3.2937 \(\int x^2 \sqrt{a+b \sqrt{c x^2}} \, dx\)

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*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))

________________________________________________________________________________________

Rubi [A]  time = 0.0397586, 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, Rules used = {368, 43} \[ \frac{2 a^2 x^3 \left (a+b \sqrt{c x^2}\right )^{3/2}}{3 b^3 \left (c x^2\right )^{3/2}}+\frac{2 x^3 \left (a+b \sqrt{c x^2}\right )^{7/2}}{7 b^3 \left (c x^2\right )^{3/2}}-\frac{4 a x^3 \left (a+b \sqrt{c x^2}\right )^{5/2}}{5 b^3 \left (c x^2\right )^{3/2}} \]

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*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))

Rule 368

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*((c_.)*(x_)^(q_))^(n_))^(p_.), x_Symbol] :> Dist[(d*x)^(m + 1)/(d*((c*x^q
)^(1/q))^(m + 1)), Subst[Int[x^m*(a + b*x^(n*q))^p, x], x, (c*x^q)^(1/q)], x] /; FreeQ[{a, b, c, d, m, n, p, q
}, x] && IntegerQ[n*q] && NeQ[x, (c*x^q)^(1/q)]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int x^2 \sqrt{a+b \sqrt{c x^2}} \, dx &=\frac{x^3 \operatorname{Subst}\left (\int x^2 \sqrt{a+b x} \, dx,x,\sqrt{c x^2}\right )}{\left (c x^2\right )^{3/2}}\\ &=\frac{x^3 \operatorname{Subst}\left (\int \left (\frac{a^2 \sqrt{a+b x}}{b^2}-\frac{2 a (a+b x)^{3/2}}{b^2}+\frac{(a+b x)^{5/2}}{b^2}\right ) \, dx,x,\sqrt{c x^2}\right )}{\left (c x^2\right )^{3/2}}\\ &=\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{4 a x^3 \left (a+b \sqrt{c x^2}\right )^{5/2}}{5 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}}\\ \end{align*}

Mathematica [A]  time = 0.0290982, size = 64, normalized size = 0.57 \[ \frac{2 x^3 \left (a+b \sqrt{c x^2}\right )^{3/2} \left (8 a^2-12 a b \sqrt{c x^2}+15 b^2 c x^2\right )}{105 b^3 \left (c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.007, size = 55, normalized size = 0.5 \begin{align*} -{\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}}}} \end{align*}

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

________________________________________________________________________________________

Maxima [B]  time = 1.06851, size = 522, normalized size = 4.62 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*(c*x^2)^(1/2))^(1/2),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 + 221616*c^2)*sqrt(c))*b^3*x^3 + (c^8 + 382*c^7 + 15946*c^6 + 1581
72*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 + 1020
60*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 + 11226
6*c^3 + 231903*c^2 + 2*(14*c^5 + 1333*c^4 + 22953*c^3 + 97011*c^2 + 91125*c + 8748)*sqrt(c) + 84564*c)*a^3)*sq
rt(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 + 10497
6*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)

________________________________________________________________________________________

Fricas [A]  time = 1.30185, size = 154, normalized size = 1.36 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*(c*x^2)^(1/2))^(1/2),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))*sqrt(sqrt(c*x^2)*b + a)/(b^3*c^2*
x)

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

Giac [A]  time = 1.17032, size = 66, normalized size = 0.58 \begin{align*} \frac{2 \,{\left (15 \,{\left (b \sqrt{c} x + a\right )}^{\frac{7}{2}} - 42 \,{\left (b \sqrt{c} x + a\right )}^{\frac{5}{2}} a + 35 \,{\left (b \sqrt{c} x + a\right )}^{\frac{3}{2}} a^{2}\right )}}{105 \, b^{3} c^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*(c*x^2)^(1/2))^(1/2),x, algorithm="giac")

[Out]

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