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

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

[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]
)

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Rubi [A]  time = 0.13103, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {1804, 778, 192, 191} \[ \frac{2 x (3 a C+4 A b)}{105 a^3 b^2 \sqrt{a+b x^2}}+\frac{x (3 a C+4 A b)}{105 a^2 b^2 \left (a+b x^2\right )^{3/2}}-\frac{x (3 a C+4 A b)+2 a B}{35 a b^2 \left (a+b x^2\right )^{5/2}}-\frac{x^2 (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}} \]

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 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]

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 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 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]

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*}

Mathematica [A]  time = 0.0945074, size = 87, normalized size = 0.63 \[ \frac{7 a^2 b^2 x^3 \left (5 A+3 C x^2\right )-21 a^3 b B x^2-6 a^4 B+2 a b^3 x^5 \left (14 A+3 C x^2\right )+8 A b^4 x^7}{105 a^3 b^2 \left (a+b x^2\right )^{7/2}} \]

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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Maple [A]  time = 0.006, size = 88, normalized size = 0.6 \begin{align*}{\frac{8\,A{b}^{4}{x}^{7}+6\,Ca{b}^{3}{x}^{7}+28\,Aa{b}^{3}{x}^{5}+21\,C{a}^{2}{b}^{2}{x}^{5}+35\,A{x}^{3}{b}^{2}{a}^{2}-21\,B{x}^{2}b{a}^{3}-6\,B{a}^{4}}{105\,{b}^{2}{a}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}} \end{align*}

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)/b^2/a^3

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Maxima [A]  time = 1.03386, size = 266, normalized size = 1.91 \begin{align*} -\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{align*}

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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Fricas [A]  time = 1.67192, size = 277, normalized size = 1.99 \begin{align*} \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{align*}

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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Sympy [B]  time = 110.567, size = 904, normalized size = 6.5 \begin{align*} \text{result too large to display} \end{align*}

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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Giac [A]  time = 1.18506, size = 127, normalized size = 0.91 \begin{align*} \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{align*}

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)