3.383 \(\int \frac{d+e x}{(a+c x^2)^{9/2}} \, dx\)
Optimal. Leaf size=91 \[ \frac{16 d x}{35 a^4 \sqrt{a+c x^2}}+\frac{8 d x}{35 a^3 \left (a+c x^2\right )^{3/2}}+\frac{6 d x}{35 a^2 \left (a+c x^2\right )^{5/2}}-\frac{a e-c d x}{7 a c \left (a+c x^2\right )^{7/2}} \]
[Out]
-(a*e - c*d*x)/(7*a*c*(a + c*x^2)^(7/2)) + (6*d*x)/(35*a^2*(a + c*x^2)^(5/2)) + (8*d*x)/(35*a^3*(a + c*x^2)^(3
/2)) + (16*d*x)/(35*a^4*Sqrt[a + c*x^2])
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Rubi [A] time = 0.0211028, antiderivative size = 91, normalized size of antiderivative = 1.,
number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used =
{639, 192, 191} \[ \frac{16 d x}{35 a^4 \sqrt{a+c x^2}}+\frac{8 d x}{35 a^3 \left (a+c x^2\right )^{3/2}}+\frac{6 d x}{35 a^2 \left (a+c x^2\right )^{5/2}}-\frac{a e-c d x}{7 a c \left (a+c x^2\right )^{7/2}} \]
Antiderivative was successfully verified.
[In]
Int[(d + e*x)/(a + c*x^2)^(9/2),x]
[Out]
-(a*e - c*d*x)/(7*a*c*(a + c*x^2)^(7/2)) + (6*d*x)/(35*a^2*(a + c*x^2)^(5/2)) + (8*d*x)/(35*a^3*(a + c*x^2)^(3
/2)) + (16*d*x)/(35*a^4*Sqrt[a + c*x^2])
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 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 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]
Rubi steps
\begin{align*} \int \frac{d+e x}{\left (a+c x^2\right )^{9/2}} \, dx &=-\frac{a e-c d x}{7 a c \left (a+c x^2\right )^{7/2}}+\frac{(6 d) \int \frac{1}{\left (a+c x^2\right )^{7/2}} \, dx}{7 a}\\ &=-\frac{a e-c d x}{7 a c \left (a+c x^2\right )^{7/2}}+\frac{6 d x}{35 a^2 \left (a+c x^2\right )^{5/2}}+\frac{(24 d) \int \frac{1}{\left (a+c x^2\right )^{5/2}} \, dx}{35 a^2}\\ &=-\frac{a e-c d x}{7 a c \left (a+c x^2\right )^{7/2}}+\frac{6 d x}{35 a^2 \left (a+c x^2\right )^{5/2}}+\frac{8 d x}{35 a^3 \left (a+c x^2\right )^{3/2}}+\frac{(16 d) \int \frac{1}{\left (a+c x^2\right )^{3/2}} \, dx}{35 a^3}\\ &=-\frac{a e-c d x}{7 a c \left (a+c x^2\right )^{7/2}}+\frac{6 d x}{35 a^2 \left (a+c x^2\right )^{5/2}}+\frac{8 d x}{35 a^3 \left (a+c x^2\right )^{3/2}}+\frac{16 d x}{35 a^4 \sqrt{a+c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0337421, size = 67, normalized size = 0.74 \[ \frac{70 a^2 c^2 d x^3+35 a^3 c d x-5 a^4 e+56 a c^3 d x^5+16 c^4 d x^7}{35 a^4 c \left (a+c x^2\right )^{7/2}} \]
Antiderivative was successfully verified.
[In]
Integrate[(d + e*x)/(a + c*x^2)^(9/2),x]
[Out]
(-5*a^4*e + 35*a^3*c*d*x + 70*a^2*c^2*d*x^3 + 56*a*c^3*d*x^5 + 16*c^4*d*x^7)/(35*a^4*c*(a + c*x^2)^(7/2))
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Maple [A] time = 0.005, size = 64, normalized size = 0.7 \begin{align*} -{\frac{-16\,{c}^{4}d{x}^{7}-56\,{c}^{3}d{x}^{5}a-70\,{c}^{2}d{x}^{3}{a}^{2}-35\,dx{a}^{3}c+5\,e{a}^{4}}{35\,{a}^{4}c} \left ( c{x}^{2}+a \right ) ^{-{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((e*x+d)/(c*x^2+a)^(9/2),x)
[Out]
-1/35*(-16*c^4*d*x^7-56*a*c^3*d*x^5-70*a^2*c^2*d*x^3-35*a^3*c*d*x+5*a^4*e)/(c*x^2+a)^(7/2)/a^4/c
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Maxima [A] time = 0.996122, size = 108, normalized size = 1.19 \begin{align*} \frac{16 \, d x}{35 \, \sqrt{c x^{2} + a} a^{4}} + \frac{8 \, d x}{35 \,{\left (c x^{2} + a\right )}^{\frac{3}{2}} a^{3}} + \frac{6 \, d x}{35 \,{\left (c x^{2} + a\right )}^{\frac{5}{2}} a^{2}} + \frac{d x}{7 \,{\left (c x^{2} + a\right )}^{\frac{7}{2}} a} - \frac{e}{7 \,{\left (c x^{2} + a\right )}^{\frac{7}{2}} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)/(c*x^2+a)^(9/2),x, algorithm="maxima")
[Out]
16/35*d*x/(sqrt(c*x^2 + a)*a^4) + 8/35*d*x/((c*x^2 + a)^(3/2)*a^3) + 6/35*d*x/((c*x^2 + a)^(5/2)*a^2) + 1/7*d*
x/((c*x^2 + a)^(7/2)*a) - 1/7*e/((c*x^2 + a)^(7/2)*c)
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Fricas [A] time = 1.48138, size = 227, normalized size = 2.49 \begin{align*} \frac{{\left (16 \, c^{4} d x^{7} + 56 \, a c^{3} d x^{5} + 70 \, a^{2} c^{2} d x^{3} + 35 \, a^{3} c d x - 5 \, a^{4} e\right )} \sqrt{c x^{2} + a}}{35 \,{\left (a^{4} c^{5} x^{8} + 4 \, a^{5} c^{4} x^{6} + 6 \, a^{6} c^{3} x^{4} + 4 \, a^{7} c^{2} x^{2} + a^{8} c\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)/(c*x^2+a)^(9/2),x, algorithm="fricas")
[Out]
1/35*(16*c^4*d*x^7 + 56*a*c^3*d*x^5 + 70*a^2*c^2*d*x^3 + 35*a^3*c*d*x - 5*a^4*e)*sqrt(c*x^2 + a)/(a^4*c^5*x^8
+ 4*a^5*c^4*x^6 + 6*a^6*c^3*x^4 + 4*a^7*c^2*x^2 + a^8*c)
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Sympy [B] time = 77.8094, size = 1360, normalized size = 14.95 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)/(c*x**2+a)**(9/2),x)
[Out]
d*(35*a**14*x/(35*a**(37/2)*sqrt(1 + c*x**2/a) + 210*a**(35/2)*c*x**2*sqrt(1 + c*x**2/a) + 525*a**(33/2)*c**2*
x**4*sqrt(1 + c*x**2/a) + 700*a**(31/2)*c**3*x**6*sqrt(1 + c*x**2/a) + 525*a**(29/2)*c**4*x**8*sqrt(1 + c*x**2
/a) + 210*a**(27/2)*c**5*x**10*sqrt(1 + c*x**2/a) + 35*a**(25/2)*c**6*x**12*sqrt(1 + c*x**2/a)) + 175*a**13*c*
x**3/(35*a**(37/2)*sqrt(1 + c*x**2/a) + 210*a**(35/2)*c*x**2*sqrt(1 + c*x**2/a) + 525*a**(33/2)*c**2*x**4*sqrt
(1 + c*x**2/a) + 700*a**(31/2)*c**3*x**6*sqrt(1 + c*x**2/a) + 525*a**(29/2)*c**4*x**8*sqrt(1 + c*x**2/a) + 210
*a**(27/2)*c**5*x**10*sqrt(1 + c*x**2/a) + 35*a**(25/2)*c**6*x**12*sqrt(1 + c*x**2/a)) + 371*a**12*c**2*x**5/(
35*a**(37/2)*sqrt(1 + c*x**2/a) + 210*a**(35/2)*c*x**2*sqrt(1 + c*x**2/a) + 525*a**(33/2)*c**2*x**4*sqrt(1 + c
*x**2/a) + 700*a**(31/2)*c**3*x**6*sqrt(1 + c*x**2/a) + 525*a**(29/2)*c**4*x**8*sqrt(1 + c*x**2/a) + 210*a**(2
7/2)*c**5*x**10*sqrt(1 + c*x**2/a) + 35*a**(25/2)*c**6*x**12*sqrt(1 + c*x**2/a)) + 429*a**11*c**3*x**7/(35*a**
(37/2)*sqrt(1 + c*x**2/a) + 210*a**(35/2)*c*x**2*sqrt(1 + c*x**2/a) + 525*a**(33/2)*c**2*x**4*sqrt(1 + c*x**2/
a) + 700*a**(31/2)*c**3*x**6*sqrt(1 + c*x**2/a) + 525*a**(29/2)*c**4*x**8*sqrt(1 + c*x**2/a) + 210*a**(27/2)*c
**5*x**10*sqrt(1 + c*x**2/a) + 35*a**(25/2)*c**6*x**12*sqrt(1 + c*x**2/a)) + 286*a**10*c**4*x**9/(35*a**(37/2)
*sqrt(1 + c*x**2/a) + 210*a**(35/2)*c*x**2*sqrt(1 + c*x**2/a) + 525*a**(33/2)*c**2*x**4*sqrt(1 + c*x**2/a) + 7
00*a**(31/2)*c**3*x**6*sqrt(1 + c*x**2/a) + 525*a**(29/2)*c**4*x**8*sqrt(1 + c*x**2/a) + 210*a**(27/2)*c**5*x*
*10*sqrt(1 + c*x**2/a) + 35*a**(25/2)*c**6*x**12*sqrt(1 + c*x**2/a)) + 104*a**9*c**5*x**11/(35*a**(37/2)*sqrt(
1 + c*x**2/a) + 210*a**(35/2)*c*x**2*sqrt(1 + c*x**2/a) + 525*a**(33/2)*c**2*x**4*sqrt(1 + c*x**2/a) + 700*a**
(31/2)*c**3*x**6*sqrt(1 + c*x**2/a) + 525*a**(29/2)*c**4*x**8*sqrt(1 + c*x**2/a) + 210*a**(27/2)*c**5*x**10*sq
rt(1 + c*x**2/a) + 35*a**(25/2)*c**6*x**12*sqrt(1 + c*x**2/a)) + 16*a**8*c**6*x**13/(35*a**(37/2)*sqrt(1 + c*x
**2/a) + 210*a**(35/2)*c*x**2*sqrt(1 + c*x**2/a) + 525*a**(33/2)*c**2*x**4*sqrt(1 + c*x**2/a) + 700*a**(31/2)*
c**3*x**6*sqrt(1 + c*x**2/a) + 525*a**(29/2)*c**4*x**8*sqrt(1 + c*x**2/a) + 210*a**(27/2)*c**5*x**10*sqrt(1 +
c*x**2/a) + 35*a**(25/2)*c**6*x**12*sqrt(1 + c*x**2/a))) + e*Piecewise((-1/(7*a**3*c*sqrt(a + c*x**2) + 21*a**
2*c**2*x**2*sqrt(a + c*x**2) + 21*a*c**3*x**4*sqrt(a + c*x**2) + 7*c**4*x**6*sqrt(a + c*x**2)), Ne(c, 0)), (x*
*2/(2*a**(9/2)), True))
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Giac [A] time = 1.34884, size = 92, normalized size = 1.01 \begin{align*} \frac{{\left (2 \,{\left (4 \,{\left (\frac{2 \, c^{3} d x^{2}}{a^{4}} + \frac{7 \, c^{2} d}{a^{3}}\right )} x^{2} + \frac{35 \, c d}{a^{2}}\right )} x^{2} + \frac{35 \, d}{a}\right )} x - \frac{5 \, e}{c}}{35 \,{\left (c x^{2} + a\right )}^{\frac{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)/(c*x^2+a)^(9/2),x, algorithm="giac")
[Out]
1/35*((2*(4*(2*c^3*d*x^2/a^4 + 7*c^2*d/a^3)*x^2 + 35*c*d/a^2)*x^2 + 35*d/a)*x - 5*e/c)/(c*x^2 + a)^(7/2)