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

Optimal. Leaf size=818 \[ \frac{5 \left (a-b x^2\right )^{2/3} x}{288 a^3 \left (b x^2+3 a\right )}-\frac{5 x}{288 a^3 \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}+\frac{\left (a-b x^2\right )^{2/3} x}{48 a^2 \left (b x^2+3 a\right )^2}+\frac{5 \tan ^{-1}\left (\frac{\sqrt{3} \sqrt{a}}{\sqrt{b} x}\right )}{144\ 2^{2/3} \sqrt{3} a^{17/6} \sqrt{b}}+\frac{5 \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}{\sqrt{b} x}\right )}{144\ 2^{2/3} \sqrt{3} a^{17/6} \sqrt{b}}-\frac{5 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{432\ 2^{2/3} a^{17/6} \sqrt{b}}+\frac{5 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}\right )}{144\ 2^{2/3} a^{17/6} \sqrt{b}}-\frac{5 \sqrt{2+\sqrt{3}} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a-b x^2} \sqrt [3]{a}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{192\ 3^{3/4} a^{8/3} b \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} x}+\frac{5 \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a-b x^2} \sqrt [3]{a}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{144 \sqrt{2} \sqrt [4]{3} a^{8/3} b \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} x} \]

[Out]

(x*(a - b*x^2)^(2/3))/(48*a^2*(3*a + b*x^2)^2) + (5*x*(a - b*x^2)^(2/3))/(288*a^
3*(3*a + b*x^2)) - (5*x)/(288*a^3*((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))) +
(5*ArcTan[(Sqrt[3]*Sqrt[a])/(Sqrt[b]*x)])/(144*2^(2/3)*Sqrt[3]*a^(17/6)*Sqrt[b]
) + (5*ArcTan[(Sqrt[3]*a^(1/6)*(a^(1/3) - 2^(1/3)*(a - b*x^2)^(1/3)))/(Sqrt[b]*x
)])/(144*2^(2/3)*Sqrt[3]*a^(17/6)*Sqrt[b]) - (5*ArcTanh[(Sqrt[b]*x)/Sqrt[a]])/(4
32*2^(2/3)*a^(17/6)*Sqrt[b]) + (5*ArcTanh[(Sqrt[b]*x)/(a^(1/6)*(a^(1/3) + 2^(1/3
)*(a - b*x^2)^(1/3)))])/(144*2^(2/3)*a^(17/6)*Sqrt[b]) - (5*Sqrt[2 + Sqrt[3]]*(a
^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x
^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticE[ArcSin[((1 +
Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3
))], -7 + 4*Sqrt[3]])/(192*3^(3/4)*a^(8/3)*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b
*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)]) + (5*(a^(1/3) - (
a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))
/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticF[ArcSin[((1 + Sqrt[3])*
a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 +
4*Sqrt[3]])/(144*Sqrt[2]*3^(1/4)*a^(8/3)*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x
^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)])

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Rubi [A]  time = 1.4196, antiderivative size = 818, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{5 \left (a-b x^2\right )^{2/3} x}{288 a^3 \left (b x^2+3 a\right )}-\frac{5 x}{288 a^3 \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}+\frac{\left (a-b x^2\right )^{2/3} x}{48 a^2 \left (b x^2+3 a\right )^2}+\frac{5 \tan ^{-1}\left (\frac{\sqrt{3} \sqrt{a}}{\sqrt{b} x}\right )}{144\ 2^{2/3} \sqrt{3} a^{17/6} \sqrt{b}}+\frac{5 \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}{\sqrt{b} x}\right )}{144\ 2^{2/3} \sqrt{3} a^{17/6} \sqrt{b}}-\frac{5 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{432\ 2^{2/3} a^{17/6} \sqrt{b}}+\frac{5 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}\right )}{144\ 2^{2/3} a^{17/6} \sqrt{b}}-\frac{5 \sqrt{2+\sqrt{3}} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a-b x^2} \sqrt [3]{a}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{192\ 3^{3/4} a^{8/3} b \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} x}+\frac{5 \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a-b x^2} \sqrt [3]{a}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{144 \sqrt{2} \sqrt [4]{3} a^{8/3} b \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} x} \]

Warning: Unable to verify antiderivative.

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

[Out]

(x*(a - b*x^2)^(2/3))/(48*a^2*(3*a + b*x^2)^2) + (5*x*(a - b*x^2)^(2/3))/(288*a^
3*(3*a + b*x^2)) - (5*x)/(288*a^3*((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))) +
(5*ArcTan[(Sqrt[3]*Sqrt[a])/(Sqrt[b]*x)])/(144*2^(2/3)*Sqrt[3]*a^(17/6)*Sqrt[b]
) + (5*ArcTan[(Sqrt[3]*a^(1/6)*(a^(1/3) - 2^(1/3)*(a - b*x^2)^(1/3)))/(Sqrt[b]*x
)])/(144*2^(2/3)*Sqrt[3]*a^(17/6)*Sqrt[b]) - (5*ArcTanh[(Sqrt[b]*x)/Sqrt[a]])/(4
32*2^(2/3)*a^(17/6)*Sqrt[b]) + (5*ArcTanh[(Sqrt[b]*x)/(a^(1/6)*(a^(1/3) + 2^(1/3
)*(a - b*x^2)^(1/3)))])/(144*2^(2/3)*a^(17/6)*Sqrt[b]) - (5*Sqrt[2 + Sqrt[3]]*(a
^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x
^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticE[ArcSin[((1 +
Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3
))], -7 + 4*Sqrt[3]])/(192*3^(3/4)*a^(8/3)*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b
*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)]) + (5*(a^(1/3) - (
a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))
/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticF[ArcSin[((1 + Sqrt[3])*
a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 +
4*Sqrt[3]])/(144*Sqrt[2]*3^(1/4)*a^(8/3)*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x
^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)])

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [C]  time = 0.386385, size = 352, normalized size = 0.43 \[ \frac{x \left (\frac{675 a^2 \left (3 a+b x^2\right ) F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )}{2 b x^2 \left (F_1\left (\frac{3}{2};\frac{4}{3},1;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )-F_1\left (\frac{3}{2};\frac{1}{3},2;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )\right )+9 a F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )}+\frac{25 a b x^2 \left (3 a+b x^2\right ) F_1\left (\frac{3}{2};\frac{1}{3},1;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )}{2 b x^2 \left (F_1\left (\frac{5}{2};\frac{4}{3},1;\frac{7}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )-F_1\left (\frac{5}{2};\frac{1}{3},2;\frac{7}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )\right )+15 a F_1\left (\frac{3}{2};\frac{1}{3},1;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )}+3 \left (a-b x^2\right ) \left (21 a+5 b x^2\right )\right )}{864 a^3 \sqrt [3]{a-b x^2} \left (3 a+b x^2\right )^2} \]

Warning: Unable to verify antiderivative.

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

[Out]

(x*(3*(a - b*x^2)*(21*a + 5*b*x^2) + (675*a^2*(3*a + b*x^2)*AppellF1[1/2, 1/3, 1
, 3/2, (b*x^2)/a, -(b*x^2)/(3*a)])/(9*a*AppellF1[1/2, 1/3, 1, 3/2, (b*x^2)/a, -(
b*x^2)/(3*a)] + 2*b*x^2*(-AppellF1[3/2, 1/3, 2, 5/2, (b*x^2)/a, -(b*x^2)/(3*a)]
+ AppellF1[3/2, 4/3, 1, 5/2, (b*x^2)/a, -(b*x^2)/(3*a)])) + (25*a*b*x^2*(3*a + b
*x^2)*AppellF1[3/2, 1/3, 1, 5/2, (b*x^2)/a, -(b*x^2)/(3*a)])/(15*a*AppellF1[3/2,
1/3, 1, 5/2, (b*x^2)/a, -(b*x^2)/(3*a)] + 2*b*x^2*(-AppellF1[5/2, 1/3, 2, 7/2,
(b*x^2)/a, -(b*x^2)/(3*a)] + AppellF1[5/2, 4/3, 1, 7/2, (b*x^2)/a, -(b*x^2)/(3*a
)]))))/(864*a^3*(a - b*x^2)^(1/3)*(3*a + b*x^2)^2)

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Maple [F]  time = 0.068, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( b{x}^{2}+3\,a \right ) ^{3}}{\frac{1}{\sqrt [3]{-b{x}^{2}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-b*x^2+a)^(1/3)/(b*x^2+3*a)^3,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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