3.6.15 \(\int \frac {1}{x^3 \sqrt {a+b x^3} \sqrt {c+d x^3}} \, dx\) [515]

Optimal. Leaf size=88 \[ -\frac {\sqrt {1+\frac {b x^3}{a}} \sqrt {1+\frac {d x^3}{c}} F_1\left (-\frac {2}{3};\frac {1}{2},\frac {1}{2};\frac {1}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{2 x^2 \sqrt {a+b x^3} \sqrt {c+d x^3}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.06, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {525, 524} \begin {gather*} -\frac {\sqrt {\frac {b x^3}{a}+1} \sqrt {\frac {d x^3}{c}+1} F_1\left (-\frac {2}{3};\frac {1}{2},\frac {1}{2};\frac {1}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{2 x^2 \sqrt {a+b x^3} \sqrt {c+d x^3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^3*Sqrt[a + b*x^3]*Sqrt[c + d*x^3]),x]

[Out]

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

Rule 524

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)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], 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 525

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[a^IntPar
t[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 {1}{x^3 \sqrt {a+b x^3} \sqrt {c+d x^3}} \, dx &=\frac {\sqrt {1+\frac {b x^3}{a}} \int \frac {1}{x^3 \sqrt {1+\frac {b x^3}{a}} \sqrt {c+d x^3}} \, dx}{\sqrt {a+b x^3}}\\ &=\frac {\left (\sqrt {1+\frac {b x^3}{a}} \sqrt {1+\frac {d x^3}{c}}\right ) \int \frac {1}{x^3 \sqrt {1+\frac {b x^3}{a}} \sqrt {1+\frac {d x^3}{c}}} \, dx}{\sqrt {a+b x^3} \sqrt {c+d x^3}}\\ &=-\frac {\sqrt {1+\frac {b x^3}{a}} \sqrt {1+\frac {d x^3}{c}} F_1\left (-\frac {2}{3};\frac {1}{2},\frac {1}{2};\frac {1}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{2 x^2 \sqrt {a+b x^3} \sqrt {c+d x^3}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(365\) vs. \(2(88)=176\).
time = 2.49, size = 365, normalized size = 4.15 \begin {gather*} \frac {b d x^6 \sqrt {1+\frac {b x^3}{a}} \sqrt {1+\frac {d x^3}{c}} F_1\left (\frac {4}{3};\frac {1}{2},\frac {1}{2};\frac {7}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+\frac {4 \left (-4 a c \left (2 a c+3 b c x^3+3 a d x^3+2 b d x^6\right ) F_1\left (\frac {1}{3};\frac {1}{2},\frac {1}{2};\frac {4}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+3 x^3 \left (a+b x^3\right ) \left (c+d x^3\right ) \left (a d F_1\left (\frac {4}{3};\frac {1}{2},\frac {3}{2};\frac {7}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+b c F_1\left (\frac {4}{3};\frac {3}{2},\frac {1}{2};\frac {7}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )\right )\right )}{8 a c F_1\left (\frac {1}{3};\frac {1}{2},\frac {1}{2};\frac {4}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )-3 x^3 \left (a d F_1\left (\frac {4}{3};\frac {1}{2},\frac {3}{2};\frac {7}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+b c F_1\left (\frac {4}{3};\frac {3}{2},\frac {1}{2};\frac {7}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )\right )}}{8 a c x^2 \sqrt {a+b x^3} \sqrt {c+d x^3}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(x^3*Sqrt[a + b*x^3]*Sqrt[c + d*x^3]),x]

[Out]

(b*d*x^6*Sqrt[1 + (b*x^3)/a]*Sqrt[1 + (d*x^3)/c]*AppellF1[4/3, 1/2, 1/2, 7/3, -((b*x^3)/a), -((d*x^3)/c)] + (4
*(-4*a*c*(2*a*c + 3*b*c*x^3 + 3*a*d*x^3 + 2*b*d*x^6)*AppellF1[1/3, 1/2, 1/2, 4/3, -((b*x^3)/a), -((d*x^3)/c)]
+ 3*x^3*(a + b*x^3)*(c + d*x^3)*(a*d*AppellF1[4/3, 1/2, 3/2, 7/3, -((b*x^3)/a), -((d*x^3)/c)] + b*c*AppellF1[4
/3, 3/2, 1/2, 7/3, -((b*x^3)/a), -((d*x^3)/c)])))/(8*a*c*AppellF1[1/3, 1/2, 1/2, 4/3, -((b*x^3)/a), -((d*x^3)/
c)] - 3*x^3*(a*d*AppellF1[4/3, 1/2, 3/2, 7/3, -((b*x^3)/a), -((d*x^3)/c)] + b*c*AppellF1[4/3, 3/2, 1/2, 7/3, -
((b*x^3)/a), -((d*x^3)/c)])))/(8*a*c*x^2*Sqrt[a + b*x^3]*Sqrt[c + d*x^3])

________________________________________________________________________________________

Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{3} \sqrt {b \,x^{3}+a}\, \sqrt {d \,x^{3}+c}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

integrate(1/(sqrt(b*x^3 + a)*sqrt(d*x^3 + c)*x^3), x)

________________________________________________________________________________________

Fricas [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(1/x^3/(b*x^3+a)^(1/2)/(d*x^3+c)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(b*x^3 + a)*sqrt(d*x^3 + c)/(b*d*x^9 + (b*c + a*d)*x^6 + a*c*x^3), x)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{3} \sqrt {a + b x^{3}} \sqrt {c + d x^{3}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(x**3*sqrt(a + b*x**3)*sqrt(c + d*x**3)), x)

________________________________________________________________________________________

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

[Out]

integrate(1/(sqrt(b*x^3 + a)*sqrt(d*x^3 + c)*x^3), x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{x^3\,\sqrt {b\,x^3+a}\,\sqrt {d\,x^3+c}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________