3.7.67 \(\int \frac {x \sqrt [3]{a+b x^3}}{c+d x^3} \, dx\) [667]

Optimal. Leaf size=234 \[ -\frac {\sqrt [3]{b} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} d}+\frac {\sqrt [3]{b c-a d} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{c} d}-\frac {\sqrt [3]{b c-a d} \log \left (c+d x^3\right )}{6 \sqrt [3]{c} d}-\frac {\sqrt [3]{b} \log \left (\sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{2 d}+\frac {\sqrt [3]{b c-a d} \log \left (\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{c} d} \]

[Out]

-1/6*(-a*d+b*c)^(1/3)*ln(d*x^3+c)/c^(1/3)/d-1/2*b^(1/3)*ln(b^(1/3)*x-(b*x^3+a)^(1/3))/d+1/2*(-a*d+b*c)^(1/3)*l
n((-a*d+b*c)^(1/3)*x/c^(1/3)-(b*x^3+a)^(1/3))/c^(1/3)/d-1/3*b^(1/3)*arctan(1/3*(1+2*b^(1/3)*x/(b*x^3+a)^(1/3))
*3^(1/2))/d*3^(1/2)+1/3*(-a*d+b*c)^(1/3)*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^(1/3)/d*3^(1/2)

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Rubi [A]
time = 0.06, antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {495, 337, 503} \begin {gather*} \frac {\sqrt [3]{b c-a d} \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 )}{\sqrt {3} \sqrt [3]{c} d}-\frac {\sqrt [3]{b} \text {ArcTan}\left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} d}-\frac {\sqrt [3]{b c-a d} \log \left (c+d x^3\right )}{6 \sqrt [3]{c} d}+\frac {\sqrt [3]{b c-a d} \log \left (\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{c} d}-\frac {\sqrt [3]{b} \log \left (\sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 337

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

Rule 495

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

Rule 503

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

Rubi steps

\begin {align*} \int \frac {x \sqrt [3]{a+b x^3}}{c+d x^3} \, dx &=\frac {\sqrt [3]{a+b x^3} \int \frac {x \sqrt [3]{1+\frac {b x^3}{a}}}{c+d x^3} \, dx}{\sqrt [3]{1+\frac {b x^3}{a}}}\\ &=\frac {x^2 \sqrt [3]{a+b x^3} F_1\left (\frac {2}{3};-\frac {1}{3},1;\frac {5}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{2 c \sqrt [3]{1+\frac {b x^3}{a}}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 2.29, size = 423, normalized size = 1.81 \begin {gather*} \frac {-4 \sqrt {3} \sqrt [3]{b} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{b} x}{\sqrt [3]{b} x+2 \sqrt [3]{a+b x^3}}\right )-\frac {2 \sqrt {-6-6 i \sqrt {3}} \sqrt [3]{b c-a d} \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]{c}}-4 \sqrt [3]{b} \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )+\frac {2 i \left (i+\sqrt {3}\right ) \sqrt [3]{b c-a d} \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]{c}}+2 \sqrt [3]{b} \log \left (b^{2/3} x^2+\sqrt [3]{b} x \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )+\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{b c-a d} \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]{c}}}{12 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*Sqrt[3]*b^(1/3)*ArcTan[(Sqrt[3]*b^(1/3)*x)/(b^(1/3)*x + 2*(a + b*x^3)^(1/3))] - (2*Sqrt[-6 - (6*I)*Sqrt[3]
]*(b*c - a*d)^(1/3)*ArcTan[(3*(b*c - a*d)^(1/3)*x)/(Sqrt[3]*(b*c - a*d)^(1/3)*x - (3*I + Sqrt[3])*c^(1/3)*(a +
b*x^3)^(1/3))])/c^(1/3) - 4*b^(1/3)*Log[-(b^(1/3)*x) + (a + b*x^3)^(1/3)] + ((2*I)*(I + Sqrt[3])*(b*c - a*d)^
(1/3)*Log[2*(b*c - a*d)^(1/3)*x + (1 + I*Sqrt[3])*c^(1/3)*(a + b*x^3)^(1/3)])/c^(1/3) + 2*b^(1/3)*Log[b^(2/3)*
x^2 + b^(1/3)*x*(a + b*x^3)^(1/3) + (a + b*x^3)^(2/3)] + ((1 - I*Sqrt[3])*(b*c - a*d)^(1/3)*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 + Sqrt[3])*c^(2/3)*(a + b*x^
3)^(2/3)])/c^(1/3))/(12*d)

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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {x \left (b \,x^{3}+a \right )^{\frac {1}{3}}}{d \,x^{3}+c}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Fricas [A]
time = 2.94, size = 330, normalized size = 1.41 \begin {gather*} \frac {2 \, \sqrt {3} \left (\frac {b c - a d}{c}\right )^{\frac {1}{3}} \arctan \left (-\frac {\sqrt {3} {\left (b c - a d\right )} x + 2 \, \sqrt {3} {\left (b x^{3} + a\right )}^{\frac {1}{3}} c \left (\frac {b c - a d}{c}\right )^{\frac {2}{3}}}{3 \, {\left (b c - a d\right )} x}\right ) - 2 \, \sqrt {3} \left (-b\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} b x + 2 \, \sqrt {3} {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-b\right )^{\frac {2}{3}}}{3 \, b x}\right ) + 2 \, \left (-b\right )^{\frac {1}{3}} \log \left (\frac {\left (-b\right )^{\frac {1}{3}} x + {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right ) + 2 \, \left (\frac {b c - a d}{c}\right )^{\frac {1}{3}} \log \left (-\frac {x \left (\frac {b c - a d}{c}\right )^{\frac {1}{3}} - {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right ) - \left (-b\right )^{\frac {1}{3}} \log \left (\frac {\left (-b\right )^{\frac {2}{3}} x^{2} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-b\right )^{\frac {1}{3}} x + {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right ) - \left (\frac {b c - a d}{c}\right )^{\frac {1}{3}} \log \left (\frac {x^{2} \left (\frac {b c - a d}{c}\right )^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} x \left (\frac {b c - a d}{c}\right )^{\frac {1}{3}} + {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right )}{6 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(2*sqrt(3)*((b*c - a*d)/c)^(1/3)*arctan(-1/3*(sqrt(3)*(b*c - a*d)*x + 2*sqrt(3)*(b*x^3 + a)^(1/3)*c*((b*c
- a*d)/c)^(2/3))/((b*c - a*d)*x)) - 2*sqrt(3)*(-b)^(1/3)*arctan(1/3*(sqrt(3)*b*x + 2*sqrt(3)*(b*x^3 + a)^(1/3)
*(-b)^(2/3))/(b*x)) + 2*(-b)^(1/3)*log(((-b)^(1/3)*x + (b*x^3 + a)^(1/3))/x) + 2*((b*c - a*d)/c)^(1/3)*log(-(x
*((b*c - a*d)/c)^(1/3) - (b*x^3 + a)^(1/3))/x) - (-b)^(1/3)*log(((-b)^(2/3)*x^2 - (b*x^3 + a)^(1/3)*(-b)^(1/3)
*x + (b*x^3 + a)^(2/3))/x^2) - ((b*c - a*d)/c)^(1/3)*log((x^2*((b*c - a*d)/c)^(2/3) + (b*x^3 + a)^(1/3)*x*((b*
c - a*d)/c)^(1/3) + (b*x^3 + a)^(2/3))/x^2))/d

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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