3.1.100 \(\int \frac {(a+b x^3)^{2/3}}{(c+d x^3)^2} \, dx\) [100]

Optimal. Leaf size=182 \[ \frac {x \left (a+b x^3\right )^{2/3}}{3 c \left (c+d x^3\right )}+\frac {2 a \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3} c^{5/3} \sqrt [3]{b c-a d}}+\frac {a \log \left (c+d x^3\right )}{9 c^{5/3} \sqrt [3]{b c-a d}}-\frac {a \log \left (\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{3 c^{5/3} \sqrt [3]{b c-a d}} \]

[Out]

1/3*x*(b*x^3+a)^(2/3)/c/(d*x^3+c)+1/9*a*ln(d*x^3+c)/c^(5/3)/(-a*d+b*c)^(1/3)-1/3*a*ln((-a*d+b*c)^(1/3)*x/c^(1/
3)-(b*x^3+a)^(1/3))/c^(5/3)/(-a*d+b*c)^(1/3)+2/9*a*arctan(1/3*(1+2*(-a*d+b*c)^(1/3)*x/c^(1/3)/(b*x^3+a)^(1/3))
*3^(1/2))/c^(5/3)/(-a*d+b*c)^(1/3)*3^(1/2)

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Rubi [A]
time = 0.04, antiderivative size = 182, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {386, 384} \begin {gather*} \frac {2 a \text {ArcTan}\left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{3 \sqrt {3} c^{5/3} \sqrt [3]{b c-a d}}+\frac {a \log \left (c+d x^3\right )}{9 c^{5/3} \sqrt [3]{b c-a d}}-\frac {a \log \left (\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{3 c^{5/3} \sqrt [3]{b c-a d}}+\frac {x \left (a+b x^3\right )^{2/3}}{3 c \left (c+d x^3\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(a + b*x^3)^(2/3))/(3*c*(c + d*x^3)) + (2*a*ArcTan[(1 + (2*(b*c - a*d)^(1/3)*x)/(c^(1/3)*(a + b*x^3)^(1/3))
)/Sqrt[3]])/(3*Sqrt[3]*c^(5/3)*(b*c - a*d)^(1/3)) + (a*Log[c + d*x^3])/(9*c^(5/3)*(b*c - a*d)^(1/3)) - (a*Log[
((b*c - a*d)^(1/3)*x)/c^(1/3) - (a + b*x^3)^(1/3)])/(3*c^(5/3)*(b*c - a*d)^(1/3))

Rule 384

Int[1/(((a_) + (b_.)*(x_)^3)^(1/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/c, 3]}, Simp[
ArcTan[(1 + (2*q*x)/(a + b*x^3)^(1/3))/Sqrt[3]]/(Sqrt[3]*c*q), x] + (-Simp[Log[q*x - (a + b*x^3)^(1/3)]/(2*c*q
), x] + Simp[Log[c + d*x^3]/(6*c*q), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 386

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Simp[(-x)*(a + b*x^n)^(p + 1)*(
(c + d*x^n)^q/(a*n*(p + 1))), x] - Dist[c*(q/(a*(p + 1))), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1), x], x]
/; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[n*(p + q + 1) + 1, 0] && GtQ[q, 0] && NeQ[p, -1]

Rubi steps

\begin {align*} \int \frac {\left (a+b x^3\right )^{2/3}}{\left (c+d x^3\right )^2} \, dx &=\frac {x \left (a+b x^3\right )^{2/3}}{3 c \left (c+d x^3\right )}+\frac {(2 a) \int \frac {1}{\sqrt [3]{a+b x^3} \left (c+d x^3\right )} \, dx}{3 c}\\ &=\frac {x \left (a+b x^3\right )^{2/3}}{3 c \left (c+d x^3\right )}+\frac {(2 a) \text {Subst}\left (\int \frac {1}{c-(b c-a d) x^3} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )}{3 c}\\ &=\frac {x \left (a+b x^3\right )^{2/3}}{3 c \left (c+d x^3\right )}+\frac {(2 a) \text {Subst}\left (\int \frac {1}{\sqrt [3]{c}-\sqrt [3]{b c-a d} x} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )}{9 c^{5/3}}+\frac {(2 a) \text {Subst}\left (\int \frac {2 \sqrt [3]{c}+\sqrt [3]{b c-a d} x}{c^{2/3}+\sqrt [3]{c} \sqrt [3]{b c-a d} x+(b c-a d)^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )}{9 c^{5/3}}\\ &=\frac {x \left (a+b x^3\right )^{2/3}}{3 c \left (c+d x^3\right )}-\frac {2 a \log \left (\sqrt [3]{c}-\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{a+b x^3}}\right )}{9 c^{5/3} \sqrt [3]{b c-a d}}+\frac {a \text {Subst}\left (\int \frac {1}{c^{2/3}+\sqrt [3]{c} \sqrt [3]{b c-a d} x+(b c-a d)^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )}{3 c^{4/3}}+\frac {a \text {Subst}\left (\int \frac {\sqrt [3]{c} \sqrt [3]{b c-a d}+2 (b c-a d)^{2/3} x}{c^{2/3}+\sqrt [3]{c} \sqrt [3]{b c-a d} x+(b c-a d)^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )}{9 c^{5/3} \sqrt [3]{b c-a d}}\\ &=\frac {x \left (a+b x^3\right )^{2/3}}{3 c \left (c+d x^3\right )}-\frac {2 a \log \left (\sqrt [3]{c}-\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{a+b x^3}}\right )}{9 c^{5/3} \sqrt [3]{b c-a d}}+\frac {a \log \left (c^{2/3}+\frac {(b c-a d)^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac {\sqrt [3]{c} \sqrt [3]{b c-a d} x}{\sqrt [3]{a+b x^3}}\right )}{9 c^{5/3} \sqrt [3]{b c-a d}}-\frac {(2 a) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}\right )}{3 c^{5/3} \sqrt [3]{b c-a d}}\\ &=\frac {x \left (a+b x^3\right )^{2/3}}{3 c \left (c+d x^3\right )}+\frac {2 a \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3} c^{5/3} \sqrt [3]{b c-a d}}-\frac {2 a \log \left (\sqrt [3]{c}-\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{a+b x^3}}\right )}{9 c^{5/3} \sqrt [3]{b c-a d}}+\frac {a \log \left (c^{2/3}+\frac {(b c-a d)^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac {\sqrt [3]{c} \sqrt [3]{b c-a d} x}{\sqrt [3]{a+b x^3}}\right )}{9 c^{5/3} \sqrt [3]{b c-a d}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 1.74, size = 319, normalized size = 1.75 \begin {gather*} \frac {\frac {6 c^{2/3} x \left (a+b x^3\right )^{2/3}}{c+d x^3}-\frac {2 \sqrt {-6+6 i \sqrt {3}} a \tan ^{-1}\left (\frac {3 \sqrt [3]{b c-a d} x}{\sqrt {3} \sqrt [3]{b c-a d} x-\left (3 i+\sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{a+b x^3}}\right )}{\sqrt [3]{b c-a d}}+\frac {2 \left (a+i \sqrt {3} a\right ) \log \left (2 \sqrt [3]{b c-a d} x+\left (1+i \sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{a+b x^3}\right )}{\sqrt [3]{b c-a d}}-\frac {i \left (-i+\sqrt {3}\right ) a \log \left (2 (b c-a d)^{2/3} x^2+\left (-1-i \sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{b c-a d} x \sqrt [3]{a+b x^3}+i \left (i+\sqrt {3}\right ) c^{2/3} \left (a+b x^3\right )^{2/3}\right )}{\sqrt [3]{b c-a d}}}{18 c^{5/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((6*c^(2/3)*x*(a + b*x^3)^(2/3))/(c + d*x^3) - (2*Sqrt[-6 + (6*I)*Sqrt[3]]*a*ArcTan[(3*(b*c - a*d)^(1/3)*x)/(S
qrt[3]*(b*c - a*d)^(1/3)*x - (3*I + Sqrt[3])*c^(1/3)*(a + b*x^3)^(1/3))])/(b*c - a*d)^(1/3) + (2*(a + I*Sqrt[3
]*a)*Log[2*(b*c - a*d)^(1/3)*x + (1 + I*Sqrt[3])*c^(1/3)*(a + b*x^3)^(1/3)])/(b*c - a*d)^(1/3) - (I*(-I + Sqrt
[3])*a*Log[2*(b*c - a*d)^(2/3)*x^2 + (-1 - I*Sqrt[3])*c^(1/3)*(b*c - a*d)^(1/3)*x*(a + b*x^3)^(1/3) + I*(I + S
qrt[3])*c^(2/3)*(a + b*x^3)^(2/3)])/(b*c - a*d)^(1/3))/(18*c^(5/3))

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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{\left (d \,x^{3}+c \right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (b\,x^3+a\right )}^{2/3}}{{\left (d\,x^3+c\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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