3.139 \(\int \frac{1}{\left (a-b x^2\right )^{7/3} \left (3 a+b x^2\right )} \, dx\)

Optimal. Leaf size=796 \[ \frac{21 x}{64 a^3 \sqrt [3]{a-b x^2}}+\frac{21 x}{64 a^3 \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}+\frac{3 x}{32 a^2 \left (a-b x^2\right )^{4/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt{a}}{\sqrt{b} x}\right )}{32\ 2^{2/3} \sqrt{3} a^{17/6} \sqrt{b}}+\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}{\sqrt{b} x}\right )}{32\ 2^{2/3} \sqrt{3} a^{17/6} \sqrt{b}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{96\ 2^{2/3} a^{17/6} \sqrt{b}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}\right )}{32\ 2^{2/3} a^{17/6} \sqrt{b}}+\frac{21 \sqrt [4]{3} \sqrt{2+\sqrt{3}} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a-b x^2} \sqrt [3]{a}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{128 a^{8/3} b \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} x}-\frac{7\ 3^{3/4} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a-b x^2} \sqrt [3]{a}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{32 \sqrt{2} a^{8/3} b \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} x} \]

[Out]

(3*x)/(32*a^2*(a - b*x^2)^(4/3)) + (21*x)/(64*a^3*(a - b*x^2)^(1/3)) + (21*x)/(6
4*a^3*((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))) + ArcTan[(Sqrt[3]*Sqrt[a])/(S
qrt[b]*x)]/(32*2^(2/3)*Sqrt[3]*a^(17/6)*Sqrt[b]) + ArcTan[(Sqrt[3]*a^(1/6)*(a^(1
/3) - 2^(1/3)*(a - b*x^2)^(1/3)))/(Sqrt[b]*x)]/(32*2^(2/3)*Sqrt[3]*a^(17/6)*Sqrt
[b]) - ArcTanh[(Sqrt[b]*x)/Sqrt[a]]/(96*2^(2/3)*a^(17/6)*Sqrt[b]) + ArcTanh[(Sqr
t[b]*x)/(a^(1/6)*(a^(1/3) + 2^(1/3)*(a - b*x^2)^(1/3)))]/(32*2^(2/3)*a^(17/6)*Sq
rt[b]) + (21*3^(1/4)*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]])/(128*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)]) - (7*3^(3/4)*(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]])/(32*Sqrt[2]*a^(8/3)*b*x*Sqr
t[-((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 = 1.29136, antiderivative size = 796, 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 \[ \frac{21 x}{64 a^3 \sqrt [3]{a-b x^2}}+\frac{21 x}{64 a^3 \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}+\frac{3 x}{32 a^2 \left (a-b x^2\right )^{4/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt{a}}{\sqrt{b} x}\right )}{32\ 2^{2/3} \sqrt{3} a^{17/6} \sqrt{b}}+\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}{\sqrt{b} x}\right )}{32\ 2^{2/3} \sqrt{3} a^{17/6} \sqrt{b}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{96\ 2^{2/3} a^{17/6} \sqrt{b}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}\right )}{32\ 2^{2/3} a^{17/6} \sqrt{b}}+\frac{21 \sqrt [4]{3} \sqrt{2+\sqrt{3}} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a-b x^2} \sqrt [3]{a}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{128 a^{8/3} b \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} x}-\frac{7\ 3^{3/4} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a-b x^2} \sqrt [3]{a}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{32 \sqrt{2} a^{8/3} b \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} x} \]

Warning: Unable to verify antiderivative.

[In]  Int[1/((a - b*x^2)^(7/3)*(3*a + b*x^2)),x]

[Out]

(3*x)/(32*a^2*(a - b*x^2)^(4/3)) + (21*x)/(64*a^3*(a - b*x^2)^(1/3)) + (21*x)/(6
4*a^3*((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))) + ArcTan[(Sqrt[3]*Sqrt[a])/(S
qrt[b]*x)]/(32*2^(2/3)*Sqrt[3]*a^(17/6)*Sqrt[b]) + ArcTan[(Sqrt[3]*a^(1/6)*(a^(1
/3) - 2^(1/3)*(a - b*x^2)^(1/3)))/(Sqrt[b]*x)]/(32*2^(2/3)*Sqrt[3]*a^(17/6)*Sqrt
[b]) - ArcTanh[(Sqrt[b]*x)/Sqrt[a]]/(96*2^(2/3)*a^(17/6)*Sqrt[b]) + ArcTanh[(Sqr
t[b]*x)/(a^(1/6)*(a^(1/3) + 2^(1/3)*(a - b*x^2)^(1/3)))]/(32*2^(2/3)*a^(17/6)*Sq
rt[b]) + (21*3^(1/4)*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]])/(128*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)]) - (7*3^(3/4)*(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]])/(32*Sqrt[2]*a^(8/3)*b*x*Sqr
t[-((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)**(7/3)/(b*x**2+3*a),x)

[Out]

Timed out

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Mathematica [C]  time = 0.327704, size = 347, normalized size = 0.44 \[ \frac{x \left (-\frac{153 a^2 F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )}{\left (3 a+b x^2\right ) \left (2 b x^2 \left (F_1\left (\frac{3}{2};\frac{4}{3},1;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )-F_1\left (\frac{3}{2};\frac{1}{3},2;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )\right )+9 a F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )\right )}-\frac{35 a b x^2 F_1\left (\frac{3}{2};\frac{1}{3},1;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )}{\left (3 a+b x^2\right ) \left (2 b x^2 \left (F_1\left (\frac{5}{2};\frac{4}{3},1;\frac{7}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )-F_1\left (\frac{5}{2};\frac{1}{3},2;\frac{7}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )\right )+15 a F_1\left (\frac{3}{2};\frac{1}{3},1;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )\right )}+\frac{3 \left (9 a-7 b x^2\right )}{a-b x^2}\right )}{64 a^3 \sqrt [3]{a-b x^2}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(x*((3*(9*a - 7*b*x^2))/(a - b*x^2) - (153*a^2*AppellF1[1/2, 1/3, 1, 3/2, (b*x^2
)/a, -(b*x^2)/(3*a)])/((3*a + b*x^2)*(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)]))) - (35*a*b*x^2*Appel
lF1[3/2, 1/3, 1, 5/2, (b*x^2)/a, -(b*x^2)/(3*a)])/((3*a + b*x^2)*(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)])))))/(64*a^3*(a - b*x^2)^(1/3))

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Maple [F]  time = 0.062, size = 0, normalized size = 0. \[ \int{\frac{1}{b{x}^{2}+3\,a} \left ( -b{x}^{2}+a \right ) ^{-{\frac{7}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-b*x^2+a)^(7/3)/(b*x^2+3*a),x)

[Out]

int(1/(-b*x^2+a)^(7/3)/(b*x^2+3*a),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + 3 \, a\right )}{\left (-b x^{2} + a\right )}^{\frac{7}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + 3*a)*(-b*x^2 + a)^(7/3)),x, algorithm="maxima")

[Out]

integrate(1/((b*x^2 + 3*a)*(-b*x^2 + a)^(7/3)), 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)*(-b*x^2 + a)^(7/3)),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a - b x^{2}\right )^{\frac{7}{3}} \left (3 a + b x^{2}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(-b*x**2+a)**(7/3)/(b*x**2+3*a),x)

[Out]

Integral(1/((a - b*x**2)**(7/3)*(3*a + b*x**2)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + 3 \, a\right )}{\left (-b x^{2} + a\right )}^{\frac{7}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + 3*a)*(-b*x^2 + a)^(7/3)),x, algorithm="giac")

[Out]

integrate(1/((b*x^2 + 3*a)*(-b*x^2 + a)^(7/3)), x)