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

Optimal. Leaf size=119 \[ \frac{8 B x}{105 a^3 b \sqrt{a+b x^2}}+\frac{4 B x}{105 a^2 b \left (a+b x^2\right )^{3/2}}-\frac{2 a C+5 A b-b B x}{35 a b^2 \left (a+b x^2\right )^{5/2}}-\frac{x (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0876955, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {1804, 639, 192, 191} \[ \frac{8 B x}{105 a^3 b \sqrt{a+b x^2}}+\frac{4 B x}{105 a^2 b \left (a+b x^2\right )^{3/2}}-\frac{2 a C+5 A b-b B x}{35 a b^2 \left (a+b x^2\right )^{5/2}}-\frac{x (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(x*(a*B - (A*b - a*C)*x))/(7*a*b*(a + b*x^2)^(7/2)) - (5*A*b + 2*a*C - b*B*x)/(35*a*b^2*(a + b*x^2)^(5/2)) +
(4*B*x)/(105*a^2*b*(a + b*x^2)^(3/2)) + (8*B*x)/(105*a^3*b*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 639

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

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 \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx &=-\frac{x (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac{\int \frac{-a B-(5 A b+2 a C) x}{\left (a+b x^2\right )^{7/2}} \, dx}{7 a b}\\ &=-\frac{x (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac{5 A b+2 a C-b B x}{35 a b^2 \left (a+b x^2\right )^{5/2}}+\frac{(4 B) \int \frac{1}{\left (a+b x^2\right )^{5/2}} \, dx}{35 a b}\\ &=-\frac{x (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac{5 A b+2 a C-b B x}{35 a b^2 \left (a+b x^2\right )^{5/2}}+\frac{4 B x}{105 a^2 b \left (a+b x^2\right )^{3/2}}+\frac{(8 B) \int \frac{1}{\left (a+b x^2\right )^{3/2}} \, dx}{105 a^2 b}\\ &=-\frac{x (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac{5 A b+2 a C-b B x}{35 a b^2 \left (a+b x^2\right )^{5/2}}+\frac{4 B x}{105 a^2 b \left (a+b x^2\right )^{3/2}}+\frac{8 B x}{105 a^3 b \sqrt{a+b x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0580431, size = 75, normalized size = 0.63 \[ \frac{-3 a^3 b \left (5 A+7 C x^2\right )+35 a^2 b^2 B x^3-6 a^4 C+28 a b^3 B x^5+8 b^4 B x^7}{105 a^3 b^2 \left (a+b x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 73, normalized size = 0.6 \begin{align*} -{\frac{-8\,B{x}^{7}{b}^{4}-28\,B{x}^{5}a{b}^{3}-35\,B{x}^{3}{a}^{2}{b}^{2}+21\,C{x}^{2}{a}^{3}b+15\,A{a}^{3}b+6\,C{a}^{4}}{105\,{a}^{3}{b}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/105*(-8*B*b^4*x^7-28*B*a*b^3*x^5-35*B*a^2*b^2*x^3+21*C*a^3*b*x^2+15*A*a^3*b+6*C*a^4)/(b*x^2+a)^(7/2)/a^3/b^
2

________________________________________________________________________________________

Maxima [A]  time = 1.17386, size = 166, normalized size = 1.39 \begin{align*} -\frac{C x^{2}}{5 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b} - \frac{B x}{7 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b} + \frac{8 \, B x}{105 \, \sqrt{b x^{2} + a} a^{3} b} + \frac{4 \, B x}{105 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{2} b} + \frac{B x}{35 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a b} - \frac{2 \, C a}{35 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b^{2}} - \frac{A}{7 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.69919, size = 250, normalized size = 2.1 \begin{align*} \frac{{\left (8 \, B b^{4} x^{7} + 28 \, B a b^{3} x^{5} + 35 \, B a^{2} b^{2} x^{3} - 21 \, C a^{3} b x^{2} - 6 \, C a^{4} - 15 \, A a^{3} b\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*(C*x^2+B*x+A)/(b*x^2+a)^(9/2),x, algorithm="fricas")

[Out]

1/105*(8*B*b^4*x^7 + 28*B*a*b^3*x^5 + 35*B*a^2*b^2*x^3 - 21*C*a^3*b*x^2 - 6*C*a^4 - 15*A*a^3*b)*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)

________________________________________________________________________________________

Sympy [A]  time = 71.2865, size = 796, normalized size = 6.69 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*Piecewise((-1/(7*a**3*b*sqrt(a + b*x**2) + 21*a**2*b**2*x**2*sqrt(a + b*x**2) + 21*a*b**3*x**4*sqrt(a + b*x*
*2) + 7*b**4*x**6*sqrt(a + b*x**2)), Ne(b, 0)), (x**2/(2*a**(9/2)), True)) + B*(35*a**5*x**3/(105*a**(19/2)*sq
rt(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*sqrt(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))) + C*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*sqrt(a + b*x**2)), Ne(b, 0)), (x**4
/(4*a**(9/2)), True))

________________________________________________________________________________________

Giac [A]  time = 1.22822, size = 111, normalized size = 0.93 \begin{align*} \frac{{\left ({\left (4 \,{\left (\frac{2 \, B b^{2} x^{2}}{a^{3}} + \frac{7 \, B b}{a^{2}}\right )} x^{2} + \frac{35 \, B}{a}\right )} x - \frac{21 \, C}{b}\right )} x^{2} - \frac{3 \,{\left (2 \, C a^{4} b + 5 \, A a^{3} b^{2}\right )}}{a^{3} b^{3}}}{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*(C*x^2+B*x+A)/(b*x^2+a)^(9/2),x, algorithm="giac")

[Out]

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