3.4.83 \(\int \frac {d+e x}{(a+c x^2)^{9/2}} \, dx\)

Optimal. Leaf size=91 \[ \frac {16 d x}{35 a^4 \sqrt {a+c x^2}}+\frac {8 d x}{35 a^3 \left (a+c x^2\right )^{3/2}}+\frac {6 d x}{35 a^2 \left (a+c x^2\right )^{5/2}}+\frac {c d x-a e}{7 a c \left (a+c x^2\right )^{7/2}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 91, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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}} \end {gather*}

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 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 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]

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 = 0.03, size = 67, normalized size = 0.74 \begin {gather*} \frac {-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 \left (a+c x^2\right )^{7/2}} \end {gather*}

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))

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.65, size = 67, normalized size = 0.74 \begin {gather*} \frac {-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 \left (a+c x^2\right )^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(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))

________________________________________________________________________________________

fricas [A]  time = 0.44, size = 108, normalized size = 1.19 \begin {gather*} \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 {gather*}

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)

________________________________________________________________________________________

giac [A]  time = 0.23, size = 68, normalized size = 0.75 \begin {gather*} \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 {gather*}

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)

________________________________________________________________________________________

maple [A]  time = 0.05, size = 64, normalized size = 0.70 \begin {gather*} -\frac {-16 c^{4} d \,x^{7}-56 c^{3} d \,x^{5} a -70 c^{2} d \,x^{3} a^{2}-35 d x \,a^{3} c +5 e \,a^{4}}{35 \left (c \,x^{2}+a \right )^{\frac {7}{2}} a^{4} c} \end {gather*}

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

________________________________________________________________________________________

maxima [A]  time = 0.61, size = 80, normalized size = 0.88 \begin {gather*} \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 {gather*}

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)

________________________________________________________________________________________

mupad [B]  time = 1.18, size = 74, normalized size = 0.81 \begin {gather*} \frac {16\,d\,x}{35\,a^4\,\sqrt {c\,x^2+a}}-\frac {\frac {e}{7\,c}-\frac {d\,x}{7\,a}}{{\left (c\,x^2+a\right )}^{7/2}}+\frac {8\,d\,x}{35\,a^3\,{\left (c\,x^2+a\right )}^{3/2}}+\frac {6\,d\,x}{35\,a^2\,{\left (c\,x^2+a\right )}^{5/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)/(a + c*x^2)^(9/2),x)

[Out]

(16*d*x)/(35*a^4*(a + c*x^2)^(1/2)) - (e/(7*c) - (d*x)/(7*a))/(a + c*x^2)^(7/2) + (8*d*x)/(35*a^3*(a + c*x^2)^
(3/2)) + (6*d*x)/(35*a^2*(a + c*x^2)^(5/2))

________________________________________________________________________________________

sympy [B]  time = 39.86, size = 1360, normalized size = 14.95

result too large to display

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))

________________________________________________________________________________________