3.250 \(\int x^5 (c \sqrt{a+b x^2})^{3/2} \, dx\)

Optimal. Leaf size=102 \[ \frac{2 a^2 \left (a+b x^2\right ) \left (c \sqrt{a+b x^2}\right )^{3/2}}{7 b^3}+\frac{2 \left (a+b x^2\right )^3 \left (c \sqrt{a+b x^2}\right )^{3/2}}{15 b^3}-\frac{4 a \left (a+b x^2\right )^2 \left (c \sqrt{a+b x^2}\right )^{3/2}}{11 b^3} \]

[Out]

(2*a^2*(c*Sqrt[a + b*x^2])^(3/2)*(a + b*x^2))/(7*b^3) - (4*a*(c*Sqrt[a + b*x^2])^(3/2)*(a + b*x^2)^2)/(11*b^3)
+ (2*(c*Sqrt[a + b*x^2])^(3/2)*(a + b*x^2)^3)/(15*b^3)

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Rubi [A]  time = 0.157332, antiderivative size = 113, normalized size of antiderivative = 1.11, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {6720, 266, 43} \[ \frac{2 a^2 c \left (a+b x^2\right )^{3/2} \sqrt{c \sqrt{a+b x^2}}}{7 b^3}+\frac{2 c \left (a+b x^2\right )^{7/2} \sqrt{c \sqrt{a+b x^2}}}{15 b^3}-\frac{4 a c \left (a+b x^2\right )^{5/2} \sqrt{c \sqrt{a+b x^2}}}{11 b^3} \]

Antiderivative was successfully verified.

[In]

Int[x^5*(c*Sqrt[a + b*x^2])^(3/2),x]

[Out]

(2*a^2*c*Sqrt[c*Sqrt[a + b*x^2]]*(a + b*x^2)^(3/2))/(7*b^3) - (4*a*c*Sqrt[c*Sqrt[a + b*x^2]]*(a + b*x^2)^(5/2)
)/(11*b^3) + (2*c*Sqrt[c*Sqrt[a + b*x^2]]*(a + b*x^2)^(7/2))/(15*b^3)

Rule 6720

Int[(u_.)*((a_.)*(v_)^(m_.))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a*v^m)^FracPart[p])/v^(m*FracPart[p]), Int
[u*v^(m*p), x], x] /; FreeQ[{a, m, p}, x] &&  !IntegerQ[p] &&  !FreeQ[v, x] &&  !(EqQ[a, 1] && EqQ[m, 1]) &&
!(EqQ[v, x] && EqQ[m, 1])

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

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^5 \left (c \sqrt{a+b x^2}\right )^{3/2} \, dx &=\frac{\left (c \sqrt{c \sqrt{a+b x^2}}\right ) \int x^5 \left (a+b x^2\right )^{3/4} \, dx}{\sqrt [4]{a+b x^2}}\\ &=\frac{\left (c \sqrt{c \sqrt{a+b x^2}}\right ) \operatorname{Subst}\left (\int x^2 (a+b x)^{3/4} \, dx,x,x^2\right )}{2 \sqrt [4]{a+b x^2}}\\ &=\frac{\left (c \sqrt{c \sqrt{a+b x^2}}\right ) \operatorname{Subst}\left (\int \left (\frac{a^2 (a+b x)^{3/4}}{b^2}-\frac{2 a (a+b x)^{7/4}}{b^2}+\frac{(a+b x)^{11/4}}{b^2}\right ) \, dx,x,x^2\right )}{2 \sqrt [4]{a+b x^2}}\\ &=\frac{2 a^2 c \sqrt{c \sqrt{a+b x^2}} \left (a+b x^2\right )^{3/2}}{7 b^3}-\frac{4 a c \sqrt{c \sqrt{a+b x^2}} \left (a+b x^2\right )^{5/2}}{11 b^3}+\frac{2 c \sqrt{c \sqrt{a+b x^2}} \left (a+b x^2\right )^{7/2}}{15 b^3}\\ \end{align*}

Mathematica [A]  time = 0.0234396, size = 52, normalized size = 0.51 \[ \frac{2 \left (a+b x^2\right ) \left (32 a^2-56 a b x^2+77 b^2 x^4\right ) \left (c \sqrt{a+b x^2}\right )^{3/2}}{1155 b^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5*(c*Sqrt[a + b*x^2])^(3/2),x]

[Out]

(2*(c*Sqrt[a + b*x^2])^(3/2)*(a + b*x^2)*(32*a^2 - 56*a*b*x^2 + 77*b^2*x^4))/(1155*b^3)

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Maple [A]  time = 0.006, size = 47, normalized size = 0.5 \begin{align*}{\frac{ \left ( 2\,b{x}^{2}+2\,a \right ) \left ( 77\,{b}^{2}{x}^{4}-56\,ab{x}^{2}+32\,{a}^{2} \right ) }{1155\,{b}^{3}} \left ( c\sqrt{b{x}^{2}+a} \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(c*(b*x^2+a)^(1/2))^(3/2),x)

[Out]

2/1155*(b*x^2+a)*(77*b^2*x^4-56*a*b*x^2+32*a^2)*(c*(b*x^2+a)^(1/2))^(3/2)/b^3

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Maxima [A]  time = 0.999604, size = 86, normalized size = 0.84 \begin{align*} \frac{2 \,{\left (165 \, \left (\sqrt{b x^{2} + a} c\right )^{\frac{7}{2}} a^{2} c^{4} - 210 \, \left (\sqrt{b x^{2} + a} c\right )^{\frac{11}{2}} a c^{2} + 77 \, \left (\sqrt{b x^{2} + a} c\right )^{\frac{15}{2}}\right )}}{1155 \, b^{3} c^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(c*(b*x^2+a)^(1/2))^(3/2),x, algorithm="maxima")

[Out]

2/1155*(165*(sqrt(b*x^2 + a)*c)^(7/2)*a^2*c^4 - 210*(sqrt(b*x^2 + a)*c)^(11/2)*a*c^2 + 77*(sqrt(b*x^2 + a)*c)^
(15/2))/(b^3*c^6)

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Fricas [A]  time = 1.72647, size = 151, normalized size = 1.48 \begin{align*} \frac{2 \,{\left (77 \, b^{3} c x^{6} + 21 \, a b^{2} c x^{4} - 24 \, a^{2} b c x^{2} + 32 \, a^{3} c\right )} \sqrt{b x^{2} + a} \sqrt{\sqrt{b x^{2} + a} c}}{1155 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(c*(b*x^2+a)^(1/2))^(3/2),x, algorithm="fricas")

[Out]

2/1155*(77*b^3*c*x^6 + 21*a*b^2*c*x^4 - 24*a^2*b*c*x^2 + 32*a^3*c)*sqrt(b*x^2 + a)*sqrt(sqrt(b*x^2 + a)*c)/b^3

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Sympy [A]  time = 117.062, size = 116, normalized size = 1.14 \begin{align*} \begin{cases} \frac{64 a^{3} c^{\frac{3}{2}} \left (a + b x^{2}\right )^{\frac{3}{4}}}{1155 b^{3}} - \frac{16 a^{2} c^{\frac{3}{2}} x^{2} \left (a + b x^{2}\right )^{\frac{3}{4}}}{385 b^{2}} + \frac{2 a c^{\frac{3}{2}} x^{4} \left (a + b x^{2}\right )^{\frac{3}{4}}}{55 b} + \frac{2 c^{\frac{3}{2}} x^{6} \left (a + b x^{2}\right )^{\frac{3}{4}}}{15} & \text{for}\: b \neq 0 \\\frac{x^{6} \left (\sqrt{a} c\right )^{\frac{3}{2}}}{6} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(c*(b*x**2+a)**(1/2))**(3/2),x)

[Out]

Piecewise((64*a**3*c**(3/2)*(a + b*x**2)**(3/4)/(1155*b**3) - 16*a**2*c**(3/2)*x**2*(a + b*x**2)**(3/4)/(385*b
**2) + 2*a*c**(3/2)*x**4*(a + b*x**2)**(3/4)/(55*b) + 2*c**(3/2)*x**6*(a + b*x**2)**(3/4)/15, Ne(b, 0)), (x**6
*(sqrt(a)*c)**(3/2)/6, True))

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Giac [A]  time = 1.19837, size = 62, normalized size = 0.61 \begin{align*} \frac{2 \,{\left (77 \,{\left (b x^{2} + a\right )}^{\frac{15}{4}} - 210 \,{\left (b x^{2} + a\right )}^{\frac{11}{4}} a + 165 \,{\left (b x^{2} + a\right )}^{\frac{7}{4}} a^{2}\right )} c^{\frac{3}{2}}}{1155 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(c*(b*x^2+a)^(1/2))^(3/2),x, algorithm="giac")

[Out]

2/1155*(77*(b*x^2 + a)^(15/4) - 210*(b*x^2 + a)^(11/4)*a + 165*(b*x^2 + a)^(7/4)*a^2)*c^(3/2)/b^3