3.269 \(\int \frac{\sqrt{c+d x^3}}{x^3 (4 c+d x^3)} \, dx\)
Optimal. Leaf size=66 \[ -\frac{\sqrt{c+d x^3} F_1\left (-\frac{2}{3};1,-\frac{1}{2};\frac{1}{3};-\frac{d x^3}{4 c},-\frac{d x^3}{c}\right )}{8 c x^2 \sqrt{\frac{d x^3}{c}+1}} \]
[Out]
-(Sqrt[c + d*x^3]*AppellF1[-2/3, 1, -1/2, 1/3, -(d*x^3)/(4*c), -((d*x^3)/c)])/(8*c*x^2*Sqrt[1 + (d*x^3)/c])
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Rubi [A] time = 0.056457, antiderivative size = 66, normalized size of antiderivative = 1.,
number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used =
{511, 510} \[ -\frac{\sqrt{c+d x^3} F_1\left (-\frac{2}{3};1,-\frac{1}{2};\frac{1}{3};-\frac{d x^3}{4 c},-\frac{d x^3}{c}\right )}{8 c x^2 \sqrt{\frac{d x^3}{c}+1}} \]
Antiderivative was successfully verified.
[In]
Int[Sqrt[c + d*x^3]/(x^3*(4*c + d*x^3)),x]
[Out]
-(Sqrt[c + d*x^3]*AppellF1[-2/3, 1, -1/2, 1/3, -(d*x^3)/(4*c), -((d*x^3)/c)])/(8*c*x^2*Sqrt[1 + (d*x^3)/c])
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{\sqrt{c+d x^3}}{x^3 \left (4 c+d x^3\right )} \, dx &=\frac{\sqrt{c+d x^3} \int \frac{\sqrt{1+\frac{d x^3}{c}}}{x^3 \left (4 c+d x^3\right )} \, dx}{\sqrt{1+\frac{d x^3}{c}}}\\ &=-\frac{\sqrt{c+d x^3} F_1\left (-\frac{2}{3};1,-\frac{1}{2};\frac{1}{3};-\frac{d x^3}{4 c},-\frac{d x^3}{c}\right )}{8 c x^2 \sqrt{1+\frac{d x^3}{c}}}\\ \end{align*}
Mathematica [B] time = 0.126923, size = 244, normalized size = 3.7 \[ \frac{\frac{2048 c^3 d x^3 F_1\left (\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right )}{\left (4 c+d x^3\right ) \left (16 c F_1\left (\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right )-3 d x^3 \left (F_1\left (\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right )+2 F_1\left (\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right )\right )\right )}-d^2 x^6 \sqrt{\frac{d x^3}{c}+1} F_1\left (\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right )-32 c \left (c+d x^3\right )}{256 c^2 x^2 \sqrt{c+d x^3}} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[Sqrt[c + d*x^3]/(x^3*(4*c + d*x^3)),x]
[Out]
(-32*c*(c + d*x^3) - d^2*x^6*Sqrt[1 + (d*x^3)/c]*AppellF1[4/3, 1/2, 1, 7/3, -((d*x^3)/c), -(d*x^3)/(4*c)] + (2
048*c^3*d*x^3*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), -(d*x^3)/(4*c)])/((4*c + d*x^3)*(16*c*AppellF1[1/3, 1/2
, 1, 4/3, -((d*x^3)/c), -(d*x^3)/(4*c)] - 3*d*x^3*(AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3)/c), -(d*x^3)/(4*c)] +
2*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), -(d*x^3)/(4*c)]))))/(256*c^2*x^2*Sqrt[c + d*x^3])
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Maple [C] time = 0.02, size = 1002, normalized size = 15.2 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d*x^3+c)^(1/2)/x^3/(d*x^3+4*c),x)
[Out]
1/4/c*(-1/2/x^2*(d*x^3+c)^(1/2)-1/2*I*3^(1/2)*(-d^2*c)^(1/3)*(I*(x+1/2/d*(-d^2*c)^(1/3)-1/2*I*3^(1/2)/d*(-d^2*
c)^(1/3))*3^(1/2)*d/(-d^2*c)^(1/3))^(1/2)*((x-1/d*(-d^2*c)^(1/3))/(-3/2/d*(-d^2*c)^(1/3)+1/2*I*3^(1/2)/d*(-d^2
*c)^(1/3)))^(1/2)*(-I*(x+1/2/d*(-d^2*c)^(1/3)+1/2*I*3^(1/2)/d*(-d^2*c)^(1/3))*3^(1/2)*d/(-d^2*c)^(1/3))^(1/2)/
(d*x^3+c)^(1/2)*EllipticF(1/3*3^(1/2)*(I*(x+1/2/d*(-d^2*c)^(1/3)-1/2*I*3^(1/2)/d*(-d^2*c)^(1/3))*3^(1/2)*d/(-d
^2*c)^(1/3))^(1/2),(I*3^(1/2)/d*(-d^2*c)^(1/3)/(-3/2/d*(-d^2*c)^(1/3)+1/2*I*3^(1/2)/d*(-d^2*c)^(1/3)))^(1/2)))
-1/4*d/c*(-2/3*I*3^(1/2)/d*(-d^2*c)^(1/3)*(I*(x+1/2/d*(-d^2*c)^(1/3)-1/2*I*3^(1/2)/d*(-d^2*c)^(1/3))*3^(1/2)*d
/(-d^2*c)^(1/3))^(1/2)*((x-1/d*(-d^2*c)^(1/3))/(-3/2/d*(-d^2*c)^(1/3)+1/2*I*3^(1/2)/d*(-d^2*c)^(1/3)))^(1/2)*(
-I*(x+1/2/d*(-d^2*c)^(1/3)+1/2*I*3^(1/2)/d*(-d^2*c)^(1/3))*3^(1/2)*d/(-d^2*c)^(1/3))^(1/2)/(d*x^3+c)^(1/2)*Ell
ipticF(1/3*3^(1/2)*(I*(x+1/2/d*(-d^2*c)^(1/3)-1/2*I*3^(1/2)/d*(-d^2*c)^(1/3))*3^(1/2)*d/(-d^2*c)^(1/3))^(1/2),
(I*3^(1/2)/d*(-d^2*c)^(1/3)/(-3/2/d*(-d^2*c)^(1/3)+1/2*I*3^(1/2)/d*(-d^2*c)^(1/3)))^(1/2))+1/3*I/d^3*2^(1/2)*s
um(1/_alpha^2*(-d^2*c)^(1/3)*(1/2*I*d*(2*x+1/d*(-I*3^(1/2)*(-d^2*c)^(1/3)+(-d^2*c)^(1/3)))/(-d^2*c)^(1/3))^(1/
2)*(d*(x-1/d*(-d^2*c)^(1/3))/(-3*(-d^2*c)^(1/3)+I*3^(1/2)*(-d^2*c)^(1/3)))^(1/2)*(-1/2*I*d*(2*x+1/d*(I*3^(1/2)
*(-d^2*c)^(1/3)+(-d^2*c)^(1/3)))/(-d^2*c)^(1/3))^(1/2)/(d*x^3+c)^(1/2)*(I*(-d^2*c)^(1/3)*_alpha*3^(1/2)*d-I*3^
(1/2)*(-d^2*c)^(2/3)+2*_alpha^2*d^2-(-d^2*c)^(1/3)*_alpha*d-(-d^2*c)^(2/3))*EllipticPi(1/3*3^(1/2)*(I*(x+1/2/d
*(-d^2*c)^(1/3)-1/2*I*3^(1/2)/d*(-d^2*c)^(1/3))*3^(1/2)*d/(-d^2*c)^(1/3))^(1/2),1/6/d*(2*I*(-d^2*c)^(1/3)*3^(1
/2)*_alpha^2*d-I*(-d^2*c)^(2/3)*3^(1/2)*_alpha+I*3^(1/2)*c*d-3*(-d^2*c)^(2/3)*_alpha-3*c*d)/c,(I*3^(1/2)/d*(-d
^2*c)^(1/3)/(-3/2/d*(-d^2*c)^(1/3)+1/2*I*3^(1/2)/d*(-d^2*c)^(1/3)))^(1/2)),_alpha=RootOf(_Z^3*d+4*c)))
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{d x^{3} + c}}{{\left (d x^{3} + 4 \, c\right )} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x^3+c)^(1/2)/x^3/(d*x^3+4*c),x, algorithm="maxima")
[Out]
integrate(sqrt(d*x^3 + c)/((d*x^3 + 4*c)*x^3), x)
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{d x^{3} + c}}{d x^{6} + 4 \, c x^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x^3+c)^(1/2)/x^3/(d*x^3+4*c),x, algorithm="fricas")
[Out]
integral(sqrt(d*x^3 + c)/(d*x^6 + 4*c*x^3), x)
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c + d x^{3}}}{x^{3} \left (4 c + d x^{3}\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x**3+c)**(1/2)/x**3/(d*x**3+4*c),x)
[Out]
Integral(sqrt(c + d*x**3)/(x**3*(4*c + d*x**3)), x)
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{d x^{3} + c}}{{\left (d x^{3} + 4 \, c\right )} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x^3+c)^(1/2)/x^3/(d*x^3+4*c),x, algorithm="giac")
[Out]
integrate(sqrt(d*x^3 + c)/((d*x^3 + 4*c)*x^3), x)