3.2.23 \(\int \frac {1}{\sqrt [3]{-a-b x^2} (\frac {9 a d}{b}+d x^2)} \, dx\)

Optimal. Leaf size=153 \[ -\frac {\sqrt {b} \tan ^{-1}\left (\frac {\left (\sqrt [3]{-a-b x^2}+\sqrt [3]{a}\right )^2}{3 \sqrt [6]{a} \sqrt {b} x}\right )}{12 a^{5/6} d}+\frac {\sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{-a-b x^2}+\sqrt [3]{a}\right )}{\sqrt {b} x}\right )}{4 \sqrt {3} a^{5/6} d}-\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} x}{3 \sqrt {a}}\right )}{12 a^{5/6} d} \]

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Rubi [A]  time = 0.03, antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {394} \begin {gather*} -\frac {\sqrt {b} \tan ^{-1}\left (\frac {\left (\sqrt [3]{-a-b x^2}+\sqrt [3]{a}\right )^2}{3 \sqrt [6]{a} \sqrt {b} x}\right )}{12 a^{5/6} d}+\frac {\sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{-a-b x^2}+\sqrt [3]{a}\right )}{\sqrt {b} x}\right )}{4 \sqrt {3} a^{5/6} d}-\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} x}{3 \sqrt {a}}\right )}{12 a^{5/6} d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((-a - b*x^2)^(1/3)*((9*a*d)/b + d*x^2)),x]

[Out]

-(Sqrt[b]*ArcTan[(Sqrt[b]*x)/(3*Sqrt[a])])/(12*a^(5/6)*d) - (Sqrt[b]*ArcTan[(a^(1/3) + (-a - b*x^2)^(1/3))^2/(
3*a^(1/6)*Sqrt[b]*x)])/(12*a^(5/6)*d) + (Sqrt[b]*ArcTanh[(Sqrt[3]*a^(1/6)*(a^(1/3) + (-a - b*x^2)^(1/3)))/(Sqr
t[b]*x)])/(4*Sqrt[3]*a^(5/6)*d)

Rule 394

Int[1/(((a_) + (b_.)*(x_)^2)^(1/3)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> With[{q = Rt[b/a, 2]}, Simp[(q*ArcTan[
(q*x)/3])/(12*Rt[a, 3]*d), x] + (Simp[(q*ArcTan[(Rt[a, 3] - (a + b*x^2)^(1/3))^2/(3*Rt[a, 3]^2*q*x)])/(12*Rt[a
, 3]*d), x] - Simp[(q*ArcTanh[(Sqrt[3]*(Rt[a, 3] - (a + b*x^2)^(1/3)))/(Rt[a, 3]*q*x)])/(4*Sqrt[3]*Rt[a, 3]*d)
, x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[b*c - 9*a*d, 0] && PosQ[b/a]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt [3]{-a-b x^2} \left (\frac {9 a d}{b}+d x^2\right )} \, dx &=-\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} x}{3 \sqrt {a}}\right )}{12 a^{5/6} d}-\frac {\sqrt {b} \tan ^{-1}\left (\frac {\left (\sqrt [3]{a}+\sqrt [3]{-a-b x^2}\right )^2}{3 \sqrt [6]{a} \sqrt {b} x}\right )}{12 a^{5/6} d}+\frac {\sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{-a-b x^2}\right )}{\sqrt {b} x}\right )}{4 \sqrt {3} a^{5/6} d}\\ \end {align*}

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Mathematica [C]  time = 0.16, size = 172, normalized size = 1.12 \begin {gather*} \frac {27 a b x F_1\left (\frac {1}{2};\frac {1}{3},1;\frac {3}{2};-\frac {b x^2}{a},-\frac {b x^2}{9 a}\right )}{d \sqrt [3]{-a-b x^2} \left (9 a+b x^2\right ) \left (27 a F_1\left (\frac {1}{2};\frac {1}{3},1;\frac {3}{2};-\frac {b x^2}{a},-\frac {b x^2}{9 a}\right )-2 b x^2 \left (F_1\left (\frac {3}{2};\frac {1}{3},2;\frac {5}{2};-\frac {b x^2}{a},-\frac {b x^2}{9 a}\right )+3 F_1\left (\frac {3}{2};\frac {4}{3},1;\frac {5}{2};-\frac {b x^2}{a},-\frac {b x^2}{9 a}\right )\right )\right )} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[1/((-a - b*x^2)^(1/3)*((9*a*d)/b + d*x^2)),x]

[Out]

(27*a*b*x*AppellF1[1/2, 1/3, 1, 3/2, -((b*x^2)/a), -1/9*(b*x^2)/a])/(d*(-a - b*x^2)^(1/3)*(9*a + b*x^2)*(27*a*
AppellF1[1/2, 1/3, 1, 3/2, -((b*x^2)/a), -1/9*(b*x^2)/a] - 2*b*x^2*(AppellF1[3/2, 1/3, 2, 5/2, -((b*x^2)/a), -
1/9*(b*x^2)/a] + 3*AppellF1[3/2, 4/3, 1, 5/2, -((b*x^2)/a), -1/9*(b*x^2)/a])))

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IntegrateAlgebraic [F]  time = 5.44, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [3]{-a-b x^2} \left (\frac {9 a d}{b}+d x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[1/((-a - b*x^2)^(1/3)*((9*a*d)/b + d*x^2)),x]

[Out]

Defer[IntegrateAlgebraic][1/((-a - b*x^2)^(1/3)*((9*a*d)/b + d*x^2)), x]

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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(1/(-b*x^2-a)^(1/3)/(9*a*d/b+d*x^2),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 {1}{{\left (-b x^{2} - a\right )}^{\frac {1}{3}} {\left (d x^{2} + \frac {9 \, a d}{b}\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-b*x^2-a)^(1/3)/(9*a*d/b+d*x^2),x, algorithm="giac")

[Out]

integrate(1/((-b*x^2 - a)^(1/3)*(d*x^2 + 9*a*d/b)), x)

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maple [F]  time = 0.32, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (-b \,x^{2}-a \right )^{\frac {1}{3}} \left (d \,x^{2}+\frac {9 a d}{b}\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-b*x^2-a)^(1/3)/(d*x^2+9*a/b*d),x)

[Out]

int(1/(-b*x^2-a)^(1/3)/(d*x^2+9*a/b*d),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (-b x^{2} - a\right )}^{\frac {1}{3}} {\left (d x^{2} + \frac {9 \, a d}{b}\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-b*x^2-a)^(1/3)/(9*a*d/b+d*x^2),x, algorithm="maxima")

[Out]

integrate(1/((-b*x^2 - a)^(1/3)*(d*x^2 + 9*a*d/b)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\left (-b\,x^2-a\right )}^{1/3}\,\left (d\,x^2+\frac {9\,a\,d}{b}\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((- a - b*x^2)^(1/3)*(d*x^2 + (9*a*d)/b)),x)

[Out]

int(1/((- a - b*x^2)^(1/3)*(d*x^2 + (9*a*d)/b)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {b \int \frac {1}{9 a \sqrt [3]{- a - b x^{2}} + b x^{2} \sqrt [3]{- a - b x^{2}}}\, dx}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-b*x**2-a)**(1/3)/(9*a*d/b+d*x**2),x)

[Out]

b*Integral(1/(9*a*(-a - b*x**2)**(1/3) + b*x**2*(-a - b*x**2)**(1/3)), x)/d

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