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

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

[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 = 0.0867765, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036 \[ -\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\sqrt{b} x}\right )}{4 \sqrt{3} a^{5/6} d}+\frac{\sqrt{b} \tanh ^{-1}\left (\frac{\left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}{3 \sqrt [6]{a} \sqrt{b} x}\right )}{12 a^{5/6} d}-\frac{\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} x}{3 \sqrt{a}}\right )}{12 a^{5/6} d} \]

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)

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Rubi in Sympy [A]  time = 38.0804, size = 53, normalized size = 0.35 \[ - \frac{b x \left (a - b x^{2}\right )^{\frac{2}{3}} \operatorname{appellf_{1}}{\left (\frac{1}{2},\frac{1}{3},1,\frac{3}{2},\frac{b x^{2}}{a},\frac{b x^{2}}{9 a} \right )}}{9 a^{2} d \left (1 - \frac{b x^{2}}{a}\right )^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b*x*(a - b*x**2)**(2/3)*appellf1(1/2, 1/3, 1, 3/2, b*x**2/a, b*x**2/(9*a))/(9*a
**2*d*(1 - b*x**2/a)**(2/3))

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Mathematica [C]  time = 0.273785, size = 167, normalized size = 1.09 \[ -\frac{27 a b x F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{b x^2}{a},\frac{b x^2}{9 a}\right )}{d \sqrt [3]{a-b x^2} \left (9 a-b x^2\right ) \left (2 b x^2 \left (F_1\left (\frac{3}{2};\frac{1}{3},2;\frac{5}{2};\frac{b x^2}{a},\frac{b x^2}{9 a}\right )+3 F_1\left (\frac{3}{2};\frac{4}{3},1;\frac{5}{2};\frac{b x^2}{a},\frac{b x^2}{9 a}\right )\right )+27 a F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{b x^2}{a},\frac{b x^2}{9 a}\right )\right )} \]

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*(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, (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 = 0.07, size = 0, normalized size = 0. \[ \int{1{\frac{1}{\sqrt [3]{-b{x}^{2}+a}}} \left ( -9\,{\frac{ad}{b}}+d{x}^{2} \right ) ^{-1}}\, dx \]

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. \[ \int \frac{1}{{\left (-b x^{2} + a\right )}^{\frac{1}{3}}{\left (d x^{2} - \frac{9 \, a d}{b}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x^2 + a)^(1/3)*(d*x^2 - 9*a*d/b)),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. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{b \int \frac{1}{- 9 a \sqrt [3]{a - b x^{2}} + b x^{2} \sqrt [3]{a - b x^{2}}}\, dx}{d} \]

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/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-b x^{2} + a\right )}^{\frac{1}{3}}{\left (d x^{2} - \frac{9 \, a d}{b}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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