3.2931 \(\int x^3 \sqrt {a+b \sqrt {c x^2}} \, dx\)

Optimal. Leaf size=116 \[ -\frac {2 a^3 \left (a+b \sqrt {c x^2}\right )^{3/2}}{3 b^4 c^2}+\frac {6 a^2 \left (a+b \sqrt {c x^2}\right )^{5/2}}{5 b^4 c^2}+\frac {2 \left (a+b \sqrt {c x^2}\right )^{9/2}}{9 b^4 c^2}-\frac {6 a \left (a+b \sqrt {c x^2}\right )^{7/2}}{7 b^4 c^2} \]

[Out]

-2/3*a^3*(a+b*(c*x^2)^(1/2))^(3/2)/b^4/c^2+6/5*a^2*(a+b*(c*x^2)^(1/2))^(5/2)/b^4/c^2-6/7*a*(a+b*(c*x^2)^(1/2))
^(7/2)/b^4/c^2+2/9*(a+b*(c*x^2)^(1/2))^(9/2)/b^4/c^2

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Rubi [A]  time = 0.05, antiderivative size = 116, 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} \[ \frac {6 a^2 \left (a+b \sqrt {c x^2}\right )^{5/2}}{5 b^4 c^2}-\frac {2 a^3 \left (a+b \sqrt {c x^2}\right )^{3/2}}{3 b^4 c^2}+\frac {2 \left (a+b \sqrt {c x^2}\right )^{9/2}}{9 b^4 c^2}-\frac {6 a \left (a+b \sqrt {c x^2}\right )^{7/2}}{7 b^4 c^2} \]

Antiderivative was successfully verified.

[In]

Int[x^3*Sqrt[a + b*Sqrt[c*x^2]],x]

[Out]

(-2*a^3*(a + b*Sqrt[c*x^2])^(3/2))/(3*b^4*c^2) + (6*a^2*(a + b*Sqrt[c*x^2])^(5/2))/(5*b^4*c^2) - (6*a*(a + b*S
qrt[c*x^2])^(7/2))/(7*b^4*c^2) + (2*(a + b*Sqrt[c*x^2])^(9/2))/(9*b^4*c^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^3 \sqrt {a+b \sqrt {c x^2}} \, dx &=\frac {\operatorname {Subst}\left (\int x^3 \sqrt {a+b x} \, dx,x,\sqrt {c x^2}\right )}{c^2}\\ &=\frac {\operatorname {Subst}\left (\int \left (-\frac {a^3 \sqrt {a+b x}}{b^3}+\frac {3 a^2 (a+b x)^{3/2}}{b^3}-\frac {3 a (a+b x)^{5/2}}{b^3}+\frac {(a+b x)^{7/2}}{b^3}\right ) \, dx,x,\sqrt {c x^2}\right )}{c^2}\\ &=-\frac {2 a^3 \left (a+b \sqrt {c x^2}\right )^{3/2}}{3 b^4 c^2}+\frac {6 a^2 \left (a+b \sqrt {c x^2}\right )^{5/2}}{5 b^4 c^2}-\frac {6 a \left (a+b \sqrt {c x^2}\right )^{7/2}}{7 b^4 c^2}+\frac {2 \left (a+b \sqrt {c x^2}\right )^{9/2}}{9 b^4 c^2}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 72, normalized size = 0.62 \[ \frac {2 \left (a+b \sqrt {c x^2}\right )^{3/2} \left (-16 a^3+24 a^2 b \sqrt {c x^2}-30 a b^2 c x^2+35 b^3 \left (c x^2\right )^{3/2}\right )}{315 b^4 c^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*Sqrt[a + b*Sqrt[c*x^2]],x]

[Out]

(2*(a + b*Sqrt[c*x^2])^(3/2)*(-16*a^3 - 30*a*b^2*c*x^2 + 24*a^2*b*Sqrt[c*x^2] + 35*b^3*(c*x^2)^(3/2)))/(315*b^
4*c^2)

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fricas [A]  time = 0.93, size = 75, normalized size = 0.65 \[ \frac {2 \, {\left (35 \, b^{4} c^{2} x^{4} - 6 \, a^{2} b^{2} c x^{2} - 16 \, a^{4} + {\left (5 \, a b^{3} c x^{2} + 8 \, a^{3} b\right )} \sqrt {c x^{2}}\right )} \sqrt {\sqrt {c x^{2}} b + a}}{315 \, b^{4} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*(c*x^2)^(1/2))^(1/2),x, algorithm="fricas")

[Out]

2/315*(35*b^4*c^2*x^4 - 6*a^2*b^2*c*x^2 - 16*a^4 + (5*a*b^3*c*x^2 + 8*a^3*b)*sqrt(c*x^2))*sqrt(sqrt(c*x^2)*b +
a)/(b^4*c^2)

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giac [A]  time = 0.17, size = 164, normalized size = 1.41 \[ \frac {2 \, {\left (\frac {9 \, {\left (5 \, {\left (b \sqrt {c} x + a\right )}^{\frac {7}{2}} \sqrt {c} - 21 \, {\left (b \sqrt {c} x + a\right )}^{\frac {5}{2}} a \sqrt {c} + 35 \, {\left (b \sqrt {c} x + a\right )}^{\frac {3}{2}} a^{2} \sqrt {c} - 35 \, \sqrt {b \sqrt {c} x + a} a^{3} \sqrt {c}\right )} a}{b^{3} c^{2}} + \frac {35 \, {\left (b \sqrt {c} x + a\right )}^{\frac {9}{2}} - 180 \, {\left (b \sqrt {c} x + a\right )}^{\frac {7}{2}} a + 378 \, {\left (b \sqrt {c} x + a\right )}^{\frac {5}{2}} a^{2} - 420 \, {\left (b \sqrt {c} x + a\right )}^{\frac {3}{2}} a^{3} + 315 \, \sqrt {b \sqrt {c} x + a} a^{4}}{b^{3} c^{\frac {3}{2}}}\right )}}{315 \, b \sqrt {c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*(c*x^2)^(1/2))^(1/2),x, algorithm="giac")

[Out]

2/315*(9*(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))*a/(b^3*c^2) + (35*(b*sqrt(c)*x + a)^(9/2) - 180*(b*sqrt(
c)*x + a)^(7/2)*a + 378*(b*sqrt(c)*x + a)^(5/2)*a^2 - 420*(b*sqrt(c)*x + a)^(3/2)*a^3 + 315*sqrt(b*sqrt(c)*x +
a)*a^4)/(b^3*c^(3/2)))/(b*sqrt(c))

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maple [A]  time = 0.00, size = 63, normalized size = 0.54 \[ \frac {2 \left (a +\sqrt {c \,x^{2}}\, b \right )^{\frac {3}{2}} \left (-30 a \,b^{2} c \,x^{2}-16 a^{3}+24 \sqrt {c \,x^{2}}\, a^{2} b +35 \left (c \,x^{2}\right )^{\frac {3}{2}} b^{3}\right )}{315 b^{4} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(a+(c*x^2)^(1/2)*b)^(3/2)*(35*(c*x^2)^(3/2)*b^3-30*c*x^2*a*b^2+24*(c*x^2)^(1/2)*a^2*b-16*a^3)/c^2/b^4

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maxima [A]  time = 0.63, size = 85, normalized size = 0.73 \[ \frac {2 \, {\left (\frac {35 \, {\left (\sqrt {c x^{2}} b + a\right )}^{\frac {9}{2}}}{b^{4}} - \frac {135 \, {\left (\sqrt {c x^{2}} b + a\right )}^{\frac {7}{2}} a}{b^{4}} + \frac {189 \, {\left (\sqrt {c x^{2}} b + a\right )}^{\frac {5}{2}} a^{2}}{b^{4}} - \frac {105 \, {\left (\sqrt {c x^{2}} b + a\right )}^{\frac {3}{2}} a^{3}}{b^{4}}\right )}}{315 \, c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*(c*x^2)^(1/2))^(1/2),x, algorithm="maxima")

[Out]

2/315*(35*(sqrt(c*x^2)*b + a)^(9/2)/b^4 - 135*(sqrt(c*x^2)*b + a)^(7/2)*a/b^4 + 189*(sqrt(c*x^2)*b + a)^(5/2)*
a^2/b^4 - 105*(sqrt(c*x^2)*b + a)^(3/2)*a^3/b^4)/c^2

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int x^3\,\sqrt {a+b\,\sqrt {c\,x^2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{3} \sqrt {a + b \sqrt {c x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**3*sqrt(a + b*sqrt(c*x**2)), x)

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