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

Optimal. Leaf size=760 \[ -\frac {d^3 x^4 \left (\frac {b x^3}{a}+1\right )^{2/3} F_1\left (\frac {4}{3};\frac {2}{3},2;\frac {7}{3};-\frac {b x^3}{a},-\frac {d^3 x^3}{c^3}\right )}{2 c^5 \left (a+b x^3\right )^{2/3}}+\frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} F_1\left (\frac {1}{3};\frac {2}{3},2;\frac {4}{3};-\frac {b x^3}{a},-\frac {d^3 x^3}{c^3}\right )}{c^2 \left (a+b x^3\right )^{2/3}}-\frac {d \left (3 b c^3-a d^3\right ) \log \left (c^3+d^3 x^3\right )}{9 c \left (b c^3-a d^3\right )^{5/3}}+\frac {d \left (3 b c^3-a d^3\right ) \log \left (\frac {x \sqrt [3]{b c^3-a d^3}}{c}-\sqrt [3]{a+b x^3}\right )}{3 c \left (b c^3-a d^3\right )^{5/3}}+\frac {2 d \left (3 b c^3-a d^3\right ) \tan ^{-1}\left (\frac {\frac {2 x \sqrt [3]{b c^3-a d^3}}{c \sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{3 \sqrt {3} c \left (b c^3-a d^3\right )^{5/3}}-\frac {a d^4 \log \left (c^3+d^3 x^3\right )}{9 c \left (b c^3-a d^3\right )^{5/3}}+\frac {a d^4 \log \left (\frac {x \sqrt [3]{b c^3-a d^3}}{c}-\sqrt [3]{a+b x^3}\right )}{3 c \left (b c^3-a d^3\right )^{5/3}}+\frac {2 a d^4 \tan ^{-1}\left (\frac {\frac {2 x \sqrt [3]{b c^3-a d^3}}{c \sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{3 \sqrt {3} c \left (b c^3-a d^3\right )^{5/3}}+\frac {d^4 x^2 \sqrt [3]{a+b x^3}}{\left (c^3+d^3 x^3\right ) \left (b c^3-a d^3\right )}-\frac {b c^2 d \log \left (c^3+d^3 x^3\right )}{3 \left (b c^3-a d^3\right )^{5/3}}+\frac {b c^2 d \log \left (\sqrt [3]{b c^3-a d^3}+d \sqrt [3]{a+b x^3}\right )}{\left (b c^3-a d^3\right )^{5/3}}-\frac {2 b c^2 d \tan ^{-1}\left (\frac {1-\frac {2 d \sqrt [3]{a+b x^3}}{\sqrt [3]{b c^3-a d^3}}}{\sqrt {3}}\right )}{\sqrt {3} \left (b c^3-a d^3\right )^{5/3}}+\frac {c^2 d^2 \sqrt [3]{a+b x^3}}{\left (c^3+d^3 x^3\right ) \left (b c^3-a d^3\right )} \]

[Out]

c^2*d^2*(b*x^3+a)^(1/3)/(-a*d^3+b*c^3)/(d^3*x^3+c^3)+d^4*x^2*(b*x^3+a)^(1/3)/(-a*d^3+b*c^3)/(d^3*x^3+c^3)+x*(1
+b*x^3/a)^(2/3)*AppellF1(1/3,2/3,2,4/3,-b*x^3/a,-d^3*x^3/c^3)/c^2/(b*x^3+a)^(2/3)-1/2*d^3*x^4*(1+b*x^3/a)^(2/3
)*AppellF1(4/3,2/3,2,7/3,-b*x^3/a,-d^3*x^3/c^3)/c^5/(b*x^3+a)^(2/3)-1/3*b*c^2*d*ln(d^3*x^3+c^3)/(-a*d^3+b*c^3)
^(5/3)-1/9*a*d^4*ln(d^3*x^3+c^3)/c/(-a*d^3+b*c^3)^(5/3)-1/9*d*(-a*d^3+3*b*c^3)*ln(d^3*x^3+c^3)/c/(-a*d^3+b*c^3
)^(5/3)+1/3*a*d^4*ln((-a*d^3+b*c^3)^(1/3)*x/c-(b*x^3+a)^(1/3))/c/(-a*d^3+b*c^3)^(5/3)+1/3*d*(-a*d^3+3*b*c^3)*l
n((-a*d^3+b*c^3)^(1/3)*x/c-(b*x^3+a)^(1/3))/c/(-a*d^3+b*c^3)^(5/3)+b*c^2*d*ln((-a*d^3+b*c^3)^(1/3)+d*(b*x^3+a)
^(1/3))/(-a*d^3+b*c^3)^(5/3)+2/9*a*d^4*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)^(5/3)*3^(1/2)+2/9*d*(-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)^(5/3)*3^(1/2)-2/3*b*c^2*d*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)^(5/3)*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 = {} \[ \int \frac {1}{(c+d x)^2 \left (a+b x^3\right )^{2/3}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 0.32, size = 0, normalized size = 0.00 \[ \int \frac {1}{(c+d x)^2 \left (a+b x^3\right )^{2/3}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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