3.144 \(\int \frac{x^3 \left (c+d x^2+e x^4+f x^6\right )}{\sqrt{a+b x^2}} \, dx\)

Optimal. Leaf size=167 \[ \frac{\left (a+b x^2\right )^{5/2} \left (6 a^2 f-3 a b e+b^2 d\right )}{5 b^5}+\frac{\left (a+b x^2\right )^{3/2} \left (-4 a^3 f+3 a^2 b e-2 a b^2 d+b^3 c\right )}{3 b^5}-\frac{a \sqrt{a+b x^2} \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{b^5}+\frac{\left (a+b x^2\right )^{7/2} (b e-4 a f)}{7 b^5}+\frac{f \left (a+b x^2\right )^{9/2}}{9 b^5} \]

[Out]

-((a*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Sqrt[a + b*x^2])/b^5) + ((b^3*c - 2*a*b
^2*d + 3*a^2*b*e - 4*a^3*f)*(a + b*x^2)^(3/2))/(3*b^5) + ((b^2*d - 3*a*b*e + 6*a
^2*f)*(a + b*x^2)^(5/2))/(5*b^5) + ((b*e - 4*a*f)*(a + b*x^2)^(7/2))/(7*b^5) + (
f*(a + b*x^2)^(9/2))/(9*b^5)

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Rubi [A]  time = 0.354669, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ \frac{\left (a+b x^2\right )^{5/2} \left (6 a^2 f-3 a b e+b^2 d\right )}{5 b^5}+\frac{\left (a+b x^2\right )^{3/2} \left (-4 a^3 f+3 a^2 b e-2 a b^2 d+b^3 c\right )}{3 b^5}-\frac{a \sqrt{a+b x^2} \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{b^5}+\frac{\left (a+b x^2\right )^{7/2} (b e-4 a f)}{7 b^5}+\frac{f \left (a+b x^2\right )^{9/2}}{9 b^5} \]

Antiderivative was successfully verified.

[In]  Int[(x^3*(c + d*x^2 + e*x^4 + f*x^6))/Sqrt[a + b*x^2],x]

[Out]

-((a*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Sqrt[a + b*x^2])/b^5) + ((b^3*c - 2*a*b
^2*d + 3*a^2*b*e - 4*a^3*f)*(a + b*x^2)^(3/2))/(3*b^5) + ((b^2*d - 3*a*b*e + 6*a
^2*f)*(a + b*x^2)^(5/2))/(5*b^5) + ((b*e - 4*a*f)*(a + b*x^2)^(7/2))/(7*b^5) + (
f*(a + b*x^2)^(9/2))/(9*b^5)

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Rubi in Sympy [A]  time = 62.4364, size = 156, normalized size = 0.93 \[ \frac{a \sqrt{a + b x^{2}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{b^{5}} + \frac{f \left (a + b x^{2}\right )^{\frac{9}{2}}}{9 b^{5}} - \frac{\left (a + b x^{2}\right )^{\frac{7}{2}} \left (4 a f - b e\right )}{7 b^{5}} + \frac{\left (a + b x^{2}\right )^{\frac{5}{2}} \left (6 a^{2} f - 3 a b e + b^{2} d\right )}{5 b^{5}} - \frac{\left (a + b x^{2}\right )^{\frac{3}{2}} \left (4 a^{3} f - 3 a^{2} b e + 2 a b^{2} d - b^{3} c\right )}{3 b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(f*x**6+e*x**4+d*x**2+c)/(b*x**2+a)**(1/2),x)

[Out]

a*sqrt(a + b*x**2)*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)/b**5 + f*(a + b*x**2)
**(9/2)/(9*b**5) - (a + b*x**2)**(7/2)*(4*a*f - b*e)/(7*b**5) + (a + b*x**2)**(5
/2)*(6*a**2*f - 3*a*b*e + b**2*d)/(5*b**5) - (a + b*x**2)**(3/2)*(4*a**3*f - 3*a
**2*b*e + 2*a*b**2*d - b**3*c)/(3*b**5)

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Mathematica [A]  time = 0.151845, size = 122, normalized size = 0.73 \[ \frac{\sqrt{a+b x^2} \left (128 a^4 f-16 a^3 b \left (9 e+4 f x^2\right )+24 a^2 b^2 \left (7 d+3 e x^2+2 f x^4\right )-2 a b^3 \left (105 c+42 d x^2+27 e x^4+20 f x^6\right )+b^4 x^2 \left (105 c+63 d x^2+45 e x^4+35 f x^6\right )\right )}{315 b^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^3*(c + d*x^2 + e*x^4 + f*x^6))/Sqrt[a + b*x^2],x]

[Out]

(Sqrt[a + b*x^2]*(128*a^4*f - 16*a^3*b*(9*e + 4*f*x^2) + 24*a^2*b^2*(7*d + 3*e*x
^2 + 2*f*x^4) - 2*a*b^3*(105*c + 42*d*x^2 + 27*e*x^4 + 20*f*x^6) + b^4*x^2*(105*
c + 63*d*x^2 + 45*e*x^4 + 35*f*x^6)))/(315*b^5)

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Maple [A]  time = 0.01, size = 145, normalized size = 0.9 \[{\frac{35\,f{x}^{8}{b}^{4}-40\,a{b}^{3}f{x}^{6}+45\,{b}^{4}e{x}^{6}+48\,{a}^{2}{b}^{2}f{x}^{4}-54\,a{b}^{3}e{x}^{4}+63\,{b}^{4}d{x}^{4}-64\,{a}^{3}bf{x}^{2}+72\,{a}^{2}{b}^{2}e{x}^{2}-84\,a{b}^{3}d{x}^{2}+105\,{b}^{4}c{x}^{2}+128\,{a}^{4}f-144\,{a}^{3}be+168\,{a}^{2}{b}^{2}d-210\,a{b}^{3}c}{315\,{b}^{5}}\sqrt{b{x}^{2}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(f*x^6+e*x^4+d*x^2+c)/(b*x^2+a)^(1/2),x)

[Out]

1/315*(b*x^2+a)^(1/2)*(35*b^4*f*x^8-40*a*b^3*f*x^6+45*b^4*e*x^6+48*a^2*b^2*f*x^4
-54*a*b^3*e*x^4+63*b^4*d*x^4-64*a^3*b*f*x^2+72*a^2*b^2*e*x^2-84*a*b^3*d*x^2+105*
b^4*c*x^2+128*a^4*f-144*a^3*b*e+168*a^2*b^2*d-210*a*b^3*c)/b^5

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^6 + e*x^4 + d*x^2 + c)*x^3/sqrt(b*x^2 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.249499, size = 181, normalized size = 1.08 \[ \frac{{\left (35 \, b^{4} f x^{8} + 5 \,{\left (9 \, b^{4} e - 8 \, a b^{3} f\right )} x^{6} - 210 \, a b^{3} c + 168 \, a^{2} b^{2} d - 144 \, a^{3} b e + 128 \, a^{4} f + 3 \,{\left (21 \, b^{4} d - 18 \, a b^{3} e + 16 \, a^{2} b^{2} f\right )} x^{4} +{\left (105 \, b^{4} c - 84 \, a b^{3} d + 72 \, a^{2} b^{2} e - 64 \, a^{3} b f\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{315 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^6 + e*x^4 + d*x^2 + c)*x^3/sqrt(b*x^2 + a),x, algorithm="fricas")

[Out]

1/315*(35*b^4*f*x^8 + 5*(9*b^4*e - 8*a*b^3*f)*x^6 - 210*a*b^3*c + 168*a^2*b^2*d
- 144*a^3*b*e + 128*a^4*f + 3*(21*b^4*d - 18*a*b^3*e + 16*a^2*b^2*f)*x^4 + (105*
b^4*c - 84*a*b^3*d + 72*a^2*b^2*e - 64*a^3*b*f)*x^2)*sqrt(b*x^2 + a)/b^5

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Sympy [A]  time = 4.04707, size = 340, normalized size = 2.04 \[ \begin{cases} \frac{128 a^{4} f \sqrt{a + b x^{2}}}{315 b^{5}} - \frac{16 a^{3} e \sqrt{a + b x^{2}}}{35 b^{4}} - \frac{64 a^{3} f x^{2} \sqrt{a + b x^{2}}}{315 b^{4}} + \frac{8 a^{2} d \sqrt{a + b x^{2}}}{15 b^{3}} + \frac{8 a^{2} e x^{2} \sqrt{a + b x^{2}}}{35 b^{3}} + \frac{16 a^{2} f x^{4} \sqrt{a + b x^{2}}}{105 b^{3}} - \frac{2 a c \sqrt{a + b x^{2}}}{3 b^{2}} - \frac{4 a d x^{2} \sqrt{a + b x^{2}}}{15 b^{2}} - \frac{6 a e x^{4} \sqrt{a + b x^{2}}}{35 b^{2}} - \frac{8 a f x^{6} \sqrt{a + b x^{2}}}{63 b^{2}} + \frac{c x^{2} \sqrt{a + b x^{2}}}{3 b} + \frac{d x^{4} \sqrt{a + b x^{2}}}{5 b} + \frac{e x^{6} \sqrt{a + b x^{2}}}{7 b} + \frac{f x^{8} \sqrt{a + b x^{2}}}{9 b} & \text{for}\: b \neq 0 \\\frac{\frac{c x^{4}}{4} + \frac{d x^{6}}{6} + \frac{e x^{8}}{8} + \frac{f x^{10}}{10}}{\sqrt{a}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(f*x**6+e*x**4+d*x**2+c)/(b*x**2+a)**(1/2),x)

[Out]

Piecewise((128*a**4*f*sqrt(a + b*x**2)/(315*b**5) - 16*a**3*e*sqrt(a + b*x**2)/(
35*b**4) - 64*a**3*f*x**2*sqrt(a + b*x**2)/(315*b**4) + 8*a**2*d*sqrt(a + b*x**2
)/(15*b**3) + 8*a**2*e*x**2*sqrt(a + b*x**2)/(35*b**3) + 16*a**2*f*x**4*sqrt(a +
b*x**2)/(105*b**3) - 2*a*c*sqrt(a + b*x**2)/(3*b**2) - 4*a*d*x**2*sqrt(a + b*x*
*2)/(15*b**2) - 6*a*e*x**4*sqrt(a + b*x**2)/(35*b**2) - 8*a*f*x**6*sqrt(a + b*x*
*2)/(63*b**2) + c*x**2*sqrt(a + b*x**2)/(3*b) + d*x**4*sqrt(a + b*x**2)/(5*b) +
e*x**6*sqrt(a + b*x**2)/(7*b) + f*x**8*sqrt(a + b*x**2)/(9*b), Ne(b, 0)), ((c*x*
*4/4 + d*x**6/6 + e*x**8/8 + f*x**10/10)/sqrt(a), True))

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GIAC/XCAS [A]  time = 0.223768, size = 296, normalized size = 1.77 \[ \frac{105 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} b^{3} c - 315 \, \sqrt{b x^{2} + a} a b^{3} c + 63 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} b^{2} d - 210 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a b^{2} d + 315 \, \sqrt{b x^{2} + a} a^{2} b^{2} d + 35 \,{\left (b x^{2} + a\right )}^{\frac{9}{2}} f - 180 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} a f + 378 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a^{2} f - 420 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{3} f + 315 \, \sqrt{b x^{2} + a} a^{4} f + 45 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b e - 189 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a b e + 315 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{2} b e - 315 \, \sqrt{b x^{2} + a} a^{3} b e}{315 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^6 + e*x^4 + d*x^2 + c)*x^3/sqrt(b*x^2 + a),x, algorithm="giac")

[Out]

1/315*(105*(b*x^2 + a)^(3/2)*b^3*c - 315*sqrt(b*x^2 + a)*a*b^3*c + 63*(b*x^2 + a
)^(5/2)*b^2*d - 210*(b*x^2 + a)^(3/2)*a*b^2*d + 315*sqrt(b*x^2 + a)*a^2*b^2*d +
35*(b*x^2 + a)^(9/2)*f - 180*(b*x^2 + a)^(7/2)*a*f + 378*(b*x^2 + a)^(5/2)*a^2*f
- 420*(b*x^2 + a)^(3/2)*a^3*f + 315*sqrt(b*x^2 + a)*a^4*f + 45*(b*x^2 + a)^(7/2
)*b*e - 189*(b*x^2 + a)^(5/2)*a*b*e + 315*(b*x^2 + a)^(3/2)*a^2*b*e - 315*sqrt(b
*x^2 + a)*a^3*b*e)/b^5