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

Optimal. Leaf size=151 \[ \frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a} \left (\sqrt [3]{b x^2-a}+\sqrt [3]{a}\right )}{\sqrt{b} x}\right )}{4 \sqrt{3} a^{5/6} d}-\frac{\sqrt{b} \tanh ^{-1}\left (\frac{\left (\sqrt [3]{b x^2-a}+\sqrt [3]{a}\right )^2}{3 \sqrt [6]{a} \sqrt{b} x}\right )}{12 a^{5/6} d}+\frac{\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} x}{3 \sqrt{a}}\right )}{12 a^{5/6} d} \]

[Out]

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

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Rubi [A]  time = 0.0282874, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.034, Rules used = {395} \[ \frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a} \left (\sqrt [3]{b x^2-a}+\sqrt [3]{a}\right )}{\sqrt{b} x}\right )}{4 \sqrt{3} a^{5/6} d}-\frac{\sqrt{b} \tanh ^{-1}\left (\frac{\left (\sqrt [3]{b x^2-a}+\sqrt [3]{a}\right )^2}{3 \sqrt [6]{a} \sqrt{b} x}\right )}{12 a^{5/6} d}+\frac{\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} x}{3 \sqrt{a}}\right )}{12 a^{5/6} d} \]

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

Rule 395

Int[1/(((a_) + (b_.)*(x_)^2)^(1/3)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> With[{q = Rt[-(b/a), 2]}, -Simp[(q*Arc
Tanh[(q*x)/3])/(12*Rt[a, 3]*d), x] + (Simp[(q*ArcTanh[(Rt[a, 3] - (a + b*x^2)^(1/3))^2/(3*Rt[a, 3]^2*q*x)])/(1
2*Rt[a, 3]*d), x] - Simp[(q*ArcTan[(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] && NegQ[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{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}+\frac{\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} x}{3 \sqrt{a}}\right )}{12 a^{5/6} d}-\frac{\sqrt{b} \tanh ^{-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}\\ \end{align*}

Mathematica [C]  time = 0.122239, size = 168, normalized size = 1.11 \[ -\frac{27 a b x F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{b x^2}{a},\frac{b x^2}{9 a}\right )}{d \left (9 a-b x^2\right ) \sqrt [3]{b x^2-a} \left (2 b x^2 \left (F_1\left (\frac{3}{2};\frac{1}{3},2;\frac{5}{2};\frac{b x^2}{a},\frac{b x^2}{9 a}\right )+3 F_1\left (\frac{3}{2};\frac{4}{3},1;\frac{5}{2};\frac{b x^2}{a},\frac{b x^2}{9 a}\right )\right )+27 a F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{b x^2}{a},\frac{b x^2}{9 a}\right )\right )} \]

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, (b*x^2)/(9*a)])/(d*(9*a - b*x^2)*(-a + b*x^2)^(1/3)*(27*a*App
ellF1[1/2, 1/3, 1, 3/2, (b*x^2)/a, (b*x^2)/(9*a)] + 2*b*x^2*(AppellF1[3/2, 1/3, 2, 5/2, (b*x^2)/a, (b*x^2)/(9*
a)] + 3*AppellF1[3/2, 4/3, 1, 5/2, (b*x^2)/a, (b*x^2)/(9*a)])))

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Maple [F]  time = 0.043, size = 0, normalized size = 0. \begin{align*} \int{{\frac{1}{\sqrt [3]{b{x}^{2}-a}}} \left ( -9\,{\frac{ad}{b}}+d{x}^{2} \right ) ^{-1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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{align*}

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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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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{align*}

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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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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{align*}

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)