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

Optimal. Leaf size=416 \[ -\frac {\log \left (2^{2/3}-\frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{3\ 2^{2/3} \sqrt [3]{a} \sqrt [3]{b} d}+\frac {\log \left (\frac {2^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{\left (a+b x^3\right )^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1\right )}{3\ 2^{2/3} \sqrt [3]{a} \sqrt [3]{b} d}-\frac {\sqrt [3]{2} \log \left (\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1\right )}{3 \sqrt [3]{a} \sqrt [3]{b} d}+\frac {\log \left (\frac {\left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{\left (a+b x^3\right )^{2/3}}+\frac {2^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+2 \sqrt [3]{2}\right )}{6\ 2^{2/3} \sqrt [3]{a} \sqrt [3]{b} d}-\frac {\sqrt [3]{2} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{b} d}-\frac {\tan ^{-1}\left (\frac {\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3} \sqrt [3]{a} \sqrt [3]{b} d} \]

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Rubi [C]  time = 0.03, antiderivative size = 61, normalized size of antiderivative = 0.15, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {430, 429} \begin {gather*} \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 {b x^3}{a}\right )}{a d \sqrt [3]{\frac {b x^3}{a}+1}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(x*(a + b*x^3)^(1/3)*AppellF1[1/3, -1/3, 1, 4/3, -((b*x^3)/a), (b*x^3)/a])/(a*d*(1 + (b*x^3)/a)^(1/3))

Rule 429

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*x*AppellF1[1/n, -p,
-q, 1 + 1/n, -((b*x^n)/a), -((d*x^n)/c)], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n
, -1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 430

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^F
racPart[p])/(1 + (b*x^n)/a)^FracPart[p], Int[(1 + (b*x^n)/a)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n,
p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] &&  !(IntegerQ[p] || GtQ[a, 0])

Rubi steps

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

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Mathematica [C]  time = 0.15, size = 154, normalized size = 0.37 \begin {gather*} \frac {4 a 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 {b x^3}{a}\right )}{d \left (a-b x^3\right ) \left (b x^3 \left (3 F_1\left (\frac {4}{3};-\frac {1}{3},2;\frac {7}{3};-\frac {b x^3}{a},\frac {b x^3}{a}\right )+F_1\left (\frac {4}{3};\frac {2}{3},1;\frac {7}{3};-\frac {b x^3}{a},\frac {b x^3}{a}\right )\right )+4 a F_1\left (\frac {1}{3};-\frac {1}{3},1;\frac {4}{3};-\frac {b x^3}{a},\frac {b x^3}{a}\right )\right )} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(4*a*x*(a + b*x^3)^(1/3)*AppellF1[1/3, -1/3, 1, 4/3, -((b*x^3)/a), (b*x^3)/a])/(d*(a - b*x^3)*(4*a*AppellF1[1/
3, -1/3, 1, 4/3, -((b*x^3)/a), (b*x^3)/a] + b*x^3*(3*AppellF1[4/3, -1/3, 2, 7/3, -((b*x^3)/a), (b*x^3)/a] + Ap
pellF1[4/3, 2/3, 1, 7/3, -((b*x^3)/a), (b*x^3)/a])))

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IntegrateAlgebraic [A]  time = 3.09, size = 533, normalized size = 1.28 \begin {gather*} \frac {\log \left (2^{2/3} a^{2/3}-\sqrt [3]{2} \sqrt [3]{b} x \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{a+b x^3}+2\ 2^{2/3} \sqrt [3]{a} \sqrt [3]{b} x+2^{2/3} b^{2/3} x^2\right )}{3\ 2^{2/3} \sqrt [3]{a} \sqrt [3]{b} d}+\frac {\log \left (2^{2/3} a^{2/3}+2 \sqrt [3]{2} \sqrt [3]{b} x \sqrt [3]{a+b x^3}+4 \left (a+b x^3\right )^{2/3}+2 \sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{a+b x^3}+2\ 2^{2/3} \sqrt [3]{a} \sqrt [3]{b} x+2^{2/3} b^{2/3} x^2\right )}{6\ 2^{2/3} \sqrt [3]{a} \sqrt [3]{b} d}-\frac {\sqrt [3]{2} \log \left (\sqrt [3]{a+b x^3}+\sqrt [3]{2} \sqrt [3]{a}+\sqrt [3]{2} \sqrt [3]{b} x\right )}{3 \sqrt [3]{a} \sqrt [3]{b} d}-\frac {\log \left (2 \sqrt [3]{a+b x^3}-\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{b} x\right )}{3\ 2^{2/3} \sqrt [3]{a} \sqrt [3]{b} d}+\frac {\sqrt [3]{2} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{a+b x^3}}{\sqrt [3]{a+b x^3}-2 \sqrt [3]{2} \sqrt [3]{a}-2 \sqrt [3]{2} \sqrt [3]{b} x}\right )}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{b} d}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{a+b x^3}}{\sqrt [3]{a+b x^3}+\sqrt [3]{2} \sqrt [3]{a}+\sqrt [3]{2} \sqrt [3]{b} x}\right )}{2^{2/3} \sqrt {3} \sqrt [3]{a} \sqrt [3]{b} d} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a + b*x^3)^(1/3)/(a*d - b*d*x^3),x]

[Out]

(2^(1/3)*ArcTan[(Sqrt[3]*(a + b*x^3)^(1/3))/(-2*2^(1/3)*a^(1/3) - 2*2^(1/3)*b^(1/3)*x + (a + b*x^3)^(1/3))])/(
Sqrt[3]*a^(1/3)*b^(1/3)*d) + ArcTan[(Sqrt[3]*(a + b*x^3)^(1/3))/(2^(1/3)*a^(1/3) + 2^(1/3)*b^(1/3)*x + (a + b*
x^3)^(1/3))]/(2^(2/3)*Sqrt[3]*a^(1/3)*b^(1/3)*d) - (2^(1/3)*Log[2^(1/3)*a^(1/3) + 2^(1/3)*b^(1/3)*x + (a + b*x
^3)^(1/3)])/(3*a^(1/3)*b^(1/3)*d) - Log[-(2^(1/3)*a^(1/3)) - 2^(1/3)*b^(1/3)*x + 2*(a + b*x^3)^(1/3)]/(3*2^(2/
3)*a^(1/3)*b^(1/3)*d) + Log[2^(2/3)*a^(2/3) + 2*2^(2/3)*a^(1/3)*b^(1/3)*x + 2^(2/3)*b^(2/3)*x^2 - 2^(1/3)*a^(1
/3)*(a + b*x^3)^(1/3) - 2^(1/3)*b^(1/3)*x*(a + b*x^3)^(1/3) + (a + b*x^3)^(2/3)]/(3*2^(2/3)*a^(1/3)*b^(1/3)*d)
+ Log[2^(2/3)*a^(2/3) + 2*2^(2/3)*a^(1/3)*b^(1/3)*x + 2^(2/3)*b^(2/3)*x^2 + 2*2^(1/3)*a^(1/3)*(a + b*x^3)^(1/
3) + 2*2^(1/3)*b^(1/3)*x*(a + b*x^3)^(1/3) + 4*(a + b*x^3)^(2/3)]/(6*2^(2/3)*a^(1/3)*b^(1/3)*d)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.63, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{-b d \,x^{3}+a d}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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