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]

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

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Rubi [A]  time = 0.09, antiderivative size = 119, normalized size of antiderivative = 1.00, 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 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 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 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 \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*}

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Mathematica [A]  time = 0.09, size = 75, normalized size = 0.63 \[ \frac {-6 a^4 C-3 a^3 b \left (5 A+7 C x^2\right )+35 a^2 b^2 B x^3+28 a b^3 B x^5+8 b^4 B x^7}{105 a^3 b^2 \left (a+b x^2\right )^{7/2}} \]

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))

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fricas [A]  time = 0.69, size = 119, normalized size = 1.00 \[ \frac {{\left (8 \, B b^{4} x^{7} + 28 \, B a b^{3} x^{5} + 35 \, B a^{2} b^{2} x^{3} - 21 \, C a^{3} b x^{2} - 6 \, C a^{4} - 15 \, A a^{3} b\right )} \sqrt {b x^{2} + a}}{105 \, {\left (a^{3} b^{6} x^{8} + 4 \, a^{4} b^{5} x^{6} + 6 \, a^{5} b^{4} x^{4} + 4 \, a^{6} b^{3} x^{2} + a^{7} b^{2}\right )}} \]

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)

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giac [A]  time = 0.60, size = 82, normalized size = 0.69 \[ \frac {{\left ({\left (4 \, {\left (\frac {2 \, B b^{2} x^{2}}{a^{3}} + \frac {7 \, B b}{a^{2}}\right )} x^{2} + \frac {35 \, B}{a}\right )} x - \frac {21 \, C}{b}\right )} x^{2} - \frac {3 \, {\left (2 \, C a^{4} b + 5 \, A a^{3} b^{2}\right )}}{a^{3} b^{3}}}{105 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} \]

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)

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maple [A]  time = 0.00, size = 73, normalized size = 0.61 \[ -\frac {-8 B \,x^{7} b^{4}-28 B \,x^{5} a \,b^{3}-35 B \,x^{3} a^{2} b^{2}+21 C \,a^{3} b \,x^{2}+15 A \,a^{3} b +6 C \,a^{4}}{105 \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{3} b^{2}} \]

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

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maxima [A]  time = 1.34, size = 123, normalized size = 1.03 \[ -\frac {C x^{2}}{5 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} - \frac {B x}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} + \frac {8 \, B x}{105 \, \sqrt {b x^{2} + a} a^{3} b} + \frac {4 \, B x}{105 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2} b} + \frac {B x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a b} - \frac {2 \, C a}{35 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}} - \frac {A}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} \]

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)

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mupad [B]  time = 1.05, size = 99, normalized size = 0.83 \[ \frac {8\,B\,x}{105\,a^3\,b\,\sqrt {b\,x^2+a}}-\frac {\frac {A}{7\,b}-\frac {C\,a}{7\,b^2}+\frac {B\,x}{7\,b}}{{\left (b\,x^2+a\right )}^{7/2}}-\frac {\frac {C}{5\,b^2}-\frac {B\,x}{35\,a\,b}}{{\left (b\,x^2+a\right )}^{5/2}}+\frac {4\,B\,x}{105\,a^2\,b\,{\left (b\,x^2+a\right )}^{3/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 85.30, size = 796, normalized size = 6.69 \[ A \left (\begin {cases} - \frac {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}}} & \text {for}\: b \neq 0 \\\frac {x^{2}}{2 a^{\frac {9}{2}}} & \text {otherwise} \end {cases}\right ) + B \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 ) + C \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 ) \]

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))

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