3.931 \(\int \frac{\sqrt [4]{a+b x^2}}{(c x)^{19/2}} \, dx\)

Optimal. Leaf size=113 \[ \frac{256 \left (a+b x^2\right )^{17/4}}{3315 a^4 c (c x)^{17/2}}-\frac{64 \left (a+b x^2\right )^{13/4}}{195 a^3 c (c x)^{17/2}}+\frac{8 \left (a+b x^2\right )^{9/4}}{15 a^2 c (c x)^{17/2}}-\frac{2 \left (a+b x^2\right )^{5/4}}{5 a c (c x)^{17/2}} \]

[Out]

(-2*(a + b*x^2)^(5/4))/(5*a*c*(c*x)^(17/2)) + (8*(a + b*x^2)^(9/4))/(15*a^2*c*(c
*x)^(17/2)) - (64*(a + b*x^2)^(13/4))/(195*a^3*c*(c*x)^(17/2)) + (256*(a + b*x^2
)^(17/4))/(3315*a^4*c*(c*x)^(17/2))

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Rubi [A]  time = 0.125991, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{256 \left (a+b x^2\right )^{17/4}}{3315 a^4 c (c x)^{17/2}}-\frac{64 \left (a+b x^2\right )^{13/4}}{195 a^3 c (c x)^{17/2}}+\frac{8 \left (a+b x^2\right )^{9/4}}{15 a^2 c (c x)^{17/2}}-\frac{2 \left (a+b x^2\right )^{5/4}}{5 a c (c x)^{17/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^(1/4)/(c*x)^(19/2),x]

[Out]

(-2*(a + b*x^2)^(5/4))/(5*a*c*(c*x)^(17/2)) + (8*(a + b*x^2)^(9/4))/(15*a^2*c*(c
*x)^(17/2)) - (64*(a + b*x^2)^(13/4))/(195*a^3*c*(c*x)^(17/2)) + (256*(a + b*x^2
)^(17/4))/(3315*a^4*c*(c*x)^(17/2))

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Rubi in Sympy [A]  time = 15.3173, size = 99, normalized size = 0.88 \[ - \frac{2 \left (a + b x^{2}\right )^{\frac{5}{4}}}{5 a c \left (c x\right )^{\frac{17}{2}}} + \frac{8 \left (a + b x^{2}\right )^{\frac{9}{4}}}{15 a^{2} c \left (c x\right )^{\frac{17}{2}}} - \frac{64 \left (a + b x^{2}\right )^{\frac{13}{4}}}{195 a^{3} c \left (c x\right )^{\frac{17}{2}}} + \frac{256 \left (a + b x^{2}\right )^{\frac{17}{4}}}{3315 a^{4} c \left (c x\right )^{\frac{17}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**(1/4)/(c*x)**(19/2),x)

[Out]

-2*(a + b*x**2)**(5/4)/(5*a*c*(c*x)**(17/2)) + 8*(a + b*x**2)**(9/4)/(15*a**2*c*
(c*x)**(17/2)) - 64*(a + b*x**2)**(13/4)/(195*a**3*c*(c*x)**(17/2)) + 256*(a + b
*x**2)**(17/4)/(3315*a**4*c*(c*x)**(17/2))

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Mathematica [A]  time = 0.047713, size = 74, normalized size = 0.65 \[ \frac{2 \sqrt [4]{a+b x^2} \left (-195 a^4-15 a^3 b x^2+20 a^2 b^2 x^4-32 a b^3 x^6+128 b^4 x^8\right )}{3315 a^4 c^9 x^8 \sqrt{c x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^(1/4)/(c*x)^(19/2),x]

[Out]

(2*(a + b*x^2)^(1/4)*(-195*a^4 - 15*a^3*b*x^2 + 20*a^2*b^2*x^4 - 32*a*b^3*x^6 +
128*b^4*x^8))/(3315*a^4*c^9*x^8*Sqrt[c*x])

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Maple [A]  time = 0.009, size = 53, normalized size = 0.5 \[ -{\frac{2\,x \left ( -128\,{b}^{3}{x}^{6}+160\,a{b}^{2}{x}^{4}-180\,{a}^{2}b{x}^{2}+195\,{a}^{3} \right ) }{3315\,{a}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{4}}} \left ( cx \right ) ^{-{\frac{19}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(1/4)/(c*x)^(19/2),x)

[Out]

-2/3315*x*(b*x^2+a)^(5/4)*(-128*b^3*x^6+160*a*b^2*x^4-180*a^2*b*x^2+195*a^3)/a^4
/(c*x)^(19/2)

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Maxima [A]  time = 1.38554, size = 97, normalized size = 0.86 \[ \frac{2 \,{\left (\frac{663 \,{\left (b x^{2} + a\right )}^{\frac{5}{4}} b^{3}}{x^{\frac{5}{2}}} - \frac{1105 \,{\left (b x^{2} + a\right )}^{\frac{9}{4}} b^{2}}{x^{\frac{9}{2}}} + \frac{765 \,{\left (b x^{2} + a\right )}^{\frac{13}{4}} b}{x^{\frac{13}{2}}} - \frac{195 \,{\left (b x^{2} + a\right )}^{\frac{17}{4}}}{x^{\frac{17}{2}}}\right )}}{3315 \, a^{4} c^{\frac{19}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(1/4)/(c*x)^(19/2),x, algorithm="maxima")

[Out]

2/3315*(663*(b*x^2 + a)^(5/4)*b^3/x^(5/2) - 1105*(b*x^2 + a)^(9/4)*b^2/x^(9/2) +
765*(b*x^2 + a)^(13/4)*b/x^(13/2) - 195*(b*x^2 + a)^(17/4)/x^(17/2))/(a^4*c^(19
/2))

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Fricas [A]  time = 0.238607, size = 92, normalized size = 0.81 \[ \frac{2 \,{\left (128 \, b^{4} x^{8} - 32 \, a b^{3} x^{6} + 20 \, a^{2} b^{2} x^{4} - 15 \, a^{3} b x^{2} - 195 \, a^{4}\right )}{\left (b x^{2} + a\right )}^{\frac{1}{4}} \sqrt{c x}}{3315 \, a^{4} c^{10} x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(1/4)/(c*x)^(19/2),x, algorithm="fricas")

[Out]

2/3315*(128*b^4*x^8 - 32*a*b^3*x^6 + 20*a^2*b^2*x^4 - 15*a^3*b*x^2 - 195*a^4)*(b
*x^2 + a)^(1/4)*sqrt(c*x)/(a^4*c^10*x^9)

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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((b*x**2+a)**(1/4)/(c*x)**(19/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.236024, size = 359, normalized size = 3.18 \[ \frac{2 \,{\left (\frac{663 \,{\left (b c^{4} x^{2} + a c^{4}\right )}^{\frac{1}{4}}{\left (b c^{2} + \frac{a c^{2}}{x^{2}}\right )} b^{3} c^{6}}{\sqrt{c x}} - \frac{1105 \,{\left (b^{2} c^{8} x^{4} + 2 \, a b c^{8} x^{2} + a^{2} c^{8}\right )}{\left (b c^{4} x^{2} + a c^{4}\right )}^{\frac{1}{4}} b^{2}}{\sqrt{c x} x^{4}} + \frac{765 \,{\left (b^{3} c^{12} x^{6} + 3 \, a b^{2} c^{12} x^{4} + 3 \, a^{2} b c^{12} x^{2} + a^{3} c^{12}\right )}{\left (b c^{4} x^{2} + a c^{4}\right )}^{\frac{1}{4}} b}{\sqrt{c x} c^{4} x^{6}} - \frac{195 \,{\left (b^{4} c^{16} x^{8} + 4 \, a b^{3} c^{16} x^{6} + 6 \, a^{2} b^{2} c^{16} x^{4} + 4 \, a^{3} b c^{16} x^{2} + a^{4} c^{16}\right )}{\left (b c^{4} x^{2} + a c^{4}\right )}^{\frac{1}{4}}}{\sqrt{c x} c^{8} x^{8}}\right )}}{3315 \, a^{4} c^{18}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(1/4)/(c*x)^(19/2),x, algorithm="giac")

[Out]

2/3315*(663*(b*c^4*x^2 + a*c^4)^(1/4)*(b*c^2 + a*c^2/x^2)*b^3*c^6/sqrt(c*x) - 11
05*(b^2*c^8*x^4 + 2*a*b*c^8*x^2 + a^2*c^8)*(b*c^4*x^2 + a*c^4)^(1/4)*b^2/(sqrt(c
*x)*x^4) + 765*(b^3*c^12*x^6 + 3*a*b^2*c^12*x^4 + 3*a^2*b*c^12*x^2 + a^3*c^12)*(
b*c^4*x^2 + a*c^4)^(1/4)*b/(sqrt(c*x)*c^4*x^6) - 195*(b^4*c^16*x^8 + 4*a*b^3*c^1
6*x^6 + 6*a^2*b^2*c^16*x^4 + 4*a^3*b*c^16*x^2 + a^4*c^16)*(b*c^4*x^2 + a*c^4)^(1
/4)/(sqrt(c*x)*c^8*x^8))/(a^4*c^18)