3.17.3 \(\int \frac {1}{\sqrt [3]{a+b x} (c+d x)^{2/3}} \, dx\) [1603]

Optimal. Leaf size=126 \[ -\frac {\sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{b} \sqrt [3]{c+d x}}\right )}{\sqrt [3]{b} d^{2/3}}-\frac {\log (c+d x)}{2 \sqrt [3]{b} d^{2/3}}-\frac {3 \log \left (-1+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}\right )}{2 \sqrt [3]{b} d^{2/3}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {61} \begin {gather*} -\frac {\sqrt {3} \text {ArcTan}\left (\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{b} \sqrt [3]{c+d x}}+\frac {1}{\sqrt {3}}\right )}{\sqrt [3]{b} d^{2/3}}-\frac {3 \log \left (\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}-1\right )}{2 \sqrt [3]{b} d^{2/3}}-\frac {\log (c+d x)}{2 \sqrt [3]{b} d^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 61

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

Rubi steps

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

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Mathematica [A]
time = 0.15, size = 155, normalized size = 1.23 \begin {gather*} \frac {2 \sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{d}-\frac {\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{a+b x}}\right )+\log \left (d^{2/3}+\frac {\sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{c+d x}}{\sqrt [3]{a+b x}}+\frac {b^{2/3} (c+d x)^{2/3}}{(a+b x)^{2/3}}\right )}{2 \sqrt [3]{b} d^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (b x +a \right )^{\frac {1}{3}} \left (d x +c \right )^{\frac {2}{3}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 238 vs. \(2 (90) = 180\).
time = 1.21, size = 521, normalized size = 4.13 \begin {gather*} \left [\frac {\sqrt {3} b d \sqrt {\frac {\left (-b d^{2}\right )^{\frac {1}{3}}}{b}} \log \left (-3 \, b d^{2} x - 2 \, b c d - a d^{2} - 3 \, \left (-b d^{2}\right )^{\frac {1}{3}} {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} d - \sqrt {3} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} b d - \left (-b d^{2}\right )^{\frac {2}{3}} {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} + \left (-b d^{2}\right )^{\frac {1}{3}} {\left (b d x + a d\right )}\right )} \sqrt {\frac {\left (-b d^{2}\right )^{\frac {1}{3}}}{b}}\right ) - 2 \, \left (-b d^{2}\right )^{\frac {2}{3}} \log \left (\frac {{\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} b d - \left (-b d^{2}\right )^{\frac {2}{3}} {\left (b x + a\right )}}{b x + a}\right ) + \left (-b d^{2}\right )^{\frac {2}{3}} \log \left (\frac {{\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} b d + \left (-b d^{2}\right )^{\frac {2}{3}} {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} - \left (-b d^{2}\right )^{\frac {1}{3}} {\left (b d x + a d\right )}}{b x + a}\right )}{2 \, b d^{2}}, \frac {2 \, \sqrt {3} b d \sqrt {-\frac {\left (-b d^{2}\right )^{\frac {1}{3}}}{b}} \arctan \left (\frac {\sqrt {3} {\left (2 \, \left (-b d^{2}\right )^{\frac {2}{3}} {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} - \left (-b d^{2}\right )^{\frac {1}{3}} {\left (b d x + a d\right )}\right )} \sqrt {-\frac {\left (-b d^{2}\right )^{\frac {1}{3}}}{b}}}{3 \, {\left (b d^{2} x + a d^{2}\right )}}\right ) - 2 \, \left (-b d^{2}\right )^{\frac {2}{3}} \log \left (\frac {{\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} b d - \left (-b d^{2}\right )^{\frac {2}{3}} {\left (b x + a\right )}}{b x + a}\right ) + \left (-b d^{2}\right )^{\frac {2}{3}} \log \left (\frac {{\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} b d + \left (-b d^{2}\right )^{\frac {2}{3}} {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} - \left (-b d^{2}\right )^{\frac {1}{3}} {\left (b d x + a d\right )}}{b x + a}\right )}{2 \, b d^{2}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(sqrt(3)*b*d*sqrt((-b*d^2)^(1/3)/b)*log(-3*b*d^2*x - 2*b*c*d - a*d^2 - 3*(-b*d^2)^(1/3)*(b*x + a)^(2/3)*(
d*x + c)^(1/3)*d - sqrt(3)*(2*(b*x + a)^(1/3)*(d*x + c)^(2/3)*b*d - (-b*d^2)^(2/3)*(b*x + a)^(2/3)*(d*x + c)^(
1/3) + (-b*d^2)^(1/3)*(b*d*x + a*d))*sqrt((-b*d^2)^(1/3)/b)) - 2*(-b*d^2)^(2/3)*log(((b*x + a)^(2/3)*(d*x + c)
^(1/3)*b*d - (-b*d^2)^(2/3)*(b*x + a))/(b*x + a)) + (-b*d^2)^(2/3)*log(((b*x + a)^(1/3)*(d*x + c)^(2/3)*b*d +
(-b*d^2)^(2/3)*(b*x + a)^(2/3)*(d*x + c)^(1/3) - (-b*d^2)^(1/3)*(b*d*x + a*d))/(b*x + a)))/(b*d^2), 1/2*(2*sqr
t(3)*b*d*sqrt(-(-b*d^2)^(1/3)/b)*arctan(1/3*sqrt(3)*(2*(-b*d^2)^(2/3)*(b*x + a)^(2/3)*(d*x + c)^(1/3) - (-b*d^
2)^(1/3)*(b*d*x + a*d))*sqrt(-(-b*d^2)^(1/3)/b)/(b*d^2*x + a*d^2)) - 2*(-b*d^2)^(2/3)*log(((b*x + a)^(2/3)*(d*
x + c)^(1/3)*b*d - (-b*d^2)^(2/3)*(b*x + a))/(b*x + a)) + (-b*d^2)^(2/3)*log(((b*x + a)^(1/3)*(d*x + c)^(2/3)*
b*d + (-b*d^2)^(2/3)*(b*x + a)^(2/3)*(d*x + c)^(1/3) - (-b*d^2)^(1/3)*(b*d*x + a*d))/(b*x + a)))/(b*d^2)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [3]{a + b x} \left (c + d x\right )^{\frac {2}{3}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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