3.14.53 \(\int \frac {1}{(a+b x) (c+d x)^{2/3}} \, dx\)

Optimal. Leaf size=140 \[ -\frac {\log (a+b x)}{2 \sqrt [3]{b} (b c-a d)^{2/3}}+\frac {3 \log \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{2 \sqrt [3]{b} (b c-a d)^{2/3}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}}+1}{\sqrt {3}}\right )}{\sqrt [3]{b} (b c-a d)^{2/3}} \]

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Rubi [A]  time = 0.07, antiderivative size = 140, 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, 617, 204, 31} \begin {gather*} -\frac {\log (a+b x)}{2 \sqrt [3]{b} (b c-a d)^{2/3}}+\frac {3 \log \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{2 \sqrt [3]{b} (b c-a d)^{2/3}}-\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 )}{\sqrt [3]{b} (b c-a d)^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x)*(c + d*x)^(2/3)),x]

[Out]

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

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_))^(2/3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/b, 3]}, -Simp[L
og[RemoveContent[a + b*x, x]]/(2*b*q^2), x] + (-Dist[3/(2*b*q), Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x
)^(1/3)], x] - Dist[3/(2*b*q^2), 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 204

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

Rule 617

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

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Mathematica [A]  time = 0.07, size = 154, normalized size = 1.10 \begin {gather*} -\frac {\log \left (\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+b^{2/3} (c+d x)^{2/3}\right )-2 \log \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )+2 \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 )}{2 \sqrt [3]{b} (b c-a d)^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(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)])/(b^(1/3)*(b*c - a*d)^(2/3))

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IntegrateAlgebraic [A]  time = 0.21, size = 190, normalized size = 1.36 \begin {gather*} -\frac {\log \left (-\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{a d-b c}+(a d-b c)^{2/3}+b^{2/3} (c+d x)^{2/3}\right )}{2 \sqrt [3]{b} (a d-b c)^{2/3}}+\frac {\log \left (\sqrt [3]{a d-b c}+\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\sqrt [3]{b} (a d-b c)^{2/3}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt {3} \sqrt [3]{a d-b c}}\right )}{\sqrt [3]{b} (a d-b c)^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 1.58, size = 900, normalized size = 6.43 \begin {gather*} \left [-\frac {\sqrt {3} {\left (b^{2} c - a b d\right )} \sqrt {-\frac {{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )}^{\frac {1}{3}}}{b}} \log \left (-\frac {3 \, b^{2} c^{2} - 4 \, a b c d + a^{2} d^{2} + 2 \, {\left (b^{2} c d - a b d^{2}\right )} x + \sqrt {3} {\left (2 \, {\left (b^{2} c - a b d\right )} {\left (d x + c\right )}^{\frac {2}{3}} - {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )}^{\frac {1}{3}} {\left (b c - a d\right )} - {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}}\right )} \sqrt {-\frac {{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )}^{\frac {1}{3}}}{b}} - 3 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )}^{\frac {1}{3}} {\left (b c - a d\right )} {\left (d x + c\right )}^{\frac {1}{3}}}{b x + a}\right ) + {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )}^{\frac {2}{3}} \log \left (-{\left (b^{2} c - a b d\right )} {\left (d x + c\right )}^{\frac {2}{3}} - {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )}^{\frac {1}{3}} {\left (b c - a d\right )} - {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}}\right ) - 2 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )}^{\frac {2}{3}} \log \left (-{\left (b^{2} c - a b d\right )} {\left (d x + c\right )}^{\frac {1}{3}} + {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )}^{\frac {2}{3}}\right )}{2 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )}}, -\frac {2 \, \sqrt {3} {\left (b^{2} c - a b d\right )} \sqrt {\frac {{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )}^{\frac {1}{3}}}{b}} \arctan \left (\frac {\sqrt {3} {\left ({\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )}^{\frac {1}{3}} {\left (b c - a d\right )} + 2 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}}\right )} \sqrt {\frac {{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )}^{\frac {1}{3}}}{b}}}{3 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}}\right ) + {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )}^{\frac {2}{3}} \log \left (-{\left (b^{2} c - a b d\right )} {\left (d x + c\right )}^{\frac {2}{3}} - {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )}^{\frac {1}{3}} {\left (b c - a d\right )} - {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}}\right ) - 2 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )}^{\frac {2}{3}} \log \left (-{\left (b^{2} c - a b d\right )} {\left (d x + c\right )}^{\frac {1}{3}} + {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )}^{\frac {2}{3}}\right )}{2 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/2*(sqrt(3)*(b^2*c - a*b*d)*sqrt(-(b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)^(1/3)/b)*log(-(3*b^2*c^2 - 4*a*b*c*d
+ a^2*d^2 + 2*(b^2*c*d - a*b*d^2)*x + sqrt(3)*(2*(b^2*c - a*b*d)*(d*x + c)^(2/3) - (b^3*c^2 - 2*a*b^2*c*d + a^
2*b*d^2)^(1/3)*(b*c - a*d) - (b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)^(2/3)*(d*x + c)^(1/3))*sqrt(-(b^3*c^2 - 2*a*b
^2*c*d + a^2*b*d^2)^(1/3)/b) - 3*(b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)^(1/3)*(b*c - a*d)*(d*x + c)^(1/3))/(b*x +
a)) + (b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)^(2/3)*log(-(b^2*c - a*b*d)*(d*x + c)^(2/3) - (b^3*c^2 - 2*a*b^2*c*d
+ a^2*b*d^2)^(1/3)*(b*c - a*d) - (b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)^(2/3)*(d*x + c)^(1/3)) - 2*(b^3*c^2 - 2*
a*b^2*c*d + a^2*b*d^2)^(2/3)*log(-(b^2*c - a*b*d)*(d*x + c)^(1/3) + (b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)^(2/3))
)/(b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2), -1/2*(2*sqrt(3)*(b^2*c - a*b*d)*sqrt((b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2
)^(1/3)/b)*arctan(1/3*sqrt(3)*((b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)^(1/3)*(b*c - a*d) + 2*(b^3*c^2 - 2*a*b^2*c*
d + a^2*b*d^2)^(2/3)*(d*x + c)^(1/3))*sqrt((b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)^(1/3)/b)/(b^2*c^2 - 2*a*b*c*d +
a^2*d^2)) + (b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)^(2/3)*log(-(b^2*c - a*b*d)*(d*x + c)^(2/3) - (b^3*c^2 - 2*a*b
^2*c*d + a^2*b*d^2)^(1/3)*(b*c - a*d) - (b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)^(2/3)*(d*x + c)^(1/3)) - 2*(b^3*c^
2 - 2*a*b^2*c*d + a^2*b*d^2)^(2/3)*log(-(b^2*c - a*b*d)*(d*x + c)^(1/3) + (b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)^
(2/3)))/(b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)]

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giac [A]  time = 0.96, size = 207, normalized size = 1.48 \begin {gather*} -\frac {3 \, {\left (b^{3} c - a b^{2} d\right )}^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (d x + c\right )}^{\frac {1}{3}} + \left (\frac {b c - a d}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {b c - a d}{b}\right )^{\frac {1}{3}}}\right )}{\sqrt {3} b^{2} c - \sqrt {3} a b d} - \frac {{\left (b^{3} c - a b^{2} d\right )}^{\frac {1}{3}} \log \left ({\left (d x + c\right )}^{\frac {2}{3}} + {\left (d x + c\right )}^{\frac {1}{3}} \left (\frac {b c - a d}{b}\right )^{\frac {1}{3}} + \left (\frac {b c - a d}{b}\right )^{\frac {2}{3}}\right )}{2 \, {\left (b^{2} c - a b d\right )}} + \frac {\left (\frac {b c - a d}{b}\right )^{\frac {1}{3}} \log \left ({\left | {\left (d x + c\right )}^{\frac {1}{3}} - \left (\frac {b c - a d}{b}\right )^{\frac {1}{3}} \right |}\right )}{b c - a d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 160, normalized size = 1.14 \begin {gather*} \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 )}{\left (\frac {a d -b c}{b}\right )^{\frac {2}{3}} b}+\frac {\ln \left (\left (d x +c \right )^{\frac {1}{3}}+\left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}\right )}{\left (\frac {a d -b c}{b}\right )^{\frac {2}{3}} b}-\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 \left (\frac {a d -b c}{b}\right )^{\frac {2}{3}} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/b/((a*d-b*c)/b)^(2/3)*ln((d*x+c)^(1/3)+((a*d-b*c)/b)^(1/3))-1/2/b/((a*d-b*c)/b)^(2/3)*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))+1/b/((a*d-b*c)/b)^(2/3)*3^(1/2)*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)^(2/3),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 details)Is a*d-b*c positive or negative?

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mupad [B]  time = 0.37, size = 206, normalized size = 1.47 \begin {gather*} \frac {\ln \left (9\,b^2\,{\left (c+d\,x\right )}^{1/3}-\frac {9\,b^3\,c-9\,a\,b^2\,d}{b^{1/3}\,{\left (a\,d-b\,c\right )}^{2/3}}\right )}{b^{1/3}\,{\left (a\,d-b\,c\right )}^{2/3}}+\frac {\ln \left (9\,b^2\,{\left (c+d\,x\right )}^{1/3}-\frac {\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,\left (9\,b^3\,c-9\,a\,b^2\,d\right )}{2\,b^{1/3}\,{\left (a\,d-b\,c\right )}^{2/3}}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,b^{1/3}\,{\left (a\,d-b\,c\right )}^{2/3}}-\frac {\ln \left (9\,b^2\,{\left (c+d\,x\right )}^{1/3}+\frac {\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,\left (9\,b^3\,c-9\,a\,b^2\,d\right )}{2\,b^{1/3}\,{\left (a\,d-b\,c\right )}^{2/3}}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,b^{1/3}\,{\left (a\,d-b\,c\right )}^{2/3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + b*x)*(c + d*x)^(2/3)),x)

[Out]

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

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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 ) \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)/(d*x+c)**(2/3),x)

[Out]

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

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