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

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

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 113, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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}} \end {gather*}

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 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])

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)]

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 = 0.03, size = 64, normalized size = 0.57 \begin {gather*} \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}} \end {gather*}

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))

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 2.65, size = 0, normalized size = 0.00 \begin {gather*} \int x^2 \sqrt {a+b \sqrt {c x^2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.62, size = 70, normalized size = 0.62 \begin {gather*} \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 {gather*}

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)

________________________________________________________________________________________

giac [A]  time = 0.17, size = 135, normalized size = 1.19 \begin {gather*} \frac {2 \, {\left (\frac {7 \, {\left (3 \, {\left (b \sqrt {c} x + a\right )}^{\frac {5}{2}} - 10 \, {\left (b \sqrt {c} x + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {b \sqrt {c} x + a} a^{2}\right )} a}{b^{2} c} + \frac {3 \, {\left (5 \, {\left (b \sqrt {c} x + a\right )}^{\frac {7}{2}} \sqrt {c} - 21 \, {\left (b \sqrt {c} x + a\right )}^{\frac {5}{2}} a \sqrt {c} + 35 \, {\left (b \sqrt {c} x + a\right )}^{\frac {3}{2}} a^{2} \sqrt {c} - 35 \, \sqrt {b \sqrt {c} x + a} a^{3} \sqrt {c}\right )}}{b^{2} c^{\frac {3}{2}}}\right )}}{105 \, b \sqrt {c}} \end {gather*}

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*(7*(3*(b*sqrt(c)*x + a)^(5/2) - 10*(b*sqrt(c)*x + a)^(3/2)*a + 15*sqrt(b*sqrt(c)*x + a)*a^2)*a/(b^2*c) +
3*(5*(b*sqrt(c)*x + a)^(7/2)*sqrt(c) - 21*(b*sqrt(c)*x + a)^(5/2)*a*sqrt(c) + 35*(b*sqrt(c)*x + a)^(3/2)*a^2*
sqrt(c) - 35*sqrt(b*sqrt(c)*x + a)*a^3*sqrt(c))/(b^2*c^(3/2)))/(b*sqrt(c))

________________________________________________________________________________________

maple [A]  time = 0.00, size = 55, normalized size = 0.49 \begin {gather*} -\frac {2 \left (a +\sqrt {c \,x^{2}}\, b \right )^{\frac {3}{2}} \left (-15 b^{2} c \,x^{2}-8 a^{2}+12 \sqrt {c \,x^{2}}\, a b \right ) x^{3}}{105 \left (c \,x^{2}\right )^{\frac {3}{2}} b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [B]  time = 0.71, size = 387, normalized size = 3.42 \begin {gather*} \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 {gather*}

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)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^2\,\sqrt {a+b\,\sqrt {c\,x^2}} \,d x \end {gather*}

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]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{2} \sqrt {a + b \sqrt {c x^{2}}}\, dx \end {gather*}

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)

________________________________________________________________________________________