3.2977 \(\int \frac{\sqrt{a+b (c x^3)^{3/2}}}{x^9} \, dx\)

Optimal. Leaf size=139 \[ -\frac{45 b^2 c^3 x \sqrt{\frac{b \left (c x^3\right )^{3/2}}{a}+1} \, _2F_1\left (\frac{2}{9},\frac{1}{2};\frac{11}{9};-\frac{b \left (c x^3\right )^{3/2}}{a}\right )}{448 a \sqrt{a+b \left (c x^3\right )^{3/2}}}-\frac{9 b c^3 x \sqrt{a+b \left (c x^3\right )^{3/2}}}{112 a \left (c x^3\right )^{3/2}}-\frac{\sqrt{a+b \left (c x^3\right )^{3/2}}}{8 x^8} \]

[Out]

-Sqrt[a + b*(c*x^3)^(3/2)]/(8*x^8) - (9*b*c^3*x*Sqrt[a + b*(c*x^3)^(3/2)])/(112*a*(c*x^3)^(3/2)) - (45*b^2*c^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)])/(448*a*Sqrt[a + b*
(c*x^3)^(3/2)])

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Rubi [A]  time = 0.0916807, antiderivative size = 141, normalized size of antiderivative = 1.01, number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {369, 341, 277, 325, 365, 364} \[ -\frac{45 b^2 c^3 x \sqrt{\frac{b \left (c x^3\right )^{3/2}}{a}+1} \, _2F_1\left (\frac{2}{9},\frac{1}{2};\frac{11}{9};-\frac{b \left (c x^3\right )^{3/2}}{a}\right )}{448 a \sqrt{a+b \left (c x^3\right )^{3/2}}}-\frac{9 b c^5 x^7 \sqrt{a+b \left (c x^3\right )^{3/2}}}{112 a \left (c x^3\right )^{7/2}}-\frac{\sqrt{a+b \left (c x^3\right )^{3/2}}}{8 x^8} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b*(c*x^3)^(3/2)]/x^9,x]

[Out]

-Sqrt[a + b*(c*x^3)^(3/2)]/(8*x^8) - (9*b*c^5*x^7*Sqrt[a + b*(c*x^3)^(3/2)])/(112*a*(c*x^3)^(7/2)) - (45*b^2*c
^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)])/(448*a*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 277

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

Rule 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && 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 \frac{\sqrt{a+b \left (c x^3\right )^{3/2}}}{x^9} \, dx &=\operatorname{Subst}\left (\int \frac{\sqrt{a+b c^{3/2} x^{9/2}}}{x^9} \, dx,\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=\operatorname{Subst}\left (2 \operatorname{Subst}\left (\int \frac{\sqrt{a+b c^{3/2} x^9}}{x^{17}} \, dx,x,\sqrt{x}\right ),\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=-\frac{\sqrt{a+b \left (c x^3\right )^{3/2}}}{8 x^8}+\operatorname{Subst}\left (\frac{1}{16} \left (9 b c^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1}{x^8 \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{\sqrt{a+b \left (c x^3\right )^{3/2}}}{8 x^8}-\frac{9 b c^5 x^7 \sqrt{a+b \left (c x^3\right )^{3/2}}}{112 a \left (c x^3\right )^{7/2}}-\operatorname{Subst}\left (\frac{\left (45 b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{a+b c^{3/2} x^9}} \, dx,x,\sqrt{x}\right )}{224 a},\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=-\frac{\sqrt{a+b \left (c x^3\right )^{3/2}}}{8 x^8}-\frac{9 b c^5 x^7 \sqrt{a+b \left (c x^3\right )^{3/2}}}{112 a \left (c x^3\right )^{7/2}}-\operatorname{Subst}\left (\frac{\left (45 b^2 c^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 )}{224 a \sqrt{a+b c^{3/2} x^{9/2}}},\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=-\frac{\sqrt{a+b \left (c x^3\right )^{3/2}}}{8 x^8}-\frac{9 b c^5 x^7 \sqrt{a+b \left (c x^3\right )^{3/2}}}{112 a \left (c x^3\right )^{7/2}}-\frac{45 b^2 c^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 )}{448 a \sqrt{a+b \left (c x^3\right )^{3/2}}}\\ \end{align*}

Mathematica [F]  time = 0.0498699, size = 0, normalized size = 0. \[ \int \frac{\sqrt{a+b \left (c x^3\right )^{3/2}}}{x^9} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [F]  time = 0.059, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{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((a+b*(c*x^3)^(3/2))^(1/2)/x^9,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\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((a+b*(c*x^3)^(3/2))^(1/2)/x^9,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((a+b*(c*x^3)^(3/2))^(1/2)/x^9,x, algorithm="fricas")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\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((a+b*(c*x^3)^(3/2))^(1/2)/x^9,x, algorithm="giac")

[Out]

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