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

Optimal. Leaf size=1513 \[ \frac {2 a^2 \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 ) d^6}{9 \sqrt {3} c^2 \left (b c^3-a d^3\right )^{7/3}}+\frac {a^2 \log \left (c^3+d^3 x^3\right ) d^6}{27 c^2 \left (b c^3-a d^3\right )^{7/3}}-\frac {a^2 \log \left (\frac {\sqrt [3]{b c^3-a d^3} x}{c}-\sqrt [3]{b x^3+a}\right ) d^6}{9 c^2 \left (b c^3-a d^3\right )^{7/3}}+\frac {6 x^5 \sqrt [3]{\frac {b x^3}{a}+1} F_1\left (\frac {5}{3};\frac {1}{3},3;\frac {8}{3};-\frac {b x^3}{a},-\frac {d^3 x^3}{c^3}\right ) d^4}{5 c^7 \sqrt [3]{b x^3+a}}+\frac {7 a \left (3 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 ) d^3}{9 \sqrt {3} c^2 \left (b c^3-a d^3\right )^{7/3}}+\frac {7 a \left (3 b c^3-a d^3\right ) \log \left (c^3+d^3 x^3\right ) d^3}{54 c^2 \left (b c^3-a d^3\right )^{7/3}}-\frac {7 a \left (3 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 ) d^3}{18 c^2 \left (b c^3-a d^3\right )^{7/3}}-\frac {7 \left (3 b c^3+a d^3\right ) x \left (b x^3+a\right )^{2/3} d^3}{18 \left (b c^3-a d^3\right )^2 \left (c^3+d^3 x^3\right )}+\frac {\left (3 b c^3-7 a d^3\right ) x \left (b x^3+a\right )^{2/3} d^3}{18 \left (b c^3-a d^3\right )^2 \left (c^3+d^3 x^3\right )}-\frac {\left (9 b c^3-5 a d^3\right ) x \left (b x^3+a\right )^{2/3} d^3}{18 \left (b c^3-a d^3\right )^2 \left (c^3+d^3 x^3\right )}-\frac {3 c^3 x \left (b x^3+a\right )^{2/3} d^3}{2 \left (b c^3-a d^3\right ) \left (c^3+d^3 x^3\right )^2}+\frac {4 b c^4 \left (b x^3+a\right )^{2/3} d^2}{3 \left (b c^3-a d^3\right )^2 \left (c^3+d^3 x^3\right )}-\frac {c \left (b c^3-3 a d^3\right ) \left (b x^3+a\right )^{2/3} d^2}{3 \left (b c^3-a d^3\right )^2 \left (c^3+d^3 x^3\right )}+\frac {3 c^4 \left (b x^3+a\right )^{2/3} d^2}{2 \left (b c^3-a d^3\right ) \left (c^3+d^3 x^3\right )^2}-\frac {3 x^2 \sqrt [3]{\frac {b x^3}{a}+1} F_1\left (\frac {2}{3};\frac {1}{3},3;\frac {5}{3};-\frac {b x^3}{a},-\frac {d^3 x^3}{c^3}\right ) d}{2 c^4 \sqrt [3]{b x^3+a}}+\frac {\left (9 b^2 c^6-12 a b d^3 c^3+5 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 )}{9 \sqrt {3} c^2 \left (b c^3-a d^3\right )^{7/3}}-\frac {4 b^2 c^4 \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 )^{7/3}}+\frac {b c \left (b c^3-3 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 )^{7/3}}+\frac {\left (9 b^2 c^6-12 a b d^3 c^3+5 a^2 d^6\right ) \log \left (c^3+d^3 x^3\right )}{54 c^2 \left (b c^3-a d^3\right )^{7/3}}+\frac {2 b^2 c^4 \log \left (c^3+d^3 x^3\right )}{9 \left (b c^3-a d^3\right )^{7/3}}-\frac {b c \left (b c^3-3 a d^3\right ) \log \left (c^3+d^3 x^3\right )}{18 \left (b c^3-a d^3\right )^{7/3}}-\frac {\left (9 b^2 c^6-12 a b d^3 c^3+5 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 )}{18 c^2 \left (b c^3-a d^3\right )^{7/3}}-\frac {2 b^2 c^4 \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 )^{7/3}}+\frac {b c \left (b c^3-3 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 )^{7/3}} \]

[Out]

3/2*c^4*d^2*(b*x^3+a)^(2/3)/(-a*d^3+b*c^3)/(d^3*x^3+c^3)^2-3/2*c^3*d^3*x*(b*x^3+a)^(2/3)/(-a*d^3+b*c^3)/(d^3*x
^3+c^3)^2+4/3*b*c^4*d^2*(b*x^3+a)^(2/3)/(-a*d^3+b*c^3)^2/(d^3*x^3+c^3)-1/3*c*d^2*(-3*a*d^3+b*c^3)*(b*x^3+a)^(2
/3)/(-a*d^3+b*c^3)^2/(d^3*x^3+c^3)+1/18*d^3*(-7*a*d^3+3*b*c^3)*x*(b*x^3+a)^(2/3)/(-a*d^3+b*c^3)^2/(d^3*x^3+c^3
)-1/18*d^3*(-5*a*d^3+9*b*c^3)*x*(b*x^3+a)^(2/3)/(-a*d^3+b*c^3)^2/(d^3*x^3+c^3)-7/18*d^3*(a*d^3+3*b*c^3)*x*(b*x
^3+a)^(2/3)/(-a*d^3+b*c^3)^2/(d^3*x^3+c^3)-3/2*d*x^2*(1+b*x^3/a)^(1/3)*AppellF1(2/3,1/3,3,5/3,-b*x^3/a,-d^3*x^
3/c^3)/c^4/(b*x^3+a)^(1/3)+6/5*d^4*x^5*(1+b*x^3/a)^(1/3)*AppellF1(5/3,1/3,3,8/3,-b*x^3/a,-d^3*x^3/c^3)/c^7/(b*
x^3+a)^(1/3)+2/9*b^2*c^4*ln(d^3*x^3+c^3)/(-a*d^3+b*c^3)^(7/3)+1/27*a^2*d^6*ln(d^3*x^3+c^3)/c^2/(-a*d^3+b*c^3)^
(7/3)-1/18*b*c*(-3*a*d^3+b*c^3)*ln(d^3*x^3+c^3)/(-a*d^3+b*c^3)^(7/3)+7/54*a*d^3*(-a*d^3+3*b*c^3)*ln(d^3*x^3+c^
3)/c^2/(-a*d^3+b*c^3)^(7/3)+1/54*(5*a^2*d^6-12*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)^(7/3)
-1/9*a^2*d^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)^(7/3)-7/18*a*d^3*(-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^2/(-a*d^3+b*c^3)^(7/3)-1/18*(5*a^2*d^6-12*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)^(7/3)-2/3*b^2*c^4*ln((-a*d^3+b*c^3)^(1/3)+d
*(b*x^3+a)^(1/3))/(-a*d^3+b*c^3)^(7/3)+1/6*b*c*(-3*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)^(7/3)+2/27*a^2*d^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)^(7/3)*3^(1/2)+7/27*a*d^3*(-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^2/(-a*d^3+b*c^3)^(7/3)*3^(1/2)+1/27*(5*a^2*d^6-12*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)^(7/3)*3^(1/2)-4/9*b^2*c^4*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)^(7/3)*3^(1/2)+1/9*b*c*(-3*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)^(7/3)*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 = {} \[ \int \frac {1}{(c+d x)^3 \sqrt [3]{a+b x^3}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 0.48, size = 0, normalized size = 0.00 \[ \int \frac {1}{(c+d x)^3 \sqrt [3]{a+b x^3}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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