3.2923 \(\int x^5 \sqrt{a+b \sqrt{c x^2}} \, dx\)

Optimal. Leaf size=174 \[ -\frac{2 a^5 \left (a+b \sqrt{c x^2}\right )^{3/2}}{3 b^6 c^3}+\frac{2 a^4 \left (a+b \sqrt{c x^2}\right )^{5/2}}{b^6 c^3}-\frac{20 a^3 \left (a+b \sqrt{c x^2}\right )^{7/2}}{7 b^6 c^3}+\frac{20 a^2 \left (a+b \sqrt{c x^2}\right )^{9/2}}{9 b^6 c^3}+\frac{2 \left (a+b \sqrt{c x^2}\right )^{13/2}}{13 b^6 c^3}-\frac{10 a \left (a+b \sqrt{c x^2}\right )^{11/2}}{11 b^6 c^3} \]

[Out]

(-2*a^5*(a + b*Sqrt[c*x^2])^(3/2))/(3*b^6*c^3) + (2*a^4*(a + b*Sqrt[c*x^2])^(5/2
))/(b^6*c^3) - (20*a^3*(a + b*Sqrt[c*x^2])^(7/2))/(7*b^6*c^3) + (20*a^2*(a + b*S
qrt[c*x^2])^(9/2))/(9*b^6*c^3) - (10*a*(a + b*Sqrt[c*x^2])^(11/2))/(11*b^6*c^3)
+ (2*(a + b*Sqrt[c*x^2])^(13/2))/(13*b^6*c^3)

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Rubi [A]  time = 0.204756, antiderivative size = 174, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ -\frac{2 a^5 \left (a+b \sqrt{c x^2}\right )^{3/2}}{3 b^6 c^3}+\frac{2 a^4 \left (a+b \sqrt{c x^2}\right )^{5/2}}{b^6 c^3}-\frac{20 a^3 \left (a+b \sqrt{c x^2}\right )^{7/2}}{7 b^6 c^3}+\frac{20 a^2 \left (a+b \sqrt{c x^2}\right )^{9/2}}{9 b^6 c^3}+\frac{2 \left (a+b \sqrt{c x^2}\right )^{13/2}}{13 b^6 c^3}-\frac{10 a \left (a+b \sqrt{c x^2}\right )^{11/2}}{11 b^6 c^3} \]

Antiderivative was successfully verified.

[In]  Int[x^5*Sqrt[a + b*Sqrt[c*x^2]],x]

[Out]

(-2*a^5*(a + b*Sqrt[c*x^2])^(3/2))/(3*b^6*c^3) + (2*a^4*(a + b*Sqrt[c*x^2])^(5/2
))/(b^6*c^3) - (20*a^3*(a + b*Sqrt[c*x^2])^(7/2))/(7*b^6*c^3) + (20*a^2*(a + b*S
qrt[c*x^2])^(9/2))/(9*b^6*c^3) - (10*a*(a + b*Sqrt[c*x^2])^(11/2))/(11*b^6*c^3)
+ (2*(a + b*Sqrt[c*x^2])^(13/2))/(13*b^6*c^3)

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Rubi in Sympy [A]  time = 25.523, size = 165, normalized size = 0.95 \[ - \frac{2 a^{5} \left (a + b \sqrt{c x^{2}}\right )^{\frac{3}{2}}}{3 b^{6} c^{3}} + \frac{2 a^{4} \left (a + b \sqrt{c x^{2}}\right )^{\frac{5}{2}}}{b^{6} c^{3}} - \frac{20 a^{3} \left (a + b \sqrt{c x^{2}}\right )^{\frac{7}{2}}}{7 b^{6} c^{3}} + \frac{20 a^{2} \left (a + b \sqrt{c x^{2}}\right )^{\frac{9}{2}}}{9 b^{6} c^{3}} - \frac{10 a \left (a + b \sqrt{c x^{2}}\right )^{\frac{11}{2}}}{11 b^{6} c^{3}} + \frac{2 \left (a + b \sqrt{c x^{2}}\right )^{\frac{13}{2}}}{13 b^{6} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5*(a+b*(c*x**2)**(1/2))**(1/2),x)

[Out]

-2*a**5*(a + b*sqrt(c*x**2))**(3/2)/(3*b**6*c**3) + 2*a**4*(a + b*sqrt(c*x**2))*
*(5/2)/(b**6*c**3) - 20*a**3*(a + b*sqrt(c*x**2))**(7/2)/(7*b**6*c**3) + 20*a**2
*(a + b*sqrt(c*x**2))**(9/2)/(9*b**6*c**3) - 10*a*(a + b*sqrt(c*x**2))**(11/2)/(
11*b**6*c**3) + 2*(a + b*sqrt(c*x**2))**(13/2)/(13*b**6*c**3)

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Mathematica [A]  time = 0.0860709, size = 103, normalized size = 0.59 \[ \frac{2 \left (a+b \sqrt{c x^2}\right )^{3/2} \left (-256 a^5+384 a^4 b \sqrt{c x^2}-480 a^3 b^2 c x^2+560 a^2 b^3 \left (c x^2\right )^{3/2}-630 a b^4 c^2 x^4+693 b^5 \left (c x^2\right )^{5/2}\right )}{9009 b^6 c^3} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5*Sqrt[a + b*Sqrt[c*x^2]],x]

[Out]

(2*(a + b*Sqrt[c*x^2])^(3/2)*(-256*a^5 - 480*a^3*b^2*c*x^2 - 630*a*b^4*c^2*x^4 +
384*a^4*b*Sqrt[c*x^2] + 560*a^2*b^3*(c*x^2)^(3/2) + 693*b^5*(c*x^2)^(5/2)))/(90
09*b^6*c^3)

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Maple [A]  time = 0.011, size = 92, normalized size = 0.5 \[{\frac{2}{9009\,{c}^{3}{b}^{6}} \left ( a+b\sqrt{c{x}^{2}} \right ) ^{{\frac{3}{2}}} \left ( 693\, \left ( c{x}^{2} \right ) ^{5/2}{b}^{5}-630\,{c}^{2}{x}^{4}a{b}^{4}+560\, \left ( c{x}^{2} \right ) ^{3/2}{a}^{2}{b}^{3}-480\,c{x}^{2}{a}^{3}{b}^{2}+384\,\sqrt{c{x}^{2}}{a}^{4}b-256\,{a}^{5} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5*(a+b*(c*x^2)^(1/2))^(1/2),x)

[Out]

2/9009*(a+b*(c*x^2)^(1/2))^(3/2)*(693*(c*x^2)^(5/2)*b^5-630*c^2*x^4*a*b^4+560*(c
*x^2)^(3/2)*a^2*b^3-480*c*x^2*a^3*b^2+384*(c*x^2)^(1/2)*a^4*b-256*a^5)/c^3/b^6

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Maxima [A]  time = 1.38157, size = 171, normalized size = 0.98 \[ \frac{2 \,{\left (\frac{693 \,{\left (\sqrt{c x^{2}} b + a\right )}^{\frac{13}{2}}}{b^{6}} - \frac{4095 \,{\left (\sqrt{c x^{2}} b + a\right )}^{\frac{11}{2}} a}{b^{6}} + \frac{10010 \,{\left (\sqrt{c x^{2}} b + a\right )}^{\frac{9}{2}} a^{2}}{b^{6}} - \frac{12870 \,{\left (\sqrt{c x^{2}} b + a\right )}^{\frac{7}{2}} a^{3}}{b^{6}} + \frac{9009 \,{\left (\sqrt{c x^{2}} b + a\right )}^{\frac{5}{2}} a^{4}}{b^{6}} - \frac{3003 \,{\left (\sqrt{c x^{2}} b + a\right )}^{\frac{3}{2}} a^{5}}{b^{6}}\right )}}{9009 \, c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(sqrt(c*x^2)*b + a)*x^5,x, algorithm="maxima")

[Out]

2/9009*(693*(sqrt(c*x^2)*b + a)^(13/2)/b^6 - 4095*(sqrt(c*x^2)*b + a)^(11/2)*a/b
^6 + 10010*(sqrt(c*x^2)*b + a)^(9/2)*a^2/b^6 - 12870*(sqrt(c*x^2)*b + a)^(7/2)*a
^3/b^6 + 9009*(sqrt(c*x^2)*b + a)^(5/2)*a^4/b^6 - 3003*(sqrt(c*x^2)*b + a)^(3/2)
*a^5/b^6)/c^3

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Fricas [A]  time = 0.208768, size = 139, normalized size = 0.8 \[ \frac{2 \,{\left (693 \, b^{6} c^{3} x^{6} - 70 \, a^{2} b^{4} c^{2} x^{4} - 96 \, a^{4} b^{2} c x^{2} - 256 \, a^{6} +{\left (63 \, a b^{5} c^{2} x^{4} + 80 \, a^{3} b^{3} c x^{2} + 128 \, a^{5} b\right )} \sqrt{c x^{2}}\right )} \sqrt{\sqrt{c x^{2}} b + a}}{9009 \, b^{6} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(sqrt(c*x^2)*b + a)*x^5,x, algorithm="fricas")

[Out]

2/9009*(693*b^6*c^3*x^6 - 70*a^2*b^4*c^2*x^4 - 96*a^4*b^2*c*x^2 - 256*a^6 + (63*
a*b^5*c^2*x^4 + 80*a^3*b^3*c*x^2 + 128*a^5*b)*sqrt(c*x^2))*sqrt(sqrt(c*x^2)*b +
a)/(b^6*c^3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x^{5} \sqrt{a + b \sqrt{c x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5*(a+b*(c*x**2)**(1/2))**(1/2),x)

[Out]

Integral(x**5*sqrt(a + b*sqrt(c*x**2)), x)

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GIAC/XCAS [A]  time = 0.219944, size = 176, normalized size = 1.01 \[ \frac{2 \,{\left (693 \,{\left (b \sqrt{c} x + a\right )}^{\frac{13}{2}} b^{60} c^{\frac{73}{2}} - 4095 \,{\left (b \sqrt{c} x + a\right )}^{\frac{11}{2}} a b^{60} c^{\frac{73}{2}} + 10010 \,{\left (b \sqrt{c} x + a\right )}^{\frac{9}{2}} a^{2} b^{60} c^{\frac{73}{2}} - 12870 \,{\left (b \sqrt{c} x + a\right )}^{\frac{7}{2}} a^{3} b^{60} c^{\frac{73}{2}} + 9009 \,{\left (b \sqrt{c} x + a\right )}^{\frac{5}{2}} a^{4} b^{60} c^{\frac{73}{2}} - 3003 \,{\left (b \sqrt{c} x + a\right )}^{\frac{3}{2}} a^{5} b^{60} c^{\frac{73}{2}}\right )}}{9009 \, b^{66} c^{\frac{79}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(sqrt(c*x^2)*b + a)*x^5,x, algorithm="giac")

[Out]

2/9009*(693*(b*sqrt(c)*x + a)^(13/2)*b^60*c^(73/2) - 4095*(b*sqrt(c)*x + a)^(11/
2)*a*b^60*c^(73/2) + 10010*(b*sqrt(c)*x + a)^(9/2)*a^2*b^60*c^(73/2) - 12870*(b*
sqrt(c)*x + a)^(7/2)*a^3*b^60*c^(73/2) + 9009*(b*sqrt(c)*x + a)^(5/2)*a^4*b^60*c
^(73/2) - 3003*(b*sqrt(c)*x + a)^(3/2)*a^5*b^60*c^(73/2))/(b^66*c^(79/2))