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

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 = 0.0586142, 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 \[ \frac{16 d x}{35 a^4 \sqrt{a+c x^2}}+\frac{8 d x}{35 a^3 \left (a+c x^2\right )^{3/2}}+\frac{6 d x}{35 a^2 \left (a+c x^2\right )^{5/2}}-\frac{a e-c d x}{7 a c \left (a+c x^2\right )^{7/2}} \]

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])

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Rubi in Sympy [A]  time = 7.98944, size = 83, normalized size = 0.91 \[ - \frac{a e - c d x}{7 a c \left (a + c x^{2}\right )^{\frac{7}{2}}} + \frac{6 d x}{35 a^{2} \left (a + c x^{2}\right )^{\frac{5}{2}}} + \frac{8 d x}{35 a^{3} \left (a + c x^{2}\right )^{\frac{3}{2}}} + \frac{16 d x}{35 a^{4} \sqrt{a + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)/(c*x**2+a)**(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))

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Mathematica [A]  time = 0.0518724, size = 67, normalized size = 0.74 \[ \frac{-5 a^4 e+35 a^3 c d x+70 a^2 c^2 d x^3+56 a c^3 d x^5+16 c^4 d x^7}{35 a^4 c \left (a+c x^2\right )^{7/2}} \]

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 = 0.007, size = 64, normalized size = 0.7 \[ -{\frac{-16\,{c}^{4}d{x}^{7}-56\,{c}^{3}d{x}^{5}a-70\,{c}^{2}d{x}^{3}{a}^{2}-35\,dx{a}^{3}c+5\,{a}^{4}e}{35\,{a}^{4}c} \left ( c{x}^{2}+a \right ) ^{-{\frac{7}{2}}}} \]

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 = 0.685484, size = 108, normalized size = 1.19 \[ \frac{16 \, d x}{35 \, \sqrt{c x^{2} + a} a^{4}} + \frac{8 \, d x}{35 \,{\left (c x^{2} + a\right )}^{\frac{3}{2}} a^{3}} + \frac{6 \, d x}{35 \,{\left (c x^{2} + a\right )}^{\frac{5}{2}} a^{2}} + \frac{d x}{7 \,{\left (c x^{2} + a\right )}^{\frac{7}{2}} a} - \frac{e}{7 \,{\left (c x^{2} + a\right )}^{\frac{7}{2}} c} \]

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 = 0.309829, size = 146, normalized size = 1.6 \[ \frac{{\left (16 \, c^{4} d x^{7} + 56 \, a c^{3} d x^{5} + 70 \, a^{2} c^{2} d x^{3} + 35 \, a^{3} c d x - 5 \, a^{4} e\right )} \sqrt{c x^{2} + a}}{35 \,{\left (a^{4} c^{5} x^{8} + 4 \, a^{5} c^{4} x^{6} + 6 \, a^{6} c^{3} x^{4} + 4 \, a^{7} c^{2} x^{2} + a^{8} c\right )}} \]

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 [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)/(c*x**2+a)**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.277991, size = 92, normalized size = 1.01 \[ \frac{{\left (2 \,{\left (4 \,{\left (\frac{2 \, c^{3} d x^{2}}{a^{4}} + \frac{7 \, c^{2} d}{a^{3}}\right )} x^{2} + \frac{35 \, c d}{a^{2}}\right )} x^{2} + \frac{35 \, d}{a}\right )} x - \frac{5 \, e}{c}}{35 \,{\left (c x^{2} + a\right )}^{\frac{7}{2}}} \]

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)