3.21.83 \(\int \frac {(d+e x)^{2/3}}{\sqrt {a d e+(c d^2+a e^2) x+c d e x^2}} \, dx\) [2083]

Optimal. Leaf size=566 \[ \frac {3 (a e+c d x) (d+e x)^{2/3}}{2 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {3^{3/4} \left (c d^2-a e^2\right )^{2/3} \sqrt {a d e+c d^2 x} (d+e x)^{2/3} \left (\sqrt [3]{c d^2-a e^2}-\sqrt [3]{c} d^{2/3} \sqrt [3]{1+\frac {e x}{d}}\right ) \sqrt {\frac {\left (c d^2-a e^2\right )^{2/3}+\sqrt [3]{c} d^{2/3} \sqrt [3]{c d^2-a e^2} \sqrt [3]{1+\frac {e x}{d}}+c^{2/3} d^{4/3} \left (1+\frac {e x}{d}\right )^{2/3}}{\left (\sqrt [3]{c d^2-a e^2}-\left (1+\sqrt {3}\right ) \sqrt [3]{c} d^{2/3} \sqrt [3]{1+\frac {e x}{d}}\right )^2}} F\left (\cos ^{-1}\left (\frac {\sqrt [3]{c d^2-a e^2}-\left (1-\sqrt {3}\right ) \sqrt [3]{c} d^{2/3} \sqrt [3]{1+\frac {e x}{d}}}{\sqrt [3]{c d^2-a e^2}-\left (1+\sqrt {3}\right ) \sqrt [3]{c} d^{2/3} \sqrt [3]{1+\frac {e x}{d}}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{4 c d e \sqrt {d (a e+c d x)} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \sqrt {-\frac {\sqrt [3]{c} d^{2/3} \sqrt [3]{1+\frac {e x}{d}} \left (\sqrt [3]{c d^2-a e^2}-\sqrt [3]{c} d^{2/3} \sqrt [3]{1+\frac {e x}{d}}\right )}{\left (\sqrt [3]{c d^2-a e^2}-\left (1+\sqrt {3}\right ) \sqrt [3]{c} d^{2/3} \sqrt [3]{1+\frac {e x}{d}}\right )^2}}} \]

[Out]

3/2*(c*d*x+a*e)*(e*x+d)^(2/3)/c/d/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+1/4*3^(3/4)*(-a*e^2+c*d^2)^(2/3)*(e*
x+d)^(2/3)*((-a*e^2+c*d^2)^(1/3)-c^(1/3)*d^(2/3)*(1+e*x/d)^(1/3))*(((-a*e^2+c*d^2)^(1/3)-c^(1/3)*d^(2/3)*(1+e*
x/d)^(1/3)*(1-3^(1/2)))^2/((-a*e^2+c*d^2)^(1/3)-c^(1/3)*d^(2/3)*(1+e*x/d)^(1/3)*(1+3^(1/2)))^2)^(1/2)/((-a*e^2
+c*d^2)^(1/3)-c^(1/3)*d^(2/3)*(1+e*x/d)^(1/3)*(1-3^(1/2)))*((-a*e^2+c*d^2)^(1/3)-c^(1/3)*d^(2/3)*(1+e*x/d)^(1/
3)*(1+3^(1/2)))*EllipticF((1-((-a*e^2+c*d^2)^(1/3)-c^(1/3)*d^(2/3)*(1+e*x/d)^(1/3)*(1-3^(1/2)))^2/((-a*e^2+c*d
^2)^(1/3)-c^(1/3)*d^(2/3)*(1+e*x/d)^(1/3)*(1+3^(1/2)))^2)^(1/2),1/4*6^(1/2)+1/4*2^(1/2))*(c*d^2*x+a*d*e)^(1/2)
*(((-a*e^2+c*d^2)^(2/3)+c^(1/3)*d^(2/3)*(-a*e^2+c*d^2)^(1/3)*(1+e*x/d)^(1/3)+c^(2/3)*d^(4/3)*(1+e*x/d)^(2/3))/
((-a*e^2+c*d^2)^(1/3)-c^(1/3)*d^(2/3)*(1+e*x/d)^(1/3)*(1+3^(1/2)))^2)^(1/2)/c/d/e/(d*(c*d*x+a*e))^(1/2)/(a*d*e
+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(-c^(1/3)*d^(2/3)*(1+e*x/d)^(1/3)*((-a*e^2+c*d^2)^(1/3)-c^(1/3)*d^(2/3)*(1+e
*x/d)^(1/3))/((-a*e^2+c*d^2)^(1/3)-c^(1/3)*d^(2/3)*(1+e*x/d)^(1/3)*(1+3^(1/2)))^2)^(1/2)

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Rubi [A]
time = 0.54, antiderivative size = 566, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.128, Rules used = {693, 691, 52, 65, 231} \begin {gather*} \frac {3^{3/4} (d+e x)^{2/3} \left (c d^2-a e^2\right )^{2/3} \sqrt {a d e+c d^2 x} \left (\sqrt [3]{c d^2-a e^2}-\sqrt [3]{c} d^{2/3} \sqrt [3]{\frac {e x}{d}+1}\right ) \sqrt {\frac {\left (c d^2-a e^2\right )^{2/3}+\sqrt [3]{c} d^{2/3} \sqrt [3]{\frac {e x}{d}+1} \sqrt [3]{c d^2-a e^2}+c^{2/3} d^{4/3} \left (\frac {e x}{d}+1\right )^{2/3}}{\left (\sqrt [3]{c d^2-a e^2}-\left (1+\sqrt {3}\right ) \sqrt [3]{c} d^{2/3} \sqrt [3]{\frac {e x}{d}+1}\right )^2}} F\left (\text {ArcCos}\left (\frac {\sqrt [3]{c d^2-a e^2}-\left (1-\sqrt {3}\right ) \sqrt [3]{c} d^{2/3} \sqrt [3]{\frac {e x}{d}+1}}{\sqrt [3]{c d^2-a e^2}-\left (1+\sqrt {3}\right ) \sqrt [3]{c} d^{2/3} \sqrt [3]{\frac {e x}{d}+1}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{4 c d e \sqrt {d (a e+c d x)} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \sqrt {-\frac {\sqrt [3]{c} d^{2/3} \sqrt [3]{\frac {e x}{d}+1} \left (\sqrt [3]{c d^2-a e^2}-\sqrt [3]{c} d^{2/3} \sqrt [3]{\frac {e x}{d}+1}\right )}{\left (\sqrt [3]{c d^2-a e^2}-\left (1+\sqrt {3}\right ) \sqrt [3]{c} d^{2/3} \sqrt [3]{\frac {e x}{d}+1}\right )^2}}}+\frac {3 (d+e x)^{2/3} (a e+c d x)}{2 c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(a*e + c*d*x)*(d + e*x)^(2/3))/(2*c*d*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (3^(3/4)*(c*d^2 - a*e^
2)^(2/3)*Sqrt[a*d*e + c*d^2*x]*(d + e*x)^(2/3)*((c*d^2 - a*e^2)^(1/3) - c^(1/3)*d^(2/3)*(1 + (e*x)/d)^(1/3))*S
qrt[((c*d^2 - a*e^2)^(2/3) + c^(1/3)*d^(2/3)*(c*d^2 - a*e^2)^(1/3)*(1 + (e*x)/d)^(1/3) + c^(2/3)*d^(4/3)*(1 +
(e*x)/d)^(2/3))/((c*d^2 - a*e^2)^(1/3) - (1 + Sqrt[3])*c^(1/3)*d^(2/3)*(1 + (e*x)/d)^(1/3))^2]*EllipticF[ArcCo
s[((c*d^2 - a*e^2)^(1/3) - (1 - Sqrt[3])*c^(1/3)*d^(2/3)*(1 + (e*x)/d)^(1/3))/((c*d^2 - a*e^2)^(1/3) - (1 + Sq
rt[3])*c^(1/3)*d^(2/3)*(1 + (e*x)/d)^(1/3))], (2 + Sqrt[3])/4])/(4*c*d*e*Sqrt[d*(a*e + c*d*x)]*Sqrt[a*d*e + (c
*d^2 + a*e^2)*x + c*d*e*x^2]*Sqrt[-((c^(1/3)*d^(2/3)*(1 + (e*x)/d)^(1/3)*((c*d^2 - a*e^2)^(1/3) - c^(1/3)*d^(2
/3)*(1 + (e*x)/d)^(1/3)))/((c*d^2 - a*e^2)^(1/3) - (1 + Sqrt[3])*c^(1/3)*d^(2/3)*(1 + (e*x)/d)^(1/3))^2)])

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 231

Int[1/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[x*(s +
r*x^2)*(Sqrt[(s^2 - r*s*x^2 + r^2*x^4)/(s + (1 + Sqrt[3])*r*x^2)^2]/(2*3^(1/4)*s*Sqrt[a + b*x^6]*Sqrt[r*x^2*(
(s + r*x^2)/(s + (1 + Sqrt[3])*r*x^2)^2)]))*EllipticF[ArcCos[(s + (1 - Sqrt[3])*r*x^2)/(s + (1 + Sqrt[3])*r*x^
2)], (2 + Sqrt[3])/4], x]] /; FreeQ[{a, b}, x]

Rule 691

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[d^m*((a + b*x + c*x^2
)^FracPart[p]/((1 + e*(x/d))^FracPart[p]*(a/d + (c*x)/e)^FracPart[p])), Int[(1 + e*(x/d))^(m + p)*(a/d + (c/e)
*x)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !Int
egerQ[p] && (IntegerQ[m] || GtQ[d, 0]) &&  !(IGtQ[m, 0] && (IntegerQ[3*p] || IntegerQ[4*p]))

Rule 693

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[d^IntPart[m]*((d + e*
x)^FracPart[m]/(1 + e*(x/d))^FracPart[m]), Int[(1 + e*(x/d))^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c,
d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] &&  !(IntegerQ[m] || GtQ
[d, 0])

Rubi steps

\begin {align*} \int \frac {(d+e x)^{2/3}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx &=\frac {(d+e x)^{2/3} \int \frac {\left (1+\frac {e x}{d}\right )^{2/3}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{\left (1+\frac {e x}{d}\right )^{2/3}}\\ &=\frac {\left (\sqrt {a d e+c d^2 x} (d+e x)^{2/3}\right ) \int \frac {\sqrt [6]{1+\frac {e x}{d}}}{\sqrt {a d e+c d^2 x}} \, dx}{\sqrt [6]{1+\frac {e x}{d}} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ &=\frac {3 (a e+c d x) (d+e x)^{2/3}}{2 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\left (\left (1-\frac {a e^2}{c d^2}\right ) \sqrt {a d e+c d^2 x} (d+e x)^{2/3}\right ) \int \frac {1}{\sqrt {a d e+c d^2 x} \left (1+\frac {e x}{d}\right )^{5/6}} \, dx}{4 \sqrt [6]{1+\frac {e x}{d}} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ &=\frac {3 (a e+c d x) (d+e x)^{2/3}}{2 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\left (3 d \left (1-\frac {a e^2}{c d^2}\right ) \sqrt {a d e+c d^2 x} (d+e x)^{2/3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-\frac {c d^3}{e}+a d e+\frac {c d^3 x^6}{e}}} \, dx,x,\sqrt [6]{1+\frac {e x}{d}}\right )}{2 e \sqrt [6]{1+\frac {e x}{d}} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ &=\frac {3 (a e+c d x) (d+e x)^{2/3}}{2 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {3^{3/4} \left (c d^2-a e^2\right )^{2/3} \sqrt {a d e+c d^2 x} (d+e x)^{2/3} \left (\sqrt [3]{c d^2-a e^2}-\sqrt [3]{c} d^{2/3} \sqrt [3]{1+\frac {e x}{d}}\right ) \sqrt {\frac {\left (c d^2-a e^2\right )^{2/3}+\sqrt [3]{c} d^{2/3} \sqrt [3]{c d^2-a e^2} \sqrt [3]{1+\frac {e x}{d}}+c^{2/3} d^{4/3} \left (1+\frac {e x}{d}\right )^{2/3}}{\left (\sqrt [3]{c d^2-a e^2}-\left (1+\sqrt {3}\right ) \sqrt [3]{c} d^{2/3} \sqrt [3]{1+\frac {e x}{d}}\right )^2}} F\left (\cos ^{-1}\left (\frac {\sqrt [3]{c d^2-a e^2}-\left (1-\sqrt {3}\right ) \sqrt [3]{c} d^{2/3} \sqrt [3]{1+\frac {e x}{d}}}{\sqrt [3]{c d^2-a e^2}-\left (1+\sqrt {3}\right ) \sqrt [3]{c} d^{2/3} \sqrt [3]{1+\frac {e x}{d}}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{4 c d e \sqrt {d (a e+c d x)} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \sqrt {-\frac {\sqrt [3]{c} d^{2/3} \sqrt [3]{1+\frac {e x}{d}} \left (\sqrt [3]{c d^2-a e^2}-\sqrt [3]{c} d^{2/3} \sqrt [3]{1+\frac {e x}{d}}\right )}{\left (\sqrt [3]{c d^2-a e^2}-\left (1+\sqrt {3}\right ) \sqrt [3]{c} d^{2/3} \sqrt [3]{1+\frac {e x}{d}}\right )^2}}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 10.05, size = 95, normalized size = 0.17 \begin {gather*} \frac {2 \sqrt {(a e+c d x) (d+e x)} \, _2F_1\left (-\frac {1}{6},\frac {1}{2};\frac {3}{2};\frac {e (a e+c d x)}{-c d^2+a e^2}\right )}{c d \sqrt [3]{d+e x} \sqrt [6]{\frac {c d (d+e x)}{c d^2-a e^2}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[(a*e + c*d*x)*(d + e*x)]*Hypergeometric2F1[-1/6, 1/2, 3/2, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/(c*d
*(d + e*x)^(1/3)*((c*d*(d + e*x))/(c*d^2 - a*e^2))^(1/6))

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (e x +d \right )^{\frac {2}{3}}}{\sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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

[Out]

integral((x*e + d)^(2/3)/sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d + e x\right )^{\frac {2}{3}}}{\sqrt {\left (d + e x\right ) \left (a e + c d x\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((d + e*x)**(2/3)/sqrt((d + e*x)*(a*e + c*d*x)), x)

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (d+e\,x\right )}^{2/3}}{\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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