3.382 \(\int \frac{d+e x}{(a+c x^2)^{7/2}} \, dx\)

Optimal. Leaf size=71 \[ \frac{8 d x}{15 a^3 \sqrt{a+c x^2}}+\frac{4 d x}{15 a^2 \left (a+c x^2\right )^{3/2}}-\frac{a e-c d x}{5 a c \left (a+c x^2\right )^{5/2}} \]

[Out]

-(a*e - c*d*x)/(5*a*c*(a + c*x^2)^(5/2)) + (4*d*x)/(15*a^2*(a + c*x^2)^(3/2)) + (8*d*x)/(15*a^3*Sqrt[a + c*x^2
])

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Rubi [A]  time = 0.0153587, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {639, 192, 191} \[ \frac{8 d x}{15 a^3 \sqrt{a+c x^2}}+\frac{4 d x}{15 a^2 \left (a+c x^2\right )^{3/2}}-\frac{a e-c d x}{5 a c \left (a+c x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)/(a + c*x^2)^(7/2),x]

[Out]

-(a*e - c*d*x)/(5*a*c*(a + c*x^2)^(5/2)) + (4*d*x)/(15*a^2*(a + c*x^2)^(3/2)) + (8*d*x)/(15*a^3*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 )^{7/2}} \, dx &=-\frac{a e-c d x}{5 a c \left (a+c x^2\right )^{5/2}}+\frac{(4 d) \int \frac{1}{\left (a+c x^2\right )^{5/2}} \, dx}{5 a}\\ &=-\frac{a e-c d x}{5 a c \left (a+c x^2\right )^{5/2}}+\frac{4 d x}{15 a^2 \left (a+c x^2\right )^{3/2}}+\frac{(8 d) \int \frac{1}{\left (a+c x^2\right )^{3/2}} \, dx}{15 a^2}\\ &=-\frac{a e-c d x}{5 a c \left (a+c x^2\right )^{5/2}}+\frac{4 d x}{15 a^2 \left (a+c x^2\right )^{3/2}}+\frac{8 d x}{15 a^3 \sqrt{a+c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0280477, size = 55, normalized size = 0.77 \[ \frac{15 a^2 c d x-3 a^3 e+20 a c^2 d x^3+8 c^3 d x^5}{15 a^3 c \left (a+c x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)/(a + c*x^2)^(7/2),x]

[Out]

(-3*a^3*e + 15*a^2*c*d*x + 20*a*c^2*d*x^3 + 8*c^3*d*x^5)/(15*a^3*c*(a + c*x^2)^(5/2))

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Maple [A]  time = 0.005, size = 52, normalized size = 0.7 \begin{align*} -{\frac{-8\,{c}^{3}d{x}^{5}-20\,{c}^{2}d{x}^{3}a-15\,dx{a}^{2}c+3\,e{a}^{3}}{15\,{a}^{3}c} \left ( c{x}^{2}+a \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)/(c*x^2+a)^(7/2),x)

[Out]

-1/15*(-8*c^3*d*x^5-20*a*c^2*d*x^3-15*a^2*c*d*x+3*a^3*e)/(c*x^2+a)^(5/2)/a^3/c

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Maxima [A]  time = 0.994232, size = 86, normalized size = 1.21 \begin{align*} \frac{8 \, d x}{15 \, \sqrt{c x^{2} + a} a^{3}} + \frac{4 \, d x}{15 \,{\left (c x^{2} + a\right )}^{\frac{3}{2}} a^{2}} + \frac{d x}{5 \,{\left (c x^{2} + a\right )}^{\frac{5}{2}} a} - \frac{e}{5 \,{\left (c x^{2} + a\right )}^{\frac{5}{2}} c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/(c*x^2+a)^(7/2),x, algorithm="maxima")

[Out]

8/15*d*x/(sqrt(c*x^2 + a)*a^3) + 4/15*d*x/((c*x^2 + a)^(3/2)*a^2) + 1/5*d*x/((c*x^2 + a)^(5/2)*a) - 1/5*e/((c*
x^2 + a)^(5/2)*c)

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Fricas [A]  time = 1.34547, size = 178, normalized size = 2.51 \begin{align*} \frac{{\left (8 \, c^{3} d x^{5} + 20 \, a c^{2} d x^{3} + 15 \, a^{2} c d x - 3 \, a^{3} e\right )} \sqrt{c x^{2} + a}}{15 \,{\left (a^{3} c^{4} x^{6} + 3 \, a^{4} c^{3} x^{4} + 3 \, a^{5} c^{2} x^{2} + a^{6} c\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/(c*x^2+a)^(7/2),x, algorithm="fricas")

[Out]

1/15*(8*c^3*d*x^5 + 20*a*c^2*d*x^3 + 15*a^2*c*d*x - 3*a^3*e)*sqrt(c*x^2 + a)/(a^3*c^4*x^6 + 3*a^4*c^3*x^4 + 3*
a^5*c^2*x^2 + a^6*c)

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Sympy [B]  time = 31.1897, size = 486, normalized size = 6.85 \begin{align*} d \left (\frac{15 a^{5} x}{15 a^{\frac{17}{2}} \sqrt{1 + \frac{c x^{2}}{a}} + 45 a^{\frac{15}{2}} c x^{2} \sqrt{1 + \frac{c x^{2}}{a}} + 45 a^{\frac{13}{2}} c^{2} x^{4} \sqrt{1 + \frac{c x^{2}}{a}} + 15 a^{\frac{11}{2}} c^{3} x^{6} \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{35 a^{4} c x^{3}}{15 a^{\frac{17}{2}} \sqrt{1 + \frac{c x^{2}}{a}} + 45 a^{\frac{15}{2}} c x^{2} \sqrt{1 + \frac{c x^{2}}{a}} + 45 a^{\frac{13}{2}} c^{2} x^{4} \sqrt{1 + \frac{c x^{2}}{a}} + 15 a^{\frac{11}{2}} c^{3} x^{6} \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{28 a^{3} c^{2} x^{5}}{15 a^{\frac{17}{2}} \sqrt{1 + \frac{c x^{2}}{a}} + 45 a^{\frac{15}{2}} c x^{2} \sqrt{1 + \frac{c x^{2}}{a}} + 45 a^{\frac{13}{2}} c^{2} x^{4} \sqrt{1 + \frac{c x^{2}}{a}} + 15 a^{\frac{11}{2}} c^{3} x^{6} \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{8 a^{2} c^{3} x^{7}}{15 a^{\frac{17}{2}} \sqrt{1 + \frac{c x^{2}}{a}} + 45 a^{\frac{15}{2}} c x^{2} \sqrt{1 + \frac{c x^{2}}{a}} + 45 a^{\frac{13}{2}} c^{2} x^{4} \sqrt{1 + \frac{c x^{2}}{a}} + 15 a^{\frac{11}{2}} c^{3} x^{6} \sqrt{1 + \frac{c x^{2}}{a}}}\right ) + e \left (\begin{cases} - \frac{1}{5 a^{2} c \sqrt{a + c x^{2}} + 10 a c^{2} x^{2} \sqrt{a + c x^{2}} + 5 c^{3} x^{4} \sqrt{a + c x^{2}}} & \text{for}\: c \neq 0 \\\frac{x^{2}}{2 a^{\frac{7}{2}}} & \text{otherwise} \end{cases}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/(c*x**2+a)**(7/2),x)

[Out]

d*(15*a**5*x/(15*a**(17/2)*sqrt(1 + c*x**2/a) + 45*a**(15/2)*c*x**2*sqrt(1 + c*x**2/a) + 45*a**(13/2)*c**2*x**
4*sqrt(1 + c*x**2/a) + 15*a**(11/2)*c**3*x**6*sqrt(1 + c*x**2/a)) + 35*a**4*c*x**3/(15*a**(17/2)*sqrt(1 + c*x*
*2/a) + 45*a**(15/2)*c*x**2*sqrt(1 + c*x**2/a) + 45*a**(13/2)*c**2*x**4*sqrt(1 + c*x**2/a) + 15*a**(11/2)*c**3
*x**6*sqrt(1 + c*x**2/a)) + 28*a**3*c**2*x**5/(15*a**(17/2)*sqrt(1 + c*x**2/a) + 45*a**(15/2)*c*x**2*sqrt(1 +
c*x**2/a) + 45*a**(13/2)*c**2*x**4*sqrt(1 + c*x**2/a) + 15*a**(11/2)*c**3*x**6*sqrt(1 + c*x**2/a)) + 8*a**2*c*
*3*x**7/(15*a**(17/2)*sqrt(1 + c*x**2/a) + 45*a**(15/2)*c*x**2*sqrt(1 + c*x**2/a) + 45*a**(13/2)*c**2*x**4*sqr
t(1 + c*x**2/a) + 15*a**(11/2)*c**3*x**6*sqrt(1 + c*x**2/a))) + e*Piecewise((-1/(5*a**2*c*sqrt(a + c*x**2) + 1
0*a*c**2*x**2*sqrt(a + c*x**2) + 5*c**3*x**4*sqrt(a + c*x**2)), Ne(c, 0)), (x**2/(2*a**(7/2)), True))

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Giac [A]  time = 1.19559, size = 72, normalized size = 1.01 \begin{align*} \frac{{\left (4 \,{\left (\frac{2 \, c^{2} d x^{2}}{a^{3}} + \frac{5 \, c d}{a^{2}}\right )} x^{2} + \frac{15 \, d}{a}\right )} x - \frac{3 \, e}{c}}{15 \,{\left (c x^{2} + a\right )}^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/(c*x^2+a)^(7/2),x, algorithm="giac")

[Out]

1/15*((4*(2*c^2*d*x^2/a^3 + 5*c*d/a^2)*x^2 + 15*d/a)*x - 3*e/c)/(c*x^2 + a)^(5/2)