3.158 \(\int \frac{c+d x^2+e x^4+f x^6}{x^{10} \sqrt{a+b x^2}} \, dx\)
Optimal. Leaf size=189 \[ -\frac{2 b \sqrt{a+b x^2} \left (84 a^2 b e-105 a^3 f-72 a b^2 d+64 b^3 c\right )}{315 a^5 x}+\frac{\sqrt{a+b x^2} \left (84 a^2 b e-105 a^3 f-72 a b^2 d+64 b^3 c\right )}{315 a^4 x^3}-\frac{\sqrt{a+b x^2} \left (21 a^2 e-18 a b d+16 b^2 c\right )}{105 a^3 x^5}+\frac{\sqrt{a+b x^2} (8 b c-9 a d)}{63 a^2 x^7}-\frac{c \sqrt{a+b x^2}}{9 a x^9} \]
[Out]
-(c*Sqrt[a + b*x^2])/(9*a*x^9) + ((8*b*c - 9*a*d)*Sqrt[a + b*x^2])/(63*a^2*x^7) - ((16*b^2*c - 18*a*b*d + 21*a
^2*e)*Sqrt[a + b*x^2])/(105*a^3*x^5) + ((64*b^3*c - 72*a*b^2*d + 84*a^2*b*e - 105*a^3*f)*Sqrt[a + b*x^2])/(315
*a^4*x^3) - (2*b*(64*b^3*c - 72*a*b^2*d + 84*a^2*b*e - 105*a^3*f)*Sqrt[a + b*x^2])/(315*a^5*x)
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Rubi [A] time = 0.254315, antiderivative size = 189, normalized size of antiderivative = 1.,
number of steps used = 6, number of rules used = 4, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used =
{1803, 12, 271, 264} \[ -\frac{2 b \sqrt{a+b x^2} \left (84 a^2 b e-105 a^3 f-72 a b^2 d+64 b^3 c\right )}{315 a^5 x}+\frac{\sqrt{a+b x^2} \left (84 a^2 b e-105 a^3 f-72 a b^2 d+64 b^3 c\right )}{315 a^4 x^3}-\frac{\sqrt{a+b x^2} \left (21 a^2 e-18 a b d+16 b^2 c\right )}{105 a^3 x^5}+\frac{\sqrt{a+b x^2} (8 b c-9 a d)}{63 a^2 x^7}-\frac{c \sqrt{a+b x^2}}{9 a x^9} \]
Antiderivative was successfully verified.
[In]
Int[(c + d*x^2 + e*x^4 + f*x^6)/(x^10*Sqrt[a + b*x^2]),x]
[Out]
-(c*Sqrt[a + b*x^2])/(9*a*x^9) + ((8*b*c - 9*a*d)*Sqrt[a + b*x^2])/(63*a^2*x^7) - ((16*b^2*c - 18*a*b*d + 21*a
^2*e)*Sqrt[a + b*x^2])/(105*a^3*x^5) + ((64*b^3*c - 72*a*b^2*d + 84*a^2*b*e - 105*a^3*f)*Sqrt[a + b*x^2])/(315
*a^4*x^3) - (2*b*(64*b^3*c - 72*a*b^2*d + 84*a^2*b*e - 105*a^3*f)*Sqrt[a + b*x^2])/(315*a^5*x)
Rule 1803
Int[(Pq_)*(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{A = Coeff[Pq, x, 0], Q = PolynomialQuotient
[Pq - Coeff[Pq, x, 0], x^2, x]}, Simp[(A*x^(m + 1)*(a + b*x^2)^(p + 1))/(a*(m + 1)), x] + Dist[1/(a*(m + 1)),
Int[x^(m + 2)*(a + b*x^2)^p*(a*(m + 1)*Q - A*b*(m + 2*(p + 1) + 1)), x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq,
x^2] && IntegerQ[m/2] && ILtQ[(m + 1)/2 + p, 0] && LtQ[m + Expon[Pq, x] + 2*p + 1, 0]
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
Rule 271
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x^(m + 1)*(a + b*x^n)^(p + 1))/(a*(m + 1)), x]
- Dist[(b*(m + n*(p + 1) + 1))/(a*(m + 1)), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]
Rule 264
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]
Rubi steps
\begin{align*} \int \frac{c+d x^2+e x^4+f x^6}{x^{10} \sqrt{a+b x^2}} \, dx &=-\frac{c \sqrt{a+b x^2}}{9 a x^9}-\frac{\int \frac{8 b c-9 a \left (d+e x^2+f x^4\right )}{x^8 \sqrt{a+b x^2}} \, dx}{9 a}\\ &=-\frac{c \sqrt{a+b x^2}}{9 a x^9}+\frac{(8 b c-9 a d) \sqrt{a+b x^2}}{63 a^2 x^7}+\frac{\int \frac{6 b (8 b c-9 a d)-7 a \left (-9 a e-9 a f x^2\right )}{x^6 \sqrt{a+b x^2}} \, dx}{63 a^2}\\ &=-\frac{c \sqrt{a+b x^2}}{9 a x^9}+\frac{(8 b c-9 a d) \sqrt{a+b x^2}}{63 a^2 x^7}-\frac{\left (16 b^2 c-18 a b d+21 a^2 e\right ) \sqrt{a+b x^2}}{105 a^3 x^5}-\frac{\int \frac{4 b \left (48 b^2 c-54 a b d+63 a^2 e\right )-315 a^3 f}{x^4 \sqrt{a+b x^2}} \, dx}{315 a^3}\\ &=-\frac{c \sqrt{a+b x^2}}{9 a x^9}+\frac{(8 b c-9 a d) \sqrt{a+b x^2}}{63 a^2 x^7}-\frac{\left (16 b^2 c-18 a b d+21 a^2 e\right ) \sqrt{a+b x^2}}{105 a^3 x^5}-\frac{\left (64 b^3 c-72 a b^2 d+84 a^2 b e-105 a^3 f\right ) \int \frac{1}{x^4 \sqrt{a+b x^2}} \, dx}{105 a^3}\\ &=-\frac{c \sqrt{a+b x^2}}{9 a x^9}+\frac{(8 b c-9 a d) \sqrt{a+b x^2}}{63 a^2 x^7}-\frac{\left (16 b^2 c-18 a b d+21 a^2 e\right ) \sqrt{a+b x^2}}{105 a^3 x^5}+\frac{\left (64 b^3 c-72 a b^2 d+84 a^2 b e-105 a^3 f\right ) \sqrt{a+b x^2}}{315 a^4 x^3}+\frac{\left (2 b \left (64 b^3 c-72 a b^2 d+84 a^2 b e-105 a^3 f\right )\right ) \int \frac{1}{x^2 \sqrt{a+b x^2}} \, dx}{315 a^4}\\ &=-\frac{c \sqrt{a+b x^2}}{9 a x^9}+\frac{(8 b c-9 a d) \sqrt{a+b x^2}}{63 a^2 x^7}-\frac{\left (16 b^2 c-18 a b d+21 a^2 e\right ) \sqrt{a+b x^2}}{105 a^3 x^5}+\frac{\left (64 b^3 c-72 a b^2 d+84 a^2 b e-105 a^3 f\right ) \sqrt{a+b x^2}}{315 a^4 x^3}-\frac{2 b \left (64 b^3 c-72 a b^2 d+84 a^2 b e-105 a^3 f\right ) \sqrt{a+b x^2}}{315 a^5 x}\\ \end{align*}
Mathematica [A] time = 0.0855189, size = 134, normalized size = 0.71 \[ -\frac{\sqrt{a+b x^2} \left (24 a^2 b^2 x^4 \left (2 c+3 d x^2+7 e x^4\right )-2 a^3 b x^2 \left (20 c+27 d x^2+42 e x^4+105 f x^6\right )+a^4 \left (35 c+45 d x^2+63 e x^4+105 f x^6\right )-16 a b^3 x^6 \left (4 c+9 d x^2\right )+128 b^4 c x^8\right )}{315 a^5 x^9} \]
Antiderivative was successfully verified.
[In]
Integrate[(c + d*x^2 + e*x^4 + f*x^6)/(x^10*Sqrt[a + b*x^2]),x]
[Out]
-(Sqrt[a + b*x^2]*(128*b^4*c*x^8 - 16*a*b^3*x^6*(4*c + 9*d*x^2) + 24*a^2*b^2*x^4*(2*c + 3*d*x^2 + 7*e*x^4) - 2
*a^3*b*x^2*(20*c + 27*d*x^2 + 42*e*x^4 + 105*f*x^6) + a^4*(35*c + 45*d*x^2 + 63*e*x^4 + 105*f*x^6)))/(315*a^5*
x^9)
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Maple [A] time = 0.006, size = 157, normalized size = 0.8 \begin{align*} -{\frac{-210\,{a}^{3}bf{x}^{8}+168\,{a}^{2}{b}^{2}e{x}^{8}-144\,a{b}^{3}d{x}^{8}+128\,{b}^{4}c{x}^{8}+105\,{a}^{4}f{x}^{6}-84\,{a}^{3}be{x}^{6}+72\,{a}^{2}{b}^{2}d{x}^{6}-64\,a{b}^{3}c{x}^{6}+63\,{a}^{4}e{x}^{4}-54\,{a}^{3}bd{x}^{4}+48\,{a}^{2}{b}^{2}c{x}^{4}+45\,{a}^{4}d{x}^{2}-40\,{a}^{3}bc{x}^{2}+35\,c{a}^{4}}{315\,{x}^{9}{a}^{5}}\sqrt{b{x}^{2}+a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((f*x^6+e*x^4+d*x^2+c)/x^10/(b*x^2+a)^(1/2),x)
[Out]
-1/315*(b*x^2+a)^(1/2)*(-210*a^3*b*f*x^8+168*a^2*b^2*e*x^8-144*a*b^3*d*x^8+128*b^4*c*x^8+105*a^4*f*x^6-84*a^3*
b*e*x^6+72*a^2*b^2*d*x^6-64*a*b^3*c*x^6+63*a^4*e*x^4-54*a^3*b*d*x^4+48*a^2*b^2*c*x^4+45*a^4*d*x^2-40*a^3*b*c*x
^2+35*a^4*c)/x^9/a^5
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((f*x^6+e*x^4+d*x^2+c)/x^10/(b*x^2+a)^(1/2),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 2.26911, size = 327, normalized size = 1.73 \begin{align*} -\frac{{\left (2 \,{\left (64 \, b^{4} c - 72 \, a b^{3} d + 84 \, a^{2} b^{2} e - 105 \, a^{3} b f\right )} x^{8} -{\left (64 \, a b^{3} c - 72 \, a^{2} b^{2} d + 84 \, a^{3} b e - 105 \, a^{4} f\right )} x^{6} + 35 \, a^{4} c + 3 \,{\left (16 \, a^{2} b^{2} c - 18 \, a^{3} b d + 21 \, a^{4} e\right )} x^{4} - 5 \,{\left (8 \, a^{3} b c - 9 \, a^{4} d\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{315 \, a^{5} x^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((f*x^6+e*x^4+d*x^2+c)/x^10/(b*x^2+a)^(1/2),x, algorithm="fricas")
[Out]
-1/315*(2*(64*b^4*c - 72*a*b^3*d + 84*a^2*b^2*e - 105*a^3*b*f)*x^8 - (64*a*b^3*c - 72*a^2*b^2*d + 84*a^3*b*e -
105*a^4*f)*x^6 + 35*a^4*c + 3*(16*a^2*b^2*c - 18*a^3*b*d + 21*a^4*e)*x^4 - 5*(8*a^3*b*c - 9*a^4*d)*x^2)*sqrt(
b*x^2 + a)/(a^5*x^9)
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Sympy [B] time = 6.78238, size = 1642, normalized size = 8.69 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((f*x**6+e*x**4+d*x**2+c)/x**10/(b*x**2+a)**(1/2),x)
[Out]
-35*a**8*b**(33/2)*c*sqrt(a/(b*x**2) + 1)/(315*a**9*b**16*x**8 + 1260*a**8*b**17*x**10 + 1890*a**7*b**18*x**12
+ 1260*a**6*b**19*x**14 + 315*a**5*b**20*x**16) - 100*a**7*b**(35/2)*c*x**2*sqrt(a/(b*x**2) + 1)/(315*a**9*b*
*16*x**8 + 1260*a**8*b**17*x**10 + 1890*a**7*b**18*x**12 + 1260*a**6*b**19*x**14 + 315*a**5*b**20*x**16) - 98*
a**6*b**(37/2)*c*x**4*sqrt(a/(b*x**2) + 1)/(315*a**9*b**16*x**8 + 1260*a**8*b**17*x**10 + 1890*a**7*b**18*x**1
2 + 1260*a**6*b**19*x**14 + 315*a**5*b**20*x**16) - 5*a**6*b**(19/2)*d*sqrt(a/(b*x**2) + 1)/(35*a**7*b**9*x**6
+ 105*a**6*b**10*x**8 + 105*a**5*b**11*x**10 + 35*a**4*b**12*x**12) - 28*a**5*b**(39/2)*c*x**6*sqrt(a/(b*x**2
) + 1)/(315*a**9*b**16*x**8 + 1260*a**8*b**17*x**10 + 1890*a**7*b**18*x**12 + 1260*a**6*b**19*x**14 + 315*a**5
*b**20*x**16) - 9*a**5*b**(21/2)*d*x**2*sqrt(a/(b*x**2) + 1)/(35*a**7*b**9*x**6 + 105*a**6*b**10*x**8 + 105*a*
*5*b**11*x**10 + 35*a**4*b**12*x**12) - 35*a**4*b**(41/2)*c*x**8*sqrt(a/(b*x**2) + 1)/(315*a**9*b**16*x**8 + 1
260*a**8*b**17*x**10 + 1890*a**7*b**18*x**12 + 1260*a**6*b**19*x**14 + 315*a**5*b**20*x**16) - 5*a**4*b**(23/2
)*d*x**4*sqrt(a/(b*x**2) + 1)/(35*a**7*b**9*x**6 + 105*a**6*b**10*x**8 + 105*a**5*b**11*x**10 + 35*a**4*b**12*
x**12) - 3*a**4*b**(9/2)*e*sqrt(a/(b*x**2) + 1)/(15*a**5*b**4*x**4 + 30*a**4*b**5*x**6 + 15*a**3*b**6*x**8) -
280*a**3*b**(43/2)*c*x**10*sqrt(a/(b*x**2) + 1)/(315*a**9*b**16*x**8 + 1260*a**8*b**17*x**10 + 1890*a**7*b**18
*x**12 + 1260*a**6*b**19*x**14 + 315*a**5*b**20*x**16) + 5*a**3*b**(25/2)*d*x**6*sqrt(a/(b*x**2) + 1)/(35*a**7
*b**9*x**6 + 105*a**6*b**10*x**8 + 105*a**5*b**11*x**10 + 35*a**4*b**12*x**12) - 2*a**3*b**(11/2)*e*x**2*sqrt(
a/(b*x**2) + 1)/(15*a**5*b**4*x**4 + 30*a**4*b**5*x**6 + 15*a**3*b**6*x**8) - 560*a**2*b**(45/2)*c*x**12*sqrt(
a/(b*x**2) + 1)/(315*a**9*b**16*x**8 + 1260*a**8*b**17*x**10 + 1890*a**7*b**18*x**12 + 1260*a**6*b**19*x**14 +
315*a**5*b**20*x**16) + 30*a**2*b**(27/2)*d*x**8*sqrt(a/(b*x**2) + 1)/(35*a**7*b**9*x**6 + 105*a**6*b**10*x**
8 + 105*a**5*b**11*x**10 + 35*a**4*b**12*x**12) - 3*a**2*b**(13/2)*e*x**4*sqrt(a/(b*x**2) + 1)/(15*a**5*b**4*x
**4 + 30*a**4*b**5*x**6 + 15*a**3*b**6*x**8) - 448*a*b**(47/2)*c*x**14*sqrt(a/(b*x**2) + 1)/(315*a**9*b**16*x*
*8 + 1260*a**8*b**17*x**10 + 1890*a**7*b**18*x**12 + 1260*a**6*b**19*x**14 + 315*a**5*b**20*x**16) + 40*a*b**(
29/2)*d*x**10*sqrt(a/(b*x**2) + 1)/(35*a**7*b**9*x**6 + 105*a**6*b**10*x**8 + 105*a**5*b**11*x**10 + 35*a**4*b
**12*x**12) - 12*a*b**(15/2)*e*x**6*sqrt(a/(b*x**2) + 1)/(15*a**5*b**4*x**4 + 30*a**4*b**5*x**6 + 15*a**3*b**6
*x**8) - 128*b**(49/2)*c*x**16*sqrt(a/(b*x**2) + 1)/(315*a**9*b**16*x**8 + 1260*a**8*b**17*x**10 + 1890*a**7*b
**18*x**12 + 1260*a**6*b**19*x**14 + 315*a**5*b**20*x**16) + 16*b**(31/2)*d*x**12*sqrt(a/(b*x**2) + 1)/(35*a**
7*b**9*x**6 + 105*a**6*b**10*x**8 + 105*a**5*b**11*x**10 + 35*a**4*b**12*x**12) - 8*b**(17/2)*e*x**8*sqrt(a/(b
*x**2) + 1)/(15*a**5*b**4*x**4 + 30*a**4*b**5*x**6 + 15*a**3*b**6*x**8) - sqrt(b)*f*sqrt(a/(b*x**2) + 1)/(3*a*
x**2) + 2*b**(3/2)*f*sqrt(a/(b*x**2) + 1)/(3*a**2)
________________________________________________________________________________________
Giac [B] time = 1.24343, size = 900, normalized size = 4.76 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((f*x^6+e*x^4+d*x^2+c)/x^10/(b*x^2+a)^(1/2),x, algorithm="giac")
[Out]
4/315*(315*(sqrt(b)*x - sqrt(b*x^2 + a))^14*b^(3/2)*f - 1995*(sqrt(b)*x - sqrt(b*x^2 + a))^12*a*b^(3/2)*f + 84
0*(sqrt(b)*x - sqrt(b*x^2 + a))^12*b^(5/2)*e + 2520*(sqrt(b)*x - sqrt(b*x^2 + a))^10*b^(7/2)*d + 5355*(sqrt(b)
*x - sqrt(b*x^2 + a))^10*a^2*b^(3/2)*f - 3780*(sqrt(b)*x - sqrt(b*x^2 + a))^10*a*b^(5/2)*e + 8064*(sqrt(b)*x -
sqrt(b*x^2 + a))^8*b^(9/2)*c - 6552*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a*b^(7/2)*d - 7875*(sqrt(b)*x - sqrt(b*x^
2 + a))^8*a^3*b^(3/2)*f + 6804*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a^2*b^(5/2)*e - 5376*(sqrt(b)*x - sqrt(b*x^2 +
a))^6*a*b^(9/2)*c + 6048*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^2*b^(7/2)*d + 6825*(sqrt(b)*x - sqrt(b*x^2 + a))^6*
a^4*b^(3/2)*f - 6216*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^3*b^(5/2)*e + 2304*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^2*
b^(9/2)*c - 2592*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^3*b^(7/2)*d - 3465*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^5*b^(3
/2)*f + 3024*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^4*b^(5/2)*e - 576*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^3*b^(9/2)*c
+ 648*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^4*b^(7/2)*d + 945*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^6*b^(3/2)*f - 756
*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^5*b^(5/2)*e + 64*a^4*b^(9/2)*c - 72*a^5*b^(7/2)*d - 105*a^7*b^(3/2)*f + 84*
a^6*b^(5/2)*e)/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^9