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

Optimal. Leaf size=761 \[ -\frac {d x^2 \sqrt [3]{\frac {b x^3}{a}+1} F_1\left (\frac {2}{3};\frac {1}{3},2;\frac {5}{3};-\frac {b x^3}{a},-\frac {d^3 x^3}{c^3}\right )}{c^3 \sqrt [3]{a+b x^3}}+\frac {d^4 x^5 \sqrt [3]{\frac {b x^3}{a}+1} F_1\left (\frac {5}{3};\frac {1}{3},2;\frac {8}{3};-\frac {b x^3}{a},-\frac {d^3 x^3}{c^3}\right )}{5 c^6 \sqrt [3]{a+b x^3}}-\frac {c d^3 x \left (a+b x^3\right )^{2/3}}{\left (c^3+d^3 x^3\right ) \left (b c^3-a d^3\right )}+\frac {a d^3 \log \left (c^3+d^3 x^3\right )}{9 c \left (b c^3-a d^3\right )^{4/3}}+\frac {\left (3 b c^3-2 a d^3\right ) \log \left (c^3+d^3 x^3\right )}{18 c \left (b c^3-a d^3\right )^{4/3}}-\frac {a d^3 \log \left (\frac {x \sqrt [3]{b c^3-a d^3}}{c}-\sqrt [3]{a+b x^3}\right )}{3 c \left (b c^3-a d^3\right )^{4/3}}-\frac {\left (3 b c^3-2 a d^3\right ) \log \left (\frac {x \sqrt [3]{b c^3-a d^3}}{c}-\sqrt [3]{a+b x^3}\right )}{6 c \left (b c^3-a d^3\right )^{4/3}}+\frac {2 a d^3 \tan ^{-1}\left (\frac {\frac {2 x \sqrt [3]{b c^3-a d^3}}{c \sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{3 \sqrt {3} c \left (b c^3-a d^3\right )^{4/3}}+\frac {\left (3 b c^3-2 a d^3\right ) \tan ^{-1}\left (\frac {\frac {2 x \sqrt [3]{b c^3-a d^3}}{c \sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{3 \sqrt {3} c \left (b c^3-a d^3\right )^{4/3}}+\frac {b c^2 \log \left (c^3+d^3 x^3\right )}{6 \left (b c^3-a d^3\right )^{4/3}}-\frac {b c^2 \log \left (\sqrt [3]{b c^3-a d^3}+d \sqrt [3]{a+b x^3}\right )}{2 \left (b c^3-a d^3\right )^{4/3}}-\frac {b c^2 \tan ^{-1}\left (\frac {1-\frac {2 d \sqrt [3]{a+b x^3}}{\sqrt [3]{b c^3-a d^3}}}{\sqrt {3}}\right )}{\sqrt {3} \left (b c^3-a d^3\right )^{4/3}}+\frac {c^2 d^2 \left (a+b x^3\right )^{2/3}}{\left (c^3+d^3 x^3\right ) \left (b c^3-a d^3\right )} \]

[Out]

c^2*d^2*(b*x^3+a)^(2/3)/(-a*d^3+b*c^3)/(d^3*x^3+c^3)-c*d^3*x*(b*x^3+a)^(2/3)/(-a*d^3+b*c^3)/(d^3*x^3+c^3)-d*x^
2*(1+b*x^3/a)^(1/3)*AppellF1(2/3,1/3,2,5/3,-b*x^3/a,-d^3*x^3/c^3)/c^3/(b*x^3+a)^(1/3)+1/5*d^4*x^5*(1+b*x^3/a)^
(1/3)*AppellF1(5/3,1/3,2,8/3,-b*x^3/a,-d^3*x^3/c^3)/c^6/(b*x^3+a)^(1/3)+1/6*b*c^2*ln(d^3*x^3+c^3)/(-a*d^3+b*c^
3)^(4/3)+1/9*a*d^3*ln(d^3*x^3+c^3)/c/(-a*d^3+b*c^3)^(4/3)+1/18*(-2*a*d^3+3*b*c^3)*ln(d^3*x^3+c^3)/c/(-a*d^3+b*
c^3)^(4/3)-1/3*a*d^3*ln((-a*d^3+b*c^3)^(1/3)*x/c-(b*x^3+a)^(1/3))/c/(-a*d^3+b*c^3)^(4/3)-1/6*(-2*a*d^3+3*b*c^3
)*ln((-a*d^3+b*c^3)^(1/3)*x/c-(b*x^3+a)^(1/3))/c/(-a*d^3+b*c^3)^(4/3)-1/2*b*c^2*ln((-a*d^3+b*c^3)^(1/3)+d*(b*x
^3+a)^(1/3))/(-a*d^3+b*c^3)^(4/3)+2/9*a*d^3*arctan(1/3*(1+2*(-a*d^3+b*c^3)^(1/3)*x/c/(b*x^3+a)^(1/3))*3^(1/2))
/c/(-a*d^3+b*c^3)^(4/3)*3^(1/2)+1/9*(-2*a*d^3+3*b*c^3)*arctan(1/3*(1+2*(-a*d^3+b*c^3)^(1/3)*x/c/(b*x^3+a)^(1/3
))*3^(1/2))/c/(-a*d^3+b*c^3)^(4/3)*3^(1/2)-1/3*b*c^2*arctan(1/3*(1-2*d*(b*x^3+a)^(1/3)/(-a*d^3+b*c^3)^(1/3))*3
^(1/2))/(-a*d^3+b*c^3)^(4/3)*3^(1/2)

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Rubi [F]  time = 0.08, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {1}{(c+d x)^2 \sqrt [3]{a+b x^3}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 0.34, size = 0, normalized size = 0.00 \[ \int \frac {1}{(c+d x)^2 \sqrt [3]{a+b x^3}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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