3.30.75 \(\int x^9 \sqrt {a+b (c x^3)^{3/2}} \, dx\) [2975]

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

[Out]

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

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Rubi [A]
time = 0.09, antiderivative size = 170, normalized size of antiderivative = 1.00, 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 = {376, 348, 285, 327, 372, 371} \begin {gather*} \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}} \end {gather*}

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 285

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 327

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 348

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 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 372

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 376

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]

Rubi steps

\begin {align*} \int x^9 \sqrt {a+b \left (c x^3\right )^{3/2}} \, dx &=\text {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 )\\ &=\text {Subst}\left (2 \text {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}}+\text {Subst}\left (\frac {1}{49} (18 a) \text {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}-\text {Subst}\left (\frac {\left (396 a^2\right ) \text {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}+\text {Subst}\left (\frac {\left (1584 a^3\right ) \text {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}+\text {Subst}\left (\frac {\left (1584 a^3 \sqrt {1+\frac {b c^{3/2} x^{9/2}}{a}}\right ) \text {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*}

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Mathematica [F]
time = 3.64, size = 0, normalized size = 0.00 \begin {gather*} \int x^9 \sqrt {a+b \left (c x^3\right )^{3/2}} \, dx \end {gather*}

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 = 0.06, size = 0, normalized size = 0.00 \[\int x^{9} \sqrt {a +b \left (c \,x^{3}\right )^{\frac {3}{2}}}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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]
time = 154.26, size = 23, normalized size = 0.14 \begin {gather*} {\rm integral}\left (\sqrt {\sqrt {c x^{3}} b c x^{3} + a} x^{9}, x\right ) \end {gather*}

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]

integral(sqrt(sqrt(c*x^3)*b*c*x^3 + a)*x^9, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{9} \sqrt {a + b \left (c x^{3}\right )^{\frac {3}{2}}}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^9\,\sqrt {a+b\,{\left (c\,x^3\right )}^{3/2}} \,d x \end {gather*}

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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