3.2960 \(\int x^5 \sqrt{a+b \sqrt{c x^3}} \, dx\)

Optimal. Leaf size=116 \[ \frac{4 a^2 \left (a+b \sqrt{c x^3}\right )^{5/2}}{5 b^4 c^2}-\frac{4 a^3 \left (a+b \sqrt{c x^3}\right )^{3/2}}{9 b^4 c^2}+\frac{4 \left (a+b \sqrt{c x^3}\right )^{9/2}}{27 b^4 c^2}-\frac{4 a \left (a+b \sqrt{c x^3}\right )^{7/2}}{7 b^4 c^2} \]

[Out]

(-4*a^3*(a + b*Sqrt[c*x^3])^(3/2))/(9*b^4*c^2) + (4*a^2*(a + b*Sqrt[c*x^3])^(5/2))/(5*b^4*c^2) - (4*a*(a + b*S
qrt[c*x^3])^(7/2))/(7*b^4*c^2) + (4*(a + b*Sqrt[c*x^3])^(9/2))/(27*b^4*c^2)

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Rubi [A]  time = 0.0686541, antiderivative size = 116, normalized size of antiderivative = 1., 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 = {369, 266, 43} \[ \frac{4 a^2 \left (a+b \sqrt{c x^3}\right )^{5/2}}{5 b^4 c^2}-\frac{4 a^3 \left (a+b \sqrt{c x^3}\right )^{3/2}}{9 b^4 c^2}+\frac{4 \left (a+b \sqrt{c x^3}\right )^{9/2}}{27 b^4 c^2}-\frac{4 a \left (a+b \sqrt{c x^3}\right )^{7/2}}{7 b^4 c^2} \]

Antiderivative was successfully verified.

[In]

Int[x^5*Sqrt[a + b*Sqrt[c*x^3]],x]

[Out]

(-4*a^3*(a + b*Sqrt[c*x^3])^(3/2))/(9*b^4*c^2) + (4*a^2*(a + b*Sqrt[c*x^3])^(5/2))/(5*b^4*c^2) - (4*a*(a + b*S
qrt[c*x^3])^(7/2))/(7*b^4*c^2) + (4*(a + b*Sqrt[c*x^3])^(9/2))/(27*b^4*c^2)

Rule 369

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*((c_.)*(x_)^(q_))^(n_))^(p_.), x_Symbol] :> With[{k = Denominator[n]}, Su
bst[Int[(d*x)^m*(a + b*c^n*x^(n*q))^p, x], x^(1/k), (c*x^q)^(1/k)/(c^(1/k)*(x^(1/k))^(q - 1))]] /; FreeQ[{a, b
, c, d, m, p, q}, x] && FractionQ[n]

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 \sqrt{a+b \sqrt{c x^3}} \, dx &=\operatorname{Subst}\left (\int x^5 \sqrt{a+b \sqrt{c} x^{3/2}} \, dx,\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=\operatorname{Subst}\left (\frac{2}{3} \operatorname{Subst}\left (\int x^3 \sqrt{a+b \sqrt{c} x} \, dx,x,x^{3/2}\right ),\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=\operatorname{Subst}\left (\frac{2}{3} \operatorname{Subst}\left (\int \left (-\frac{a^3 \sqrt{a+b \sqrt{c} x}}{b^3 c^{3/2}}+\frac{3 a^2 \left (a+b \sqrt{c} x\right )^{3/2}}{b^3 c^{3/2}}-\frac{3 a \left (a+b \sqrt{c} x\right )^{5/2}}{b^3 c^{3/2}}+\frac{\left (a+b \sqrt{c} x\right )^{7/2}}{b^3 c^{3/2}}\right ) \, dx,x,x^{3/2}\right ),\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=-\frac{4 a^3 \left (a+b \sqrt{c x^3}\right )^{3/2}}{9 b^4 c^2}+\frac{4 a^2 \left (a+b \sqrt{c x^3}\right )^{5/2}}{5 b^4 c^2}-\frac{4 a \left (a+b \sqrt{c x^3}\right )^{7/2}}{7 b^4 c^2}+\frac{4 \left (a+b \sqrt{c x^3}\right )^{9/2}}{27 b^4 c^2}\\ \end{align*}

Mathematica [A]  time = 0.0828636, size = 72, normalized size = 0.62 \[ \frac{4 \left (a+b \sqrt{c x^3}\right )^{3/2} \left (24 a^2 b \sqrt{c x^3}-16 a^3-30 a b^2 c x^3+35 b^3 \left (c x^3\right )^{3/2}\right )}{945 b^4 c^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5*Sqrt[a + b*Sqrt[c*x^3]],x]

[Out]

(4*(a + b*Sqrt[c*x^3])^(3/2)*(-16*a^3 - 30*a*b^2*c*x^3 + 24*a^2*b*Sqrt[c*x^3] + 35*b^3*(c*x^3)^(3/2)))/(945*b^
4*c^2)

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Maple [A]  time = 0.192, size = 103, normalized size = 0.9 \begin{align*}{\frac{4}{945\,{c}^{2}{b}^{4}}\sqrt{a+b\sqrt{c{x}^{3}}} \left ( 35\,{x}^{6}{c}^{2}{b}^{4} \left ( c{x}^{3} \right ) ^{3/2}+5\,a{x}^{9}{c}^{3}{b}^{3}+8\,{a}^{3}{x}^{6}{c}^{2}b-6\,{a}^{2}c{x}^{3}{b}^{2} \left ( c{x}^{3} \right ) ^{3/2}-16\,{a}^{4} \left ( c{x}^{3} \right ) ^{3/2} \right ) \left ( c{x}^{3} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/945/c^2*(a+b*(c*x^3)^(1/2))^(1/2)*(35*x^6*c^2*b^4*(c*x^3)^(3/2)+5*a*x^9*c^3*b^3+8*a^3*x^6*c^2*b-6*a^2*c*x^3*
b^2*(c*x^3)^(3/2)-16*a^4*(c*x^3)^(3/2))/b^4/(c*x^3)^(3/2)

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Maxima [A]  time = 0.959779, size = 115, normalized size = 0.99 \begin{align*} \frac{4 \,{\left (\frac{35 \,{\left (\sqrt{c x^{3}} b + a\right )}^{\frac{9}{2}}}{b^{4}} - \frac{135 \,{\left (\sqrt{c x^{3}} b + a\right )}^{\frac{7}{2}} a}{b^{4}} + \frac{189 \,{\left (\sqrt{c x^{3}} b + a\right )}^{\frac{5}{2}} a^{2}}{b^{4}} - \frac{105 \,{\left (\sqrt{c x^{3}} b + a\right )}^{\frac{3}{2}} a^{3}}{b^{4}}\right )}}{945 \, c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/945*(35*(sqrt(c*x^3)*b + a)^(9/2)/b^4 - 135*(sqrt(c*x^3)*b + a)^(7/2)*a/b^4 + 189*(sqrt(c*x^3)*b + a)^(5/2)*
a^2/b^4 - 105*(sqrt(c*x^3)*b + a)^(3/2)*a^3/b^4)/c^2

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Fricas [A]  time = 2.21135, size = 169, normalized size = 1.46 \begin{align*} \frac{4 \,{\left (35 \, b^{4} c^{2} x^{6} - 6 \, a^{2} b^{2} c x^{3} - 16 \, a^{4} +{\left (5 \, a b^{3} c x^{3} + 8 \, a^{3} b\right )} \sqrt{c x^{3}}\right )} \sqrt{\sqrt{c x^{3}} b + a}}{945 \, b^{4} c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/945*(35*b^4*c^2*x^6 - 6*a^2*b^2*c*x^3 - 16*a^4 + (5*a*b^3*c*x^3 + 8*a^3*b)*sqrt(c*x^3))*sqrt(sqrt(c*x^3)*b +
a)/(b^4*c^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{5} \sqrt{a + b \sqrt{c x^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**5*sqrt(a + b*sqrt(c*x**3)), x)

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Giac [A]  time = 2.17042, size = 155, normalized size = 1.34 \begin{align*} \frac{4 \,{\left (\frac{16 \, \sqrt{a c} a^{4}}{b^{4} c} - \frac{105 \,{\left (\sqrt{c x} b c x + a c\right )}^{\frac{3}{2}} a^{3} c^{3} - 189 \,{\left (\sqrt{c x} b c x + a c\right )}^{\frac{5}{2}} a^{2} c^{2} + 135 \,{\left (\sqrt{c x} b c x + a c\right )}^{\frac{7}{2}} a c - 35 \,{\left (\sqrt{c x} b c x + a c\right )}^{\frac{9}{2}}}{b^{4} c^{5}}\right )}{\left | c \right |}}{945 \, c^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/945*(16*sqrt(a*c)*a^4/(b^4*c) - (105*(sqrt(c*x)*b*c*x + a*c)^(3/2)*a^3*c^3 - 189*(sqrt(c*x)*b*c*x + a*c)^(5/
2)*a^2*c^2 + 135*(sqrt(c*x)*b*c*x + a*c)^(7/2)*a*c - 35*(sqrt(c*x)*b*c*x + a*c)^(9/2))/(b^4*c^5))*abs(c)/c^(5/
2)