3.144 \(\int \frac{a+b x^2+c x^4}{x^8 \sqrt{d-e x} \sqrt{d+e x}} \, dx\)
Optimal. Leaf size=226 \[ -\frac{2 e^2 \left (d^2-e^2 x^2\right ) \left (24 a e^4+28 b d^2 e^2+35 c d^4\right )}{105 d^8 x \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (d^2-e^2 x^2\right ) \left (24 a e^4+28 b d^2 e^2+35 c d^4\right )}{105 d^6 x^3 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (d^2-e^2 x^2\right ) \left (6 a e^2+7 b d^2\right )}{35 d^4 x^5 \sqrt{d-e x} \sqrt{d+e x}}-\frac{a \left (d^2-e^2 x^2\right )}{7 d^2 x^7 \sqrt{d-e x} \sqrt{d+e x}} \]
[Out]
-(a*(d^2 - e^2*x^2))/(7*d^2*x^7*Sqrt[d - e*x]*Sqrt[d + e*x]) - ((7*b*d^2 + 6*a*e^2)*(d^2 - e^2*x^2))/(35*d^4*x
^5*Sqrt[d - e*x]*Sqrt[d + e*x]) - ((35*c*d^4 + 28*b*d^2*e^2 + 24*a*e^4)*(d^2 - e^2*x^2))/(105*d^6*x^3*Sqrt[d -
e*x]*Sqrt[d + e*x]) - (2*e^2*(35*c*d^4 + 28*b*d^2*e^2 + 24*a*e^4)*(d^2 - e^2*x^2))/(105*d^8*x*Sqrt[d - e*x]*S
qrt[d + e*x])
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Rubi [A] time = 0.178386, antiderivative size = 226, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 5, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used =
{520, 1265, 453, 271, 264} \[ -\frac{2 e^2 \left (d^2-e^2 x^2\right ) \left (24 a e^4+28 b d^2 e^2+35 c d^4\right )}{105 d^8 x \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (d^2-e^2 x^2\right ) \left (24 a e^4+28 b d^2 e^2+35 c d^4\right )}{105 d^6 x^3 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (d^2-e^2 x^2\right ) \left (6 a e^2+7 b d^2\right )}{35 d^4 x^5 \sqrt{d-e x} \sqrt{d+e x}}-\frac{a \left (d^2-e^2 x^2\right )}{7 d^2 x^7 \sqrt{d-e x} \sqrt{d+e x}} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*x^2 + c*x^4)/(x^8*Sqrt[d - e*x]*Sqrt[d + e*x]),x]
[Out]
-(a*(d^2 - e^2*x^2))/(7*d^2*x^7*Sqrt[d - e*x]*Sqrt[d + e*x]) - ((7*b*d^2 + 6*a*e^2)*(d^2 - e^2*x^2))/(35*d^4*x
^5*Sqrt[d - e*x]*Sqrt[d + e*x]) - ((35*c*d^4 + 28*b*d^2*e^2 + 24*a*e^4)*(d^2 - e^2*x^2))/(105*d^6*x^3*Sqrt[d -
e*x]*Sqrt[d + e*x]) - (2*e^2*(35*c*d^4 + 28*b*d^2*e^2 + 24*a*e^4)*(d^2 - e^2*x^2))/(105*d^8*x*Sqrt[d - e*x]*S
qrt[d + e*x])
Rule 520
Int[(u_.)*((c_) + (d_.)*(x_)^(n_.) + (e_.)*(x_)^(n2_.))^(q_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b
2_.)*(x_)^(non2_.))^(p_.), x_Symbol] :> Dist[((a1 + b1*x^(n/2))^FracPart[p]*(a2 + b2*x^(n/2))^FracPart[p])/(a1
*a2 + b1*b2*x^n)^FracPart[p], Int[u*(a1*a2 + b1*b2*x^n)^p*(c + d*x^n + e*x^(2*n))^q, x], x] /; FreeQ[{a1, b1,
a2, b2, c, d, e, n, p, q}, x] && EqQ[non2, n/2] && EqQ[n2, 2*n] && EqQ[a2*b1 + a1*b2, 0]
Rule 1265
Int[((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Wit
h[{Qx = PolynomialQuotient[(a + b*x^2 + c*x^4)^p, f*x, x], R = PolynomialRemainder[(a + b*x^2 + c*x^4)^p, f*x,
x]}, Simp[(R*(f*x)^(m + 1)*(d + e*x^2)^(q + 1))/(d*f*(m + 1)), x] + Dist[1/(d*f^2*(m + 1)), Int[(f*x)^(m + 2)
*(d + e*x^2)^q*ExpandToSum[(d*f*(m + 1)*Qx)/x - e*R*(m + 2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e, f, q},
x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && LtQ[m, -1]
Rule 453
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(c*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(a*e*(m + 1)), x] + Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)), In
t[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[n] ||
GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) && !ILtQ[p, -1]
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}{x^8 \sqrt{d-e x} \sqrt{d+e x}} \, dx &=\frac{\sqrt{d^2-e^2 x^2} \int \frac{a+b x^2+c x^4}{x^8 \sqrt{d^2-e^2 x^2}} \, dx}{\sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{a \left (d^2-e^2 x^2\right )}{7 d^2 x^7 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\sqrt{d^2-e^2 x^2} \int \frac{-7 b d^2-6 a e^2-7 c d^2 x^2}{x^6 \sqrt{d^2-e^2 x^2}} \, dx}{7 d^2 \sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{a \left (d^2-e^2 x^2\right )}{7 d^2 x^7 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (7 b d^2+6 a e^2\right ) \left (d^2-e^2 x^2\right )}{35 d^4 x^5 \sqrt{d-e x} \sqrt{d+e x}}+\frac{\left (\left (35 c d^4-4 e^2 \left (-7 b d^2-6 a e^2\right )\right ) \sqrt{d^2-e^2 x^2}\right ) \int \frac{1}{x^4 \sqrt{d^2-e^2 x^2}} \, dx}{35 d^4 \sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{a \left (d^2-e^2 x^2\right )}{7 d^2 x^7 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (7 b d^2+6 a e^2\right ) \left (d^2-e^2 x^2\right )}{35 d^4 x^5 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (35 c d^4+28 b d^2 e^2+24 a e^4\right ) \left (d^2-e^2 x^2\right )}{105 d^6 x^3 \sqrt{d-e x} \sqrt{d+e x}}+\frac{\left (2 e^2 \left (35 c d^4-4 e^2 \left (-7 b d^2-6 a e^2\right )\right ) \sqrt{d^2-e^2 x^2}\right ) \int \frac{1}{x^2 \sqrt{d^2-e^2 x^2}} \, dx}{105 d^6 \sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{a \left (d^2-e^2 x^2\right )}{7 d^2 x^7 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (7 b d^2+6 a e^2\right ) \left (d^2-e^2 x^2\right )}{35 d^4 x^5 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (35 c d^4+28 b d^2 e^2+24 a e^4\right ) \left (d^2-e^2 x^2\right )}{105 d^6 x^3 \sqrt{d-e x} \sqrt{d+e x}}-\frac{2 e^2 \left (35 c d^4+28 b d^2 e^2+24 a e^4\right ) \left (d^2-e^2 x^2\right )}{105 d^8 x \sqrt{d-e x} \sqrt{d+e x}}\\ \end{align*}
Mathematica [A] time = 0.149583, size = 124, normalized size = 0.55 \[ -\frac{\sqrt{d-e x} \sqrt{d+e x} \left (3 a \left (6 d^4 e^2 x^2+8 d^2 e^4 x^4+5 d^6+16 e^6 x^6\right )+7 b \left (4 d^4 e^2 x^4+8 d^2 e^4 x^6+3 d^6 x^2\right )+35 c d^4 x^4 \left (d^2+2 e^2 x^2\right )\right )}{105 d^8 x^7} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + b*x^2 + c*x^4)/(x^8*Sqrt[d - e*x]*Sqrt[d + e*x]),x]
[Out]
-(Sqrt[d - e*x]*Sqrt[d + e*x]*(35*c*d^4*x^4*(d^2 + 2*e^2*x^2) + 7*b*(3*d^6*x^2 + 4*d^4*e^2*x^4 + 8*d^2*e^4*x^6
) + 3*a*(5*d^6 + 6*d^4*e^2*x^2 + 8*d^2*e^4*x^4 + 16*e^6*x^6)))/(105*d^8*x^7)
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Maple [A] time = 0.005, size = 118, normalized size = 0.5 \begin{align*} -{\frac{48\,a{e}^{6}{x}^{6}+56\,b{d}^{2}{e}^{4}{x}^{6}+70\,c{d}^{4}{e}^{2}{x}^{6}+24\,a{d}^{2}{e}^{4}{x}^{4}+28\,b{d}^{4}{e}^{2}{x}^{4}+35\,c{d}^{6}{x}^{4}+18\,a{d}^{4}{e}^{2}{x}^{2}+21\,b{d}^{6}{x}^{2}+15\,a{d}^{6}}{105\,{x}^{7}{d}^{8}}\sqrt{ex+d}\sqrt{-ex+d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((c*x^4+b*x^2+a)/x^8/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x)
[Out]
-1/105*(e*x+d)^(1/2)*(-e*x+d)^(1/2)*(48*a*e^6*x^6+56*b*d^2*e^4*x^6+70*c*d^4*e^2*x^6+24*a*d^2*e^4*x^4+28*b*d^4*
e^2*x^4+35*c*d^6*x^4+18*a*d^4*e^2*x^2+21*b*d^6*x^2+15*a*d^6)/x^7/d^8
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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((c*x^4+b*x^2+a)/x^8/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 1.54043, size = 251, normalized size = 1.11 \begin{align*} -\frac{{\left (15 \, a d^{6} + 2 \,{\left (35 \, c d^{4} e^{2} + 28 \, b d^{2} e^{4} + 24 \, a e^{6}\right )} x^{6} +{\left (35 \, c d^{6} + 28 \, b d^{4} e^{2} + 24 \, a d^{2} e^{4}\right )} x^{4} + 3 \,{\left (7 \, b d^{6} + 6 \, a d^{4} e^{2}\right )} x^{2}\right )} \sqrt{e x + d} \sqrt{-e x + d}}{105 \, d^{8} x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c*x^4+b*x^2+a)/x^8/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x, algorithm="fricas")
[Out]
-1/105*(15*a*d^6 + 2*(35*c*d^4*e^2 + 28*b*d^2*e^4 + 24*a*e^6)*x^6 + (35*c*d^6 + 28*b*d^4*e^2 + 24*a*d^2*e^4)*x
^4 + 3*(7*b*d^6 + 6*a*d^4*e^2)*x^2)*sqrt(e*x + d)*sqrt(-e*x + d)/(d^8*x^7)
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c*x**4+b*x**2+a)/x**8/(-e*x+d)**(1/2)/(e*x+d)**(1/2),x)
[Out]
Timed out
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Giac [B] time = 2.50619, size = 2048, normalized size = 9.06 \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)/x^8/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x, algorithm="giac")
[Out]
-4/105*(105*c*d^4*((sqrt(2)*sqrt(d) - sqrt(-x*e + d))/sqrt(x*e + d) - sqrt(x*e + d)/(sqrt(2)*sqrt(d) - sqrt(-x
*e + d)))^13*e^4 + 105*b*d^2*((sqrt(2)*sqrt(d) - sqrt(-x*e + d))/sqrt(x*e + d) - sqrt(x*e + d)/(sqrt(2)*sqrt(d
) - sqrt(-x*e + d)))^13*e^6 - 1960*c*d^4*((sqrt(2)*sqrt(d) - sqrt(-x*e + d))/sqrt(x*e + d) - sqrt(x*e + d)/(sq
rt(2)*sqrt(d) - sqrt(-x*e + d)))^11*e^4 + 105*a*((sqrt(2)*sqrt(d) - sqrt(-x*e + d))/sqrt(x*e + d) - sqrt(x*e +
d)/(sqrt(2)*sqrt(d) - sqrt(-x*e + d)))^13*e^8 - 1400*b*d^2*((sqrt(2)*sqrt(d) - sqrt(-x*e + d))/sqrt(x*e + d)
- sqrt(x*e + d)/(sqrt(2)*sqrt(d) - sqrt(-x*e + d)))^11*e^6 + 16240*c*d^4*((sqrt(2)*sqrt(d) - sqrt(-x*e + d))/s
qrt(x*e + d) - sqrt(x*e + d)/(sqrt(2)*sqrt(d) - sqrt(-x*e + d)))^9*e^4 - 840*a*((sqrt(2)*sqrt(d) - sqrt(-x*e +
d))/sqrt(x*e + d) - sqrt(x*e + d)/(sqrt(2)*sqrt(d) - sqrt(-x*e + d)))^11*e^8 + 12656*b*d^2*((sqrt(2)*sqrt(d)
- sqrt(-x*e + d))/sqrt(x*e + d) - sqrt(x*e + d)/(sqrt(2)*sqrt(d) - sqrt(-x*e + d)))^9*e^6 - 80640*c*d^4*((sqrt
(2)*sqrt(d) - sqrt(-x*e + d))/sqrt(x*e + d) - sqrt(x*e + d)/(sqrt(2)*sqrt(d) - sqrt(-x*e + d)))^7*e^4 + 14448*
a*((sqrt(2)*sqrt(d) - sqrt(-x*e + d))/sqrt(x*e + d) - sqrt(x*e + d)/(sqrt(2)*sqrt(d) - sqrt(-x*e + d)))^9*e^8
- 69888*b*d^2*((sqrt(2)*sqrt(d) - sqrt(-x*e + d))/sqrt(x*e + d) - sqrt(x*e + d)/(sqrt(2)*sqrt(d) - sqrt(-x*e +
d)))^7*e^6 + 259840*c*d^4*((sqrt(2)*sqrt(d) - sqrt(-x*e + d))/sqrt(x*e + d) - sqrt(x*e + d)/(sqrt(2)*sqrt(d)
- sqrt(-x*e + d)))^5*e^4 - 40704*a*((sqrt(2)*sqrt(d) - sqrt(-x*e + d))/sqrt(x*e + d) - sqrt(x*e + d)/(sqrt(2)*
sqrt(d) - sqrt(-x*e + d)))^7*e^8 + 202496*b*d^2*((sqrt(2)*sqrt(d) - sqrt(-x*e + d))/sqrt(x*e + d) - sqrt(x*e +
d)/(sqrt(2)*sqrt(d) - sqrt(-x*e + d)))^5*e^6 - 501760*c*d^4*((sqrt(2)*sqrt(d) - sqrt(-x*e + d))/sqrt(x*e + d)
- sqrt(x*e + d)/(sqrt(2)*sqrt(d) - sqrt(-x*e + d)))^3*e^4 + 231168*a*((sqrt(2)*sqrt(d) - sqrt(-x*e + d))/sqrt
(x*e + d) - sqrt(x*e + d)/(sqrt(2)*sqrt(d) - sqrt(-x*e + d)))^5*e^8 - 358400*b*d^2*((sqrt(2)*sqrt(d) - sqrt(-x
*e + d))/sqrt(x*e + d) - sqrt(x*e + d)/(sqrt(2)*sqrt(d) - sqrt(-x*e + d)))^3*e^6 + 430080*c*d^4*((sqrt(2)*sqrt
(d) - sqrt(-x*e + d))/sqrt(x*e + d) - sqrt(x*e + d)/(sqrt(2)*sqrt(d) - sqrt(-x*e + d)))*e^4 - 215040*a*((sqrt(
2)*sqrt(d) - sqrt(-x*e + d))/sqrt(x*e + d) - sqrt(x*e + d)/(sqrt(2)*sqrt(d) - sqrt(-x*e + d)))^3*e^8 + 430080*
b*d^2*((sqrt(2)*sqrt(d) - sqrt(-x*e + d))/sqrt(x*e + d) - sqrt(x*e + d)/(sqrt(2)*sqrt(d) - sqrt(-x*e + d)))*e^
6 + 430080*a*((sqrt(2)*sqrt(d) - sqrt(-x*e + d))/sqrt(x*e + d) - sqrt(x*e + d)/(sqrt(2)*sqrt(d) - sqrt(-x*e +
d)))*e^8)*e^(-1)/((((sqrt(2)*sqrt(d) - sqrt(-x*e + d))/sqrt(x*e + d) - sqrt(x*e + d)/(sqrt(2)*sqrt(d) - sqrt(-
x*e + d)))^2 - 4)^7*d^8)