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

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

[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)

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Rubi [A]  time = 0.0524936, antiderivative size = 116, 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} \[ \frac{6 a^2 \left (a+b \sqrt{c x^2}\right )^{5/2}}{5 b^4 c^2}-\frac{2 a^3 \left (a+b \sqrt{c x^2}\right )^{3/2}}{3 b^4 c^2}+\frac{2 \left (a+b \sqrt{c x^2}\right )^{9/2}}{9 b^4 c^2}-\frac{6 a \left (a+b \sqrt{c x^2}\right )^{7/2}}{7 b^4 c^2} \]

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 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^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*}

Mathematica [A]  time = 0.0345468, size = 72, normalized size = 0.62 \[ \frac{2 \left (a+b \sqrt{c x^2}\right )^{3/2} \left (24 a^2 b \sqrt{c x^2}-16 a^3-30 a b^2 c x^2+35 b^3 \left (c x^2\right )^{3/2}\right )}{315 b^4 c^2} \]

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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Maple [A]  time = 0.006, size = 63, normalized size = 0.5 \begin{align*}{\frac{2}{315\,{c}^{2}{b}^{4}} \left ( a+b\sqrt{c{x}^{2}} \right ) ^{{\frac{3}{2}}} \left ( 35\, \left ( c{x}^{2} \right ) ^{3/2}{b}^{3}-30\,c{x}^{2}a{b}^{2}+24\,\sqrt{c{x}^{2}}{a}^{2}b-16\,{a}^{3} \right ) } \end{align*}

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]

2/315*(a+b*(c*x^2)^(1/2))^(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 = 0.93811, size = 115, normalized size = 0.99 \begin{align*} \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}} \end{align*}

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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Fricas [A]  time = 1.34172, size = 169, normalized size = 1.46 \begin{align*} \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}} \end{align*}

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \sqrt{a + b \sqrt{c x^{2}}}\, dx \end{align*}

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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Giac [A]  time = 1.18594, size = 103, normalized size = 0.89 \begin{align*} \frac{2 \,{\left (35 \,{\left (b \sqrt{c} x + a\right )}^{\frac{9}{2}} \sqrt{c} - 135 \,{\left (b \sqrt{c} x + a\right )}^{\frac{7}{2}} a \sqrt{c} + 189 \,{\left (b \sqrt{c} x + a\right )}^{\frac{5}{2}} a^{2} \sqrt{c} - 105 \,{\left (b \sqrt{c} x + a\right )}^{\frac{3}{2}} a^{3} \sqrt{c}\right )}}{315 \, b^{4} c^{\frac{5}{2}}} \end{align*}

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