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

Optimal. Leaf size=435 \[ \frac {x \sqrt [3]{a+b x^3} F_1\left (\frac {1}{3};-\frac {1}{3},1;\frac {4}{3};-\frac {b x^3}{a},-\frac {d^3 x^3}{c^3}\right )}{c \sqrt [3]{\frac {b x^3}{a}+1}}+\frac {\sqrt [3]{b c^3-a d^3} \log \left (c^3+d^3 x^3\right )}{3 d^2}-\frac {\sqrt [3]{b c^3-a d^3} \log \left (\frac {x \sqrt [3]{b c^3-a d^3}}{c}-\sqrt [3]{a+b x^3}\right )}{2 d^2}-\frac {\sqrt [3]{b c^3-a d^3} \log \left (\sqrt [3]{b c^3-a d^3}+d \sqrt [3]{a+b x^3}\right )}{2 d^2}-\frac {\sqrt [3]{b c^3-a d^3} \tan ^{-1}\left (\frac {\frac {2 x \sqrt [3]{b c^3-a d^3}}{c \sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} d^2}+\frac {\sqrt [3]{b c^3-a d^3} \tan ^{-1}\left (\frac {1-\frac {2 d \sqrt [3]{a+b x^3}}{\sqrt [3]{b c^3-a d^3}}}{\sqrt {3}}\right )}{\sqrt {3} d^2}+\frac {\sqrt [3]{b} c \log \left (\sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{2 d^2}+\frac {\sqrt [3]{b} c \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} d^2}+\frac {\sqrt [3]{a+b x^3}}{d} \]

[Out]

(b*x^3+a)^(1/3)/d+x*(b*x^3+a)^(1/3)*AppellF1(1/3,-1/3,1,4/3,-b*x^3/a,-d^3*x^3/c^3)/c/(1+b*x^3/a)^(1/3)+1/3*(-a
*d^3+b*c^3)^(1/3)*ln(d^3*x^3+c^3)/d^2+1/2*b^(1/3)*c*ln(b^(1/3)*x-(b*x^3+a)^(1/3))/d^2-1/2*(-a*d^3+b*c^3)^(1/3)
*ln((-a*d^3+b*c^3)^(1/3)*x/c-(b*x^3+a)^(1/3))/d^2-1/2*(-a*d^3+b*c^3)^(1/3)*ln((-a*d^3+b*c^3)^(1/3)+d*(b*x^3+a)
^(1/3))/d^2+1/3*b^(1/3)*c*arctan(1/3*(1+2*b^(1/3)*x/(b*x^3+a)^(1/3))*3^(1/2))/d^2*3^(1/2)-1/3*(-a*d^3+b*c^3)^(
1/3)*arctan(1/3*(1+2*(-a*d^3+b*c^3)^(1/3)*x/c/(b*x^3+a)^(1/3))*3^(1/2))/d^2*3^(1/2)+1/3*(-a*d^3+b*c^3)^(1/3)*a
rctan(1/3*(1-2*d*(b*x^3+a)^(1/3)/(-a*d^3+b*c^3)^(1/3))*3^(1/2))/d^2*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 = {} \[ \int \frac {\sqrt [3]{a+b x^3}}{c+d x} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*x^3)^(1/3)/(c + d*x),x]

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 0.33, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt [3]{a+b x^3}}{c+d x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.08, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{d x +c}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^3)^(1/3)/(c + d*x),x)

[Out]

int((a + b*x^3)^(1/3)/(c + d*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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