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

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

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 71, normalized size of antiderivative = 1.00, 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 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 )^{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*}

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Mathematica [A]  time = 0.03, size = 55, normalized size = 0.77 \[ \frac {-3 a^3 e+15 a^2 c d x+20 a c^2 d x^3+8 c^3 d x^5}{15 a^3 c \left (a+c x^2\right )^{5/2}} \]

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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fricas [A]  time = 0.90, size = 85, normalized size = 1.20 \[ \frac {{\left (8 \, c^{3} d x^{5} + 20 \, a c^{2} d x^{3} + 15 \, a^{2} c d x - 3 \, a^{3} e\right )} \sqrt {c x^{2} + a}}{15 \, {\left (a^{3} c^{4} x^{6} + 3 \, a^{4} c^{3} x^{4} + 3 \, a^{5} c^{2} x^{2} + a^{6} c\right )}} \]

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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giac [A]  time = 0.25, size = 53, normalized size = 0.75 \[ \frac {{\left (4 \, {\left (\frac {2 \, c^{2} d x^{2}}{a^{3}} + \frac {5 \, c d}{a^{2}}\right )} x^{2} + \frac {15 \, d}{a}\right )} x - \frac {3 \, e}{c}}{15 \, {\left (c x^{2} + a\right )}^{\frac {5}{2}}} \]

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)

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maple [A]  time = 0.05, size = 52, normalized size = 0.73 \[ -\frac {-8 c^{3} d \,x^{5}-20 c^{2} d \,x^{3} a -15 d x \,a^{2} c +3 a^{3} e}{15 \left (c \,x^{2}+a \right )^{\frac {5}{2}} a^{3} c} \]

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 = 0.53, size = 64, normalized size = 0.90 \[ \frac {8 \, d x}{15 \, \sqrt {c x^{2} + a} a^{3}} + \frac {4 \, d x}{15 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} a^{2}} + \frac {d x}{5 \, {\left (c x^{2} + a\right )}^{\frac {5}{2}} a} - \frac {e}{5 \, {\left (c x^{2} + a\right )}^{\frac {5}{2}} c} \]

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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mupad [B]  time = 1.18, size = 59, normalized size = 0.83 \[ \frac {8\,c\,d\,x\,{\left (c\,x^2+a\right )}^2-3\,a^3\,e+3\,a^2\,c\,d\,x+4\,a\,c\,d\,x\,\left (c\,x^2+a\right )}{15\,a^3\,c\,{\left (c\,x^2+a\right )}^{5/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)/(a + c*x^2)^(7/2),x)

[Out]

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

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sympy [B]  time = 23.37, size = 486, normalized size = 6.85 \[ d \left (\frac {15 a^{5} x}{15 a^{\frac {17}{2}} \sqrt {1 + \frac {c x^{2}}{a}} + 45 a^{\frac {15}{2}} c x^{2} \sqrt {1 + \frac {c x^{2}}{a}} + 45 a^{\frac {13}{2}} c^{2} x^{4} \sqrt {1 + \frac {c x^{2}}{a}} + 15 a^{\frac {11}{2}} c^{3} x^{6} \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {35 a^{4} c x^{3}}{15 a^{\frac {17}{2}} \sqrt {1 + \frac {c x^{2}}{a}} + 45 a^{\frac {15}{2}} c x^{2} \sqrt {1 + \frac {c x^{2}}{a}} + 45 a^{\frac {13}{2}} c^{2} x^{4} \sqrt {1 + \frac {c x^{2}}{a}} + 15 a^{\frac {11}{2}} c^{3} x^{6} \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {28 a^{3} c^{2} x^{5}}{15 a^{\frac {17}{2}} \sqrt {1 + \frac {c x^{2}}{a}} + 45 a^{\frac {15}{2}} c x^{2} \sqrt {1 + \frac {c x^{2}}{a}} + 45 a^{\frac {13}{2}} c^{2} x^{4} \sqrt {1 + \frac {c x^{2}}{a}} + 15 a^{\frac {11}{2}} c^{3} x^{6} \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {8 a^{2} c^{3} x^{7}}{15 a^{\frac {17}{2}} \sqrt {1 + \frac {c x^{2}}{a}} + 45 a^{\frac {15}{2}} c x^{2} \sqrt {1 + \frac {c x^{2}}{a}} + 45 a^{\frac {13}{2}} c^{2} x^{4} \sqrt {1 + \frac {c x^{2}}{a}} + 15 a^{\frac {11}{2}} c^{3} x^{6} \sqrt {1 + \frac {c x^{2}}{a}}}\right ) + e \left (\begin {cases} - \frac {1}{5 a^{2} c \sqrt {a + c x^{2}} + 10 a c^{2} x^{2} \sqrt {a + c x^{2}} + 5 c^{3} x^{4} \sqrt {a + c x^{2}}} & \text {for}\: c \neq 0 \\\frac {x^{2}}{2 a^{\frac {7}{2}}} & \text {otherwise} \end {cases}\right ) \]

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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