3.2975 \(\int x^9 \sqrt{a+b (c x^3)^{3/2}} \, dx\)

Optimal. Leaf size=170 \[ \frac{792 a^3 x \sqrt{\frac{b \left (c x^3\right )^{3/2}}{a}+1} \, _2F_1\left (\frac{2}{9},\frac{1}{2};\frac{11}{9};-\frac{b \left (c x^3\right )^{3/2}}{a}\right )}{19747 b^2 c^3 \sqrt{a+b \left (c x^3\right )^{3/2}}}-\frac{792 a^2 x \sqrt{a+b \left (c x^3\right )^{3/2}}}{19747 b^2 c^3}+\frac{36 a x \left (c x^3\right )^{3/2} \sqrt{a+b \left (c x^3\right )^{3/2}}}{1519 b c^3}+\frac{4}{49} x^{10} \sqrt{a+b \left (c x^3\right )^{3/2}} \]

[Out]

(-792*a^2*x*Sqrt[a + b*(c*x^3)^(3/2)])/(19747*b^2*c^3) + (4*x^10*Sqrt[a + b*(c*x^3)^(3/2)])/49 + (36*a*x*(c*x^
3)^(3/2)*Sqrt[a + b*(c*x^3)^(3/2)])/(1519*b*c^3) + (792*a^3*x*Sqrt[1 + (b*(c*x^3)^(3/2))/a]*Hypergeometric2F1[
2/9, 1/2, 11/9, -((b*(c*x^3)^(3/2))/a)])/(19747*b^2*c^3*Sqrt[a + b*(c*x^3)^(3/2)])

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Rubi [A]  time = 0.128102, antiderivative size = 170, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {369, 341, 279, 321, 365, 364} \[ \frac{792 a^3 x \sqrt{\frac{b \left (c x^3\right )^{3/2}}{a}+1} \, _2F_1\left (\frac{2}{9},\frac{1}{2};\frac{11}{9};-\frac{b \left (c x^3\right )^{3/2}}{a}\right )}{19747 b^2 c^3 \sqrt{a+b \left (c x^3\right )^{3/2}}}-\frac{792 a^2 x \sqrt{a+b \left (c x^3\right )^{3/2}}}{19747 b^2 c^3}+\frac{36 a x \left (c x^3\right )^{3/2} \sqrt{a+b \left (c x^3\right )^{3/2}}}{1519 b c^3}+\frac{4}{49} x^{10} \sqrt{a+b \left (c x^3\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[x^9*Sqrt[a + b*(c*x^3)^(3/2)],x]

[Out]

(-792*a^2*x*Sqrt[a + b*(c*x^3)^(3/2)])/(19747*b^2*c^3) + (4*x^10*Sqrt[a + b*(c*x^3)^(3/2)])/49 + (36*a*x*(c*x^
3)^(3/2)*Sqrt[a + b*(c*x^3)^(3/2)])/(1519*b*c^3) + (792*a^3*x*Sqrt[1 + (b*(c*x^3)^(3/2))/a]*Hypergeometric2F1[
2/9, 1/2, 11/9, -((b*(c*x^3)^(3/2))/a)])/(19747*b^2*c^3*Sqrt[a + b*(c*x^3)^(3/2)])

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 341

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

Rule 279

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m +
n*p + 1)), x] + Dist[(a*n*p)/(m + n*p + 1), Int[(c*x)^m*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x]
&& IGtQ[n, 0] && GtQ[p, 0] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 365

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^FracPart[p])
/(1 + (b*x^n)/a)^FracPart[p], Int[(c*x)^m*(1 + (b*x^n)/a)^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[
p, 0] &&  !(ILtQ[p, 0] || GtQ[a, 0])

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin{align*} \int x^9 \sqrt{a+b \left (c x^3\right )^{3/2}} \, dx &=\operatorname{Subst}\left (\int x^9 \sqrt{a+b c^{3/2} x^{9/2}} \, dx,\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=\operatorname{Subst}\left (2 \operatorname{Subst}\left (\int x^{19} \sqrt{a+b c^{3/2} x^9} \, dx,x,\sqrt{x}\right ),\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=\frac{4}{49} x^{10} \sqrt{a+b \left (c x^3\right )^{3/2}}+\operatorname{Subst}\left (\frac{1}{49} (18 a) \operatorname{Subst}\left (\int \frac{x^{19}}{\sqrt{a+b c^{3/2} x^9}} \, dx,x,\sqrt{x}\right ),\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=\frac{4}{49} x^{10} \sqrt{a+b \left (c x^3\right )^{3/2}}+\frac{36 a x \left (c x^3\right )^{3/2} \sqrt{a+b \left (c x^3\right )^{3/2}}}{1519 b c^3}-\operatorname{Subst}\left (\frac{\left (396 a^2\right ) \operatorname{Subst}\left (\int \frac{x^{10}}{\sqrt{a+b c^{3/2} x^9}} \, dx,x,\sqrt{x}\right )}{1519 b c^{3/2}},\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=-\frac{792 a^2 x \sqrt{a+b \left (c x^3\right )^{3/2}}}{19747 b^2 c^3}+\frac{4}{49} x^{10} \sqrt{a+b \left (c x^3\right )^{3/2}}+\frac{36 a x \left (c x^3\right )^{3/2} \sqrt{a+b \left (c x^3\right )^{3/2}}}{1519 b c^3}+\operatorname{Subst}\left (\frac{\left (1584 a^3\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{a+b c^{3/2} x^9}} \, dx,x,\sqrt{x}\right )}{19747 b^2 c^3},\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=-\frac{792 a^2 x \sqrt{a+b \left (c x^3\right )^{3/2}}}{19747 b^2 c^3}+\frac{4}{49} x^{10} \sqrt{a+b \left (c x^3\right )^{3/2}}+\frac{36 a x \left (c x^3\right )^{3/2} \sqrt{a+b \left (c x^3\right )^{3/2}}}{1519 b c^3}+\operatorname{Subst}\left (\frac{\left (1584 a^3 \sqrt{1+\frac{b c^{3/2} x^{9/2}}{a}}\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{1+\frac{b c^{3/2} x^9}{a}}} \, dx,x,\sqrt{x}\right )}{19747 b^2 c^3 \sqrt{a+b c^{3/2} x^{9/2}}},\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=-\frac{792 a^2 x \sqrt{a+b \left (c x^3\right )^{3/2}}}{19747 b^2 c^3}+\frac{4}{49} x^{10} \sqrt{a+b \left (c x^3\right )^{3/2}}+\frac{36 a x \left (c x^3\right )^{3/2} \sqrt{a+b \left (c x^3\right )^{3/2}}}{1519 b c^3}+\frac{792 a^3 x \sqrt{1+\frac{b \left (c x^3\right )^{3/2}}{a}} \, _2F_1\left (\frac{2}{9},\frac{1}{2};\frac{11}{9};-\frac{b \left (c x^3\right )^{3/2}}{a}\right )}{19747 b^2 c^3 \sqrt{a+b \left (c x^3\right )^{3/2}}}\\ \end{align*}

Mathematica [F]  time = 0.0608777, size = 0, normalized size = 0. \[ \int x^9 \sqrt{a+b \left (c x^3\right )^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[x^9*Sqrt[a + b*(c*x^3)^(3/2)],x]

[Out]

Integrate[x^9*Sqrt[a + b*(c*x^3)^(3/2)], x]

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Maple [F]  time = 0.058, size = 0, normalized size = 0. \begin{align*} \int{x}^{9}\sqrt{a+b \left ( c{x}^{3} \right ) ^{{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^9*(a+b*(c*x^3)^(3/2))^(1/2),x)

[Out]

int(x^9*(a+b*(c*x^3)^(3/2))^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\left (c x^{3}\right )^{\frac{3}{2}} b + a} x^{9}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9*(a+b*(c*x^3)^(3/2))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt((c*x^3)^(3/2)*b + a)*x^9, x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9*(a+b*(c*x^3)^(3/2))^(1/2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{9} \sqrt{a + b \left (c x^{3}\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**9*(a+b*(c*x**3)**(3/2))**(1/2),x)

[Out]

Integral(x**9*sqrt(a + b*(c*x**3)**(3/2)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\left (c x^{3}\right )^{\frac{3}{2}} b + a} x^{9}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9*(a+b*(c*x^3)^(3/2))^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt((c*x^3)^(3/2)*b + a)*x^9, x)