3.881 \(\int \frac{x^3}{(a+b x^6)^2 \sqrt{c+d x^6}} \, dx\)

Optimal. Leaf size=64 \[ \frac{x^4 \sqrt{\frac{d x^6}{c}+1} F_1\left (\frac{2}{3};2,\frac{1}{2};\frac{5}{3};-\frac{b x^6}{a},-\frac{d x^6}{c}\right )}{4 a^2 \sqrt{c+d x^6}} \]

[Out]

(x^4*Sqrt[1 + (d*x^6)/c]*AppellF1[2/3, 2, 1/2, 5/3, -((b*x^6)/a), -((d*x^6)/c)])/(4*a^2*Sqrt[c + d*x^6])

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Rubi [A]  time = 0.0730317, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {465, 511, 510} \[ \frac{x^4 \sqrt{\frac{d x^6}{c}+1} F_1\left (\frac{2}{3};2,\frac{1}{2};\frac{5}{3};-\frac{b x^6}{a},-\frac{d x^6}{c}\right )}{4 a^2 \sqrt{c+d x^6}} \]

Antiderivative was successfully verified.

[In]

Int[x^3/((a + b*x^6)^2*Sqrt[c + d*x^6]),x]

[Out]

(x^4*Sqrt[1 + (d*x^6)/c]*AppellF1[2/3, 2, 1/2, 5/3, -((b*x^6)/a), -((d*x^6)/c)])/(4*a^2*Sqrt[c + d*x^6])

Rule 465

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = GCD[m + 1,
n]}, Dist[1/k, Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p*(c + d*x^(n/k))^q, x], x, x^k], x] /; k != 1] /;
FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IntegerQ[m]

Rule 511

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[(a^IntPa
rt[p]*(a + b*x^n)^FracPart[p])/(1 + (b*x^n)/a)^FracPart[p], Int[(e*x)^m*(1 + (b*x^n)/a)^p*(c + d*x^n)^q, x], x
] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] &&  !(IntegerQ[
p] || GtQ[a, 0])

Rule 510

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(a^p*c^q
*(e*x)^(m + 1)*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, -((b*x^n)/a), -((d*x^n)/c)])/(e*(m + 1)), x] /; Free
Q[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a
, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rubi steps

\begin{align*} \int \frac{x^3}{\left (a+b x^6\right )^2 \sqrt{c+d x^6}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x}{\left (a+b x^3\right )^2 \sqrt{c+d x^3}} \, dx,x,x^2\right )\\ &=\frac{\sqrt{1+\frac{d x^6}{c}} \operatorname{Subst}\left (\int \frac{x}{\left (a+b x^3\right )^2 \sqrt{1+\frac{d x^3}{c}}} \, dx,x,x^2\right )}{2 \sqrt{c+d x^6}}\\ &=\frac{x^4 \sqrt{1+\frac{d x^6}{c}} F_1\left (\frac{2}{3};2,\frac{1}{2};\frac{5}{3};-\frac{b x^6}{a},-\frac{d x^6}{c}\right )}{4 a^2 \sqrt{c+d x^6}}\\ \end{align*}

Mathematica [B]  time = 0.16944, size = 168, normalized size = 2.62 \[ -\frac{x^4 \left (b d x^6 \left (a+b x^6\right ) \sqrt{\frac{d x^6}{c}+1} F_1\left (\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )-5 \left (a+b x^6\right ) \sqrt{\frac{d x^6}{c}+1} (b c-3 a d) F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )-10 a b \left (c+d x^6\right )\right )}{60 a^2 \left (a+b x^6\right ) \sqrt{c+d x^6} (b c-a d)} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^3/((a + b*x^6)^2*Sqrt[c + d*x^6]),x]

[Out]

-(x^4*(-10*a*b*(c + d*x^6) - 5*(b*c - 3*a*d)*(a + b*x^6)*Sqrt[1 + (d*x^6)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x
^6)/c), -((b*x^6)/a)] + b*d*x^6*(a + b*x^6)*Sqrt[1 + (d*x^6)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^6)/c), -((b*
x^6)/a)]))/(60*a^2*(b*c - a*d)*(a + b*x^6)*Sqrt[c + d*x^6])

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Maple [F]  time = 0.04, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{3}}{ \left ( b{x}^{6}+a \right ) ^{2}}{\frac{1}{\sqrt{d{x}^{6}+c}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(b*x^6+a)^2/(d*x^6+c)^(1/2),x)

[Out]

int(x^3/(b*x^6+a)^2/(d*x^6+c)^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{{\left (b x^{6} + a\right )}^{2} \sqrt{d x^{6} + c}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(b*x^6+a)^2/(d*x^6+c)^(1/2),x, algorithm="maxima")

[Out]

integrate(x^3/((b*x^6 + a)^2*sqrt(d*x^6 + c)), 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^3/(b*x^6+a)^2/(d*x^6+c)^(1/2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [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**3/(b*x**6+a)**2/(d*x**6+c)**(1/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{{\left (b x^{6} + a\right )}^{2} \sqrt{d x^{6} + c}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(b*x^6+a)^2/(d*x^6+c)^(1/2),x, algorithm="giac")

[Out]

integrate(x^3/((b*x^6 + a)^2*sqrt(d*x^6 + c)), x)