3.2930 \(\int x^2 \sqrt{a+b \sqrt{c x^2}} \, dx\)

Optimal. Leaf size=113 \[ \frac{2 a^2 x^3 \left (a+b \sqrt{c x^2}\right )^{3/2}}{3 b^3 \left (c x^2\right )^{3/2}}+\frac{2 x^3 \left (a+b \sqrt{c x^2}\right )^{7/2}}{7 b^3 \left (c x^2\right )^{3/2}}-\frac{4 a x^3 \left (a+b \sqrt{c x^2}\right )^{5/2}}{5 b^3 \left (c x^2\right )^{3/2}} \]

[Out]

(2*a^2*x^3*(a + b*Sqrt[c*x^2])^(3/2))/(3*b^3*(c*x^2)^(3/2)) - (4*a*x^3*(a + b*Sq
rt[c*x^2])^(5/2))/(5*b^3*(c*x^2)^(3/2)) + (2*x^3*(a + b*Sqrt[c*x^2])^(7/2))/(7*b
^3*(c*x^2)^(3/2))

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Rubi [A]  time = 0.113564, antiderivative size = 113, 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 \[ \frac{2 a^2 x^3 \left (a+b \sqrt{c x^2}\right )^{3/2}}{3 b^3 \left (c x^2\right )^{3/2}}+\frac{2 x^3 \left (a+b \sqrt{c x^2}\right )^{7/2}}{7 b^3 \left (c x^2\right )^{3/2}}-\frac{4 a x^3 \left (a+b \sqrt{c x^2}\right )^{5/2}}{5 b^3 \left (c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*a^2*x^3*(a + b*Sqrt[c*x^2])^(3/2))/(3*b^3*(c*x^2)^(3/2)) - (4*a*x^3*(a + b*Sq
rt[c*x^2])^(5/2))/(5*b^3*(c*x^2)^(3/2)) + (2*x^3*(a + b*Sqrt[c*x^2])^(7/2))/(7*b
^3*(c*x^2)^(3/2))

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Rubi in Sympy [A]  time = 12.5006, size = 105, normalized size = 0.93 \[ \frac{2 a^{2} x^{3} \left (a + b \sqrt{c x^{2}}\right )^{\frac{3}{2}}}{3 b^{3} \left (c x^{2}\right )^{\frac{3}{2}}} - \frac{4 a x^{3} \left (a + b \sqrt{c x^{2}}\right )^{\frac{5}{2}}}{5 b^{3} \left (c x^{2}\right )^{\frac{3}{2}}} + \frac{2 x^{3} \left (a + b \sqrt{c x^{2}}\right )^{\frac{7}{2}}}{7 b^{3} \left (c x^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a**2*x**3*(a + b*sqrt(c*x**2))**(3/2)/(3*b**3*(c*x**2)**(3/2)) - 4*a*x**3*(a +
b*sqrt(c*x**2))**(5/2)/(5*b**3*(c*x**2)**(3/2)) + 2*x**3*(a + b*sqrt(c*x**2))**
(7/2)/(7*b**3*(c*x**2)**(3/2))

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Mathematica [A]  time = 0.177544, size = 0, normalized size = 0. \[ \int x^2 \sqrt{a+b \sqrt{c x^2}} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [A]  time = 0.008, size = 55, normalized size = 0.5 \[ -{\frac{2\,{x}^{3}}{105\,{b}^{3}} \left ( a+b\sqrt{c{x}^{2}} \right ) ^{{\frac{3}{2}}} \left ( -15\,{x}^{2}{b}^{2}c+12\,\sqrt{c{x}^{2}}ab-8\,{a}^{2} \right ) \left ( c{x}^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/105*x^3*(a+b*(c*x^2)^(1/2))^(3/2)*(-15*x^2*b^2*c+12*(c*x^2)^(1/2)*a*b-8*a^2)/
(c*x^2)^(3/2)/b^3

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Maxima [A]  time = 1.45874, size = 522, normalized size = 4.62 \[ \frac{{\left ({\left (31 \, c^{8} + 3784 \, c^{7} + 91078 \, c^{6} + 622632 \, c^{5} + 1266003 \, c^{4} + 635688 \, c^{3} + 34992 \, c^{2} +{\left (c^{8} + 440 \, c^{7} + 21986 \, c^{6} + 276544 \, c^{5} + 1038501 \, c^{4} + 1095120 \, c^{3} + 221616 \, c^{2}\right )} \sqrt{c}\right )} b^{3} x^{3} +{\left (c^{8} + 382 \, c^{7} + 15946 \, c^{6} + 158172 \, c^{5} + 425925 \, c^{4} + 266814 \, c^{3} + 17496 \, c^{2} +{\left (29 \, c^{7} + 3020 \, c^{6} + 59186 \, c^{5} + 306288 \, c^{4} + 414153 \, c^{3} + 102060 \, c^{2}\right )} \sqrt{c}\right )} a b^{2} x^{2} - 2 \,{\left (c^{7} + 354 \, c^{6} + 13280 \, c^{5} + 112266 \, c^{4} + 231903 \, c^{3} + 84564 \, c^{2} + 2 \,{\left (14 \, c^{6} + 1333 \, c^{5} + 22953 \, c^{4} + 97011 \, c^{3} + 91125 \, c^{2} + 8748 \, c\right )} \sqrt{c}\right )} a^{2} b x + 2 \,{\left (c^{6} + 354 \, c^{5} + 13280 \, c^{4} + 112266 \, c^{3} + 231903 \, c^{2} + 2 \,{\left (14 \, c^{5} + 1333 \, c^{4} + 22953 \, c^{3} + 97011 \, c^{2} + 91125 \, c + 8748\right )} \sqrt{c} + 84564 \, c\right )} a^{3}\right )} \sqrt{b \sqrt{c} x + a}}{{\left (c^{9} + 533 \, c^{8} + 33338 \, c^{7} + 549778 \, c^{6} + 2906397 \, c^{5} + 4893129 \, c^{4} + 2128680 \, c^{3} + 104976 \, c^{2} + 2 \,{\left (17 \, c^{8} + 2552 \, c^{7} + 78518 \, c^{6} + 726132 \, c^{5} + 2190753 \, c^{4} + 1960524 \, c^{3} + 349920 \, c^{2}\right )} \sqrt{c}\right )} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

((31*c^8 + 3784*c^7 + 91078*c^6 + 622632*c^5 + 1266003*c^4 + 635688*c^3 + 34992*
c^2 + (c^8 + 440*c^7 + 21986*c^6 + 276544*c^5 + 1038501*c^4 + 1095120*c^3 + 2216
16*c^2)*sqrt(c))*b^3*x^3 + (c^8 + 382*c^7 + 15946*c^6 + 158172*c^5 + 425925*c^4
+ 266814*c^3 + 17496*c^2 + (29*c^7 + 3020*c^6 + 59186*c^5 + 306288*c^4 + 414153*
c^3 + 102060*c^2)*sqrt(c))*a*b^2*x^2 - 2*(c^7 + 354*c^6 + 13280*c^5 + 112266*c^4
+ 231903*c^3 + 84564*c^2 + 2*(14*c^6 + 1333*c^5 + 22953*c^4 + 97011*c^3 + 91125
*c^2 + 8748*c)*sqrt(c))*a^2*b*x + 2*(c^6 + 354*c^5 + 13280*c^4 + 112266*c^3 + 23
1903*c^2 + 2*(14*c^5 + 1333*c^4 + 22953*c^3 + 97011*c^2 + 91125*c + 8748)*sqrt(c
) + 84564*c)*a^3)*sqrt(b*sqrt(c)*x + a)/((c^9 + 533*c^8 + 33338*c^7 + 549778*c^6
+ 2906397*c^5 + 4893129*c^4 + 2128680*c^3 + 104976*c^2 + 2*(17*c^8 + 2552*c^7 +
78518*c^6 + 726132*c^5 + 2190753*c^4 + 1960524*c^3 + 349920*c^2)*sqrt(c))*b^3)

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Fricas [A]  time = 0.207894, size = 95, normalized size = 0.84 \[ \frac{2 \,{\left (15 \, b^{3} c^{2} x^{4} - 4 \, a^{2} b c x^{2} +{\left (3 \, a b^{2} c x^{2} + 8 \, a^{3}\right )} \sqrt{c x^{2}}\right )} \sqrt{\sqrt{c x^{2}} b + a}}{105 \, b^{3} c^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/105*(15*b^3*c^2*x^4 - 4*a^2*b*c*x^2 + (3*a*b^2*c*x^2 + 8*a^3)*sqrt(c*x^2))*sqr
t(sqrt(c*x^2)*b + a)/(b^3*c^2*x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.21708, size = 90, normalized size = 0.8 \[ \frac{2 \,{\left (15 \,{\left (b \sqrt{c} x + a\right )}^{\frac{7}{2}} b^{12} c^{6} - 42 \,{\left (b \sqrt{c} x + a\right )}^{\frac{5}{2}} a b^{12} c^{6} + 35 \,{\left (b \sqrt{c} x + a\right )}^{\frac{3}{2}} a^{2} b^{12} c^{6}\right )}}{105 \, b^{15} c^{\frac{15}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/105*(15*(b*sqrt(c)*x + a)^(7/2)*b^12*c^6 - 42*(b*sqrt(c)*x + a)^(5/2)*a*b^12*c
^6 + 35*(b*sqrt(c)*x + a)^(3/2)*a^2*b^12*c^6)/(b^15*c^(15/2))