3.283 \(\int \frac{a+b x^2+c x^4}{(d+e x^2)^{9/2}} \, dx\)
Optimal. Leaf size=126 \[ \frac{2 e x^7 \left (4 e (6 a e+b d)+3 c d^2\right )}{105 d^4 \left (d+e x^2\right )^{7/2}}+\frac{x^5 \left (4 e (6 a e+b d)+3 c d^2\right )}{15 d^3 \left (d+e x^2\right )^{7/2}}+\frac{x^3 (6 a e+b d)}{3 d^2 \left (d+e x^2\right )^{7/2}}+\frac{a x}{d \left (d+e x^2\right )^{7/2}} \]
[Out]
(a*x)/(d*(d + e*x^2)^(7/2)) + ((b*d + 6*a*e)*x^3)/(3*d^2*(d + e*x^2)^(7/2)) + ((3*c*d^2 + 4*e*(b*d + 6*a*e))*x
^5)/(15*d^3*(d + e*x^2)^(7/2)) + (2*e*(3*c*d^2 + 4*e*(b*d + 6*a*e))*x^7)/(105*d^4*(d + e*x^2)^(7/2))
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Rubi [A] time = 0.146389, antiderivative size = 126, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used =
{1155, 1803, 12, 271, 264} \[ \frac{2 e x^7 \left (4 e (6 a e+b d)+3 c d^2\right )}{105 d^4 \left (d+e x^2\right )^{7/2}}+\frac{x^5 \left (4 e (6 a e+b d)+3 c d^2\right )}{15 d^3 \left (d+e x^2\right )^{7/2}}+\frac{x^3 (6 a e+b d)}{3 d^2 \left (d+e x^2\right )^{7/2}}+\frac{a x}{d \left (d+e x^2\right )^{7/2}} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*x^2 + c*x^4)/(d + e*x^2)^(9/2),x]
[Out]
(a*x)/(d*(d + e*x^2)^(7/2)) + ((b*d + 6*a*e)*x^3)/(3*d^2*(d + e*x^2)^(7/2)) + ((3*c*d^2 + 4*e*(b*d + 6*a*e))*x
^5)/(15*d^3*(d + e*x^2)^(7/2)) + (2*e*(3*c*d^2 + 4*e*(b*d + 6*a*e))*x^7)/(105*d^4*(d + e*x^2)^(7/2))
Rule 1155
Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp[(a^p*x*(d + e*x^2
)^(q + 1))/d, x] + Dist[1/d, Int[x^2*(d + e*x^2)^q*(d*PolynomialQuotient[(a + b*x^2 + c*x^4)^p - a^p, x^2, x]
- e*a^p*(2*q + 3)), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0
] && IGtQ[p, 0] && ILtQ[q + 1/2, 0] && LtQ[4*p + 2*q + 1, 0]
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{a+b x^2+c x^4}{\left (d+e x^2\right )^{9/2}} \, dx &=\frac{a x}{d \left (d+e x^2\right )^{7/2}}+\frac{\int \frac{x^2 \left (6 a e+d \left (b+c x^2\right )\right )}{\left (d+e x^2\right )^{9/2}} \, dx}{d}\\ &=\frac{a x}{d \left (d+e x^2\right )^{7/2}}+\frac{(b d+6 a e) x^3}{3 d^2 \left (d+e x^2\right )^{7/2}}+\frac{\int \frac{\left (3 c d^2+4 e (b d+6 a e)\right ) x^4}{\left (d+e x^2\right )^{9/2}} \, dx}{3 d^2}\\ &=\frac{a x}{d \left (d+e x^2\right )^{7/2}}+\frac{(b d+6 a e) x^3}{3 d^2 \left (d+e x^2\right )^{7/2}}+\frac{1}{3} \left (3 c+\frac{4 e (b d+6 a e)}{d^2}\right ) \int \frac{x^4}{\left (d+e x^2\right )^{9/2}} \, dx\\ &=\frac{a x}{d \left (d+e x^2\right )^{7/2}}+\frac{(b d+6 a e) x^3}{3 d^2 \left (d+e x^2\right )^{7/2}}+\frac{\left (3 c d^2+4 e (b d+6 a e)\right ) x^5}{15 d^3 \left (d+e x^2\right )^{7/2}}+\frac{\left (2 e \left (3 c d^2+4 e (b d+6 a e)\right )\right ) \int \frac{x^6}{\left (d+e x^2\right )^{9/2}} \, dx}{15 d^3}\\ &=\frac{a x}{d \left (d+e x^2\right )^{7/2}}+\frac{(b d+6 a e) x^3}{3 d^2 \left (d+e x^2\right )^{7/2}}+\frac{\left (3 c d^2+4 e (b d+6 a e)\right ) x^5}{15 d^3 \left (d+e x^2\right )^{7/2}}+\frac{2 e \left (3 c d^2+4 e (b d+6 a e)\right ) x^7}{105 d^4 \left (d+e x^2\right )^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.094898, size = 101, normalized size = 0.8 \[ \frac{3 a \left (70 d^2 e x^3+35 d^3 x+56 d e^2 x^5+16 e^3 x^7\right )+d x^3 \left (b \left (35 d^2+28 d e x^2+8 e^2 x^4\right )+3 c d x^2 \left (7 d+2 e x^2\right )\right )}{105 d^4 \left (d+e x^2\right )^{7/2}} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + b*x^2 + c*x^4)/(d + e*x^2)^(9/2),x]
[Out]
(3*a*(35*d^3*x + 70*d^2*e*x^3 + 56*d*e^2*x^5 + 16*e^3*x^7) + d*x^3*(3*c*d*x^2*(7*d + 2*e*x^2) + b*(35*d^2 + 28
*d*e*x^2 + 8*e^2*x^4)))/(105*d^4*(d + e*x^2)^(7/2))
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Maple [A] time = 0.006, size = 100, normalized size = 0.8 \begin{align*}{\frac{x \left ( 48\,a{e}^{3}{x}^{6}+8\,bd{e}^{2}{x}^{6}+6\,c{d}^{2}e{x}^{6}+168\,ad{e}^{2}{x}^{4}+28\,b{d}^{2}e{x}^{4}+21\,c{d}^{3}{x}^{4}+210\,a{d}^{2}e{x}^{2}+35\,b{d}^{3}{x}^{2}+105\,a{d}^{3} \right ) }{105\,{d}^{4}} \left ( e{x}^{2}+d \right ) ^{-{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((c*x^4+b*x^2+a)/(e*x^2+d)^(9/2),x)
[Out]
1/105*x*(48*a*e^3*x^6+8*b*d*e^2*x^6+6*c*d^2*e*x^6+168*a*d*e^2*x^4+28*b*d^2*e*x^4+21*c*d^3*x^4+210*a*d^2*e*x^2+
35*b*d^3*x^2+105*a*d^3)/(e*x^2+d)^(7/2)/d^4
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Maxima [B] time = 0.99571, size = 306, normalized size = 2.43 \begin{align*} -\frac{c x^{3}}{4 \,{\left (e x^{2} + d\right )}^{\frac{7}{2}} e} + \frac{16 \, a x}{35 \, \sqrt{e x^{2} + d} d^{4}} + \frac{8 \, a x}{35 \,{\left (e x^{2} + d\right )}^{\frac{3}{2}} d^{3}} + \frac{6 \, a x}{35 \,{\left (e x^{2} + d\right )}^{\frac{5}{2}} d^{2}} + \frac{a x}{7 \,{\left (e x^{2} + d\right )}^{\frac{7}{2}} d} + \frac{3 \, c x}{140 \,{\left (e x^{2} + d\right )}^{\frac{5}{2}} e^{2}} + \frac{2 \, c x}{35 \, \sqrt{e x^{2} + d} d^{2} e^{2}} + \frac{c x}{35 \,{\left (e x^{2} + d\right )}^{\frac{3}{2}} d e^{2}} - \frac{3 \, c d x}{28 \,{\left (e x^{2} + d\right )}^{\frac{7}{2}} e^{2}} - \frac{b x}{7 \,{\left (e x^{2} + d\right )}^{\frac{7}{2}} e} + \frac{8 \, b x}{105 \, \sqrt{e x^{2} + d} d^{3} e} + \frac{4 \, b x}{105 \,{\left (e x^{2} + d\right )}^{\frac{3}{2}} d^{2} e} + \frac{b x}{35 \,{\left (e x^{2} + d\right )}^{\frac{5}{2}} d e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c*x^4+b*x^2+a)/(e*x^2+d)^(9/2),x, algorithm="maxima")
[Out]
-1/4*c*x^3/((e*x^2 + d)^(7/2)*e) + 16/35*a*x/(sqrt(e*x^2 + d)*d^4) + 8/35*a*x/((e*x^2 + d)^(3/2)*d^3) + 6/35*a
*x/((e*x^2 + d)^(5/2)*d^2) + 1/7*a*x/((e*x^2 + d)^(7/2)*d) + 3/140*c*x/((e*x^2 + d)^(5/2)*e^2) + 2/35*c*x/(sqr
t(e*x^2 + d)*d^2*e^2) + 1/35*c*x/((e*x^2 + d)^(3/2)*d*e^2) - 3/28*c*d*x/((e*x^2 + d)^(7/2)*e^2) - 1/7*b*x/((e*
x^2 + d)^(7/2)*e) + 8/105*b*x/(sqrt(e*x^2 + d)*d^3*e) + 4/105*b*x/((e*x^2 + d)^(3/2)*d^2*e) + 1/35*b*x/((e*x^2
+ d)^(5/2)*d*e)
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Fricas [A] time = 5.13565, size = 294, normalized size = 2.33 \begin{align*} \frac{{\left (2 \,{\left (3 \, c d^{2} e + 4 \, b d e^{2} + 24 \, a e^{3}\right )} x^{7} + 7 \,{\left (3 \, c d^{3} + 4 \, b d^{2} e + 24 \, a d e^{2}\right )} x^{5} + 105 \, a d^{3} x + 35 \,{\left (b d^{3} + 6 \, a d^{2} e\right )} x^{3}\right )} \sqrt{e x^{2} + d}}{105 \,{\left (d^{4} e^{4} x^{8} + 4 \, d^{5} e^{3} x^{6} + 6 \, d^{6} e^{2} x^{4} + 4 \, d^{7} e x^{2} + d^{8}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c*x^4+b*x^2+a)/(e*x^2+d)^(9/2),x, algorithm="fricas")
[Out]
1/105*(2*(3*c*d^2*e + 4*b*d*e^2 + 24*a*e^3)*x^7 + 7*(3*c*d^3 + 4*b*d^2*e + 24*a*d*e^2)*x^5 + 105*a*d^3*x + 35*
(b*d^3 + 6*a*d^2*e)*x^3)*sqrt(e*x^2 + d)/(d^4*e^4*x^8 + 4*d^5*e^3*x^6 + 6*d^6*e^2*x^4 + 4*d^7*e*x^2 + d^8)
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Sympy [B] time = 145.542, size = 1989, normalized size = 15.79 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c*x**4+b*x**2+a)/(e*x**2+d)**(9/2),x)
[Out]
a*(35*d**14*x/(35*d**(37/2)*sqrt(1 + e*x**2/d) + 210*d**(35/2)*e*x**2*sqrt(1 + e*x**2/d) + 525*d**(33/2)*e**2*
x**4*sqrt(1 + e*x**2/d) + 700*d**(31/2)*e**3*x**6*sqrt(1 + e*x**2/d) + 525*d**(29/2)*e**4*x**8*sqrt(1 + e*x**2
/d) + 210*d**(27/2)*e**5*x**10*sqrt(1 + e*x**2/d) + 35*d**(25/2)*e**6*x**12*sqrt(1 + e*x**2/d)) + 175*d**13*e*
x**3/(35*d**(37/2)*sqrt(1 + e*x**2/d) + 210*d**(35/2)*e*x**2*sqrt(1 + e*x**2/d) + 525*d**(33/2)*e**2*x**4*sqrt
(1 + e*x**2/d) + 700*d**(31/2)*e**3*x**6*sqrt(1 + e*x**2/d) + 525*d**(29/2)*e**4*x**8*sqrt(1 + e*x**2/d) + 210
*d**(27/2)*e**5*x**10*sqrt(1 + e*x**2/d) + 35*d**(25/2)*e**6*x**12*sqrt(1 + e*x**2/d)) + 371*d**12*e**2*x**5/(
35*d**(37/2)*sqrt(1 + e*x**2/d) + 210*d**(35/2)*e*x**2*sqrt(1 + e*x**2/d) + 525*d**(33/2)*e**2*x**4*sqrt(1 + e
*x**2/d) + 700*d**(31/2)*e**3*x**6*sqrt(1 + e*x**2/d) + 525*d**(29/2)*e**4*x**8*sqrt(1 + e*x**2/d) + 210*d**(2
7/2)*e**5*x**10*sqrt(1 + e*x**2/d) + 35*d**(25/2)*e**6*x**12*sqrt(1 + e*x**2/d)) + 429*d**11*e**3*x**7/(35*d**
(37/2)*sqrt(1 + e*x**2/d) + 210*d**(35/2)*e*x**2*sqrt(1 + e*x**2/d) + 525*d**(33/2)*e**2*x**4*sqrt(1 + e*x**2/
d) + 700*d**(31/2)*e**3*x**6*sqrt(1 + e*x**2/d) + 525*d**(29/2)*e**4*x**8*sqrt(1 + e*x**2/d) + 210*d**(27/2)*e
**5*x**10*sqrt(1 + e*x**2/d) + 35*d**(25/2)*e**6*x**12*sqrt(1 + e*x**2/d)) + 286*d**10*e**4*x**9/(35*d**(37/2)
*sqrt(1 + e*x**2/d) + 210*d**(35/2)*e*x**2*sqrt(1 + e*x**2/d) + 525*d**(33/2)*e**2*x**4*sqrt(1 + e*x**2/d) + 7
00*d**(31/2)*e**3*x**6*sqrt(1 + e*x**2/d) + 525*d**(29/2)*e**4*x**8*sqrt(1 + e*x**2/d) + 210*d**(27/2)*e**5*x*
*10*sqrt(1 + e*x**2/d) + 35*d**(25/2)*e**6*x**12*sqrt(1 + e*x**2/d)) + 104*d**9*e**5*x**11/(35*d**(37/2)*sqrt(
1 + e*x**2/d) + 210*d**(35/2)*e*x**2*sqrt(1 + e*x**2/d) + 525*d**(33/2)*e**2*x**4*sqrt(1 + e*x**2/d) + 700*d**
(31/2)*e**3*x**6*sqrt(1 + e*x**2/d) + 525*d**(29/2)*e**4*x**8*sqrt(1 + e*x**2/d) + 210*d**(27/2)*e**5*x**10*sq
rt(1 + e*x**2/d) + 35*d**(25/2)*e**6*x**12*sqrt(1 + e*x**2/d)) + 16*d**8*e**6*x**13/(35*d**(37/2)*sqrt(1 + e*x
**2/d) + 210*d**(35/2)*e*x**2*sqrt(1 + e*x**2/d) + 525*d**(33/2)*e**2*x**4*sqrt(1 + e*x**2/d) + 700*d**(31/2)*
e**3*x**6*sqrt(1 + e*x**2/d) + 525*d**(29/2)*e**4*x**8*sqrt(1 + e*x**2/d) + 210*d**(27/2)*e**5*x**10*sqrt(1 +
e*x**2/d) + 35*d**(25/2)*e**6*x**12*sqrt(1 + e*x**2/d))) + b*(35*d**5*x**3/(105*d**(19/2)*sqrt(1 + e*x**2/d) +
420*d**(17/2)*e*x**2*sqrt(1 + e*x**2/d) + 630*d**(15/2)*e**2*x**4*sqrt(1 + e*x**2/d) + 420*d**(13/2)*e**3*x**
6*sqrt(1 + e*x**2/d) + 105*d**(11/2)*e**4*x**8*sqrt(1 + e*x**2/d)) + 63*d**4*e*x**5/(105*d**(19/2)*sqrt(1 + e*
x**2/d) + 420*d**(17/2)*e*x**2*sqrt(1 + e*x**2/d) + 630*d**(15/2)*e**2*x**4*sqrt(1 + e*x**2/d) + 420*d**(13/2)
*e**3*x**6*sqrt(1 + e*x**2/d) + 105*d**(11/2)*e**4*x**8*sqrt(1 + e*x**2/d)) + 36*d**3*e**2*x**7/(105*d**(19/2)
*sqrt(1 + e*x**2/d) + 420*d**(17/2)*e*x**2*sqrt(1 + e*x**2/d) + 630*d**(15/2)*e**2*x**4*sqrt(1 + e*x**2/d) + 4
20*d**(13/2)*e**3*x**6*sqrt(1 + e*x**2/d) + 105*d**(11/2)*e**4*x**8*sqrt(1 + e*x**2/d)) + 8*d**2*e**3*x**9/(10
5*d**(19/2)*sqrt(1 + e*x**2/d) + 420*d**(17/2)*e*x**2*sqrt(1 + e*x**2/d) + 630*d**(15/2)*e**2*x**4*sqrt(1 + e*
x**2/d) + 420*d**(13/2)*e**3*x**6*sqrt(1 + e*x**2/d) + 105*d**(11/2)*e**4*x**8*sqrt(1 + e*x**2/d))) + c*(7*d*x
**5/(35*d**(11/2)*sqrt(1 + e*x**2/d) + 105*d**(9/2)*e*x**2*sqrt(1 + e*x**2/d) + 105*d**(7/2)*e**2*x**4*sqrt(1
+ e*x**2/d) + 35*d**(5/2)*e**3*x**6*sqrt(1 + e*x**2/d)) + 2*e*x**7/(35*d**(11/2)*sqrt(1 + e*x**2/d) + 105*d**(
9/2)*e*x**2*sqrt(1 + e*x**2/d) + 105*d**(7/2)*e**2*x**4*sqrt(1 + e*x**2/d) + 35*d**(5/2)*e**3*x**6*sqrt(1 + e*
x**2/d)))
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Giac [A] time = 1.17566, size = 153, normalized size = 1.21 \begin{align*} \frac{{\left ({\left (x^{2}{\left (\frac{2 \,{\left (3 \, c d^{2} e^{4} + 4 \, b d e^{5} + 24 \, a e^{6}\right )} x^{2} e^{\left (-3\right )}}{d^{4}} + \frac{7 \,{\left (3 \, c d^{3} e^{3} + 4 \, b d^{2} e^{4} + 24 \, a d e^{5}\right )} e^{\left (-3\right )}}{d^{4}}\right )} + \frac{35 \,{\left (b d^{3} e^{3} + 6 \, a d^{2} e^{4}\right )} e^{\left (-3\right )}}{d^{4}}\right )} x^{2} + \frac{105 \, a}{d}\right )} x}{105 \,{\left (x^{2} e + d\right )}^{\frac{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c*x^4+b*x^2+a)/(e*x^2+d)^(9/2),x, algorithm="giac")
[Out]
1/105*((x^2*(2*(3*c*d^2*e^4 + 4*b*d*e^5 + 24*a*e^6)*x^2*e^(-3)/d^4 + 7*(3*c*d^3*e^3 + 4*b*d^2*e^4 + 24*a*d*e^5
)*e^(-3)/d^4) + 35*(b*d^3*e^3 + 6*a*d^2*e^4)*e^(-3)/d^4)*x^2 + 105*a/d)*x/(x^2*e + d)^(7/2)