3.15.58 \(\int \frac {1}{(a+b x) \sqrt [3]{c+d x}} \, dx\) [1458]

Optimal. Leaf size=139 \[ \frac {\sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}}}{\sqrt {3}}\right )}{b^{2/3} \sqrt [3]{b c-a d}}-\frac {\log (a+b x)}{2 b^{2/3} \sqrt [3]{b c-a d}}+\frac {3 \log \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{2 b^{2/3} \sqrt [3]{b c-a d}} \]

[Out]

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

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Rubi [A]
time = 0.09, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {57, 631, 210, 31} \begin {gather*} -\frac {\log (a+b x)}{2 b^{2/3} \sqrt [3]{b c-a d}}+\frac {3 \log \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{2 b^{2/3} \sqrt [3]{b c-a d}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}}+1}{\sqrt {3}}\right )}{b^{2/3} \sqrt [3]{b c-a d}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x)*(c + d*x)^(1/3)),x]

[Out]

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 57

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/b, 3]}, Simp[-L
og[RemoveContent[a + b*x, x]]/(2*b*q), x] + (Dist[3/(2*b), Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1/
3)], x] - Dist[3/(2*b*q), Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x] && PosQ
[(b*c - a*d)/b]

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rubi steps

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

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Mathematica [A]
time = 0.19, size = 154, normalized size = 1.11 \begin {gather*} \frac {-2 \sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{-b c+a d}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{-b c+a d}+\sqrt [3]{b} \sqrt [3]{c+d x}\right )+\log \left ((-b c+a d)^{2/3}-\sqrt [3]{b} \sqrt [3]{-b c+a d} \sqrt [3]{c+d x}+b^{2/3} (c+d x)^{2/3}\right )}{2 b^{2/3} \sqrt [3]{-b c+a d}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: maximum recursion depth exceeded while calling a Python object} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

cought exception: maximum recursion depth exceeded while calling a Python object

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Maple [A]
time = 0.17, size = 161, normalized size = 1.16

method	result	size
derivativedivides	\(-\frac {\ln \left (\left (d x +c \right )^{\frac {1}{3}}+\left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}\right )}{b \left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (\left (d x +c \right )^{\frac {2}{3}}-\left (\frac {a d -b c}{b}\right )^{\frac {1}{3}} \left (d x +c \right )^{\frac {1}{3}}+\left (\frac {a d -b c}{b}\right )^{\frac {2}{3}}\right )}{2 b \left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (d x +c \right )^{\frac {1}{3}}}{\left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{b \left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}}\)	\(161\)
default	\(-\frac {\ln \left (\left (d x +c \right )^{\frac {1}{3}}+\left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}\right )}{b \left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (\left (d x +c \right )^{\frac {2}{3}}-\left (\frac {a d -b c}{b}\right )^{\frac {1}{3}} \left (d x +c \right )^{\frac {1}{3}}+\left (\frac {a d -b c}{b}\right )^{\frac {2}{3}}\right )}{2 b \left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (d x +c \right )^{\frac {1}{3}}}{\left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{b \left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}}\)	\(161\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for
more detail

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 239 vs. \(2 (108) = 216\).
time = 0.32, size = 570, normalized size = 4.10 \begin {gather*} \left [\frac {\sqrt {3} {\left (b^{2} c - a b d\right )} \sqrt {-\frac {{\left (b^{3} c - a b^{2} d\right )}^{\frac {1}{3}}}{b c - a d}} \log \left (\frac {2 \, b^{2} d x + 3 \, b^{2} c - a b d - \sqrt {3} {\left ({\left (b^{3} c - a b^{2} d\right )}^{\frac {1}{3}} {\left (b c - a d\right )} + {\left (b^{2} c - a b d\right )} {\left (d x + c\right )}^{\frac {1}{3}} - 2 \, {\left (b^{3} c - a b^{2} d\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {2}{3}}\right )} \sqrt {-\frac {{\left (b^{3} c - a b^{2} d\right )}^{\frac {1}{3}}}{b c - a d}} - 3 \, {\left (b^{3} c - a b^{2} d\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}}}{b x + a}\right ) - {\left (b^{3} c - a b^{2} d\right )}^{\frac {2}{3}} \log \left ({\left (d x + c\right )}^{\frac {2}{3}} b^{2} + {\left (b^{3} c - a b^{2} d\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {1}{3}} b + {\left (b^{3} c - a b^{2} d\right )}^{\frac {2}{3}}\right ) + 2 \, {\left (b^{3} c - a b^{2} d\right )}^{\frac {2}{3}} \log \left ({\left (d x + c\right )}^{\frac {1}{3}} b - {\left (b^{3} c - a b^{2} d\right )}^{\frac {1}{3}}\right )}{2 \, {\left (b^{3} c - a b^{2} d\right )}}, \frac {2 \, \sqrt {3} {\left (b^{2} c - a b d\right )} \sqrt {\frac {{\left (b^{3} c - a b^{2} d\right )}^{\frac {1}{3}}}{b c - a d}} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (d x + c\right )}^{\frac {1}{3}} b + {\left (b^{3} c - a b^{2} d\right )}^{\frac {1}{3}}\right )} \sqrt {\frac {{\left (b^{3} c - a b^{2} d\right )}^{\frac {1}{3}}}{b c - a d}}}{3 \, b}\right ) - {\left (b^{3} c - a b^{2} d\right )}^{\frac {2}{3}} \log \left ({\left (d x + c\right )}^{\frac {2}{3}} b^{2} + {\left (b^{3} c - a b^{2} d\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {1}{3}} b + {\left (b^{3} c - a b^{2} d\right )}^{\frac {2}{3}}\right ) + 2 \, {\left (b^{3} c - a b^{2} d\right )}^{\frac {2}{3}} \log \left ({\left (d x + c\right )}^{\frac {1}{3}} b - {\left (b^{3} c - a b^{2} d\right )}^{\frac {1}{3}}\right )}{2 \, {\left (b^{3} c - a b^{2} d\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.01, size = 280, normalized size = 2.01 \begin {gather*} 3 \left (\frac {\left (\left (-a b^{2} d+b^{3} c\right )^{\frac {1}{3}}\right )^{2} \ln \left (\left (\left (c+d x\right )^{\frac {1}{3}}\right )^{2}+\left (\frac {-a d+b c}{b}\right )^{\frac {1}{3}} \left (c+d x\right )^{\frac {1}{3}}+\left (\frac {-a d+b c}{b}\right )^{\frac {1}{3}} \left (\frac {-a d+b c}{b}\right )^{\frac {1}{3}}\right )}{6 a b^{2} d-6 b^{3} c}-\frac {4 \left (\left (-a b^{2} d+b^{3} c\right )^{\frac {1}{3}}\right )^{2} \arctan \left (\frac {2 \left (\left (c+d x\right )^{\frac {1}{3}}+\frac {\left (\frac {-a d+b c}{b}\right )^{\frac {1}{3}}}{2}\right )}{\sqrt {3} \left (\frac {-a d+b c}{b}\right )^{\frac {1}{3}}}\right )}{4 \sqrt {3} a b^{2} d-4 \sqrt {3} b^{3} c}-\frac {\left (\frac {-a d+b c}{b}\right )^{\frac {1}{3}} \left (\frac {-a d+b c}{b}\right )^{\frac {1}{3}} \ln \left |\left (c+d x\right )^{\frac {1}{3}}-\left (\frac {-a d+b c}{b}\right )^{\frac {1}{3}}\right |}{3 \left (-b c+a d\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.21, size = 204, normalized size = 1.47 \begin {gather*} \frac {\ln \left (9\,b\,{\left (c+d\,x\right )}^{1/3}-\frac {9\,b^3\,c-9\,a\,b^2\,d}{b^{4/3}\,{\left (b\,c-a\,d\right )}^{2/3}}\right )}{b^{2/3}\,{\left (b\,c-a\,d\right )}^{1/3}}+\frac {\ln \left (9\,b\,{\left (c+d\,x\right )}^{1/3}-\frac {{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,\left (9\,b^3\,c-9\,a\,b^2\,d\right )}{4\,b^{4/3}\,{\left (b\,c-a\,d\right )}^{2/3}}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,b^{2/3}\,{\left (b\,c-a\,d\right )}^{1/3}}-\frac {\ln \left (9\,b\,{\left (c+d\,x\right )}^{1/3}-\frac {{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,\left (9\,b^3\,c-9\,a\,b^2\,d\right )}{4\,b^{4/3}\,{\left (b\,c-a\,d\right )}^{2/3}}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,b^{2/3}\,{\left (b\,c-a\,d\right )}^{1/3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + b*x)*(c + d*x)^(1/3)),x)

[Out]

log(9*b*(c + d*x)^(1/3) - (9*b^3*c - 9*a*b^2*d)/(b^(4/3)*(b*c - a*d)^(2/3)))/(b^(2/3)*(b*c - a*d)^(1/3)) + (lo
g(9*b*(c + d*x)^(1/3) - ((3^(1/2)*1i - 1)^2*(9*b^3*c - 9*a*b^2*d))/(4*b^(4/3)*(b*c - a*d)^(2/3)))*(3^(1/2)*1i
- 1))/(2*b^(2/3)*(b*c - a*d)^(1/3)) - (log(9*b*(c + d*x)^(1/3) - ((3^(1/2)*1i + 1)^2*(9*b^3*c - 9*a*b^2*d))/(4
*b^(4/3)*(b*c - a*d)^(2/3)))*(3^(1/2)*1i + 1))/(2*b^(2/3)*(b*c - a*d)^(1/3))

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