3.925 \(\int \frac {x^2}{(a+b x^8)^2 \sqrt {c+d x^8}} \, dx\)

Optimal. Leaf size=64 \[ \frac {x^3 \sqrt {\frac {d x^8}{c}+1} F_1\left (\frac {3}{8};2,\frac {1}{2};\frac {11}{8};-\frac {b x^8}{a},-\frac {d x^8}{c}\right )}{3 a^2 \sqrt {c+d x^8}} \]

[Out]

1/3*x^3*AppellF1(3/8,2,1/2,11/8,-b*x^8/a,-d*x^8/c)*(1+d*x^8/c)^(1/2)/a^2/(d*x^8+c)^(1/2)

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Rubi [A]  time = 0.05, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {511, 510} \[ \frac {x^3 \sqrt {\frac {d x^8}{c}+1} F_1\left (\frac {3}{8};2,\frac {1}{2};\frac {11}{8};-\frac {b x^8}{a},-\frac {d x^8}{c}\right )}{3 a^2 \sqrt {c+d x^8}} \]

Antiderivative was successfully verified.

[In]

Int[x^2/((a + b*x^8)^2*Sqrt[c + d*x^8]),x]

[Out]

(x^3*Sqrt[1 + (d*x^8)/c]*AppellF1[3/8, 2, 1/2, 11/8, -((b*x^8)/a), -((d*x^8)/c)])/(3*a^2*Sqrt[c + d*x^8])

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])

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])

Rubi steps

\begin {align*} \int \frac {x^2}{\left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx &=\frac {\sqrt {1+\frac {d x^8}{c}} \int \frac {x^2}{\left (a+b x^8\right )^2 \sqrt {1+\frac {d x^8}{c}}} \, dx}{\sqrt {c+d x^8}}\\ &=\frac {x^3 \sqrt {1+\frac {d x^8}{c}} F_1\left (\frac {3}{8};2,\frac {1}{2};\frac {11}{8};-\frac {b x^8}{a},-\frac {d x^8}{c}\right )}{3 a^2 \sqrt {c+d x^8}}\\ \end {align*}

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Mathematica [B]  time = 0.22, size = 170, normalized size = 2.66 \[ \frac {x^3 \left (3 b d x^8 \left (a+b x^8\right ) \sqrt {\frac {d x^8}{c}+1} F_1\left (\frac {11}{8};\frac {1}{2},1;\frac {19}{8};-\frac {d x^8}{c},-\frac {b x^8}{a}\right )+11 \left (a+b x^8\right ) \sqrt {\frac {d x^8}{c}+1} (5 b c-8 a d) F_1\left (\frac {3}{8};\frac {1}{2},1;\frac {11}{8};-\frac {d x^8}{c},-\frac {b x^8}{a}\right )+33 a b \left (c+d x^8\right )\right )}{264 a^2 \left (a+b x^8\right ) \sqrt {c+d x^8} (b c-a d)} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^2/((a + b*x^8)^2*Sqrt[c + d*x^8]),x]

[Out]

(x^3*(33*a*b*(c + d*x^8) + 11*(5*b*c - 8*a*d)*(a + b*x^8)*Sqrt[1 + (d*x^8)/c]*AppellF1[3/8, 1/2, 1, 11/8, -((d
*x^8)/c), -((b*x^8)/a)] + 3*b*d*x^8*(a + b*x^8)*Sqrt[1 + (d*x^8)/c]*AppellF1[11/8, 1/2, 1, 19/8, -((d*x^8)/c),
-((b*x^8)/a)]))/(264*a^2*(b*c - a*d)*(a + b*x^8)*Sqrt[c + d*x^8])

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(b*x^8+a)^2/(d*x^8+c)^(1/2),x, algorithm="fricas")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{{\left (b x^{8} + a\right )}^{2} \sqrt {d x^{8} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(b*x^8+a)^2/(d*x^8+c)^(1/2),x, algorithm="giac")

[Out]

integrate(x^2/((b*x^8 + a)^2*sqrt(d*x^8 + c)), x)

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maple [F]  time = 0.64, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\left (b \,x^{8}+a \right )^{2} \sqrt {d \,x^{8}+c}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(b*x^8+a)^2/(d*x^8+c)^(1/2),x)

[Out]

int(x^2/(b*x^8+a)^2/(d*x^8+c)^(1/2),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{{\left (b x^{8} + a\right )}^{2} \sqrt {d x^{8} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(b*x^8+a)^2/(d*x^8+c)^(1/2),x, algorithm="maxima")

[Out]

integrate(x^2/((b*x^8 + a)^2*sqrt(d*x^8 + c)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x^2}{{\left (b\,x^8+a\right )}^2\,\sqrt {d\,x^8+c}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/((a + b*x^8)^2*(c + d*x^8)^(1/2)),x)

[Out]

int(x^2/((a + b*x^8)^2*(c + d*x^8)^(1/2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\left (a + b x^{8}\right )^{2} \sqrt {c + d x^{8}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(b*x**8+a)**2/(d*x**8+c)**(1/2),x)

[Out]

Integral(x**2/((a + b*x**8)**2*sqrt(c + d*x**8)), x)

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