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

Optimal. Leaf size=1357 \[ \frac {d^6 \left (\frac {b x^3}{a}+1\right )^{2/3} F_1\left (\frac {7}{3};\frac {2}{3},3;\frac {10}{3};-\frac {b x^3}{a},-\frac {d^3 x^3}{c^3}\right ) x^7}{7 c^9 \left (b x^3+a\right )^{2/3}}-\frac {7 d^3 \left (\frac {b x^3}{a}+1\right )^{2/3} F_1\left (\frac {4}{3};\frac {2}{3},3;\frac {7}{3};-\frac {b x^3}{a},-\frac {d^3 x^3}{c^3}\right ) x^4}{4 c^6 \left (b x^3+a\right )^{2/3}}+\frac {d^4 \left (3 b c^3+2 a d^3\right ) \sqrt [3]{b x^3+a} x^2}{3 c \left (b c^3-a d^3\right )^2 \left (c^3+d^3 x^3\right )}+\frac {d^4 \left (9 b c^3-4 a d^3\right ) \sqrt [3]{b x^3+a} x^2}{6 c \left (b c^3-a d^3\right )^2 \left (c^3+d^3 x^3\right )}+\frac {3 c^2 d^4 \sqrt [3]{b x^3+a} x^2}{2 \left (b c^3-a d^3\right ) \left (c^3+d^3 x^3\right )^2}+\frac {\left (\frac {b x^3}{a}+1\right )^{2/3} F_1\left (\frac {1}{3};\frac {2}{3},3;\frac {4}{3};-\frac {b x^3}{a},-\frac {d^3 x^3}{c^3}\right ) x}{c^3 \left (b x^3+a\right )^{2/3}}+\frac {2 a d^4 \left (6 b c^3-a d^3\right ) \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{b c^3-a d^3} x}{c \sqrt [3]{b x^3+a}}+1}{\sqrt {3}}\right )}{3 \sqrt {3} c^2 \left (b c^3-a d^3\right )^{8/3}}+\frac {d \left (9 b^2 c^6-6 a b d^3 c^3+2 a^2 d^6\right ) \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{b c^3-a d^3} x}{c \sqrt [3]{b x^3+a}}+1}{\sqrt {3}}\right )}{3 \sqrt {3} c^2 \left (b c^3-a d^3\right )^{8/3}}-\frac {10 b^2 c^4 d \tan ^{-1}\left (\frac {1-\frac {2 d \sqrt [3]{b x^3+a}}{\sqrt [3]{b c^3-a d^3}}}{\sqrt {3}}\right )}{3 \sqrt {3} \left (b c^3-a d^3\right )^{8/3}}+\frac {b c d \left (b c^3-6 a d^3\right ) \tan ^{-1}\left (\frac {1-\frac {2 d \sqrt [3]{b x^3+a}}{\sqrt [3]{b c^3-a d^3}}}{\sqrt {3}}\right )}{3 \sqrt {3} \left (b c^3-a d^3\right )^{8/3}}-\frac {a d^4 \left (6 b c^3-a d^3\right ) \log \left (c^3+d^3 x^3\right )}{9 c^2 \left (b c^3-a d^3\right )^{8/3}}-\frac {d \left (9 b^2 c^6-6 a b d^3 c^3+2 a^2 d^6\right ) \log \left (c^3+d^3 x^3\right )}{18 c^2 \left (b c^3-a d^3\right )^{8/3}}-\frac {5 b^2 c^4 d \log \left (c^3+d^3 x^3\right )}{9 \left (b c^3-a d^3\right )^{8/3}}+\frac {b c d \left (b c^3-6 a d^3\right ) \log \left (c^3+d^3 x^3\right )}{18 \left (b c^3-a d^3\right )^{8/3}}+\frac {a d^4 \left (6 b c^3-a d^3\right ) \log \left (\frac {\sqrt [3]{b c^3-a d^3} x}{c}-\sqrt [3]{b x^3+a}\right )}{3 c^2 \left (b c^3-a d^3\right )^{8/3}}+\frac {d \left (9 b^2 c^6-6 a b d^3 c^3+2 a^2 d^6\right ) \log \left (\frac {\sqrt [3]{b c^3-a d^3} x}{c}-\sqrt [3]{b x^3+a}\right )}{6 c^2 \left (b c^3-a d^3\right )^{8/3}}+\frac {5 b^2 c^4 d \log \left (\sqrt [3]{b x^3+a} d+\sqrt [3]{b c^3-a d^3}\right )}{3 \left (b c^3-a d^3\right )^{8/3}}-\frac {b c d \left (b c^3-6 a d^3\right ) \log \left (\sqrt [3]{b x^3+a} d+\sqrt [3]{b c^3-a d^3}\right )}{6 \left (b c^3-a d^3\right )^{8/3}}+\frac {5 b c^4 d^2 \sqrt [3]{b x^3+a}}{3 \left (b c^3-a d^3\right )^2 \left (c^3+d^3 x^3\right )}-\frac {c d^2 \left (b c^3-6 a d^3\right ) \sqrt [3]{b x^3+a}}{6 \left (b c^3-a d^3\right )^2 \left (c^3+d^3 x^3\right )}+\frac {3 c^4 d^2 \sqrt [3]{b x^3+a}}{2 \left (b c^3-a d^3\right ) \left (c^3+d^3 x^3\right )^2} \]

[Out]

3/2*c^4*d^2*(b*x^3+a)^(1/3)/(-a*d^3+b*c^3)/(d^3*x^3+c^3)^2+3/2*c^2*d^4*x^2*(b*x^3+a)^(1/3)/(-a*d^3+b*c^3)/(d^3
*x^3+c^3)^2+5/3*b*c^4*d^2*(b*x^3+a)^(1/3)/(-a*d^3+b*c^3)^2/(d^3*x^3+c^3)-1/6*c*d^2*(-6*a*d^3+b*c^3)*(b*x^3+a)^
(1/3)/(-a*d^3+b*c^3)^2/(d^3*x^3+c^3)+1/6*d^4*(-4*a*d^3+9*b*c^3)*x^2*(b*x^3+a)^(1/3)/c/(-a*d^3+b*c^3)^2/(d^3*x^
3+c^3)+1/3*d^4*(2*a*d^3+3*b*c^3)*x^2*(b*x^3+a)^(1/3)/c/(-a*d^3+b*c^3)^2/(d^3*x^3+c^3)+x*(1+b*x^3/a)^(2/3)*Appe
llF1(1/3,2/3,3,4/3,-b*x^3/a,-d^3*x^3/c^3)/c^3/(b*x^3+a)^(2/3)-7/4*d^3*x^4*(1+b*x^3/a)^(2/3)*AppellF1(4/3,2/3,3
,7/3,-b*x^3/a,-d^3*x^3/c^3)/c^6/(b*x^3+a)^(2/3)+1/7*d^6*x^7*(1+b*x^3/a)^(2/3)*AppellF1(7/3,2/3,3,10/3,-b*x^3/a
,-d^3*x^3/c^3)/c^9/(b*x^3+a)^(2/3)-5/9*b^2*c^4*d*ln(d^3*x^3+c^3)/(-a*d^3+b*c^3)^(8/3)+1/18*b*c*d*(-6*a*d^3+b*c
^3)*ln(d^3*x^3+c^3)/(-a*d^3+b*c^3)^(8/3)-1/9*a*d^4*(-a*d^3+6*b*c^3)*ln(d^3*x^3+c^3)/c^2/(-a*d^3+b*c^3)^(8/3)-1
/18*d*(2*a^2*d^6-6*a*b*c^3*d^3+9*b^2*c^6)*ln(d^3*x^3+c^3)/c^2/(-a*d^3+b*c^3)^(8/3)+1/3*a*d^4*(-a*d^3+6*b*c^3)*
ln((-a*d^3+b*c^3)^(1/3)*x/c-(b*x^3+a)^(1/3))/c^2/(-a*d^3+b*c^3)^(8/3)+1/6*d*(2*a^2*d^6-6*a*b*c^3*d^3+9*b^2*c^6
)*ln((-a*d^3+b*c^3)^(1/3)*x/c-(b*x^3+a)^(1/3))/c^2/(-a*d^3+b*c^3)^(8/3)+5/3*b^2*c^4*d*ln((-a*d^3+b*c^3)^(1/3)+
d*(b*x^3+a)^(1/3))/(-a*d^3+b*c^3)^(8/3)-1/6*b*c*d*(-6*a*d^3+b*c^3)*ln((-a*d^3+b*c^3)^(1/3)+d*(b*x^3+a)^(1/3))/
(-a*d^3+b*c^3)^(8/3)+2/9*a*d^4*(-a*d^3+6*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^2/(-a*d^3+b*c^3)^(8/3)*3^(1/2)+1/9*d*(2*a^2*d^6-6*a*b*c^3*d^3+9*b^2*c^6)*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^2/(-a*d^3+b*c^3)^(8/3)*3^(1/2)-10/9*b^2*c^4*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)^(8/3)*3^(1/2)+1/9*b*c*d*(-6*a*d^3+b*c^3)*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)^(8/3)*3^(1/2)

________________________________________________________________________________________

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 = {} \[ \int \frac {1}{(c+d x)^3 \left (a+b x^3\right )^{2/3}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [F]  time = 0.44, size = 0, normalized size = 0.00 \[ \int \frac {1}{(c+d x)^3 \left (a+b x^3\right )^{2/3}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [F]  time = 0.07, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (d x +c \right )^{3} \left (b \,x^{3}+a \right )^{\frac {2}{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________