3.880 \(\int \frac {x^4}{(a+b x^6)^2 \sqrt {c+d x^6}} \, dx\)

Optimal. Leaf size=64 \[ \frac {x^5 \sqrt {\frac {d x^6}{c}+1} F_1\left (\frac {5}{6};2,\frac {1}{2};\frac {11}{6};-\frac {b x^6}{a},-\frac {d x^6}{c}\right )}{5 a^2 \sqrt {c+d x^6}} \]

[Out]

1/5*x^5*AppellF1(5/6,2,1/2,11/6,-b*x^6/a,-d*x^6/c)*(1+d*x^6/c)^(1/2)/a^2/(d*x^6+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} \[ \frac {x^5 \sqrt {\frac {d x^6}{c}+1} F_1\left (\frac {5}{6};2,\frac {1}{2};\frac {11}{6};-\frac {b x^6}{a},-\frac {d x^6}{c}\right )}{5 a^2 \sqrt {c+d x^6}} \]

Antiderivative was successfully verified.

[In]

Int[x^4/((a + b*x^6)^2*Sqrt[c + d*x^6]),x]

[Out]

(x^5*Sqrt[1 + (d*x^6)/c]*AppellF1[5/6, 2, 1/2, 11/6, -((b*x^6)/a), -((d*x^6)/c)])/(5*a^2*Sqrt[c + d*x^6])

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^4}{\left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx &=\frac {\sqrt {1+\frac {d x^6}{c}} \int \frac {x^4}{\left (a+b x^6\right )^2 \sqrt {1+\frac {d x^6}{c}}} \, dx}{\sqrt {c+d x^6}}\\ &=\frac {x^5 \sqrt {1+\frac {d x^6}{c}} F_1\left (\frac {5}{6};2,\frac {1}{2};\frac {11}{6};-\frac {b x^6}{a},-\frac {d x^6}{c}\right )}{5 a^2 \sqrt {c+d x^6}}\\ \end {align*}

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Mathematica [B]  time = 0.22, size = 169, normalized size = 2.64 \[ \frac {x^5 \left (-10 b d x^6 \left (a+b x^6\right ) \sqrt {\frac {d x^6}{c}+1} F_1\left (\frac {11}{6};\frac {1}{2},1;\frac {17}{6};-\frac {d x^6}{c},-\frac {b x^6}{a}\right )+11 \left (a+b x^6\right ) \sqrt {\frac {d x^6}{c}+1} (b c-6 a d) F_1\left (\frac {5}{6};\frac {1}{2},1;\frac {11}{6};-\frac {d x^6}{c},-\frac {b x^6}{a}\right )+55 a b \left (c+d x^6\right )\right )}{330 a^2 \left (a+b x^6\right ) \sqrt {c+d x^6} (b c-a d)} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^4/((a + b*x^6)^2*Sqrt[c + d*x^6]),x]

[Out]

(x^5*(55*a*b*(c + d*x^6) + 11*(b*c - 6*a*d)*(a + b*x^6)*Sqrt[1 + (d*x^6)/c]*AppellF1[5/6, 1/2, 1, 11/6, -((d*x
^6)/c), -((b*x^6)/a)] - 10*b*d*x^6*(a + b*x^6)*Sqrt[1 + (d*x^6)/c]*AppellF1[11/6, 1/2, 1, 17/6, -((d*x^6)/c),
-((b*x^6)/a)]))/(330*a^2*(b*c - a*d)*(a + b*x^6)*Sqrt[c + d*x^6])

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fricas [F]  time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {d x^{6} + c} x^{4}}{b^{2} d x^{18} + {\left (b^{2} c + 2 \, a b d\right )} x^{12} + {\left (2 \, a b c + a^{2} d\right )} x^{6} + a^{2} c}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x^6+a)^2/(d*x^6+c)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(d*x^6 + c)*x^4/(b^2*d*x^18 + (b^2*c + 2*a*b*d)*x^12 + (2*a*b*c + a^2*d)*x^6 + a^2*c), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4}}{{\left (b x^{6} + a\right )}^{2} \sqrt {d x^{6} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x^6+a)^2/(d*x^6+c)^(1/2),x, algorithm="giac")

[Out]

integrate(x^4/((b*x^6 + a)^2*sqrt(d*x^6 + c)), x)

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maple [F]  time = 0.63, size = 0, normalized size = 0.00 \[ \int \frac {x^{4}}{\left (b \,x^{6}+a \right )^{2} \sqrt {d \,x^{6}+c}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(b*x^6+a)^2/(d*x^6+c)^(1/2),x)

[Out]

int(x^4/(b*x^6+a)^2/(d*x^6+c)^(1/2),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4}}{{\left (b x^{6} + a\right )}^{2} \sqrt {d x^{6} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x^6+a)^2/(d*x^6+c)^(1/2),x, algorithm="maxima")

[Out]

integrate(x^4/((b*x^6 + a)^2*sqrt(d*x^6 + c)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x^4}{{\left (b\,x^6+a\right )}^2\,\sqrt {d\,x^6+c}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/((a + b*x^6)^2*(c + d*x^6)^(1/2)),x)

[Out]

int(x^4/((a + b*x^6)^2*(c + d*x^6)^(1/2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4}}{\left (a + b x^{6}\right )^{2} \sqrt {c + d x^{6}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/(b*x**6+a)**2/(d*x**6+c)**(1/2),x)

[Out]

Integral(x**4/((a + b*x**6)**2*sqrt(c + d*x**6)), x)

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