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

Optimal. Leaf size=398 \[ -\frac {\log \left (2^{2/3}-\frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{3\ 2^{2/3} \sqrt [3]{a} \sqrt [3]{b}}+\frac {\log \left (\frac {2^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{\left (a+b x^3\right )^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1\right )}{3\ 2^{2/3} \sqrt [3]{a} \sqrt [3]{b}}-\frac {\sqrt [3]{2} \log \left (\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1\right )}{3 \sqrt [3]{a} \sqrt [3]{b}}+\frac {\log \left (\frac {\left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{\left (a+b x^3\right )^{2/3}}+\frac {2^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+2 \sqrt [3]{2}\right )}{6\ 2^{2/3} \sqrt [3]{a} \sqrt [3]{b}}-\frac {\sqrt [3]{2} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{b}}-\frac {\tan ^{-1}\left (\frac {\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3} \sqrt [3]{a} \sqrt [3]{b}} \]

[Out]

-1/6*ln(2^(2/3)+(-a^(1/3)-b^(1/3)*x)/(b*x^3+a)^(1/3))*2^(1/3)/a^(1/3)/b^(1/3)+1/6*ln(1+2^(2/3)*(a^(1/3)+b^(1/3
)*x)^2/(b*x^3+a)^(2/3)-2^(1/3)*(a^(1/3)+b^(1/3)*x)/(b*x^3+a)^(1/3))*2^(1/3)/a^(1/3)/b^(1/3)-1/3*2^(1/3)*ln(1+2
^(1/3)*(a^(1/3)+b^(1/3)*x)/(b*x^3+a)^(1/3))/a^(1/3)/b^(1/3)+1/12*ln(2*2^(1/3)+(a^(1/3)+b^(1/3)*x)^2/(b*x^3+a)^
(2/3)+2^(2/3)*(a^(1/3)+b^(1/3)*x)/(b*x^3+a)^(1/3))*2^(1/3)/a^(1/3)/b^(1/3)-1/3*2^(1/3)*arctan(1/3*(1-2*2^(1/3)
*(a^(1/3)+b^(1/3)*x)/(b*x^3+a)^(1/3))*3^(1/2))/a^(1/3)/b^(1/3)*3^(1/2)-1/6*arctan(1/3*(1+2^(1/3)*(a^(1/3)+b^(1
/3)*x)/(b*x^3+a)^(1/3))*3^(1/2))*2^(1/3)/a^(1/3)/b^(1/3)*3^(1/2)

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Rubi [C]  time = 0.03, antiderivative size = 58, normalized size of antiderivative = 0.15, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {430, 429} \[ \frac {x \sqrt [3]{a+b x^3} F_1\left (\frac {1}{3};1,-\frac {1}{3};\frac {4}{3};\frac {b x^3}{a},-\frac {b x^3}{a}\right )}{a \sqrt [3]{\frac {b x^3}{a}+1}} \]

Warning: Unable to verify antiderivative.

[In]

Int[(a + b*x^3)^(1/3)/(a - b*x^3),x]

[Out]

(x*(a + b*x^3)^(1/3)*AppellF1[1/3, 1, -1/3, 4/3, (b*x^3)/a, -((b*x^3)/a)])/(a*(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-b x^3} \, dx &=\frac {\sqrt [3]{a+b x^3} \int \frac {\sqrt [3]{1+\frac {b x^3}{a}}}{a-b 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};1,-\frac {1}{3};\frac {4}{3};\frac {b x^3}{a},-\frac {b x^3}{a}\right )}{a \sqrt [3]{1+\frac {b x^3}{a}}}\\ \end {align*}

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Mathematica [C]  time = 0.13, size = 151, normalized size = 0.38 \[ \frac {4 a x \sqrt [3]{a+b x^3} F_1\left (\frac {1}{3};-\frac {1}{3},1;\frac {4}{3};-\frac {b x^3}{a},\frac {b x^3}{a}\right )}{\left (a-b x^3\right ) \left (b x^3 \left (3 F_1\left (\frac {4}{3};-\frac {1}{3},2;\frac {7}{3};-\frac {b x^3}{a},\frac {b x^3}{a}\right )+F_1\left (\frac {4}{3};\frac {2}{3},1;\frac {7}{3};-\frac {b x^3}{a},\frac {b x^3}{a}\right )\right )+4 a F_1\left (\frac {1}{3};-\frac {1}{3},1;\frac {4}{3};-\frac {b x^3}{a},\frac {b x^3}{a}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*x^3)^(1/3)/(a - b*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])/((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] + Appe
llF1[4/3, 2/3, 1, 7/3, -((b*x^3)/a), (b*x^3)/a])))

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fricas [B]  time = 29.48, size = 644, normalized size = 1.62 \[ -\frac {1}{18} \, \sqrt {3} 2^{\frac {1}{3}} \left (-\frac {1}{a b}\right )^{\frac {1}{3}} \arctan \left (\frac {6 \, \sqrt {3} 2^{\frac {2}{3}} {\left (a b^{6} x^{16} + 33 \, a^{2} b^{5} x^{13} + 110 \, a^{3} b^{4} x^{10} + 110 \, a^{4} b^{3} x^{7} + 33 \, a^{5} b^{2} x^{4} + a^{6} b x\right )} {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-\frac {1}{a b}\right )^{\frac {2}{3}} + 24 \, \sqrt {3} 2^{\frac {1}{3}} {\left (a b^{5} x^{14} + 2 \, a^{2} b^{4} x^{11} - 6 \, a^{3} b^{3} x^{8} + 2 \, a^{4} b^{2} x^{5} + a^{5} b x^{2}\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}} \left (-\frac {1}{a b}\right )^{\frac {1}{3}} - \sqrt {3} {\left (b^{6} x^{18} - 42 \, a b^{5} x^{15} - 417 \, a^{2} b^{4} x^{12} - 812 \, a^{3} b^{3} x^{9} - 417 \, a^{4} b^{2} x^{6} - 42 \, a^{5} b x^{3} + a^{6}\right )}}{3 \, {\left (b^{6} x^{18} + 102 \, a b^{5} x^{15} + 447 \, a^{2} b^{4} x^{12} + 628 \, a^{3} b^{3} x^{9} + 447 \, a^{4} b^{2} x^{6} + 102 \, a^{5} b x^{3} + a^{6}\right )}}\right ) - \frac {1}{36} \cdot 2^{\frac {1}{3}} \left (-\frac {1}{a b}\right )^{\frac {1}{3}} \log \left (\frac {12 \cdot 2^{\frac {2}{3}} {\left (a b^{3} x^{8} + 4 \, a^{2} b^{2} x^{5} + a^{3} b x^{2}\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}} \left (-\frac {1}{a b}\right )^{\frac {2}{3}} - 2^{\frac {1}{3}} {\left (b^{4} x^{12} + 32 \, a b^{3} x^{9} + 78 \, a^{2} b^{2} x^{6} + 32 \, a^{3} b x^{3} + a^{4}\right )} \left (-\frac {1}{a b}\right )^{\frac {1}{3}} + 6 \, {\left (b^{3} x^{10} + 11 \, a b^{2} x^{7} + 11 \, a^{2} b x^{4} + a^{3} x\right )} {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{b^{4} x^{12} - 4 \, a b^{3} x^{9} + 6 \, a^{2} b^{2} x^{6} - 4 \, a^{3} b x^{3} + a^{4}}\right ) + \frac {1}{18} \cdot 2^{\frac {1}{3}} \left (-\frac {1}{a b}\right )^{\frac {1}{3}} \log \left (-\frac {12 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} x^{2} + 2^{\frac {2}{3}} {\left (b^{2} x^{6} - 2 \, a b x^{3} + a^{2}\right )} \left (-\frac {1}{a b}\right )^{\frac {2}{3}} + 6 \cdot 2^{\frac {1}{3}} {\left (b x^{4} + a x\right )} {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-\frac {1}{a b}\right )^{\frac {1}{3}}}{b^{2} x^{6} - 2 \, a b x^{3} + a^{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/18*sqrt(3)*2^(1/3)*(-1/(a*b))^(1/3)*arctan(1/3*(6*sqrt(3)*2^(2/3)*(a*b^6*x^16 + 33*a^2*b^5*x^13 + 110*a^3*b
^4*x^10 + 110*a^4*b^3*x^7 + 33*a^5*b^2*x^4 + a^6*b*x)*(b*x^3 + a)^(1/3)*(-1/(a*b))^(2/3) + 24*sqrt(3)*2^(1/3)*
(a*b^5*x^14 + 2*a^2*b^4*x^11 - 6*a^3*b^3*x^8 + 2*a^4*b^2*x^5 + a^5*b*x^2)*(b*x^3 + a)^(2/3)*(-1/(a*b))^(1/3) -
sqrt(3)*(b^6*x^18 - 42*a*b^5*x^15 - 417*a^2*b^4*x^12 - 812*a^3*b^3*x^9 - 417*a^4*b^2*x^6 - 42*a^5*b*x^3 + a^6
))/(b^6*x^18 + 102*a*b^5*x^15 + 447*a^2*b^4*x^12 + 628*a^3*b^3*x^9 + 447*a^4*b^2*x^6 + 102*a^5*b*x^3 + a^6)) -
1/36*2^(1/3)*(-1/(a*b))^(1/3)*log((12*2^(2/3)*(a*b^3*x^8 + 4*a^2*b^2*x^5 + a^3*b*x^2)*(b*x^3 + a)^(2/3)*(-1/(
a*b))^(2/3) - 2^(1/3)*(b^4*x^12 + 32*a*b^3*x^9 + 78*a^2*b^2*x^6 + 32*a^3*b*x^3 + a^4)*(-1/(a*b))^(1/3) + 6*(b^
3*x^10 + 11*a*b^2*x^7 + 11*a^2*b*x^4 + a^3*x)*(b*x^3 + a)^(1/3))/(b^4*x^12 - 4*a*b^3*x^9 + 6*a^2*b^2*x^6 - 4*a
^3*b*x^3 + a^4)) + 1/18*2^(1/3)*(-1/(a*b))^(1/3)*log(-(12*(b*x^3 + a)^(2/3)*x^2 + 2^(2/3)*(b^2*x^6 - 2*a*b*x^3
+ a^2)*(-1/(a*b))^(2/3) + 6*2^(1/3)*(b*x^4 + a*x)*(b*x^3 + a)^(1/3)*(-1/(a*b))^(1/3))/(b^2*x^6 - 2*a*b*x^3 +
a^2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.70, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{-b \,x^{3}+a}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^3)^(1/3)/(a - b*x^3),x)

[Out]

int((a + b*x^3)^(1/3)/(a - b*x^3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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