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

Optimal. Leaf size=127 \[ \frac {8 x (a C+6 A b)}{105 a^4 b \sqrt {a+b x^2}}+\frac {4 x (a C+6 A b)}{105 a^3 b \left (a+b x^2\right )^{3/2}}+\frac {x (a C+6 A b)}{35 a^2 b \left (a+b x^2\right )^{5/2}}-\frac {a B-x (A b-a C)}{7 a b \left (a+b x^2\right )^{7/2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.07, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1814, 12, 192, 191} \[ \frac {8 x (a C+6 A b)}{105 a^4 b \sqrt {a+b x^2}}+\frac {4 x (a C+6 A b)}{105 a^3 b \left (a+b x^2\right )^{3/2}}+\frac {x (a C+6 A b)}{35 a^2 b \left (a+b x^2\right )^{5/2}}-\frac {a B-x (A b-a C)}{7 a b \left (a+b x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 1814

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

Rubi steps

\begin {align*} \int \frac {A+B x+C x^2}{\left (a+b x^2\right )^{9/2}} \, dx &=-\frac {a B-(A b-a C) x}{7 a b \left (a+b x^2\right )^{7/2}}-\frac {\int \frac {-6 A-\frac {a C}{b}}{\left (a+b x^2\right )^{7/2}} \, dx}{7 a}\\ &=-\frac {a B-(A b-a C) x}{7 a b \left (a+b x^2\right )^{7/2}}+\frac {(6 A b+a C) \int \frac {1}{\left (a+b x^2\right )^{7/2}} \, dx}{7 a b}\\ &=-\frac {a B-(A b-a C) x}{7 a b \left (a+b x^2\right )^{7/2}}+\frac {(6 A b+a C) x}{35 a^2 b \left (a+b x^2\right )^{5/2}}+\frac {(4 (6 A b+a C)) \int \frac {1}{\left (a+b x^2\right )^{5/2}} \, dx}{35 a^2 b}\\ &=-\frac {a B-(A b-a C) x}{7 a b \left (a+b x^2\right )^{7/2}}+\frac {(6 A b+a C) x}{35 a^2 b \left (a+b x^2\right )^{5/2}}+\frac {4 (6 A b+a C) x}{105 a^3 b \left (a+b x^2\right )^{3/2}}+\frac {(8 (6 A b+a C)) \int \frac {1}{\left (a+b x^2\right )^{3/2}} \, dx}{105 a^3 b}\\ &=-\frac {a B-(A b-a C) x}{7 a b \left (a+b x^2\right )^{7/2}}+\frac {(6 A b+a C) x}{35 a^2 b \left (a+b x^2\right )^{5/2}}+\frac {4 (6 A b+a C) x}{105 a^3 b \left (a+b x^2\right )^{3/2}}+\frac {8 (6 A b+a C) x}{105 a^4 b \sqrt {a+b x^2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.09, size = 92, normalized size = 0.72 \[ \frac {-15 a^4 B+35 a^3 b x \left (3 A+C x^2\right )+14 a^2 b^2 x^3 \left (15 A+2 C x^2\right )+8 a b^3 x^5 \left (21 A+C x^2\right )+48 A b^4 x^7}{105 a^4 b \left (a+b x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-15*a^4*B + 48*A*b^4*x^7 + 35*a^3*b*x*(3*A + C*x^2) + 8*a*b^3*x^5*(21*A + C*x^2) + 14*a^2*b^2*x^3*(15*A + 2*C
*x^2))/(105*a^4*b*(a + b*x^2)^(7/2))

________________________________________________________________________________________

fricas [A]  time = 0.71, size = 137, normalized size = 1.08 \[ \frac {{\left (8 \, {\left (C a b^{3} + 6 \, A b^{4}\right )} x^{7} + 105 \, A a^{3} b x + 28 \, {\left (C a^{2} b^{2} + 6 \, A a b^{3}\right )} x^{5} - 15 \, B a^{4} + 35 \, {\left (C a^{3} b + 6 \, A a^{2} b^{2}\right )} x^{3}\right )} \sqrt {b x^{2} + a}}{105 \, {\left (a^{4} b^{5} x^{8} + 4 \, a^{5} b^{4} x^{6} + 6 \, a^{6} b^{3} x^{4} + 4 \, a^{7} b^{2} x^{2} + a^{8} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.50, size = 112, normalized size = 0.88 \[ \frac {{\left ({\left (4 \, x^{2} {\left (\frac {2 \, {\left (C a b^{5} + 6 \, A b^{6}\right )} x^{2}}{a^{4} b^{3}} + \frac {7 \, {\left (C a^{2} b^{4} + 6 \, A a b^{5}\right )}}{a^{4} b^{3}}\right )} + \frac {35 \, {\left (C a^{3} b^{3} + 6 \, A a^{2} b^{4}\right )}}{a^{4} b^{3}}\right )} x^{2} + \frac {105 \, A}{a}\right )} x - \frac {15 \, B}{b}}{105 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.01, size = 96, normalized size = 0.76 \[ \frac {48 A \,b^{4} x^{7}+8 C a \,b^{3} x^{7}+168 A \,x^{5} a \,b^{3}+28 C \,a^{2} b^{2} x^{5}+210 A \,x^{3} a^{2} b^{2}+35 C \,a^{3} b \,x^{3}+105 A x \,a^{3} b -15 B \,a^{4}}{105 \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{4} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/105*(48*A*b^4*x^7+8*C*a*b^3*x^7+168*A*a*b^3*x^5+28*C*a^2*b^2*x^5+210*A*a^2*b^2*x^3+35*C*a^3*b*x^3+105*A*a^3*
b*x-15*B*a^4)/(b*x^2+a)^(7/2)/a^4/b

________________________________________________________________________________________

maxima [A]  time = 1.33, size = 153, normalized size = 1.20 \[ \frac {16 \, A x}{35 \, \sqrt {b x^{2} + a} a^{4}} + \frac {8 \, A x}{35 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{3}} + \frac {6 \, A x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{2}} + \frac {A x}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a} - \frac {C x}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} + \frac {8 \, C x}{105 \, \sqrt {b x^{2} + a} a^{3} b} + \frac {4 \, C x}{105 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2} b} + \frac {C x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a b} - \frac {B}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

16/35*A*x/(sqrt(b*x^2 + a)*a^4) + 8/35*A*x/((b*x^2 + a)^(3/2)*a^3) + 6/35*A*x/((b*x^2 + a)^(5/2)*a^2) + 1/7*A*
x/((b*x^2 + a)^(7/2)*a) - 1/7*C*x/((b*x^2 + a)^(7/2)*b) + 8/105*C*x/(sqrt(b*x^2 + a)*a^3*b) + 4/105*C*x/((b*x^
2 + a)^(3/2)*a^2*b) + 1/35*C*x/((b*x^2 + a)^(5/2)*a*b) - 1/7*B/((b*x^2 + a)^(7/2)*b)

________________________________________________________________________________________

mupad [B]  time = 1.03, size = 115, normalized size = 0.91 \[ \frac {x\,\left (6\,A\,b+C\,a\right )}{35\,a^2\,b\,{\left (b\,x^2+a\right )}^{5/2}}-\frac {\frac {B}{7\,b}-x\,\left (\frac {A}{7\,a}-\frac {C}{7\,b}\right )}{{\left (b\,x^2+a\right )}^{7/2}}+\frac {x\,\left (24\,A\,b+4\,C\,a\right )}{105\,a^3\,b\,{\left (b\,x^2+a\right )}^{3/2}}+\frac {x\,\left (48\,A\,b+8\,C\,a\right )}{105\,a^4\,b\,\sqrt {b\,x^2+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x*(6*A*b + C*a))/(35*a^2*b*(a + b*x^2)^(5/2)) - (B/(7*b) - x*(A/(7*a) - C/(7*b)))/(a + b*x^2)^(7/2) + (x*(24*
A*b + 4*C*a))/(105*a^3*b*(a + b*x^2)^(3/2)) + (x*(48*A*b + 8*C*a))/(105*a^4*b*(a + b*x^2)^(1/2))

________________________________________________________________________________________

sympy [B]  time = 94.22, size = 1880, normalized size = 14.80 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*(35*a**14*x/(35*a**(37/2)*sqrt(1 + b*x**2/a) + 210*a**(35/2)*b*x**2*sqrt(1 + b*x**2/a) + 525*a**(33/2)*b**2*
x**4*sqrt(1 + b*x**2/a) + 700*a**(31/2)*b**3*x**6*sqrt(1 + b*x**2/a) + 525*a**(29/2)*b**4*x**8*sqrt(1 + b*x**2
/a) + 210*a**(27/2)*b**5*x**10*sqrt(1 + b*x**2/a) + 35*a**(25/2)*b**6*x**12*sqrt(1 + b*x**2/a)) + 175*a**13*b*
x**3/(35*a**(37/2)*sqrt(1 + b*x**2/a) + 210*a**(35/2)*b*x**2*sqrt(1 + b*x**2/a) + 525*a**(33/2)*b**2*x**4*sqrt
(1 + b*x**2/a) + 700*a**(31/2)*b**3*x**6*sqrt(1 + b*x**2/a) + 525*a**(29/2)*b**4*x**8*sqrt(1 + b*x**2/a) + 210
*a**(27/2)*b**5*x**10*sqrt(1 + b*x**2/a) + 35*a**(25/2)*b**6*x**12*sqrt(1 + b*x**2/a)) + 371*a**12*b**2*x**5/(
35*a**(37/2)*sqrt(1 + b*x**2/a) + 210*a**(35/2)*b*x**2*sqrt(1 + b*x**2/a) + 525*a**(33/2)*b**2*x**4*sqrt(1 + b
*x**2/a) + 700*a**(31/2)*b**3*x**6*sqrt(1 + b*x**2/a) + 525*a**(29/2)*b**4*x**8*sqrt(1 + b*x**2/a) + 210*a**(2
7/2)*b**5*x**10*sqrt(1 + b*x**2/a) + 35*a**(25/2)*b**6*x**12*sqrt(1 + b*x**2/a)) + 429*a**11*b**3*x**7/(35*a**
(37/2)*sqrt(1 + b*x**2/a) + 210*a**(35/2)*b*x**2*sqrt(1 + b*x**2/a) + 525*a**(33/2)*b**2*x**4*sqrt(1 + b*x**2/
a) + 700*a**(31/2)*b**3*x**6*sqrt(1 + b*x**2/a) + 525*a**(29/2)*b**4*x**8*sqrt(1 + b*x**2/a) + 210*a**(27/2)*b
**5*x**10*sqrt(1 + b*x**2/a) + 35*a**(25/2)*b**6*x**12*sqrt(1 + b*x**2/a)) + 286*a**10*b**4*x**9/(35*a**(37/2)
*sqrt(1 + b*x**2/a) + 210*a**(35/2)*b*x**2*sqrt(1 + b*x**2/a) + 525*a**(33/2)*b**2*x**4*sqrt(1 + b*x**2/a) + 7
00*a**(31/2)*b**3*x**6*sqrt(1 + b*x**2/a) + 525*a**(29/2)*b**4*x**8*sqrt(1 + b*x**2/a) + 210*a**(27/2)*b**5*x*
*10*sqrt(1 + b*x**2/a) + 35*a**(25/2)*b**6*x**12*sqrt(1 + b*x**2/a)) + 104*a**9*b**5*x**11/(35*a**(37/2)*sqrt(
1 + b*x**2/a) + 210*a**(35/2)*b*x**2*sqrt(1 + b*x**2/a) + 525*a**(33/2)*b**2*x**4*sqrt(1 + b*x**2/a) + 700*a**
(31/2)*b**3*x**6*sqrt(1 + b*x**2/a) + 525*a**(29/2)*b**4*x**8*sqrt(1 + b*x**2/a) + 210*a**(27/2)*b**5*x**10*sq
rt(1 + b*x**2/a) + 35*a**(25/2)*b**6*x**12*sqrt(1 + b*x**2/a)) + 16*a**8*b**6*x**13/(35*a**(37/2)*sqrt(1 + b*x
**2/a) + 210*a**(35/2)*b*x**2*sqrt(1 + b*x**2/a) + 525*a**(33/2)*b**2*x**4*sqrt(1 + b*x**2/a) + 700*a**(31/2)*
b**3*x**6*sqrt(1 + b*x**2/a) + 525*a**(29/2)*b**4*x**8*sqrt(1 + b*x**2/a) + 210*a**(27/2)*b**5*x**10*sqrt(1 +
b*x**2/a) + 35*a**(25/2)*b**6*x**12*sqrt(1 + b*x**2/a))) + B*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)) + C*(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*sq
rt(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) + 1
05*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)))

________________________________________________________________________________________