 35*A/a)*x - 21*
B/b)*x^2 - 6*B*a/b^2)/(b*x^2 + a)^(7/2)

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maple [A]  time = 0.01, size = 88, normalized size = 0.63 \begin {gather*} \frac {8 A \,b^{4} x^{7}+6 C a \,b^{3} x^{7}+28 A \,x^{5} a \,b^{3}+21 C \,a^{2} b^{2} x^{5}+35 A \,x^{3} a^{2} b^{2}-21 B \,a^{3} b \,x^{2}-6 B \,a^{4}}{105 \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{3} b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(C*x^2+B*x+A)/(b*x^2+a)^(9/2),x)

[Out]

1/105*(8*A*b^4*x^7+6*C*a*b^3*x^7+28*A*a*b^3*x^5+21*C*a^2*b^2*x^5+35*A*a^2*b^2*x^3-21*B*a^3*b*x^2-6*B*a^4)/(b*x
^2+a)^(7/2)/a^3/b^2

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maxima [A]  time = 1.42, size = 197, normalized size = 1.42 \begin {gather*} -\frac {C x^{3}}{4 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} - \frac {B x^{2}}{5 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} + \frac {3 \, C x}{140 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{2}} + \frac {2 \, C x}{35 \, \sqrt {b x^{2} + a} a^{2} b^{2}} + \frac {C x}{35 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a b^{2}} - \frac {3 \, C a x}{28 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}} - \frac {A x}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} + \frac {8 \, A x}{105 \, \sqrt {b x^{2} + a} a^{3} b} + \frac {4 \, A x}{105 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2} b} + \frac {A x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a b} - \frac {2 \, B a}{35 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(C*x^2+B*x+A)/(b*x^2+a)^(9/2),x, algorithm="maxima")

[Out]

-1/4*C*x^3/((b*x^2 + a)^(7/2)*b) - 1/5*B*x^2/((b*x^2 + a)^(7/2)*b) + 3/140*C*x/((b*x^2 + a)^(5/2)*b^2) + 2/35*
C*x/(sqrt(b*x^2 + a)*a^2*b^2) + 1/35*C*x/((b*x^2 + a)^(3/2)*a*b^2) - 3/28*C*a*x/((b*x^2 + a)^(7/2)*b^2) - 1/7*
A*x/((b*x^2 + a)^(7/2)*b) + 8/105*A*x/(sqrt(b*x^2 + a)*a^3*b) + 4/105*A*x/((b*x^2 + a)^(3/2)*a^2*b) + 1/35*A*x
/((b*x^2 + a)^(5/2)*a*b) - 2/35*B*a/((b*x^2 + a)^(7/2)*b^2)

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mupad [B]  time = 1.09, size = 133, normalized size = 0.96 \begin {gather*} \frac {x\,\left (4\,A\,b+3\,C\,a\right )}{105\,a^2\,b^2\,{\left (b\,x^2+a\right )}^{3/2}}-\frac {\frac {B}{5\,b^2}+x\,\left (\frac {C}{5\,b^2}-\frac {A\,b-C\,a}{35\,a\,b^2}\right )}{{\left (b\,x^2+a\right )}^{5/2}}-\frac {x\,\left (\frac {A}{7\,b}-\frac {C\,a}{7\,b^2}\right )-\frac {B\,a}{7\,b^2}}{{\left (b\,x^2+a\right )}^{7/2}}+\frac {x\,\left (8\,A\,b+6\,C\,a\right )}{105\,a^3\,b^2\,\sqrt {b\,x^2+a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2*(A + B*x + C*x^2))/(a + b*x^2)^(9/2),x)

[Out]

(x*(4*A*b + 3*C*a))/(105*a^2*b^2*(a + b*x^2)^(3/2)) - (B/(5*b^2) + x*(C/(5*b^2) - (A*b - C*a)/(35*a*b^2)))/(a
+ b*x^2)^(5/2) - (x*(A/(7*b) - (C*a)/(7*b^2)) - (B*a)/(7*b^2))/(a + b*x^2)^(7/2) + (x*(8*A*b + 6*C*a))/(105*a^
3*b^2*(a + b*x^2)^(1/2))

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sympy [B]  time = 118.65, size = 904, normalized size = 6.50 \begin {gather*} A \left (\frac {35 a^{5} x^{3}}{105 a^{\frac {19}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 420 a^{\frac {17}{2}} b x^{2} \sqrt {1 + \frac {b x^{2}}{a}} + 630 a^{\frac {15}{2}} b^{2} x^{4} \sqrt {1 + \frac {b x^{2}}{a}} + 420 a^{\frac {13}{2}} b^{3} x^{6} \sqrt {1 + \frac {b x^{2}}{a}} + 105 a^{\frac {11}{2}} b^{4} x^{8} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {63 a^{4} b x^{5}}{105 a^{\frac {19}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 420 a^{\frac {17}{2}} b x^{2} \sqrt {1 + \frac {b x^{2}}{a}} + 630 a^{\frac {15}{2}} b^{2} x^{4} \sqrt {1 + \frac {b x^{2}}{a}} + 420 a^{\frac {13}{2}} b^{3} x^{6} \sqrt {1 + \frac {b x^{2}}{a}} + 105 a^{\frac {11}{2}} b^{4} x^{8} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {36 a^{3} b^{2} x^{7}}{105 a^{\frac {19}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 420 a^{\frac {17}{2}} b x^{2} \sqrt {1 + \frac {b x^{2}}{a}} + 630 a^{\frac {15}{2}} b^{2} x^{4} \sqrt {1 + \frac {b x^{2}}{a}} + 420 a^{\frac {13}{2}} b^{3} x^{6} \sqrt {1 + \frac {b x^{2}}{a}} + 105 a^{\frac {11}{2}} b^{4} x^{8} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {8 a^{2} b^{3} x^{9}}{105 a^{\frac {19}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 420 a^{\frac {17}{2}} b x^{2} \sqrt {1 + \frac {b x^{2}}{a}} + 630 a^{\frac {15}{2}} b^{2} x^{4} \sqrt {1 + \frac {b x^{2}}{a}} + 420 a^{\frac {13}{2}} b^{3} x^{6} \sqrt {1 + \frac {b x^{2}}{a}} + 105 a^{\frac {11}{2}} b^{4} x^{8} \sqrt {1 + \frac {b x^{2}}{a}}}\right ) + B \left (\begin {cases} - \frac {2 a}{35 a^{3} b^{2} \sqrt {a + b x^{2}} + 105 a^{2} b^{3} x^{2} \sqrt {a + b x^{2}} + 105 a b^{4} x^{4} \sqrt {a + b x^{2}} + 35 b^{5} x^{6} \sqrt {a + b x^{2}}} - \frac {7 b x^{2}}{35 a^{3} b^{2} \sqrt {a + b x^{2}} + 105 a^{2} b^{3} x^{2} \sqrt {a + b x^{2}} + 105 a b^{4} x^{4} \sqrt {a + b x^{2}} + 35 b^{5} x^{6} \sqrt {a + b x^{2}}} & \text {for}\: b \neq 0 \\\frac {x^{4}}{4 a^{\frac {9}{2}}} & \text {otherwise} \end {cases}\right ) + C \left (\frac {7 a x^{5}}{35 a^{\frac {11}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 105 a^{\frac {9}{2}} b x^{2} \sqrt {1 + \frac {b x^{2}}{a}} + 105 a^{\frac {7}{2}} b^{2} x^{4} \sqrt {1 + \frac {b x^{2}}{a}} + 35 a^{\frac {5}{2}} b^{3} x^{6} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {2 b x^{7}}{35 a^{\frac {11}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 105 a^{\frac {9}{2}} b x^{2} \sqrt {1 + \frac {b x^{2}}{a}} + 105 a^{\frac {7}{2}} b^{2} x^{4} \sqrt {1 + \frac {b x^{2}}{a}} + 35 a^{\frac {5}{2}} b^{3} x^{6} \sqrt {1 + \frac {b x^{2}}{a}}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(C*x**2+B*x+A)/(b*x**2+a)**(9/2),x)

[Out]

A*(35*a**5*x**3/(105*a**(19/2)*sqrt(1 + b*x**2/a) + 420*a**(17/2)*b*x**2*sqrt(1 + b*x**2/a) + 630*a**(15/2)*b*
*2*x**4*sqrt(1 + b*x**2/a) + 420*a**(13/2)*b**3*x**6*sqrt(1 + b*x**2/a) + 105*a**(11/2)*b**4*x**8*sqrt(1 + b*x
**2/a)) + 63*a**4*b*x**5/(105*a**(19/2)*sqrt(1 + b*x**2/a) + 420*a**(17/2)*b*x**2*sqrt(1 + b*x**2/a) + 630*a**
(15/2)*b**2*x**4*sqrt(1 + b*x**2/a) + 420*a**(13/2)*b**3*x**6*sqrt(1 + b*x**2/a) + 105*a**(11/2)*b**4*x**8*sqr
t(1 + b*x**2/a)) + 36*a**3*b**2*x**7/(105*a**(19/2)*sqrt(1 + b*x**2/a) + 420*a**(17/2)*b*x**2*sqrt(1 + b*x**2/
a) + 630*a**(15/2)*b**2*x**4*sqrt(1 + b*x**2/a) + 420*a**(13/2)*b**3*x**6*sqrt(1 + b*x**2/a) + 105*a**(11/2)*b
**4*x**8*sqrt(1 + b*x**2/a)) + 8*a**2*b**3*x**9/(105*a**(19/2)*sqrt(1 + b*x**2/a) + 420*a**(17/2)*b*x**2*sqrt(
1 + b*x**2/a) + 630*a**(15/2)*b**2*x**4*sqrt(1 + b*x**2/a) + 420*a**(13/2)*b**3*x**6*sqrt(1 + b*x**2/a) + 105*
a**(11/2)*b**4*x**8*sqrt(1 + b*x**2/a))) + B*Piecewise((-2*a/(35*a**3*b**2*sqrt(a + b*x**2) + 105*a**2*b**3*x*
*2*sqrt(a + b*x**2) + 105*a*b**4*x**4*sqrt(a + b*x**2) + 35*b**5*x**6*sqrt(a + b*x**2)) - 7*b*x**2/(35*a**3*b*
*2*sqrt(a + b*x**2) + 105*a**2*b**3*x**2*sqrt(a + b*x**2) + 105*a*b**4*x**4*sqrt(a + b*x**2) + 35*b**5*x**6*sq
rt(a + b*x**2)), Ne(b, 0)), (x**4/(4*a**(9/2)), True)) + C*(7*a*x**5/(35*a**(11/2)*sqrt(1 + b*x**2/a) + 105*a*
*(9/2)*b*x**2*sqrt(1 + b*x**2/a) + 105*a**(7/2)*b**2*x**4*sqrt(1 + b*x**2/a) + 35*a**(5/2)*b**3*x**6*sqrt(1 +
b*x**2/a)) + 2*b*x**7/(35*a**(11/2)*sqrt(1 + b*x**2/a) + 105*a**(9/2)*b*x**2*sqrt(1 + b*x**2/a) + 105*a**(7/2)
*b**2*x**4*sqrt(1 + b*x**2/a) + 35*a**(5/2)*b**3*x**6*sqrt(1 + b*x**2/a)))

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